Maxima 5.42.455.gc5fa8d39c Manual

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Maxima is a computer algebra system, implemented in Lisp.

Maxima is derived from the Macsyma system, developed at MIT in the years 1968 through 1982 as part of Project MAC. MIT turned over a copy of the Macsyma source code to the Department of Energy in 1982; that version is now known as DOE Macsyma. A copy of DOE Macsyma was maintained by Professor William F. Schelter of the University of Texas from 1982 until his death in 2001. In 1998, Schelter obtained permission from the Department of Energy to release the DOE Macsyma source code under the GNU Public License, and in 2000 he initiated the Maxima project at SourceForge to maintain and develop DOE Macsyma, now called Maxima.


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1 Introduction to Maxima

Start Maxima with the command "maxima". Maxima will display version information and a prompt. End each Maxima command with a semicolon. End the session with the command "quit();". Here’s a sample session:

[wfs@chromium]$ maxima
Maxima 5.9.1 http://maxima.sourceforge.net
Using Lisp CMU Common Lisp 19a
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
This is a development version of Maxima. The function bug_report()
provides bug reporting information.
(%i1) factor(10!);
                            8  4  2
(%o1)                      2  3  5  7
(%i2) expand ((x + y)^6);
       6        5       2  4       3  3       4  2      5      6
(%o2) y  + 6 x y  + 15 x  y  + 20 x  y  + 15 x  y  + 6 x  y + x
(%i3) factor (x^6 - 1);
                              2            2
(%o3)       (x - 1) (x + 1) (x  - x + 1) (x  + x + 1)
(%i4) quit();
[wfs@chromium]$

Maxima can search the info pages. Use the describe command to show information about the command or all the commands and variables containing a string. The question mark ? (exact search) and double question mark ?? (inexact search) are abbreviations for describe:

(%i1) ?? integ
 0: Functions and Variables for Elliptic Integrals
 1: Functions and Variables for Integration
 2: Introduction to Elliptic Functions and Integrals
 3: Introduction to Integration
 4: askinteger  (Functions and Variables for Simplification)
 5: integerp  (Functions and Variables for Miscellaneous Options)
 6: integer_partitions  (Functions and Variables for Sets)
 7: integrate  (Functions and Variables for Integration)
 8: integrate_use_rootsof  (Functions and Variables for Integration)
 9: integration_constant_counter  (Functions and Variables for
    Integration)
 10: nonnegintegerp  (Functions and Variables for linearalgebra)
Enter space-separated numbers, `all' or `none': 5 4

 -- Function: integerp (<expr>)
     Returns `true' if <expr> is a literal numeric integer, otherwise
     `false'.

     `integerp' returns false if its argument is a symbol, even if the
     argument is declared integer.

     Examples:

          (%i1) integerp (0);
          (%o1)                         true
          (%i2) integerp (1);
          (%o2)                         true
          (%i3) integerp (-17);
          (%o3)                         true
          (%i4) integerp (0.0);
          (%o4)                         false
          (%i5) integerp (1.0);
          (%o5)                         false
          (%i6) integerp (%pi);
          (%o6)                         false
          (%i7) integerp (n);
          (%o7)                         false
          (%i8) declare (n, integer);
          (%o8)                         done
          (%i9) integerp (n);
          (%o9)                         false

 -- Function: askinteger (<expr>, integer)
 -- Function: askinteger (<expr>)
 -- Function: askinteger (<expr>, even)
 -- Function: askinteger (<expr>, odd)
     `askinteger (<expr>, integer)' attempts to determine from the
     `assume' database whether <expr> is an integer.  `askinteger'
     prompts the user if it cannot tell otherwise, and attempt to
     install the information in the database if possible.  `askinteger
     (<expr>)' is equivalent to `askinteger (<expr>, integer)'.

     `askinteger (<expr>, even)' and `askinteger (<expr>, odd)'
     likewise attempt to determine if <expr> is an even integer or odd
     integer, respectively.

(%o1)                                true

To use a result in later calculations, you can assign it to a variable or refer to it by its automatically supplied label. In addition, % refers to the most recent calculated result:

(%i1) u: expand ((x + y)^6);
       6        5       2  4       3  3       4  2      5      6
(%o1) y  + 6 x y  + 15 x  y  + 20 x  y  + 15 x  y  + 6 x  y + x
(%i2) diff (u, x);
         5         4       2  3       3  2       4        5
(%o2) 6 y  + 30 x y  + 60 x  y  + 60 x  y  + 30 x  y + 6 x
(%i3) factor (%o2);
                                    5
(%o3)                      6 (y + x)

Maxima knows about complex numbers and numerical constants:

(%i1) cos(%pi);
(%o1)                          - 1
(%i2) exp(%i*%pi);
(%o2)                          - 1

Maxima can do differential and integral calculus:

(%i1) u: expand ((x + y)^6);
       6        5       2  4       3  3       4  2      5      6
(%o1) y  + 6 x y  + 15 x  y  + 20 x  y  + 15 x  y  + 6 x  y + x
(%i2) diff (%, x);
         5         4       2  3       3  2       4        5
(%o2) 6 y  + 30 x y  + 60 x  y  + 60 x  y  + 30 x  y + 6 x
(%i3) integrate (1/(1 + x^3), x);
                                  2 x - 1
                2            atan(-------)
           log(x  - x + 1)        sqrt(3)    log(x + 1)
(%o3)    - --------------- + ------------- + ----------
                  6             sqrt(3)          3

Maxima can solve linear systems and cubic equations:

(%i1) linsolve ([3*x + 4*y = 7, 2*x + a*y = 13], [x, y]);
                        7 a - 52        25
(%o1)              [x = --------, y = -------]
                        3 a - 8       3 a - 8
(%i2) solve (x^3 - 3*x^2 + 5*x = 15, x);
(%o2)       [x = - sqrt(5) %i, x = sqrt(5) %i, x = 3]

Maxima can solve nonlinear sets of equations. Note that if you don’t want a result printed, you can finish your command with $ instead of ;.

(%i1) eq_1: x^2 + 3*x*y + y^2 = 0$
(%i2) eq_2: 3*x + y = 1$
(%i3) solve ([eq_1, eq_2]);
              3 sqrt(5) + 7      sqrt(5) + 3
(%o3) [[y = - -------------, x = -----------], 
                    2                 2

                               3 sqrt(5) - 7        sqrt(5) - 3
                          [y = -------------, x = - -----------]]
                                     2                   2

Maxima can generate plots of one or more functions:

(%i1) plot2d (sin(x)/x, [x, -20, 20])$
figures/introduction1
(%i2) plot2d ([atan(x), erf(x), tanh(x)], [x, -5, 5], [y, -1.5, 2])$
figures/introduction2
(%i3) plot3d (sin(sqrt(x^2 + y^2))/sqrt(x^2 + y^2), 
         [x, -12, 12], [y, -12, 12])$
figures/introduction3
·

Category: Help


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2 Bug Detection and Reporting


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2.1 Functions and Variables for Bug Detection and Reporting

Function: run_testsuite ([options])

Run the Maxima test suite. Tests producing the desired answer are considered “passes,” as are tests that do not produce the desired answer, but are marked as known bugs.

run_testsuite takes the following optional keyword arguments

display_all

Display all tests. Normally, the tests are not displayed, unless the test fails. (Defaults to false).

display_known_bugs

Displays tests that are marked as known bugs. (Default is false).

tests

This is a single test or a list of tests that should be run. Each test can be specified by either a string or a symbol. By default, all tests are run. The complete set of tests is specified by testsuite_files.

time

Display time information. If true, the time taken for each test file is displayed. If all, the time for each individual test is shown if display_all is true. The default is false, so no timing information is shown.

share_tests

Load additional tests for the share directory. If true, these additional tests are run as a part of the testsuite. If false, no tests from the share directory are run. If only, only the tests from the share directory are run. Of course, the actual set of test that are run can be controlled by the tests option. The default is false.

For example run_testsuite(display_known_bugs = true, tests=[rtest5]) runs just test rtest5 and displays the test that are marked as known bugs.

run_testsuite(display_all = true, tests=["rtest1", rtest1a]) will run tests rtest1 and rtest2, and displays each test.

run_testsuite changes the Maxima environment. Typically a test script executes kill to establish a known environment (namely one without user-defined functions and variables) and then defines functions and variables appropriate to the test.

run_testsuite returns done.

Option variable: testsuite_files

testsuite_files is the set of tests to be run by run_testsuite. It is a list of names of the files containing the tests to run. If some of the tests in a file are known to fail, then instead of listing the name of the file, a list containing the file name and the test numbers that fail is used.

For example, this is a part of the default set of tests:

 ["rtest13s", ["rtest14", 57, 63]]

This specifies the testsuite consists of the files "rtest13s" and "rtest14", but "rtest14" contains two tests that are known to fail: 57 and 63.

Option variable: share_testsuite_files

share_testsuite_files is the set of tests from the share directory that is run as a part of the test suite by run_testsuite..

Function: bug_report ()

Prints out Maxima and Lisp version numbers, and gives a link to the Maxima project bug report web page. The version information is the same as reported by build_info.

When a bug is reported, it is helpful to copy the Maxima and Lisp version information into the bug report.

bug_report returns an empty string "".

Function: build_info ()

Returns a summary of the parameters of the Maxima build, as a Maxima structure (defined by defstruct). The fields of the structure are: version, timestamp, host, lisp_name, and lisp_version. When the pretty-printer is enabled (via display2d), the structure is displayed as a short table.

See also bug_report.

Examples:

(%i1) build_info ();
(%o1) 
Maxima version: "5.36.1"
Maxima build date: "2015-06-02 11:26:48"
Host type: "x86_64-unknown-linux-gnu"
Lisp implementation type: "GNU Common Lisp (GCL)"
Lisp implementation version: "GCL 2.6.12"
(%i2) x : build_info ()$
(%i3) x@version;
(%o3)                               5.36.1
(%i4) x@timestamp;
(%o4)                         2015-06-02 11:26:48
(%i5) x@host;
(%o5)                      x86_64-unknown-linux-gnu
(%i6) x@lisp_name;
(%o6)                        GNU Common Lisp (GCL)
(%i7) x@lisp_version;
(%o7)                             GCL 2.6.12
(%i8) x;
(%o8) 
Maxima version: "5.36.1"
Maxima build date: "2015-06-02 11:26:48"
Host type: "x86_64-unknown-linux-gnu"
Lisp implementation type: "GNU Common Lisp (GCL)"
Lisp implementation version: "GCL 2.6.12"

The Maxima version string can (here 5.36.1) can look very different:

(%i1) build_info();
(%o1) 
Maxima version: "branch_5_37_base_331_g8322940_dirty"
Maxima build date: "2016-01-01 15:37:35"
Host type: "x86_64-unknown-linux-gnu"
Lisp implementation type: "CLISP"
Lisp implementation version: "2.49 (2010-07-07) (built 3605577779) (memory 3660647857)"

In that case, Maxima was not build from a released sourcecode, but directly from the GIT-checkout of the sourcecode. In the example, the checkout is 331 commits after the latest GIT tag (usually a Maxima (major) release (5.37 in our example)) and the abbreviated commit hash of the last commit was "8322940".


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3 Help


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3.1 Documentation

The Maxima on-line user’s manual can be viewed in different forms. From the Maxima interactive prompt, the user’s manual is viewed as plain text by the ? command (i.e., the describe function). The user’s manual is viewed as info hypertext by the info viewer program and as a web page by any ordinary web browser.

example displays examples for many Maxima functions. For example,

(%i1) example (integrate);

yields

(%i2) test(f):=block([u],u:integrate(f,x),ratsimp(f-diff(u,x)))
(%o2) test(f) := block([u], u : integrate(f, x),
                                         ratsimp(f - diff(u, x)))
(%i3) test(sin(x))
(%o3)                           0
(%i4) test(1/(x+1))
(%o4)                           0
(%i5) test(1/(x^2+1))
(%o5)                           0

and additional output.

·

Category: Console interaction


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3.2 Functions and Variables for Help

Function: apropos (string)

Searches for Maxima names which have string appearing anywhere within them. Thus, apropos (exp) returns a list of all the flags and functions which have exp as part of their names, such as expand, exp, and exponentialize. Thus if you can only remember part of the name of something you can use this command to find the rest of the name. Similarly, you could say apropos (tr_) to find a list of many of the switches relating to the translator, most of which begin with tr_.

apropos("") returns a list with all Maxima names.

apropos returns the empty list [], if no name is found.

Example:

Show all Maxima symbols which have "gamma" in the name:

(%i1) apropos("gamma");
(%o1) [%gamma, gamma, gammalim, gamma_expand, gamma_incomplete_lower, 
gamma_incomplete, gamma_incomplete_generalized, 
gamma_incomplete_regularized, Gamma, log_gamma, makegamma, 
prefer_gamma_incomplete, gamma-incomplete, 
gamma_incomplete_generalized_regularized]

Function: demo (filename)

Evaluates Maxima expressions in filename and displays the results. demo pauses after evaluating each expression and continues after the user enters a carriage return. (If running in Xmaxima, demo may need to see a semicolon ; followed by a carriage return.)

demo searches the list of directories file_search_demo to find filename. If the file has the suffix dem, the suffix may be omitted. See also file_search.

demo evaluates its argument. demo returns the name of the demonstration file.

Example:

(%i1) demo ("disol");

batching /home/wfs/maxima/share/simplification/disol.dem
 At the _ prompt, type ';' followed by enter to get next demo
(%i2)                      load("disol")

_
(%i3)           exp1 : a (e (g + f) + b (d + c))
(%o3)               a (e (g + f) + b (d + c))

_
(%i4)                disolate(exp1, a, b, e)
(%t4)                         d + c

(%t5)                         g + f

(%o5)                   a (%t5 e + %t4 b)

_

Function: describe
    describe (string)
    describe (string, exact)
    describe (string, inexact)

describe(string) is equivalent to describe(string, exact).

describe(string, exact) finds an item with title equal (case-insensitive) to string, if there is any such item.

describe(string, inexact) finds all documented items which contain string in their titles. If there is more than one such item, Maxima asks the user to select an item or items to display.

At the interactive prompt, ? foo (with a space between ? and foo) is equivalent to describe("foo", exact), and ?? foo is equivalent to describe("foo", inexact).

describe("", inexact) yields a list of all topics documented in the on-line manual.

describe quotes its argument. describe returns true if some documentation is found, otherwise false.

See also Documentation.

Example:

(%i1) ?? integ
 0: Functions and Variables for Elliptic Integrals
 1: Functions and Variables for Integration
 2: Introduction to Elliptic Functions and Integrals
 3: Introduction to Integration
 4: askinteger  (Functions and Variables for Simplification)
 5: integerp  (Functions and Variables for Miscellaneous Options)
 6: integer_partitions  (Functions and Variables for Sets)
 7: integrate  (Functions and Variables for Integration)
 8: integrate_use_rootsof  (Functions and Variables for
    Integration)
 9: integration_constant_counter  (Functions and Variables for
    Integration)
 10: nonnegintegerp  (Functions and Variables for linearalgebra)
Enter space-separated numbers, `all' or `none': 7 8

 -- Function: integrate (<expr>, <x>)
 -- Function: integrate (<expr>, <x>, <a>, <b>)
     Attempts to symbolically compute the integral of <expr> with
     respect to <x>.  `integrate (<expr>, <x>)' is an indefinite
     integral, while `integrate (<expr>, <x>, <a>, <b>)' is a
     definite integral, [...]
     
 -- Option variable: integrate_use_rootsof
     Default value: `false'

     When `integrate_use_rootsof' is `true' and the denominator of
     a rational function cannot be factored, `integrate' returns
     the integral in a form which is a sum over the roots (not yet
     known) of the denominator.
     [...]

In this example, items 7 and 8 were selected (output is shortened as indicated by [...]). All or none of the items could have been selected by entering all or none, which can be abbreviated a or n, respectively.

Function: example
    example (topic)
    example ()

example (topic) displays some examples of topic, which is a symbol or a string. To get examples for operators like if, do, or lambda the argument must be a string, e.g. example ("do"). example is not case sensitive. Most topics are function names.

example () returns the list of all recognized topics.

The name of the file containing the examples is given by the global option variable manual_demo, which defaults to "manual.demo".

example quotes its argument. example returns done unless no examples are found or there is no argument, in which case example returns the list of all recognized topics.

Examples:

(%i1) example(append);
(%i2) append([y+x,0,-3.2],[2.5e+20,x])
(%o2)             [y + x, 0, - 3.2, 2.5e+20, x]
(%o2)                         done
(%i3) example("lambda");
(%i4) lambda([x,y,z],x^2+y^2+z^2)
                                    2    2    2
(%o4)            lambda([x, y, z], x  + y  + z )
(%i5) %(1,2,a)
                              2
(%o5)                        a  + 5
(%i6) 1+2+a
(%o6)                         a + 3
(%o6)                         done

Option variable: manual_demo

Default value: "manual.demo"

manual_demo specifies the name of the file containing the examples for the function example. See example.


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4 Command Line


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4.1 Introduction to Command Line


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4.2 Functions and Variables for Command Line

System variable: __

__ is the input expression currently being evaluated. That is, while an input expression expr is being evaluated, __ is expr.

__ is assigned the input expression before the input is simplified or evaluated. However, the value of __ is simplified (but not evaluated) when it is displayed.

__ is recognized by batch and load. In a file processed by batch, __ has the same meaning as at the interactive prompt. In a file processed by load, __ is bound to the input expression most recently entered at the interactive prompt or in a batch file; __ is not bound to the input expressions in the file being processed. In particular, when load (filename) is called from the interactive prompt, __ is bound to load (filename) while the file is being processed.

See also _ and %.

Examples:

(%i1) print ("I was called as", __);
I was called as print(I was called as, __) 
(%o1)              print(I was called as, __)
(%i2) foo (__);
(%o2)                     foo(foo(__))
(%i3) g (x) := (print ("Current input expression =", __), 0);
(%o3) g(x) := (print("Current input expression =", __), 0)
(%i4) [aa : 1, bb : 2, cc : 3];
(%o4)                       [1, 2, 3]
(%i5) (aa + bb + cc)/(dd + ee + g(x));
                            cc + bb + aa
Current input expression = -------------- 
                           g(x) + ee + dd
                                6
(%o5)                        -------
                             ee + dd

System variable: _

_ is the most recent input expression (e.g., %i1, %i2, %i3, …).

_ is assigned the input expression before the input is simplified or evaluated. However, the value of _ is simplified (but not evaluated) when it is displayed.

_ is recognized by batch and load. In a file processed by batch, _ has the same meaning as at the interactive prompt. In a file processed by load, _ is bound to the input expression most recently evaluated at the interactive prompt or in a batch file; _ is not bound to the input expressions in the file being processed.

See also __ and %.

Examples:

(%i1) 13 + 29;
(%o1)                          42
(%i2) :lisp $_
((MPLUS) 13 29)
(%i2) _;
(%o2)                          42
(%i3) sin (%pi/2);
(%o3)                           1
(%i4) :lisp $_
((%SIN) ((MQUOTIENT) $%PI 2))
(%i4) _;
(%o4)                           1
(%i5) a: 13$
(%i6) b: 29$
(%i7) a + b;
(%o7)                          42
(%i8) :lisp $_
((MPLUS) $A $B)
(%i8) _;
(%o8)                         b + a
(%i9) a + b;
(%o9)                          42
(%i10) ev (_);
(%o10)                         42

System variable: %

% is the output expression (e.g., %o1, %o2, %o3, …) most recently computed by Maxima, whether or not it was displayed.

% is recognized by batch and load. In a file processed by batch, % has the same meaning as at the interactive prompt. In a file processed by load, % is bound to the output expression most recently computed at the interactive prompt or in a batch file; % is not bound to output expressions in the file being processed.

See also _, %%, and %th.

System variable: %%

In compound statements, namely block, lambda, or (s_1, ..., s_n), %% is the value of the previous statement.

At the first statement in a compound statement, or outside of a compound statement, %% is undefined.

%% is recognized by batch and load, and it has the same meaning as at the interactive prompt.

See also %.

Examples:

The following two examples yield the same result.

(%i1) block (integrate (x^5, x), ev (%%, x=2) - ev (%%, x=1));
                               21
(%o1)                          --
                               2
(%i2) block ([prev], prev: integrate (x^5, x),
               ev (prev, x=2) - ev (prev, x=1));
                               21
(%o2)                          --
                               2

A compound statement may comprise other compound statements. Whether a statement be simple or compound, %% is the value of the previous statement.

(%i3) block (block (a^n, %%*42), %%/6);
                                 n
(%o3)                         7 a

Within a compound statement, the value of %% may be inspected at a break prompt, which is opened by executing the break function. For example, entering %%; in the following example yields 42.

(%i4) block (a: 42, break ())$

Entering a Maxima break point. Type 'exit;' to resume.
_%%;
42
_

Function: %th (i)

The value of the i’th previous output expression. That is, if the next expression to be computed is the n’th output, %th (m) is the (n - m)’th output.

%th is recognized by batch and load. In a file processed by batch, %th has the same meaning as at the interactive prompt. In a file processed by load, %th refers to output expressions most recently computed at the interactive prompt or in a batch file; %th does not refer to output expressions in the file being processed.

See also % and %%.

Example:

%th is useful in batch files or for referring to a group of output expressions. This example sets s to the sum of the last five output expressions.

(%i1) 1;2;3;4;5;
(%o1)                           1
(%o2)                           2
(%o3)                           3
(%o4)                           4
(%o5)                           5
(%i6) block (s: 0, for i:1 thru 5 do s: s + %th(i), s);
(%o6)                          15

Special symbol: ?

As prefix to a function or variable name, ? signifies that the name is a Lisp name, not a Maxima name. For example, ?round signifies the Lisp function ROUND. See Lisp and Maxima for more on this point.

The notation ? word (a question mark followed a word, separated by whitespace) is equivalent to describe("word"). The question mark must occur at the beginning of an input line; otherwise it is not recognized as a request for documentation. See also describe.

Special symbol: ??

The notation ?? word (?? followed a word, separated by whitespace) is equivalent to describe("word", inexact). The question mark must occur at the beginning of an input line; otherwise it is not recognized as a request for documentation. See also describe.

Input terminator: $

The dollar sign $ terminates an input expression, and the most recent output % and an output label, e.g. %o1, are assigned the result, but the result is not displayed.

See also ;.

Example:

(%i1) 1 + 2 + 3 $
(%i2) %;
(%o2)                           6
(%i3) %o1;
(%o3)                           6
Input terminator: ;

The semicolon ; terminates an input expression, and the resulting output is displayed.

See also $.

Example:

(%i1) 1 + 2 + 3;
(%o1)                           6
Option variable: inchar

Default value: %i

inchar is the prefix of the labels of expressions entered by the user. Maxima automatically constructs a label for each input expression by concatenating inchar and linenum.

inchar may be assigned any string or symbol, not necessarily a single character. Because Maxima internally takes into account only the first char of the prefix, the prefixes inchar, outchar, and linechar should have a different first char. Otherwise some commands like kill(inlabels) do not work as expected.

See also labels.

Example:

(%i1) inchar: "input";
(%o1)                         input
(input2) expand((a+b)^3);
                     3        2      2      3
(%o2)               b  + 3 a b  + 3 a  b + a

System variable: infolists

Default value: []

infolists is a list of the names of all of the information lists in Maxima. These are:

labels

All bound %i, %o, and %t labels.

values

All bound atoms which are user variables, not Maxima options or switches, created by : or :: or functional binding.

functions

All user-defined functions, created by := or define.

arrays

All arrays, hashed arrays and memoizing functions.

macros

All user-defined macro functions, created by ::=.

myoptions

All options ever reset by the user (whether or not they are later reset to their default values).

rules

All user-defined pattern matching and simplification rules, created by tellsimp, tellsimpafter, defmatch, or defrule.

aliases

All atoms which have a user-defined alias, created by the alias, ordergreat, orderless functions or by declaring the atom as a noun with declare.

dependencies

All atoms which have functional dependencies, created by the depends, dependencies, or gradef functions.

gradefs

All functions which have user-defined derivatives, created by the gradef function.

props

All atoms which have any property other than those mentioned above, such as properties established by atvalue or matchdeclare, etc., as well as properties established in the declare function.

let_rule_packages

All user-defined let rule packages plus the special package default_let_rule_package. (default_let_rule_package is the name of the rule package used when one is not explicitly set by the user.)

Function: kill
    kill (a_1, …, a_n)
    kill (labels)
    kill (inlabels, outlabels, linelabels)
    kill (n)
    kill ([m, n])
    kill (values, functions, arrays, …)
    kill (all)
    kill (allbut (a_1, …, a_n))

Removes all bindings (value, function, array, or rule) from the arguments a_1, …, a_n. An argument a_k may be a symbol or a single array element. When a_k is a single array element, kill unbinds that element without affecting any other elements of the array.

Several special arguments are recognized. Different kinds of arguments may be combined, e.g., kill (inlabels, functions, allbut (foo, bar)).

kill (labels) unbinds all input, output, and intermediate expression labels created so far. kill (inlabels) unbinds only input labels which begin with the current value of inchar. Likewise, kill (outlabels) unbinds only output labels which begin with the current value of outchar, and kill (linelabels) unbinds only intermediate expression labels which begin with the current value of linechar.

kill (n), where n is an integer, unbinds the n most recent input and output labels.

kill ([m, n]) unbinds input and output labels m through n.

kill (infolist), where infolist is any item in infolists (such as values, functions, or arrays) unbinds all items in infolist. See also infolists.

kill (all) unbinds all items on all infolists. kill (all) does not reset global variables to their default values; see reset on this point.

kill (allbut (a_1, ..., a_n)) unbinds all items on all infolists except for a_1, …, a_n. kill (allbut (infolist)) unbinds all items except for the ones on infolist, where infolist is values, functions, arrays, etc.

The memory taken up by a bound property is not released until all symbols are unbound from it. In particular, to release the memory taken up by the value of a symbol, one unbinds the output label which shows the bound value, as well as unbinding the symbol itself.

kill quotes its arguments. The quote-quote operator '' defeats quotation.

kill (symbol) unbinds all properties of symbol. In contrast, the functions remvalue, remfunction, remarray, and remrule unbind a specific property. Note that facts declared by assume don’t require a symbol they apply to, therefore aren’t stored as properties of symbols and therefore aren’t affected by kill.

kill always returns done, even if an argument has no binding.

Function: labels (symbol)

Returns the list of input, output, or intermediate expression labels which begin with symbol. Typically symbol is the value of inchar, outchar, or linechar. If no labels begin with symbol, labels returns an empty list.

By default, Maxima displays the result of each user input expression, giving the result an output label. The output display is suppressed by terminating the input with $ (dollar sign) instead of ; (semicolon). An output label is constructed and bound to the result, but not displayed, and the label may be referenced in the same way as displayed output labels. See also %, %%, and %th.

Intermediate expression labels can be generated by some functions. The option variable programmode controls whether solve and some other functions generate intermediate expression labels instead of returning a list of expressions. Some other functions, such as ldisplay, always generate intermediate expression labels.

See also inchar, outchar, linechar, and infolists.

System variable: labels

The variable labels is the list of input, output, and intermediate expression labels, including all previous labels if inchar, outchar, or linechar were redefined.

Option variable: linechar

Default value: %t

linechar is the prefix of the labels of intermediate expressions generated by Maxima. Maxima constructs a label for each intermediate expression (if displayed) by concatenating linechar and linenum.

linechar may be assigned any string or symbol, not necessarily a single character. Because Maxima internally takes into account only the first char of the prefix, the prefixes inchar, outchar, and linechar should have a different first char. Otherwise some commands like kill(inlabels) do not work as expected.

Intermediate expressions might or might not be displayed. See programmode and labels.

System variable: linenum

The line number of the current pair of input and output expressions.

System variable: myoptions

Default value: []

myoptions is the list of all options ever reset by the user, whether or not they get reset to their default value.

Option variable: nolabels

Default value: false

When nolabels is true, input and output result labels (%i and %o, respectively) are displayed, but the labels are not bound to results, and the labels are not appended to the labels list. Since labels are not bound to results, garbage collection can recover the memory taken up by the results.

Otherwise input and output result labels are bound to results, and the labels are appended to the labels list.

Intermediate expression labels (%t) are not affected by nolabels; whether nolabels is true or false, intermediate expression labels are bound and appended to the labels list.

See also batch, load, and labels.

Option variable: optionset

Default value: false

When optionset is true, Maxima prints out a message whenever a Maxima option is reset. This is useful if the user is doubtful of the spelling of some option and wants to make sure that the variable he assigned a value to was truly an option variable.

Example:

(%i1) optionset:true;
assignment: assigning to option optionset
(%o1)                         true
(%i2) gamma_expand:true;
assignment: assigning to option gamma_expand
(%o2)                         true

Option variable: outchar

Default value: %o

outchar is the prefix of the labels of expressions computed by Maxima. Maxima automatically constructs a label for each computed expression by concatenating outchar and linenum.

outchar may be assigned any string or symbol, not necessarily a single character. Because Maxima internally takes into account only the first char of the prefix, the prefixes inchar, outchar and linechar should have a different first char. Otherwise some commands like kill(inlabels) do not work as expected.

See also labels.

Example:

(%i1) outchar: "output";
(output1)                    output
(%i2) expand((a+b)^3);
                     3        2      2      3
(output2)           b  + 3 a b  + 3 a  b + a

Function: playback
    playback ()
    playback (n)
    playback ([m, n])
    playback ([m])
    playback (input)
    playback (slow)
    playback (time)
    playback (grind)

Displays input, output, and intermediate expressions, without recomputing them. playback only displays the expressions bound to labels; any other output (such as text printed by print or describe, or error messages) is not displayed. See also labels.

playback quotes its arguments. The quote-quote operator '' defeats quotation. playback always returns done.

playback () (with no arguments) displays all input, output, and intermediate expressions generated so far. An output expression is displayed even if it was suppressed by the $ terminator when it was originally computed.

playback (n) displays the most recent n expressions. Each input, output, and intermediate expression counts as one.

playback ([m, n]) displays input, output, and intermediate expressions with numbers from m through n, inclusive.

playback ([m]) is equivalent to playback ([m, m]); this usually prints one pair of input and output expressions.

playback (input) displays all input expressions generated so far.

playback (slow) pauses between expressions and waits for the user to press enter. This behavior is similar to demo. playback (slow) is useful in conjunction with save or stringout when creating a secondary-storage file in order to pick out useful expressions.

playback (time) displays the computation time for each expression.

playback (grind) displays input expressions in the same format as the grind function. Output expressions are not affected by the grind option. See grind.

Arguments may be combined, e.g., playback ([5, 10], grind, time, slow).

Option variable: prompt

Default value: _

prompt is the prompt symbol of the demo function, playback (slow) mode, and the Maxima break loop (as invoked by break).

Function: quit ()

Terminates the Maxima session. Note that the function must be invoked as quit(); or quit()$, not quit by itself.

To stop a lengthy computation, type control-C. The default action is to return to the Maxima prompt. If *debugger-hook* is nil, control-C opens the Lisp debugger. See also Debugging.

Function: read (expr_1, …, expr_n)

Prints expr_1, …, expr_n, then reads one expression from the console and returns the evaluated expression. The expression is terminated with a semicolon ; or dollar sign $.

See also readonly

Example:

(%i1) foo: 42$ 
(%i2) foo: read ("foo is", foo, " -- enter new value.")$
foo is 42  -- enter new value. 
(a+b)^3;
(%i3) foo;
                                     3
(%o3)                         (b + a)

Function: readonly (expr_1, …, expr_n)

Prints expr_1, …, expr_n, then reads one expression from the console and returns the expression (without evaluation). The expression is terminated with a ; (semicolon) or $ (dollar sign).

See also read.

Examples:

(%i1) aa: 7$
(%i2) foo: readonly ("Enter an expression:");
Enter an expression: 
2^aa;
                                  aa
(%o2)                            2
(%i3) foo: read ("Enter an expression:");
Enter an expression: 
2^aa;
(%o3)                            128

Function: reset ()

Resets many global variables and options, and some other variables, to their default values.

reset processes the variables on the Lisp list *variable-initial-values*. The Lisp macro defmvar puts variables on this list (among other actions). Many, but not all, global variables and options are defined by defmvar, and some variables defined by defmvar are not global variables or options.

Option variable: showtime

Default value: false

When showtime is true, the computation time and elapsed time is printed with each output expression.

The computation time is always recorded, so time and playback can display the computation time even when showtime is false.

See also timer.

Function: to_lisp ()

Enters the Lisp system under Maxima. (to-maxima) returns to Maxima.

Example:

Define a function and enter the Lisp system under Maxima. The definition is inspected on the property list, then the function definition is extracted, factored and stored in the variable $result. The variable can be used in Maxima after returning to Maxima.

(%i1) f(x):=x^2+x;
                                  2
(%o1)                    f(x) := x  + x
(%i2) to_lisp();
Type (to-maxima) to restart, ($quit) to quit Maxima.
MAXIMA> (symbol-plist '$f)
(MPROPS (NIL MEXPR ((LAMBDA) ((MLIST) $X) 
                             ((MPLUS) ((MEXPT) $X 2) $X))))
MAXIMA> (setq $result ($factor (caddr (mget '$f 'mexpr))))
((MTIMES SIMP FACTORED) $X ((MPLUS SIMP IRREDUCIBLE) 1 $X))
MAXIMA> (to-maxima)
Returning to Maxima
(%o2)                         true
(%i3) result;
(%o3)                       x (x + 1)

System variable: values

Initial value: []

values is a list of all bound user variables (not Maxima options or switches). The list comprises symbols bound by :, or ::.

If the value of a variable is removed with the commands kill, remove, or remvalue the variable is deleted from values.

See functions for a list of user defined functions.

Examples:

First, values shows the symbols a, b, and c, but not d, it is not bound to a value, and not the user function f. The values are removed from the variables. values is the empty list.

(%i1) [a:99, b:: a-90, c:a-b, d, f(x):=x^2];
                                           2
(%o1)              [99, 9, 90, d, f(x) := x ]
(%i2) values;
(%o2)                       [a, b, c]
(%i3) [kill(a), remove(b,value), remvalue(c)];
(%o3)                   [done, done, [c]]
(%i4) values;
(%o4)                          []


Previous: , Up: Command Line   [Contents][Index]

4.3 Functions and Variables for Display

Option variable: %edispflag

Default value: false

When %edispflag is true, Maxima displays %e to a negative exponent as a quotient. For example, %e^-x is displayed as 1/%e^x. See also exptdispflag.

Example:

(%i1) %e^-10;
                               - 10
(%o1)                        %e
(%i2) %edispflag:true$
(%i3) %e^-10;
                               1
(%o3)                         ----
                                10
                              %e

Option variable: absboxchar

Default value: !

absboxchar is the character used to draw absolute value signs around expressions which are more than one line tall.

Example:

(%i1) abs((x^3+1));
                            ! 3    !
(%o1)                       !x  + 1!

Function: disp (expr_1, expr_2, …)

is like display but only the value of the arguments are displayed rather than equations. This is useful for complicated arguments which don’t have names or where only the value of the argument is of interest and not the name.

See also ldisp and print.

Example:

(%i1) b[1,2]:x-x^2$
(%i2) x:123$
(%i3) disp(x, b[1,2], sin(1.0));
                               123

                                  2
                             x - x

                       0.8414709848078965

(%o3)                         done

Function: display (expr_1, expr_2, …)

Displays equations whose left side is expr_i unevaluated, and whose right side is the value of the expression centered on the line. This function is useful in blocks and for statements in order to have intermediate results displayed. The arguments to display are usually atoms, subscripted variables, or function calls.

See also ldisplay, disp, and ldisp.

Example:

(%i1) b[1,2]:x-x^2$
(%i2) x:123$
(%i3) display(x, b[1,2], sin(1.0));
                             x = 123

                                      2
                         b     = x - x
                          1, 2

                  sin(1.0) = 0.8414709848078965

(%o3)                         done

Option variable: display2d

Default value: true

When display2d is true, the console display is an attempt to present mathematical expressions as they might appear in books and articles, using only letters, numbers, and some punctuation characters. This display is sometimes called the "pretty printer" display.

When display2d is false, the console display is a 1-dimensional or linear form which is the same as the output produced by grind.

When display2d is false, the value of stringdisp is ignored, and strings are always displayed with quote marks.

See also leftjust to switch between a left justified and a centered display of equations.

Example:

(%i1) x/(x^2+1);
                               x
(%o1)                        ------
                              2
                             x  + 1
(%i2) display2d:false$
(%i3) x/(x^2+1);
(%o3) x/(x^2+1)

Option variable: display_format_internal

Default value: false

When display_format_internal is true, expressions are displayed without being transformed in ways that hide the internal mathematical representation. The display then corresponds to what inpart returns rather than part.

Examples:

User     part       inpart
a-b;      a - b     a + (- 1) b

           a            - 1
a/b;       -         a b
           b
                       1/2
sqrt(x);   sqrt(x)    x

          4 X        4
X*4/3;    ---        - X
           3         3

Function: dispterms (expr)

Displays expr in parts one below the other. That is, first the operator of expr is displayed, then each term in a sum, or factor in a product, or part of a more general expression is displayed separately. This is useful if expr is too large to be otherwise displayed. For example if P1, P2, … are very large expressions then the display program may run out of storage space in trying to display P1 + P2 + ... all at once. However, dispterms (P1 + P2 + ...) displays P1, then below it P2, etc. When not using dispterms, if an exponential expression is too wide to be displayed as A^B it appears as expt (A, B) (or as ncexpt (A, B) in the case of A^^B).

Example:

(%i1) dispterms(2*a*sin(x)+%e^x);

+

2 a sin(x)

  x
%e

(%o1)                         done

Special symbol: expt (a, b)
Special symbol: ncexpt (a, b)

If an exponential expression is too wide to be displayed as a^b it appears as expt (a, b) (or as ncexpt (a, b) in the case of a^^b).

expt and ncexpt are not recognized in input.

Option variable: exptdispflag

Default value: true

When exptdispflag is true, Maxima displays expressions with negative exponents using quotients. See also %edispflag.

Example:

(%i1) exptdispflag:true;
(%o1)                         true
(%i2) 10^-x;
                                1
(%o2)                          ---
                                 x
                               10
(%i3) exptdispflag:false;
(%o3)                         false
(%i4) 10^-x;
                                - x
(%o4)                         10

Function: grind (expr)

The function grind prints expr to the console in a form suitable for input to Maxima. grind always returns done.

When expr is the name of a function or macro, grind prints the function or macro definition instead of just the name.

See also string, which returns a string instead of printing its output. grind attempts to print the expression in a manner which makes it slightly easier to read than the output of string.

grind evaluates its argument.

Examples:

(%i1) aa + 1729;
(%o1)                       aa + 1729
(%i2) grind (%);
aa+1729$
(%o2)                         done
(%i3) [aa, 1729, aa + 1729];
(%o3)                 [aa, 1729, aa + 1729]
(%i4) grind (%);
[aa,1729,aa+1729]$
(%o4)                         done
(%i5) matrix ([aa, 17], [29, bb]);
                           [ aa  17 ]
(%o5)                      [        ]
                           [ 29  bb ]
(%i6) grind (%);
matrix([aa,17],[29,bb])$
(%o6)                         done
(%i7) set (aa, 17, 29, bb);
(%o7)                   {17, 29, aa, bb}
(%i8) grind (%);
{17,29,aa,bb}$
(%o8)                         done
(%i9) exp (aa / (bb + 17)^29);
                                aa
                            -----------
                                     29
                            (bb + 17)
(%o9)                     %e
(%i10) grind (%);
%e^(aa/(bb+17)^29)$
(%o10)                        done
(%i11) expr: expand ((aa + bb)^10);
         10           9        2   8         3   7         4   6
(%o11) bb   + 10 aa bb  + 45 aa  bb  + 120 aa  bb  + 210 aa  bb
         5   5         6   4         7   3        8   2
 + 252 aa  bb  + 210 aa  bb  + 120 aa  bb  + 45 aa  bb
        9        10
 + 10 aa  bb + aa
(%i12) grind (expr);
bb^10+10*aa*bb^9+45*aa^2*bb^8+120*aa^3*bb^7+210*aa^4*bb^6
     +252*aa^5*bb^5+210*aa^6*bb^4+120*aa^7*bb^3+45*aa^8*bb^2
     +10*aa^9*bb+aa^10$
(%o12)                        done
(%i13) string (expr);
(%o13) bb^10+10*aa*bb^9+45*aa^2*bb^8+120*aa^3*bb^7+210*aa^4*bb^6\
+252*aa^5*bb^5+210*aa^6*bb^4+120*aa^7*bb^3+45*aa^8*bb^2+10*aa^9*\
bb+aa^10
(%i14) cholesky (A):= block ([n : length (A), L : copymatrix (A),
  p : makelist (0, i, 1, length (A))],
  for i thru n do for j : i thru n do
  (x : L[i, j], x : x - sum (L[j, k] * L[i, k], k, 1, i - 1),
  if i = j then p[i] : 1 / sqrt(x) else L[j, i] : x * p[i]),
  for i thru n do L[i, i] : 1 / p[i],
  for i thru n do for j : i + 1 thru n do L[i, j] : 0, L)$
define: warning: redefining the built-in function cholesky
(%i15) grind (cholesky);
cholesky(A):=block(
         [n:length(A),L:copymatrix(A),
          p:makelist(0,i,1,length(A))],
         for i thru n do
             (for j from i thru n do
                  (x:L[i,j],x:x-sum(L[j,k]*L[i,k],k,1,i-1),
                   if i = j then p[i]:1/sqrt(x)
                       else L[j,i]:x*p[i])),
         for i thru n do L[i,i]:1/p[i],
         for i thru n do (for j from i+1 thru n do L[i,j]:0),L)$
(%o15)                        done
(%i16) string (fundef (cholesky));
(%o16) cholesky(A):=block([n:length(A),L:copymatrix(A),p:makelis\
t(0,i,1,length(A))],for i thru n do (for j from i thru n do (x:L\
[i,j],x:x-sum(L[j,k]*L[i,k],k,1,i-1),if i = j then p[i]:1/sqrt(x\
) else L[j,i]:x*p[i])),for i thru n do L[i,i]:1/p[i],for i thru \
n do (for j from i+1 thru n do L[i,j]:0),L)

Option variable: grind

When the variable grind is true, the output of string and stringout has the same format as that of grind; otherwise no attempt is made to specially format the output of those functions. The default value of the variable grind is false.

grind can also be specified as an argument of playback. When grind is present, playback prints input expressions in the same format as the grind function. Otherwise, no attempt is made to specially format input expressions.

Option variable: ibase

Default value: 10

ibase is the base for integers read by Maxima.

ibase may be assigned any integer between 2 and 36 (decimal), inclusive. When ibase is greater than 10, the numerals comprise the decimal numerals 0 through 9 plus letters of the alphabet A, B, C, …, as needed to make ibase digits in all. Letters are interpreted as digits only if the first digit is 0 through 9.

Uppercase and lowercase letters are not distinguished. The numerals for base 36, the largest acceptable base, comprise 0 through 9 and A through Z.

Whatever the value of ibase, when an integer is terminated by a decimal point, it is interpreted in base 10.

See also obase.

Examples:

ibase less than 10 (for example binary numbers).

(%i1) ibase : 2 $
(%i2) obase;
(%o2)                          10
(%i3) 1111111111111111;
(%o3)                         65535

ibase greater than 10. Letters are interpreted as digits only if the first digit is 0 through 9 which means that hexadecimal numbers might need to be prepended by a 0.

(%i1) ibase : 16 $
(%i2) obase;
(%o2)                          10
(%i3) 1000;
(%o3)                         4096
(%i4) abcd;
(%o4)                         abcd
(%i5) symbolp (abcd);
(%o5)                         true
(%i6) 0abcd;
(%o6)                         43981
(%i7) symbolp (0abcd);
(%o7)                         false

When an integer is terminated by a decimal point, it is interpreted in base 10.

(%i1) ibase : 36 $
(%i2) obase;
(%o2)                          10
(%i3) 1234;
(%o3)                         49360
(%i4) 1234.;
(%o4)                         1234

Function: ldisp (expr_1, …, expr_n)

Displays expressions expr_1, …, expr_n to the console as printed output. ldisp assigns an intermediate expression label to each argument and returns the list of labels.

See also disp, display, and ldisplay.

Examples:

(%i1) e: (a+b)^3;
                                   3
(%o1)                       (b + a)
(%i2) f: expand (e);
                     3        2      2      3
(%o2)               b  + 3 a b  + 3 a  b + a
(%i3) ldisp (e, f);
                                   3
(%t3)                       (b + a)

                     3        2      2      3
(%t4)               b  + 3 a b  + 3 a  b + a

(%o4)                      [%t3, %t4]
(%i4) %t3;
                                   3
(%o4)                       (b + a)
(%i5) %t4;
                     3        2      2      3
(%o5)               b  + 3 a b  + 3 a  b + a

Function: ldisplay (expr_1, …, expr_n)

Displays expressions expr_1, …, expr_n to the console as printed output. Each expression is printed as an equation of the form lhs = rhs in which lhs is one of the arguments of ldisplay and rhs is its value. Typically each argument is a variable. ldisp assigns an intermediate expression label to each equation and returns the list of labels.

See also display, disp, and ldisp.

Examples:

(%i1) e: (a+b)^3;
                                   3
(%o1)                       (b + a)
(%i2) f: expand (e);
                     3        2      2      3
(%o2)               b  + 3 a b  + 3 a  b + a
(%i3) ldisplay (e, f);
                                     3
(%t3)                     e = (b + a)

                       3        2      2      3
(%t4)             f = b  + 3 a b  + 3 a  b + a

(%o4)                      [%t3, %t4]
(%i4) %t3;
                                     3
(%o4)                     e = (b + a)
(%i5) %t4;
                       3        2      2      3
(%o5)             f = b  + 3 a b  + 3 a  b + a

Option variable: leftjust

Default value: false

When leftjust is true, equations in 2D-display are drawn left justified rather than centered.

See also display2d to switch between 1D- and 2D-display.

Example:

(%i1) expand((x+1)^3);
                        3      2
(%o1)                  x  + 3 x  + 3 x + 1
(%i2) leftjust:true$
(%i3) expand((x+1)^3);
       3      2
(%o3) x  + 3 x  + 3 x + 1

Option variable: linel

Default value: 79

linel is the assumed width (in characters) of the console display for the purpose of displaying expressions. linel may be assigned any value by the user, although very small or very large values may be impractical. Text printed by built-in Maxima functions, such as error messages and the output of describe, is not affected by linel.

Option variable: lispdisp

Default value: false

When lispdisp is true, Lisp symbols are displayed with a leading question mark ?. Otherwise, Lisp symbols are displayed with no leading mark. This has the same effect for 1-d and 2-d display.

Examples:

(%i1) lispdisp: false$
(%i2) ?foo + ?bar;
(%o2)                       foo + bar
(%i3) lispdisp: true$
(%i4) ?foo + ?bar;
(%o4)                      ?foo + ?bar

Option variable: negsumdispflag

Default value: true

When negsumdispflag is true, x - y displays as x - y instead of as - y + x. Setting it to false causes the special check in display for the difference of two expressions to not be done. One application is that thus a + %i*b and a - %i*b may both be displayed the same way.

Option variable: obase

Default value: 10

obase is the base for integers displayed by Maxima.

obase may be assigned any integer between 2 and 36 (decimal), inclusive. When obase is greater than 10, the numerals comprise the decimal numerals 0 through 9 plus capital letters of the alphabet A, B, C, …, as needed. A leading 0 digit is displayed if the leading digit is otherwise a letter. The numerals for base 36, the largest acceptable base, comprise 0 through 9, and A through Z.

See also ibase.

Examples:

(%i1) obase : 2;
(%o1)                          10
(%i10) 2^8 - 1;
(%o10)                      11111111
(%i11) obase : 8;
(%o3)                          10
(%i4) 8^8 - 1;
(%o4)                       77777777
(%i5) obase : 16;
(%o5)                          10
(%i6) 16^8 - 1;
(%o6)                       0FFFFFFFF
(%i7) obase : 36;
(%o7)                          10
(%i8) 36^8 - 1;
(%o8)                       0ZZZZZZZZ

Option variable: pfeformat

Default value: false

When pfeformat is true, a ratio of integers is displayed with the solidus (forward slash) character, and an integer denominator n is displayed as a leading multiplicative term 1/n.

Examples:

(%i1) pfeformat: false$
(%i2) 2^16/7^3;
                              65536
(%o2)                         -----
                               343
(%i3) (a+b)/8;
                              b + a
(%o3)                         -----
                                8
(%i4) pfeformat: true$ 
(%i5) 2^16/7^3;
(%o5)                       65536/343
(%i6) (a+b)/8;
(%o6)                      1/8 (b + a)

Option variable: powerdisp

Default value: false

When powerdisp is true, a sum is displayed with its terms in order of increasing power. Thus a polynomial is displayed as a truncated power series, with the constant term first and the highest power last.

By default, terms of a sum are displayed in order of decreasing power.

Example:

(%i1) powerdisp:true;
(%o1)                         true
(%i2) x^2+x^3+x^4;
                           2    3    4
(%o2)                     x  + x  + x
(%i3) powerdisp:false;
(%o3)                         false
(%i4) x^2+x^3+x^4;
                           4    3    2
(%o4)                     x  + x  + x

Function: print (expr_1, …, expr_n)

Evaluates and displays expr_1, …, expr_n one after another, from left to right, starting at the left edge of the console display.

The value returned by print is the value of its last argument. print does not generate intermediate expression labels.

See also display, disp, ldisplay, and ldisp. Those functions display one expression per line, while print attempts to display two or more expressions per line.

To display the contents of a file, see printfile.

Examples:

(%i1) r: print ("(a+b)^3 is", expand ((a+b)^3), "log (a^10/b) is",
      radcan (log (a^10/b)))$
            3        2      2      3
(a+b)^3 is b  + 3 a b  + 3 a  b + a  log (a^10/b) is 

                                              10 log(a) - log(b) 
(%i2) r;
(%o2)                  10 log(a) - log(b)
(%i3) disp ("(a+b)^3 is", expand ((a+b)^3), "log (a^10/b) is",
      radcan (log (a^10/b)))$
                           (a+b)^3 is

                     3        2      2      3
                    b  + 3 a b  + 3 a  b + a

                         log (a^10/b) is

                       10 log(a) - log(b)

Option variable: sqrtdispflag

Default value: true

When sqrtdispflag is false, causes sqrt to display with exponent 1/2.

Option variable: stardisp

Default value: false

When stardisp is true, multiplication is displayed with an asterisk * between operands.

Option variable: ttyoff

Default value: false

When ttyoff is true, output expressions are not displayed. Output expressions are still computed and assigned labels. See labels.

Text printed by built-in Maxima functions, such as error messages and the output of describe, is not affected by ttyoff.


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5 Data Types and Structures


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5.1 Numbers


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5.1.1 Introduction to Numbers

Complex numbers

A complex expression is specified in Maxima by adding the real part of the expression to %i times the imaginary part. Thus the roots of the equation x^2 - 4*x + 13 = 0 are 2 + 3*%i and 2 - 3*%i. Note that simplification of products of complex expressions can be effected by expanding the product. Simplification of quotients, roots, and other functions of complex expressions can usually be accomplished by using the realpart, imagpart, rectform, polarform, abs, carg functions.

·

Category: Complex variables


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5.1.2 Functions and Variables for Numbers

Function: bfloat (expr)

Converts all numbers and functions of numbers in expr to bigfloat numbers. The number of significant digits in the resulting bigfloats is specified by the global variable fpprec.

When float2bf is false a warning message is printed when a floating point number is converted into a bigfloat number (since this may lead to loss of precision).

Function: bfloatp (expr)

Returns true if expr is a bigfloat number, otherwise false.

Option variable: bftorat

Default value: false

bftorat controls the conversion of bfloats to rational numbers. When bftorat is false, ratepsilon will be used to control the conversion (this results in relatively small rational numbers). When bftorat is true, the rational number generated will accurately represent the bfloat.

Note: bftorat has no effect on the transformation to rational numbers with the function rationalize.

Example:

(%i1) ratepsilon:1e-4;
(%o1)                         1.e-4
(%i2) rat(bfloat(11111/111111)), bftorat:false;
`rat' replaced 9.99990999991B-2 by 1/10 = 1.0B-1
                               1
(%o2)/R/                       --
                               10
(%i3) rat(bfloat(11111/111111)), bftorat:true;
`rat' replaced 9.99990999991B-2 by 11111/111111 = 9.99990999991B-2
                             11111
(%o3)/R/                     ------
                             111111

Option variable: bftrunc

Default value: true

bftrunc causes trailing zeroes in non-zero bigfloat numbers not to be displayed. Thus, if bftrunc is false, bfloat (1) displays as 1.000000000000000B0. Otherwise, this is displayed as 1.0B0.

Function: evenp (expr)

Returns true if expr is a literal even integer, otherwise false.

evenp returns false if expr is a symbol, even if expr is declared even.

Function: float (expr)

Converts integers, rational numbers and bigfloats in expr to floating point numbers. It is also an evflag, float causes non-integral rational numbers and bigfloat numbers to be converted to floating point.

Option variable: float2bf

Default value: true

When float2bf is false, a warning message is printed when a floating point number is converted into a bigfloat number (since this may lead to loss of precision).

Function: floatnump (expr)

Returns true if expr is a floating point number, otherwise false.

Option variable: fpprec

Default value: 16

fpprec is the number of significant digits for arithmetic on bigfloat numbers. fpprec does not affect computations on ordinary floating point numbers.

See also bfloat and fpprintprec.

Option variable: fpprintprec

Default value: 0

fpprintprec is the number of digits to print when printing an ordinary float or bigfloat number.

For ordinary floating point numbers, when fpprintprec has a value between 2 and 16 (inclusive), the number of digits printed is equal to fpprintprec. Otherwise, fpprintprec is 0, or greater than 16, and the number of digits printed is 16.

For bigfloat numbers, when fpprintprec has a value between 2 and fpprec (inclusive), the number of digits printed is equal to fpprintprec. Otherwise, fpprintprec is 0, or greater than fpprec, and the number of digits printed is equal to fpprec.

For both ordinary floats and bigfloats, trailing zero digits are suppressed. The actual number of digits printed is less than fpprintprec if there are trailing zero digits.

fpprintprec cannot be 1.

Function: integerp (expr)

Returns true if expr is a literal numeric integer, otherwise false.

integerp returns false if expr is a symbol, even if expr is declared integer.

Examples:

(%i1) integerp (0);
(%o1)                         true
(%i2) integerp (1);
(%o2)                         true
(%i3) integerp (-17);
(%o3)                         true
(%i4) integerp (0.0);
(%o4)                         false
(%i5) integerp (1.0);
(%o5)                         false
(%i6) integerp (%pi);
(%o6)                         false
(%i7) integerp (n);
(%o7)                         false
(%i8) declare (n, integer);
(%o8)                         done
(%i9) integerp (n);
(%o9)                         false

Option variable: m1pbranch

Default value: false

m1pbranch is the principal branch for -1 to a power. Quantities such as (-1)^(1/3) (that is, an "odd" rational exponent) and (-1)^(1/4) (that is, an "even" rational exponent) are handled as follows:

              domain:real
                            
(-1)^(1/3):      -1         
(-1)^(1/4):   (-1)^(1/4)   

             domain:complex              
m1pbranch:false          m1pbranch:true
(-1)^(1/3)               1/2+%i*sqrt(3)/2
(-1)^(1/4)              sqrt(2)/2+%i*sqrt(2)/2

Function: nonnegintegerp (n)

Return true if and only if n >= 0 and n is an integer.

Function: numberp (expr)

Returns true if expr is a literal integer, rational number, floating point number, or bigfloat, otherwise false.

numberp returns false if expr is a symbol, even if expr is a symbolic number such as %pi or %i, or declared to be even, odd, integer, rational, irrational, real, imaginary, or complex.

Examples:

e, irrational, f, real, g, imaginary, h, complex);

(%i1) numberp (42);
(%o1)                         true
(%i2) numberp (-13/19);
(%o2)                         true
(%i3) numberp (3.14159);
(%o3)                         true
(%i4) numberp (-1729b-4);
(%o4)                         true
(%i5) map (numberp, [%e, %pi, %i, %phi, inf, minf]);
(%o5)      [false, false, false, false, false, false]
(%i6) declare (a, even, b, odd, c, integer, d, rational,
     e, irrational, f, real, g, imaginary, h, complex);
(%o6)                         done
(%i7) map (numberp, [a, b, c, d, e, f, g, h]);
(%o7) [false, false, false, false, false, false, false, false]

Option variable: numer

numer causes some mathematical functions (including exponentiation) with numerical arguments to be evaluated in floating point. It causes variables in expr which have been given numerals to be replaced by their values. It also sets the float switch on.

See also %enumer.

Examples:

(%i1) [sqrt(2), sin(1), 1/(1+sqrt(3))];
                                        1
(%o1)            [sqrt(2), sin(1), -----------]
                                   sqrt(3) + 1
(%i2) [sqrt(2), sin(1), 1/(1+sqrt(3))],numer;
(%o2) [1.414213562373095, 0.8414709848078965, 0.3660254037844387]

Option variable: numer_pbranch

Default value: false

The option variable numer_pbranch controls the numerical evaluation of the power of a negative integer, rational, or floating point number. When numer_pbranch is true and the exponent is a floating point number or the option variable numer is true too, Maxima evaluates the numerical result using the principal branch. Otherwise a simplified, but not an evaluated result is returned.

Examples:

(%i1) (-2)^0.75;
                                 0.75
(%o1)                       (- 2)
(%i2) (-2)^0.75,numer_pbranch:true;
(%o2)       1.189207115002721 %i - 1.189207115002721
(%i3) (-2)^(3/4);
                               3/4  3/4
(%o3)                     (- 1)    2
(%i4) (-2)^(3/4),numer;
                                          0.75
(%o4)              1.681792830507429 (- 1)
(%i5) (-2)^(3/4),numer,numer_pbranch:true;
(%o5)       1.189207115002721 %i - 1.189207115002721

Function: numerval (x_1, expr_1, …, var_n, expr_n)

Declares the variables x_1, …, x_n to have numeric values equal to expr_1, …, expr_n. The numeric value is evaluated and substituted for the variable in any expressions in which the variable occurs if the numer flag is true. See also ev.

The expressions expr_1, …, expr_n can be any expressions, not necessarily numeric.

Function: oddp (expr)

Returns true if expr is a literal odd integer, otherwise false.

oddp returns false if expr is a symbol, even if expr is declared odd.

Option variable: ratepsilon

Default value: 2.0e-15

ratepsilon is the tolerance used in the conversion of floating point numbers to rational numbers, when the option variable bftorat has the value false. See bftorat for an example.

Function: rationalize (expr)

Convert all double floats and big floats in the Maxima expression expr to their exact rational equivalents. If you are not familiar with the binary representation of floating point numbers, you might be surprised that rationalize (0.1) does not equal 1/10. This behavior isn’t special to Maxima – the number 1/10 has a repeating, not a terminating, binary representation.

(%i1) rationalize (0.5);
                                1
(%o1)                           -
                                2
(%i2) rationalize (0.1);
                        3602879701896397
(%o2)                   -----------------
                        36028797018963968
(%i3) fpprec : 5$
(%i4) rationalize (0.1b0);
                             209715
(%o4)                        -------
                             2097152
(%i5) fpprec : 20$
(%i6) rationalize (0.1b0);
                     236118324143482260685
(%o6)                ----------------------
                     2361183241434822606848
(%i7) rationalize (sin (0.1*x + 5.6));
               3602879701896397 x   3152519739159347
(%o7)      sin(------------------ + ----------------)
               36028797018963968    562949953421312

Function: ratnump (expr)

Returns true if expr is a literal integer or ratio of literal integers, otherwise false.


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5.2 Strings


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5.2.1 Introduction to Strings

Strings (quoted character sequences) are enclosed in double quote marks " for input, and displayed with or without the quote marks, depending on the global variable stringdisp.

Strings may contain any characters, including embedded tab, newline, and carriage return characters. The sequence \" is recognized as a literal double quote, and \\ as a literal backslash. When backslash appears at the end of a line, the backslash and the line termination (either newline or carriage return and newline) are ignored, so that the string continues with the next line. No other special combinations of backslash with another character are recognized; when backslash appears before any character other than ", \, or a line termination, the backslash is ignored. There is no way to represent a special character (such as tab, newline, or carriage return) except by embedding the literal character in the string.

There is no character type in Maxima; a single character is represented as a one-character string.

The stringproc add-on package contains many functions for working with strings.

Examples:

(%i1) s_1 : "This is a string.";
(%o1)                   This is a string.
(%i2) s_2 : "Embedded \"double quotes\" and backslash \\ characters.";
(%o2) Embedded "double quotes" and backslash \ characters.
(%i3) s_3 : "Embedded line termination
(%o3) Embedded line termination
in this string.
(%i4) in this string.";
(%o4) Ignore the line termination characters in this string.
(%i5) s_4 : "Ignore the \
(%o5)                         false
(%i6) line termination \
(%o6)                   This is a string.
(%i7) characters in \
(%o7)                         true
(%i8) this string.";
(%o8)                  "This is a string."
(%i9) stringdisp : false;
·

Category: Syntax


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5.2.2 Functions and Variables for Strings

Function: concat (arg_1, arg_2, …)

Concatenates its arguments. The arguments must evaluate to atoms. The return value is a symbol if the first argument is a symbol and a string otherwise.

concat evaluates its arguments. The single quote ' prevents evaluation.

See also sconcat and eval_string.

(%i1) y: 7$
(%i2) z: 88$
(%i3) concat (y, z/2);
(%o3)                          744
(%i4) concat ('y, z/2);
(%o4)                          y44

A symbol constructed by concat may be assigned a value and appear in expressions. The :: (double colon) assignment operator evaluates its left-hand side.

(%i5) a: concat ('y, z/2);
(%o5)                          y44
(%i6) a:: 123;
(%o6)                          123
(%i7) y44;
(%o7)                          123
(%i8) b^a;
                               y44
(%o8)                         b
(%i9) %, numer;
                               123
(%o9)                         b

Note that although concat (1, 2) looks like a number, it is a string.

(%i10) concat (1, 2) + 3;
(%o10)                       12 + 3

Function: sconcat (arg_1, arg_2, …)

Concatenates its arguments into a string. Unlike concat, the arguments do not need to be atoms.

See also concat, and eval_string. For complex string conversions see also printf.

(%i1) sconcat ("xx[", 3, "]:", expand ((x+y)^3));
(%o1)             xx[3]:y^3+3*x*y^2+3*x^2*y+x^3

Another purpose for sconcat is to convert arbitrary objects to strings.

(%i1) sconcat (x);
(%o1)                           x
(%i2) stringp(%);
(%o2)                         true

Function: string (expr)

Converts expr to Maxima’s linear notation just as if it had been typed in.

The return value of string is a string, and thus it cannot be used in a computation.

Option variable: stringdisp

Default value: false

When stringdisp is true, strings are displayed enclosed in double quote marks. Otherwise, quote marks are not displayed.

stringdisp is always true when displaying a function definition.

Examples:

(%i1) stringdisp: false$
(%i2) "This is an example string.";
(%o2)              This is an example string.
(%i3) foo () :=
      print ("This is a string in a function definition.");
(%o3) foo() := 
              print("This is a string in a function definition.")
(%i4) stringdisp: true$
(%i5) "This is an example string.";
(%o5)             "This is an example string."


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5.3 Constants


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5.3.1 Functions and Variables for Constants

Constant: %e

%e represents the base of the natural logarithm, also known as Euler’s number. The numeric value of %e is the double-precision floating-point value 2.718281828459045d0.

Constant: %i

%i represents the imaginary unit, sqrt(- 1).

Constant: false

false represents the Boolean constant of the same name. Maxima implements false by the value NIL in Lisp.

Constant: %gamma

The Euler-Mascheroni constant, 0.5772156649015329 ....

Constant: ind

ind represents a bounded, indefinite result.

See also limit.

Example:

(%i1) limit (sin(1/x), x, 0);
(%o1)                          ind

Constant: inf

inf represents real positive infinity.

Constant: infinity

infinity represents complex infinity.

Constant: minf

minf represents real minus (i.e., negative) infinity.

Constant: %phi

%phi represents the so-called golden mean, (1 + sqrt(5))/2. The numeric value of %phi is the double-precision floating-point value 1.618033988749895d0.

fibtophi expresses Fibonacci numbers fib(n) in terms of %phi.

By default, Maxima does not know the algebraic properties of %phi. After evaluating tellrat(%phi^2 - %phi - 1) and algebraic: true, ratsimp can simplify some expressions containing %phi.

Examples:

fibtophi expresses Fibonacci numbers fib(n) in terms of %phi.

(%i1) fibtophi (fib (n));
                           n             n
                       %phi  - (1 - %phi)
(%o1)                  -------------------
                           2 %phi - 1
(%i2) fib (n-1) + fib (n) - fib (n+1);
(%o2)          - fib(n + 1) + fib(n) + fib(n - 1)
(%i3) fibtophi (%);
            n + 1             n + 1       n             n
        %phi      - (1 - %phi)        %phi  - (1 - %phi)
(%o3) - --------------------------- + -------------------
                2 %phi - 1                2 %phi - 1
                                          n - 1             n - 1
                                      %phi      - (1 - %phi)
                                    + ---------------------------
                                              2 %phi - 1
(%i4) ratsimp (%);
(%o4)                           0

By default, Maxima does not know the algebraic properties of %phi. After evaluating tellrat (%phi^2 - %phi - 1) and algebraic: true, ratsimp can simplify some expressions containing %phi.

(%i1) e : expand ((%phi^2 - %phi - 1) * (A + 1));
                 2                      2
(%o1)        %phi  A - %phi A - A + %phi  - %phi - 1
(%i2) ratsimp (e);
                  2                     2
(%o2)        (%phi  - %phi - 1) A + %phi  - %phi - 1
(%i3) tellrat (%phi^2 - %phi - 1);
                            2
(%o3)                  [%phi  - %phi - 1]
(%i4) algebraic : true;
(%o4)                         true
(%i5) ratsimp (e);
(%o5)                           0

Constant: %pi

%pi represents the ratio of the perimeter of a circle to its diameter. The numeric value of %pi is the double-precision floating-point value 3.141592653589793d0.

Constant: true

true represents the Boolean constant of the same name. Maxima implements true by the value T in Lisp.

Constant: und

und represents an undefined result.

See also limit.

Example:

(%i1) limit (x*sin(x), x, inf);
(%o1)                          und

Constant: zeroa

zeroa represents an infinitesimal above zero. zeroa can be used in expressions. limit simplifies expressions which contain infinitesimals.

See also zerob and limit.

Example:

limit simplifies expressions which contain infinitesimals:

(%i1) limit(zeroa);
(%o1)                           0
(%i2) limit(x+zeroa);
(%o2)                           x

Constant: zerob

zerob represents an infinitesimal below zero. zerob can be used in expressions. limit simplifies expressions which contain infinitesimals.

See also zeroa and limit.


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5.4 Lists


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5.4.1 Introduction to Lists

Lists are the basic building block for Maxima and Lisp. All data types other than arrays, hashed arrays and numbers are represented as Lisp lists, These Lisp lists have the form

((MPLUS) $A 2)

to indicate an expression a+2. At Maxima level one would see the infix notation a+2. Maxima also has lists which are printed as

[1, 2, 7, x+y]

for a list with 4 elements. Internally this corresponds to a Lisp list of the form

((MLIST) 1 2 7 ((MPLUS) $X $Y))

The flag which denotes the type field of the Maxima expression is a list itself, since after it has been through the simplifier the list would become

((MLIST SIMP) 1 2 7 ((MPLUS SIMP) $X $Y))

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5.4.2 Functions and Variables for Lists

Operator: [
Operator: ]

[ and ] mark the beginning and end, respectively, of a list.

[ and ] also enclose the subscripts of a list, array, hashed array, or memoizing function. Note that other than for arrays accessing the nth element of a list may need an amount of time that is roughly proportional to n, See Performance considerations for Lists.

Note that if an element of a subscripted variable is written to before a list or an array of this name is declared a hashed array (see Arrays) is created, not a list.

Examples:

(%i1) x: [a, b, c];
(%o1)                       [a, b, c]
(%i2) x[3];
(%o2)                           c
(%i3) array (y, fixnum, 3);
(%o3)                           y
(%i4) y[2]: %pi;
(%o4)                          %pi
(%i5) y[2];
(%o5)                          %pi
(%i6) z['foo]: 'bar;
(%o6)                          bar
(%i7) z['foo];
(%o7)                          bar
(%i8) g[k] := 1/(k^2+1);
                                  1
(%o8)                     g  := ------
                           k     2
                                k  + 1
(%i9) g[10];
                                1
(%o9)                          ---
                               101

Function: append (list_1, …, list_n)

Returns a single list of the elements of list_1 followed by the elements of list_2, … append also works on general expressions, e.g. append (f(a,b), f(c,d,e)); yields f(a,b,c,d,e).

See also addrow, addcol and join.

Do example(append); for an example.

Function: assoc
    assoc (key, list, default)
    assoc (key, list)

This function searches for key in the left hand side of the input list. The list argument should be a list, each of whose elements is an expression with exactly two parts. Most usually, the elements of list are themselves lists, each with two elements.

The assoc function iterates along list, checking the first part of each element for equality with key. If an element is found where the comparison is true, assoc returns the second part of that element. If there is no such element in the list, assoc returns either false or default, if given.

For example, in the expression assoc (y, [[x,1], [y,2], [z,3]]), the assoc function searches for x in the left hand side of the list [[y,1],[x,2]] and finds it at the second term, returning 2. In assoc (z, [[x,1], [z,2], [z,3]]), the search stops at the first term starting with z and returns 2. In assoc(x, [[y,1]]), there is no matching element, so assoc returns false.

(%i1) assoc (y, [[x,1], [y,2],[z,3]]);
(%o1)                           2
(%i2) assoc (z, [[x,1], [z,2], [z,3]]);
(%o2)                           2
(%i3) assoc (x, [[y,1]]);
(%o3)                         false

Function: cons
    cons (expr, list)
    cons (expr_1, expr_2)

cons (expr, list) returns a new list constructed of the element expr as its first element, followed by the elements of list. This is analogous to the Lisp language construction operation "cons".

The Maxima function cons can also be used where the second argument is other than a list and this might be useful. In this case, cons (expr_1, expr_2) returns an expression with same operator as expr_2 but with argument cons(expr_1, args(expr_2)). Examples:

(%i1) cons(a,[b,c,d]);
(%o1)                     [a, b, c, d]
(%i2) cons(a,f(b,c,d));
(%o2)                     f(a, b, c, d)

In general, cons applied to a nonlist doesn’t make sense. For instance, cons(a,b^c) results in an illegal expression, since ’^’ cannot take three arguments.

When inflag is true, cons operates on the internal structure of an expression, otherwise cons operates on the displayed form. Especially when inflag is true, cons applied to a nonlist sometimes gives a surprising result; for example

(%i1) cons(a,-a), inflag : true;
                                 2
(%o1)                         - a
(%i2) cons(a,-a), inflag : false;
(%o2)                           0

Function: copylist (list)

Returns a copy of the list list.

Function: create_list (form, x_1, list_1, …, x_n, list_n)

Create a list by evaluating form with x_1 bound to each element of list_1, and for each such binding bind x_2 to each element of list_2, … The number of elements in the result will be the product of the number of elements in each list. Each variable x_i must actually be a symbol – it will not be evaluated. The list arguments will be evaluated once at the beginning of the iteration.

(%i1) create_list (x^i, i, [1, 3, 7]);
                                3   7
(%o1)                      [x, x , x ]

With a double iteration:

(%i1) create_list ([i, j], i, [a, b], j, [e, f, h]);
(%o1)   [[a, e], [a, f], [a, h], [b, e], [b, f], [b, h]]

Instead of list_i two args may be supplied each of which should evaluate to a number. These will be the inclusive lower and upper bounds for the iteration.

(%i1) create_list ([i, j], i, [1, 2, 3], j, 1, i);
(%o1)   [[1, 1], [2, 1], [2, 2], [3, 1], [3, 2], [3, 3]]

Note that the limits or list for the j variable can depend on the current value of i.

Function: delete
    delete (expr_1, expr_2)
    delete (expr_1, expr_2, n)

delete(expr_1, expr_2) removes from expr_2 any arguments of its top-level operator which are the same (as determined by "=") as expr_1. Note that "=" tests for formal equality, not equivalence. Note also that arguments of subexpressions are not affected.

expr_1 may be an atom or a non-atomic expression. expr_2 may be any non-atomic expression. delete returns a new expression; it does not modify expr_2.

delete(expr_1, expr_2, n) removes from expr_2 the first n arguments of the top-level operator which are the same as expr_1. If there are fewer than n such arguments, then all such arguments are removed.

Examples:

Removing elements from a list.

(%i1) delete (y, [w, x, y, z, z, y, x, w]);
(%o1)                  [w, x, z, z, x, w]

Removing terms from a sum.

(%i1) delete (sin(x), x + sin(x) + y);
(%o1)                         y + x

Removing factors from a product.

(%i1) delete (u - x, (u - w)*(u - x)*(u - y)*(u - z));
(%o1)                (u - w) (u - y) (u - z)

Removing arguments from an arbitrary expression.

(%i1) delete (a, foo (a, b, c, d, a));
(%o1)                     foo(b, c, d)

Limit the number of removed arguments.

(%i1) delete (a, foo (a, b, a, c, d, a), 2);
(%o1)                    foo(b, c, d, a)

Whether arguments are the same as expr_1 is determined by "=". Arguments which are equal but not "=" are not removed.

(%i1) [is (equal (0, 0)), is (equal (0, 0.0)), is (equal (0, 0b0))];
(%o1)                  [true, true, true]
(%i2) [is (0 = 0), is (0 = 0.0), is (0 = 0b0)];
(%o2)                 [true, false, false]
(%i3) delete (0, [0, 0.0, 0b0]);
(%o3)                     [0.0, 0.0b0]
(%i4) is (equal ((x + y)*(x - y), x^2 - y^2));
(%o4)                         true
(%i5) is ((x + y)*(x - y) = x^2 - y^2);
(%o5)                         false
(%i6) delete ((x + y)*(x - y), [(x + y)*(x - y), x^2 - y^2]);
                              2    2
(%o6)                       [x  - y ]

Function: eighth (expr)

Returns the 8’th item of expression or list expr. See first for more details.

Function: endcons
    endcons (expr, list)
    endcons (expr_1, expr_2)

endcons (expr, list) returns a new list constructed of the elements of list followed by expr. The Maxima function endcons can also be used where the second argument is other than a list and this might be useful. In this case, endcons (expr_1, expr_2) returns an expression with same operator as expr_2 but with argument endcons(expr_1, args(expr_2)). Examples:

(%i1) endcons(a,[b,c,d]);
(%o1)                     [b, c, d, a]
(%i2) endcons(a,f(b,c,d));
(%o2)                     f(b, c, d, a)

In general, endcons applied to a nonlist doesn’t make sense. For instance, endcons(a,b^c) results in an illegal expression, since ’^’ cannot take three arguments.

When inflag is true, endcons operates on the internal structure of an expression, otherwise endcons operates on the displayed form. Especially when inflag is true, endcons applied to a nonlist sometimes gives a surprising result; for example

(%i1) endcons(a,-a), inflag : true;
                                 2
(%o1)                         - a
(%i2) endcons(a,-a), inflag : false;
(%o2)                           0

Function: fifth (expr)

Returns the 5’th item of expression or list expr. See first for more details.

Function: first (expr)

Returns the first part of expr which may result in the first element of a list, the first row of a matrix, the first term of a sum, etc.:

(%i1) matrix([1,2],[3,4]);
                                   [ 1  2 ]
(%o1)                              [      ]
                                   [ 3  4 ]
(%i2) first(%);
(%o2)                              [1,2]
(%i3) first(%);
(%o3)                              1
(%i4) first(a*b/c+d+e/x);
                                   a b
(%o4)                              ---
                                    c
(%i5) first(a=b/c+d+e/x);
(%o5)                              a

Note that first and its related functions, rest and last, work on the form of expr which is displayed not the form which is typed on input. If the variable inflag is set to true however, these functions will look at the internal form of expr. One reason why this may make a difference is that the simplifier re-orders expressions:

(%i1) x+y;
(%o1)                              y+1
(%i2) first(x+y),inflag : true;
(%o2)                              x
(%i3) first(x+y),inflag : false;
(%o3)                              y

The functions secondtenth yield the second through the tenth part of their input argument.

See also firstn and part.

Function: firstn (expr, count)

Returns the first count arguments of expr, if expr has at least count arguments. Returns expr if expr has less than count arguments.

expr may be any nonatomic expression. When expr is something other than a list, firstn returns an expression which has the same operator as expr. count must be a nonnegative integer.

firstn honors the global flag inflag, which governs whether the internal form of an expression is processed (when inflag is true) or the displayed form (when inflag is false).

Note that firstn(expr, 1), which returns a nonatomic expression containing the first argument, is not the same as first(expr), which returns the first argument by itself.

See also lastn and rest.

Examples:

firstn returns the first count elements of expr, if expr has at least count elements.

(%i1) mylist : [1, a, 2, b, 3, x, 4 - y, 2*z + sin(u)];
(%o1)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]
(%i2) firstn (mylist, 0);
(%o2)                          []
(%i3) firstn (mylist, 1);
(%o3)                          [1]
(%i4) firstn (mylist, 2);
(%o4)                        [1, a]
(%i5) firstn (mylist, 7);
(%o5)               [1, a, 2, b, 3, x, 4 - y]

firstn returns expr if expr has less than count elements.

(%i1) mylist : [1, a, 2, b, 3, x, 4 - y, 2*z + sin(u)];
(%o1)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]
(%i2) firstn (mylist, 100);
(%o2)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

expr may be any nonatomic expression.

(%i1) myfoo : foo(1, a, 2, b, 3, x, 4 - y, 2*z + sin(u));
(%o1)      foo(1, a, 2, b, 3, x, 4 - y, 2 z + sin(u))
(%i2) firstn (myfoo, 4);
(%o2)                    foo(1, a, 2, b)
(%i3) mybar : bar[m, n](1, a, 2, b, 3, x, 4 - y, 2*z + sin(u));
(%o3)    bar    (1, a, 2, b, 3, x, 4 - y, 2 z + sin(u))
            m, n
(%i4) firstn (mybar, 4);
(%o4)                  bar    (1, a, 2, b)
                          m, n
(%i5) mymatrix : genmatrix (lambda ([i, j], 10*i + j), 10, 4) $
(%i6) firstn (mymatrix, 3);
                       [ 11  12  13  14 ]
                       [                ]
(%o6)                  [ 21  22  23  24 ]
                       [                ]
                       [ 31  32  33  34 ]

firstn honors the global flag inflag.

(%i1) myexpr : a + b + c + d + e;
(%o1)                   e + d + c + b + a
(%i2) firstn (myexpr, 3), inflag=true;
(%o2)                       c + b + a
(%i3) firstn (myexpr, 3), inflag=false;
(%o3)                       e + d + c

Note that firstn(expr, 1) is not the same as first(expr).

(%i1) firstn ([w, x, y, z], 1);
(%o1)                          [w]
(%i2) first ([w, x, y, z]);
(%o2)                           w

Function: fourth (expr)

Returns the 4’th item of expression or list expr. See first for more details.

Function: join (l, m)

Creates a new list containing the elements of lists l and m, interspersed. The result has elements [l[1], m[1], l[2], m[2], ...]. The lists l and m may contain any type of elements.

If the lists are different lengths, join ignores elements of the longer list.

Maxima complains if l or m is not a list.

See also append.

Examples:

(%i1) L1: [a, sin(b), c!, d - 1];
(%o1)                [a, sin(b), c!, d - 1]
(%i2) join (L1, [1, 2, 3, 4]);
(%o2)          [a, 1, sin(b), 2, c!, 3, d - 1, 4]
(%i3) join (L1, [aa, bb, cc, dd, ee, ff]);
(%o3)        [a, aa, sin(b), bb, c!, cc, d - 1, dd]

Function: last (expr)

Returns the last part (term, row, element, etc.) of the expr.

See also lastn.

Function: lastn (expr, count)

Returns the last count arguments of expr, if expr has at least count arguments. Returns expr if expr has less than count arguments.

expr may be any nonatomic expression. When expr is something other than a list, lastn returns an expression which has the same operator as expr. count must be a nonnegative integer.

lastn honors the global flag inflag, which governs whether the internal form of an expression is processed (when inflag is true) or the displayed form (when inflag is false).

Note that lastn(expr, 1), which returns a nonatomic expression containing the last argument, is not the same as last(expr), which returns the last argument by itself.

See also firstn and rest.

Examples:

lastn returns the last count elements of expr, if expr has at least count elements.

(%i1) mylist : [1, a, 2, b, 3, x, 4 - y, 2*z + sin(u)];
(%o1)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]
(%i2) lastn (mylist, 0);
(%o2)                          []
(%i3) lastn (mylist, 1);
(%o3)                    [2 z + sin(u)]
(%i4) lastn (mylist, 2);
(%o4)                 [4 - y, 2 z + sin(u)]
(%i5) lastn (mylist, 7);
(%o5)         [a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

lastn returns expr if expr has less than count elements.

(%i1) mylist : [1, a, 2, b, 3, x, 4 - y, 2*z + sin(u)];
(%o1)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]
(%i2) lastn (mylist, 100);
(%o2)        [1, a, 2, b, 3, x, 4 - y, 2 z + sin(u)]

expr may be any nonatomic expression.

(%i1) myfoo : foo(1, a, 2, b, 3, x, 4 - y, 2*z + sin(u));
(%o1)      foo(1, a, 2, b, 3, x, 4 - y, 2 z + sin(u))
(%i2) lastn (myfoo, 4);
(%o2)            foo(3, x, 4 - y, 2 z + sin(u))
(%i3) mybar : bar[m, n](1, a, 2, b, 3, x, 4 - y, 2*z + sin(u));
(%o3)    bar    (1, a, 2, b, 3, x, 4 - y, 2 z + sin(u))
            m, n
(%i4) lastn (mybar, 4);
(%o4)          bar    (3, x, 4 - y, 2 z + sin(u))
                  m, n
(%i5) mymatrix : genmatrix (lambda ([i, j], 10*i + j), 10, 4) $
(%i6) lastn (mymatrix, 3);
                     [ 81   82   83   84  ]
                     [                    ]
(%o6)                [ 91   92   93   94  ]
                     [                    ]
                     [ 101  102  103  104 ]

lastn honors the global flag inflag.

(%i1) myexpr : a + b + c + d + e;
(%o1)                   e + d + c + b + a
(%i2) lastn (myexpr, 3), inflag=true;
(%o2)                       e + d + c
(%i3) lastn (myexpr, 3), inflag=false;
(%o3)                       c + b + a

Note that lastn(expr, 1) is not the same as last(expr).

(%i1) lastn ([w, x, y, z], 1);
(%o1)                          [z]
(%i2) last ([w, x, y, z]);
(%o2)                           z

Function: length (expr)

Returns (by default) the number of parts in the external (displayed) form of expr. For lists this is the number of elements, for matrices it is the number of rows, and for sums it is the number of terms (see dispform).

The length command is affected by the inflag switch. So, e.g. length(a/(b*c)); gives 2 if inflag is false (Assuming exptdispflag is true), but 3 if inflag is true (the internal representation is essentially a*b^-1*c^-1).

Determining a list’s length typically needs an amount of time proportional to the number of elements in the list. If the length of a list is used inside a loop it therefore might drastically increase the performance if the length is calculated outside the loop instead.

Option variable: listarith

Default value: true

If false causes any arithmetic operations with lists to be suppressed; when true, list-matrix operations are contagious causing lists to be converted to matrices yielding a result which is always a matrix. However, list-list operations should return lists.

Function: listp (expr)

Returns true if expr is a list else false.

Function: lreduce
    lreduce (F, s)
    lreduce (F, s, s_0)

Extends the binary function F to an n-ary function by composition, where s is a list.

lreduce(F, s) returns F(... F(F(s_1, s_2), s_3), ... s_n). When the optional argument s_0 is present, the result is equivalent to lreduce(F, cons(s_0, s)).

The function F is first applied to the leftmost list elements, thus the name "lreduce".

See also rreduce, xreduce, and tree_reduce.

Examples:

lreduce without the optional argument.

(%i1) lreduce (f, [1, 2, 3]);
(%o1)                     f(f(1, 2), 3)
(%i2) lreduce (f, [1, 2, 3, 4]);
(%o2)                  f(f(f(1, 2), 3), 4)

lreduce with the optional argument.

(%i1) lreduce (f, [1, 2, 3], 4);
(%o1)                  f(f(f(4, 1), 2), 3)

lreduce applied to built-in binary operators. / is the division operator.

(%i1) lreduce ("^", args ({a, b, c, d}));
                               b c d
(%o1)                       ((a ) )
(%i2) lreduce ("/", args ({a, b, c, d}));
                                a
(%o2)                         -----
                              b c d

Function: makelist
    makelist ()
    makelist (expr, n)
    makelist (expr, i, i_max)
    makelist (expr, i, i_0, i_max)
    makelist (expr, i, i_0, i_max, step)
    makelist (expr, x, list)

The first form, makelist (), creates an empty list. The second form, makelist (expr), creates a list with expr as its single element. makelist (expr, n) creates a list of n elements generated from expr.

The most general form, makelist (expr, i, i_0, i_max, step), returns the list of elements obtained when ev (expr, i=j) is applied to the elements j of the sequence: i_0, i_0 + step, i_0 + 2*step, ..., with |j| less than or equal to |i_max|.

The increment step can be a number (positive or negative) or an expression. If it is omitted, the default value 1 will be used. If both i_0 and step are omitted, they will both have a default value of 1.

makelist (expr, x, list) returns a list, the j’th element of which is equal to ev (expr, x=list[j]) for j equal to 1 through length (list).

Examples:

(%i1) makelist (concat (x,i), i, 6);
(%o1)               [x1, x2, x3, x4, x5, x6]
(%i2) makelist (x=y, y, [a, b, c]);
(%o2)                 [x = a, x = b, x = c]
(%i3) makelist (x^2, x, 3, 2*%pi, 2);
(%o3)                        [9, 25]
(%i4) makelist (random(6), 4);
(%o4)                     [2, 0, 2, 5]
(%i5) flatten (makelist (makelist (i^2, 3), i, 4));
(%o5)        [1, 1, 1, 4, 4, 4, 9, 9, 9, 16, 16, 16]
(%i6) flatten (makelist (makelist (i^2, i, 3), 4));
(%o6)         [1, 4, 9, 1, 4, 9, 1, 4, 9, 1, 4, 9]

Function: member (expr_1, expr_2)

Returns true if is(expr_1 = a) for some element a in args(expr_2), otherwise returns false.

expr_2 is typically a list, in which case args(expr_2) = expr_2 and is(expr_1 = a) for some element a in expr_2 is the test.

member does not inspect parts of the arguments of expr_2, so it may return false even if expr_1 is a part of some argument of expr_2.

See also elementp.

Examples:

(%i1) member (8, [8, 8.0, 8b0]);
(%o1)                         true
(%i2) member (8, [8.0, 8b0]);
(%o2)                         false
(%i3) member (b, [a, b, c]);
(%o3)                         true
(%i4) member (b, [[a, b], [b, c]]);
(%o4)                         false
(%i5) member ([b, c], [[a, b], [b, c]]);
(%o5)                         true
(%i6) F (1, 1/2, 1/4, 1/8);
                               1  1  1
(%o6)                     F(1, -, -, -)
                               2  4  8
(%i7) member (1/8, %);
(%o7)                         true
(%i8) member ("ab", ["aa", "ab", sin(1), a + b]);
(%o8)                         true

Function: ninth (expr)

Returns the 9’th item of expression or list expr. See first for more details.

Function: pop (list)

pop removes and returns the first element from the list list. The argument list must be a mapatom that is bound to a nonempty list. If the argument list is not bound to a nonempty list, Maxima signals an error. For examples, see push.

Function: push (item, list)

push prepends the item item to the list list and returns a copy of the new list. The second argument list must be a mapatom that is bound to a list. The first argument item can be any Maxima symbol or expression. If the argument list is not bound to a list, Maxima signals an error.

To remove the first item from a list, see pop.

Examples:

(%i1) ll: [];
(%o1)                          []
(%i2) push (x, ll);
(%o2)                          [x]
(%i3) push (x^2+y, ll);
                                 2
(%o3)                      [y + x , x]
(%i4) a: push ("string", ll);
                                     2
(%o4)                  [string, y + x , x]
(%i5) pop (ll);
(%o5)                        string
(%i6) pop (ll);
                                  2
(%o6)                        y + x
(%i7) pop (ll);
(%o7)                           x
(%i8) ll;
(%o8)                          []
(%i9) a;
                                     2
(%o9)                  [string, y + x , x]

Function: rest
    rest (expr, n)
    rest (expr)

Returns expr with its first n elements removed if n is positive and its last - n elements removed if n is negative. If n is 1 it may be omitted. The first argument expr may be a list, matrix, or other expression. When expr is an atom, rest signals an error; when expr is an empty list and partswitch is false, rest signals an error. When expr is an empty list and partswitch is true, rest returns end.

Applying rest to expression such as f(a,b,c) returns f(b,c). In general, applying rest to an nonlist doesn’t make sense. For example, because ’^’ requires two arguments, rest(a^b) results in an error message. The functions args and op may be useful as well, since args(a^b) returns [a,b] and op(a^b) returns ^.

See also firstn and lastn.

(%i1) rest(a+b+c);
(%o1) b+a
(%i2) rest(a+b+c,2);
(%o2) a
(%i3) rest(a+b+c,-2);
(%o3) c

Function: reverse (list)

Reverses the order of the members of the list (not the members themselves). reverse also works on general expressions, e.g. reverse(a=b); gives b=a.

Function: rreduce
    rreduce (F, s)
    rreduce (F, s, s_{n + 1})

Extends the binary function F to an n-ary function by composition, where s is a list.

rreduce(F, s) returns F(s_1, ... F(s_{n - 2}, F(s_{n - 1}, s_n))). When the optional argument s_{n + 1} is present, the result is equivalent to rreduce(F, endcons(s_{n + 1}, s)).

The function F is first applied to the rightmost list elements, thus the name "rreduce".

See also lreduce, tree_reduce, and xreduce.

Examples:

rreduce without the optional argument.

(%i1) rreduce (f, [1, 2, 3]);
(%o1)                     f(1, f(2, 3))
(%i2) rreduce (f, [1, 2, 3, 4]);
(%o2)                  f(1, f(2, f(3, 4)))

rreduce with the optional argument.

(%i1) rreduce (f, [1, 2, 3], 4);
(%o1)                  f(1, f(2, f(3, 4)))

rreduce applied to built-in binary operators. / is the division operator.

(%i1) rreduce ("^", args ({a, b, c, d}));
                                 d
                                c
                               b
(%o1)                         a
(%i2) rreduce ("/", args ({a, b, c, d}));
                               a c
(%o2)                          ---
                               b d

Function: second (expr)

Returns the 2’nd item of expression or list expr. See first for more details.

Function: seventh (expr)

Returns the 7’th item of expression or list expr. See first for more details.

Function: sixth (expr)

Returns the 6’th item of expression or list expr. See first for more details.

Function: sort
    sort (L, P)
    sort (L)

sort(L, P) sorts a list L according to a predicate P of two arguments which defines a strict weak order on the elements of L. If P(a, b) is true, then a appears before b in the result. If neither P(a, b) nor P(b, a) are true, then a and b are equivalent, and appear in the result in the same order as in the input. That is, sort is a stable sort.

If P(a, b) and P(b, a) are both true for some elements of L, then P is not a valid sort predicate, and the result is undefined. If P(a, b) is something other than true or false, sort signals an error.

The predicate may be specified as the name of a function or binary infix operator, or as a lambda expression. If specified as the name of an operator, the name must be enclosed in double quotes.

The sorted list is returned as a new object; the argument L is not modified.

sort(L) is equivalent to sort(L, orderlessp).

The default sorting order is ascending, as determined by orderlessp. The predicate ordergreatp sorts a list in descending order.

All Maxima atoms and expressions are comparable under orderlessp and ordergreatp.

Operators < and > order numbers, constants, and constant expressions by magnitude. Note that orderlessp and ordergreatp do not order numbers, constants, and constant expressions by magnitude.

ordermagnitudep orders numbers, constants, and constant expressions the same as <, and all other elements the same as orderlessp.

Examples:

sort sorts a list according to a predicate of two arguments which defines a strict weak order on the elements of the list.

(%i1) sort ([1, a, b, 2, 3, c], 'orderlessp);
(%o1)                  [1, 2, 3, a, b, c]
(%i2) sort ([1, a, b, 2, 3, c], 'ordergreatp);
(%o2)                  [c, b, a, 3, 2, 1]

The predicate may be specified as the name of a function or binary infix operator, or as a lambda expression. If specified as the name of an operator, the name must be enclosed in double quotes.

(%i1) L : [[1, x], [3, y], [4, w], [2, z]];
(%o1)           [[1, x], [3, y], [4, w], [2, z]]
(%i2) foo (a, b) := a[1] > b[1];
(%o2)                 foo(a, b) := a  > b
                                    1    1
(%i3) sort (L, 'foo);
(%o3)           [[4, w], [3, y], [2, z], [1, x]]
(%i4) infix (">>");
(%o4)                          >>
(%i5) a >> b := a[1] > b[1];
(%o5)                  (a >> b) := a  > b
                                    1    1
(%i6) sort (L, ">>");
(%o6)           [[4, w], [3, y], [2, z], [1, x]]
(%i7) sort (L, lambda ([a, b], a[1] > b[1]));
(%o7)           [[4, w], [3, y], [2, z], [1, x]]

sort(L) is equivalent to sort(L, orderlessp).

(%i1) L : [a, 2*b, -5, 7, 1 + %e, %pi];
(%o1)             [a, 2 b, - 5, 7, %e + 1, %pi]
(%i2) sort (L);
(%o2)             [- 5, 7, %e + 1, %pi, a, 2 b]
(%i3) sort (L, 'orderlessp);
(%o3)             [- 5, 7, %e + 1, %pi, a, 2 b]

The default sorting order is ascending, as determined by orderlessp. The predicate ordergreatp sorts a list in descending order.

(%i1) L : [a, 2*b, -5, 7, 1 + %e, %pi];
(%o1)             [a, 2 b, - 5, 7, %e + 1, %pi]
(%i2) sort (L);
(%o2)             [- 5, 7, %e + 1, %pi, a, 2 b]
(%i3) sort (L, 'ordergreatp);
(%o3)             [2 b, a, %pi, %e + 1, 7, - 5]

All Maxima atoms and expressions are comparable under orderlessp and ordergreatp.

(%i1) L : [11, -17, 29b0, 9*c, 7.55, foo(x, y), -5/2, b + a];
                                                 5
(%o1)  [11, - 17, 2.9b1, 9 c, 7.55, foo(x, y), - -, b + a]
                                                 2
(%i2) sort (L, orderlessp);
                5
(%o2)  [- 17, - -, 7.55, 11, 2.9b1, b + a, 9 c, foo(x, y)]
                2
(%i3) sort (L, ordergreatp);
                                                  5
(%o3)  [foo(x, y), 9 c, b + a, 2.9b1, 11, 7.55, - -, - 17]
                                                  2

Operators < and > order numbers, constants, and constant expressions by magnitude. Note that orderlessp and ordergreatp do not order numbers, constants, and constant expressions by magnitude.

(%i1) L : [%pi, 3, 4, %e, %gamma];
(%o1)                [%pi, 3, 4, %e, %gamma]
(%i2) sort (L, ">");
(%o2)                [4, %pi, 3, %e, %gamma]
(%i3) sort (L, ordergreatp);
(%o3)                [%pi, %gamma, %e, 4, 3]

ordermagnitudep orders numbers, constants, and constant expressions the same as <, and all other elements the same as orderlessp.

(%i1) L : [%i, 1+%i, 2*x, minf, inf, %e, sin(1), 0, 1, 2, 3, 1.0, 1.0b0];
(%o1) [%i, %i + 1, 2 x, minf, inf, %e, sin(1), 0, 1, 2, 3, 1.0, 
                                                           1.0b0]
(%i2) sort (L, ordermagnitudep);
(%o2) [minf, 0, sin(1), 1, 1.0, 1.0b0, 2, %e, 3, inf, %i, 
                                                     %i + 1, 2 x]
(%i3) sort (L, orderlessp);
(%o3) [0, 1, 1.0, 2, 3, sin(1), 1.0b0, %e, %i, %i + 1, inf, 
                                                       minf, 2 x]

Function: sublist (list, p)

Returns the list of elements of list for which the predicate p returns true.

Example:

(%i1) L: [1, 2, 3, 4, 5, 6];
(%o1)                  [1, 2, 3, 4, 5, 6]
(%i2) sublist (L, evenp);
(%o2)                       [2, 4, 6]

Function: sublist_indices (L, P)

Returns the indices of the elements x of the list L for which the predicate maybe(P(x)) returns true; this excludes unknown as well as false. P may be the name of a function or a lambda expression. L must be a literal list.

Examples:

(%i1) sublist_indices ('[a, b, b, c, 1, 2, b, 3, b],
                       lambda ([x], x='b));
(%o1)                     [2, 3, 7, 9]
(%i2) sublist_indices ('[a, b, b, c, 1, 2, b, 3, b], symbolp);
(%o2)                  [1, 2, 3, 4, 7, 9]
(%i3) sublist_indices ([1 > 0, 1 < 0, 2 < 1, 2 > 1, 2 > 0],
                       identity);
(%o3)                       [1, 4, 5]
(%i4) assume (x < -1);
(%o4)                       [x < - 1]
(%i5) map (maybe, [x > 0, x < 0, x < -2]);
(%o5)                [false, true, unknown]
(%i6) sublist_indices ([x > 0, x < 0, x < -2], identity);
(%o6)                          [2]

Function: tenth (expr)

Returns the 10’th item of expression or list expr. See first for more details.

Function: third (expr)

Returns the 3’rd item of expression or list expr. See first for more details.

Function: tree_reduce
    tree_reduce (F, s)
    tree_reduce (F, s, s_0)

Extends the binary function F to an n-ary function by composition, where s is a set or list.

tree_reduce is equivalent to the following: Apply F to successive pairs of elements to form a new list [F(s_1, s_2), F(s_3, s_4), ...], carrying the final element unchanged if there are an odd number of elements. Then repeat until the list is reduced to a single element, which is the return value.

When the optional argument s_0 is present, the result is equivalent tree_reduce(F, cons(s_0, s)).

For addition of floating point numbers, tree_reduce may return a sum that has a smaller rounding error than either rreduce or lreduce.

The elements of s and the partial results may be arranged in a minimum-depth binary tree, thus the name "tree_reduce".

Examples:

tree_reduce applied to a list with an even number of elements.

(%i1) tree_reduce (f, [a, b, c, d]);
(%o1)                  f(f(a, b), f(c, d))

tree_reduce applied to a list with an odd number of elements.

(%i1) tree_reduce (f, [a, b, c, d, e]);
(%o1)               f(f(f(a, b), f(c, d)), e)

Function: unique (L)

Returns the unique elements of the list L.

When all the elements of L are unique, unique returns a shallow copy of L, not L itself.

If L is not a list, unique returns L.

Example:

(%i1) unique ([1, %pi, a + b, 2, 1, %e, %pi, a + b, [1]]);
(%o1)              [1, 2, %e, %pi, [1], b + a]
Function: xreduce
    xreduce (F, s)
    xreduce (F, s, s_0)

Extends the function F to an n-ary function by composition, or, if F is already n-ary, applies F to s. When F is not n-ary, xreduce is the same as lreduce. The argument s is a list.

Functions known to be n-ary include addition +, multiplication *, and, or, max, min, and append. Functions may also be declared n-ary by declare(F, nary). For these functions, xreduce is expected to be faster than either rreduce or lreduce.

When the optional argument s_0 is present, the result is equivalent to xreduce(s, cons(s_0, s)).

Floating point addition is not exactly associative; be that as it may, xreduce applies Maxima’s n-ary addition when s contains floating point numbers.

Examples:

xreduce applied to a function known to be n-ary. F is called once, with all arguments.

(%i1) declare (F, nary);
(%o1)                         done
(%i2) F ([L]) := L;
(%o2)                      F([L]) := L
(%i3) xreduce (F, [a, b, c, d, e]);
(%o3)                    [a, b, c, d, e]

xreduce applied to a function not known to be n-ary. G is called several times, with two arguments each time.

(%i1) G ([L]) := L;
(%o1)                      G([L]) := L
(%i2) xreduce (G, [a, b, c, d, e]);
(%o2)                 [[[[a, b], c], d], e]
(%i3) lreduce (G, [a, b, c, d, e]);
(%o3)                 [[[[a, b], c], d], e]


Previous: , Up: Lists   [Contents][Index]

5.4.3 Performance considerations for Lists

Lists provide efficient ways of appending and removing elements. They can be created without knowing their final dimensions. Lisp provides efficient means of copying and handling lists. Also nested lists do not need to be strictly rectangular. These advantages over declared arrays come with the drawback that the amount of time needed for accessing a random element within a list may be roughly proportional to the element’s distance from its beginning. Efficient traversal of lists is still possible, though, by using the list as a stack or a fifo:

(%i1) l:[Test,1,2,3,4];
(%o1)                  [Test, 1, 2, 3, 4]
(%i2) while l # [] do
   disp(pop(l));
                              Test

                                1

                                2

                                3

                                4

(%o2)                         done

Another even faster example would be:

(%i1) l:[Test,1,2,3,4];
(%o1)                  [Test, 1, 2, 3, 4]
(%i2) for i in l do
   disp(pop(l));
                              Test

                                1

                                2

                                3

                                4

(%o2)                         done

Beginning traversal with the last element of a list is possible after reversing the list using reverse (). If the elements of a long list need to be processed in a different order performance might be increased by converting the list into a declared array first.

It is also to note that the ending condition of for loops is tested for every iteration which means that the result of a length should be cached if it is used in the ending condition:

(%i1) l:makelist(i,i,1,100000)$
(%i2) lngth:length(l);
(%o2)                        100000
(%i3) x:1;
(%o3)                           1
(%i4) for i:1 thru lngth do
    x:x+1$
(%i5) x;
(%o5)                        100001

Next: , Previous: , Up: Data Types and Structures   [Contents][Index]

5.5 Arrays

Maxima supports 3 array-like constructs:


Previous: , Up: Arrays   [Contents][Index]

5.5.1 Functions and Variables for Arrays

Function: array
    array (name, dim_1, …, dim_n)
    array (name, type, dim_1, …, dim_n)
    array ([name_1, …, name_m], dim_1, …, dim_n)

Creates an n-dimensional array. n may be less than or equal to 5. The subscripts for the i’th dimension are the integers running from 0 to dim_i.

array (name, dim_1, ..., dim_n) creates a general array.

array (name, type, dim_1, ..., dim_n) creates an array, with elements of a specified type. type can be fixnum for integers of limited size or flonum for floating-point numbers.

array ([name_1, ..., name_m], dim_1, ..., dim_n) creates m arrays, all of the same dimensions.

See also arraymake, arrayinfo and make_array.

Function: arrayapply (A, [i_1, …, i_n])

Evaluates A [i_1, ..., i_n], where A is an array and i_1, …, i_n are integers.

This is reminiscent of apply, except the first argument is an array instead of a function.

Function: arrayinfo (A)

Returns information about the array A. The argument A may be a declared array, a hashed array, a memoizing function, or a subscripted function.

For declared arrays, arrayinfo returns a list comprising the atom declared, the number of dimensions, and the size of each dimension. The elements of the array, both bound and unbound, are returned by listarray.

For undeclared arrays (hashed arrays), arrayinfo returns a list comprising the atom hashed, the number of subscripts, and the subscripts of every element which has a value. The values are returned by listarray.

For memoizing functions, arrayinfo returns a list comprising the atom hashed, the number of subscripts, and any subscript values for which there are stored function values. The stored function values are returned by listarray.

For subscripted functions, arrayinfo returns a list comprising the atom hashed, the number of subscripts, and any subscript values for which there are lambda expressions. The lambda expressions are returned by listarray.

See also listarray.

Examples:

arrayinfo and listarray applied to a declared array.

(%i1) array (aa, 2, 3);
(%o1)                          aa
(%i2) aa [2, 3] : %pi;
(%o2)                          %pi
(%i3) aa [1, 2] : %e;
(%o3)                          %e
(%i4) arrayinfo (aa);
(%o4)                 [declared, 2, [2, 3]]
(%i5) listarray (aa);
(%o5) [#####, #####, #####, #####, #####, #####, %e, #####, 
                                        #####, #####, #####, %pi]

arrayinfo and listarray applied to a undeclared array (hashed array.).

(%i1) bb [FOO] : (a + b)^2;
                                   2
(%o1)                       (b + a)
(%i2) bb [BAR] : (c - d)^3;
                                   3
(%o2)                       (c - d)
(%i3) arrayinfo (bb);
(%o3)               [hashed, 1, [BAR], [FOO]]
(%i4) listarray (bb);
                              3         2
(%o4)                 [(c - d) , (b + a) ]

arrayinfo and listarray applied to a memoizing function.

(%i1) cc [x, y] := y / x;
                                     y
(%o1)                      cc     := -
                             x, y    x
(%i2) cc [u, v];
                                v
(%o2)                           -
                                u
(%i3) cc [4, z];
                                z
(%o3)                           -
                                4
(%i4) arrayinfo (cc);
(%o4)              [hashed, 2, [4, z], [u, v]]
(%i5) listarray (cc);
                              z  v
(%o5)                        [-, -]
                              4  u

Using arrayinfo in order to convert an undeclared array to a declared array:

(%i1) for i:0 thru 10 do a[i]:i^2$
(%i2) indices:map(first,rest(rest(arrayinfo(a))));
(%o2)          [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
(%i3) array(A,fixnum,length(indices)-1)$
(%i4) fillarray(A,map(lambda([x],a[x]),indices))$
(%i5) listarray(A);
(%o5)       [0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100]

arrayinfo and listarray applied to a subscripted function.

(%i1) dd [x] (y) := y ^ x;
                                     x
(%o1)                     dd (y) := y
                            x
(%i2) dd [a + b];
                                    b + a
(%o2)                  lambda([y], y     )
(%i3) dd [v - u];
                                    v - u
(%o3)                  lambda([y], y     )
(%i4) arrayinfo (dd);
(%o4)             [hashed, 1, [b + a], [v - u]]
(%i5) listarray (dd);
                         b + a                v - u
(%o5)      [lambda([y], y     ), lambda([y], y     )]

Function: arraymake (A, [i_1, …, i_n])

Returns the expression A[i_1, ..., i_n]. The result is an unevaluated array reference.

arraymake is reminiscent of funmake, except the return value is an unevaluated array reference instead of an unevaluated function call.

Examples:

(%i1) arraymake (A, [1]);
(%o1)                          A
                                1
(%i2) arraymake (A, [k]);
(%o2)                          A
                                k
(%i3) arraymake (A, [i, j, 3]);
(%o3)                       A
                             i, j, 3
(%i4) array (A, fixnum, 10);
(%o4)                           A
(%i5) fillarray (A, makelist (i^2, i, 1, 11));
(%o5)                           A
(%i6) arraymake (A, [5]);
(%o6)                          A
                                5
(%i7) ''%;
(%o7)                          36
(%i8) L : [a, b, c, d, e];
(%o8)                    [a, b, c, d, e]
(%i9) arraymake ('L, [n]);
(%o9)                          L
                                n
(%i10) ''%, n = 3;
(%o10)                          c
(%i11) A2 : make_array (fixnum, 10);
(%o11)        {Lisp Array: #(0 0 0 0 0 0 0 0 0 0)}
(%i12) fillarray (A2, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
(%o12)        {Lisp Array: #(1 2 3 4 5 6 7 8 9 10)}
(%i13) arraymake ('A2, [8]);
(%o13)                         A2
                                 8
(%i14) ''%;
(%o14)                          9

System variable: arrays

Default value: []

arrays is a list of arrays that have been allocated. These comprise arrays declared by array, hashed arrays that can be constructed by implicit definition (assigning something to an element that isn’t yet declared as a list or an array), and memoizing functions defined by := and define. Arrays defined by make_array are not included.

See also array, arrayapply, arrayinfo, arraymake, fillarray, listarray, and rearray.

Examples:

(%i1) array (aa, 5, 7);
(%o1)                          aa
(%i2) bb [FOO] : (a + b)^2;
                                   2
(%o2)                       (b + a)
(%i3) cc [x] := x/100;
                                   x
(%o3)                      cc  := ---
                             x    100
(%i4) dd : make_array ('any, 7);
(%o4)     {Lisp Array: #(NIL NIL NIL NIL NIL NIL NIL)}
(%i5) arrays;
(%o5)                     [aa, bb, cc]

Function: arraysetapply (A, [i_1, …, i_n], x)

Assigns x to A[i_1, ..., i_n], where A is an array and i_1, …, i_n are integers.

arraysetapply evaluates its arguments.

Function: fillarray (A, B)

Fills array A from B, which is a list or an array.

If a specific type was declared for A when it was created, it can only be filled with elements of that same type; it is an error if an attempt is made to copy an element of a different type.

If the dimensions of the arrays A and B are different, A is filled in row-major order. If there are not enough elements in B the last element is used to fill out the rest of A. If there are too many, the remaining ones are ignored.

fillarray returns its first argument.

Examples:

Create an array of 9 elements and fill it from a list.

(%i1) array (a1, fixnum, 8);
(%o1)                          a1
(%i2) listarray (a1);
(%o2)              [0, 0, 0, 0, 0, 0, 0, 0, 0]
(%i3) fillarray (a1, [1, 2, 3, 4, 5, 6, 7, 8, 9]);
(%o3)                          a1
(%i4) listarray (a1);
(%o4)              [1, 2, 3, 4, 5, 6, 7, 8, 9]

When there are too few elements to fill the array, the last element is repeated. When there are too many elements, the extra elements are ignored.

(%i1) a2 : make_array (fixnum, 8);
(%o1)           {Lisp Array: #(0 0 0 0 0 0 0 0)}
(%i2) fillarray (a2, [1, 2, 3, 4, 5]);
(%o2)           {Lisp Array: #(1 2 3 4 5 5 5 5)}
(%i3) fillarray (a2, [4]);
(%o3)           {Lisp Array: #(4 4 4 4 4 4 4 4)}
(%i4) fillarray (a2, makelist (i, i, 1, 100));
(%o4)           {Lisp Array: #(1 2 3 4 5 6 7 8)}

Multiple-dimension arrays are filled in row-major order.

(%i1) a3 : make_array (fixnum, 2, 5);
(%o1)      {Lisp Array: #2A((0 0 0 0 0) (0 0 0 0 0))}
(%i2) fillarray (a3, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]);
(%o2)      {Lisp Array: #2A((1 2 3 4 5) (6 7 8 9 10))}
(%i3) a4 : make_array (fixnum, 5, 2);
(%o3)   {Lisp Array: #2A((0 0) (0 0) (0 0) (0 0) (0 0))}
(%i4) fillarray (a4, a3);
(%o4)   {Lisp Array: #2A((1 2) (3 4) (5 6) (7 8) (9 10))}

Function: listarray (A)

Returns a list of the elements of the array A. The argument A may be an array, an undeclared array (hashed array), a memoizing function, or a subscripted function.

Elements are listed in row-major order. That is, elements are sorted according to the first index, then according to the second index, and so on. The sorting order of index values is the same as the order established by orderless.

For undeclared arrays (hashed arrays), memoizing functions, and subscripted functions, the elements correspond to the index values returned by arrayinfo.

Unbound elements of general arrays (that is, not fixnum and not flonum) are returned as #####. Unbound elements of fixnum or flonum arrays are returned as 0 or 0.0, respectively. Unbound elements of hashed arrays, memoizing functions, and subscripted functions are not returned.

Examples:

listarray and arrayinfo applied to a declared array.

(%i1) array (aa, 2, 3);
(%o1)                          aa
(%i2) aa [2, 3] : %pi;
(%o2)                          %pi
(%i3) aa [1, 2] : %e;
(%o3)                          %e
(%i4) listarray (aa);
(%o4) [#####, #####, #####, #####, #####, #####, %e, #####, 
                                        #####, #####, #####, %pi]
(%i5) arrayinfo (aa);
(%o5)                 [declared, 2, [2, 3]]

listarray and arrayinfo applied to a undeclared array (hashed array).

(%i1) bb [FOO] : (a + b)^2;
                                   2
(%o1)                       (b + a)
(%i2) bb [BAR] : (c - d)^3;
                                   3
(%o2)                       (c - d)
(%i3) listarray (bb);
                              3         2
(%o3)                 [(c - d) , (b + a) ]
(%i4) arrayinfo (bb);
(%o4)               [hashed, 1, [BAR], [FOO]]

listarray and arrayinfo applied to an memoizing function.

(%i1) cc [x, y] := y / x;
                                     y
(%o1)                      cc     := -
                             x, y    x
(%i2) cc [u, v];
                                v
(%o2)                           -
                                u
(%i3) cc [4, z];
                                z
(%o3)                           -
                                4
(%i4) listarray (cc);
                              z  v
(%o4)                        [-, -]
                              4  u
(%i5) arrayinfo (cc);
(%o5)              [hashed, 2, [4, z], [u, v]]

listarray and arrayinfo applied to a subscripted function.

(%i1) dd [x] (y) := y ^ x;
                                     x
(%o1)                     dd (y) := y
                            x
(%i2) dd [a + b];
                                    b + a
(%o2)                  lambda([y], y     )
(%i3) dd [v - u];
                                    v - u
(%o3)                  lambda([y], y     )
(%i4) listarray (dd);
                         b + a                v - u
(%o4)      [lambda([y], y     ), lambda([y], y     )]
(%i5) arrayinfo (dd);
(%o5)             [hashed, 1, [b + a], [v - u]]

Function: make_array (type, dim_1, …, dim_n)

Creates and returns a Lisp array. type may be any, flonum, fixnum, hashed or functional. There are n indices, and the i’th index runs from 0 to dim_i - 1.

The advantage of make_array over array is that the return value doesn’t have a name, and once a pointer to it goes away, it will also go away. For example, if y: make_array (...) then y points to an object which takes up space, but after y: false, y no longer points to that object, so the object can be garbage collected.

Examples:

(%i1) A1 : make_array (fixnum, 10);
(%o1)         {Lisp Array: #(0 0 0 0 0 0 0 0 0 0)}
(%i2) A1 [8] : 1729;
(%o2)                         1729
(%i3) A1;
(%o3)        {Lisp Array: #(0 0 0 0 0 0 0 0 1729 0)}
(%i4) A2 : make_array (flonum, 10);
(%o4) {Lisp Array: #(0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0)}
(%i5) A2 [2] : 2.718281828;
(%o5)                      2.718281828
(%i6) A2;
(%o6) 
 {Lisp Array: #(0.0 0.0 2.718281828 0.0 0.0 0.0 0.0 0.0 0.0 0.0)}
(%i7) A3 : make_array (any, 10);
(%o7) {Lisp Array: #(NIL NIL NIL NIL NIL NIL NIL NIL NIL NIL)}
(%i8) A3 [4] : x - y - z;
(%o8)                     (- z) - y + x
(%i9) A3;
(%o9) {Lisp Array: #(NIL NIL NIL NIL
               ((MPLUS SIMP) $X ((MTIMES SIMP) -1 $Y) ((MTIMES S\
IMP) -1 $Z))
               NIL NIL NIL NIL NIL)}
(%i10) A4 : make_array (fixnum, 2, 3, 5);
(%o10) {Lisp Array: #3A(((0 0 0 0 0) (0 0 0 0 0) (0 0 0 0 0))
                 ((0 0 0 0 0) (0 0 0 0 0) (0 0 0 0 0)))}
(%i11) fillarray (A4, makelist (i, i, 1, 2*3*5));
(%o11) {Lisp Array: #3A(((1 2 3 4 5) (6 7 8 9 10) (11 12 13 14 1\
5))
                 ((16 17 18 19 20) (21 22 23 24 25) (26 27 28 29\
 30)))}
(%i12) A4 [0, 2, 1];
(%o12)                         12

Function: rearray (A, dim_1, …, dim_n)

Changes the dimensions of an array. The new array will be filled with the elements of the old one in row-major order. If the old array was too small, the remaining elements are filled with false, 0.0 or 0, depending on the type of the array. The type of the array cannot be changed.

Function: remarray
    remarray (A_1, …, A_n)
    remarray (all)

Removes arrays and array associated functions and frees the storage occupied. The arguments may be declared arrays, hashed arrays, array functions, and subscripted functions.

remarray (all) removes all items in the global list arrays.

It may be necessary to use this function if it is desired to clear the cache of a memoizing function.

remarray returns the list of arrays removed.

remarray quotes its arguments.

Function: subvar (x, i)

Evaluates the subscripted expression x[i].

subvar evaluates its arguments.

arraymake (x, [i]) constructs the expression x[i], but does not evaluate it.

Examples:

(%i1) x : foo $
(%i2) i : 3 $
(%i3) subvar (x, i);
(%o3)                         foo
                                 3
(%i4) foo : [aa, bb, cc, dd, ee]$
(%i5) subvar (x, i);
(%o5)                          cc
(%i6) arraymake (x, [i]);
(%o6)                         foo
                                 3
(%i7) ''%;
(%o7)                          cc

Function: subvarp (expr)

Returns true if expr is a subscripted variable, for example a[i].

Option variable: use_fast_arrays

Default value: false

When use_fast_arrays is true, arrays declared by array are values instead of properties, and undeclared arrays (hashed arrays) are implemented as Lisp hashed arrays.

When use_fast_arrays is false, arrays declared by array are properties, and undeclared arrays are implemented with Maxima’s own hashed array implementation.

Note that the code use_fast_arrays switches to is not necessarily faster than the default one; Arrays created by make_array are not affected by use_fast_arrays.

See also translate_fast_arrays.

Option variable: translate_fast_arrays

Default value: false

When translate_fast_arrays is true, the Maxima-to-Lisp translator generates code that assumes arrays are values instead of properties, as if use_fast_arrays were true.

When translate_fast_arrays is false, the Maxima-to-Lisp translator generates code that assumes arrays are properties, as if use_fast_arrays were false.


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5.6 Structures


Next: , Previous: , Up: Structures   [Contents][Index]

5.6.1 Introduction to Structures

Maxima provides a simple data aggregate called a structure. A structure is an expression in which arguments are identified by name (the field name) and the expression as a whole is identified by its operator (the structure name). A field value can be any expression.

A structure is defined by the defstruct function; the global variable structures is the list of user-defined structures. The function new creates instances of structures. The @ operator refers to fields. kill(S) removes the structure definition S, and kill(x@ a) unbinds the field a of the structure instance x.

In the pretty-printing console display (with display2d equal to true), structure instances are displayed with the value of each field represented as an equation, with the field name on the left-hand side and the value on the right-hand side. (The equation is only a display construct; only the value is actually stored.) In 1-dimensional display (via grind or with display2d equal to false), structure instances are displayed without the field names.

There is no way to use a field name as a function name, although a field value can be a lambda expression. Nor can the values of fields be restricted to certain types; any field can be assigned any kind of expression. There is no way to make some fields accessible or inaccessible in different contexts; all fields are always visible.


Previous: , Up: Structures   [Contents][Index]

5.6.2 Functions and Variables for Structures

Global variable: structures

structures is the list of user-defined structures defined by defstruct.

Function: defstruct
    defstruct (S(a_1, …, a_n))
    defstruct (S(a_1 = v_1, …, a_n = v_n))

Define a structure, which is a list of named fields a_1, …, a_n associated with a symbol S. An instance of a structure is just an expression which has operator S and exactly n arguments. new(S) creates a new instance of structure S.

An argument which is just a symbol a specifies the name of a field. An argument which is an equation a = v specifies the field name a and its default value v. The default value can be any expression.

defstruct puts S on the list of user-defined structures, structures.

kill(S) removes S from the list of user-defined structures, and removes the structure definition.

Examples:

(%i1) defstruct (foo (a, b, c));
(%o1)                    [foo(a, b, c)]
(%i2) structures;
(%o2)                    [foo(a, b, c)]
(%i3) new (foo);
(%o3)                     foo(a, b, c)
(%i4) defstruct (bar (v, w, x = 123, y = %pi));
(%o4)             [bar(v, w, x = 123, y = %pi)]
(%i5) structures;
(%o5)      [foo(a, b, c), bar(v, w, x = 123, y = %pi)]
(%i6) new (bar);
(%o6)              bar(v, w, x = 123, y = %pi)
(%i7) kill (foo);
(%o7)                         done
(%i8) structures;
(%o8)             [bar(v, w, x = 123, y = %pi)]

Function: new
    new (S)
    new (S (v_1, …, v_n))

new creates new instances of structures.

new(S) creates a new instance of structure S in which each field is assigned its default value, if any, or no value at all if no default was specified in the structure definition.

new(S(v_1, ..., v_n)) creates a new instance of S in which fields are assigned the values v_1, …, v_n.

Examples:

(%i1) defstruct (foo (w, x = %e, y = 42, z));
(%o1)              [foo(w, x = %e, y = 42, z)]
(%i2) new (foo);
(%o2)               foo(w, x = %e, y = 42, z)
(%i3) new (foo (1, 2, 4, 8));
(%o3)            foo(w = 1, x = 2, y = 4, z = 8)

Operator: @

@ is the structure field access operator. The expression x@ a refers to the value of field a of the structure instance x. The field name is not evaluated.

If the field a in x has not been assigned a value, x@ a evaluates to itself.

kill(x@ a) removes the value of field a in x.

Examples:

(%i1) defstruct (foo (x, y, z));
(%o1)                    [foo(x, y, z)]
(%i2) u : new (foo (123, a - b, %pi));
(%o2)           foo(x = 123, y = a - b, z = %pi)
(%i3) u@z;
(%o3)                          %pi
(%i4) u@z : %e;
(%o4)                          %e
(%i5) u;
(%o5)            foo(x = 123, y = a - b, z = %e)
(%i6) kill (u@z);
(%o6)                         done
(%i7) u;
(%o7)              foo(x = 123, y = a - b, z)
(%i8) u@z;
(%o8)                          u@z

The field name is not evaluated.

(%i1) defstruct (bar (g, h));
(%o1)                      [bar(g, h)]
(%i2) x : new (bar);
(%o2)                       bar(g, h)
(%i3) x@h : 42;
(%o3)                          42
(%i4) h : 123;
(%o4)                          123
(%i5) x@h;
(%o5)                          42
(%i6) x@h : 19;
(%o6)                          19
(%i7) x;
(%o7)                    bar(g, h = 19)
(%i8) h;
(%o8)                          123


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6 Expressions


Next: , Previous: , Up: Expressions   [Contents][Index]

6.1 Introduction to Expressions

There are a number of reserved words which should not be used as variable names. Their use would cause a possibly cryptic syntax error.

integrate            next           from                 diff            
in                   at             limit                sum             
for                  and            elseif               then            
else                 do             or                   if              
unless               product        while                thru            
step                                                                     

Most things in Maxima are expressions. A sequence of expressions can be made into an expression by separating them by commas and putting parentheses around them. This is similar to the C comma expression.

(%i1) x: 3$
(%i2) (x: x+1, x: x^2);
(%o2)                          16
(%i3) (if (x > 17) then 2 else 4);
(%o3)                           4
(%i4) (if (x > 17) then x: 2 else y: 4, y+x);
(%o4)                          20

Even loops in Maxima are expressions, although the value they return is the not too useful done.

(%i1) y: (x: 1, for i from 1 thru 10 do (x: x*i))$
(%i2) y;
(%o2)                         done

Whereas what you really want is probably to include a third term in the comma expression which actually gives back the value.

(%i3) y: (x: 1, for i from 1 thru 10 do (x: x*i), x)$
(%i4) y;
(%o4)                        3628800

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6.2 Nouns and Verbs

Maxima distinguishes between operators which are "nouns" and operators which are "verbs". A verb is an operator which can be executed. A noun is an operator which appears as a symbol in an expression, without being executed. By default, function names are verbs. A verb can be changed into a noun by quoting the function name or applying the nounify function. A noun can be changed into a verb by applying the verbify function. The evaluation flag nouns causes ev to evaluate nouns in an expression.

The verb form is distinguished by a leading dollar sign $ on the corresponding Lisp symbol. In contrast, the noun form is distinguished by a leading percent sign % on the corresponding Lisp symbol. Some nouns have special display properties, such as 'integrate and 'derivative (returned by diff), but most do not. By default, the noun and verb forms of a function are identical when displayed. The global flag noundisp causes Maxima to display nouns with a leading quote mark '.

See also noun, nouns, nounify, and verbify.

Examples:

(%i1) foo (x) := x^2;
                                     2
(%o1)                     foo(x) := x
(%i2) foo (42);
(%o2)                         1764
(%i3) 'foo (42);
(%o3)                        foo(42)
(%i4) 'foo (42), nouns;
(%o4)                         1764
(%i5) declare (bar, noun);
(%o5)                         done
(%i6) bar (x) := x/17;
                                    x
(%o6)                     bar(x) := --
                                    17
(%i7) bar (52);
(%o7)                        bar(52)
(%i8) bar (52), nouns;
(%o8)                        bar(52)
(%i9) integrate (1/x, x, 1, 42);
(%o9)                        log(42)
(%i10) 'integrate (1/x, x, 1, 42);
                             42
                            /
                            [   1
(%o10)                      I   - dx
                            ]   x
                            /
                             1
(%i11) ev (%, nouns);
(%o11)                       log(42)
·

Category: Evaluation · Category: Nouns and verbs


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6.3 Identifiers

Maxima identifiers may comprise alphabetic characters, plus the numerals 0 through 9, plus any special character preceded by the backslash \ character.

A numeral may be the first character of an identifier if it is preceded by a backslash. Numerals which are the second or later characters need not be preceded by a backslash.

Characters may be declared alphabetic by the declare function. If so declared, they need not be preceded by a backslash in an identifier. The alphabetic characters are initially A through Z, a through z, %, and _.

Maxima is case-sensitive. The identifiers foo, FOO, and Foo are distinct. See Lisp and Maxima for more on this point.

A Maxima identifier is a Lisp symbol which begins with a dollar sign $. Any other Lisp symbol is preceded by a question mark ? when it appears in Maxima. See Lisp and Maxima for more on this point.

Examples:

(%i1) %an_ordinary_identifier42;
(%o1)               %an_ordinary_identifier42
(%i2) embedded\ spaces\ in\ an\ identifier;
(%o2)           embedded spaces in an identifier
(%i3) symbolp (%);
(%o3)                         true
(%i4) [foo+bar, foo\+bar];
(%o4)                 [foo + bar, foo+bar]
(%i5) [1729, \1729];
(%o5)                     [1729, 1729]
(%i6) [symbolp (foo\+bar), symbolp (\1729)];
(%o6)                     [true, true]
(%i7) [is (foo\+bar = foo+bar), is (\1729 = 1729)];
(%o7)                    [false, false]
(%i8) baz\~quux;
(%o8)                       baz~quux
(%i9) declare ("~", alphabetic);
(%o9)                         done
(%i10) baz~quux;
(%o10)                      baz~quux
(%i11) [is (foo = FOO), is (FOO = Foo), is (Foo = foo)];
(%o11)                [false, false, false]
(%i12) :lisp (defvar *my-lisp-variable* '$foo)
*MY-LISP-VARIABLE*
(%i12) ?\*my\-lisp\-variable\*;
(%o12)                         foo
·

Category: Syntax


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6.4 Inequality

Maxima has the inequality operators <, <=, >=, >, #, and notequal. See if for a description of conditional expressions.


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6.5 Functions and Variables for Expressions

Function: alias (new_name_1, old_name_1, …, new_name_n, old_name_n)

provides an alternate name for a (user or system) function, variable, array, etc. Any even number of arguments may be used.

System variable: aliases

Default value: []

aliases is the list of atoms which have a user defined alias (set up by the alias, ordergreat, orderless functions or by declaring the atom a noun with declare.)

Keyword: allbut

works with the part commands (i.e. part, inpart, substpart, substinpart, dpart, and lpart). For example,

(%i1) expr : e + d + c + b + a;
(%o1)                   e + d + c + b + a
(%i2) part (expr, [2, 5]);
(%o2)                         d + a

while

(%i1) expr : e + d + c + b + a;
(%o1)                   e + d + c + b + a
(%i2) part (expr, allbut (2, 5));
(%o2)                       e + c + b

allbut is also recognized by kill.

(%i1) [aa : 11, bb : 22, cc : 33, dd : 44, ee : 55];
(%o1)                 [11, 22, 33, 44, 55]
(%i2) kill (allbut (cc, dd));
(%o0)                         done
(%i1) [aa, bb, cc, dd];
(%o1)                   [aa, bb, 33, 44]

kill(allbut(a_1, a_2, ...)) has the effect of kill(all) except that it does not kill the symbols a_1, a_2, …

Function: args (expr)

Returns the list of arguments of expr, which may be any kind of expression other than an atom. Only the arguments of the top-level operator are extracted; subexpressions of expr appear as elements or subexpressions of elements of the list of arguments.

The order of the items in the list may depend on the global flag inflag.

args (expr) is equivalent to substpart ("[", expr, 0). See also substpart, apply, funmake, and op.

How to convert a matrix to a nested list:

(%i1) M:matrix([1,2],[3,4]);
                            [ 1  2 ]
(%o1)                       [      ]
                            [ 3  4 ]
(%i2) args(M);
(%o2)                   [[1, 2], [3, 4]]

Since maxima internally treats a sum of n terms as a summation command with n arguments args() can extract the list of terms in a sum:

(%i1) a+b+c;
(%o1)                       c + b + a
(%i2) args(%);
(%o2)                       [c, b, a]

Function: atom (expr)

Returns true if expr is atomic (i.e. a number, name or string) else false. Thus atom(5) is true while atom(a[1]) and atom(sin(x)) are false (assuming a[1] and x are unbound).

Function: box
    box (expr)
    box (expr, a)

Returns expr enclosed in a box. The return value is an expression with box as the operator and expr as the argument. A box is drawn on the display when display2d is true.

box (expr, a) encloses expr in a box labelled by the symbol a. The label is truncated if it is longer than the width of the box.

box evaluates its argument. However, a boxed expression does not evaluate to its content, so boxed expressions are effectively excluded from computations. rembox removes the box again.

boxchar is the character used to draw the box in box and in the dpart and lpart functions.

See also rembox, dpart and lpart.

Examples:

(%i1) box (a^2 + b^2);
                            """""""""
                            " 2    2"
(%o1)                       "b  + a "
                            """""""""
(%i2) a : 1234;
(%o2)                         1234
(%i3) b : c - d;
(%o3)                         c - d
(%i4) box (a^2 + b^2);
                      """"""""""""""""""""
                      "       2          "
(%o4)                 "(c - d)  + 1522756"
                      """"""""""""""""""""
(%i5) box (a^2 + b^2, term_1);
                      term_1""""""""""""""
                      "       2          "
(%o5)                 "(c - d)  + 1522756"
                      """"""""""""""""""""
(%i6) 1729 - box (1729);
                                 """"""
(%o6)                     1729 - "1729"
                                 """"""
(%i7) boxchar: "-";
(%o7)                           -
(%i8) box (sin(x) + cos(y));
                        -----------------
(%o8)                   -cos(y) + sin(x)-
                        -----------------

Option variable: boxchar

Default value: "

boxchar is the character used to draw the box in the box and in the dpart and lpart functions.

All boxes in an expression are drawn with the current value of boxchar; the drawing character is not stored with the box expression.

Function: collapse (expr)

Collapses expr by causing all of its common (i.e., equal) subexpressions to share (i.e., use the same cells), thereby saving space. (collapse is a subroutine used by the optimize command.) Thus, calling collapse may be useful after loading in a save file. You can collapse several expressions together by using collapse ([expr_1, ..., expr_n]). Similarly, you can collapse the elements of the array A by doing collapse (listarray ('A)).

Function: copy (e)

Return a copy of the Maxima expression e. Although e can be any Maxima expression, the copy function is the most useful when e is either a list or a matrix; consider:

(%i1) m : [1,[2,3]]$
(%i2) mm : m$
(%i3) mm[2][1] : x$
(%i4) m;
(%o4)                      [1, [x, 3]]
(%i5) mm;
(%o5)                      [1, [x, 3]]

Let’s try the same experiment, but this time let mm be a copy of m

(%i1) m : [1,[2,3]]$
(%i2) mm : copy(m)$
(%i3) mm[2][1] : x$
(%i4) m;
(%o4)                      [1, [2, 3]]
(%i5) mm;
(%o5)                      [1, [x, 3]]

This time, the assignment to mm does not change the value of m.

Function: disolate (expr, x_1, …, x_n)

is similar to isolate (expr, x) except that it enables the user to isolate more than one variable simultaneously. This might be useful, for example, if one were attempting to change variables in a multiple integration, and that variable change involved two or more of the integration variables. This function is autoloaded from simplification/disol.mac. A demo is available by demo("disol")$.

Function: dispform
    dispform (expr)
    dispform (expr, all)

Returns the external representation of expr.

dispform(expr) returns the external representation with respect to the main (top-level) operator. dispform(expr, all) returns the external representation with respect to all operators in expr.

See also part, inpart, and inflag.

Examples:

The internal representation of - x is "negative one times x" while the external representation is "minus x".

(%i1) - x;
(%o1)                          - x
(%i2) ?format (true, "~S~%", %);
((MTIMES SIMP) -1 $X)
(%o2)                         false
(%i3) dispform (- x);
(%o3)                          - x
(%i4) ?format (true, "~S~%", %);
((MMINUS SIMP) $X)
(%o4)                         false

The internal representation of sqrt(x) is "x to the power 1/2" while the external representation is "square root of x".

(%i1) sqrt (x);
(%o1)                        sqrt(x)
(%i2) ?format (true, "~S~%", %);
((MEXPT SIMP) $X ((RAT SIMP) 1 2))
(%o2)                         false
(%i3) dispform (sqrt (x));
(%o3)                        sqrt(x)
(%i4) ?format (true, "~S~%", %);
((%SQRT SIMP) $X)
(%o4)                         false

Use of the optional argument all.

(%i1) expr : sin (sqrt (x));
(%o1)                     sin(sqrt(x))
(%i2) freeof (sqrt, expr);
(%o2)                         true
(%i3) freeof (sqrt, dispform (expr));
(%o3)                         true
(%i4) freeof (sqrt, dispform (expr, all));
(%o4)                         false

Function: dpart (expr, n_1, …, n_k)

Selects the same subexpression as part, but instead of just returning that subexpression as its value, it returns the whole expression with the selected subexpression displayed inside a box. The box is actually part of the expression.

(%i1) dpart (x+y/z^2, 1, 2, 1);
                             y
(%o1)                       ---- + x
                               2
                            """
                            "z"
                            """

Option variable: exptisolate

Default value: false

exptisolate, when true, causes isolate (expr, var) to examine exponents of atoms (such as %e) which contain var.

Option variable: exptsubst

Default value: false

exptsubst, when true, permits substitutions such as y for %e^x in %e^(a x).

(%i1) %e^(a*x);
                                a x
(%o1)                         %e
(%i2) exptsubst;
(%o2)                         false
(%i3) subst(y, %e^x, %e^(a*x));
                                a x
(%o3)                         %e
(%i4) exptsubst: not exptsubst;
(%o4)                         true
(%i5) subst(y, %e^x, %e^(a*x));
                                a
(%o5)                          y

Function: freeof (x_1, …, x_n, expr)

freeof (x_1, expr) returns true if no subexpression of expr is equal to x_1 or if x_1 occurs only as a dummy variable in expr, or if x_1 is neither the noun nor verb form of any operator in expr, and returns false otherwise.

freeof (x_1, ..., x_n, expr) is equivalent to freeof (x_1, expr) and ... and freeof (x_n, expr).

The arguments x_1, …, x_n may be names of functions and variables, subscripted names, operators (enclosed in double quotes), or general expressions. freeof evaluates its arguments.

freeof operates only on expr as it stands (after simplification and evaluation) and does not attempt to determine if some equivalent expression would give a different result. In particular, simplification may yield an equivalent but different expression which comprises some different elements than the original form of expr.

A variable is a dummy variable in an expression if it has no binding outside of the expression. Dummy variables recognized by freeof are the index of a sum or product, the limit variable in limit, the integration variable in the definite integral form of integrate, the original variable in laplace, formal variables in at expressions, and arguments in lambda expressions.

The indefinite form of integrate is not free of its variable of integration.

Examples:

Arguments are names of functions, variables, subscripted names, operators, and expressions. freeof (a, b, expr) is equivalent to freeof (a, expr) and freeof (b, expr).

(%i1) expr: z^3 * cos (a[1]) * b^(c+d);
                                 d + c  3
(%o1)                   cos(a ) b      z
                             1
(%i2) freeof (z, expr);
(%o2)                         false
(%i3) freeof (cos, expr);
(%o3)                         false
(%i4) freeof (a[1], expr);
(%o4)                         false
(%i5) freeof (cos (a[1]), expr);
(%o5)                         false
(%i6) freeof (b^(c+d), expr);
(%o6)                         false
(%i7) freeof ("^", expr);
(%o7)                         false
(%i8) freeof (w, sin, a[2], sin (a[2]), b*(c+d), expr);
(%o8)                         true

freeof evaluates its arguments.

(%i1) expr: (a+b)^5$
(%i2) c: a$
(%i3) freeof (c, expr);
(%o3)                         false

freeof does not consider equivalent expressions. Simplification may yield an equivalent but different expression.

(%i1) expr: (a+b)^5$
(%i2) expand (expr);
          5        4       2  3       3  2      4      5
(%o2)    b  + 5 a b  + 10 a  b  + 10 a  b  + 5 a  b + a
(%i3) freeof (a+b, %);
(%o3)                         true
(%i4) freeof (a+b, expr);
(%o4)                         false
(%i5) exp (x);
                                 x
(%o5)                          %e
(%i6) freeof (exp, exp (x));
(%o6)                         true

A summation or definite integral is free of its dummy variable. An indefinite integral is not free of its variable of integration.

(%i1) freeof (i, 'sum (f(i), i, 0, n));
(%o1)                         true
(%i2) freeof (x, 'integrate (x^2, x, 0, 1));
(%o2)                         true
(%i3) freeof (x, 'integrate (x^2, x));
(%o3)                         false

Option variable: inflag

Default value: false

When inflag is true, functions for part extraction inspect the internal form of expr.

Note that the simplifier re-orders expressions. Thus first (x + y) returns x if inflag is true and y if inflag is false. (first (y + x) gives the same results.)

Also, setting inflag to true and calling part or substpart is the same as calling inpart or substinpart.

Functions affected by the setting of inflag are: part, substpart, first, rest, last, length, the forin construct, map, fullmap, maplist, reveal and pickapart.

Function: inpart (expr, n_1, …, n_k)

is similar to part but works on the internal representation of the expression rather than the displayed form and thus may be faster since no formatting is done. Care should be taken with respect to the order of subexpressions in sums and products (since the order of variables in the internal form is often different from that in the displayed form) and in dealing with unary minus, subtraction, and division (since these operators are removed from the expression). part (x+y, 0) or inpart (x+y, 0) yield +, though in order to refer to the operator it must be enclosed in "s. For example ... if inpart (%o9,0) = "+" then ....

Examples:

(%i1) x + y + w*z;
(%o1)                      w z + y + x
(%i2) inpart (%, 3, 2);
(%o2)                           z
(%i3) part (%th (2), 1, 2);
(%o3)                           z
(%i4) 'limit (f(x)^g(x+1), x, 0, minus);
                                  g(x + 1)
(%o4)                 limit   f(x)
                      x -> 0-
(%i5) inpart (%, 1, 2);
(%o5)                       g(x + 1)

Function: isolate (expr, x)

Returns expr with subexpressions which are sums and which do not contain var replaced by intermediate expression labels (these being atomic symbols like %t1, %t2, …). This is often useful to avoid unnecessary expansion of subexpressions which don’t contain the variable of interest. Since the intermediate labels are bound to the subexpressions they can all be substituted back by evaluating the expression in which they occur.

exptisolate (default value: false) if true will cause isolate to examine exponents of atoms (like %e) which contain var.

isolate_wrt_times if true, then isolate will also isolate with respect to products. See isolate_wrt_times. See also disolate.

Do example (isolate) for examples.

Option variable: isolate_wrt_times

Default value: false

When isolate_wrt_times is true, isolate will also isolate with respect to products. E.g. compare both settings of the switch on

(%i1) isolate_wrt_times: true$
(%i2) isolate (expand ((a+b+c)^2), c);

(%t2)                          2 a


(%t3)                          2 b

                          2            2
(%t4)                    b  + 2 a b + a

                     2
(%o4)               c  + %t3 c + %t2 c + %t4
(%i4) isolate_wrt_times: false$
(%i5) isolate (expand ((a+b+c)^2), c);
                     2
(%o5)               c  + 2 b c + 2 a c + %t4

Option variable: listconstvars

Default value: false

When listconstvars is true the list returned by listofvars contains constant variables, such as %e, %pi, %i or any variables declared as constant that occur in expr. A variable is declared as constant type via declare, and constantp returns true for all variables declared as constant. The default is to omit constant variables from listofvars return value.

Option variable: listdummyvars

Default value: true

When listdummyvars is false, "dummy variables" in the expression will not be included in the list returned by listofvars. (The meaning of "dummy variables" is as given in freeof. "Dummy variables" are mathematical things like the index of a sum or product, the limit variable, and the definite integration variable.)

Example:

(%i1) listdummyvars: true$
(%i2) listofvars ('sum(f(i), i, 0, n));
(%o2)                        [i, n]
(%i3) listdummyvars: false$
(%i4) listofvars ('sum(f(i), i, 0, n));
(%o4)                          [n]

Function: listofvars (expr)

Returns a list of the variables in expr.

listconstvars if true causes listofvars to include %e, %pi, %i, and any variables declared constant in the list it returns if they appear in expr. The default is to omit these.

See also the option variable listdummyvars to exclude or include "dummy variables" in the list of variables.

(%i1) listofvars (f (x[1]+y) / g^(2+a));
(%o1)                     [g, a, x , y]
                                  1

Function: lfreeof (list, expr)

For each member m of list, calls freeof (m, expr). It returns false if any call to freeof does and true otherwise.

Example:

(%i1) lfreeof ([ a, x], x^2+b);
(%o1)                         false
(%i2) lfreeof ([ b, x], x^2+b);
(%o2)                         false
(%i3) lfreeof ([ a, y], x^2+b);
(%o3)                         true

Function: lpart (label, expr, n_1, …, n_k)

is similar to dpart but uses a labelled box. A labelled box is similar to the one produced by dpart but it has a name in the top line.

Property: mainvar

You may declare variables to be mainvar. The ordering scale for atoms is essentially: numbers < constants (e.g., %e, %pi) < scalars < other variables < mainvars. E.g., compare expand ((X+Y)^4) with (declare (x, mainvar), expand ((x+y)^4)). (Note: Care should be taken if you elect to use the above feature. E.g., if you subtract an expression in which x is a mainvar from one in which x isn’t a mainvar, resimplification e.g. with ev (expr, simp) may be necessary if cancellation is to occur. Also, if you save an expression in which x is a mainvar, you probably should also save x.)

Property: noun

noun is one of the options of the declare command. It makes a function so declared a "noun", meaning that it won’t be evaluated automatically.

Example:

(%i1) factor (12345678);
                             2
(%o1)                     2 3  47 14593
(%i2) declare (factor, noun);
(%o2)                         done
(%i3) factor (12345678);
(%o3)                   factor(12345678)
(%i4) ''%, nouns;
                             2
(%o4)                     2 3  47 14593

Option variable: noundisp

Default value: false

When noundisp is true, nouns display with a single quote. This switch is always true when displaying function definitions.

Function: nounify (f)

Returns the noun form of the function name f. This is needed if one wishes to refer to the name of a verb function as if it were a noun. Note that some verb functions will return their noun forms if they can’t be evaluated for certain arguments. This is also the form returned if a function call is preceded by a quote.

See also verbify.

Function: nterms (expr)

Returns the number of terms that expr would have if it were fully expanded out and no cancellations or combination of terms occurred. Note that expressions like sin (expr), sqrt (expr), exp (expr), etc. count as just one term regardless of how many terms expr has (if it is a sum).

Function: op (expr)

Returns the main operator of the expression expr. op (expr) is equivalent to part (expr, 0).

op returns a string if the main operator is a built-in or user-defined prefix, binary or n-ary infix, postfix, matchfix, or nofix operator. Otherwise, if expr is a subscripted function expression, op returns the subscripted function; in this case the return value is not an atom. Otherwise, expr is a memoizing function or ordinary function expression, and op returns a symbol.

op observes the value of the global flag inflag.

op evaluates it argument.

See also args.

Examples:

(%i1) stringdisp: true$
(%i2) op (a * b * c);
(%o2)                          "*"
(%i3) op (a * b + c);
(%o3)                          "+"
(%i4) op ('sin (a + b));
(%o4)                          sin
(%i5) op (a!);
(%o5)                          "!"
(%i6) op (-a);
(%o6)                          "-"
(%i7) op ([a, b, c]);
(%o7)                          "["
(%i8) op ('(if a > b then c else d));
(%o8)                         "if"
(%i9) op ('foo (a));
(%o9)                          foo
(%i10) prefix (foo);
(%o10)                        "foo"
(%i11) op (foo a);
(%o11)                        "foo"
(%i12) op (F [x, y] (a, b, c));
(%o12)                        F
                               x, y
(%i13) op (G [u, v, w]);
(%o13)                          G

Function: operatorp
    operatorp (expr, op)
    operatorp (expr, [op_1, …, op_n])

operatorp (expr, op) returns true if op is equal to the operator of expr.

operatorp (expr, [op_1, ..., op_n]) returns true if some element op_1, …, op_n is equal to the operator of expr.

Option variable: opsubst

Default value: true

When opsubst is false, subst does not attempt to substitute into the operator of an expression. E.g., (opsubst: false, subst (x^2, r, r+r[0])) will work.

(%i1) r+r[0];
(%o1)                        r + r
                                  0
(%i2) opsubst;
(%o2)                         true
(%i3) subst (x^2, r, r+r[0]);
                            2     2
(%o3)                      x  + (x )
                                    0
(%i4) opsubst: not opsubst;
(%o4)                         false
(%i5) subst (x^2, r, r+r[0]);
                              2
(%o5)                        x  + r
                                   0

Function: optimize (expr)

Returns an expression that produces the same value and side effects as expr but does so more efficiently by avoiding the recomputation of common subexpressions. optimize also has the side effect of "collapsing" its argument so that all common subexpressions are shared. Do example (optimize) for examples.

Option variable: optimprefix

Default value: %

optimprefix is the prefix used for generated symbols by the optimize command.

Function: ordergreat (v_1, …, v_n)
Function: orderless (v_1, …, v_n)

ordergreat changes the canonical ordering of Maxima expressions such that v_1 succeeds v_2 succeeds … succeeds v_n, and v_n succeeds any other symbol not mentioned as an argument.

orderless changes the canonical ordering of Maxima expressions such that v_1 precedes v_2 precedes … precedes v_n, and v_n precedes any other variable not mentioned as an argument.

The order established by ordergreat and orderless is dissolved by unorder. ordergreat and orderless can be called only once each, unless unorder is called; only the last call to ordergreat and orderless has any effect.

See also ordergreatp.

Function: ordergreatp (expr_1, expr_2)
Function: orderlessp (expr_1, expr_2)

ordergreatp returns true if expr_1 succeeds expr_2 in the canonical ordering of Maxima expressions, and false otherwise.

orderlessp returns true if expr_1 precedes expr_2 in the canonical ordering of Maxima expressions, and false otherwise.

All Maxima atoms and expressions are comparable under ordergreatp and orderlessp, although there are isolated examples of expressions for which these predicates are not transitive; that is a bug.

The canonical ordering of atoms (symbols, literal numbers, and strings) is the following.

(integers and floats) precede (bigfloats) precede (declared constants) precede (strings) precede (declared scalars) precede (first argument to orderless) precedes … precedes (last argument to orderless) precedes (other symbols) precede (last argument to ordergreat) precedes … precedes (first argument to ordergreat) precedes (declared main variables)

For non-atomic expressions, the canonical ordering is derived from the ordering for atoms. For the built-in + * and ^ operators, the ordering is not easily summarized. For other built-in operators and all other functions and operators, expressions are ordered by their arguments (beginning with the first argument), then by the name of the operator or function. In the case of subscripted expressions, the subscripted symbol is considered the operator and the subscript is considered an argument.

The canonical ordering of expressions is modified by the functions ordergreat and orderless, and the mainvar, constant, and scalar declarations.

See also sort.

Examples:

Ordering ordinary symbols and constants. Note that %pi is not ordered according to its numerical value.

(%i1) stringdisp : true;
(%o1)                         true
(%i2) sort ([%pi, 3b0, 3.0, x, X, "foo", 3, a, 4, "bar", 4.0, 4b0]);
(%o2) [3, 3.0, 4, 4.0, 3.0b0, 4.0b0, %pi, "bar", "foo", X, a, x]

Effect of ordergreat and orderless functions.

(%i1) sort ([M, H, K, T, E, W, G, A, P, J, S]);
(%o1)           [A, E, G, H, J, K, M, P, S, T, W]
(%i2) ordergreat (S, J);
(%o2)                         done
(%i3) orderless (M, H);
(%o3)                         done
(%i4) sort ([M, H, K, T, E, W, G, A, P, J, S]);
(%o4)           [M, H, A, E, G, K, P, T, W, J, S]

Effect of mainvar, constant, and scalar declarations.

(%i1) sort ([aa, foo, bar, bb, baz, quux, cc, dd, A1, B1, C1]);
(%o1)   [A1, B1, C1, aa, bar, baz, bb, cc, dd, foo, quux]
(%i2) declare (aa, mainvar);
(%o2)                         done
(%i3) declare ([baz, quux], constant);
(%o3)                         done
(%i4) declare ([A1, B1], scalar);
(%o4)                         done
(%i5) sort ([aa, foo, bar, bb, baz, quux, cc, dd, A1, B1, C1]);
(%o5)   [baz, quux, A1, B1, C1, bar, bb, cc, dd, foo, aa]

Ordering non-atomic expressions.

(%i1) sort ([1, 2, n, f(1), f(2), f(2, 1), g(1), g(1, 2), g(n),
       f(n, 1)]);
(%o1) [1, 2, f(1), g(1), g(1, 2), f(2), f(2, 1), n, g(n), 
                                                         f(n, 1)]
(%i2) sort ([foo(1), X[1], X[k], foo(k), 1, k]);
(%o2)            [1, X , foo(1), k, X , foo(k)]
                      1              k

Function: part (expr, n_1, …, n_k)

Returns parts of the displayed form of expr. It obtains the part of expr as specified by the indices n_1, …, n_k. First part n_1 of expr is obtained, then part n_2 of that, etc. The result is part n_k of … part n_2 of part n_1 of expr. If no indices are specified expr is returned.

part can be used to obtain an element of a list, a row of a matrix, etc.

If the last argument to a part function is a list of indices then several subexpressions are picked out, each one corresponding to an index of the list. Thus part (x + y + z, [1, 3]) is z+x.

piece holds the last expression selected when using the part functions. It is set during the execution of the function and thus may be referred to in the function itself as shown below.

If partswitch is set to true then end is returned when a selected part of an expression doesn’t exist, otherwise an error message is given.

See also inpart, substpart, substinpart, dpart, and lpart.

Examples:

(%i1) part(z+2*y+a,2);
(%o1)                          2 y
(%i2) part(z+2*y+a,[1,3]);
(%o2)                         z + a
(%i3) part(z+2*y+a,2,1);
(%o3)                           2

example (part) displays additional examples.

Function: partition (expr, x)

Returns a list of two expressions. They are (1) the factors of expr (if it is a product), the terms of expr (if it is a sum), or the list (if it is a list) which don’t contain x and, (2) the factors, terms, or list which do.

Examples:

(%i1) partition (2*a*x*f(x), x);
(%o1)                     [2 a, x f(x)]
(%i2) partition (a+b, x);
(%o2)                      [b + a, 0]
(%i3) partition ([a, b, f(a), c], a); 
(%o3)                  [[b, c], [a, f(a)]]

Option variable: partswitch

Default value: false

When partswitch is true, end is returned when a selected part of an expression doesn’t exist, otherwise an error message is given.

Function: pickapart (expr, n)

Assigns intermediate expression labels to subexpressions of expr at depth n, an integer. Subexpressions at greater or lesser depths are not assigned labels. pickapart returns an expression in terms of intermediate expressions equivalent to the original expression expr.

See also part, dpart, lpart, inpart, and reveal.

Examples:

(%i1) expr: (a+b)/2 + sin (x^2)/3 - log (1 + sqrt(x+1));
                                          2
                                     sin(x )   b + a
(%o1)       - log(sqrt(x + 1) + 1) + ------- + -----
                                        3        2
(%i2) pickapart (expr, 0);
                                          2
                                     sin(x )   b + a
(%t2)       - log(sqrt(x + 1) + 1) + ------- + -----
                                        3        2
(%o2)                          %t2
(%i3) pickapart (expr, 1);

(%t3)                - log(sqrt(x + 1) + 1)


                                  2
                             sin(x )
(%t4)                        -------
                                3


                              b + a
(%t5)                         -----
                                2

(%o5)                    %t5 + %t4 + %t3
(%i5) pickapart (expr, 2);

(%t6)                 log(sqrt(x + 1) + 1)


                                  2
(%t7)                        sin(x )


(%t8)                         b + a

                         %t8   %t7
(%o8)                    --- + --- - %t6
                          2     3
(%i8) pickapart (expr, 3);

(%t9)                    sqrt(x + 1) + 1


                                2
(%t10)                         x

                  b + a              sin(%t10)
(%o10)            ----- - log(%t9) + ---------
                    2                    3
(%i10) pickapart (expr, 4);

(%t11)                     sqrt(x + 1)
                      2
                 sin(x )   b + a
(%o11)           ------- + ----- - log(%t11 + 1)
                    3        2
(%i11) pickapart (expr, 5);

(%t12)                        x + 1

                   2
              sin(x )   b + a
(%o12)        ------- + ----- - log(sqrt(%t12) + 1)
                 3        2
(%i12) pickapart (expr, 6);
                  2
             sin(x )   b + a
(%o12)       ------- + ----- - log(sqrt(x + 1) + 1)
                3        2

System variable: piece

Holds the last expression selected when using the part functions. It is set during the execution of the function and thus may be referred to in the function itself.

Function: psubst
    psubst (list, expr)
    psubst (a, b, expr)

psubst(a, b, expr) is simliar to subst. See subst.

In distinction from subst the function psubst makes parallel substitutions, if the first argument list is a list of equations.

See also sublis for making parallel substitutions and let and letsimp for others ways to do substitutions.

Example:

The first example shows parallel substitution with psubst. The second example shows the result for the function subst, which does a serial substitution.

(%i1) psubst ([a^2=b, b=a], sin(a^2) + sin(b));
(%o1)                    sin(b) + sin(a)
(%i2) subst ([a^2=b, b=a], sin(a^2) + sin(b));
(%o2)                       2 sin(a)

Function: rembox
    rembox (expr, unlabelled)
    rembox (expr, label)
    rembox (expr)

Removes boxes from expr.

rembox (expr, unlabelled) removes all unlabelled boxes from expr.

rembox (expr, label) removes only boxes bearing label.

rembox (expr) removes all boxes, labelled and unlabelled.

Boxes are drawn by the box, dpart, and lpart functions.

Examples:

(%i1) expr: (a*d - b*c)/h^2 + sin(%pi*x);
                                  a d - b c
(%o1)                sin(%pi x) + ---------
                                      2
                                     h
(%i2) dpart (dpart (expr, 1, 1), 2, 2);
dpart: fell off the end.
 -- an error. To debug this try: debugmode(true);
(%i3) expr2: lpart (BAR, lpart (FOO, %, 1), 2);
                                  BAR""""""""
                   FOO"""""""""   "a d - b c"
(%o3)              "sin(%pi x)" + "---------"
                   """"""""""""   "    2    "
                                  "   h     "
                                  """""""""""
(%i4) rembox (expr2, unlabelled);
                                  BAR""""""""
                   FOO"""""""""   "a d - b c"
(%o4)              "sin(%pi x)" + "---------"
                   """"""""""""   "    2    "
                                  "   h     "
                                  """""""""""
(%i5) rembox (expr2, FOO);
                                 BAR""""""""
                                 "a d - b c"
(%o5)               sin(%pi x) + "---------"
                                 "    2    "
                                 "   h     "
                                 """""""""""
(%i6) rembox (expr2, BAR);
                    FOO"""""""""   a d - b c
(%o6)               "sin(%pi x)" + ---------
                    """"""""""""       2
                                      h
(%i7) rembox (expr2);
                                  a d - b c
(%o7)                sin(%pi x) + ---------
                                      2
                                     h

Function: reveal (expr, depth)

Replaces parts of expr at the specified integer depth with descriptive summaries.

When depth is greater than or equal to the maximum depth of expr, reveal (expr, depth) returns expr unmodified.

reveal evaluates its arguments. reveal returns the summarized expression.

Example:

(%i1) e: expand ((a - b)^2)/expand ((exp(a) + exp(b))^2);
                          2            2
                         b  - 2 a b + a
(%o1)               -------------------------
                        b + a     2 b     2 a
                    2 %e      + %e    + %e
(%i2) reveal (e, 1);
(%o2)                       Quotient
(%i3) reveal (e, 2);
                             Sum(3)
(%o3)                        ------
                             Sum(3)
(%i4) reveal (e, 3);
                     Expt + Negterm + Expt
(%o4)               ------------------------
                    Product(2) + Expt + Expt
(%i5) reveal (e, 4);
                       2                 2
                      b  - Product(3) + a
(%o5)         ------------------------------------
                         Product(2)     Product(2)
              2 Expt + %e           + %e
(%i6) reveal (e, 5);
                         2            2
                        b  - 2 a b + a
(%o6)              --------------------------
                       Sum(2)     2 b     2 a
                   2 %e       + %e    + %e
(%i7) reveal (e, 6);
                          2            2
                         b  - 2 a b + a
(%o7)               -------------------------
                        b + a     2 b     2 a
                    2 %e      + %e    + %e

Function: sublis (list, expr)

Makes multiple parallel substitutions into an expression. list is a list of equations. The left hand side of the equations must be an atom.

The variable sublis_apply_lambda controls simplification after sublis.

See also psubst for making parallel substitutions.

Example:

(%i1) sublis ([a=b, b=a], sin(a) + cos(b));
(%o1)                    sin(b) + cos(a)

Option variable: sublis_apply_lambda

Default value: true

Controls whether lambda’s substituted are applied in simplification after sublis is used or whether you have to do an ev to get things to apply. true means do the application.

Option variable: subnumsimp

Default value: false

If true then the functions subst and psubst can substitute a subscripted variable f[x] with a number, when only the symbol f is given.

See also subst.

(%i1) subst(100,g,g[x]+2);

subst: cannot substitute 100 for operator g in expression g
                                                           x
 -- an error. To debug this try: debugmode(true);

(%i2) subst(100,g,g[x]+2),subnumsimp:true;
(%o2)                          102

Function: subst (a, b, c)

Substitutes a for b in c. b must be an atom or a complete subexpression of c. For example, x+y+z is a complete subexpression of 2*(x+y+z)/w while x+y is not. When b does not have these characteristics, one may sometimes use substpart or ratsubst (see below). Alternatively, if b is of the form e/f then one could use subst (a*f, e, c) while if b is of the form e^(1/f) then one could use subst (a^f, e, c). The subst command also discerns the x^y in x^-y so that subst (a, sqrt(x), 1/sqrt(x)) yields 1/a. a and b may also be operators of an expression enclosed in double-quotes " or they may be function names. If one wishes to substitute for the independent variable in derivative forms then the at function (see below) should be used.

subst is an alias for substitute.

The commands subst (eq_1, expr) or subst ([eq_1, ..., eq_k], expr) are other permissible forms. The eq_i are equations indicating substitutions to be made. For each equation, the right side will be substituted for the left in the expression expr. The equations are substituted in serial from left to right in expr. See the functions sublis and psubst for making parallel substitutions.

exptsubst if true permits substitutions like y for %e^x in %e^(a*x) to take place.

When opsubst is false, subst will not attempt to substitute into the operator of an expression. E.g. (opsubst: false, subst (x^2, r, r+r[0])) will work.

See also at, ev and psubst, as well as let and letsimp.

Examples:

(%i1) subst (a, x+y, x + (x+y)^2 + y);
                                    2
(%o1)                      y + x + a
(%i2) subst (-%i, %i, a + b*%i);
(%o2)                       a - %i b

The substitution is done in serial for a list of equations. Compare this with a parallel substitution:

(%i1) subst([a=b, b=c], a+b);
(%o1)                          2 c
(%i2) sublis([a=b, b=c], a+b);
(%o2)                         c + b

Single-character Operators like + and - have to be quoted in order to be replaced by subst. It is to note, though, that a+b-c might be expressed as a+b+(-1*c) internally.

(%i3) subst(["+"="-"],a+b-c);
(%o3)                                 c-b+a

The difference between subst and at can be seen in the following example:

(%i1) g1:y(t)=a*x(t)+b*diff(x(t),t);
                            d
(%o1)             y(t) = b (-- (x(t))) + a x(t)
                            dt
(%i2) subst('diff(x(t),t)=1,g1);
(%o2)                   y(t) = a x(t) + b
(%i3) at(g1,'diff(x(t),t)=1);
                              !
                     d        !
(%o3)      y(t) = b (-- (x(t))!             ) + a x(t)
                     dt       !d
                              !-- (x(t)) = 1
                               dt

For further examples, do example (subst).

Function: substinpart (x, expr, n_1, …, n_k)

Similar to substpart, but substinpart works on the internal representation of expr.

Examples:

(%i1) x . 'diff (f(x), x, 2);
                              2
                             d
(%o1)                   x . (--- (f(x)))
                               2
                             dx
(%i2) substinpart (d^2, %, 2);
                                  2
(%o2)                        x . d
(%i3) substinpart (f1, f[1](x + 1), 0);
(%o3)                       f1(x + 1)

If the last argument to a part function is a list of indices then several subexpressions are picked out, each one corresponding to an index of the list. Thus

(%i1) part (x + y + z, [1, 3]);
(%o1)                         z + x

piece holds the value of the last expression selected when using the part functions. It is set during the execution of the function and thus may be referred to in the function itself as shown below. If partswitch is set to true then end is returned when a selected part of an expression doesn’t exist, otherwise an error message is given.

(%i1) expr: 27*y^3 + 54*x*y^2 + 36*x^2*y + y + 8*x^3 + x + 1;
              3         2       2            3
(%o1)     27 y  + 54 x y  + 36 x  y + y + 8 x  + x + 1
(%i2) part (expr, 2, [1, 3]);
                                  2
(%o2)                         54 y
(%i3) sqrt (piece/54);
(%o3)                        abs(y)
(%i4) substpart (factor (piece), expr, [1, 2, 3, 5]);
                               3
(%o4)               (3 y + 2 x)  + y + x + 1
(%i5) expr: 1/x + y/x - 1/z;
                             1    y   1
(%o5)                     (- -) + - + -
                             z    x   x
(%i6) substpart (xthru (piece), expr, [2, 3]);
                            y + 1   1
(%o6)                       ----- - -
                              x     z

Also, setting the option inflag to true and calling part or substpart is the same as calling inpart or substinpart.

Function: substpart (x, expr, n_1, …, n_k)

Substitutes x for the subexpression picked out by the rest of the arguments as in part. It returns the new value of expr. x may be some operator to be substituted for an operator of expr. In some cases x needs to be enclosed in double-quotes " (e.g. substpart ("+", a*b, 0) yields b + a).

Example:

(%i1) 1/(x^2 + 2);
                               1
(%o1)                        ------
                              2
                             x  + 2
(%i2) substpart (3/2, %, 2, 1, 2);
                               1
(%o2)                       --------
                             3/2
                            x    + 2
(%i3) a*x + f(b, y);
(%o3)                     a x + f(b, y)
(%i4) substpart ("+", %, 1, 0);
(%o4)                    x + f(b, y) + a

Also, setting the option inflag to true and calling part or substpart is the same as calling inpart or substinpart.

Function: symbolp (expr)

Returns true if expr is a symbol, else false.

See also Identifiers.

Function: unorder ()

Disables the aliasing created by the last use of the ordering commands ordergreat and orderless. ordergreat and orderless may not be used more than one time each without calling unorder. unorder does not substitute back in expressions the original symbols for the aliases introduced by ordergreat and orderless. Therefore, after execution of unorder the aliases appear in previous expressions.

See also ordergreat and orderless.

Examples:

ordergreat(a) introduces an alias for the symbol a. Therefore, the difference of %o2 and %o4 does not vanish. unorder does not substitute back the symbol a and the alias appears in the output %o7.

(%i1) unorder();
(%o1)                          []
(%i2) b*x + a^2;
                                   2
(%o2)                       b x + a
(%i3) ordergreat (a);
(%o3)                         done
(%i4) b*x + a^2;
 %th(1) - %th(3);
                             2
(%o4)                       a  + b x
(%i5) unorder();
                              2    2
(%o5)                        a  - a
(%i6) %th(2);
(%o6)                          [a]

Function: verbify (f)

Returns the verb form of the function name f. See also verb, noun, and nounify.

Examples:

(%i1) verbify ('foo);
(%o1)                          foo
(%i2) :lisp $%
$FOO
(%i2) nounify (foo);
(%o2)                          foo
(%i3) :lisp $%
%FOO


Next: , Previous: , Up: Top   [Contents][Index]

7 Operators


Next: , Previous: , Up: Operators   [Contents][Index]

7.1 Introduction to operators

It is possible to define new operators with specified precedence, to undefine existing operators, or to redefine the precedence of existing operators. An operator may be unary prefix or unary postfix, binary infix, n-ary infix, matchfix, or nofix. "Matchfix" means a pair of symbols which enclose their argument or arguments, and "nofix" means an operator which takes no arguments. As examples of the different types of operators, there are the following.

unary prefix

negation - a

unary postfix

factorial a!

binary infix

exponentiation a^b

n-ary infix

addition a + b

matchfix

list construction [a, b]

(There are no built-in nofix operators; for an example of such an operator, see nofix.)

The mechanism to define a new operator is straightforward. It is only necessary to declare a function as an operator; the operator function might or might not be defined.

An example of user-defined operators is the following. Note that the explicit function call "dd" (a) is equivalent to dd a, likewise "<-" (a, b) is equivalent to a <- b. Note also that the functions "dd" and "<-" are undefined in this example.

(%i1) prefix ("dd");
(%o1)                          dd
(%i2) dd a;
(%o2)                         dd a
(%i3) "dd" (a);
(%o3)                         dd a
(%i4) infix ("<-");
(%o4)                          <-
(%i5) a <- dd b;
(%o5)                      a <- dd b
(%i6) "<-" (a, "dd" (b));
(%o6)                      a <- dd b

The Maxima functions which define new operators are summarized in this table, stating the default left and right binding powers (lbp and rbp, respectively). (Binding power determines operator precedence. However, since left and right binding powers can differ, binding power is somewhat more complicated than precedence.) Some of the operation definition functions take additional arguments; see the function descriptions for details.

prefix

rbp=180

postfix

lbp=180

infix

lbp=180, rbp=180

nary

lbp=180, rbp=180

matchfix

(binding power not applicable)

nofix

(binding power not applicable)

For comparison, here are some built-in operators and their left and right binding powers.

Operator   lbp     rbp

  :        180     20 
  ::       180     20 
  :=       180     20 
  ::=      180     20 
  !        160
  !!       160
  ^        140     139 
  .        130     129 
  *        120
  /        120     120 
  +        100     100 
  -        100     134 
  =        80      80 
  #        80      80 
  >        80      80 
  >=       80      80 
  <        80      80 
  <=       80      80 
  not              70 
  and      65
  or       60
  ,        10
  $        -1
  ;        -1

remove and kill remove operator properties from an atom. remove ("a", op) removes only the operator properties of a. kill ("a") removes all properties of a, including the operator properties. Note that the name of the operator must be enclosed in quotation marks.

(%i1) infix ("##");
(%o1)                          ##
(%i2) "##" (a, b) := a^b;
                                     b
(%o2)                     a ## b := a
(%i3) 5 ## 3;
(%o3)                          125
(%i4) remove ("##", op);
(%o4)                         done
(%i5) 5 ## 3;
Incorrect syntax: # is not a prefix operator
5 ##
  ^
(%i5) "##" (5, 3);
(%o5)                          125
(%i6) infix ("##");
(%o6)                          ##
(%i7) 5 ## 3;
(%o7)                          125
(%i8) kill ("##");
(%o8)                         done
(%i9) 5 ## 3;
Incorrect syntax: # is not a prefix operator
5 ##
  ^
(%i9) "##" (5, 3);
(%o9)                       ##(5, 3)
·

Category: Operators · Category: Syntax


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7.2 Arithmetic operators

Operator: +
Operator: -
Operator: *
Operator: /
Operator: ^

The symbols + * / and ^ represent addition, multiplication, division, and exponentiation, respectively. The names of these operators are "+" "*" "/" and "^", which may appear where the name of a function or operator is required.

The symbols + and - represent unary addition and negation, respectively, and the names of these operators are "+" and "-", respectively.

Subtraction a - b is represented within Maxima as addition, a + (- b). Expressions such as a + (- b) are displayed as subtraction. Maxima recognizes "-" only as the name of the unary negation operator, and not as the name of the binary subtraction operator.

Division a / b is represented within Maxima as multiplication, a * b^(- 1). Expressions such as a * b^(- 1) are displayed as division. Maxima recognizes "/" as the name of the division operator.

Addition and multiplication are n-ary, commutative operators. Division and exponentiation are binary, noncommutative operators.

Maxima sorts the operands of commutative operators to construct a canonical representation. For internal storage, the ordering is determined by orderlessp. For display, the ordering for addition is determined by ordergreatp, and for multiplication, it is the same as the internal ordering.

Arithmetic computations are carried out on literal numbers (integers, rationals, ordinary floats, and bigfloats). Except for exponentiation, all arithmetic operations on numbers are simplified to numbers. Exponentiation is simplified to a number if either operand is an ordinary float or bigfloat or if the result is an exact integer or rational; otherwise an exponentiation may be simplified to sqrt or another exponentiation or left unchanged.

Floating-point contagion applies to arithmetic computations: if any operand is a bigfloat, the result is a bigfloat; otherwise, if any operand is an ordinary float, the result is an ordinary float; otherwise, the operands are rationals or integers and the result is a rational or integer.

Arithmetic computations are a simplification, not an evaluation. Thus arithmetic is carried out in quoted (but simplified) expressions.

Arithmetic operations are applied element-by-element to lists when the global flag listarith is true, and always applied element-by-element to matrices. When one operand is a list or matrix and another is an operand of some other type, the other operand is combined with each of the elements of the list or matrix.

Examples:

Addition and multiplication are n-ary, commutative operators. Maxima sorts the operands to construct a canonical representation. The names of these operators are "+" and "*".

(%i1) c + g + d + a + b + e + f;
(%o1)               g + f + e + d + c + b + a
(%i2) [op (%), args (%)];
(%o2)              [+, [g, f, e, d, c, b, a]]
(%i3) c * g * d * a * b * e * f;
(%o3)                     a b c d e f g
(%i4) [op (%), args (%)];
(%o4)              [*, [a, b, c, d, e, f, g]]
(%i5) apply ("+", [a, 8, x, 2, 9, x, x, a]);
(%o5)                    3 x + 2 a + 19
(%i6) apply ("*", [a, 8, x, 2, 9, x, x, a]);
                                 2  3
(%o6)                       144 a  x

Division and exponentiation are binary, noncommutative operators. The names of these operators are "/" and "^".

(%i1) [a / b, a ^ b];
                              a   b
(%o1)                        [-, a ]
                              b
(%i2) [map (op, %), map (args, %)];
(%o2)              [[/, ^], [[a, b], [a, b]]]
(%i3) [apply ("/", [a, b]), apply ("^", [a, b])];
                              a   b
(%o3)                        [-, a ]
                              b

Subtraction and division are represented internally in terms of addition and multiplication, respectively.

(%i1) [inpart (a - b, 0), inpart (a - b, 1), inpart (a - b, 2)];
(%o1)                      [+, a, - b]
(%i2) [inpart (a / b, 0), inpart (a / b, 1), inpart (a / b, 2)];
                                   1
(%o2)                       [*, a, -]
                                   b

Computations are carried out on literal numbers. Floating-point contagion applies.

(%i1) 17 + b - (1/2)*29 + 11^(2/4);
                                       5
(%o1)                   b + sqrt(11) + -
                                       2
(%i2) [17 + 29, 17 + 29.0, 17 + 29b0];
(%o2)                   [46, 46.0, 4.6b1]

Arithmetic computations are a simplification, not an evaluation.

(%i1) simp : false;
(%o1)                         false
(%i2) '(17 + 29*11/7 - 5^3);
                              29 11    3
(%o2)                    17 + ----- - 5
                                7
(%i3) simp : true;
(%o3)                         true
(%i4) '(17 + 29*11/7 - 5^3);
                                437
(%o4)                         - ---
                                 7

Arithmetic is carried out element-by-element for lists (depending on listarith) and matrices.

(%i1) matrix ([a, x], [h, u]) - matrix ([1, 2], [3, 4]);
                        [ a - 1  x - 2 ]
(%o1)                   [              ]
                        [ h - 3  u - 4 ]
(%i2) 5 * matrix ([a, x], [h, u]);
                          [ 5 a  5 x ]
(%o2)                     [          ]
                          [ 5 h  5 u ]
(%i3) listarith : false;
(%o3)                         false
(%i4) [a, c, m, t] / [1, 7, 2, 9];
                          [a, c, m, t]
(%o4)                     ------------
                          [1, 7, 2, 9]
(%i5) [a, c, m, t] ^ x;
                                      x
(%o5)                     [a, c, m, t]
(%i6) listarith : true;
(%o6)                         true
(%i7) [a, c, m, t] / [1, 7, 2, 9];
                              c  m  t
(%o7)                     [a, -, -, -]
                              7  2  9
(%i8) [a, c, m, t] ^ x;
                          x   x   x   x
(%o8)                   [a , c , m , t ]

Operator: **

Exponentiation operator. Maxima recognizes ** as the same operator as ^ in input, and it is displayed as ^ in 1-dimensional output, or by placing the exponent as a superscript in 2-dimensional output.

The fortran function displays the exponentiation operator as **, whether it was input as ** or ^.

Examples:

(%i1) is (a**b = a^b);
(%o1)                         true
(%i2) x**y + x^z;
                              z    y
(%o2)                        x  + x
(%i3) string (x**y + x^z);
(%o3)                        x^z+x^y
(%i4) fortran (x**y + x^z);
      x**z+x**y
(%o4)                         done

Operator: ^^

Noncommutative exponentiation operator. ^^ is the exponentiation operator corresponding to noncommutative multiplication ., just as the ordinary exponentiation operator ^ corresponds to commutative multiplication *.

Noncommutative exponentiation is displayed by ^^ in 1-dimensional output, and by placing the exponent as a superscript within angle brackets < > in 2-dimensional output.

Examples:

(%i1) a . a . b . b . b + a * a * a * b * b;
                        3  2    <2>    <3>
(%o1)                  a  b  + a    . b
(%i2) string (a . a . b . b . b + a * a * a * b * b);
(%o2)                  a^3*b^2+a^^2 . b^^3

Operator: .

The dot operator, for matrix (non-commutative) multiplication. When "." is used in this way, spaces should be left on both sides of it, e.g. A . B This distinguishes it plainly from a decimal point in a floating point number.

See also Dot, dot0nscsimp, dot0simp, dot1simp, dotassoc, dotconstrules, dotdistrib, dotexptsimp, dotident, and dotscrules.


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7.3 Relational operators

Operator: <
Operator: <=
Operator: >=
Operator: >

The symbols < <= >= and > represent less than, less than or equal, greater than or equal, and greater than, respectively. The names of these operators are "<" "<=" ">=" and ">", which may appear where the name of a function or operator is required.

These relational operators are all binary operators; constructs such as a < b < c are not recognized by Maxima.

Relational expressions are evaluated to Boolean values by the functions is and maybe, and the programming constructs if, while, and unless. Relational expressions are not otherwise evaluated or simplified to Boolean values, although the arguments of relational expressions are evaluated (when evaluation is not otherwise prevented by quotation).

When a relational expression cannot be evaluated to true or false, the behavior of is and if are governed by the global flag prederror. When prederror is true, is and if trigger an error. When prederror is false, is returns unknown, and if returns a partially-evaluated conditional expression.

maybe always behaves as if prederror were false, and while and unless always behave as if prederror were true.

Relational operators do not distribute over lists or other aggregates.

See also =, #, equal, and notequal.

Examples:

Relational expressions are evaluated to Boolean values by some functions and programming constructs.

(%i1) [x, y, z] : [123, 456, 789];
(%o1)                    [123, 456, 789]
(%i2) is (x < y);
(%o2)                         true
(%i3) maybe (y > z);
(%o3)                         false
(%i4) if x >= z then 1 else 0;
(%o4)                           0
(%i5) block ([S], S : 0, for i:1 while i <= 100 do S : S + i, 
             return (S));
(%o5)                         5050

Relational expressions are not otherwise evaluated or simplified to Boolean values, although the arguments of relational expressions are evaluated.

(%o1)                    [123, 456, 789]
(%i2) [x < y, y <= z, z >= y, y > z];
(%o2)    [123 < 456, 456 <= 789, 789 >= 456, 456 > 789]
(%i3) map (is, %);
(%o3)               [true, true, true, false]


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7.4 Logical operators

Operator: and

The logical conjunction operator. and is an n-ary infix operator; its operands are Boolean expressions, and its result is a Boolean value.

and forces evaluation (like is) of one or more operands, and may force evaluation of all operands.

Operands are evaluated in the order in which they appear. and evaluates only as many of its operands as necessary to determine the result. If any operand is false, the result is false and no further operands are evaluated.

The global flag prederror governs the behavior of and when an evaluated operand cannot be determined to be true or false. and prints an error message when prederror is true. Otherwise, operands which do not evaluate to true or false are accepted, and the result is a Boolean expression.

and is not commutative: a and b might not be equal to b and a due to the treatment of indeterminate operands.

Operator: not

The logical negation operator. not is a prefix operator; its operand is a Boolean expression, and its result is a Boolean value.

not forces evaluation (like is) of its operand.

The global flag prederror governs the behavior of not when its operand cannot be determined to be true or false. not prints an error message when prederror is true. Otherwise, operands which do not evaluate to true or false are accepted, and the result is a Boolean expression.

Operator: or

The logical disjunction operator. or is an n-ary infix operator; its operands are Boolean expressions, and its result is a Boolean value.

or forces evaluation (like is) of one or more operands, and may force evaluation of all operands.

Operands are evaluated in the order in which they appear. or evaluates only as many of its operands as necessary to determine the result. If any operand is true, the result is true and no further operands are evaluated.

The global flag prederror governs the behavior of or when an evaluated operand cannot be determined to be true or false. or prints an error message when prederror is true. Otherwise, operands which do not evaluate to true or false are accepted, and the result is a Boolean expression.

or is not commutative: a or b might not be equal to b or a due to the treatment of indeterminate operands.


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7.5 Operators for Equations

Operator: #

Represents the negation of syntactic equality =.

Note that because of the rules for evaluation of predicate expressions (in particular because not expr causes evaluation of expr), not a = b is equivalent to is(a # b), instead of a # b.

Examples:

(%i1) a = b;
(%o1)                         a = b
(%i2) is (a = b);
(%o2)                         false
(%i3) a # b;
(%o3)                         a # b
(%i4) not a = b;
(%o4)                         true
(%i5) is (a # b);
(%o5)                         true
(%i6) is (not a = b);
(%o6)                         true

Operator: =

The equation operator.

An expression a = b, by itself, represents an unevaluated equation, which might or might not hold. Unevaluated equations may appear as arguments to solve and algsys or some other functions.

The function is evaluates = to a Boolean value. is(a = b) evaluates a = b to true when a and b are identical. That is, a and b are atoms which are identical, or they are not atoms and their operators are identical and their arguments are identical. Otherwise, is(a = b) evaluates to false; it never evaluates to unknown. When is(a = b) is true, a and b are said to be syntactically equal, in contrast to equivalent expressions, for which is(equal(a, b)) is true. Expressions can be equivalent and not syntactically equal.

The negation of = is represented by #. As with =, an expression a # b, by itself, is not evaluated. is(a # b) evaluates a # b to true or false.

In addition to is, some other operators evaluate = and # to true or false, namely if, and, or, and not.

Note that because of the rules for evaluation of predicate expressions (in particular because not expr causes evaluation of expr), not a = b is equivalent to is(a # b), instead of a # b.

rhs and lhs return the right-hand and left-hand sides, respectively, of an equation or inequation.

See also equal and notequal.

Examples:

An expression a = b, by itself, represents an unevaluated equation, which might or might not hold.

(%i1) eq_1 : a * x - 5 * y = 17;
(%o1)                    a x - 5 y = 17
(%i2) eq_2 : b * x + 3 * y = 29;
(%o2)                    3 y + b x = 29
(%i3) solve ([eq_1, eq_2], [x, y]);
                        196         29 a - 17 b
(%o3)          [[x = ---------, y = -----------]]
                     5 b + 3 a       5 b + 3 a
(%i4) subst (%, [eq_1, eq_2]);
         196 a     5 (29 a - 17 b)
(%o4) [--------- - --------------- = 17, 
       5 b + 3 a      5 b + 3 a
                                  196 b     3 (29 a - 17 b)
                                --------- + --------------- = 29]
                                5 b + 3 a      5 b + 3 a
(%i5) ratsimp (%);
(%o5)                  [17 = 17, 29 = 29]

is(a = b) evaluates a = b to true when a and b are syntactically equal (that is, identical). Expressions can be equivalent and not syntactically equal.

(%i1) a : (x + 1) * (x - 1);
(%o1)                    (x - 1) (x + 1)
(%i2) b : x^2 - 1;
                              2
(%o2)                        x  - 1
(%i3) [is (a = b), is (a # b)];
(%o3)                     [false, true]
(%i4) [is (equal (a, b)), is (notequal (a, b))];
(%o4)                     [true, false]

Some operators evaluate = and # to true or false.

(%i1) if expand ((x + y)^2) = x^2 + 2 * x * y + y^2 then FOO else
      BAR;
(%o1)                          FOO
(%i2) eq_3 : 2 * x = 3 * x;
(%o2)                       2 x = 3 x
(%i3) eq_4 : exp (2) = %e^2;
                              2     2
(%o3)                       %e  = %e
(%i4) [eq_3 and eq_4, eq_3 or eq_4, not eq_3];
(%o4)                  [false, true, true]

Because not expr causes evaluation of expr, not a = b is equivalent to is(a # b).

(%i1) [2 * x # 3 * x, not (2 * x = 3 * x)];
(%o1)                   [2 x # 3 x, true]
(%i2) is (2 * x # 3 * x);
(%o2)                         true


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7.6 Assignment operators

Operator: :

Assignment operator.

When the left-hand side is a simple variable (not subscripted), : evaluates its right-hand side and associates that value with the left-hand side.

When the left-hand side is a subscripted element of a list, matrix, declared Maxima array, or Lisp array, the right-hand side is assigned to that element. The subscript must name an existing element; such objects cannot be extended by naming nonexistent elements.

When the left-hand side is a subscripted element of a hashed array, the right-hand side is assigned to that element, if it already exists, or a new element is allocated, if it does not already exist.

When the left-hand side is a list of simple and/or subscripted variables, the right-hand side must evaluate to a list, and the elements of the right-hand side are assigned to the elements of the left-hand side, in parallel.

See also kill and remvalue, which undo the association between the left-hand side and its value.

Examples:

Assignment to a simple variable.

(%i1) a;
(%o1)                           a
(%i2) a : 123;
(%o2)                          123
(%i3) a;
(%o3)                          123

Assignment to an element of a list.

(%i1) b : [1, 2, 3];
(%o1)                       [1, 2, 3]
(%i2) b[3] : 456;
(%o2)                          456
(%i3) b;
(%o3)                      [1, 2, 456]

Assignment to a variable that neither is the name of a list nor of an array creates a hashed array.

(%i1) c[99] : 789;
(%o1)                          789
(%i2) c[99];
(%o2)                          789
(%i3) c;
(%o3)                           c
(%i4) arrayinfo (c);
(%o4)                   [hashed, 1, [99]]
(%i5) listarray (c);
(%o5)                         [789]

Multiple assignment.

(%i1) [a, b, c] : [45, 67, 89];
(%o1)                     [45, 67, 89]
(%i2) a;
(%o2)                          45
(%i3) b;
(%o3)                          67
(%i4) c;
(%o4)                          89

Multiple assignment is carried out in parallel. The values of a and b are exchanged in this example.

(%i1) [a, b] : [33, 55];
(%o1)                       [33, 55]
(%i2) [a, b] : [b, a];
(%o2)                       [55, 33]
(%i3) a;
(%o3)                          55
(%i4) b;
(%o4)                          33

Operator: ::

Assignment operator.

:: is the same as : (which see) except that :: evaluates its left-hand side as well as its right-hand side.

Examples:

(%i1) x : 'foo;
(%o1)                          foo
(%i2) x :: 123;
(%o2)                          123
(%i3) foo;
(%o3)                          123
(%i4) x : '[a, b, c];
(%o4)                       [a, b, c]
(%i5) x :: [11, 22, 33];
(%o5)                     [11, 22, 33]
(%i6) a;
(%o6)                          11
(%i7) b;
(%o7)                          22
(%i8) c;
(%o8)                          33

Operator: ::=

Macro function definition operator. ::= defines a function (called a "macro" for historical reasons) which quotes its arguments, and the expression which it returns (called the "macro expansion") is evaluated in the context from which the macro was called. A macro function is otherwise the same as an ordinary function.

macroexpand returns a macro expansion (without evaluating it). macroexpand (foo (x)) followed by ''% is equivalent to foo (x) when foo is a macro function.

::= puts the name of the new macro function onto the global list macros. kill, remove, and remfunction unbind macro function definitions and remove names from macros.

fundef or dispfun return a macro function definition or assign it to a label, respectively.

Macro functions commonly contain buildq and splice expressions to construct an expression, which is then evaluated.

Examples

A macro function quotes its arguments, so message (1) shows y - z, not the value of y - z. The macro expansion (the quoted expression '(print ("(2) x is equal to", x))) is evaluated in the context from which the macro was called, printing message (2).

(%i1) x: %pi$
(%i2) y: 1234$
(%i3) z: 1729 * w$
(%i4) printq1 (x) ::= block (print ("(1) x is equal to", x),
      '(print ("(2) x is equal to", x)))$
(%i5) printq1 (y - z);
(1) x is equal to y - z
(2) x is equal to %pi
(%o5)                                 %pi

An ordinary function evaluates its arguments, so message (1) shows the value of y - z. The return value is not evaluated, so message (2) is not printed until the explicit evaluation ''%.

(%i1) x: %pi$
(%i2) y: 1234$
(%i3) z: 1729 * w$
(%i4) printe1 (x) := block (print ("(1) x is equal to", x),
      '(print ("(2) x is equal to", x)))$
(%i5) printe1 (y - z);
(1) x is equal to 1234 - 1729 w
(%o5)                     print((2) x is equal to, x)
(%i6) ''%;
(2) x is equal to %pi
(%o6)                                 %pi

macroexpand returns a macro expansion. macroexpand (foo (x)) followed by ''% is equivalent to foo (x) when foo is a macro function.

(%i1) x: %pi$
(%i2) y: 1234$
(%i3) z: 1729 * w$
(%i4) g (x) ::= buildq ([x], print ("x is equal to", x))$
(%i5) macroexpand (g (y - z));
(%o5)                     print(x is equal to, y - z)
(%i6) ''%;
x is equal to 1234 - 1729 w
(%o6)                            1234 - 1729 w
(%i7) g (y - z);
x is equal to 1234 - 1729 w
(%o7)                            1234 - 1729 w

Operator: :=

The function definition operator.

f(x_1, ..., x_n) := expr defines a function named f with arguments x_1, …, x_n and function body expr. := never evaluates the function body (unless explicitly evaluated by quote-quote ''). The function body is evaluated every time the function is called.

f[x_1, ..., x_n] := expr defines a so-called memoizing function. Its function body is evaluated just once for each distinct value of its arguments, and that value is returned, without evaluating the function body, whenever the arguments have those values again. (A function of this kind is also known as a “array function”.)

f[x_1, ..., x_n](y_1, ..., y_m) := expr is a special case of a memoizing function. f[x_1, ..., x_n] is a memoizing function which returns a lambda expression with arguments y_1, ..., y_m. The function body is evaluated once for each distinct value of x_1, ..., x_n, and the body of the lambda expression is that value.

When the last or only function argument x_n is a list of one element, the function defined by := accepts a variable number of arguments. Actual arguments are assigned one-to-one to formal arguments x_1, …, x_(n - 1), and any further actual arguments, if present, are assigned to x_n as a list.

All function definitions appear in the same namespace; defining a function f within another function g does not automatically limit the scope of f to g. However, local(f) makes the definition of function f effective only within the block or other compound expression in which local appears.

If some formal argument x_k is a quoted symbol, the function defined by := does not evaluate the corresponding actual argument. Otherwise all actual arguments are evaluated.

See also define and ::=.

Examples:

:= never evaluates the function body (unless explicitly evaluated by quote-quote).

(%i1) expr : cos(y) - sin(x);
(%o1)                    cos(y) - sin(x)
(%i2) F1 (x, y) := expr;
(%o2)                   F1(x, y) := expr
(%i3) F1 (a, b);
(%o3)                    cos(y) - sin(x)
(%i4) F2 (x, y) := ''expr;
(%o4)              F2(x, y) := cos(y) - sin(x)
(%i5) F2 (a, b);
(%o5)                    cos(b) - sin(a)

f(x_1, ..., x_n) := ... defines an ordinary function.

(%i1) G1(x, y) := (print ("Evaluating G1 for x=", x, "and y=", y), x.y - y.x);
(%o1) G1(x, y) := (print("Evaluating G1 for x=", x, "and y=", 
                                               y), x . y - y . x)
(%i2) G1([1, a], [2, b]);
Evaluating G1 for x= [1, a] and y= [2, b] 
(%o2)                           0
(%i3) G1([1, a], [2, b]);
Evaluating G1 for x= [1, a] and y= [2, b] 
(%o3)                           0

f[x_1, ..., x_n] := ... defines a memoizing function.

(%i1) G2[a] := (print ("Evaluating G2 for a=", a), a^2);
                                                     2
(%o1)     G2  := (print("Evaluating G2 for a=", a), a )
            a
(%i2) G2[1234];
Evaluating G2 for a= 1234 
(%o2)                        1522756
(%i3) G2[1234];
(%o3)                        1522756
(%i4) G2[2345];
Evaluating G2 for a= 2345 
(%o4)                        5499025
(%i5) arrayinfo (G2);
(%o5)              [hashed, 1, [1234], [2345]]
(%i6) listarray (G2);
(%o6)                  [1522756, 5499025]

f[x_1, ..., x_n](y_1, ..., y_m) := expr is a special case of a memoizing function.

(%i1) G3[n](x) := (print ("Evaluating G3 for n=", n), diff (sin(x)^2, x, n));
(%o1) G3 (x) := (print("Evaluating G3 for n=", n), 
        n
                                                     2
                                             diff(sin (x), x, n))
(%i2) G3[2];
Evaluating G3 for n= 2 
                                2           2
(%o2)          lambda([x], 2 cos (x) - 2 sin (x))
(%i3) G3[2];
                                2           2
(%o3)          lambda([x], 2 cos (x) - 2 sin (x))
(%i4) G3[2](1);
                           2           2
(%o4)                 2 cos (1) - 2 sin (1)
(%i5) arrayinfo (G3);
(%o5)                   [hashed, 1, [2]]
(%i6) listarray (G3);
                                2           2
(%o6)         [lambda([x], 2 cos (x) - 2 sin (x))]

When the last or only function argument x_n is a list of one element, the function defined by := accepts a variable number of arguments.

(%i1) H ([L]) := apply ("+", L);
(%o1)                H([L]) := apply("+", L)
(%i2) H (a, b, c);
(%o2)                       c + b + a

local makes a local function definition.

(%i1) foo (x) := 1 - x;
(%o1)                    foo(x) := 1 - x
(%i2) foo (100);
(%o2)                         - 99
(%i3) block (local (foo), foo (x) := 2 * x, foo (100));
(%o3)                          200
(%i4) foo (100);
(%o4)                         - 99


Previous: , Up: Operators   [Contents][Index]

7.7 User defined operators

Function: infix
    infix (op)
    infix (op, lbp, rbp)
    infix (op, lbp, rbp, lpos, rpos, pos)

Declares op to be an infix operator. An infix operator is a function of two arguments, with the name of the function written between the arguments. For example, the subtraction operator - is an infix operator.

infix (op) declares op to be an infix operator with default binding powers (left and right both equal to 180) and parts of speech (left and right both equal to any).

infix (op, lbp, rbp) declares op to be an infix operator with stated left and right binding powers and default parts of speech (left and right both equal to any).

infix (op, lbp, rbp, lpos, rpos, pos) declares op to be an infix operator with stated left and right binding powers and parts of speech lpos, rpos, and pos for the left operand, the right operand, and the operator result, respectively.

"Part of speech", in reference to operator declarations, means expression type. Three types are recognized: expr, clause, and any, indicating an algebraic expression, a Boolean expression, or any kind of expression, respectively. Maxima can detect some syntax errors by comparing the declared part of speech to an actual expression.

The precedence of op with respect to other operators derives from the left and right binding powers of the operators in question. If the left and right binding powers of op are both greater the left and right binding powers of some other operator, then op takes precedence over the other operator. If the binding powers are not both greater or less, some more complicated relation holds.

The associativity of op depends on its binding powers. Greater left binding power (lbp) implies an instance of op is evaluated before other operators to its left in an expression, while greater right binding power (rbp) implies an instance of op is evaluated before other operators to its right in an expression. Thus greater lbp makes op right-associative, while greater rbp makes op left-associative. If lbp is equal to rbp, op is left-associative.

See also Introduction to operators.

Examples:

If the left and right binding powers of op are both greater the left and right binding powers of some other operator, then op takes precedence over the other operator.

(%i1) :lisp (get '$+ 'lbp)
100
(%i1) :lisp (get '$+ 'rbp)
100
(%i1) infix ("##", 101, 101);
(%o1)                          ##
(%i2) "##"(a, b) := sconcat("(", a, ",", b, ")");
(%o2)       (a ## b) := sconcat("(", a, ",", b, ")")
(%i3) 1 + a ## b + 2;
(%o3)                       (a,b) + 3
(%i4) infix ("##", 99, 99);
(%o4)                          ##
(%i5) 1 + a ## b + 2;
(%o5)                       (a+1,b+2)

Greater lbp makes op right-associative, while greater rbp makes op left-associative.

(%i1) infix ("##", 100, 99);
(%o1)                          ##
(%i2) "##"(a, b) := sconcat("(", a, ",", b, ")")$
(%i3) foo ## bar ## baz;
(%o3)                    (foo,(bar,baz))
(%i4) infix ("##", 100, 101);
(%o4)                          ##
(%i5) foo ## bar ## baz;
(%o5)                    ((foo,bar),baz)

Maxima can detect some syntax errors by comparing the declared part of speech to an actual expression.

(%i1) infix ("##", 100, 99, expr, expr, expr);
(%o1)                          ##
(%i2) if x ## y then 1 else 0;
Incorrect syntax: Found algebraic expression where logical
expression expected
if x ## y then 
             ^
(%i2) infix ("##", 100, 99, expr, expr, clause);
(%o2)                          ##
(%i3) if x ## y then 1 else 0;
(%o3)                if x ## y then 1 else 0

Function: matchfix
    matchfix (ldelimiter, rdelimiter)
    matchfix (ldelimiter, rdelimiter, arg_pos, pos)

Declares a matchfix operator with left and right delimiters ldelimiter and rdelimiter. The delimiters are specified as strings.

A "matchfix" operator is a function of any number of arguments, such that the arguments occur between matching left and right delimiters. The delimiters may be any strings, so long as the parser can distinguish the delimiters from the operands and other expressions and operators. In practice this rules out unparseable delimiters such as %, ,, $ and ;, and may require isolating the delimiters with white space. The right delimiter can be the same or different from the left delimiter.

A left delimiter can be associated with only one right delimiter; two different matchfix operators cannot have the same left delimiter.

An existing operator may be redeclared as a matchfix operator without changing its other properties. In particular, built-in operators such as addition + can be declared matchfix, but operator functions cannot be defined for built-in operators.

The command matchfix (ldelimiter, rdelimiter, arg_pos, pos) declares the argument part-of-speech arg_pos and result part-of-speech pos, and the delimiters ldelimiter and rdelimiter.

"Part of speech", in reference to operator declarations, means expression type. Three types are recognized: expr, clause, and any, indicating an algebraic expression, a Boolean expression, or any kind of expression, respectively. Maxima can detect some syntax errors by comparing the declared part of speech to an actual expression.

The function to carry out a matchfix operation is an ordinary user-defined function. The operator function is defined in the usual way with the function definition operator := or define. The arguments may be written between the delimiters, or with the left delimiter as a quoted string and the arguments following in parentheses. dispfun (ldelimiter) displays the function definition.

The only built-in matchfix operator is the list constructor [ ]. Parentheses ( ) and double-quotes " " act like matchfix operators, but are not treated as such by the Maxima parser.

matchfix evaluates its arguments. matchfix returns its first argument, ldelimiter.

Examples:

Delimiters may be almost any strings.

(%i1) matchfix ("@@", "~");
(%o1)                          @@
(%i2) @@ a, b, c ~;
(%o2)                      @@a, b, c~
(%i3) matchfix (">>", "<<");
(%o3)                          >>
(%i4) >> a, b, c <<;
(%o4)                      >>a, b, c<<
(%i5) matchfix ("foo", "oof");
(%o5)                          foo
(%i6) foo a, b, c oof;
(%o6)                     fooa, b, coof
(%i7) >> w + foo x, y oof + z << / @@ p, q ~;
                     >>z + foox, yoof + w<<
(%o7)                ----------------------
                            @@p, q~

Matchfix operators are ordinary user-defined functions.

(%i1) matchfix ("!-", "-!");
(%o1)                         "!-"
(%i2) !- x, y -! := x/y - y/x;
                                    x   y
(%o2)                   !-x, y-! := - - -
                                    y   x
(%i3) define (!-x, y-!, x/y - y/x);
                                    x   y
(%o3)                   !-x, y-! := - - -
                                    y   x
(%i4) define ("!-" (x, y), x/y - y/x);
                                    x   y
(%o4)                   !-x, y-! := - - -
                                    y   x
(%i5) dispfun ("!-");
                                    x   y
(%t5)                   !-x, y-! := - - -
                                    y   x

(%o5)                         done
(%i6) !-3, 5-!;
                                16
(%o6)                         - --
                                15
(%i7) "!-" (3, 5);
                                16
(%o7)                         - --
                                15

Function: nary
    nary (op)
    nary (op, bp, arg_pos, pos)

An nary operator is used to denote a function of any number of arguments, each of which is separated by an occurrence of the operator, e.g. A+B or A+B+C. The nary("x") function is a syntax extension function to declare x to be an nary operator. Functions may be declared to be nary. If declare(j,nary); is done, this tells the simplifier to simplify, e.g. j(j(a,b),j(c,d)) to j(a, b, c, d).

See also Introduction to operators.

Function: nofix
    nofix (op)
    nofix (op, pos)

nofix operators are used to denote functions of no arguments. The mere presence of such an operator in a command will cause the corresponding function to be evaluated. For example, when one types "exit;" to exit from a Maxima break, "exit" is behaving similar to a nofix operator. The function nofix("x") is a syntax extension function which declares x to be a nofix operator.

See also Introduction to operators.

Function: postfix
    postfix (op)
    postfix (op, lbp, lpos, pos)

postfix operators like the prefix variety denote functions of a single argument, but in this case the argument immediately precedes an occurrence of the operator in the input string, e.g. 3!. The postfix("x") function is a syntax extension function to declare x to be a postfix operator.

See also Introduction to operators.

Function: prefix
    prefix (op)
    prefix (op, rbp, rpos, pos)

A prefix operator is one which signifies a function of one argument, which argument immediately follows an occurrence of the operator. prefix("x") is a syntax extension function to declare x to be a prefix operator.

See also Introduction to operators.


Next: , Previous: , Up: Top   [Contents][Index]

8 Evaluation


Previous: , Up: Evaluation   [Contents][Index]

8.1 Functions and Variables for Evaluation

Operator: '

The single quote operator ' prevents evaluation.

Applied to a symbol, the single quote prevents evaluation of the symbol.

Applied to a function call, the single quote prevents evaluation of the function call, although the arguments of the function are still evaluated (if evaluation is not otherwise prevented). The result is the noun form of the function call.

Applied to a parenthesized expression, the single quote prevents evaluation of all symbols and function calls in the expression. E.g., '(f(x)) means do not evaluate the expression f(x). 'f(x) (with the single quote applied to f instead of f(x)) means return the noun form of f applied to [x].

The single quote does not prevent simplification.

When the global flag noundisp is true, nouns display with a single quote. This switch is always true when displaying function definitions.

See also the quote-quote operator '' and nouns.

Examples:

Applied to a symbol, the single quote prevents evaluation of the symbol.

(%i1) aa: 1024;
(%o1)                         1024
(%i2) aa^2;
(%o2)                        1048576
(%i3) 'aa^2;
                                 2
(%o3)                          aa
(%i4) ''%;
(%o4)                        1048576

Applied to a function call, the single quote prevents evaluation of the function call. The result is the noun form of the function call.

(%i1) x0: 5;
(%o1)                           5
(%i2) x1: 7;
(%o2)                           7
(%i3) integrate (x^2, x, x0, x1);
                               218
(%o3)                          ---
                                3
(%i4) 'integrate (x^2, x, x0, x1);
                             7
                            /
                            [   2
(%o4)                       I  x  dx
                            ]
                            /
                             5
(%i5) %, nouns;
                               218
(%o5)                          ---
                                3

Applied to a parenthesized expression, the single quote prevents evaluation of all symbols and function calls in the expression.

(%i1) aa: 1024;
(%o1)                         1024
(%i2) bb: 19;
(%o2)                          19
(%i3) sqrt(aa) + bb;
(%o3)                          51
(%i4) '(sqrt(aa) + bb);
(%o4)                     bb + sqrt(aa)
(%i5) ''%;
(%o5)                          51

The single quote does not prevent simplification.

(%i1) sin (17 * %pi) + cos (17 * %pi);
(%o1)                          - 1
(%i2) '(sin (17 * %pi) + cos (17 * %pi));
(%o2)                          - 1

Maxima considers floating point operations by its in-built mathematical functions to be a simplification.

(%i1) sin(1.0);
(%o1)                          .8414709848078965
(%i2) '(sin(1.0));
(%o2)                          .8414709848078965

When the global flag noundisp is true, nouns display with a single quote.

(%i1) x:%pi;
(%o1)                                 %pi
(%i2) bfloat(x);
(%o2)                         3.141592653589793b0
(%i3) sin(x);
(%o3)                                  0
(%i4) noundisp;
(%o4)                                false
(%i5) 'bfloat(x);
(%o5)                             bfloat(%pi)
(%i6) bfloat('x);
(%o6)                                  x
(%i7) 'sin(x);
(%o7)                                  0
(%i8) sin('x);
(%o8)                               sin(x)
(%i9) noundisp : not noundisp;
(%o9)                                true
(%i10) 'bfloat(x);
(%o10)                           'bfloat(%pi)
(%i11) bfloat('x);
(%o11)                                 x
(%i12) 'sin(x);
(%o12)                                 0
(%i13) sin('x);
(%o13)                              sin(x)
(%i14)

Operator: ''

The quote-quote operator '' (two single quote marks) modifies evaluation in input expressions.

Applied to a general expression expr, quote-quote causes the value of expr to be substituted for expr in the input expression.

Applied to the operator of an expression, quote-quote changes the operator from a noun to a verb (if it is not already a verb).

The quote-quote operator is applied by the input parser; it is not stored as part of a parsed input expression. The quote-quote operator is always applied as soon as it is parsed, and cannot be quoted. Thus quote-quote causes evaluation when evaluation is otherwise suppressed, such as in function definitions, lambda expressions, and expressions quoted by single quote '.

Quote-quote is recognized by batch and load.

See also ev, the single-quote operator ' and nouns.

Examples:

Applied to a general expression expr, quote-quote causes the value of expr to be substituted for expr in the input expression.

(%i1) expand ((a + b)^3);
                     3        2      2      3
(%o1)               b  + 3 a b  + 3 a  b + a
(%i2) [_, ''_];
                         3    3        2      2      3
(%o2)     [expand((b + a) ), b  + 3 a b  + 3 a  b + a ]
(%i3) [%i1, ''%i1];
                         3    3        2      2      3
(%o3)     [expand((b + a) ), b  + 3 a b  + 3 a  b + a ]
(%i4) [aa : cc, bb : dd, cc : 17, dd : 29];
(%o4)                   [cc, dd, 17, 29]
(%i5) foo_1 (x) := aa - bb * x;
(%o5)                 foo_1(x) := aa - bb x
(%i6) foo_1 (10);
(%o6)                      cc - 10 dd
(%i7) ''%;
(%o7)                         - 273
(%i8) ''(foo_1 (10));
(%o8)                         - 273
(%i9) foo_2 (x) := ''aa - ''bb * x;
(%o9)                 foo_2(x) := cc - dd x
(%i10) foo_2 (10);
(%o10)                        - 273
(%i11) [x0 : x1, x1 : x2, x2 : x3];
(%o11)                    [x1, x2, x3]
(%i12) x0;
(%o12)                         x1
(%i13) ''x0;
(%o13)                         x2
(%i14) '' ''x0;
(%o14)                         x3

Applied to the operator of an expression, quote-quote changes the operator from a noun to a verb (if it is not already a verb).

(%i1) declare (foo, noun);
(%o1)                         done
(%i2) foo (x) := x - 1729;
(%o2)                 ''foo(x) := x - 1729
(%i3) foo (100);
(%o3)                       foo(100)
(%i4) ''foo (100);
(%o4)                        - 1629

The quote-quote operator is applied by the input parser; it is not stored as part of a parsed input expression.

(%i1) [aa : bb, cc : dd, bb : 1234, dd : 5678];
(%o1)                 [bb, dd, 1234, 5678]
(%i2) aa + cc;
(%o2)                        dd + bb
(%i3) display (_, op (_), args (_));
                           _ = cc + aa

                         op(cc + aa) = +

                    args(cc + aa) = [cc, aa]

(%o3)                         done
(%i4) ''(aa + cc);
(%o4)                         6912
(%i5) display (_, op (_), args (_));
                           _ = dd + bb

                         op(dd + bb) = +

                    args(dd + bb) = [dd, bb]

(%o5)                         done

Quote-quote causes evaluation when evaluation is otherwise suppressed, such as in function definitions, lambda expressions, and expressions quoted by single quote '.

(%i1) foo_1a (x) := ''(integrate (log (x), x));
(%o1)               foo_1a(x) := x log(x) - x
(%i2) foo_1b (x) := integrate (log (x), x);
(%o2)           foo_1b(x) := integrate(log(x), x)
(%i3) dispfun (foo_1a, foo_1b);
(%t3)               foo_1a(x) := x log(x) - x

(%t4)           foo_1b(x) := integrate(log(x), x)

(%o4)                      [%t3, %t4]
(%i5) integrate (log (x), x);
(%o5)                     x log(x) - x
(%i6) foo_2a (x) := ''%;
(%o6)               foo_2a(x) := x log(x) - x
(%i7) foo_2b (x) := %;
(%o7)                    foo_2b(x) := %
(%i8) dispfun (foo_2a, foo_2b);
(%t8)               foo_2a(x) := x log(x) - x

(%t9)                    foo_2b(x) := %

(%o9)                      [%t7, %t8]
(%i10) F : lambda ([u], diff (sin (u), u));
(%o10)             lambda([u], diff(sin(u), u))
(%i11) G : lambda ([u], ''(diff (sin (u), u)));
(%o11)                  lambda([u], cos(u))
(%i12) '(sum (a[k], k, 1, 3) + sum (b[k], k, 1, 3));
(%o12)         sum(b , k, 1, 3) + sum(a , k, 1, 3)
                    k                  k
(%i13) '(''(sum (a[k], k, 1, 3)) + ''(sum (b[k], k, 1, 3)));
(%o13)             b  + a  + b  + a  + b  + a
                    3    3    2    2    1    1

Function: ev (expr, arg_1, …, arg_n)

Evaluates the expression expr in the environment specified by the arguments arg_1, …, arg_n. The arguments are switches (Boolean flags), assignments, equations, and functions. ev returns the result (another expression) of the evaluation.

The evaluation is carried out in steps, as follows.

  1. First the environment is set up by scanning the arguments which may be any or all of the following.
    • simp causes expr to be simplified regardless of the setting of the switch simp which inhibits simplification if false.
    • noeval suppresses the evaluation phase of ev (see step (4) below). This is useful in conjunction with the other switches and in causing expr to be resimplified without being reevaluated.
    • nouns causes the evaluation of noun forms (typically unevaluated functions such as 'integrate or 'diff) in expr.
    • expand causes expansion.
    • expand (m, n) causes expansion, setting the values of maxposex and maxnegex to m and n respectively.
    • detout causes any matrix inverses computed in expr to have their determinant kept outside of the inverse rather than dividing through each element.
    • diff causes all differentiations indicated in expr to be performed.
    • derivlist (x, y, z, ...) causes only differentiations with respect to the indicated variables. See also derivlist.
    • risch causes integrals in expr to be evaluated using the Risch algorithm. See risch. The standard integration routine is invoked when using the special symbol nouns.
    • float causes non-integral rational numbers to be converted to floating point.
    • numer causes some mathematical functions (including exponentiation) with numerical arguments to be evaluated in floating point. It causes variables in expr which have been given numervals to be replaced by their values. It also sets the float switch on.
    • pred causes predicates (expressions which evaluate to true or false) to be evaluated.
    • eval causes an extra post-evaluation of expr to occur. (See step (5) below.) eval may occur multiple times. For each instance of eval, the expression is evaluated again.
    • A where A is an atom declared to be an evaluation flag evflag causes A to be bound to true during the evaluation of expr.
    • V: expression (or alternately V=expression) causes V to be bound to the value of expression during the evaluation of expr. Note that if V is a Maxima option, then expression is used for its value during the evaluation of expr. If more than one argument to ev is of this type then the binding is done in parallel. If V is a non-atomic expression then a substitution rather than a binding is performed.
    • F where F, a function name, has been declared to be an evaluation function evfun causes F to be applied to expr.
    • Any other function names, e.g. sum, cause evaluation of occurrences of those names in expr as though they were verbs.
    • In addition a function occurring in expr (say F(x)) may be defined locally for the purpose of this evaluation of expr by giving F(x) := expression as an argument to ev.
    • If an atom not mentioned above or a subscripted variable or subscripted expression was given as an argument, it is evaluated and if the result is an equation or assignment then the indicated binding or substitution is performed. If the result is a list then the members of the list are treated as if they were additional arguments given to ev. This permits a list of equations to be given (e.g. [X=1, Y=A**2]) or a list of names of equations (e.g., [%t1, %t2] where %t1 and %t2 are equations) such as that returned by solve.

    The arguments of ev may be given in any order with the exception of substitution equations which are handled in sequence, left to right, and evaluation functions which are composed, e.g., ev (expr, ratsimp, realpart) is handled as realpart (ratsimp (expr)).

    The simp, numer, and float switches may also be set locally in a block, or globally in Maxima so that they will remain in effect until being reset.

    If expr is a canonical rational expression (CRE), then the expression returned by ev is also a CRE, provided the numer and float switches are not both true.

  2. During step (1), a list is made of the non-subscripted variables appearing on the left side of equations in the arguments or in the value of some arguments if the value is an equation. The variables (subscripted variables which do not have associated memoizing functions as well as non-subscripted variables) in the expression expr are replaced by their global values, except for those appearing in this list. Usually, expr is just a label or % (as in %i2 in the example below), so this step simply retrieves the expression named by the label, so that ev may work on it.
  3. If any substitutions are indicated by the arguments, they are carried out now.
  4. The resulting expression is then re-evaluated (unless one of the arguments was noeval) and simplified according to the arguments. Note that any function calls in expr will be carried out after the variables in it are evaluated and that ev(F(x)) thus may behave like F(ev(x)).
  5. For each instance of eval in the arguments, steps (3) and (4) are repeated.

See also '', at and subst.

Examples:

(%i1) sin(x) + cos(y) + (w+1)^2 + 'diff (sin(w), w);
                                     d                    2
(%o1)              cos(y) + sin(x) + -- (sin(w)) + (w + 1)
                                     dw
(%i2) ev (%, numer, expand, diff, x=2, y=1);
                               2
(%o2)                cos(w) + w  + 2 w + 2.449599732693821

An alternate top level syntax has been provided for ev, whereby one may just type in its arguments, without the ev(). That is, one may write simply

expr, arg_1, ..., arg_n

This is not permitted as part of another expression, e.g., in functions, blocks, etc.

Notice the parallel binding process in the following example.

(%i3) programmode: false;
(%o3)                                false
(%i4) x+y, x: a+y, y: 2;
(%o4)                              y + a + 2
(%i5) 2*x - 3*y = 3$
(%i6) -3*x + 2*y = -4$
(%i7) solve ([%o5, %o6]);
Solution

                                          1
(%t7)                               y = - -
                                          5

                                         6
(%t8)                                x = -
                                         5
(%o8)                            [[%t7, %t8]]
(%i8) %o6, %o8;
(%o8)                              - 4 = - 4
(%i9) x + 1/x > gamma (1/2);
                                   1
(%o9)                          x + - > sqrt(%pi)
                                   x
(%i10) %, numer, x=1/2;
(%o10)                      2.5 > 1.772453850905516
(%i11) %, pred;
(%o11)                               true

Special symbol: eval

As an argument in a call to ev (expr), eval causes an extra evaluation of expr. See ev.

Example:

(%i1) [a:b,b:c,c:d,d:e];
(%o1)                            [b, c, d, e]
(%i2) a;
(%o2)                                  b
(%i3) ev(a);
(%o3)                                  c
(%i4) ev(a),eval;
(%o4)                                  e
(%i5) a,eval,eval;
(%o5)                                  e

Property: evflag

When a symbol x has the evflag property, the expressions ev(expr, x) and expr, x (at the interactive prompt) are equivalent to ev(expr, x = true). That is, x is bound to true while expr is evaluated.

The expression declare(x, evflag) gives the evflag property to the variable x.

The flags which have the evflag property by default are the following:

   algebraic          cauchysum       demoivre
   dotscrules         %emode          %enumer
   exponentialize     exptisolate     factorflag
   float              halfangles      infeval
   isolate_wrt_times  keepfloat       letrat
   listarith          logabs          logarc 
   logexpand          lognegint       
   m1pbranch          numer_pbranch   programmode 
   radexpand          ratalgdenom     ratfac 
   ratmx              ratsimpexpons   simp 
   simpproduct        simpsum         sumexpand
   trigexpand

Examples:

(%i1) sin (1/2);
                                 1
(%o1)                        sin(-)
                                 2
(%i2) sin (1/2), float;
(%o2)                   0.479425538604203
(%i3) sin (1/2), float=true;
(%o3)                   0.479425538604203
(%i4) simp : false;
(%o4)                         false
(%i5) 1 + 1;
(%o5)                         1 + 1
(%i6) 1 + 1, simp;
(%o6)                           2
(%i7) simp : true;
(%o7)                         true
(%i8) sum (1/k^2, k, 1, inf);
                            inf
                            ====
                            \     1
(%o8)                        >    --
                            /      2
                            ====  k
                            k = 1
(%i9) sum (1/k^2, k, 1, inf), simpsum;
                                 2
                              %pi
(%o9)                         ----
                               6
(%i10) declare (aa, evflag);
(%o10)                        done
(%i11) if aa = true then YES else NO;
(%o11)                         NO
(%i12) if aa = true then YES else NO, aa;
(%o12)                         YES

Property: evfun

When a function F has the evfun property, the expressions ev(expr, F) and expr, F (at the interactive prompt) are equivalent to F(ev(expr)).

If two or more evfun functions F, G, etc., are specified, the functions are applied in the order that they are specified.

The expression declare(F, evfun) gives the evfun property to the function F. The functions which have the evfun property by default are the following:

   bfloat          factor       fullratsimp
   logcontract     polarform    radcan
   ratexpand       ratsimp      rectform
   rootscontract   trigexpand   trigreduce

Examples:

(%i1) x^3 - 1;
                              3
(%o1)                        x  - 1
(%i2) x^3 - 1, factor;
                                2
(%o2)                 (x - 1) (x  + x + 1)
(%i3) factor (x^3 - 1);
                                2
(%o3)                 (x - 1) (x  + x + 1)
(%i4) cos(4 * x) / sin(x)^4;
                            cos(4 x)
(%o4)                       --------
                               4
                            sin (x)
(%i5) cos(4 * x) / sin(x)^4, trigexpand;
                 4           2       2         4
              sin (x) - 6 cos (x) sin (x) + cos (x)
(%o5)         -------------------------------------
                                4
                             sin (x)
(%i6) cos(4 * x) / sin(x)^4, trigexpand, ratexpand;
                           2         4
                      6 cos (x)   cos (x)
(%o6)               - --------- + ------- + 1
                          2          4
                       sin (x)    sin (x)
(%i7) ratexpand (trigexpand (cos(4 * x) / sin(x)^4));
                           2         4
                      6 cos (x)   cos (x)
(%o7)               - --------- + ------- + 1
                          2          4
                       sin (x)    sin (x)
(%i8) declare ([F, G], evfun);
(%o8)                         done
(%i9) (aa : bb, bb : cc, cc : dd);
(%o9)                          dd
(%i10) aa;
(%o10)                         bb
(%i11) aa, F;
(%o11)                        F(cc)
(%i12) F (aa);
(%o12)                        F(bb)
(%i13) F (ev (aa));
(%o13)                        F(cc)
(%i14) aa, F, G;
(%o14)                      G(F(cc))
(%i15) G (F (ev (aa)));
(%o15)                      G(F(cc))

Option variable: infeval

Enables "infinite evaluation" mode. ev repeatedly evaluates an expression until it stops changing. To prevent a variable, say X, from being evaluated away in this mode, simply include X='X as an argument to ev. Of course expressions such as ev (X, X=X+1, infeval) will generate an infinite loop.

Special symbol: noeval

noeval suppresses the evaluation phase of ev. This is useful in conjunction with other switches and in causing expressions to be resimplified without being reevaluated.

Special symbol: nouns

nouns is an evflag. When used as an option to the ev command, nouns converts all "noun" forms occurring in the expression being ev’d to "verbs", i.e., evaluates them. See also noun, nounify, verb, and verbify.

Special symbol: pred

As an argument in a call to ev (expr), pred causes predicates (expressions which evaluate to true or false) to be evaluated. See ev.

Example:

(%i1) 1<2;
(%o1)                                1 < 2
(%i2) 1<2,pred;
(%o2)                                true


Next: , Previous: , Up: Top   [Contents][Index]

9 Simplification


Next: , Up: Simplification   [Contents][Index]

9.1 Introduction to Simplification

Maxima performs a cycle of actions in response to each new user-typed command. This consists of four steps: reading or "parsing" the input, evaluation, simplification and output. Parsing converts a syntactically valid sequence of typed characters into a data structure to be used for the rest of the operations. Evaluation includes replacing names with their assigned values. Simplification means rewriting an expression to be easier for the user or other programs to understand. Output includes displaying computational results in a variety of different formats and notations.

Evaluation and simplification sometimes appear to have similar functionality as they both have the goal of removing "complexity" and system designers have sometimes divided a task so that it is performed partly in each. For example, integrate(x,x) evaluates the answer as x*x/2, which is then simplified to x^2/2.

Evaluation is always present: it is the consequence of having a programming system with functions, subroutines, variables, values, loops, assignments and so on. In the evaluation step, built-in or user-defined function names are replaced by their definitions, variables are replaced by their values. This is largely the same as activities of a conventional programming language, but extended to work with symbolic mathematical data. Because of the generality of the mathematics at hand, there are different possible models of evaluation and so the systems has optional "flags" that can steer the process of evaluation. See Functions and Variables for Evaluation.

By contrast, the intent of simplification is to maintain the value of an expression while re-formulating its representation to be smaller, simpler to understand, or to conform to particular specifications (like factored, expanded). For example, sin(0) to 0 or x+x to 2*x. There are several powerful tools to alter the results of simplification, since it is largely in this part of the system that a user can incorporate knowledge of newly introduced functions or symbolic notation into Maxima.

Simplification is generally done at four different levels:

The internal simplifier belongs to the heart of Maxima. It is a large and complicated collection of programs, and it has been refined over many years and by thousands of users. Nevertheless, especially if you are trying out novel ideas or unconventional notation, you may find it helpful to make small (or large) changes to the program yourself. For details see for example the paper at the end of https://people.eecs.berkeley.edu/~fateman/papers/intro5.txt.

Maxima internally represents expressions as "trees" with operators or "roots" like +, * , = and operands ("leaves") which are variables like x, y, z, functions or sub-trees, like x*y. Each operator has a simplification program associated with it. + (which also covers binary - since a-b = a+(-1)*b) and * (which also covers / since a/b = a*b^(-1)) have rather elaborate simplification programs. These simplification programs (simplus, simptimes, simpexpt, etc.) are called whenever the simplifier encounters the respective arithmetic operators in an expression tree to be analyzed.

The structure of the simplifier dates back to 1965, and many hands have worked on it through the years. The structure turns out to be, in modern jargon, data- directed, or object-oriented. The program dispatches to the appropriate routine depending on the root of some sub-tree of the expression, recursively. This general notion means you can make modifications to the simplification process by very local changes to the program. In many cases it is conceptually straightforward to add an operator and add its simplification routine without disturbing existing code.

We note that in addition to this general simplifier operating on algebraic expression trees, there are several other representations of expressions in Maxima which have separate methods and simplifiers. For example, the rat function converts polynomials to vectors of coefficients to assist in rapid manipulation of such forms. Other representations include Taylor series and the (rarely used) Poisson series.

All operators introduced by the user initially have no simplification programs associated with them. Maxima does not know anything about function "f" and so typing f(a,b) will result in simplifying a,b, but not f. Even some built-in operators have no simplifications. For example, = does not "simplify" – it is a place-holder with no simplification semantics other than to simplify its two arguments, in this case referred to as the left and right sides. Other parts of Maxima such as the solve program take special note of equations, that is, trees with = as the root. (Note – in Maxima, the assignment operation is : . That is, q: 4 sets the value of the symbol q to 4. Function definition is done with :=. )

The general simplifier returns results with an internal flag indicating the expression and each sub-expression has been simplified. This does not guarantee that it is unique over all possible equivalent expressions. That’s too hard (theoretically, not possible given the generality of what can be expressed in Maxima). However, some aspects of the expression, such as the ordering of terms in a sum or product, are made uniform. This is important for the other programs to work properly.

You can set a number of option variables which direct Maxima’s processing to favor particular kinds of patterns as being goals. You can even use the most extreme option which is to turn the simplifier off by simp:false. We do not recommend this since many internal routines expect their arguments to be simplified. (About the only time it seems plausible to turn off the simplifier is in the rare case that you want to over-ride a built-in simplification. In that case you might temporarily disable the simplifier, put in the new transformation via tellsimp, and then re-enable the simplifier by simp:true.)

It is more plausible for you to associate user-defined symbolic function names or operators with properties (additive, lassociative, oddfun, antisymmetric, linear, outative, commutative, multiplicative, rassociative, evenfun, nary and symmetric). These options steer the simplifier processing in systematic directions.

For example, declare(f,oddfun) specifies that f is an odd function. Maxima will simplify f(-x) to -f(x). In the case of an even function, that is declare(g,evenfun), Maxima will simplify g(-x) to g(x). You can also associate a programming function with a name such as h(x):=x^2+1. In that case the evaluator will immediately replace h(3) by 10, and h(a+1) by (a+1)^2+1, so any properties of h will be ignored.

In addition to these directly related properties set up by the user, facts and properties from the actual context may have an impact on the simplifier’s behavior, too. See Introduction to Maximas Database.

Example: sin(n*%pi) is simplified to zero, if n is an integer.

(%i1) sin(n*%pi);
(%o1)                      sin(%pi n)
(%i2) declare(n, integer);
(%o2)                         done
(%i3) sin(n*%pi);
(%o3)                           0

If automated simplification is not sufficient, you can consider a variety of built-in, but explicitly called simplfication functions (ratsimp, expand, factor, radcan and others). There are also flags that will push simplification into one or another direction. Given demoivre:true the simplifier rewrites complex exponentials as trigonometric forms. Given exponentialize:true the simplifier tries to do the reverse: rewrite trigonometric forms as complex exponentials.

As everywhere in Maxima, by writing your own functions (be it in the Maxima user language or in the implementation language Lisp) and explicitly calling them at selected places in the program, you can respond to your individual simplification needs. Lisp gives you a handle on all the internal mechanisms, but you rarely need this full generality. "Tellsimp" is designed to generate much of the Lisp internal interface into the simplifier automatically. See See Rules and Patterns.

Over the years (Maxima/Macsyma’s origins date back to about 1966!) users have contributed numerous application packages and tools to extend or alter its functional behavior. Various non-standard and "share" packages exist to modify or extend simplification as well. You are invited to look into this more experimental material where work is still in progress See simplification-pkg.

The following appended material is optional on a first reading, and reading it is not necessary for productive use of Maxima. It is for the curious user who wants to understand what is going on, or the ambitious programmer who might wish to change the (open-source) code. Experimentation with redefining Maxima Lisp code is easily possible: to change the definition of a Lisp program (say the one that simplifies cos(), named simp%cos), you simply load into Maxima a text file that will overwrite the simp%cos function from the maxima package.


Previous: , Up: Simplification   [Contents][Index]

9.2 Functions and Variables for Simplification

Property: additive

If declare(f,additive) has been executed, then:

(1) If f is univariate, whenever the simplifier encounters f applied to a sum, f will be distributed over that sum. I.e. f(y+x) will simplify to f(y)+f(x).

(2) If f is a function of 2 or more arguments, additivity is defined as additivity in the first argument to f, as in the case of sum or integrate, i.e. f(h(x)+g(x),x) will simplify to f(h(x),x)+f(g(x),x). This simplification does not occur when f is applied to expressions of the form sum(x[i],i,lower-limit,upper-limit).

Example:

(%i1) F3 (a + b + c);
(%o1)                     F3(c + b + a)
(%i2) declare (F3, additive);
(%o2)                         done
(%i3) F3 (a + b + c);
(%o3)                 F3(c) + F3(b) + F3(a)

Property: antisymmetric

If declare(h,antisymmetric) is done, this tells the simplifier that h is antisymmetric. E.g. h(x,z,y) will simplify to - h(x, y, z). That is, it will give (-1)^n times the result given by symmetric or commutative, where n is the number of interchanges of two arguments necessary to convert it to that form.

Examples:

(%i1) S (b, a);
(%o1)                        S(b, a)
(%i2) declare (S, symmetric);
(%o2)                         done
(%i3) S (b, a);
(%o3)                        S(a, b)
(%i4) S (a, c, e, d, b);
(%o4)                   S(a, b, c, d, e)
(%i5) T (b, a);
(%o5)                        T(b, a)
(%i6) declare (T, antisymmetric);
(%o6)                         done
(%i7) T (b, a);
(%o7)                       - T(a, b)
(%i8) T (a, c, e, d, b);
(%o8)                   T(a, b, c, d, e)

Function: combine (expr)

Simplifies the sum expr by combining terms with the same denominator into a single term.

Example:

(%i1) 1*f/2*b + 2*c/3*a + 3*f/4*b +c/5*b*a;
                      5 b f   a b c   2 a c
(%o1)                 ----- + ----- + -----
                        4       5       3
(%i2) combine (%);
                  75 b f + 4 (3 a b c + 10 a c)
(%o2)             -----------------------------
                               60

Property: commutative

If declare(h, commutative) is done, this tells the simplifier that h is a commutative function. E.g. h(x, z, y) will simplify to h(x, y, z). This is the same as symmetric.

Exemplo:

(%i1) S (b, a);
(%o1)                        S(b, a)
(%i2) S (a, b) + S (b, a);
(%o2)                   S(b, a) + S(a, b)
(%i3) declare (S, commutative);
(%o3)                         done
(%i4) S (b, a);
(%o4)                        S(a, b)
(%i5) S (a, b) + S (b, a);
(%o5)                       2 S(a, b)
(%i6) S (a, c, e, d, b);
(%o6)                   S(a, b, c, d, e)

Function: demoivre (expr)
Option variable: demoivre

The function demoivre (expr) converts one expression without setting the global variable demoivre.

When the variable demoivre is true, complex exponentials are converted into equivalent expressions in terms of circular functions: exp (a + b*%i) simplifies to %e^a * (cos(b) + %i*sin(b)) if b is free of %i. a and b are not expanded.

The default value of demoivre is false.

exponentialize converts circular and hyperbolic functions to exponential form. demoivre and exponentialize cannot both be true at the same time.

Function: distrib (expr)

Distributes sums over products. It differs from expand in that it works at only the top level of an expression, i.e., it doesn’t recurse and it is faster than expand. It differs from multthru in that it expands all sums at that level.

Examples:

(%i1) distrib ((a+b) * (c+d));
(%o1)                 b d + a d + b c + a c
(%i2) multthru ((a+b) * (c+d));
(%o2)                 (b + a) d + (b + a) c
(%i3) distrib (1/((a+b) * (c+d)));
                                1
(%o3)                    ---------------
                         (b + a) (d + c)
(%i4) expand (1/((a+b) * (c+d)), 1, 0);
                                1
(%o4)                 ---------------------
                      b d + a d + b c + a c

Option variable: distribute_over

Default value: true

distribute_over controls the mapping of functions over bags like lists, matrices, and equations. At this time not all Maxima functions have this property. It is possible to look up this property with the command properties.

The mapping of functions is switched off, when setting distribute_over to the value false.

Examples:

The sin function maps over a list:

(%i1) sin([x,1,1.0]);
(%o1)         [sin(x), sin(1), 0.8414709848078965]

mod is a function with two arguments which maps over lists. Mapping over nested lists is possible too:

(%i1) mod([x,11,2*a],10);
(%o1)             [mod(x, 10), 1, 2 mod(a, 5)]
(%i2) mod([[x,y,z],11,2*a],10);
(%o2) [[mod(x, 10), mod(y, 10), mod(z, 10)], 1, 2 mod(a, 5)]

Mapping of the floor function over a matrix and an equation:

(%i1) floor(matrix([a,b],[c,d]));
                     [ floor(a)  floor(b) ]
(%o1)                [                    ]
                     [ floor(c)  floor(d) ]
(%i2) floor(a=b);
(%o2)                  floor(a) = floor(b)

Functions with more than one argument map over any of the arguments or all arguments:

(%i1) expintegral_e([1,2],[x,y]);
(%o1) [[expintegral_e(1, x), expintegral_e(1, y)], 
                      [expintegral_e(2, x), expintegral_e(2, y)]]

Check if a function has the property distribute_over:

(%i1) properties(abs);
(%o1) [integral, rule, distributes over bags, noun, gradef, 
                                                 system function]

The mapping of functions is switched off, when setting distribute_over to the value false.

(%i1) distribute_over;
(%o1)                         true
(%i2) sin([x,1,1.0]);
(%o2)         [sin(x), sin(1), 0.8414709848078965]
(%i3) distribute_over : not distribute_over;
(%o3)                         false
(%i4) sin([x,1,1.0]);
(%o4)                   sin([x, 1, 1.0])

Option variable: domain

Default value: real

When domain is set to complex, sqrt (x^2) will remain sqrt (x^2) instead of returning abs(x).

Property: evenfun
Property: oddfun

declare(f, evenfun) or declare(f, oddfun) tells Maxima to recognize the function f as an even or odd function.

Examples:

(%i1) o (- x) + o (x);
(%o1)                     o(x) + o(- x)
(%i2) declare (o, oddfun);
(%o2)                         done
(%i3) o (- x) + o (x);
(%o3)                           0
(%i4) e (- x) - e (x);
(%o4)                     e(- x) - e(x)
(%i5) declare (e, evenfun);
(%o5)                         done
(%i6) e (- x) - e (x);
(%o6)                           0
Function: expand
    expand (expr)
    expand (expr, p, n)

Expand expression expr. Products of sums and exponentiated sums are multiplied out, numerators of rational expressions which are sums are split into their respective terms, and multiplication (commutative and non-commutative) are distributed over addition at all levels of expr.

For polynomials one should usually use ratexpand which uses a more efficient algorithm.

maxnegex and maxposex control the maximum negative and positive exponents, respectively, which will expand.

expand (expr, p, n) expands expr, using p for maxposex and n for maxnegex. This is useful in order to expand part but not all of an expression.

expon - the exponent of the largest negative power which is automatically expanded (independent of calls to expand). For example if expon is 4 then (x+1)^(-5) will not be automatically expanded.

expop - the highest positive exponent which is automatically expanded. Thus (x+1)^3, when typed, will be automatically expanded only if expop is greater than or equal to 3. If it is desired to have (x+1)^n expanded where n is greater than expop then executing expand ((x+1)^n) will work only if maxposex is not less than n.

expand(expr, 0, 0) causes a resimplification of expr. expr is not reevaluated. In distinction from ev(expr, noeval) a special representation (e. g. a CRE form) is removed. See also ev.

The expand flag used with ev causes expansion.

The file share/simplification/facexp.mac contains several related functions (in particular facsum, factorfacsum and collectterms, which are autoloaded) and variables (nextlayerfactor and facsum_combine) that provide the user with the ability to structure expressions by controlled expansion. Brief function descriptions are available in simplification/facexp.usg. A demo is available by doing demo("facexp").

Examples:

(%i1) expr:(x+1)^2*(y+1)^3;
                               2        3
(%o1)                   (x + 1)  (y + 1)
(%i2) expand(expr);
       2  3        3    3      2  2        2      2      2
(%o2) x  y  + 2 x y  + y  + 3 x  y  + 6 x y  + 3 y  + 3 x  y
                                                      2
                                     + 6 x y + 3 y + x  + 2 x + 1
(%i3) expand(expr,2);
               2        3              3          3
(%o3)         x  (y + 1)  + 2 x (y + 1)  + (y + 1)
(%i4) expr:(x+1)^-2*(y+1)^3;
                                   3
                            (y + 1)
(%o4)                       --------
                                   2
                            (x + 1)
(%i5) expand(expr);
            3               2
           y             3 y            3 y             1
(%o5) ------------ + ------------ + ------------ + ------------
       2              2              2              2
      x  + 2 x + 1   x  + 2 x + 1   x  + 2 x + 1   x  + 2 x + 1
(%i6) expand(expr,2,2);
                                   3
                            (y + 1)
(%o6)                     ------------
                           2
                          x  + 2 x + 1

Resimplify an expression without expansion:

(%i1) expr:(1+x)^2*sin(x);
                                2
(%o1)                    (x + 1)  sin(x)
(%i2) exponentialize:true;
(%o2)                         true
(%i3) expand(expr,0,0);
                            2    %i x     - %i x
                  %i (x + 1)  (%e     - %e      )
(%o3)           - -------------------------------
                                 2

Function: expandwrt (expr, x_1, …, x_n)

Expands expression expr with respect to the variables x_1, …, x_n. All products involving the variables appear explicitly. The form returned will be free of products of sums of expressions that are not free of the variables. x_1, …, x_n may be variables, operators, or expressions.

By default, denominators are not expanded, but this can be controlled by means of the switch expandwrt_denom.

This function is autoloaded from simplification/stopex.mac.

Option variable: expandwrt_denom

Default value: false

expandwrt_denom controls the treatment of rational expressions by expandwrt. If true, then both the numerator and denominator of the expression will be expanded according to the arguments of expandwrt, but if expandwrt_denom is false, then only the numerator will be expanded in that way.

Function: expandwrt_factored (expr, x_1, …, x_n)

is similar to expandwrt, but treats expressions that are products somewhat differently. expandwrt_factored expands only on those factors of expr that contain the variables x_1, …, x_n.

This function is autoloaded from simplification/stopex.mac.

Option variable: expon

Default value: 0

expon is the exponent of the largest negative power which is automatically expanded (independent of calls to expand). For example, if expon is 4 then (x+1)^(-5) will not be automatically expanded.

Function: exponentialize (expr)
Option variable: exponentialize

The function exponentialize (expr) converts circular and hyperbolic functions in expr to exponentials, without setting the global variable exponentialize.

When the variable exponentialize is true, all circular and hyperbolic functions are converted to exponential form. The default value is false.

demoivre converts complex exponentials into circular functions. exponentialize and demoivre cannot both be true at the same time.

Option variable: expop

Default value: 0

expop is the highest positive exponent which is automatically expanded. Thus (x + 1)^3, when typed, will be automatically expanded only if expop is greater than or equal to 3. If it is desired to have (x + 1)^n expanded where n is greater than expop then executing expand ((x + 1)^n) will work only if maxposex is not less than n.

Property: lassociative

declare (g, lassociative) tells the Maxima simplifier that g is left-associative. E.g., g (g (a, b), g (c, d)) will simplify to g (g (g (a, b), c), d).

Property: linear

One of Maxima’s operator properties. For univariate f so declared, "expansion" f(x + y) yields f(x) + f(y), f(a*x) yields a*f(x) takes place where a is a "constant". For functions of two or more arguments, "linearity" is defined to be as in the case of sum or integrate, i.e., f (a*x + b, x) yields a*f(x,x) + b*f(1,x) for a and b free of x.

Example:

(%i1) declare (f, linear);
(%o1)                         done
(%i2) f(x+y);
(%o2)                      f(y) + f(x)
(%i3) declare (a, constant);
(%o3)                         done
(%i4) f(a*x);
(%o4)                        a f(x)

linear is equivalent to additive and outative. See also opproperties.

Example:

(%i1) 'sum (F(k) + G(k), k, 1, inf);
                       inf
                       ====
                       \
(%o1)                   >    (G(k) + F(k))
                       /
                       ====
                       k = 1
(%i2) declare (nounify (sum), linear);
(%o2)                         done
(%i3) 'sum (F(k) + G(k), k, 1, inf);
                     inf          inf
                     ====         ====
                     \            \
(%o3)                 >    G(k) +  >    F(k)
                     /            /
                     ====         ====
                     k = 1        k = 1

Option variable: maxnegex

Default value: 1000

maxnegex is the largest negative exponent which will be expanded by the expand command, see also maxposex.

Option variable: maxposex

Default value: 1000

maxposex is the largest exponent which will be expanded with the expand command, see also maxnegex.

Property: multiplicative

declare(f, multiplicative) tells the Maxima simplifier that f is multiplicative.

  1. If f is univariate, whenever the simplifier encounters f applied to a product, f distributes over that product. E.g., f(x*y) simplifies to f(x)*f(y). This simplification is not applied to expressions of the form f('product(...)).
  2. If f is a function of 2 or more arguments, multiplicativity is defined as multiplicativity in the first argument to f, e.g., f (g(x) * h(x), x) simplifies to f (g(x) ,x) * f (h(x), x).

declare(nounify(product), multiplicative) tells Maxima to simplify symbolic products.

Example:

(%i1) F2 (a * b * c);
(%o1)                       F2(a b c)
(%i2) declare (F2, multiplicative);
(%o2)                         done
(%i3) F2 (a * b * c);
(%o3)                   F2(a) F2(b) F2(c)

declare(nounify(product), multiplicative) tells Maxima to simplify symbolic products.

(%i1) product (a[i] * b[i], i, 1, n);
                             n
                           /===\
                            ! !
(%o1)                       ! !  a  b
                            ! !   i  i
                           i = 1
(%i2) declare (nounify (product), multiplicative);
(%o2)                         done
(%i3) product (a[i] * b[i], i, 1, n);
                          n         n
                        /===\     /===\
                         ! !       ! !
(%o3)                  ( ! !  a )  ! !  b
                         ! !   i   ! !   i
                        i = 1     i = 1

Function: multthru
    multthru (expr)
    multthru (expr_1, expr_2)

Multiplies a factor (which should be a sum) of expr by the other factors of expr. That is, expr is f_1 f_2 ... f_n where at least one factor, say f_i, is a sum of terms. Each term in that sum is multiplied by the other factors in the product. (Namely all the factors except f_i). multthru does not expand exponentiated sums. This function is the fastest way to distribute products (commutative or noncommutative) over sums. Since quotients are represented as products multthru can be used to divide sums by products as well.

multthru (expr_1, expr_2) multiplies each term in expr_2 (which should be a sum or an equation) by expr_1. If expr_1 is not itself a sum then this form is equivalent to multthru (expr_1*expr_2).

(%i1) x/(x-y)^2 - 1/(x-y) - f(x)/(x-y)^3;
                      1        x         f(x)
(%o1)             - ----- + -------- - --------
                    x - y          2          3
                            (x - y)    (x - y)
(%i2) multthru ((x-y)^3, %);
                           2
(%o2)             - (x - y)  + x (x - y) - f(x)
(%i3) ratexpand (%);
                           2
(%o3)                   - y  + x y - f(x)
(%i4) ((a+b)^10*s^2 + 2*a*b*s + (a*b)^2)/(a*b*s^2);
                        10  2              2  2
                 (b + a)   s  + 2 a b s + a  b
(%o4)            ------------------------------
                                  2
                             a b s
(%i5) multthru (%);  /* note that this does not expand (b+a)^10 */
                                        10
                       2   a b   (b + a)
(%o5)                  - + --- + ---------
                       s    2       a b
                           s
(%i6) multthru (a.(b+c.(d+e)+f));
(%o6)            a . f + a . c . (e + d) + a . b
(%i7) expand (a.(b+c.(d+e)+f));
(%o7)         a . f + a . c . e + a . c . d + a . b

Property: nary

declare(f, nary) tells Maxima to recognize the function f as an n-ary function.

The nary declaration is not the same as calling the nary function. The sole effect of declare(f, nary) is to instruct the Maxima simplifier to flatten nested expressions, for example, to simplify foo(x, foo(y, z)) to foo(x, y, z). See also declare.

Example:

(%i1) H (H (a, b), H (c, H (d, e)));
(%o1)               H(H(a, b), H(c, H(d, e)))
(%i2) declare (H, nary);
(%o2)                         done
(%i3) H (H (a, b), H (c, H (d, e)));
(%o3)                   H(a, b, c, d, e)
Option variable: negdistrib

Default value: true

When negdistrib is true, -1 distributes over an expression. E.g., -(x + y) becomes - y - x. Setting it to false will allow - (x + y) to be displayed like that. This is sometimes useful but be very careful: like the simp flag, this is one flag you do not want to set to false as a matter of course or necessarily for other than local use in your Maxima.

Example:

(%i1) negdistrib;
(%o1)                         true
(%i2) -(x+y);
(%o2)                       (- y) - x
(%i3) negdistrib : not negdistrib ;
(%o3)                         false
(%i4) -(x+y);
(%o4)                       - (y + x)

System variable: opproperties

opproperties is the list of the special operator properties recognized by the Maxima simplifier.

Items are added to the opproperties list by the function define_opproperty.

Example:

(%i1) opproperties;
(%o1) [linear, additive, multiplicative, outative, evenfun, 
oddfun, commutative, symmetric, antisymmetric, nary, 
lassociative, rassociative]

Function: define_opproperty (property_name, simplifier_fn)

Declares the symbol property_name to be an operator property, which is simplified by simplifier_fn, which may be the name of a Maxima or Lisp function or a lambda expression. After define_opproperty is called, functions and operators may be declared to have the property_name property, and simplifier_fn is called to simplify them.

simplifier_fn must be a function of one argument, which is an expression in which the main operator is declared to have the property_name property.

simplifier_fn is called with the global flag simp disabled. Therefore simplifier_fn must be able to carry out its simplification without making use of the general simplifier.

define_opproperty appends property_name to the global list opproperties.

define_opproperty returns done.

Example:

Declare a new property, identity, which is simplified by simplify_identity. Declare that f and g have the new property.

(%i1) define_opproperty (identity, simplify_identity);
(%o1)                         done
(%i2) simplify_identity(e) := first(e);
(%o2)           simplify_identity(e) := first(e)
(%i3) declare ([f, g], identity);
(%o3)                         done
(%i4) f(10 + t);
(%o4)                        t + 10
(%i5) g(3*u) - f(2*u);
(%o5)                           u

Property: outative

declare(f, outative) tells the Maxima simplifier that constant factors in the argument of f can be pulled out.

  1. If f is univariate, whenever the simplifier encounters f applied to a product, that product will be partitioned into factors that are constant and factors that are not and the constant factors will be pulled out. E.g., f(a*x) will simplify to a*f(x) where a is a constant. Non-atomic constant factors will not be pulled out.
  2. If f is a function of 2 or more arguments, outativity is defined as in the case of sum or integrate, i.e., f (a*g(x), x) will simplify to a * f(g(x), x) for a free of x.

sum, integrate, and limit are all outative.

Example:

(%i1) F1 (100 * x);
(%o1)                       F1(100 x)
(%i2) declare (F1, outative);
(%o2)                         done
(%i3) F1 (100 * x);
(%o3)                       100 F1(x)
(%i4) declare (zz, constant);
(%o4)                         done
(%i5) F1 (zz * y);
(%o5)                       zz F1(y)

Function: radcan (expr)

Simplifies expr, which can contain logs, exponentials, and radicals, by converting it into a form which is canonical over a large class of expressions and a given ordering of variables; that is, all functionally equivalent forms are mapped into a unique form. For a somewhat larger class of expressions, radcan produces a regular form. Two equivalent expressions in this class do not necessarily have the same appearance, but their difference can be simplified by radcan to zero.

For some expressions radcan is quite time consuming. This is the cost of exploring certain relationships among the components of the expression for simplifications based on factoring and partial-fraction expansions of exponents.

Examples:

(%i1) radcan((log(x+x^2)-log(x))^a/log(1+x)^(a/2));
                                    a/2
(%o1)                     log(x + 1)
(%i2) radcan((log(1+2*a^x+a^(2*x))/log(1+a^x)));
(%o2)                           2
(%i3) radcan((%e^x-1)/(1+%e^(x/2)));
                              x/2
(%o3)                       %e    - 1

Option variable: radexpand

Default value: true

radexpand controls some simplifications of radicals.

When radexpand is all, causes nth roots of factors of a product which are powers of n to be pulled outside of the radical. E.g. if radexpand is all, sqrt (16*x^2) simplifies to 4*x.

More particularly, consider sqrt (x^2).

Note that domain only matters when radexpand is true.

Property: rassociative

declare (g, rassociative) tells the Maxima simplifier that g is right-associative. E.g., g(g(a, b), g(c, d)) simplifies to g(a, g(b, g(c, d))).

Function: scsimp (expr, rule_1, …, rule_n)

Sequential Comparative Simplification (method due to Stoute). scsimp attempts to simplify expr according to the rules rule_1, …, rule_n. If a smaller expression is obtained, the process repeats. Otherwise after all simplifications are tried, it returns the original answer.

example (scsimp) displays some examples.

Option variable: simp

Default value: true

simp enables simplification. This is the default. simp is also an evflag, which is recognized by the function ev. See ev.

When simp is used as an evflag with a value false, the simplification is suppressed only during the evaluation phase of an expression. The flag does not suppress the simplification which follows the evaluation phase.

Many Maxima functions and operations require simplification to be enabled to work normally. When simplification is disabled, many results will be incomplete, and in addition there may be incorrect results or program errors.

Examples:

The simplification is switched off globally. The expression sin(1.0) is not simplified to its numerical value. The simp-flag switches the simplification on.

(%i1) simp:false;
(%o1)                         false
(%i2) sin(1.0);
(%o2)                       sin(1.0)
(%i3) sin(1.0),simp;
(%o3)                  0.8414709848078965

The simplification is switched on again. The simp-flag cannot suppress the simplification completely. The output shows a simplified expression, but the variable x has an unsimplified expression as a value, because the assignment has occurred during the evaluation phase of the expression.

(%i1) simp:true;
(%o1)                         true
(%i2) x:sin(1.0),simp:false;
(%o2)                  0.8414709848078965
(%i3) :lisp $x
((%SIN) 1.0)

Property: symmetric

declare (h, symmetric) tells the Maxima simplifier that h is a symmetric function. E.g., h (x, z, y) simplifies to h (x, y, z).

commutative is synonymous with symmetric.

Function: xthru (expr)

Combines all terms of expr (which should be a sum) over a common denominator without expanding products and exponentiated sums as ratsimp does. xthru cancels common factors in the numerator and denominator of rational expressions but only if the factors are explicit.

Sometimes it is better to use xthru before ratsimping an expression in order to cause explicit factors of the gcd of the numerator and denominator to be canceled thus simplifying the expression to be ratsimped.

Examples:

(%i1) ((x+2)^20 - 2*y)/(x+y)^20 + (x+y)^(-19) - x/(x+y)^20;
                                20
                 1       (x + 2)   - 2 y       x
(%o1)        --------- + --------------- - ---------
                    19             20             20
             (y + x)        (y + x)        (y + x)
(%i2) xthru (%);
                                 20
                          (x + 2)   - y
(%o2)                     -------------
                                   20
                            (y + x)


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10 Mathematical Functions


Next: , Previous: , Up: Mathematical Functions   [Contents][Index]

10.1 Functions for Numbers

Function: abs (z)

The abs function represents the mathematical absolute value function and works for both numerical and symbolic values. If the argument, z, is a real or complex number, abs returns the absolute value of z. If possible, symbolic expressions using the absolute value function are also simplified.

Maxima can differentiate, integrate and calculate limits for expressions containing abs. The abs_integrate package further extends Maxima’s ability to calculate integrals involving the abs function. See (%i12) in the examples below.

When applied to a list or matrix, abs automatically distributes over the terms. Similarly, it distributes over both sides of an equation. To alter this behaviour, see the variable distribute_over.

See also cabs.

Examples:

Calculation of abs for real and complex numbers, including numerical constants and various infinities. The first example shows how abs distributes over the elements of a list.

(%i1) abs([-4, 0, 1, 1+%i]);
(%o1)                  [4, 0, 1, sqrt(2)]

(%i2) abs((1+%i)*(1-%i));
(%o2)                           2
(%i3) abs(%e+%i);
                                2
(%o3)                    sqrt(%e  + 1)
(%i4) abs([inf, infinity, minf]);
(%o4)                   [inf, inf, inf]

Simplification of expressions containing abs:

(%i5) abs(x^2);
                                2
(%o5)                          x
(%i6) abs(x^3);
                             2
(%o6)                       x  abs(x)

(%i7) abs(abs(x));
(%o7)                       abs(x)
(%i8) abs(conjugate(x));
(%o8)                       abs(x)

Integrating and differentiating with the abs function. Note that more integrals involving the abs function can be performed, if the abs_integrate package is loaded. The last example shows the Laplace transform of abs: see laplace.

(%i9) diff(x*abs(x),x),expand;
(%o9)                       2 abs(x)

(%i10) integrate(abs(x),x);
                             x abs(x)
(%o10)                       --------
                                2

(%i11) integrate(x*abs(x),x);
                           /
                           [
(%o11)                     I x abs(x) dx
                           ]
                           /

(%i12) load("abs_integrate")$
(%i13) integrate(x*abs(x),x);
                      2           3
                     x  abs(x)   x  signum(x)
(%o13)               --------- - ------------
                         2            6

(%i14) integrate(abs(x),x,-2,%pi);
                               2
                            %pi
(%o14)                      ---- + 2
                             2

(%i15) laplace(abs(x),x,s);
                               1
(%o15)                         --
                                2
                               s

Function: ceiling (x)

When x is a real number, return the least integer that is greater than or equal to x.

If x is a constant expression (10 * %pi, for example), ceiling evaluates x using big floating point numbers, and applies ceiling to the resulting big float. Because ceiling uses floating point evaluation, it’s possible, although unlikely, that ceiling could return an erroneous value for constant inputs. To guard against errors, the floating point evaluation is done using three values for fpprec.

For non-constant inputs, ceiling tries to return a simplified value. Here are examples of the simplifications that ceiling knows about:

(%i1) ceiling (ceiling (x));
(%o1)                      ceiling(x)
(%i2) ceiling (floor (x));
(%o2)                       floor(x)
(%i3) declare (n, integer)$
(%i4) [ceiling (n), ceiling (abs (n)), ceiling (max (n, 6))];
(%o4)                [n, abs(n), max(6, n)]
(%i5) assume (x > 0, x < 1)$
(%i6) ceiling (x);
(%o6)                           1
(%i7) tex (ceiling (a));
$$\left \lceil a \right \rceil$$
(%o7)                         false

The ceiling function distributes over lists, matrices and equations. See distribute_over.

Finally, for all inputs that are manifestly complex, ceiling returns a noun form.

If the range of a function is a subset of the integers, it can be declared to be integervalued. Both the ceiling and floor functions can use this information; for example:

(%i1) declare (f, integervalued)$
(%i2) floor (f(x));
(%o2)                         f(x)
(%i3) ceiling (f(x) - 1);
(%o3)                       f(x) - 1

Example use:

(%i1) unitfrac(r) := block([uf : [], q],
    if not(ratnump(r)) then
       error("unitfrac: argument must be a rational number"),
    while r # 0 do (
        uf : cons(q : 1/ceiling(1/r), uf),
        r : r - q),
    reverse(uf));
(%o1) unitfrac(r) := block([uf : [], q], 
if not ratnump(r) then error("unitfrac: argument must be a rational number"
                                     1
), while r # 0 do (uf : cons(q : ----------, uf), r : r - q), 
                                         1
                                 ceiling(-)
                                         r
reverse(uf))
(%i2) unitfrac (9/10);
                            1  1  1
(%o2)                      [-, -, --]
                            2  3  15
(%i3) apply ("+", %);
                               9
(%o3)                          --
                               10
(%i4) unitfrac (-9/10);
                                  1
(%o4)                       [- 1, --]
                                  10
(%i5) apply ("+", %);
                                9
(%o5)                         - --
                                10
(%i6) unitfrac (36/37);
                        1  1  1  1    1
(%o6)                  [-, -, -, --, ----]
                        2  3  8  69  6808
(%i7) apply ("+", %);
                               36
(%o7)                          --
                               37

Function: entier (x)

Returns the largest integer less than or equal to x where x is numeric. fix (as in fixnum) is a synonym for this, so fix(x) is precisely the same.

Function: floor (x)

When x is a real number, return the largest integer that is less than or equal to x.

If x is a constant expression (10 * %pi, for example), floor evaluates x using big floating point numbers, and applies floor to the resulting big float. Because floor uses floating point evaluation, it’s possible, although unlikely, that floor could return an erroneous value for constant inputs. To guard against errors, the floating point evaluation is done using three values for fpprec.

For non-constant inputs, floor tries to return a simplified value. Here are examples of the simplifications that floor knows about:

(%i1) floor (ceiling (x));
(%o1)                      ceiling(x)
(%i2) floor (floor (x));
(%o2)                       floor(x)
(%i3) declare (n, integer)$
(%i4) [floor (n), floor (abs (n)), floor (min (n, 6))];
(%o4)                [n, abs(n), min(6, n)]
(%i5) assume (x > 0, x < 1)$
(%i6) floor (x);
(%o6)                           0
(%i7) tex (floor (a));
$$\left \lfloor a \right \rfloor$$
(%o7)                         false

The floor function distributes over lists, matrices and equations. See distribute_over.

Finally, for all inputs that are manifestly complex, floor returns a noun form.

If the range of a function is a subset of the integers, it can be declared to be integervalued. Both the ceiling and floor functions can use this information; for example:

(%i1) declare (f, integervalued)$
(%i2) floor (f(x));
(%o2)                         f(x)
(%i3) ceiling (f(x) - 1);
(%o3)                       f(x) - 1

Function: fix (x)

A synonym for entier (x).

Function: lmax (L)

When L is a list or a set, return apply ('max, args (L)). When L is not a list or a set, signal an error. See also lmin and max.

Function: lmin (L)

When L is a list or a set, return apply ('min, args (L)). When L is not a list or a set, signal an error. See also lmax and min.

Function: max (x_1, …, x_n)

Return a simplified value for the maximum of the expressions x_1 through x_n. When get (trylevel, maxmin), is 2 or greater, max uses the simplification max (e, -e) --> |e|. When get (trylevel, maxmin) is 3 or greater, max tries to eliminate expressions that are between two other arguments; for example, max (x, 2*x, 3*x) --> max (x, 3*x). To set the value of trylevel to 2, use put (trylevel, 2, maxmin).

See also min and lmax.

Function: min (x_1, …, x_n)

Return a simplified value for the minimum of the expressions x_1 through x_n. When get (trylevel, maxmin), is 2 or greater, min uses the simplification min (e, -e) --> -|e|. When get (trylevel, maxmin) is 3 or greater, min tries to eliminate expressions that are between two other arguments; for example, min (x, 2*x, 3*x) --> min (x, 3*x). To set the value of trylevel to 2, use put (trylevel, 2, maxmin).

See also max and lmin.

Function: round (x)

When x is a real number, returns the closest integer to x. Multiples of 1/2 are rounded to the nearest even integer. Evaluation of x is similar to floor and ceiling.

The round function distributes over lists, matrices and equations. See distribute_over.

Function: signum (x)

For either real or complex numbers x, the signum function returns 0 if x is zero; for a nonzero numeric input x, the signum function returns x/abs(x).

For non-numeric inputs, Maxima attempts to determine the sign of the input. When the sign is negative, zero, or positive, signum returns -1,0, 1, respectively. For all other values for the sign, signum a simplified but equivalent form. The simplifications include reflection (signum(-x) gives -signum(x)) and multiplicative identity (signum(x*y) gives signum(x) * signum(y)).

The signum function distributes over a list, a matrix, or an equation. See distribute_over.

Function: truncate (x)

When x is a real number, return the closest integer to x not greater in absolute value than x. Evaluation of x is similar to floor and ceiling.

The truncate function distributes over lists, matrices and equations. See distribute_over.


Next: , Previous: , Up: Mathematical Functions   [Contents][Index]

10.2 Functions for Complex Numbers

Function: cabs (expr)

Calculates the absolute value of an expression representing a complex number. Unlike the function abs, the cabs function always decomposes its argument into a real and an imaginary part. If x and y represent real variables or expressions, the cabs function calculates the absolute value of x + %i*y as

(%i1) cabs (1);
(%o1)                           1
(%i2) cabs (1 + %i);
(%o2)                        sqrt(2)
(%i3) cabs (exp (%i));
(%o3)                           1
(%i4) cabs (exp (%pi * %i));
(%o4)                           1
(%i5) cabs (exp (3/2 * %pi * %i));
(%o5)                           1
(%i6) cabs (17 * exp (2 * %i));
(%o6)                          17

If cabs returns a noun form this most commonly is caused by some properties of the variables involved not being known:

(%i1) cabs (a+%i*b);
                                2    2
(%o1)                     sqrt(b  + a )
(%i2) declare(a,real,b,real);
(%o2)                         done
(%i3) cabs (a+%i*b);
                                2    2
(%o3)                     sqrt(b  + a )
(%i4) assume(a>0,b>0);
(%o4)                    [a > 0, b > 0]
(%i5) cabs (a+%i*b);
                                2    2
(%o5)                     sqrt(b  + a )

The cabs function can use known properties like symmetry properties of complex functions to help it calculate the absolute value of an expression. If such identities exist, they can be advertised to cabs using function properties. The symmetries that cabs understands are: mirror symmetry, conjugate function and complex characteristic.

cabs is a verb function and is not suitable for symbolic calculations. For such calculations (including integration, differentiation and taking limits of expressions containing absolute values), use abs.

The result of cabs can include the absolute value function, abs, and the arc tangent, atan2.

When applied to a list or matrix, cabs automatically distributes over the terms. Similarly, it distributes over both sides of an equation.

For further ways to compute with complex numbers, see the functions rectform, realpart, imagpart, carg, conjugate and polarform.

Examples:

Examples with sqrt and sin.

(%i1) cabs(sqrt(1+%i*x));
                             2     1/4
(%o1)                      (x  + 1)
(%i2) cabs(sin(x+%i*y));
                    2        2         2        2
(%o2)       sqrt(cos (x) sinh (y) + sin (x) cosh (y))

The error function, erf, has mirror symmetry, which is used here in the calculation of the absolute value with a complex argument:

(%i3) cabs(erf(x+%i*y));
                                          2
           (erf(%i y + x) - erf(%i y - x))
(%o3) sqrt(--------------------------------
                          4
                                                               2
                                (erf(%i y + x) + erf(%i y - x))
                              - --------------------------------)
                                               4

Maxima knows complex identities for the Bessel functions, which allow it to compute the absolute value for complex arguments. Here is an example for bessel_j.

(%i4) cabs(bessel_j(1,%i));
(%o4)                 abs(bessel_j(1, %i))

Function: carg (z)

Returns the complex argument of z. The complex argument is an angle theta in (-%pi, %pi] such that r exp (theta %i) = z where r is the magnitude of z.

carg is a computational function, not a simplifying function.

See also abs (complex magnitude), polarform, rectform, realpart, and imagpart.

Examples:

(%i1) carg (1);
(%o1)                           0
(%i2) carg (1 + %i);
                               %pi
(%o2)                          ---
                                4
(%i3) carg (exp (%i));
                               sin(1)
(%o3)                     atan(------)
                               cos(1)
(%i4) carg (exp (%pi * %i));
(%o4)                          %pi
(%i5) carg (exp (3/2 * %pi * %i));
                                %pi
(%o5)                         - ---
                                 2
(%i6) carg (17 * exp (2 * %i));
                            sin(2)
(%o6)                  atan(------) + %pi
                            cos(2)

If carg returns a noun form this most communly is caused by some properties of the variables involved not being known:

(%i1) carg (a+%i*b);
(%o1)                      atan2(b, a)
(%i2) declare(a,real,b,real);
(%o2)                         done
(%i3) carg (a+%i*b);
(%o3)                      atan2(b, a)
(%i4) assume(a>0,b>0);
(%o4)                    [a > 0, b > 0]
(%i5) carg (a+%i*b);
                                  b
(%o5)                        atan(-)
                                  a

Function: conjugate (x)

Returns the complex conjugate of x.

(%i1) declare ([aa, bb], real, cc, complex, ii, imaginary);
(%o1)                         done
(%i2) conjugate (aa + bb*%i);
(%o2)                      aa - %i bb
(%i3) conjugate (cc);
(%o3)                     conjugate(cc)
(%i4) conjugate (ii);
(%o4)                         - ii
(%i5) conjugate (xx + yy);
(%o5)                        yy + xx

Function: imagpart (expr)

Returns the imaginary part of the expression expr.

imagpart is a computational function, not a simplifying function.

See also abs, carg, polarform, rectform, and realpart.

Example:

(%i1) imagpart (a+b*%i);
(%o1)                           b
(%i2) imagpart (1+sqrt(2)*%i);
(%o2)                        sqrt(2)
(%i3) imagpart (1);
(%o3)                           0
(%i4) imagpart (sqrt(2)*%i);
(%o4)                        sqrt(2)

Function: polarform (expr)

Returns an expression r %e^(%i theta) equivalent to expr, such that r and theta are purely real.

Example:

(%i1) polarform(a+b*%i);
                       2    2    %i atan2(b, a)
(%o1)            sqrt(b  + a ) %e
(%i2) polarform(1+%i);
                                  %i %pi
                                  ------
                                    4
(%o2)                   sqrt(2) %e
(%i3) polarform(1+2*%i);
                                %i atan(2)
(%o3)                 sqrt(5) %e

Function: realpart (expr)

Returns the real part of expr. realpart and imagpart will work on expressions involving trigonometric and hyperbolic functions, as well as square root, logarithm, and exponentiation.

Example:

(%i1) realpart (a+b*%i);
(%o1)                           a
(%i2) realpart (1+sqrt(2)*%i);
(%o2)                           1
(%i3) realpart (sqrt(2)*%i);
(%o3)                           0
(%i4) realpart (1);
(%o4)                           1

Function: rectform (expr)

Returns an expression a + b %i equivalent to expr, such that a and b are purely real.

Example:

(%i1) rectform(sqrt(2)*%e^(%i*%pi/4));
(%o1)                        %i + 1
(%i2) rectform(sqrt(b^2+a^2)*%e^(%i*atan2(b, a)));
(%o2)                       %i b + a
(%i3) rectform(sqrt(5)*%e^(%i*atan(2)));
(%o3)                       2 %i + 1


Next: , Previous: , Up: Mathematical Functions   [Contents][Index]

10.3 Combinatorial Functions

Operator: !!

The double factorial operator.

For an integer, float, or rational number n, n!! evaluates to the product n (n-2) (n-4) (n-6) ... (n - 2 (k-1)) where k is equal to entier (n/2), that is, the largest integer less than or equal to n/2. Note that this definition does not coincide with other published definitions for arguments which are not integers.

For an even (or odd) integer n, n!! evaluates to the product of all the consecutive even (or odd) integers from 2 (or 1) through n inclusive.

For an argument n which is not an integer, float, or rational, n!! yields a noun form genfact (n, n/2, 2).

Function: binomial (x, y)

The binomial coefficient x!/(y! (x - y)!). If x and y are integers, then the numerical value of the binomial coefficient is computed. If y, or x - y, is an integer, the binomial coefficient is expressed as a polynomial.

Examples:

(%i1) binomial (11, 7);
(%o1)                          330
(%i2) 11! / 7! / (11 - 7)!;
(%o2)                          330
(%i3) binomial (x, 7);
        (x - 6) (x - 5) (x - 4) (x - 3) (x - 2) (x - 1) x
(%o3)   -------------------------------------------------
                              5040
(%i4) binomial (x + 7, x);
      (x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (x + 6) (x + 7)
(%o4) -------------------------------------------------------
                               5040
(%i5) binomial (11, y);
(%o5)                    binomial(11, y)

Function: factcomb (expr)

Tries to combine the coefficients of factorials in expr with the factorials themselves by converting, for example, (n + 1)*n! into (n + 1)!.

sumsplitfact if set to false will cause minfactorial to be applied after a factcomb.

Example:

(%i1) sumsplitfact;
(%o1)                         true
(%i2) (n + 1)*(n + 1)*n!;
                                  2
(%o2)                      (n + 1)  n!
(%i3) factcomb (%);
(%o3)                  (n + 2)! - (n + 1)!
(%i4) sumsplitfact: not sumsplitfact;
(%o4)                         false
(%i5) (n + 1)*(n + 1)*n!;
                                  2
(%o5)                      (n + 1)  n!
(%i6) factcomb (%);
(%o6)                 n (n + 1)! + (n + 1)!

Function: factorial
Operator: !

Represents the factorial function. Maxima treats factorial (x) the same as x!.

For any complex number x, except for negative integers, x! is defined as gamma(x+1).

For an integer x, x! simplifies to the product of the integers from 1 to x inclusive. 0! simplifies to 1. For a real or complex number in float or bigfloat precision x, x! simplifies to the value of gamma (x+1). For x equal to n/2 where n is an odd integer, x! simplifies to a rational factor times sqrt (%pi) (since gamma (1/2) is equal to sqrt (%pi)).

The option variables factlim and gammalim control the numerical evaluation of factorials for integer and rational arguments. The functions minfactorial and factcomb simplifies expressions containing factorials.

The functions gamma, bffac, and cbffac are varieties of the gamma function. bffac and cbffac are called internally by gamma to evaluate the gamma function for real and complex numbers in bigfloat precision.

makegamma substitutes gamma for factorials and related functions.

Maxima knows the derivative of the factorial function and the limits for specific values like negative integers.

The option variable factorial_expand controls the simplification of expressions like (n+x)!, where n is an integer.

See also binomial.

The factorial of an integer is simplified to an exact number unless the operand is greater than factlim. The factorial for real and complex numbers is evaluated in float or bigfloat precision.

(%i1) factlim : 10;
(%o1)                          10
(%i2) [0!, (7/2)!, 8!, 20!];
                     105 sqrt(%pi)
(%o2)            [1, -------------, 40320, 20!]
                          16
(%i3) [4,77!, (1.0+%i)!];
(%o3) [4, 77!, 0.3430658398165453 %i + 0.6529654964201667]
(%i4) [2.86b0!, 1.0b0+%i)!];
incorrect syntax: Missing ]
[2.86b0!, 1.0b0+%i)
                 ^

The factorial of a known constant, or general expression is not simplified. Even so it may be possible to simplify the factorial after evaluating the operand.

(%i1) [(%i + 1)!, %pi!, %e!, (cos(1) + sin(1))!];
(%o1)      [(%i + 1)!, %pi!, %e!, (sin(1) + cos(1))!]
(%i2) ev (%, numer, %enumer);
(%o2) [0.3430658398165453 %i + 0.6529654964201667, 
         7.188082728976031, 4.260820476357003, 1.227580202486819]

Factorials are simplified, not evaluated. Thus x! may be replaced even in a quoted expression.

(%i1) '([0!, (7/2)!, 4.77!, 8!, 20!]);
          105 sqrt(%pi)
(%o1) [1, -------------, 81.44668037931197, 40320, 
               16
                                             2432902008176640000]

Maxima knows the derivative of the factorial function.

(%i1) diff(x!,x);
(%o1)                    x! psi (x + 1)
                               0

The option variable factorial_expand controls expansion and simplification of expressions with the factorial function.

(%i1) (n+1)!/n!,factorial_expand:true;
(%o1)                         n + 1

Option variable: factlim

Default value: 100000

factlim specifies the highest factorial which is automatically expanded. If it is -1 then all integers are expanded.

Option variable: factorial_expand

Default value: false

The option variable factorial_expand controls the simplification of expressions like (n+1)!, where n is an integer. See factorial for an example.

Function: genfact (x, y, z)

Returns the generalized factorial, defined as x (x-z) (x - 2 z) ... (x - (y - 1) z). Thus, when x is an integer, genfact (x, x, 1) = x! and genfact (x, x/2, 2) = x!!.

Function: minfactorial (expr)

Examines expr for occurrences of two factorials which differ by an integer. minfactorial then turns one into a polynomial times the other.

(%i1) n!/(n+2)!;
                               n!
(%o1)                       --------
                            (n + 2)!
(%i2) minfactorial (%);
                                1
(%o2)                    ---------------
                         (n + 1) (n + 2)

Option variable: sumsplitfact

Default value: true

When sumsplitfact is false, minfactorial is applied after a factcomb.

(%i1) sumsplitfact;
(%o1)                         true
(%i2) n!/(n+2)!;
                               n!
(%o2)                       --------
                            (n + 2)!
(%i3) factcomb(%);
                               n!
(%o3)                       --------
                            (n + 2)!
(%i4) sumsplitfact: not sumsplitfact ;
(%o4)                         false
(%i5) n!/(n+2)!;
                               n!
(%o5)                       --------
                            (n + 2)!
(%i6) factcomb(%);
                                1
(%o6)                    ---------------
                         (n + 1) (n + 2)


Next: , Previous: , Up: Mathematical Functions   [Contents][Index]

10.4 Root, Exponential and Logarithmic Functions

Option variable: %e_to_numlog

Default value: false

When true, r some rational number, and x some expression, %e^(r*log(x)) will be simplified into x^r . It should be noted that the radcan command also does this transformation, and more complicated transformations of this ilk as well. The logcontract command "contracts" expressions containing log.

Option variable: %emode

Default value: true

When %emode is true, %e^(%pi %i x) is simplified as follows.

%e^(%pi %i x) simplifies to cos (%pi x) + %i sin (%pi x) if x is a floating point number, an integer, or a multiple of 1/2, 1/3, 1/4, or 1/6, and then further simplified.

For other numerical x, %e^(%pi %i x) simplifies to %e^(%pi %i y) where y is x - 2 k for some integer k such that abs(y) < 1.

When %emode is false, no special simplification of %e^(%pi %i x) is carried out.

(%i1) %emode;
(%o1)                         true
(%i2) %e^(%pi*%i*1);
(%o2)                          - 1
(%i3) %e^(%pi*%i*216/144);
(%o3)                         - %i
(%i4) %e^(%pi*%i*192/144);
                          sqrt(3) %i    1
(%o4)                  (- ----------) - -
                              2         2
(%i5) %e^(%pi*%i*180/144);
                           %i          1
(%o5)                 (- -------) - -------
                         sqrt(2)    sqrt(2)
(%i6) %e^(%pi*%i*120/144);
                          %i   sqrt(3)
(%o6)                     -- - -------
                          2       2
(%i7) %e^(%pi*%i*121/144);
                            121 %i %pi
                            ----------
                               144
(%o7)                     %e

Option variable: %enumer

Default value: false

When %enumer is true, %e is replaced by its numeric value 2.718… whenever numer is true.

When %enumer is false, this substitution is carried out only if the exponent in %e^x evaluates to a number.

See also ev and numer.

(%i1) %enumer;
(%o1)                         false
(%i2) numer;
(%o2)                         false
(%i3) 2*%e;
(%o3)                         2 %e
(%i4) %enumer: not %enumer;
(%o4)                         true
(%i5) 2*%e;
(%o5)                         2 %e
(%i6) numer: not numer;
(%o6)                         true
(%i7) 2*%e;
(%o7)                   5.43656365691809
(%i8) 2*%e^1;
(%o8)                   5.43656365691809
(%i9) 2*%e^x;
                                         x
(%o9)                 2 2.718281828459045

Function: exp (x)

Represents the exponential function. Instances of exp (x) in input are simplified to %e^x; exp does not appear in simplified expressions.

demoivre if true causes %e^(a + b %i) to simplify to %e^(a (cos(b) + %i sin(b))) if b is free of %i. See demoivre.

%emode, when true, causes %e^(%pi %i x) to be simplified. See %emode.

%enumer, when true causes %e to be replaced by 2.718… whenever numer is true. See %enumer.

(%i1) demoivre;
(%o1)                         false
(%i2) %e^(a + b*%i);
                             %i b + a
(%o2)                      %e
(%i3) demoivre: not demoivre;
(%o3)                         true
(%i4) %e^(a + b*%i);
                      a
(%o4)               %e  (%i sin(b) + cos(b))

Function: li [s] (z)

Represents the polylogarithm function of order s and argument z, defined by the infinite series

                                 inf
                                 ====   k
                                 \     z
                        Li (z) =  >    --
                          s      /      s
                                 ====  k
                                 k = 1

li [1] is - log (1 - z). li [2] and li [3] are the dilogarithm and trilogarithm functions, respectively.

When the order is 1, the polylogarithm simplifies to - log (1 - z), which in turn simplifies to a numerical value if z is a real or complex floating point number or the numer evaluation flag is present.

When the order is 2 or 3, the polylogarithm simplifies to a numerical value if z is a real floating point number or the numer evaluation flag is present.

Examples:

(%i1) assume (x > 0);
(%o1)                        [x > 0]
(%i2) integrate ((log (1 - t)) / t, t, 0, x);
(%o2)                       - li (x)
                                2
(%i3) li [2] (7);
(%o3)                        li (7)
                               2
(%i4) li [2] (7), numer;
(%o4)       1.248273182099423 - 6.113257028817991 %i
(%i5) li [3] (7);
(%o5)                        li (7)
                               3
(%i6) li [2] (7), numer;
(%o6)       1.248273182099423 - 6.113257028817991 %i
(%i7) L : makelist (i / 4.0, i, 0, 8);
(%o7)   [0.0, 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0]
(%i8) map (lambda ([x], li [2] (x)), L);
(%o8) [0.0, 0.2676526390827326, 0.5822405264650125, 
0.978469392930306, 1.644934066848226, 
2.190177011441645 - 0.7010261415046585 %i, 
2.37439527027248 - 1.2738062049196 %i, 
2.448686765338205 - 1.758084848210787 %i, 
2.467401100272339 - 2.177586090303601 %i]
(%i9) map (lambda ([x], li [3] (x)), L);
(%o9) [0.0, 0.2584613953442624, 0.537213192678042, 
0.8444258046482203, 1.2020569, 1.642866878950322
 - 0.07821473130035025 %i, 2.060877505514697
 - 0.2582419849982037 %i, 2.433418896388322
 - 0.4919260182322965 %i, 2.762071904015935
 - 0.7546938285978846 %i]

Function: log (x)

Represents the natural (base e) logarithm of x.

Maxima does not have a built-in function for the base 10 logarithm or other bases. log10(x) := log(x) / log(10) is a useful definition.

Simplification and evaluation of logarithms is governed by several global flags:

logexpand

causes log(a^b) to become b*log(a). If it is set to all, log(a*b) will also simplify to log(a)+log(b). If it is set to super, then log(a/b) will also simplify to log(a)-log(b) for rational numbers a/b, a#1. (log(1/b), for b integer, always simplifies.) If it is set to false, all of these simplifications will be turned off.

logsimp

if false then no simplification of %e to a power containing log’s is done.

lognegint

if true implements the rule log(-n) -> log(n)+%i*%pi for n a positive integer.

%e_to_numlog

when true, r some rational number, and x some expression, the expression %e^(r*log(x)) will be simplified into x^r. It should be noted that the radcan command also does this transformation, and more complicated transformations of this as well. The logcontract command "contracts" expressions containing log.

Option variable: logabs

Default value: false

When doing indefinite integration where logs are generated, e.g. integrate(1/x,x), the answer is given in terms of log(abs(...)) if logabs is true, but in terms of log(...) if logabs is false. For definite integration, the logabs:true setting is used, because here "evaluation" of the indefinite integral at the endpoints is often needed.

Option variable: logarc
Function: logarc (expr)

When the global variable logarc is true, inverse circular and hyperbolic functions are replaced by equivalent logarithmic functions. The default value of logarc is false.

The function logarc(expr) carries out that replacement for an expression expr without setting the global variable logarc.

Option variable: logconcoeffp

Default value: false

Controls which coefficients are contracted when using logcontract. It may be set to the name of a predicate function of one argument. E.g. if you like to generate SQRTs, you can do logconcoeffp:'logconfun$ logconfun(m):=featurep(m,integer) or ratnump(m)$ . Then logcontract(1/2*log(x)); will give log(sqrt(x)).

Function: logcontract (expr)

Recursively scans the expression expr, transforming subexpressions of the form a1*log(b1) + a2*log(b2) + c into log(ratsimp(b1^a1 * b2^a2)) + c

(%i1) 2*(a*log(x) + 2*a*log(y))$
(%i2) logcontract(%);
                                 2  4
(%o2)                     a log(x  y )

The declaration declare(n,integer) causes logcontract(2*a*n*log(x)) to simplify to a*log(x^(2*n)). The coefficients that "contract" in this manner are those such as the 2 and the n here which satisfy featurep(coeff,integer). The user can control which coefficients are contracted by setting the option logconcoeffp to the name of a predicate function of one argument. E.g. if you like to generate SQRTs, you can do logconcoeffp:'logconfun$ logconfun(m):=featurep(m,integer) or ratnump(m)$ . Then logcontract(1/2*log(x)); will give log(sqrt(x)).

Option variable: logexpand

Default value: true

If true, that is the default value, causes log(a^b) to become b*log(a). If it is set to all, log(a*b) will also simplify to log(a)+log(b). If it is set to super, then log(a/b) will also simplify to log(a)-log(b) for rational numbers a/b, a#1. (log(1/b), for integer b, always simplifies.) If it is set to false, all of these simplifications will be turned off.

When logexpand is set to all or super, the logarithm of a product expression simplifies to a summation of logarithms.

Examples:

When logexpand is true, log(a^b) simplifies to b*log(a).

(%i1) log(n^2), logexpand=true;
(%o1)                       2 log(n)

When logexpand is all, log(a*b) simplifies to log(a)+log(b).

(%i1) log(10*x), logexpand=all;
(%o1)                   log(x) + log(10)

When logexpand is super, log(a/b) simplifies to log(a)-log(b) for rational numbers a/b with a#1.

(%i1) log(a/(n + 1)), logexpand=super;
(%o1)                  log(a) - log(n + 1)

When logexpand is set to all or super, the logarithm of a product expression simplifies to a summation of logarithms.

(%i1) my_product : product (X(i), i, 1, n);
                             n
                           /===\
                            ! !
(%o1)                       ! !  X(i)
                            ! !
                           i = 1
(%i2) log(my_product), logexpand=all;
                          n
                         ====
                         \
(%o2)                     >    log(X(i))
                         /
                         ====
                         i = 1
(%i3) log(my_product), logexpand=super;
                          n
                         ====
                         \
(%o3)                     >    log(X(i))
                         /
                         ====
                         i = 1

When logexpand is false, these simplifications are disabled.

(%i1) logexpand : false $
(%i2) log(n^2);
                                  2
(%o2)                        log(n )
(%i3) log(10*x);
(%o3)                       log(10 x)
(%i4) log(a/(n + 1));
                                 a
(%o4)                      log(-----)
                               n + 1
(%i5) log ('product (X(i), i, 1, n));
                               n
                             /===\
                              ! !
(%o5)                    log( ! !  X(i))
                              ! !
                             i = 1

Option variable: lognegint

Default value: false

If true implements the rule log(-n) -> log(n)+%i*%pi for n a positive integer.

Option variable: logsimp

Default value: true

If false then no simplification of %e to a power containing log’s is done.

Function: plog (x)

Represents the principal branch of the complex-valued natural logarithm with -%pi < carg(x) <= +%pi .

Function: sqrt (x)

The square root of x. It is represented internally by x^(1/2). See also rootscontract and radexpand.


Next: , Previous: , Up: Mathematical Functions   [Contents][Index]

10.5 Trigonometric Functions


Next: , Previous: , Up: Trigonometric Functions   [Contents][Index]

10.5.1 Introduction to Trigonometric

Maxima has many trigonometric functions defined. Not all trigonometric identities are programmed, but it is possible for the user to add many of them using the pattern matching capabilities of the system. The trigonometric functions defined in Maxima are: acos, acosh, acot, acoth, acsc, acsch, asec, asech, asin, asinh, atan, atanh, cos, cosh, cot, coth, csc, csch, sec, sech, sin, sinh, tan, and tanh. There are a number of commands especially for handling trigonometric functions, see trigexpand, trigreduce, and the switch trigsign. Two share packages extend the simplification rules built into Maxima, ntrig and atrig1. Do describe(command) for details.

·

Category: Trigonometric functions


Previous: , Up: Trigonometric Functions   [Contents][Index]

10.5.2 Functions and Variables for Trigonometric

Option variable: %piargs

Default value: true

When %piargs is true, trigonometric functions are simplified to algebraic constants when the argument is an integer multiple of %pi, %pi/2, %pi/3, %pi/4, or %pi/6.

Maxima knows some identities which can be applied when %pi, etc., are multiplied by an integer variable (that is, a symbol declared to be integer).

Examples:

(%i1) %piargs : false$
(%i2) [sin (%pi), sin (%pi/2), sin (%pi/3)];
                                %pi       %pi
(%o2)            [sin(%pi), sin(---), sin(---)]
                                 2         3
(%i3) [sin (%pi/4), sin (%pi/5), sin (%pi/6)];
                      %pi       %pi       %pi
(%o3)            [sin(---), sin(---), sin(---)]
                       4         5         6
(%i4) %piargs : true$
(%i5) [sin (%pi), sin (%pi/2), sin (%pi/3)];
                                sqrt(3)
(%o5)                    [0, 1, -------]
                                   2
(%i6) [sin (%pi/4), sin (%pi/5), sin (%pi/6)];
                         1         %pi   1
(%o6)                [-------, sin(---), -]
                      sqrt(2)       5    2
(%i7) [cos (%pi/3), cos (10*%pi/3), tan (10*%pi/3),
       cos (sqrt(2)*%pi/3)];
                1    1               sqrt(2) %pi
(%o7)          [-, - -, sqrt(3), cos(-----------)]
                2    2                    3

Some identities are applied when %pi and %pi/2 are multiplied by an integer variable.

(%i1) declare (n, integer, m, even)$
(%i2) [sin (%pi * n), cos (%pi * m), sin (%pi/2 * m),
       cos (%pi/2 * m)];
                                      m/2
(%o2)                  [0, 1, 0, (- 1)   ]

Option variable: %iargs

Default value: true

When %iargs is true, trigonometric functions are simplified to hyperbolic functions when the argument is apparently a multiple of the imaginary unit %i.

Even when the argument is demonstrably real, the simplification is applied; Maxima considers only whether the argument is a literal multiple of %i.

Examples:

(%i1) %iargs : false$
(%i2) [sin (%i * x), cos (%i * x), tan (%i * x)];
(%o2)           [sin(%i x), cos(%i x), tan(%i x)]
(%i3) %iargs : true$
(%i4) [sin (%i * x), cos (%i * x), tan (%i * x)];
(%o4)           [%i sinh(x), cosh(x), %i tanh(x)]

Even when the argument is demonstrably real, the simplification is applied.

(%i1) declare (x, imaginary)$
(%i2) [featurep (x, imaginary), featurep (x, real)];
(%o2)                     [true, false]
(%i3) sin (%i * x);
(%o3)                      %i sinh(x)

Function: acos (x)

– Arc Cosine.

Function: acosh (x)

– Hyperbolic Arc Cosine.

Function: acot (x)

– Arc Cotangent.

Function: acoth (x)

– Hyperbolic Arc Cotangent.

Function: acsc (x)

– Arc Cosecant.

Function: acsch (x)

– Hyperbolic Arc Cosecant.

Function: asec (x)

– Arc Secant.

Function: asech (x)

– Hyperbolic Arc Secant.

Function: asin (x)

– Arc Sine.

Function: asinh (x)

– Hyperbolic Arc Sine.

Function: atan (x)

– Arc Tangent.

See also atan2.

Function: atan2 (y, x)

– yields the value of atan(y/x) in the interval -%pi to %pi.

See also atan.

Function: atanh (x)

– Hyperbolic Arc Tangent.

Package: atrig1

The atrig1 package contains several additional simplification rules for inverse trigonometric functions. Together with rules already known to Maxima, the following angles are fully implemented: 0, %pi/6, %pi/4, %pi/3, and %pi/2. Corresponding angles in the other three quadrants are also available. Do load("atrig1"); to use them.

Function: cos (x)

– Cosine.

Function: cosh (x)

– Hyperbolic Cosine.

Function: cot (x)

– Cotangent.

Function: coth (x)

– Hyperbolic Cotangent.

Function: csc (x)

– Cosecant.

Function: csch (x)

– Hyperbolic Cosecant.

Option variable: halfangles

Default value: false

When halfangles is true, trigonometric functions of arguments expr/2 are simplified to functions of expr.

For a real argument x in the interval 0 < x < 2*%pi the sine of the half-angle simplifies to a simple formula:

                         sqrt(1 - cos(x))
                         ----------------
                             sqrt(2)

A complicated factor is needed to make this formula correct for all complex arguments z:

           realpart(z)
     floor(-----------)
              2 %pi
(- 1)                   (1 - unit_step(- imagpart(z))

                            realpart(z)            realpart(z)
                      floor(-----------) - ceiling(-----------)
                               2 %pi                  2 %pi
                ((- 1)                                          + 1))

Maxima knows this factor and similar factors for the functions sin, cos, sinh, and cosh. For special values of the argument z these factors simplify accordingly.

Examples:

(%i1) halfangles : false$
(%i2) sin (x / 2);
                                 x
(%o2)                        sin(-)
                                 2
(%i3) halfangles : true$
(%i4) sin (x / 2);
                            x
                    floor(-----)
                          2 %pi
               (- 1)             sqrt(1 - cos(x))
(%o4)          ----------------------------------
                            sqrt(2)
(%i5) assume(x>0, x<2*%pi)$
(%i6) sin(x / 2);
                        sqrt(1 - cos(x))
(%o6)                   ----------------
                            sqrt(2)

Package: ntrig

The ntrig package contains a set of simplification rules that are used to simplify trigonometric function whose arguments are of the form f(n %pi/10) where f is any of the functions sin, cos, tan, csc, sec and cot.

Function: sec (x)

– Secant.

Function: sech (x)

– Hyperbolic Secant.

Function: sin (x)

– Sine.

Function: sinh (x)

– Hyperbolic Sine.

Function: tan (x)

– Tangent.

Function: tanh (x)

– Hyperbolic Tangent.

Function: trigexpand (expr)

Expands trigonometric and hyperbolic functions of sums of angles and of multiple angles occurring in expr. For best results, expr should be expanded. To enhance user control of simplification, this function expands only one level at a time, expanding sums of angles or multiple angles. To obtain full expansion into sines and cosines immediately, set the switch trigexpand: true.

trigexpand is governed by the following global flags:

trigexpand

If true causes expansion of all expressions containing sin’s and cos’s occurring subsequently.

halfangles

If true causes half-angles to be simplified away.

trigexpandplus

Controls the "sum" rule for trigexpand, expansion of sums (e.g. sin(x + y)) will take place only if trigexpandplus is true.

trigexpandtimes

Controls the "product" rule for trigexpand, expansion of products (e.g. sin(2 x)) will take place only if trigexpandtimes is true.

Examples:

(%i1) x+sin(3*x)/sin(x),trigexpand=true,expand;
                         2            2
(%o1)              (- sin (x)) + 3 cos (x) + x
(%i2) trigexpand(sin(10*x+y));
(%o2)          cos(10 x) sin(y) + sin(10 x) cos(y)

Option variable: trigexpandplus

Default value: true

trigexpandplus controls the "sum" rule for trigexpand. Thus, when the trigexpand command is used or the trigexpand switch set to true, expansion of sums (e.g. sin(x+y)) will take place only if trigexpandplus is true.

Option variable: trigexpandtimes

Default value: true

trigexpandtimes controls the "product" rule for trigexpand. Thus, when the trigexpand command is used or the trigexpand switch set to true, expansion of products (e.g. sin(2*x)) will take place only if trigexpandtimes is true.

Option variable: triginverses

Default value: true

triginverses controls the simplification of the composition of trigonometric and hyperbolic functions with their inverse functions.

If all, both e.g. atan(tan(x)) and tan(atan(x)) simplify to x.

If true, the arcfun(fun(x)) simplification is turned off.

If false, both the arcfun(fun(x)) and fun(arcfun(x)) simplifications are turned off.

Function: trigreduce
    trigreduce (expr, x)
    trigreduce (expr)

Combines products and powers of trigonometric and hyperbolic sin’s and cos’s of x into those of multiples of x. It also tries to eliminate these functions when they occur in denominators. If x is omitted then all variables in expr are used.

See also poissimp.

(%i1) trigreduce(-sin(x)^2+3*cos(x)^2+x);
               cos(2 x)      cos(2 x)   1        1
(%o1)          -------- + 3 (-------- + -) + x - -
                  2             2       2        2

Option variable: trigsign

Default value: true

When trigsign is true, it permits simplification of negative arguments to trigonometric functions. E.g., sin(-x) will become -sin(x) only if trigsign is true.

Function: trigsimp (expr)

Employs the identities sin(x)^2 + cos(x)^2 = 1 and cosh(x)^2 - sinh(x)^2 = 1 to simplify expressions containing tan, sec, etc., to sin, cos, sinh, cosh.

trigreduce, ratsimp, and radcan may be able to further simplify the result.

demo ("trgsmp.dem") displays some examples of trigsimp.

Function: trigrat (expr)

Gives a canonical simplified quasilinear form of a trigonometrical expression; expr is a rational fraction of several sin, cos or tan, the arguments of them are linear forms in some variables (or kernels) and %pi/n (n integer) with integer coefficients. The result is a simplified fraction with numerator and denominator linear in sin and cos. Thus trigrat linearize always when it is possible.

(%i1) trigrat(sin(3*a)/sin(a+%pi/3));
(%o1)            sqrt(3) sin(2 a) + cos(2 a) - 1

The following example is taken from Davenport, Siret, and Tournier, Calcul Formel, Masson (or in English, Addison-Wesley), section 1.5.5, Morley theorem.

(%i1) c : %pi/3 - a - b$
(%i2) bc : sin(a)*sin(3*c)/sin(a+b);
                                           %pi
                 sin(a) sin(3 ((- b) - a + ---))
                                            3
(%o2)            -------------------------------
                           sin(b + a)
(%i3) ba : bc, c=a, a=c;
                                         %pi
                    sin(3 a) sin(b + a - ---)
                                          3
(%o3)               -------------------------
                                  %pi
                          sin(a - ---)
                                   3
(%i4) ac2 : ba^2 + bc^2 - 2*bc*ba*cos(b);
         2         2         %pi
      sin (3 a) sin (b + a - ---)
                              3
(%o4) ---------------------------
                2     %pi
             sin (a - ---)
                       3
                                         %pi
 - (2 sin(a) sin(3 a) sin(3 ((- b) - a + ---)) cos(b)
                                          3
             %pi            %pi
 sin(b + a - ---))/(sin(a - ---) sin(b + a))
              3              3
      2       2                %pi
   sin (a) sin (3 ((- b) - a + ---))
                                3
 + ---------------------------------
                 2
              sin (b + a)
(%i5) trigrat (ac2);
(%o5) - (sqrt(3) sin(4 b + 4 a) - cos(4 b + 4 a)
 - 2 sqrt(3) sin(4 b + 2 a) + 2 cos(4 b + 2 a)
 - 2 sqrt(3) sin(2 b + 4 a) + 2 cos(2 b + 4 a)
 + 4 sqrt(3) sin(2 b + 2 a) - 8 cos(2 b + 2 a) - 4 cos(2 b - 2 a)
 + sqrt(3) sin(4 b) - cos(4 b) - 2 sqrt(3) sin(2 b) + 10 cos(2 b)
 + sqrt(3) sin(4 a) - cos(4 a) - 2 sqrt(3) sin(2 a) + 10 cos(2 a)
 - 9)/4


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10.6 Random Numbers

Function: make_random_state
    make_random_state (n)
    make_random_state (s)
    make_random_state (true)
    make_random_state (false)

A random state object represents the state of the random number generator. The state comprises 627 32-bit words.

make_random_state (n) returns a new random state object created from an integer seed value equal to n modulo 2^32. n may be negative.

make_random_state (s) returns a copy of the random state s.

make_random_state (true) returns a new random state object, using the current computer clock time as the seed.

make_random_state (false) returns a copy of the current state of the random number generator.

Function: set_random_state (s)

Copies s to the random number generator state.

set_random_state always returns done.

Function: random (x)

Returns a pseudorandom number. If x is an integer, random (x) returns an integer from 0 through x - 1 inclusive. If x is a floating point number, random (x) returns a nonnegative floating point number less than x. random complains with an error if x is neither an integer nor a float, or if x is not positive.

The functions make_random_state and set_random_state maintain the state of the random number generator.

The Maxima random number generator is an implementation of the Mersenne twister MT 19937.

Examples:

(%i1) s1: make_random_state (654321)$
(%i2) set_random_state (s1);
(%o2)                         done
(%i3) random (1000);
(%o3)                          768
(%i4) random (9573684);
(%o4)                        7657880
(%i5) random (2^75);
(%o5)                11804491615036831636390
(%i6) s2: make_random_state (false)$
(%i7) random (1.0);
(%o7)                  0.2310127244107132
(%i8) random (10.0);
(%o8)                   4.394553645870825
(%i9) random (100.0);
(%o9)                   32.28666704056853
(%i10) set_random_state (s2);
(%o10)                        done
(%i11) random (1.0);
(%o11)                 0.2310127244107132
(%i12) random (10.0);
(%o12)                  4.394553645870825
(%i13) random (100.0);
(%o13)                  32.28666704056853


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11 Maximas Database


Next: , Previous: , Up: Maximas Database   [Contents][Index]

11.1 Introduction to Maximas Database


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11.2 Functions and Variables for Properties

Property: alphabetic

alphabetic is a property type recognized by declare. The expression declare(s, alphabetic) tells Maxima to recognize as alphabetic all of the characters in s, which must be a string.

See also Identifiers.

Example:

(%i1) xx\~yy\`\@ : 1729;
(%o1)                         1729
(%i2) declare ("~`@", alphabetic);
(%o2)                         done
(%i3) xx~yy`@ + @yy`xx + `xx@@yy~;
(%o3)               `xx@@yy~ + @yy`xx + 1729
(%i4) listofvars (%);
(%o4)                  [@yy`xx, `xx@@yy~]

Property: bindtest

The command declare(x, bindtest) tells Maxima to trigger an error when the symbol x is evaluated unbound.

(%i1) aa + bb;
(%o1)                        bb + aa
(%i2) declare (aa, bindtest);
(%o2)                         done
(%i3) aa + bb;
aa unbound variable
 -- an error.  Quitting.  To debug this try debugmode(true);
(%i4) aa : 1234;
(%o4)                         1234
(%i5) aa + bb;
(%o5)                       bb + 1234
Property: constant

declare(a, constant) declares a to be a constant. The declaration of a symbol to be constant does not prevent the assignment of a nonconstant value to the symbol.

See constantp and declare.

Example:

(%i1) declare(c, constant);
(%o1)                         done
(%i2) constantp(c);
(%o2)                         true
(%i3) c : x;
(%o3)                           x
(%i4) constantp(c);
(%o4)                         false

Function: constantp (expr)

Returns true if expr is a constant expression, otherwise returns false.

An expression is considered a constant expression if its arguments are numbers (including rational numbers, as displayed with /R/), symbolic constants such as %pi, %e, and %i, variables bound to a constant or declared constant by declare, or functions whose arguments are constant.

constantp evaluates its arguments.

See the property constant which declares a symbol to be constant.

Examples:

(%i1) constantp (7 * sin(2));
(%o1)                                true
(%i2) constantp (rat (17/29));
(%o2)                                true
(%i3) constantp (%pi * sin(%e));
(%o3)                                true
(%i4) constantp (exp (x));
(%o4)                                false
(%i5) declare (x, constant);
(%o5)                                done
(%i6) constantp (exp (x));
(%o6)                                true
(%i7) constantp (foo (x) + bar (%e) + baz (2));
(%o7)                                false
(%i8) 

Function: declare (a_1, p_1, a_2, p_2, …)

Assigns the atom or list of atoms a_i the property or list of properties p_i. When a_i and/or p_i are lists, each of the atoms gets all of the properties.

declare quotes its arguments. declare always returns done.

As noted in the description for each declaration flag, for some flags featurep(object, feature) returns true if object has been declared to have feature.

For more information about the features system, see features. To remove a property from an atom, use remove.

declare recognizes the following properties:

additive

Tells Maxima to simplify a_i expressions by the substitution a_i(x + y + z + ...) --> a_i(x) + a_i(y) + a_i(z) + .... The substitution is carried out on the first argument only.

alphabetic

Tells Maxima to recognize all characters in a_i (which must be a string) as alphabetic characters.

antisymmetric, commutative, symmetric

Tells Maxima to recognize a_i as a symmetric or antisymmetric function. commutative is the same as symmetric.

bindtest

Tells Maxima to trigger an error when a_i is evaluated unbound.

constant

Tells Maxima to consider a_i a symbolic constant.

even, odd

Tells Maxima to recognize a_i as an even or odd integer variable.

evenfun, oddfun

Tells Maxima to recognize a_i as an odd or even function.

evflag

Makes a_i known to the ev function so that a_i is bound to true during the execution of ev when a_i appears as a flag argument of ev. See evflag.

evfun

Makes a_i known to ev so that the function named by a_i is applied when a_i appears as a flag argument of ev. See evfun.

feature

Tells Maxima to recognize a_i as the name of a feature. Other atoms may then be declared to have the a_i property.

increasing, decreasing

Tells Maxima to recognize a_i as an increasing or decreasing function.

integer, noninteger

Tells Maxima to recognize a_i as an integer or noninteger variable.

integervalued

Tells Maxima to recognize a_i as an integer-valued function.

lassociative, rassociative

Tells Maxima to recognize a_i as a right-associative or left-associative function.

linear

Equivalent to declaring a_i both outative and additive.

mainvar

Tells Maxima to consider a_i a "main variable". A main variable succeeds all other constants and variables in the canonical ordering of Maxima expressions, as determined by ordergreatp.

multiplicative

Tells Maxima to simplify a_i expressions by the substitution a_i(x * y * z * ...) --> a_i(x) * a_i(y) * a_i(z) * .... The substitution is carried out on the first argument only.

nary

Tells Maxima to recognize a_i as an n-ary function.

The nary declaration is not the same as calling the nary function. The sole effect of declare(foo, nary) is to instruct the Maxima simplifier to flatten nested expressions, for example, to simplify foo(x, foo(y, z)) to foo(x, y, z).

nonarray

Tells Maxima to consider a_i not an array. This declaration prevents multiple evaluation of a subscripted variable name.

nonscalar

Tells Maxima to consider a_i a nonscalar variable. The usual application is to declare a variable as a symbolic vector or matrix.

noun

Tells Maxima to parse a_i as a noun. The effect of this is to replace instances of a_i with 'a_i or nounify(a_i), depending on the context.

outative

Tells Maxima to simplify a_i expressions by pulling constant factors out of the first argument.

When a_i has one argument, a factor is considered constant if it is a literal or declared constant.

When a_i has two or more arguments, a factor is considered constant if the second argument is a symbol and the factor is free of the second argument.

posfun

Tells Maxima to recognize a_i as a positive function.

rational, irrational

Tells Maxima to recognize a_i as a rational or irrational real variable.

real, imaginary, complex

Tells Maxima to recognize a_i as a real, pure imaginary, or complex variable.

scalar

Tells Maxima to consider a_i a scalar variable.

Examples of the usage of the properties are available in the documentation for each separate description of a property.

Property: decreasing
Property: increasing

The commands declare(f, decreasing) or declare(f, increasing) tell Maxima to recognize the function f as an decreasing or increasing function.

See also declare for more properties.

Example:

(%i1) assume(a > b);
(%o1)                        [a > b]
(%i2) is(f(a) > f(b));
(%o2)                        unknown
(%i3) declare(f, increasing);
(%o3)                         done
(%i4) is(f(a) > f(b));
(%o4)                         true

Property: even
Property: odd

declare(a, even) or declare(a, odd) tells Maxima to recognize the symbol a as an even or odd integer variable. The properties even and odd are not recognized by the functions evenp, oddp, and integerp.

See also declare and askinteger.

Example:

(%i1) declare(n, even);
(%o1)                         done
(%i2) askinteger(n, even);
(%o2)                          yes
(%i3) askinteger(n);
(%o3)                          yes
(%i4) evenp(n);
(%o4)                         false

Property: feature

Maxima understands two distinct types of features, system features and features which apply to mathematical expressions. See also status for information about system features. See also features and featurep for information about mathematical features.

feature itself is not the name of a function or variable.

Function: featurep (a, f)

Attempts to determine whether the object a has the feature f on the basis of the facts in the current database. If so, it returns true, else false.

Note that featurep returns false when neither f nor the negation of f can be established.

featurep evaluates its argument.

See also declare and features.

(%i1) declare (j, even)$
(%i2) featurep (j, integer);
(%o2)                           true

Declaration: features

Maxima recognizes certain mathematical properties of functions and variables. These are called "features".

declare (x, foo) gives the property foo to the function or variable x.

declare (foo, feature) declares a new feature foo. For example, declare ([red, green, blue], feature) declares three new features, red, green, and blue.

The predicate featurep (x, foo) returns true if x has the foo property, and false otherwise.

The infolist features is a list of known features. These are

   integer        noninteger      even
   odd            rational        irrational
   real           imaginary       complex
   analytic       increasing      decreasing
   oddfun         evenfun         posfun
   constant       commutative     lassociative
   rassociative   symmetric       antisymmetric
   integervalued

plus any user-defined features.

features is a list of mathematical features. There is also a list of non-mathematical, system-dependent features. See status.

Example:

(%i1) declare (FOO, feature);
(%o1)                         done
(%i2) declare (x, FOO);
(%o2)                         done
(%i3) featurep (x, FOO);
(%o3)                         true

Function: get (a, i)

Retrieves the user property indicated by i associated with atom a or returns false if a doesn’t have property i.

get evaluates its arguments.

See also put and qput.

(%i1) put (%e, 'transcendental, 'type);
(%o1)                    transcendental
(%i2) put (%pi, 'transcendental, 'type)$
(%i3) put (%i, 'algebraic, 'type)$
(%i4) typeof (expr) := block ([q],
        if numberp (expr)
        then return ('algebraic),
        if not atom (expr)
        then return (maplist ('typeof, expr)),
        q: get (expr, 'type),
        if q=false
        then errcatch (error(expr,"is not numeric.")) else q)$
(%i5) typeof (2*%e + x*%pi);
x is not numeric.
(%o5)  [[transcendental, []], [algebraic, transcendental]]
(%i6) typeof (2*%e + %pi);
(%o6)     [transcendental, [algebraic, transcendental]]

Property: integer
Property: noninteger

declare(a, integer) or declare(a, noninteger) tells Maxima to recognize a as an integer or noninteger variable.

See also declare.

Example:

(%i1) declare(n, integer, x, noninteger);
(%o1)                         done
(%i2) askinteger(n);
(%o2)                          yes
(%i3) askinteger(x);
(%o3)                          no

Property: integervalued

declare(f, integervalued) tells Maxima to recognize f as an integer-valued function.

See also declare.

Example:

(%i1) exp(%i)^f(x);
                              %i f(x)
(%o1)                      (%e  )
(%i2) declare(f, integervalued);
(%o2)                         done
(%i3) exp(%i)^f(x);
                              %i f(x)
(%o3)                       %e

Property: nonarray

The command declare(a, nonarray) tells Maxima to consider a not an array. This declaration prevents multiple evaluation, if a is a subscripted variable.

See also declare.

Example:

(%i1) a:'b$ b:'c$ c:'d$

(%i4) a[x];
(%o4)                          d
                                x
(%i5) declare(a, nonarray);
(%o5)                         done
(%i6) a[x];
(%o6)                          a
                                x

Property: nonscalar

Makes atoms behave as does a list or matrix with respect to the dot operator.

See also declare.

Function: nonscalarp (expr)

Returns true if expr is a non-scalar, i.e., it contains atoms declared as non-scalars, lists, or matrices.

See also the predicate function scalarp and declare.

Property: posfun

declare (f, posfun) declares f to be a positive function. is (f(x) > 0) yields true.

See also declare.

Function: printprops
    printprops (a, i)
    printprops ([a_1, …, a_n], i)
    printprops (all, i)

Displays the property with the indicator i associated with the atom a. a may also be a list of atoms or the atom all in which case all of the atoms with the given property will be used. For example, printprops ([f, g], atvalue). printprops is for properties that cannot otherwise be displayed, i.e. for atvalue, atomgrad, gradef, and matchdeclare.

Function: properties (a)

Returns a list of the names of all the properties associated with the atom a.

System variable: props

Default value: []

props are atoms which have any property other than those explicitly mentioned in infolists, such as specified by atvalue, matchdeclare, etc., as well as properties specified in the declare function.

Function: propvars (prop)

Returns a list of those atoms on the props list which have the property indicated by prop. Thus propvars (atvalue) returns a list of atoms which have atvalues.

Function: put (atom, value, indicator)

Assigns value to the property (specified by indicator) of atom. indicator may be the name of any property, not just a system-defined property.

rem reverses the effect of put.

put evaluates its arguments. put returns value.

See also qput and get.

Examples:

(%i1) put (foo, (a+b)^5, expr);
                                   5
(%o1)                       (b + a)
(%i2) put (foo, "Hello", str);
(%o2)                         Hello
(%i3) properties (foo);
(%o3)            [[user properties, str, expr]]
(%i4) get (foo, expr);
                                   5
(%o4)                       (b + a)
(%i5) get (foo, str);
(%o5)                         Hello

Function: qput (atom, value, indicator)

Assigns value to the property (specified by indicator) of atom. This is the same as put, except that the arguments are quoted.

See also get.

Example:

(%i1) foo: aa$ 
(%i2) bar: bb$
(%i3) baz: cc$
(%i4) put (foo, bar, baz);
(%o4)                          bb
(%i5) properties (aa);
(%o5)                [[user properties, cc]]
(%i6) get (aa, cc);
(%o6)                          bb
(%i7) qput (foo, bar, baz);
(%o7)                          bar
(%i8) properties (foo);
(%o8)            [value, [user properties, baz]]
(%i9) get ('foo, 'baz);
(%o9)                          bar

Property: rational
Property: irrational

declare(a, rational) or declare(a, irrational) tells Maxima to recognize a as a rational or irrational real variable.

See also declare.

Property: real
Property: imaginary
Property: complex

declare(a, real), declare(a, imaginary), or declare(a, complex) tells Maxima to recognize a as a real, pure imaginary, or complex variable.

See also declare.

Function: rem (atom, indicator)

Removes the property indicated by indicator from atom. rem reverses the effect of put.

rem returns done if atom had an indicator property when rem was called, or false if it had no such property.

Function: remove
    remove (a_1, p_1, …, a_n, p_n)
    remove ([a_1, …, a_m], [p_1, …, p_n], …)
    remove ("a", operator)
    remove (a, transfun)
    remove (all, p)

Removes properties associated with atoms.

remove (a_1, p_1, ..., a_n, p_n) removes property p_k from atom a_k.

remove ([a_1, ..., a_m], [p_1, ..., p_n], ...) removes properties p_1, ..., p_n from atoms a_1, …, a_m. There may be more than one pair of lists.

remove (all, p) removes the property p from all atoms which have it.

The removed properties may be system-defined properties such as function, macro, or mode_declare. remove does not remove properties defined by put.

A property may be transfun to remove the translated Lisp version of a function. After executing this, the Maxima version of the function is executed rather than the translated version.

remove ("a", operator) or, equivalently, remove ("a", op) removes from a the operator properties declared by prefix, infix, nary, postfix, matchfix, or nofix. Note that the name of the operator must be written as a quoted string.

remove always returns done whether or not an atom has a specified property. This behavior is unlike the more specific remove functions remvalue, remarray, remfunction, and remrule.

remove quotes its arguments.

Property: scalar

declare(a, scalar) tells Maxima to consider a a scalar variable.

See also declare.

Function: scalarp (expr)

Returns true if expr is a number, constant, or variable declared scalar with declare, or composed entirely of numbers, constants, and such variables, but not containing matrices or lists.

See also the predicate function nonscalarp.


Next: , Previous: , Up: Maximas Database   [Contents][Index]

11.3 Functions and Variables for Facts

Function: activate (context_1, …, context_n)

Activates the contexts context_1, …, context_n. The facts in these contexts are then available to make deductions and retrieve information. The facts in these contexts are not listed by facts ().

The variable activecontexts is the list of contexts which are active by way of the activate function.

System variable: activecontexts

Default value: []

activecontexts is a list of the contexts which are active by way of the activate function, as opposed to being active because they are subcontexts of the current context.

Function: askinteger
    askinteger (expr, integer)
    askinteger (expr)
    askinteger (expr, even)
    askinteger (expr, odd)

askinteger (expr, integer) attempts to determine from the assume database whether expr is an integer. askinteger prompts the user if it cannot tell otherwise, and attempt to install the information in the database if possible. askinteger (expr) is equivalent to askinteger (expr, integer).

askinteger (expr, even) and askinteger (expr, odd) likewise attempt to determine if expr is an even integer or odd integer, respectively.

Function: asksign (expr)

First attempts to determine whether the specified expression is positive, negative, or zero. If it cannot, it asks the user the necessary questions to complete its deduction. The user’s answer is recorded in the data base for the duration of the current computation. The return value of asksign is one of pos, neg, or zero.

Function: assume (pred_1, …, pred_n)

Adds predicates pred_1, …, pred_n to the current context. If a predicate is inconsistent or redundant with the predicates in the current context, it is not added to the context. The context accumulates predicates from each call to assume.

assume returns a list whose elements are the predicates added to the context or the atoms redundant or inconsistent where applicable.

The predicates pred_1, …, pred_n can only be expressions with the relational operators < <= equal notequal >= and >. Predicates cannot be literal equality = or literal inequality # expressions, nor can they be predicate functions such as integerp.

Compound predicates of the form pred_1 and ... and pred_n are recognized, but not pred_1 or ... or pred_n. not pred_k is recognized if pred_k is a relational predicate. Expressions of the form not (pred_1 and pred_2) and not (pred_1 or pred_2) are not recognized.

Maxima’s deduction mechanism is not very strong; there are many obvious consequences which cannot be determined by is. This is a known weakness.

assume does not handle predicates with complex numbers. If a predicate contains a complex number assume returns inconsistent or redunant.

assume evaluates its arguments.

See also is, facts, forget, context, and declare.

Examples:

(%i1) assume (xx > 0, yy < -1, zz >= 0);
(%o1)              [xx > 0, yy < - 1, zz >= 0]
(%i2) assume (aa < bb and bb < cc);
(%o2)                  [bb > aa, cc > bb]
(%i3) facts ();
(%o3)     [xx > 0, - 1 > yy, zz >= 0, bb > aa, cc > bb]
(%i4) is (xx > yy);
(%o4)                         true
(%i5) is (yy < -yy);
(%o5)                         true
(%i6) is (sinh (bb - aa) > 0);
(%o6)                         true
(%i7) forget (bb > aa);
(%o7)                       [bb > aa]
(%i8) prederror : false;
(%o8)                         false
(%i9) is (sinh (bb - aa) > 0);
(%o9)                        unknown
(%i10) is (bb^2 < cc^2);
(%o10)                       unknown

Option variable: assumescalar

Default value: true

assumescalar helps govern whether expressions expr for which nonscalarp (expr) is false are assumed to behave like scalars for certain transformations.

Let expr represent any expression other than a list or a matrix, and let [1, 2, 3] represent any list or matrix. Then expr . [1, 2, 3] yields [expr, 2 expr, 3 expr] if assumescalar is true, or scalarp (expr) is true, or constantp (expr) is true.

If assumescalar is true, such expressions will behave like scalars only for commutative operators, but not for noncommutative multiplication ..

When assumescalar is false, such expressions will behave like non-scalars.

When assumescalar is all, such expressions will behave like scalars for all the operators listed above.

Option variable: assume_pos

Default value: false

When assume_pos is true and the sign of a parameter x cannot be determined from the current context or other considerations, sign and asksign (x) return true. This may forestall some automatically-generated asksign queries, such as may arise from integrate or other computations.

By default, a parameter is x such that symbolp (x) or subvarp (x). The class of expressions considered parameters can be modified to some extent via the variable assume_pos_pred.

sign and asksign attempt to deduce the sign of expressions from the sign of operands within the expression. For example, if a and b are both positive, then a + b is also positive.

However, there is no way to bypass all asksign queries. In particular, when the asksign argument is a difference x - y or a logarithm log(x), asksign always requests an input from the user, even when assume_pos is true and assume_pos_pred is a function which returns true for all arguments.

Option variable: assume_pos_pred

Default value: false

When assume_pos_pred is assigned the name of a function or a lambda expression of one argument x, that function is called to determine whether x is considered a parameter for the purpose of assume_pos. assume_pos_pred is ignored when assume_pos is false.

The assume_pos_pred function is called by sign and asksign with an argument x which is either an atom, a subscripted variable, or a function call expression. If the assume_pos_pred function returns true, x is considered a parameter for the purpose of assume_pos.

By default, a parameter is x such that symbolp (x) or subvarp (x).

See also assume and assume_pos.

Examples:

(%i1) assume_pos: true$
(%i2) assume_pos_pred: symbolp$
(%i3) sign (a);
(%o3)                          pos
(%i4) sign (a[1]);
(%o4)                          pnz
(%i5) assume_pos_pred: lambda ([x], display (x), true)$
(%i6) asksign (a);
                              x = a

(%o6)                          pos
(%i7) asksign (a[1]);
                             x = a
                                  1

(%o7)                          pos
(%i8) asksign (foo (a));
                           x = foo(a)

(%o8)                          pos
(%i9) asksign (foo (a) + bar (b));
                           x = foo(a)

                           x = bar(b)

(%o9)                          pos
(%i10) asksign (log (a));
                              x = a

Is  a - 1  positive, negative, or zero?

p;
(%o10)                         pos
(%i11) asksign (a - b);
                              x = a

                              x = b

                              x = a

                              x = b

Is  b - a  positive, negative, or zero?

p;
(%o11)                         neg

Option variable: context

Default value: initial

context names the collection of facts maintained by assume and forget. assume adds facts to the collection named by context, while forget removes facts.

Binding context to a name foo changes the current context to foo. If the specified context foo does not yet exist, it is created automatically by a call to newcontext. The specified context is activated automatically.

See contexts for a general description of the context mechanism.

Option variable: contexts

Default value: [initial, global]

contexts is a list of the contexts which currently exist, including the currently active context.

The context mechanism makes it possible for a user to bind together and name a collection of facts, called a context. Once this is done, the user can have Maxima assume or forget large numbers of facts merely by activating or deactivating their context.

Any symbolic atom can be a context, and the facts contained in that context will be retained in storage until destroyed one by one by calling forget or destroyed as a whole by calling kill to destroy the context to which they belong.

Contexts exist in a hierarchy, with the root always being the context global, which contains information about Maxima that some functions need. When in a given context, all the facts in that context are "active" (meaning that they are used in deductions and retrievals) as are all the facts in any context which is a subcontext of the active context.

When a fresh Maxima is started up, the user is in a context called initial, which has global as a subcontext.

See also facts, newcontext, supcontext, killcontext, activate, deactivate, assume, and forget.

Function: deactivate (context_1, …, context_n)

Deactivates the specified contexts context_1, …, context_n.

Function: facts
    facts (item)
    facts ()

If item is the name of a context, facts (item) returns a list of the facts in the specified context.

If item is not the name of a context, facts (item) returns a list of the facts known about item in the current context. Facts that are active, but in a different context, are not listed.

facts () (i.e., without an argument) lists the current context.

Function: forget
    forget (pred_1, …, pred_n)
    forget (L)

Removes predicates established by assume. The predicates may be expressions equivalent to (but not necessarily identical to) those previously assumed.

forget (L), where L is a list of predicates, forgets each item on the list.

Function: is (expr)

Attempts to determine whether the predicate expr is provable from the facts in the assume database.

If the predicate is provably true or false, is returns true or false, respectively. Otherwise, the return value is governed by the global flag prederror. When prederror is true, is complains with an error message. Otherwise, is returns unknown.

ev(expr, pred) (which can be written expr, pred at the interactive prompt) is equivalent to is(expr).

See also assume, facts, and maybe.

Examples:

is causes evaluation of predicates.

(%i1) %pi > %e;
(%o1)                       %pi > %e
(%i2) is (%pi > %e);
(%o2)                         true

is attempts to derive predicates from the assume database.

(%i1) assume (a > b);
(%o1)                        [a > b]
(%i2) assume (b > c);
(%o2)                        [b > c]
(%i3) is (a < b);
(%o3)                         false
(%i4) is (a > c);
(%o4)                         true
(%i5) is (equal (a, c));
(%o5)                         false

If is can neither prove nor disprove a predicate from the assume database, the global flag prederror governs the behavior of is.

(%i1) assume (a > b);
(%o1)                        [a > b]
(%i2) prederror: true$
(%i3) is (a > 0);
Maxima was unable to evaluate the predicate:
a > 0
 -- an error.  Quitting.  To debug this try debugmode(true);
(%i4) prederror: false$
(%i5) is (a > 0);
(%o5)                        unknown

Function: killcontext (context_1, …, context_n)

Kills the contexts context_1, …, context_n.

If one of the contexts is the current context, the new current context will become the first available subcontext of the current context which has not been killed. If the first available unkilled context is global then initial is used instead. If the initial context is killed, a new, empty initial context is created.

killcontext refuses to kill a context which is currently active, either because it is a subcontext of the current context, or by use of the function activate.

killcontext evaluates its arguments. killcontext returns done.

Function: maybe (expr)

Attempts to determine whether the predicate expr is provable from the facts in the assume database.

If the predicate is provably true or false, maybe returns true or false, respectively. Otherwise, maybe returns unknown.

maybe is functionally equivalent to is with prederror: false, but the result is computed without actually assigning a value to prederror.

See also assume, facts, and is.

Examples:

(%i1) maybe (x > 0);
(%o1)                        unknown
(%i2) assume (x > 1);
(%o2)                        [x > 1]
(%i3) maybe (x > 0);
(%o3)                         true

Function: newcontext
    newcontext (name)
    newcontext ()

Creates a new, empty context, called name, which has global as its only subcontext. The newly-created context becomes the currently active context.

If name is not specified, a new name is created (via gensym) and returned.

newcontext evaluates its argument. newcontext returns name (if specified) or the new context name.

Function: sign (expr)

Attempts to determine the sign of expr on the basis of the facts in the current data base. It returns one of the following answers: pos (positive), neg (negative), zero, pz (positive or zero), nz (negative or zero), pn (positive or negative), or pnz (positive, negative, or zero, i.e. nothing known).

Function: supcontext
    supcontext (name, context)
    supcontext (name)
    supcontext ()

Creates a new context, called name, which has context as a subcontext. context must exist.

If context is not specified, the current context is assumed.

If name is not specified, a new name is created (via gensym) and returned.

supcontext evaluates its argument. supcontext returns name (if specified) or the new context name.


Previous: , Up: Maximas Database   [Contents][Index]

11.4 Functions and Variables for Predicates

Function: charfun (p)

Return 0 when the predicate p evaluates to false; return 1 when the predicate evaluates to true. When the predicate evaluates to something other than true or false (unknown), return a noun form.

Examples:

(%i1) charfun (x < 1);
(%o1)                    charfun(x < 1)
(%i2) subst (x = -1, %);
(%o2)                           1
(%i3) e : charfun ('"and" (-1 < x, x < 1))$
(%i4) [subst (x = -1, e), subst (x = 0, e), subst (x = 1, e)];
(%o4)                       [0, 1, 0]

Function: compare (x, y)

Return a comparison operator op (<, <=, >, >=, =, or #) such that is (x op y) evaluates to true; when either x or y depends on %i and x # y, return notcomparable; when there is no such operator or Maxima isn’t able to determine the operator, return unknown.

Examples:

(%i1) compare (1, 2);
(%o1)                           <
(%i2) compare (1, x);
(%o2)                        unknown
(%i3) compare (%i, %i);
(%o3)                           =
(%i4) compare (%i, %i + 1);
(%o4)                     notcomparable
(%i5) compare (1/x, 0);
(%o5)                           #
(%i6) compare (x, abs(x));
(%o6)                          <=

The function compare doesn’t try to determine whether the real domains of its arguments are nonempty; thus

(%i1) compare (acos (x^2 + 1), acos (x^2 + 1) + 1);
(%o1)                           <

The real domain of acos (x^2 + 1) is empty.

Function: equal (a, b)

Represents equivalence, that is, equal value.

By itself, equal does not evaluate or simplify. The function is attempts to evaluate equal to a Boolean value. is(equal(a, b)) returns true (or false) if and only if a and b are equal (or not equal) for all possible values of their variables, as determined by evaluating ratsimp(a - b); if ratsimp returns 0, the two expressions are considered equivalent. Two expressions may be equivalent even if they are not syntactically equal (i.e., identical).

When is fails to reduce equal to true or false, the result is governed by the global flag prederror. When prederror is true, is complains with an error message. Otherwise, is returns unknown.

In addition to is, some other operators evaluate equal and notequal to true or false, namely if, and, or, and not.

The negation of equal is notequal.

Examples:

By itself, equal does not evaluate or simplify.

(%i1) equal (x^2 - 1, (x + 1) * (x - 1));
                        2
(%o1)            equal(x  - 1, (x - 1) (x + 1))
(%i2) equal (x, x + 1);
(%o2)                    equal(x, x + 1)
(%i3) equal (x, y);
(%o3)                      equal(x, y)

The function is attempts to evaluate equal to a Boolean value. is(equal(a, b)) returns true when ratsimp(a - b) returns 0. Two expressions may be equivalent even if they are not syntactically equal (i.e., identical).

(%i1) ratsimp (x^2 - 1 - (x + 1) * (x - 1));
(%o1)                           0
(%i2) is (equal (x^2 - 1, (x + 1) * (x - 1)));
(%o2)                         true
(%i3) is (x^2 - 1 = (x + 1) * (x - 1));
(%o3)                         false
(%i4) ratsimp (x - (x + 1));
(%o4)                          - 1
(%i5) is (equal (x, x + 1));
(%o5)                         false
(%i6) is (x = x + 1);
(%o6)                         false
(%i7) ratsimp (x - y);
(%o7)                         x - y
(%i8) is (equal (x, y));
(%o8)                        unknown
(%i9) is (x = y);
(%o9)                         false

When is fails to reduce equal to true or false, the result is governed by the global flag prederror.

(%i1) [aa : x^2 + 2*x + 1, bb : x^2 - 2*x - 1];
                    2             2
(%o1)             [x  + 2 x + 1, x  - 2 x - 1]
(%i2) ratsimp (aa - bb);
(%o2)                        4 x + 2
(%i3) prederror : true;
(%o3)                         true
(%i4) is (equal (aa, bb));
Maxima was unable to evaluate the predicate:
       2             2
equal(x  + 2 x + 1, x  - 2 x - 1)
 -- an error.  Quitting.  To debug this try debugmode(true);
(%i5) prederror : false;
(%o5)                         false
(%i6) is (equal (aa, bb));
(%o6)                        unknown

Some operators evaluate equal and notequal to true or false.

(%i1) if equal (y, y - 1) then FOO else BAR;
(%o1)                          BAR
(%i2) eq_1 : equal (x, x + 1);
(%o2)                    equal(x, x + 1)
(%i3) eq_2 : equal (y^2 + 2*y + 1, (y + 1)^2);
                         2                   2
(%o3)             equal(y  + 2 y + 1, (y + 1) )
(%i4) [eq_1 and eq_2, eq_1 or eq_2, not eq_1];
(%o4)                  [false, true, true]

Because not expr causes evaluation of expr, not equal(a, b) is equivalent to is(notequal(a, b)).

(%i1) [notequal (2*z, 2*z - 1), not equal (2*z, 2*z - 1)];
(%o1)            [notequal(2 z, 2 z - 1), true]
(%i2) is (notequal (2*z, 2*z - 1));
(%o2)                         true

Function: notequal (a, b)

Represents the negation of equal(a, b).

Examples:

(%i1) equal (a, b);
(%o1)                      equal(a, b)
(%i2) maybe (equal (a, b));
(%o2)                        unknown
(%i3) notequal (a, b);
(%o3)                    notequal(a, b)
(%i4) not equal (a, b);
(%o4)                    notequal(a, b)
(%i5) maybe (notequal (a, b));
(%o5)                        unknown
(%i6) assume (a > b);
(%o6)                        [a > b]
(%i7) equal (a, b);
(%o7)                      equal(a, b)
(%i8) maybe (equal (a, b));
(%o8)                         false
(%i9) notequal (a, b);
(%o9)                    notequal(a, b)
(%i10) maybe (notequal (a, b));
(%o10)                        true

Function: unknown (expr)

Returns true if and only if expr contains an operator or function not recognized by the Maxima simplifier.

Function: zeroequiv (expr, v)

Tests whether the expression expr in the variable v is equivalent to zero, returning true, false, or dontknow.

zeroequiv has these restrictions:

  1. Do not use functions that Maxima does not know how to differentiate and evaluate.
  2. If the expression has poles on the real line, there may be errors in the result (but this is unlikely to occur).
  3. If the expression contains functions which are not solutions to first order differential equations (e.g. Bessel functions) there may be incorrect results.
  4. The algorithm uses evaluation at randomly chosen points for carefully selected subexpressions. This is always a somewhat hazardous business, although the algorithm tries to minimize the potential for error.

For example zeroequiv (sin(2 * x) - 2 * sin(x) * cos(x), x) returns true and zeroequiv (%e^x + x, x) returns false. On the other hand zeroequiv (log(a * b) - log(a) - log(b), a) returns dontknow because of the presence of an extra parameter b.


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12 Plotting


Next: , Previous: , Up: Plotting   [Contents][Index]

12.1 Introduction to Plotting

Maxima uses an external plotting package to make the plots (see the section on Plotting Formats). The plotting functions calculate a set of points and pass them to the plotting package together with a set of commands. That information can be passed to the external program either through a pipe or by calling the program with the name of a file where the data has been saved. The data file is given the name maxout_xxx.format, where xxx is a number that is unique to every concurrently-running instance of Maxima and format is the name of the plotting format being used (gnuplot, xmaxima, mgnuplot, gnuplot_pipes or geomview).

There are to save the plot in a graphic format file. In those cases, the file maxout_xxx.format created by Maxima includes commands that will make the external plotting program save the result in a graphic file. The default name for that graphic file is maxplot.extension, where extension is the extension normally used for the kind of graphic file selected.

The maxout_xxx.format and maxplot.extension files are created in the directory specified by the system variable maxima_tempdir. That location can be changed by assigning to that variable (or to the environment variable MAXIMA_TEMPDIR) a string that represents a valid directory where Maxima can create new files. The output of the Maxima plotting command will be a list with the names of the file(s) created, including their complete path.

If the format used is either gnuplot or xmaxima, the external programs gnuplot or xmaxima can be run, giving it the file maxout_xxx.format as argument, in order to view again a plot previously created in Maxima. Thus, when a Maxima plotting command fails, the format can be set to gnuplot or xmaxima and the plain-text file maxout_xxx.gnuplot (or maxout_xxx.xmaxima) can be inspected to look for the source of the problem.

The additional package draw provides functions similar to the ones described in this section with some extra features. Note that some plotting options have the same name in both plotting packages, but their syntax and behavior is different. To view the documentation for a graphic option opt, type ?? opt in order to choose the information for either of those two packages.

·

Category: Plotting


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12.2 Plotting Formats

Maxima can use either Gnuplot, Xmaxima or Geomview as graphics program. Gnuplot and Geomview are external programs which must be installed separately, while Xmaxima is distributed with Maxima. There are various different formats for those programs, which can be selected with the option plot_format (see also the Plotting Options section).

The plotting formats are the following:

·

Category: Plotting


Next: , Previous: , Up: Plotting   [Contents][Index]

12.3 Functions and Variables for Plotting

Function: contour_plot (expr, x_range, y_range, options, …)

It plots the contours (curves of equal value) of expr over the region x_range by y_range. Any additional arguments are treated the same as in plot3d.

This function only works when the plot format is either gnuplot or gnuplot_pipes. The additional package implicit_plot, which works in any graphic format, can also be used to plot contours but a separate expression must be given for each contour.

Examples:

(%i1) contour_plot (x^2 + y^2, [x, -4, 4], [y, -4, 4])$
figures/plotting1

You can add any options accepted by plot3d; for instance, the option legend with a value of false, to remove the legend. By default, Gnuplot chooses and displays 3 contours. To increase the number of contours, it is necessary to use a custom gnuplot_preamble, as in the next example:

(%i1) contour_plot (u^3 + v^2, [u, -4, 4], [v, -4, 4],
              [legend,false],
              [gnuplot_preamble, "set cntrparam levels 12"])$
figures/plotting2

System variable: geomview_command

This variable stores the name of the command used to run the geomview program when the plot format is geomview. Its default value is "geomview". If the geomview program is not found unless you give its complete path or if you want to try a different version of it, you may change the value of this variable. For instance,

(%i1) geomview_command: "/usr/local/bin/my_geomview"$

Function: get_plot_option (keyword, index)

Returns the current default value of the option named keyword, which is a list. The optional argument index must be a positive integer which can be used to extract only one element from the list (element 1 is the name of the option).

See also set_plot_option, remove_plot_option and the section on Plotting Options.

System variable: gnuplot_command

This variable stores the name of the command used to run the gnuplot program when the plot format is gnuplot. Its default value is "gnuplot". If the gnuplot program is not found unless you give its complete path or if you want to try a different version of it, you may change the value of this variable. For instance,

(%i1) gnuplot_command: "/usr/local/bin/my_gnuplot"$

System variable: gnuplot_file_args

When a graphic file is going to be created using gnuplot, this variable is used to specify the way the file name should be passed to gnuplot. Its default value is "~s", which means that the name of the file will be passed directly. The contents of this variable can be changed in order to add options for the gnuplot program, adding those options before the format directive "~s".

System variable: gnuplot_view_args

This variable is used to parse the argument that will be passed to the gnuplot program when the plot format is gnuplot. Its default value is "-persist ~s", where "~s" will be replaced with the name of the file where the gnuplot commands have been written (usually "maxout_xxx.gnuplot"). The option -persist tells gnuplot to exit after the commands in the file have been executed, without closing the window that displays the plot.

Those familiar with gnuplot, might want to change the value of this variable. For example, by changing it to:

(%i1) gnuplot_view_args: "~s -"$

gnuplot will not be closed after the commands in the file have been executed; thus, the window with the plot will remain, as well as the gnuplot interactive shell where other commands can be issued in order to modify the plot.

In Windows versions of Gnuplot older than 4.6.3 the behavior of "~s -" and "-persist ~s" were the opposite; namely, "-persist ~s" made the plot window and the gnuplot interactive shell remain, while "~s -" closed the gnuplot shell keeping the plot window. Therefore, when older gnuplot versions are used in Windows, it might be necessary to adjust the value of gnuplot_view_args.

Function: implicit_plot
    implicit_plot (expr, x_range, y_range)
    implicit_plot ([expr_1, …, expr_n], x_range, y_range)

Displays a plot of a function on the real plane, defined implicitly by the expression expr. The domain in the plane is defined by x_range and y_range. Several functions can be represented on the same plot, giving a list [expr_1, …, expr_n] of expressions that define them. This function uses the global format options set up with the set_plot_option. Additional options can also be given as extra arguments for the implicit_plot command.

The method used by implicit_plot consists of tracking sign changes on the domain given and it can fail for complicated expressions.

load(implicit_plot) loads this function.

Example:

(%i1) load(implicit_plot)$
(%i2) implicit_plot (x^2 = y^3 - 3*y + 1, [x, -4, 4], [y, -4, 4])$
figures/plotting3

Function: julia (x, y, ...options...)

Creates a graphic representation of the Julia set for the complex number (x + i y). The two mandatory parameters x and y must be real. This program is part of the additional package dynamics, but that package does not have to be loaded; the first time julia is used, it will be loaded automatically.

Each pixel in the grid is given a color corresponding to the number of iterations it takes the sequence that starts at that point to move out of the convergence circle of radius 2 centered at the origin. The number of pixels in the grid is controlled by the grid plot option (default 30 by 30). The maximum number of iterations is set with the option iterations. The program uses its own default palette: magenta,violet, blue, cyan, green, yellow, orange, red, brown and black, but it can be changed by adding an explicit palette option in the command.

The default domain used goes from -2 to 2 in both axes and can be changed with the x and y options. By default, the two axes are shown with the same scale, unless the option yx_ratio is used or the option same_xy is disabled. Other general plot options are also accepted.

The following example shows a region of the Julia set for the number -0.55 + i0.6. The option color_bar_tics is used to prevent Gnuplot from adjusting the color box up to 40, in which case the points corresponding the maximum 36 iterations would not be black.

(%i1) julia (-0.55, 0.6, [iterations, 36], [x, -0.3, 0.2],
      [y, 0.3, 0.9], [grid, 400, 400], [color_bar_tics, 0, 6, 36])$
figures/plotting4

Function: make_transform ([var1, var2, var3], fx, fy, fz)

Returns a function suitable to be used in the option transform_xy of plot3d. The three variables var1, var2, var3 are three dummy variable names, which represent the 3 variables given by the plot3d command (first the two independent variables and then the function that depends on those two variables). The three functions fx, fy, fz must depend only on those 3 variables, and will give the corresponding x, y and z coordinates that should be plotted. There are two transformations defined by default: polar_to_xy and spherical_to_xyz. See the documentation for those two transformations.

Function: mandelbrot (options)

Creates a graphic representation of the Mandelbrot set. This program is part of the additional package dynamics, but that package does not have to be loaded; the first time mandelbrot is used, the package will be loaded automatically.

This program can be called without any arguments, in which case it will use a default value of 9 iterations per point, a grid with dimensions set by the grid plot option (default 30 by 30) and a region that extends from -2 to 2 in both axes. The options are all the same that plot2d accepts, plus an option iterations to change the number of iterations.

Each pixel in the grid is given a color corresponding to the number of iterations it takes the sequence starting at zero to move out of the convergence circle of radius 2, centered at the origin. The maximum number of iterations is set by the option iterations. The program uses its own default palette: magenta,violet, blue, cyan, green, yellow, orange, red, brown and black, but it can be changed by adding an explicit palette option in the command. By default, the two axes are shown with the same scale, unless the option yx_ratio is used or the option same_xy is disabled.

Example:

[grid,400,400])$

(%i1) mandelbrot ([iterations, 30], [x, -2, 1], [y, -1.2, 1.2],
            [grid,400,400])$
figures/plotting5

System function: polar_to_xy

It can be given as value for the transform_xy option of plot3d. Its effect will be to interpret the two independent variables in plot3d as the distance from the z axis and the azimuthal angle (polar coordinates), and transform them into x and y coordinates.

Function: plot2d
    plot2d (plot, x_range, …, options, …)
    plot2d ([plot_1, …, plot_n], …, options, …)
    plot2d ([plot_1, …, plot_n], x_range, …, options, …)

Where plot, plot_1, …, plot_n can be either expressions, function names or a list with the any of the forms: [discrete, [x1, ..., xn], [y1, ..., yn]], [discrete, [[x1, y1], ..., [xn, ..., yn]]] or [parametric, x_expr, y_expr, t_range].

Displays a plot of one or more expressions as a function of one variable or parameter.

plot2d displays one or several plots in two dimensions. When expressions or function name are used to define the plots, they should all depend on only one variable var and the use of x_range will be mandatory, to provide the name of the variable and its minimum and maximum values; the syntax for x_range is: [variable, min, max].

A plot can also be defined in the discrete or parametric forms. The discrete form is used to plot a set of points with given coordinates. A discrete plot is defined by a list starting with the keyword discrete, followed by one or two lists of values. If two lists are given, they must have the same length; the first list will be interpreted as the x coordinates of the points to be plotted and the second list as the y coordinates. If only one list is given after the discrete keyword, each element on the list could also be a list with two values that correspond to the x and y coordinates of a point, or it could be a sequence of numerical values which will be plotted at consecutive integer values (1,2,3,...) on the x axis.

A parametric plot is defined by a list starting with the keyword parametric, followed by two expressions or function names and a range for the parameter. The range for the parameter must be a list with the name of the parameter followed by its minimum and maximum values: [param, min, max]. The plot will show the path traced out by the point with coordinates given by the two expressions or functions, as param increases from min to max.

A range for the vertical axis is an optional argument with the form: [y, min, max] (the keyword y is always used for the vertical axis). If that option is used, the plot will show that exact vertical range, independently of the values reached by the plot. If the vertical range is not specified, it will be set up according to the minimum and maximum values of the second coordinate of the plot points.

All other options should also be lists, starting with a keyword and followed by one or more values. See plot_options.

If there are several plots to be plotted, a legend will be written to identity each of the expressions. The labels that should be used in that legend can be given with the option legend. If that option is not used, Maxima will create labels from the expressions or function names.

Examples:

Plot of a common function:

(%i1) plot2d (sin(x), [x, -%pi, %pi])$
figures/plotting6

If the function grows too fast, it might be necessary to limit the values in the vertical axis using the y option:

(%i1) plot2d (sec(x), [x, -2, 2], [y, -20, 20])$
figures/plotting7

When the plot box is disabled, no labels are created for the axes. In that case, instead of using xlabel and ylabel to set the names of the axes, it is better to use option label, which allows more flexibility. Option yx_ratio is used to change the default rectangular shape of the plot; in this example the plot will fill a square.

(%i1) plot2d ( x^2 - 1, [x, -3, 3], [box, false], grid2d,
      [yx_ratio, 1], [axes, solid], [xtics, -2, 4, 2],
      [ytics, 2, 2, 6], [label, ["x", 2.9, -0.3],
      ["x^2-1", 0.1, 8]], [title, "A parabola"])$
figures/plotting8

A plot with a logarithmic scale in the vertical axis:

(%i1) plot2d (exp(3*s), [s, -2, 2], logy)$
figures/plotting9

Plotting functions by name:

(%i1) F(x) := x^2 $
(%i2) :lisp (defun |$g| (x) (m* x x x))
$g
(%i2) H(x) := if x < 0 then x^4 - 1 else 1 - x^5 $
(%i3) plot2d ([F, G, H], [u, -1, 1], [y, -1.5, 1.5])$
figures/plotting10

A plot of the butterfly curve, defined parametrically:

(%i1) r: (exp(cos(t))-2*cos(4*t)-sin(t/12)^5)$
(%i2) plot2d([parametric, r*sin(t), r*cos(t), [t, -8*%pi, 8*%pi]])$
figures/plotting11

Plot of a circle, using its parametric representation, together with the function -|x|. The circle will only look like a circle if the scale in the two axes is the same, which is done with the option same_xy.

(%i1) plot2d([[parametric, cos(t), sin(t), [t,0,2*%pi]], -abs(x)],
         [x, -sqrt(2), sqrt(2)], same_xy)$
figures/plotting12

A plot of 200 random numbers between 0 and 9:

(%i1) plot2d ([discrete, makelist ( random(10), 200)])$
figures/plotting13

A plot of a discrete set of points, defining x and y coordinates separately:

(%i1) plot2d ([discrete, makelist(i*%pi, i, 1, 5),
                            [0.6, 0.9, 0.2, 1.3, 1]])$
figures/plotting14

In the next example a table with three columns is saved in a file “data.txt” which is then read and the second and third column are plotted on the two axes:

(%i1) with_stdout ("data.txt", for x:0 thru 10 do
                             print (x, x^2, x^3))$
(%i2) data: read_matrix ("data.txt")$
(%i3) plot2d ([discrete, transpose(data)[2], transpose(data)[3]],
  [style,points], [point_type,diamond], [color,red])$
figures/plotting15

A plot of discrete data points together with a continuous function:

(%i1) xy: [[10, .6], [20, .9], [30, 1.1], [40, 1.3], [50, 1.4]]$
(%i2) plot2d([[discrete, xy], 2*%pi*sqrt(l/980)], [l,0,50],
        [style, points, lines], [color, red, blue],
        [point_type, asterisk],
        [legend, "experiment", "theory"],
        [xlabel, "pendulum's length (cm)"],
        [ylabel, "period (s)"])$
figures/plotting16

See also the section about Plotting Options.

Function: plot3d
    plot3d (expr, x_range, y_range, …, options, …)
    plot3d ([expr_1, …, expr_n], x_range, y_range, …, options, …)

Displays a plot of one or more surfaces defined as functions of two variables or in parametric form.

The functions to be plotted may be specified as expressions or function names. The mouse can be used to rotate the plot looking at the surface from different sides.

Examples:

Plot of a function of two variables:

(%i1) plot3d (u^2 - v^2, [u, -2, 2], [v, -3, 3], [grid, 100, 100],
        [mesh_lines_color,false])$
figures/plotting17

Use of the z option to limit a function that goes to infinity (in this case the function is minus infinity on the x and y axes); this also shows how to plot with only lines and no shading:

(%i1) plot3d ( log ( x^2*y^2 ), [x, -2, 2], [y, -2, 2], [z, -8, 4],
         [palette, false], [color, magenta])$
figures/plotting18

The infinite values of z can also be avoided by choosing a grid that does not fall on any points where the function is undefined, as in the next example, which also shows how to change the palette and how to include a color bar that relates colors to values of the z variable:

(%i1) plot3d (log (x^2*y^2), [x, -2, 2], [y, -2, 2],[grid, 29, 29],
       [palette, [gradient, red, orange, yellow, green]],
       color_bar, [xtics, 1], [ytics, 1], [ztics, 4],
       [color_bar_tics, 4])$
figures/plotting19

Two surfaces in the same plot. Ranges specific to one of the surfaces can be given by placing each expression and its ranges in a separate list; global ranges for the complete plot are also given after the functions definitions.

(%i1) plot3d ([[-3*x - y, [x, -2, 2], [y, -2, 2]],
   4*sin(3*(x^2 + y^2))/(x^2 + y^2), [x, -3, 3], [y, -3, 3]],
   [x, -4, 4], [y, -4, 4])$
figures/plotting20

Plot of a Klein bottle, defined parametrically:

(%i1) expr_1: 5*cos(x)*(cos(x/2)*cos(y)+sin(x/2)*sin(2*y)+3)-10$
(%i2) expr_2: -5*sin(x)*(cos(x/2)*cos(y)+sin(x/2)*sin(2*y)+3)$
(%i3) expr_3: 5*(-sin(x/2)*cos(y)+cos(x/2)*sin(2*y))$
(%i4) plot3d ([expr_1, expr_2, expr_3], [x, -%pi, %pi],
        [y, -%pi, %pi], [grid, 50, 50])$
figures/plotting21

Plot of a “spherical harmonic” function, using the predefined transformation, spherical_to_xyz to transform from spherical coordinates to rectangular coordinates. See the documentation for spherical_to_xyz.

(%i1) plot3d (sin(2*theta)*cos(phi), [theta, 0, %pi],
        [phi, 0, 2*%pi],
        [transform_xy, spherical_to_xyz], [grid,30,60],
   [legend,false])$
figures/plotting22

Use of the pre-defined function polar_to_xy to transform from cylindrical to rectangular coordinates. See the documentation for polar_to_xy.

(%i1) plot3d (r^.33*cos(th/3), [r,0,1], [th,0,6*%pi], [box, false],
   [grid, 12, 80], [transform_xy, polar_to_xy], [legend, false])$
figures/plotting23

Plot of a sphere using the transformation from spherical to rectangular coordinates. Option same_xyz is used to get the three axes scaled in the same proportion. When transformations are used, it is not convenient to eliminate the mesh lines, because Gnuplot will not show the surface correctly.

(%i1) plot3d ( 5, [theta, 0, %pi], [phi, 0, 2*%pi], same_xyz,
  [transform_xy, spherical_to_xyz], [mesh_lines_color,blue],
  [palette,[gradient,"#1b1b4e", "#8c8cf8"]], [legend, false])$
figures/plotting24

Definition of a function of two-variables using a matrix. Notice the single quote in the definition of the function, to prevent plot3d from failing when it realizes that the matrix will require integer indices.

(%i1) M: matrix([1,2,3,4], [1,2,3,2], [1,2,3,4], [1,2,3,3])$
(%i2) f(x, y) := float('M [round(x), round(y)])$
(%i3) plot3d (f(x,y), [x,1,4],[y,1,4],[grid,3,3],[legend,false])$
figures/plotting25

By setting the elevation equal to zero, a surface can be seen as a map in which each color represents a different level.

(%i1) plot3d (cos (-x^2 + y^3/4), [x,-4,4], [y,-4,4], [zlabel,""],
       [mesh_lines_color,false], [elevation,0], [azimuth,0],
       color_bar, [grid,80,80], [ztics,false], [color_bar_tics,1])$
figures/plotting26

See also the section about Plotting Options.

System variable: plot_options

This option is being kept for compatibility with older versions, but its use is deprecated. To set global plotting options, see their current values or remove options, use set_plot_option, get_plot_option and remove_plot_option.

Function: remove_plot_option (name)

Removes the default value of an option. The name of the option must be given.

See also set_plot_option, get_plot_option and the section on Plotting Options.

Function: set_plot_option (option)

Accepts any of the options listed in the section Plotting Options, and saves them for use in plotting commands. The values of the options set in each plotting command will have precedence, but if those options are not given, the default values set with this function will be used.

set_plot_option evaluates its argument and returns the complete list of options (after modifying the option given). If called without any arguments, it will simply show the list of current default options.

See also remove_plot_option, get_plot_option and the section on Plotting Options.

Example:

Modification of the grid values.

(%i1) set_plot_option ([grid, 30, 40]);
(%o1) [[plot_format, gnuplot_pipes], [grid, 30, 40], 
[run_viewer, true], [axes, true], [nticks, 29], [adapt_depth, 5], 
[color, blue, red, green, magenta, black, cyan], 
[point_type, bullet, box, triangle, plus, times, asterisk], 
[palette, [gradient, green, cyan, blue, violet], 
[gradient, magenta, violet, blue, cyan, green, yellow, orange, 
red, brown, black]], [gnuplot_preamble, ], [gnuplot_term, default]]

System function: spherical_to_xyz

It can be given as value for the transform_xy option of plot3d. Its effect will be to interpret the two independent variables and the function in plot3d as the spherical coordinates of a point (first, the angle with the z axis, then the angle of the xy projection with the x axis and finally the distance from the origin) and transform them into x, y and z coordinates.


Next: , Previous: , Up: Plotting   [Contents][Index]

12.4 Plotting Options

All options consist of a list starting with one of the keywords in this section, followed by one or more values. Some of the options may have different effects in different plotting commands as it will be pointed out in the following list. The options that accept among their possible values true or false, can also be set to true by simply writing their names. For instance, typing logx as an option is equivalent to writing [logx, true].

Plot option: adapt_depth [adapt_depth, integer]

Default value: 5

The maximum number of splittings used by the adaptive plotting routine.

Plot option: axes [axes, symbol]

Default value: true

Where symbol can be either true, false, x, y or solid. If false, no axes are shown; if equal to x or y only the x or y axis will be shown; if it is equal to true, both axes will be shown and solid will show the two axes with a solid line, rather than the default broken line. This option does not have any effect in the 3 dimensional plots.

Plot option: azimuth [azimuth, number]

Default value: 30

A plot3d plot can be thought of as starting with the x and y axis in the horizontal and vertical axis, as in plot2d, and the z axis coming out of the screen. The z axis is then rotated around the x axis through an angle equal to elevation and then the new xy plane is rotated around the new z axis through an angle azimuth. This option sets the value for the azimuth, in degrees.

See also elevation.

Plot option: box [box, symbol]

Default value: true

If set to true, a bounding box will be drawn for the plot; if set to false, no box will be drawn.

Plot option: color [color, color_1, …, color_n]

In 2d plots it defines the color (or colors) for the various curves. In plot3d, it defines the colors used for the mesh lines of the surfaces, when no palette is being used.

If there are more curves or surfaces than colors, the colors will be repeated in sequence. The valid colors are red, green, blue, magenta, cyan, yellow, orange, violet, brown, gray, black, white, or a string starting with the character # and followed by six hexadecimal digits: two for the red component, two for green component and two for the blue component. If the name of a given color is unknown color, black will be used instead.

Plot option: color_bar [color_bar, symbol]

Default value: false in plot3d, true in mandelbrot and julia

Where symbol can be either true or false. If true, whenever plot3d, mandelbrot or julia use a palette to represent different values, a box will be shown on the right, showing the corresponding between colors and values.

Plot option: color_bar_tics [color_bar_tics, x1, x2, x3]

Defines the values at which a mark and a number will be placed in the color bar. The first number is the initial value, the second the increments and the third is the last value where a mark is placed. The second and third numbers can be omitted. When only one number is given, it will be used as the increment from an initial value that will be chosen automatically.

Plot option: elevation [elevation, number]

Default value: 60

A plot3d plot can be thought of as starting with the x and y axis in the horizontal and vertical axis, as in plot2d, and the z axis coming out of the screen. The z axis is then rotated around the x axis through an angle equal to elevation and then the new xy plane is rotated around the new z axis through an angle azimuth. This option sets the value for the azimuth, in degrees.

See also azimuth.

Plot option: grid [grid, integer, integer]

Default value: 30, 30

Sets the number of grid points to use in the x- and y-directions for three-dimensional plotting or for the julia and mandelbrot programs.

For a way to actually draw a grid See grid2d.

Plot option: grid2d [grid, value]

Default value: false

Shows a grid of lines on the xy plane. The points where the grid lines are placed are the same points where tics are marked in the x and y axes, which can be controlled with the xtics and ytics options.

See also grid.

Plot option: iterations [grid, value]

Default value: 9

Number of iterations made by the programs mandelbrot and julia.

Plot option: label [label, [string, x, y], …]

Writes one or several labels in the points with x, y coordinates indicated after each label.

Plot option: legend
    legend [legend, string_1, …, string_n]
    legend [legend, false]

It specifies the labels for the plots when various plots are shown. If there are more plots than the number of labels given, they will be repeated. If given the value false, no legends will be shown. By default, the names of the expressions or functions will be used, or the words discrete1, discrete2, …, for discrete sets of points.

Plot option: logx [logx, value]

Makes the horizontal axes to be scaled logarithmically. It can be either true or false.

Plot option: logy [logy, value]

Makes the vertical axes to be scaled logarithmically. It can be either true or false.

Plot option: mesh_lines_color [mesh_lines_color, color]

Default value: black

It sets the color used by plot3d to draw the mesh lines, when a palette is being used. It accepts the same colors as for the option color (see the list of allowed colors in color). It can also be given a value false to eliminate completely the mesh lines.

Plot option: nticks [nticks, integer]

Default value: 29

When plotting functions with plot2d, it is gives the initial number of points used by the adaptive plotting routine for plotting functions. When plotting parametric functions with plot3d, it sets the number of points that will be shown for the plot.

Plot option: palette
    palette [palette, [palette_1], …, [palette_n]]
    palette [palette, false]

It can consist of one palette or a list of several palettes. Each palette is a list with a keyword followed by values. If the keyword is gradient, it should be followed by a list of valid colors.

If the keyword is hue, saturation or value, it must be followed by 4 numbers. The first three numbers, which must be between 0 and 1, define the hue, saturation and value of a basic color to be assigned to the minimum value of z. The keyword specifies which of the three attributes (hue, saturation or value) will be increased according to the values of z. The last number indicates the increase corresponding to the maximum value of z. That last number can be bigger than 1 or negative; the corresponding values of the modified attribute will be rounded modulo 1.

Gnuplot only uses the first palette in the list; xmaxima will use the palettes in the list sequentially, when several surfaces are plotted together; if the number of palettes is exhausted, they will be repeated sequentially.

The color of the mesh lines will be given by the option mesh_lines_color. If palette is given the value false, the surfaces will not be shaded but represented with a mesh of curves only. In that case, the colors of the lines will be determined by the option color.

Plot option: plot_format [plot_format, format]

Default value: gnuplot, in Windows systems, or gnuplot_pipes in other systems.

Where format is one of the following: gnuplot, xmaxima, mgnuplot, gnuplot_pipes or geomview.

It sets the format to be used for plotting.

Plot option: plot_realpart [plot_realpart, symbol]

Default value: false

If set to true, the functions to be plotted will be considered as complex functions whose real value should be plotted; this is equivalent to plotting realpart(function). If set to false, nothing will be plotted when the function does not give a real value. For instance, when x is negative, log(x) gives a complex value, with real value equal to log(abs(x)); if plot_realpart were true, log(-5) would be plotted as log(5), while nothing would be plotted if plot_realpart were false.

Plot option: point_type [point_type, type_1, …, type_n]

In gnuplot, each set of points to be plotted with the style “points” or “linespoints” will be represented with objects taken from this list, in sequential order. If there are more sets of points than objects in this list, they will be repeated sequentially. The possible objects that can be used are: bullet, circle, plus, times, asterisk, box, square, triangle, delta, wedge, nabla, diamond, lozenge.

Plot option: pdf_file [pdf_file, file_name]

Saves the plot into a PDF file with name equal to file_name, rather than showing it in the screen. By default, the file will be created in the directory defined by the variable maxima_tempdir, unless file_name contains the character “/”, in which case it will be assumed to contain the complete path where the file should be created. The value of maxima_tempdir can be changed to save the file in a different directory. When the option gnuplot_pdf_term_command is also given, it will be used to set up Gnuplot’s PDF terminal; otherwise, Gnuplot’s pdfcairo terminal will be used with solid colored lines of width 3, plot size of 17.2 cm by 12.9 cm and font of 18 points.

Plot option: png_file [png_file, file_name]

Saves the plot into a PNG graphics file with name equal to file_name, rather than showing it in the screen. By default, the file will be created in the directory defined by the variable maxima_tempdir, unless file_name contains the character “/”, in which case it will be assumed to contain the complete path where the file should be created. The value of maxima_tempdir can be changed to save the file in a different directory. When the option gnuplot_png_term_command is also given, it will be used to set up Gnuplot’s PNG terminal; otherwise, Gnuplot’s pngcairo terminal will be used, with a font of size 12.

Plot option: ps_file [ps_file, file_name]

Saves the plot into a Postscript file with name equal to file_name, rather than showing it in the screen. By default, the file will be created in the directory defined by the variable maxima_tempdir, unless file_name contains the character “/”, in which case it will be assumed to contain the complete path where the file should be created. The value of maxima_tempdir can be changed to save the file in a different directory. When the option gnuplot_ps_term_command is also given, it will be used to set up Gnuplot’s Postscript terminal; otherwise, Gnuplot’s postscript terminal will be used with the EPS option, solid colored lines of width 2, plot size of 16.4 cm by 12.3 cm and font of 24 points.

Plot option: run_viewer [run_viewer, symbol]

This option is only used when the plot format is gnuplot and the terminal is default or when the Gnuplot terminal is set to dumb (see gnuplot_term) and can have a true or false value.

If the terminal is default, a file maxout_xxx.gnuplot (or other name specified with gnuplot_out_file) is created with the gnuplot commands necessary to generate the plot. Option run_viewer controls whether or not Gnuplot will be launched to execute those commands and show the plot.

If the terminal is default, gnuplot is run to execute the commands in maxout_xxx.gnuplot, producing another file maxplot.txt (or other name specified with gnuplot_out_file). Option run_viewer controls whether or not that file, with an ASCII representation of the plot, will be shown in the Maxima or Xmaxima console.

The default value for this option is true, making the plots to be shown in either the console or a separate graphics window.

Plot option: same_xy [same_xy , value]

It can be either true or false. If true, the scales used in the x and y axes will be the same, in either 2d or 3d plots. See also yx_ratio.

Plot option: same_xyz [same_xyz , value]

It can be either true or false. If true, the scales used in the 3 axes of a 3d plot will be the same.

Plot option: style
    style [style, type_1, …, type_n]
    style [style, [style_1], …, [style_n]]

The styles that will be used for the various functions or sets of data in a 2d plot. The word style must be followed by one or more styles. If there are more functions and data sets than the styles given, the styles will be repeated. Each style can be either lines for line segments, points for isolated points, linespoints for segments and points, or dots for small isolated dots. Gnuplot accepts also an impulses style.

Each of the styles can be enclosed inside a list with some additional parameters. lines accepts one or two numbers: the width of the line and an integer that identifies a color. The default color codes are: 1: blue, 2: red, 3: magenta, 4: orange, 5: brown, 6: lime and 7: aqua. If you use Gnuplot with a terminal different than X11, those colors might be different; for example, if you use the option [gnuplot_term, ps], color index 4 will correspond to black, instead of orange.

points accepts one two or three parameters; the first parameter is the radius of the points, the second parameter is an integer that selects the color, using the same code used for lines and the third parameter is currently used only by Gnuplot and it corresponds to several objects instead of points. The default types of objects are: 1: filled circles, 2: open circles, 3: plus signs, 4: x, 5: *, 6: filled squares, 7: open squares, 8: filled triangles, 9: open triangles, 10: filled inverted triangles, 11: open inverted triangles, 12: filled lozenges and 13: open lozenges.

linespoints accepts up to four parameters: line width, points radius, color and type of object to replace the points.

See also color and point_type.

Plot option: svg_file [svg_file, file_name]

Saves the plot into an SVG file with name equal to file_name, rather than showing it in the screen. By default, the file will be created in the directory defined by the variable maxima_tempdir, unless file_name contains the character “/”, in which case it will be assumed to contain the complete path where the file should be created. The value of maxima_tempdir can be changed to save the file in a different directory. When the option gnuplot_svg_term_command is also given, it will be used to set up Gnuplot’s SVG terminal; otherwise, Gnuplot’s svg terminal will be used with font of 14 points.

Plot option: t [t, min, max]

Default range for parametric plots.

Plot option: title [title, text]

Defines a title that will be written at the top of the plot.

Plot option: transform_xy [transform_xy, symbol]

Where symbol is either false or the result obtained by using the function transform_xy. If different from false, it will be used to transform the 3 coordinates in plot3d.

See make_transform, polar_to_xy and spherical_to_xyz.

Plot option: x [x, min, max]

When used as the first option in a plot2d command (or any of the first two in plot3d), it indicates that the first independent variable is x and it sets its range. It can also be used again after the first option (or after the second option in plot3d) to define the effective horizontal domain that will be shown in the plot.

Plot option: xlabel [xlabel, string]

Specifies the string that will label the first axis; if this option is not used, that label will be the name of the independent variable, when plotting functions with plot2d or implicit_plot, or the name of the first variable, when plotting surfaces with plot3d or contours with contour_plot, or the first expression in the case of a parametric plot. It can not be used with set_plot_option.

Plot option: xtics [xtics, x1, x2, x3]

Defines the values at which a mark and a number will be placed in the x axis. The first number is the initial value, the second the increments and the third is the last value where a mark is placed. The second and third numbers can be omitted. When only one number is given, it will be used as the increment from an initial value that will be chosen automatically.

Plot option: xy_scale [xy_scale, sx, sy]

In a 2d plot, it defines the ratio of the total size of the Window to the size that will be used for the plot. The two numbers given as arguments are the scale factors for the x and y axes.

This option does not change the size of the graphic window or the placement of the graph in the window. If you want to change the aspect ratio of the plot, it is better to use option yx_ratio. For instance, [yx_ratio, 10] instead of [xy_scale, 0.1, 1].

Plot option: y [y, min, max]

When used as one of the first two options in plot3d, it indicates that one of the independent variables is y and it sets its range. Otherwise, it defines the effective domain of the second variable that will be shown in the plot.

Plot option: ylabel [ylabel, string]

Specifies the string that will label the second axis; if this option is not used, that label will be “y”, when plotting functions with plot2d or implicit_plot, or the name of the second variable, when plotting surfaces with plot3d or contours with contour_plot, or the second expression in the case of a parametric plot. It can not be used with set_plot_option.

Plot option: ytics [ytics, y1, y2, y3]

Defines the values at which a mark and a number will be placed in the y axis. The first number is the initial value, the second the increments and the third is the last value where a mark is placed. The second and third numbers can be omitted. When only one number is given, it will be used as the increment from an initial value that will be chosen automatically

Plot option: yx_ratio [yx_ratio, r]

In a 2d plot, the ratio between the vertical and the horizontal sides of the rectangle used to make the plot. See also same_xy.

Plot option: z [z, min, max]

Used in plot3d to set the effective range of values of z that will be shown in the plot.

Plot option: zlabel [zlabel, string]

Specifies the string that will label the third axis, when using plot3d. If this option is not used, that label will be “z”, when plotting surfaces, or the third expression in the case of a parametric plot. It can not be used with set_plot_option and it will be ignored by plot2d and implicit_plot.

Plot option: zmin [zmin, z]

In 3d plots, the value of z that will be at the bottom of the plot box.


Next: , Previous: , Up: Plotting   [Contents][Index]

12.5 Gnuplot Options

There are several plot options specific to gnuplot. All of them consist of a keyword (the name of the option), followed by a string that should be a valid gnuplot command, to be passed directly to gnuplot. In most cases, there exist a corresponding plotting option that will produce a similar result and whose use is more recommended than the gnuplot specific option.

Plot option: gnuplot_term [gnuplot_term, terminal_name]

Sets the output terminal type for gnuplot. The argument terminal_name can be a string or one of the following 3 special symbols

Plot option: gnuplot_out_file [gnuplot_out_file, file_name]

It can be used to replace the default name for the file that contains the commands that will interpreted by gnuplot, when the terminal is set to default, or to replace the default name of the graphic file that gnuplot creates, when the terminal is different from default. If it contains one or more slashes, “/”, the name of the file will be left as it is; otherwise, it will be appended to the path of the temporary directory. The complete name of the files created by the plotting commands is always sent as output of those commands so they can be seen if the command is ended by semi-colon.

When used in conjunction with the gnuplot_term option, it can be used to save the plot in a file, in one of the graphic formats supported by Gnuplot. To create PNG, PDF, Postscript or SVG, it is easier to use options png_file, pdf_file, ps_file, or svg_file.

Plot option: gnuplot_pm3d [gnuplot_pm3d, value]

With a value of false, it can be used to disable the use of PM3D mode, which is enabled by default.

Plot option: gnuplot_preamble [gnuplot_preamble, string]

This option inserts gnuplot commands before any other commands sent to Gnuplot. Any valid gnuplot commands may be used. Multiple commands should be separated with a semi-colon. See also gnuplot_postamble.

Plot option: gnuplot_postamble [gnuplot_postamble, string]

This option inserts gnuplot commands after other commands sent to Gnuplot and right before the plot command is sent. Any valid gnuplot commands may be used. Multiple commands should be separated with a semi-colon. See also gnuplot_preamble.

Plot option: gnuplot_default_term_command

[gnuplot_default_term_command, command]

The gnuplot command to set the terminal type for the default terminal. It this option is not set, the command used will be: "set term wxt size 640,480 font \",12\"; set term pop".

Plot option: gnuplot_dumb_term_command

[gnuplot_dumb_term_command, command]

The gnuplot command to set the terminal type for the dumb terminal. It this option is not set, the command used will be: "set term dumb 79 22", which makes the text output 79 characters by 22 characters.

Plot option: gnuplot_pdf_term_command [gnuplot_pdf_term_command, command]

The gnuplot command to set the terminal type for the PDF terminal. If this option is not set, the command used will be: "set term pdfcairo color solid lw 3 size 17.2 cm, 12.9 cm font \",18\"". See the gnuplot documentation for more information.

Plot option: gnuplot_png_term_command [gnuplot_png_term_command, command]

The gnuplot command to set the terminal type for the PNG terminal. If this option is not set, the command used will be: "set term pngcairo font \",12\"". See the gnuplot documentation for more information.

Plot option: gnuplot_ps_term_command [gnuplot_ps_term_command, command]

The gnuplot command to set the terminal type for the PostScript terminal. If this option is not set, the command used will be: "set term postscript eps color solid lw 2 size 16.4 cm, 12.3 cm font \",24\"". See the gnuplot documentation for set term postscript for more information.

Plot option: gnuplot_svg_term_command [gnuplot_svg_term_command, command]

The gnuplot command to set the terminal type for the SVG terminal. If this option is not set, the command used will be: "set term svg font \",14\"". See the gnuplot documentation for more information.

Plot option: gnuplot_curve_titles

This is an obsolete option that has been replaced legend described above.

Plot option: gnuplot_curve_styles

This is an obsolete option that has been replaced by style.


Previous: , Up: Plotting   [Contents][Index]

12.6 Gnuplot_pipes Format Functions

Function: gnuplot_start ()

Opens the pipe to gnuplot used for plotting with the gnuplot_pipes format. Is not necessary to manually open the pipe before plotting.

Function: gnuplot_close ()

Closes the pipe to gnuplot which is used with the gnuplot_pipes format.

Function: gnuplot_restart ()

Closes the pipe to gnuplot which is used with the gnuplot_pipes format and opens a new pipe.

Function: gnuplot_replot
    gnuplot_replot ()
    gnuplot_replot (s)

Updates the gnuplot window. If gnuplot_replot is called with a gnuplot command in a string s, then s is sent to gnuplot before reploting the window.

Function: gnuplot_reset ()

Resets the state of gnuplot used with the gnuplot_pipes format. To update the gnuplot window call gnuplot_replot after gnuplot_reset.


Next: , Previous: , Up: Top   [Contents][Index]

13 File Input and Output


Next: , Previous: , Up: File Input and Output   [Contents][Index]

13.1 Comments

A comment in Maxima input is any text between /* and */.

The Maxima parser treats a comment as whitespace for the purpose of finding tokens in the input stream; a token always ends at a comment. An input such as a/* foo */b contains two tokens, a and b, and not a single token ab. Comments are otherwise ignored by Maxima; neither the content nor the location of comments is stored in parsed input expressions.

Comments can be nested to arbitrary depth. The /* and */ delimiters form matching pairs. There must be the same number of /* as there are */.

Examples:

(%i1) /* aa is a variable of interest */  aa : 1234;
(%o1)                         1234
(%i2) /* Value of bb depends on aa */  bb : aa^2;
(%o2)                        1522756
(%i3) /* User-defined infix operator */  infix ("b");
(%o3)                           b
(%i4) /* Parses same as a b c, not abc */  a/* foo */b/* bar */c;
(%o4)                         a b c
(%i5) /* Comments /* can be nested /* to any depth */ */ */  1 + xyz;
(%o5)                        xyz + 1
·

Category: Syntax


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13.2 Files

A file is simply an area on a particular storage device which contains data or text. Files on the disks are figuratively grouped into "directories". A directory is just a list of files. Commands which deal with files are:

   appendfile           batch                 batchload
   closefile            file_output_append    filename_merge
   file_search          file_search_maxima    file_search_lisp
   file_search_demo     file_search_usage     file_search_tests
   file_type            file_type_lisp        file_type_maxima
   load                 load_pathname         loadfile
   loadprint            pathname_directory    pathname_name
   pathname_type        printfile             save
   stringout            with_stdout           writefile

When a file name is passed to functions like plot2d, save, or writefile and the file name does not include a path, Maxima stores the file in the current working directory. The current working directory depends on the system like Windows or Linux and on the installation.


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13.3 Functions and Variables for File Input and Output

Function: appendfile (filename)

Appends a console transcript to filename. appendfile is the same as writefile, except that the transcript file, if it exists, is always appended.

closefile closes the transcript file opened by appendfile or writefile.

Function: batch
    batch (filename)
    batch (filename, option)

batch(filename) reads Maxima expressions from filename and evaluates them. batch searches for filename in the list file_search_maxima. See also file_search.

batch(filename, demo) is like demo(filename). In this case batch searches for filename in the list file_search_demo. See demo.

batch(filename, test) is like run_testsuite with the option display_all=true. For this case batch searches filename in the list file_search_maxima and not in the list file_search_tests like run_testsuite. Furthermore, run_testsuite runs tests which are in the list testsuite_files. With batch it is possible to run any file in a test mode, which can be found in the list file_search_maxima. This is useful, when writing a test file.

filename comprises a sequence of Maxima expressions, each terminated with ; or $. The special variable % and the function %th refer to previous results within the file. The file may include :lisp constructs. Spaces, tabs, and newlines in the file are ignored. A suitable input file may be created by a text editor or by the stringout function.

batch reads each input expression from filename, displays the input to the console, computes the corresponding output expression, and displays the output expression. Input labels are assigned to the input expressions and output labels are assigned to the output expressions. batch evaluates every input expression in the file unless there is an error. If user input is requested (by asksign or askinteger, for example) batch pauses to collect the requisite input and then continue.

It may be possible to halt batch by typing control-C at the console. The effect of control-C depends on the underlying Lisp implementation.

batch has several uses, such as to provide a reservoir for working command lines, to give error-free demonstrations, or to help organize one’s thinking in solving complex problems.

batch evaluates its argument. batch returns the path of filename as a string, when called with no second argument or with the option demo. When called with the option test, the return value is a an empty list [] or a list with filename and the numbers of the tests which have failed.

See also load, batchload, and demo.

Function: batchload (filename)

Reads Maxima expressions from filename and evaluates them, without displaying the input or output expressions and without assigning labels to output expressions. Printed output (such as produced by print or describe)) is displayed, however.

The special variable % and the function %th refer to previous results from the interactive interpreter, not results within the file. The file cannot include :lisp constructs.

batchload returns the path of filename, as a string. batchload evaluates its argument.

See also batch, and load.

Function: closefile ()

Closes the transcript file opened by writefile or appendfile.

Option variable: file_output_append

Default value: false

file_output_append governs whether file output functions append or truncate their output file. When file_output_append is true, such functions append to their output file. Otherwise, the output file is truncated.

save, stringout, and with_stdout respect file_output_append. Other functions which write output files do not respect file_output_append. In particular, plotting and translation functions always truncate their output file, and tex and appendfile always append.

Function: filename_merge (path, filename)

Constructs a modified path from path and filename. If the final component of path is of the form ###.something, the component is replaced with filename.something. Otherwise, the final component is simply replaced by filename.

The result is a Lisp pathname object.

Function: file_search
    file_search (filename)
    file_search (filename, pathlist)

file_search searches for the file filename and returns the path to the file (as a string) if it can be found; otherwise file_search returns false. file_search (filename) searches in the default search directories, which are specified by the file_search_maxima, file_search_lisp, and file_search_demo variables.

file_search first checks if the actual name passed exists, before attempting to match it to “wildcard” file search patterns. See file_search_maxima concerning file search patterns.

The argument filename can be a path and file name, or just a file name, or, if a file search directory includes a file search pattern, just the base of the file name (without an extension). For example,

file_search ("/home/wfs/special/zeta.mac");
file_search ("zeta.mac");
file_search ("zeta");

all find the same file, assuming the file exists and /home/wfs/special/###.mac is in file_search_maxima.

file_search (filename, pathlist) searches only in the directories specified by pathlist, which is a list of strings. The argument pathlist supersedes the default search directories, so if the path list is given, file_search searches only the ones specified, and not any of the default search directories. Even if there is only one directory in pathlist, it must still be given as a one-element list.

The user may modify the default search directories. See file_search_maxima.

file_search is invoked by load with file_search_maxima and file_search_lisp as the search directories.

Option variable: file_search_maxima
Option variable: file_search_lisp
Option variable: file_search_demo
Option variable: file_search_usage
Option variable: file_search_tests

These variables specify lists of directories to be searched by load, demo, and some other Maxima functions. The default values of these variables name various directories in the Maxima installation.

The user can modify these variables, either to replace the default values or to append additional directories. For example,

file_search_maxima: ["/usr/local/foo/###.mac",
    "/usr/local/bar/###.mac"]$

replaces the default value of file_search_maxima, while

file_search_maxima: append (file_search_maxima,
    ["/usr/local/foo/###.mac", "/usr/local/bar/###.mac"])$

appends two additional directories. It may be convenient to put such an expression in the file maxima-init.mac so that the file search path is assigned automatically when Maxima starts. See also Introduction for Runtime Environment.

Multiple filename extensions and multiple paths can be specified by special “wildcard” constructions. The string ### expands into the sought-after name, while a comma-separated list enclosed in curly braces {foo,bar,baz} expands into multiple strings. For example, supposing the sought-after name is neumann,

"/home/{wfs,gcj}/###.{lisp,mac}"

expands into /home/wfs/neumann.lisp, /home/gcj/neumann.lisp, /home/wfs/neumann.mac, and /home/gcj/neumann.mac.

Function: file_type (filename)

Returns a guess about the content of filename, based on the filename extension. filename need not refer to an actual file; no attempt is made to open the file and inspect the content.

The return value is a symbol, either object, lisp, or maxima. If the extension is matches one of the values in file_type_maxima, file_type returns maxima. If the extension matches one of the values in file_type_lisp, file_type returns lisp. If none of the above, file_type returns object.

See also pathname_type.

See file_type_maxima and file_type_lisp for the default values.

Examples:

(%i2) map('file_type,
          ["test.lisp", "test.mac", "test.dem", "test.txt"]);
(%o2)            [lisp, maxima, maxima, object]

Option variable: file_type_lisp

Default value: [l, lsp, lisp]

file_type_lisp is a list of file extensions that maxima recognizes as denoting a Lisp source file.

See also file_type.

Option variable: file_type_maxima

Default value: [mac, mc, demo, dem, dm1, dm2, dm3, dmt, wxm]

file_type_maxima is a list of file extensions that maxima recognizes as denoting a Maxima source file.

See also file_type.

Function: load (filename)

Evaluates expressions in filename, thus bringing variables, functions, and other objects into Maxima. The binding of any existing object is clobbered by the binding recovered from filename. To find the file, load calls file_search with file_search_maxima and file_search_lisp as the search directories. If load succeeds, it returns the name of the file. Otherwise load prints an error message.

load works equally well for Lisp code and Maxima code. Files created by save, translate_file, and compile_file, which create Lisp code, and stringout, which creates Maxima code, can all be processed by load. load calls loadfile to load Lisp files and batchload to load Maxima files.

load does not recognize :lisp constructs in Maxima files, and while processing filename, the global variables _, __, %, and %th have whatever bindings they had when load was called.

It is also to note that structures will only be read back as structures if they have been defined by defstruct before the load command is called.

See also loadfile, batch, batchload, and demo. loadfile processes Lisp files; batch, batchload, and demo process Maxima files.

See file_search for more detail about the file search mechanism.

load evaluates its argument.

System variable: load_pathname

Default value: false

When a file is loaded with the functions load, loadfile or batchload the system variable load_pathname is bound to the pathname of the file which is processed.

The variable load_pathname can be accessed from the file during the loading.

Example:

Suppose we have a batchfile test.mac in the directory

"/home/dieter/workspace/mymaxima/temp/" with the following commands

print("The value of load_pathname is: ", load_pathname)$
print("End of batchfile")$

then we get the following output

(%i1) load("/home/dieter/workspace/mymaxima/temp/test.mac")$
The value of load_pathname is:  
                   /home/dieter/workspace/mymaxima/temp/test.mac 
End of batchfile

Function: loadfile (filename)

Evaluates Lisp expressions in filename. loadfile does not invoke file_search, so filename must include the file extension and as much of the path as needed to find the file.

loadfile can process files created by save, translate_file, and compile_file. The user may find it more convenient to use load instead of loadfile.

Option variable: loadprint

Default value: true

loadprint tells whether to print a message when a file is loaded.

Function: directory (path)

Returns a list of the files and directories found in path in the file system.

path may contain wildcard characters (i.e., characters which represent unspecified parts of the path), which include at least the asterisk on most systems, and possibly other characters, depending on the system.

directory relies on the Lisp function DIRECTORY, which may have implementation-specific behavior.

Function: pathname_directory (pathname)
Function: pathname_name (pathname)
Function: pathname_type (pathname)

These functions return the components of pathname.

Examples:

(%i1) pathname_directory("/home/dieter/maxima/changelog.txt");
(%o1)                 /home/dieter/maxima/
(%i2) pathname_name("/home/dieter/maxima/changelog.txt");
(%o2)                       changelog
(%i3) pathname_type("/home/dieter/maxima/changelog.txt");
(%o3)                          txt

Function: printfile (path)

Prints the file named by path to the console. path may be a string or a symbol; if it is a symbol, it is converted to a string.

If path names a file which is accessible from the current working directory, that file is printed to the console. Otherwise, printfile attempts to locate the file by appending path to each of the elements of file_search_usage via filename_merge.

printfile returns path if it names an existing file, or otherwise the result of a successful filename merge.

Function: save
    save (filename, name_1, name_2, name_3, …)
    save (filename, values, functions, labels, …)
    save (filename, [m, n])
    save (filename, name_1=expr_1, …)
    save (filename, all)
    save (filename, name_1=expr_1, name_2=expr_2, …)

Stores the current values of name_1, name_2, name_3, …, in filename. The arguments are the names of variables, functions, or other objects. If a name has no value or function associated with it, it is ignored. save returns filename.

save stores data in the form of Lisp expressions. If filename ends in .lisp the data stored by save may be recovered by load (filename). See load.

The global flag file_output_append governs whether save appends or truncates the output file. When file_output_append is true, save appends to the output file. Otherwise, save truncates the output file. In either case, save creates the file if it does not yet exist.

The special form save (filename, values, functions, labels, ...) stores the items named by values, functions, labels, etc. The names may be any specified by the variable infolists. values comprises all user-defined variables.

The special form save (filename, [m, n]) stores the values of input and output labels m through n. Note that m and n must be literal integers. Input and output labels may also be stored one by one, e.g., save ("foo.1", %i42, %o42). save (filename, labels) stores all input and output labels. When the stored labels are recovered, they clobber existing labels.

The special form save (filename, name_1=expr_1, name_2=expr_2, ...) stores the values of expr_1, expr_2, …, with names name_1, name_2, … It is useful to apply this form to input and output labels, e.g., save ("foo.1", aa=%o88). The right-hand side of the equality in this form may be any expression, which is evaluated. This form does not introduce the new names into the current Maxima environment, but only stores them in filename.

These special forms and the general form of save may be mixed at will. For example, save (filename, aa, bb, cc=42, functions, [11, 17]).

The special form save (filename, all) stores the current state of Maxima. This includes all user-defined variables, functions, arrays, etc., as well as some automatically defined items. The saved items include system variables, such as file_search_maxima or showtime, if they have been assigned new values by the user; see myoptions.

save evaluates filename and quotes all other arguments.

Function: stringout
    stringout (filename, expr_1, expr_2, expr_3, …)
    stringout (filename, [m, n])
    stringout (filename, input)
    stringout (filename, functions)
    stringout (filename, values)

stringout writes expressions to a file in the same form the expressions would be typed for input. The file can then be used as input for the batch or demo commands, and it may be edited for any purpose. stringout can be executed while writefile is in progress.

The global flag file_output_append governs whether stringout appends or truncates the output file. When file_output_append is true, stringout appends to the output file. Otherwise, stringout truncates the output file. In either case, stringout creates the file if it does not yet exist.

The general form of stringout writes the values of one or more expressions to the output file. Note that if an expression is a variable, only the value of the variable is written and not the name of the variable. As a useful special case, the expressions may be input labels (%i1, %i2, %i3, …) or output labels (%o1, %o2, %o3, …).

If grind is true, stringout formats the output using the grind format. Otherwise the string format is used. See grind and string.

The special form stringout (filename, [m, n]) writes the values of input labels m through n, inclusive.

The special form stringout (filename, input) writes all input labels to the file.

The special form stringout (filename, functions) writes all user-defined functions (named by the global list functions)) to the file.

The special form stringout (filename, values) writes all user-assigned variables (named by the global list values)) to the file. Each variable is printed as an assignment statement, with the name of the variable, a colon, and its value. Note that the general form of stringout does not print variables as assignment statements.

Function: with_stdout
    with_stdout (f, expr_1, expr_2, expr_3, …)
    with_stdout (s, expr_1, expr_2, expr_3, …)

Evaluates expr_1, expr_2, expr_3, … and writes any output thus generated to a file f or output stream s. The evaluated expressions are not written to the output. Output may be generated by print, display, grind, among other functions.

The global flag file_output_append governs whether with_stdout appends or truncates the output file f. When file_output_append is true, with_stdout appends to the output file. Otherwise, with_stdout truncates the output file. In either case, with_stdout creates the file if it does not yet exist.

with_stdout returns the value of its final argument.

See also writefile.

(%i1) with_stdout ("tmp.out", for i:5 thru 10 do
      print (i, "! yields", i!))$
(%i2) printfile ("tmp.out")$
5 ! yields 120 
6 ! yields 720 
7 ! yields 5040 
8 ! yields 40320 
9 ! yields 362880 
10 ! yields 3628800

Function: writefile (filename)

Begins writing a transcript of the Maxima session to filename. All interaction between the user and Maxima is then recorded in this file, just as it appears on the console.

As the transcript is printed in the console output format, it cannot be reloaded into Maxima. To make a file containing expressions which can be reloaded, see save and stringout. save stores expressions in Lisp form, while stringout stores expressions in Maxima form.

The effect of executing writefile when filename already exists depends on the underlying Lisp implementation; the transcript file may be clobbered, or the file may be appended. appendfile always appends to the transcript file.

It may be convenient to execute playback after writefile to save the display of previous interactions. As playback displays only the input and output variables (%i1, %o1, etc.), any output generated by a print statement in a function (as opposed to a return value) is not displayed by playback.

closefile closes the transcript file opened by writefile or appendfile.


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13.4 Functions and Variables for TeX Output

Note that the built-in TeX output functionality of wxMaxima makes no use of the functions described here but uses its own implementation instead.

Function: tex
    tex (expr)
    tex (expr, destination)
    tex (expr, false)
    tex (label)
    tex (label, destination)
    tex (label, false)

Prints a representation of an expression suitable for the TeX document preparation system. The result is a fragment of a document, which can be copied into a larger document but not processed by itself.

tex (expr) prints a TeX representation of expr on the console.

tex (label) prints a TeX representation of the expression named by label and assigns it an equation label (to be displayed to the left of the expression). The TeX equation label is the same as the Maxima label.

destination may be an output stream or file name. When destination is a file name, tex appends its output to the file. The functions openw and opena create output streams.

tex (expr, false) and tex (label, false) return their TeX output as a string.

tex evaluates its first argument after testing it to see if it is a label. Quote-quote '' forces evaluation of the argument, thereby defeating the test and preventing the label.

See also texput.

Examples:

(%i1) integrate (1/(1+x^3), x);
                                    2 x - 1
                  2            atan(-------)
             log(x  - x + 1)        sqrt(3)    log(x + 1)
(%o1)      - --------------- + ------------- + ----------
                    6             sqrt(3)          3
(%i2) tex (%o1);
$$-{{\log \left(x^2-x+1\right)}\over{6}}+{{\arctan \left({{2\,x-1
 }\over{\sqrt{3}}}\right)}\over{\sqrt{3}}}+{{\log \left(x+1\right)
 }\over{3}}\leqno{\tt (\%o1)}$$
(%o2)                          (\%o1)
(%i3) tex (integrate (sin(x), x));
$$-\cos x$$
(%o3)                           false
(%i4) tex (%o1, "foo.tex");
(%o4)                          (\%o1)

tex (expr, false) returns its TeX output as a string.

(%i1) S : tex (x * y * z, false);
(%o1) $$x\,y\,z$$
(%i2) S;
(%o2) $$x\,y\,z$$

Function: tex1 (e)

Returns a string which represents the TeX output for the expressions e. The TeX output is not enclosed in delimiters for an equation or any other environment.

Examples:

(%i1) tex1 (sin(x) + cos(x));
(%o1)                     \sin x+\cos x
Function: texput
    texput (a, s)
    texput (a, f)
    texput (a, s, operator_type)
    texput (a, [s_1, s_2], matchfix)
    texput (a, [s_1, s_2, s_3], matchfix)

Assign the TeX output for the atom a, which can be a symbol or the name of an operator.

texput (a, s) causes the tex function to interpolate the string s into the TeX output in place of a.

texput (a, f) causes the tex function to call the function f to generate TeX output. f must accept one argument, which is an expression which has operator a, and must return a string (the TeX output). f may call tex1 to generate TeX output for the arguments of the input expression.

texput (a, s, operator_type), where operator_type is prefix, infix, postfix, nary, or nofix, causes the tex function to interpolate s into the TeX output in place of a, and to place the interpolated text in the appropriate position.

texput (a, [s_1, s_2], matchfix) causes the tex function to interpolate s_1 and s_2 into the TeX output on either side of the arguments of a. The arguments (if more than one) are separated by commas.

texput (a, [s_1, s_2, s_3], matchfix) causes the tex function to interpolate s_1 and s_2 into the TeX output on either side of the arguments of a, with s_3 separating the arguments.

Examples:

Assign TeX output for a variable.

(%i1) texput (me,"\\mu_e");
(%o1)                         \mu_e
(%i2) tex (me);
$$\mu_e$$
(%o2)                         false

Assign TeX output for an ordinary function (not an operator).

(%i1) texput (lcm, "\\mathrm{lcm}");
(%o1)                     \mathrm{lcm}
(%i2) tex (lcm (a, b));
$$\mathrm{lcm}\left(a , b\right)$$
(%o2)                         false

Call a function to generate TeX output.

(%i1) texfoo (e) := block ([a, b], [a, b] : args (e),
  concat("\\left[\\stackrel{",tex1(b),"}{",tex1(a),"}\\right]"))$
(%i2) texput (foo, texfoo);
(%o2)                        texfoo
(%i3) tex (foo (2^x, %pi));
$$\left[\stackrel{\pi}{2^{x}}\right]$$
(%o3)                         false

Assign TeX output for a prefix operator.

(%i1) prefix ("grad");
(%o1)                         grad
(%i2) texput ("grad", " \\nabla ", prefix);
(%o2)                        \nabla 
(%i3) tex (grad f);
$$ \nabla f$$
(%o3)                         false

Assign TeX output for an infix operator.

(%i1) infix ("~");
(%o1)                           ~
(%i2) texput ("~", " \\times ", infix);
(%o2)                        \times 
(%i3) tex (a ~ b);
$$a \times b$$
(%o3)                         false

Assign TeX output for a postfix operator.

(%i1) postfix ("##");
(%o1)                          ##
(%i2) texput ("##", "!!", postfix);
(%o2)                          !!
(%i3) tex (x ##);
$$x!!$$
(%o3)                         false

Assign TeX output for a nary operator.

(%i1) nary ("@@");
(%o1)                          @@
(%i2) texput ("@@", " \\circ ", nary);
(%o2)                         \circ 
(%i3) tex (a @@ b @@ c @@ d);
$$a \circ b \circ c \circ d$$
(%o3)                         false

Assign TeX output for a nofix operator.

(%i1) nofix ("foo");
(%o1)                          foo
(%i2) texput ("foo", "\\mathsc{foo}", nofix);
(%o2)                     \mathsc{foo}
(%i3) tex (foo);
$$\mathsc{foo}$$
(%o3)                         false

Assign TeX output for a matchfix operator.

(%i1) matchfix ("<<", ">>");
(%o1)                          <<
(%i2) texput ("<<", [" \\langle ", " \\rangle "], matchfix);
(%o2)                [ \langle ,  \rangle ]
(%i3) tex (<<a>>);
$$ \langle a \rangle $$
(%o3)                         false
(%i4) tex (<<a, b>>);
$$ \langle a , b \rangle $$
(%o4)                         false
(%i5) texput ("<<", [" \\langle ", " \\rangle ", " \\, | \\,"],
      matchfix);
(%o5)           [ \langle ,  \rangle ,  \, | \,]
(%i6) tex (<<a>>);
$$ \langle a \rangle $$
(%o6)                         false
(%i7) tex (<<a, b>>);
$$ \langle a \, | \,b \rangle $$
(%o7)                         false

Function: get_tex_environment (op)
Function: set_tex_environment (op, before, after)

Customize the TeX environment output by tex. As maintained by these functions, the TeX environment comprises two strings: one is printed before any other TeX output, and the other is printed after.

Only the TeX environment of the top-level operator in an expression is output; TeX environments associated with other operators are ignored.

get_tex_environment returns the TeX environment which is applied to the operator op; returns the default if no other environment has been assigned.

set_tex_environment assigns the TeX environment for the operator op.

Examples:

(%i1) get_tex_environment (":=");
(%o1) [
\begin{verbatim}
, ;
\end{verbatim}
]
(%i2) tex (f (x) := 1 - x);

\begin{verbatim}
f(x):=1-x;
\end{verbatim}

(%o2)                         false
(%i3) set_tex_environment (":=", "$$", "$$");
(%o3)                       [$$, $$]
(%i4) tex (f (x) := 1 - x);
$$f(x):=1-x$$
(%o4)                         false

Function: get_tex_environment_default ()
Function: set_tex_environment_default (before, after)

Customize the TeX environment output by tex. As maintained by these functions, the TeX environment comprises two strings: one is printed before any other TeX output, and the other is printed after.

get_tex_environment_default returns the TeX environment which is applied to expressions for which the top-level operator has no specific TeX environment (as assigned by set_tex_environment).

set_tex_environment_default assigns the default TeX environment.

Examples:

(%i1) get_tex_environment_default ();
(%o1)                       [$$, $$]
(%i2) tex (f(x) + g(x));
$$g\left(x\right)+f\left(x\right)$$
(%o2)                         false
(%i3) set_tex_environment_default ("\\begin{equation}
", "
\\end{equation}");
(%o3) [\begin{equation}
, 
\end{equation}]
(%i4) tex (f(x) + g(x));
\begin{equation}
g\left(x\right)+f\left(x\right)
\end{equation}
(%o4)                         false


Previous: , Up: File Input and Output   [Contents][Index]

13.5 Functions and Variables for Fortran Output

Option variable: fortindent

Default value: 0

fortindent controls the left margin indentation of expressions printed out by the fortran command. 0 gives normal printout (i.e., 6 spaces), and positive values will causes the expressions to be printed farther to the right.

Function: fortran (expr)

Prints expr as a Fortran statement. The output line is indented with spaces. If the line is too long, fortran prints continuation lines. fortran prints the exponentiation operator ^ as **, and prints a complex number a + b %i in the form (a,b).

expr may be an equation. If so, fortran prints an assignment statement, assigning the right-hand side of the equation to the left-hand side. In particular, if the right-hand side of expr is the name of a matrix, then fortran prints an assignment statement for each element of the matrix.

If expr is not something recognized by fortran, the expression is printed in grind format without complaint. fortran does not know about lists, arrays, or functions.

fortindent controls the left margin of the printed lines. 0 is the normal margin (i.e., indented 6 spaces). Increasing fortindent causes expressions to be printed further to the right.

When fortspaces is true, fortran fills out each printed line with spaces to 80 columns.

fortran evaluates its arguments; quoting an argument defeats evaluation. fortran always returns done.

See also the function f90 for printing one or more expressions as a Fortran 90 program.

Examples:

(%i1) expr: (a + b)^12$
(%i2) fortran (expr);
      (b+a)**12                                                                 
(%o2)                         done
(%i3) fortran ('x=expr);
      x = (b+a)**12                                                             
(%o3)                         done
(%i4) fortran ('x=expand (expr));
      x = b**12+12*a*b**11+66*a**2*b**10+220*a**3*b**9+495*a**4*b**8+792
     1   *a**5*b**7+924*a**6*b**6+792*a**7*b**5+495*a**8*b**4+220*a**9*b
     2   **3+66*a**10*b**2+12*a**11*b+a**12
(%o4)                         done
(%i5) fortran ('x=7+5*%i);
      x = (7,5)                                                                 
(%o5)                         done
(%i6) fortran ('x=[1,2,3,4]);
      x = [1,2,3,4]                                                             
(%o6)                         done
(%i7) f(x) := x^2$
(%i8) fortran (f);
      f                                                                         
(%o8)                         done

Option variable: fortspaces

Default value: false

When fortspaces is true, fortran fills out each printed line with spaces to 80 columns.


Next: , Previous: , Up: Top   [Contents][Index]

14 Polynomials


Next: , Previous: , Up: Polynomials   [Contents][Index]

14.1 Introduction to Polynomials

Polynomials are stored in Maxima either in General Form or as Canonical Rational Expressions (CRE) form. The latter is a standard form, and is used internally by operations such as factor, ratsimp, and so on.

Canonical Rational Expressions constitute a kind of representation which is especially suitable for expanded polynomials and rational functions (as well as for partially factored polynomials and rational functions when ratfac is set to true). In this CRE form an ordering of variables (from most to least main) is assumed for each expression.

Polynomials are represented recursively by a list consisting of the main variable followed by a series of pairs of expressions, one for each term of the polynomial. The first member of each pair is the exponent of the main variable in that term and the second member is the coefficient of that term which could be a number or a polynomial in another variable again represented in this form. Thus the principal part of the CRE form of 3*x^2-1 is (X 2 3 0 -1) and that of 2*x*y+x-3 is (Y 1 (X 1 2) 0 (X 1 1 0 -3)) assuming y is the main variable, and is (X 1 (Y 1 2 0 1) 0 -3) assuming x is the main variable. "Main"-ness is usually determined by reverse alphabetical order.

The "variables" of a CRE expression needn’t be atomic. In fact any subexpression whose main operator is not +, -, *, / or ^ with integer power will be considered a "variable" of the expression (in CRE form) in which it occurs. For example the CRE variables of the expression x+sin(x+1)+2*sqrt(x)+1 are x, sqrt(X), and sin(x+1). If the user does not specify an ordering of variables by using the ratvars function Maxima will choose an alphabetic one.

In general, CRE’s represent rational expressions, that is, ratios of polynomials, where the numerator and denominator have no common factors, and the denominator is positive. The internal form is essentially a pair of polynomials (the numerator and denominator) preceded by the variable ordering list. If an expression to be displayed is in CRE form or if it contains any subexpressions in CRE form, the symbol /R/ will follow the line label.

See the rat function for converting an expression to CRE form.

An extended CRE form is used for the representation of Taylor series. The notion of a rational expression is extended so that the exponents of the variables can be positive or negative rational numbers rather than just positive integers and the coefficients can themselves be rational expressions as described above rather than just polynomials. These are represented internally by a recursive polynomial form which is similar to and is a generalization of CRE form, but carries additional information such as the degree of truncation. As with CRE form, the symbol /T/ follows the line label of such expressions.

·

Category: Polynomials · Category: Rational expressions


Previous: , Up: Polynomials   [Contents][Index]

14.2 Functions and Variables for Polynomials

Option variable: algebraic

Default value: false

algebraic must be set to true in order for the simplification of algebraic integers to take effect.

Option variable: berlefact

Default value: true

When berlefact is false then the Kronecker factoring algorithm will be used otherwise the Berlekamp algorithm, which is the default, will be used.

Function: bezout (p1, p2, x)

an alternative to the resultant command. It returns a matrix. determinant of this matrix is the desired resultant.

Examples:

(%i1) bezout(a*x+b, c*x^2+d, x);
                         [ b c  - a d ]
(%o1)                    [            ]
                         [  a     b   ]
(%i2) determinant(%);
                            2      2
(%o2)                      a  d + b  c
(%i3) resultant(a*x+b, c*x^2+d, x);
                            2      2
(%o3)                      a  d + b  c

Function: bothcoef (expr, x)

Returns a list whose first member is the coefficient of x in expr (as found by ratcoef if expr is in CRE form otherwise by coeff) and whose second member is the remaining part of expr. That is, [A, B] where expr = A*x + B.

Example:

(%i1) islinear (expr, x) := block ([c],
        c: bothcoef (rat (expr, x), x),
        is (freeof (x, c) and c[1] # 0))$
(%i2) islinear ((r^2 - (x - r)^2)/x, x);
(%o2)                         true

Function: coeff
    coeff (expr, x, n)
    coeff (expr, x)

Returns the coefficient of x^n in expr, where expr is a polynomial or a monomial term in x. Other than ratcoef coeff is a strictly syntactical operation and will only find literal instances of x^n in the internal representation of expr.

coeff(expr, x^n) is equivalent to coeff(expr, x, n). coeff(expr, x, 0) returns the remainder of expr which is free of x. If omitted, n is assumed to be 1.

x may be a simple variable or a subscripted variable, or a subexpression of expr which comprises an operator and all of its arguments.

It may be possible to compute coefficients of expressions which are equivalent to expr by applying expand or factor. coeff itself does not apply expand or factor or any other function.

coeff distributes over lists, matrices, and equations.

See also ratcoef.

Examples:

coeff returns the coefficient x^n in expr.

(%i1) coeff (b^3*a^3 + b^2*a^2 + b*a + 1, a^3);
                                3
(%o1)                          b

coeff(expr, x^n) is equivalent to coeff(expr, x, n).

(%i1) coeff (c[4]*z^4 - c[3]*z^3 - c[2]*z^2 + c[1]*z, z, 3);
(%o1)                         - c
                                 3
(%i2) coeff (c[4]*z^4 - c[3]*z^3 - c[2]*z^2 + c[1]*z, z^3);
(%o2)                         - c
                                 3

coeff(expr, x, 0) returns the remainder of expr which is free of x.

(%i1) coeff (a*u + b^2*u^2 + c^3*u^3, b, 0);
                            3  3
(%o1)                      c  u  + a u

x may be a simple variable or a subscripted variable, or a subexpression of expr which comprises an operator and all of its arguments.

(%i1) coeff (h^4 - 2*%pi*h^2 + 1, h, 2);
(%o1)                        - 2 %pi
(%i2) coeff (v[1]^4 - 2*%pi*v[1]^2 + 1, v[1], 2);
(%o2)                        - 2 %pi
(%i3) coeff (sin(1+x)*sin(x) + sin(1+x)^3*sin(x)^3, sin(1+x)^3);
                                3
(%o3)                        sin (x)
(%i4) coeff ((d - a)^2*(b + c)^3 + (a + b)^4*(c - d), a + b, 4);
(%o4)                         c - d

coeff itself does not apply expand or factor or any other function.

(%i1) coeff (c*(a + b)^3, a);
(%o1)                           0
(%i2) expand (c*(a + b)^3);
                 3          2        2        3
(%o2)           b  c + 3 a b  c + 3 a  b c + a  c
(%i3) coeff (%, a);
                                2
(%o3)                        3 b  c
(%i4) coeff (b^3*c + 3*a*b^2*c + 3*a^2*b*c + a^3*c, (a + b)^3);
(%o4)                           0
(%i5) factor (b^3*c + 3*a*b^2*c + 3*a^2*b*c + a^3*c);
                                  3
(%o5)                      (b + a)  c
(%i6) coeff (%, (a + b)^3);
(%o6)                           c

coeff distributes over lists, matrices, and equations.

(%i1) coeff ([4*a, -3*a, 2*a], a);
(%o1)                      [4, - 3, 2]
(%i2) coeff (matrix ([a*x, b*x], [-c*x, -d*x]), x);
                          [  a    b  ]
(%o2)                     [          ]
                          [ - c  - d ]
(%i3) coeff (a*u - b*v = 7*u + 3*v, u);
(%o3)                         a = 7

Function: content (p_1, x_1, …, x_n)

Returns a list whose first element is the greatest common divisor of the coefficients of the terms of the polynomial p_1 in the variable x_n (this is the content) and whose second element is the polynomial p_1 divided by the content.

Examples:

(%i1) content (2*x*y + 4*x^2*y^2, y);
                                   2
(%o1)                   [2 x, 2 x y  + y]

Function: denom (expr)

Returns the denominator of the rational expression expr.

See also num

(%i1) g1:(x+2)*(x+1)/((x+3)^2);
                         (x + 1) (x + 2)
(%o1)                    ---------------
                                   2
                            (x + 3)
(%i2) denom(g1);
                                   2
(%o2)                       (x + 3)
(%i3) g2:sin(x)/10*cos(x)/y;
                          cos(x) sin(x)
(%o3)                     -------------
                              10 y
(%i4) denom(g2);
(%o4)                         10 y

Function: divide (p_1, p_2, x_1, …, x_n)

computes the quotient and remainder of the polynomial p_1 divided by the polynomial p_2, in a main polynomial variable, x_n. The other variables are as in the ratvars function. The result is a list whose first element is the quotient and whose second element is the remainder.

Examples:

(%i1) divide (x + y, x - y, x);
(%o1)                       [1, 2 y]
(%i2) divide (x + y, x - y);
(%o2)                      [- 1, 2 x]

Note that y is the main variable in the second example.

Function: eliminate ([eqn_1, …, eqn_n], [x_1, …, x_k])

Eliminates variables from equations (or expressions assumed equal to zero) by taking successive resultants. This returns a list of n - k expressions with the k variables x_1, …, x_k eliminated. First x_1 is eliminated yielding n - 1 expressions, then x_2 is eliminated, etc. If k = n then a single expression in a list is returned free of the variables x_1, …, x_k. In this case solve is called to solve the last resultant for the last variable.

Example:

(%i1) expr1: 2*x^2 + y*x + z;
                                      2
(%o1)                    z + x y + 2 x
(%i2) expr2: 3*x + 5*y - z - 1;
(%o2)                  - z + 5 y + 3 x - 1
(%i3) expr3: z^2 + x - y^2 + 5;
                          2    2
(%o3)                    z  - y  + x + 5
(%i4) eliminate ([expr3, expr2, expr1], [y, z]);
             8         7         6          5          4
(%o4) [7425 x  - 1170 x  + 1299 x  + 12076 x  + 22887 x

                                    3         2
                            - 5154 x  - 1291 x  + 7688 x + 15376]

Function: ezgcd (p_1, p_2, p_3, …)

Returns a list whose first element is the greatest common divisor of the polynomials p_1, p_2, p_3, … and whose remaining elements are the polynomials divided by the greatest common divisor. This always uses the ezgcd algorithm.

See also gcd, gcdex, gcdivide, and poly_gcd.

Examples:

The three polynomials have the greatest common divisor 2*x-3. The gcd is first calculated with the function gcd and then with the function ezgcd.

(%i1) p1 : 6*x^3-17*x^2+14*x-3;
                        3       2
(%o1)                6 x  - 17 x  + 14 x - 3
(%i2) p2 : 4*x^4-14*x^3+12*x^2+2*x-3;
                    4       3       2
(%o2)            4 x  - 14 x  + 12 x  + 2 x - 3
(%i3) p3 : -8*x^3+14*x^2-x-3;
                          3       2
(%o3)                - 8 x  + 14 x  - x - 3

(%i4) gcd(p1, gcd(p2, p3));
(%o4)                        2 x - 3

(%i5) ezgcd(p1, p2, p3);
                   2               3      2           2
(%o5) [2 x - 3, 3 x  - 4 x + 1, 2 x  - 4 x  + 1, - 4 x  + x + 1]

Option variable: facexpand

Default value: true

facexpand controls whether the irreducible factors returned by factor are in expanded (the default) or recursive (normal CRE) form.

Function: factor
    factor (expr)
    factor (expr, p)

Factors the expression expr, containing any number of variables or functions, into factors irreducible over the integers. factor (expr, p) factors expr over the field of rationals with an element adjoined whose minimum polynomial is p.

factor uses ifactors function for factoring integers.

factorflag if false suppresses the factoring of integer factors of rational expressions.

dontfactor may be set to a list of variables with respect to which factoring is not to occur. (It is initially empty). Factoring also will not take place with respect to any variables which are less important (using the variable ordering assumed for CRE form) than those on the dontfactor list.

savefactors if true causes the factors of an expression which is a product of factors to be saved by certain functions in order to speed up later factorizations of expressions containing some of the same factors.

berlefact if false then the Kronecker factoring algorithm will be used otherwise the Berlekamp algorithm, which is the default, will be used.

intfaclim if true maxima will give up factorization of integers if no factor is found after trial divisions and Pollard’s rho method. If set to false (this is the case when the user calls factor explicitly), complete factorization of the integer will be attempted. The user’s setting of intfaclim is used for internal calls to factor. Thus, intfaclim may be reset to prevent Maxima from taking an inordinately long time factoring large integers.

factor_max_degree if set to a positive integer n will prevent certain polynomials from being factored if their degree in any variable exceeds n.

See also collectterms.

Examples:

(%i1) factor (2^63 - 1);
                    2
(%o1)              7  73 127 337 92737 649657
(%i2) factor (-8*y - 4*x + z^2*(2*y + x));
(%o2)               (2 y + x) (z - 2) (z + 2)
(%i3) -1 - 2*x - x^2 + y^2 + 2*x*y^2 + x^2*y^2;
                2  2        2    2    2
(%o3)          x  y  + 2 x y  + y  - x  - 2 x - 1
(%i4) block ([dontfactor: [x]], factor (%/36/(1 + 2*y + y^2)));
                       2
                     (x  + 2 x + 1) (y - 1)
(%o4)                ----------------------
                           36 (y + 1)
(%i5) factor (1 + %e^(3*x));
                      x         2 x     x
(%o5)              (%e  + 1) (%e    - %e  + 1)
(%i6) factor (1 + x^4, a^2 - 2);
                    2              2
(%o6)             (x  - a x + 1) (x  + a x + 1)
(%i7) factor (-y^2*z^2 - x*z^2 + x^2*y^2 + x^3);
                       2
(%o7)              - (y  + x) (z - x) (z + x)
(%i8) (2 + x)/(3 + x)/(b + x)/(c + x)^2;
                             x + 2
(%o8)               ------------------------
                                           2
                    (x + 3) (x + b) (x + c)
(%i9) ratsimp (%);
                4                  3
(%o9) (x + 2)/(x  + (2 c + b + 3) x

     2                       2             2                   2
 + (c  + (2 b + 6) c + 3 b) x  + ((b + 3) c  + 6 b c) x + 3 b c )
(%i10) partfrac (%, x);
           2                   4                3
(%o10) - (c  - 4 c - b + 6)/((c  + (- 2 b - 6) c

     2              2         2                2
 + (b  + 12 b + 9) c  + (- 6 b  - 18 b) c + 9 b ) (x + c))

                 c - 2
 - ---------------------------------
     2                             2
   (c  + (- b - 3) c + 3 b) (x + c)

                         b - 2
 + -------------------------------------------------
             2             2       3      2
   ((b - 3) c  + (6 b - 2 b ) c + b  - 3 b ) (x + b)

                         1
 - ----------------------------------------------
             2
   ((b - 3) c  + (18 - 6 b) c + 9 b - 27) (x + 3)
(%i11) map ('factor, %);
              2
             c  - 4 c - b + 6                 c - 2
(%o11) - ------------------------- - ------------------------
                2        2                                  2
         (c - 3)  (c - b)  (x + c)   (c - 3) (c - b) (x + c)

                       b - 2                        1
            + ------------------------ - ------------------------
                             2                          2
              (b - 3) (c - b)  (x + b)   (b - 3) (c - 3)  (x + 3)
(%i12) ratsimp ((x^5 - 1)/(x - 1));
                       4    3    2
(%o12)                x  + x  + x  + x + 1
(%i13) subst (a, x, %);
                       4    3    2
(%o13)                a  + a  + a  + a + 1
(%i14) factor (%th(2), %);
                       2        3        3    2
(%o14)   (x - a) (x - a ) (x - a ) (x + a  + a  + a + 1)
(%i15) factor (1 + x^12);
                       4        8    4
(%o15)               (x  + 1) (x  - x  + 1)
(%i16) factor (1 + x^99);
                 2            6    3
(%o16) (x + 1) (x  - x + 1) (x  - x  + 1)

   10    9    8    7    6    5    4    3    2
 (x   - x  + x  - x  + x  - x  + x  - x  + x  - x + 1)

   20    19    17    16    14    13    11    10    9    7    6
 (x   + x   - x   - x   + x   + x   - x   - x   - x  + x  + x

    4    3            60    57    51    48    42    39    33
 - x  - x  + x + 1) (x   + x   - x   - x   + x   + x   - x

    30    27    21    18    12    9    3
 - x   - x   + x   + x   - x   - x  + x  + 1)

Option variable: factor_max_degree

Default value: 1000

When factor_max_degree is set to a positive integer n, it will prevent Maxima from attempting to factor certain polynomials whose degree in any variable exceeds n. If factor_max_degree_print_warning is true, a warning message will be printed. factor_max_degree can be used to prevent excessive memory usage and/or computation time and stack overflows. Note that "obvious" factoring of polynomials such as x^2000+x^2001 to x^2000*(x+1) will still take place. To disable this behavior, set factor_max_degree to 0.

Example:

(%i1) factor_max_degree : 100$
(%i2) factor(x^100-1);
                        2        4    3    2
(%o2) (x - 1) (x + 1) (x  + 1) (x  - x  + x  - x + 1)
   4    3    2            8    6    4    2
 (x  + x  + x  + x + 1) (x  - x  + x  - x  + 1)
   20    15    10    5        20    15    10    5
 (x   - x   + x   - x  + 1) (x   + x   + x   + x  + 1)
   40    30    20    10
 (x   - x   + x   - x   + 1)
(%i3) factor(x^101-1);
                               101
Refusing to factor polynomial x    - 1
               because its degree exceeds factor_max_degree (100)
                             101
(%o3)                       x    - 1

See also: factor_max_degree_print_warning

Option variable: factor_max_degree_print_warning

Default value: true

When factor_max_degree_print_warning is true, then Maxima will print a warning message when the factoring of a polynomial is prevented because its degree exceeds the value of factor_max_degree.

See also: factor_max_degree

Option variable: factorflag

Default value: false

When factorflag is false, suppresses the factoring of integer factors of rational expressions.

Function: factorout (expr, x_1, x_2, …)

Rearranges the sum expr into a sum of terms of the form f (x_1, x_2, …)*g where g is a product of expressions not containing any x_i and f is factored.

Note that the option variable keepfloat is ignored by factorout.

Example:

(%i1) expand (a*(x+1)*(x-1)*(u+1)^2);
             2  2          2      2      2
(%o1)     a u  x  + 2 a u x  + a x  - a u  - 2 a u - a
(%i2) factorout(%,x);
         2
(%o2) a u  (x - 1) (x + 1) + 2 a u (x - 1) (x + 1)
                                              + a (x - 1) (x + 1)

Function: factorsum (expr)

Tries to group terms in factors of expr which are sums into groups of terms such that their sum is factorable. factorsum can recover the result of expand ((x + y)^2 + (z + w)^2) but it can’t recover expand ((x + 1)^2 + (x + y)^2) because the terms have variables in common.

Example:

(%i1) expand ((x + 1)*((u + v)^2 + a*(w + z)^2));
           2      2                            2      2
(%o1) a x z  + a z  + 2 a w x z + 2 a w z + a w  x + v  x

                                     2        2    2            2
                        + 2 u v x + u  x + a w  + v  + 2 u v + u
(%i2) factorsum (%);
                                   2          2
(%o2)            (x + 1) (a (z + w)  + (v + u) )

Function: fasttimes (p_1, p_2)

Returns the product of the polynomials p_1 and p_2 by using a special algorithm for multiplication of polynomials. p_1 and p_2 should be multivariate, dense, and nearly the same size. Classical multiplication is of order n_1 n_2 where n_1 is the degree of p_1 and n_2 is the degree of p_2. fasttimes is of order max (n_1, n_2)^1.585.

Function: fullratsimp (expr)

fullratsimp repeatedly applies ratsimp followed by non-rational simplification to an expression until no further change occurs, and returns the result.

When non-rational expressions are involved, one call to ratsimp followed as is usual by non-rational ("general") simplification may not be sufficient to return a simplified result. Sometimes, more than one such call may be necessary. fullratsimp makes this process convenient.

fullratsimp (expr, x_1, ..., x_n) takes one or more arguments similar to ratsimp and rat.

Example:

(%i1) expr: (x^(a/2) + 1)^2*(x^(a/2) - 1)^2/(x^a - 1);
                       a/2     2   a/2     2
                     (x    - 1)  (x    + 1)
(%o1)                -----------------------
                              a
                             x  - 1
(%i2) ratsimp (expr);
                          2 a      a
                         x    - 2 x  + 1
(%o2)                    ---------------
                              a
                             x  - 1
(%i3) fullratsimp (expr);
                              a
(%o3)                        x  - 1
(%i4) rat (expr);
                       a/2 4       a/2 2
                     (x   )  - 2 (x   )  + 1
(%o4)/R/             -----------------------
                              a
                             x  - 1

Function: fullratsubst (a, b, c)

is the same as ratsubst except that it calls itself recursively on its result until that result stops changing. This function is useful when the replacement expression and the replaced expression have one or more variables in common.

fullratsubst will also accept its arguments in the format of lratsubst. That is, the first argument may be a single substitution equation or a list of such equations, while the second argument is the expression being processed.

load ("lrats") loads fullratsubst and lratsubst.

Examples:

(%i1) load ("lrats")$
(%i2) subst ([a = b, c = d], a + c);
(%o2)                         d + b
(%i3) lratsubst ([a^2 = b, c^2 = d], (a + e)*c*(a + c));
(%o3)                (d + a c) e + a d + b c
(%i4) lratsubst (a^2 = b, a^3);
(%o4)                          a b
(%i5) ratsubst (b*a, a^2, a^3);
                               2
(%o5)                         a  b
(%i6) fullratsubst (b*a, a^2, a^3);
                                 2
(%o6)                         a b
(%i7) fullratsubst ([a^2 = b, b^2 = c, c^2 = a], a^3*b*c);
(%o7)                           b
(%i8) fullratsubst (a^2 = b*a, a^3);
                                 2
(%o8)                         a b
(%i9) errcatch (fullratsubst (b*a^2, a^2, a^3));

*** - Lisp stack overflow. RESET

Function: gcd (p_1, p_2, x_1, …)

Returns the greatest common divisor of p_1 and p_2. The flag gcd determines which algorithm is employed. Setting gcd to ez, subres, red, or spmod selects the ezgcd, subresultant prs, reduced, or modular algorithm, respectively. If gcd false then gcd (p_1, p_2, x) always returns 1 for all x. Many functions (e.g. ratsimp, factor, etc.) cause gcd’s to be taken implicitly. For homogeneous polynomials it is recommended that gcd equal to subres be used. To take the gcd when an algebraic is present, e.g., gcd (x^2 - 2*sqrt(2)* x + 2, x - sqrt(2)), the option variable algebraic must be true and gcd must not be ez.

The gcd flag, default: spmod, if false will also prevent the greatest common divisor from being taken when expressions are converted to canonical rational expression (CRE) form. This will sometimes speed the calculation if gcds are not required.

See also ezgcd, gcdex, gcdivide, and poly_gcd.

Example:

(%i1) p1:6*x^3+19*x^2+19*x+6; 
                        3       2
(%o1)                6 x  + 19 x  + 19 x + 6
(%i2) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x;
                  5       4       3       2
(%o2)          6 x  + 13 x  + 12 x  + 13 x  + 6 x
(%i3) gcd(p1, p2);
                            2
(%o3)                    6 x  + 13 x + 6
(%i4) p1/gcd(p1, p2), ratsimp;
(%o4)                         x + 1
(%i5) p2/gcd(p1, p2), ratsimp;
                              3
(%o5)                        x  + x

ezgcd returns a list whose first element is the greatest common divisor of the polynomials p_1 and p_2, and whose remaining elements are the polynomials divided by the greatest common divisor.

(%i6) ezgcd(p1, p2);
                    2                     3
(%o6)           [6 x  + 13 x + 6, x + 1, x  + x]

Function: gcdex
    gcdex (f, g)
    gcdex (f, g, x)

Returns a list [a, b, u] where u is the greatest common divisor (gcd) of f and g, and u is equal to a f + b g. The arguments f and g should be univariate polynomials, or else polynomials in x a supplied main variable since we need to be in a principal ideal domain for this to work. The gcd means the gcd regarding f and g as univariate polynomials with coefficients being rational functions in the other variables.

gcdex implements the Euclidean algorithm, where we have a sequence of L[i]: [a[i], b[i], r[i]] which are all perpendicular to [f, g, -1] and the next one is built as if q = quotient(r[i]/r[i+1]) then L[i+2]: L[i] - q L[i+1], and it terminates at L[i+1] when the remainder r[i+2] is zero.

The arguments f and g can be integers. For this case the function igcdex is called by gcdex.

See also ezgcd, gcd, gcdivide, and poly_gcd.

Examples:

(%i1) gcdex (x^2 + 1, x^3 + 4);
                       2
                      x  + 4 x - 1  x + 4
(%o1)/R/           [- ------------, -----, 1]
                           17        17
(%i2) % . [x^2 + 1, x^3 + 4, -1];
(%o2)/R/                        0

Note that the gcd in the following is 1 since we work in k(y)[x], not the y+1 we would expect in k[y, x].

(%i1) gcdex (x*(y + 1), y^2 - 1, x);
                               1
(%o1)/R/                 [0, ------, 1]
                              2
                             y  - 1

Function: gcfactor (n)

Factors the Gaussian integer n over the Gaussian integers, i.e., numbers of the form a + b %i where a and b are rational integers (i.e., ordinary integers). Factors are normalized by making a and b non-negative.

Function: gfactor (expr)

Factors the polynomial expr over the Gaussian integers (that is, the integers with the imaginary unit %i adjoined). This is like factor (expr, a^2+1) where a is %i.

Example:

(%i1) gfactor (x^4 - 1);
(%o1)           (x - 1) (x + 1) (x - %i) (x + %i)

Function: gfactorsum (expr)

is similar to factorsum but applies gfactor instead of factor.

Function: hipow (expr, x)

Returns the highest explicit exponent of x in expr. x may be a variable or a general expression. If x does not appear in expr, hipow returns 0.

hipow does not consider expressions equivalent to expr. In particular, hipow does not expand expr, so hipow (expr, x) and hipow (expand (expr, x)) may yield different results.

Examples:

(%i1) hipow (y^3 * x^2 + x * y^4, x);
(%o1)                           2
(%i2) hipow ((x + y)^5, x);
(%o2)                           1
(%i3) hipow (expand ((x + y)^5), x);
(%o3)                           5
(%i4) hipow ((x + y)^5, x + y);
(%o4)                           5
(%i5) hipow (expand ((x + y)^5), x + y);
(%o5)                           0

Option variable: intfaclim

Default value: true

If true, maxima will give up factorization of integers if no factor is found after trial divisions and Pollard’s rho method and factorization will not be complete.

When intfaclim is false (this is the case when the user calls factor explicitly), complete factorization will be attempted. intfaclim is set to false when factors are computed in divisors, divsum and totient.

Internal calls to factor respect the user-specified value of intfaclim. Setting intfaclim to true may reduce the time spent factoring large integers.

Option variable: keepfloat

Default value: false

When keepfloat is true, prevents floating point numbers from being rationalized when expressions which contain them are converted to canonical rational expression (CRE) form.

Note that the function solve and those functions calling it (eigenvalues, for example) currently ignore this flag, converting floating point numbers anyway.

Examples:

(%i1) rat(x/2.0);

rat: replaced 0.5 by 1/2 = 0.5
                                x
(%o1)/R/                        -
                                2
(%i2) rat(x/2.0), keepfloat;
(%o2)/R/                      0.5 x

solve ignores keepfloat:

(%i1) solve(1.0-x,x), keepfloat;

rat: replaced 1.0 by 1/1 = 1.0
(%o1)                        [x = 1]

Function: lopow (expr, x)

Returns the lowest exponent of x which explicitly appears in expr. Thus

(%i1) lopow ((x+y)^2 + (x+y)^a, x+y);
(%o1)                       min(a, 2)

Function: lratsubst (L, expr)

is analogous to subst (L, expr) except that it uses ratsubst instead of subst.

The first argument of lratsubst is an equation or a list of equations identical in format to that accepted by subst. The substitutions are made in the order given by the list of equations, that is, from left to right.

load ("lrats") loads fullratsubst and lratsubst.

Examples:

(%i1) load ("lrats")$
(%i2) subst ([a = b, c = d], a + c);
(%o2)                         d + b
(%i3) lratsubst ([a^2 = b, c^2 = d], (a + e)*c*(a + c));
(%o3)                (d + a c) e + a d + b c
(%i4) lratsubst (a^2 = b, a^3);
(%o4)                          a b

Option variable: modulus

Default value: false

When modulus is a positive number p, operations on canonical rational expressions (CREs, as returned by rat and related functions) are carried out modulo p, using the so-called "balanced" modulus system in which n modulo p is defined as an integer k in [-(p-1)/2, ..., 0, ..., (p-1)/2] when p is odd, or [-(p/2 - 1), ..., 0, ...., p/2] when p is even, such that a p + k equals n for some integer a.

If expr is already in canonical rational expression (CRE) form when modulus is reset, then you may need to re-rat expr, e.g., expr: rat (ratdisrep (expr)), in order to get correct results.

Typically modulus is set to a prime number. If modulus is set to a positive non-prime integer, this setting is accepted, but a warning message is displayed. Maxima signals an error, when zero or a negative integer is assigned to modulus.

Examples:

(%i1) modulus:7;
(%o1)                           7
(%i2) polymod([0,1,2,3,4,5,6,7]);
(%o2)            [0, 1, 2, 3, - 3, - 2, - 1, 0]
(%i3) modulus:false;
(%o3)                         false
(%i4) poly:x^6+x^2+1;
                            6    2
(%o4)                      x  + x  + 1
(%i5) factor(poly);
                            6    2
(%o5)                      x  + x  + 1
(%i6) modulus:13;
(%o6)                          13
(%i7) factor(poly);
                      2        4      2
(%o7)               (x  + 6) (x  - 6 x  - 2)
(%i8) polymod(%);
                            6    2
(%o8)                      x  + x  + 1

Function: num (expr)

Returns the numerator of expr if it is a ratio. If expr is not a ratio, expr is returned.

num evaluates its argument.

See also denom

(%i1) g1:(x+2)*(x+1)/((x+3)^2);
                         (x + 1) (x + 2)
(%o1)                    ---------------
                                   2
                            (x + 3)
(%i2) num(g1);
(%o2)                    (x + 1) (x + 2)
(%i3) g2:sin(x)/10*cos(x)/y;
                          cos(x) sin(x)
(%o3)                     -------------
                              10 y
(%i4) num(g2);
(%o4)                     cos(x) sin(x)

Function: polydecomp (p, x)

Decomposes the polynomial p in the variable x into the functional composition of polynomials in x. polydecomp returns a list [p_1, ..., p_n] such that

lambda ([x], p_1) (lambda ([x], p_2) (... (lambda ([x], p_n) (x))
  ...))

is equal to p. The degree of p_i is greater than 1 for i less than n.

Such a decomposition is not unique.

Examples:

(%i1) polydecomp (x^210, x);
                          7   5   3   2
(%o1)                   [x , x , x , x ]
(%i2) p : expand (subst (x^3 - x - 1, x, x^2 - a));
                6      4      3    2
(%o2)          x  - 2 x  - 2 x  + x  + 2 x - a + 1
(%i3) polydecomp (p, x);
                        2       3
(%o3)                 [x  - a, x  - x - 1]

The following function composes L = [e_1, ..., e_n] as functions in x; it is the inverse of polydecomp:

(%i1) compose (L, x) :=
  block ([r : x], for e in L do r : subst (e, x, r), r) $

Re-express above example using compose:

(%i1) polydecomp (compose ([x^2 - a, x^3 - x - 1], x), x);
                          2       3
(%o1)          [compose([x  - a, x  - x - 1], x)]

Note that though compose (polydecomp (p, x), x) always returns p (unexpanded), polydecomp (compose ([p_1, ..., p_n], x), x) does not necessarily return [p_1, ..., p_n]:

(%i1) polydecomp (compose ([x^2 + 2*x + 3, x^2], x), x);
                           2             2
(%o1)           [compose([x  + 2 x + 3, x ], x)]
(%i2) polydecomp (compose ([x^2 + x + 1, x^2 + x + 1], x), x);
                        2           2
(%o2)        [compose([x  + x + 1, x  + x + 1], x)]

Function: polymod
    polymod (p)
    polymod (p, m)

Converts the polynomial p to a modular representation with respect to the current modulus which is the value of the variable modulus.

polymod (p, m) specifies a modulus m to be used instead of the current value of modulus.

See modulus.

Function: quotient
    quotient (p_1, p_2)
    quotient (p_1, p_2, x_1, …, x_n)

Returns the polynomial p_1 divided by the polynomial p_2. The arguments x_1, …, x_n are interpreted as in ratvars.

quotient returns the first element of the two-element list returned by divide.

Function: rat
    rat (expr)
    rat (expr, x_1, …, x_n)

Converts expr to canonical rational expression (CRE) form by expanding and combining all terms over a common denominator and cancelling out the greatest common divisor of the numerator and denominator, as well as converting floating point numbers to rational numbers within a tolerance of ratepsilon. The variables are ordered according to the x_1, …, x_n, if specified, as in ratvars.

rat does not generally simplify functions other than addition +, subtraction -, multiplication *, division /, and exponentiation to an integer power, whereas ratsimp does handle those cases. Note that atoms (numbers and variables) in CRE form are not the same as they are in the general form. For example, rat(x)- x yields rat(0) which has a different internal representation than 0.

When ratfac is true, rat yields a partially factored form for CRE. During rational operations the expression is maintained as fully factored as possible without an actual call to the factor package. This should always save space and may save some time in some computations. The numerator and denominator are still made relatively prime (e.g., rat((x^2 - 1)^4/(x + 1)^2) yields (x - 1)^4 (x + 1)^2 when ratfac is true), but the factors within each part may not be relatively prime.

ratprint if false suppresses the printout of the message informing the user of the conversion of floating point numbers to rational numbers.

keepfloat if true prevents floating point numbers from being converted to rational numbers.

See also ratexpand and ratsimp.

Examples:

(%i1) ((x - 2*y)^4/(x^2 - 4*y^2)^2 + 1)*(y + a)*(2*y + x) /
      (4*y^2 + x^2);
                                           4
                                  (x - 2 y)
              (y + a) (2 y + x) (------------ + 1)
                                   2      2 2
                                 (x  - 4 y )
(%o1)         ------------------------------------
                              2    2
                           4 y  + x
(%i2) rat (%, y, a, x);
                            2 a + 2 y
(%o2)/R/                    ---------
                             x + 2 y

Option variable: ratalgdenom

Default value: true

When ratalgdenom is true, allows rationalization of denominators with respect to radicals to take effect. ratalgdenom has an effect only when canonical rational expressions (CRE) are used in algebraic mode.

Function: ratcoef
    ratcoef (expr, x, n)
    ratcoef (expr, x)

Returns the coefficient of the expression x^n in the expression expr. If omitted, n is assumed to be 1.

The return value is free (except possibly in a non-rational sense) of the variables in x. If no coefficient of this type exists, 0 is returned.

ratcoef expands and rationally simplifies its first argument and thus it may produce answers different from those of coeff which is purely syntactic. Thus ratcoef ((x + 1)/y + x, x) returns (y + 1)/y whereas coeff returns 1.

ratcoef (expr, x, 0), viewing expr as a sum, returns a sum of those terms which do not contain x. Therefore if x occurs to any negative powers, ratcoef should not be used.

Since expr is rationally simplified before it is examined, coefficients may not appear quite the way they were envisioned.

Example:

(%i1) s: a*x + b*x + 5$
(%i2) ratcoef (s, a + b);
(%o2)                           x

Function: ratdenom (expr)

Returns the denominator of expr, after coercing expr to a canonical rational expression (CRE). The return value is a CRE.

expr is coerced to a CRE by rat if it is not already a CRE. This conversion may change the form of expr by putting all terms over a common denominator.

denom is similar, but returns an ordinary expression instead of a CRE. Also, denom does not attempt to place all terms over a common denominator, and thus some expressions which are considered ratios by ratdenom are not considered ratios by denom.

Option variable: ratdenomdivide

Default value: true

When ratdenomdivide is true, ratexpand expands a ratio in which the numerator is a sum into a sum of ratios, all having a common denominator. Otherwise, ratexpand collapses a sum of ratios into a single ratio, the numerator of which is the sum of the numerators of each ratio.

Examples:

(%i1) expr: (x^2 + x + 1)/(y^2 + 7);
                            2
                           x  + x + 1
(%o1)                      ----------
                              2
                             y  + 7
(%i2) ratdenomdivide: true$
(%i3) ratexpand (expr);
                       2
                      x        x        1
(%o3)               ------ + ------ + ------
                     2        2        2
                    y  + 7   y  + 7   y  + 7
(%i4) ratdenomdivide: false$
(%i5) ratexpand (expr);
                            2
                           x  + x + 1
(%o5)                      ----------
                              2
                             y  + 7
(%i6) expr2: a^2/(b^2 + 3) + b/(b^2 + 3);
                                     2
                           b        a
(%o6)                    ------ + ------
                          2        2
                         b  + 3   b  + 3
(%i7) ratexpand (expr2);
                                  2
                             b + a
(%o7)                        ------
                              2
                             b  + 3

Function: ratdiff (expr, x)

Differentiates the rational expression expr with respect to x. expr must be a ratio of polynomials or a polynomial in x. The argument x may be a variable or a subexpression of expr.

The result is equivalent to diff, although perhaps in a different form. ratdiff may be faster than diff, for rational expressions.

ratdiff returns a canonical rational expression (CRE) if expr is a CRE. Otherwise, ratdiff returns a general expression.

ratdiff considers only the dependence of expr on x, and ignores any dependencies established by depends.

Example:

(%i1) expr: (4*x^3 + 10*x - 11)/(x^5 + 5);
                           3
                        4 x  + 10 x - 11
(%o1)                   ----------------
                              5
                             x  + 5
(%i2) ratdiff (expr, x);
                    7       5       4       2
                 8 x  + 40 x  - 55 x  - 60 x  - 50
(%o2)          - ---------------------------------
                          10       5
                         x   + 10 x  + 25
(%i3) expr: f(x)^3 - f(x)^2 + 7;
                         3       2
(%o3)                   f (x) - f (x) + 7
(%i4) ratdiff (expr, f(x));
                           2
(%o4)                   3 f (x) - 2 f(x)
(%i5) expr: (a + b)^3 + (a + b)^2;
                              3          2
(%o5)                  (b + a)  + (b + a)
(%i6) ratdiff (expr, a + b);
                    2                    2
(%o6)            3 b  + (6 a + 2) b + 3 a  + 2 a

Function: ratdisrep (expr)

Returns its argument as a general expression. If expr is a general expression, it is returned unchanged.

Typically ratdisrep is called to convert a canonical rational expression (CRE) into a general expression. This is sometimes convenient if one wishes to stop the "contagion", or use rational functions in non-rational contexts.

See also totaldisrep.

Function: ratexpand (expr)
Option variable: ratexpand

Expands expr by multiplying out products of sums and exponentiated sums, combining fractions over a common denominator, cancelling the greatest common divisor of the numerator and denominator, then splitting the numerator (if a sum) into its respective terms divided by the denominator.

The return value of ratexpand is a general expression, even if expr is a canonical rational expression (CRE).

The switch ratexpand if true will cause CRE expressions to be fully expanded when they are converted back to general form or displayed, while if it is false then they will be put into a recursive form. See also ratsimp.

When ratdenomdivide is true, ratexpand expands a ratio in which the numerator is a sum into a sum of ratios, all having a common denominator. Otherwise, ratexpand collapses a sum of ratios into a single ratio, the numerator of which is the sum of the numerators of each ratio.

When keepfloat is true, prevents floating point numbers from being rationalized when expressions which contain them are converted to canonical rational expression (CRE) form.

Examples:

(%i1) ratexpand ((2*x - 3*y)^3);
                     3         2       2        3
(%o1)          - 27 y  + 54 x y  - 36 x  y + 8 x
(%i2) expr: (x - 1)/(x + 1)^2 + 1/(x - 1);
                         x - 1       1
(%o2)                   -------- + -----
                               2   x - 1
                        (x + 1)
(%i3) expand (expr);
                    x              1           1
(%o3)          ------------ - ------------ + -----
                2              2             x - 1
               x  + 2 x + 1   x  + 2 x + 1
(%i4) ratexpand (expr);
                        2
                     2 x                 2
(%o4)           --------------- + ---------------
                 3    2            3    2
                x  + x  - x - 1   x  + x  - x - 1

Option variable: ratfac

Default value: false

When ratfac is true, canonical rational expressions (CRE) are manipulated in a partially factored form.

During rational operations the expression is maintained as fully factored as possible without calling factor. This should always save space and may save time in some computations. The numerator and denominator are made relatively prime, for example factor ((x^2 - 1)^4/(x + 1)^2) yields (x - 1)^4 (x + 1)^2, but the factors within each part may not be relatively prime.

In the ctensr (Component Tensor Manipulation) package, Ricci, Einstein, Riemann, and Weyl tensors and the scalar curvature are factored automatically when ratfac is true. ratfac should only be set for cases where the tensorial components are known to consist of few terms.

The ratfac and ratweight schemes are incompatible and may not both be used at the same time.

Function: ratnumer (expr)

Returns the numerator of expr, after coercing expr to a canonical rational expression (CRE). The return value is a CRE.

expr is coerced to a CRE by rat if it is not already a CRE. This conversion may change the form of expr by putting all terms over a common denominator.

num is similar, but returns an ordinary expression instead of a CRE. Also, num does not attempt to place all terms over a common denominator, and thus some expressions which are considered ratios by ratnumer are not considered ratios by num.

Function: ratp (expr)

Returns true if expr is a canonical rational expression (CRE) or extended CRE, otherwise false.

CRE are created by rat and related functions. Extended CRE are created by taylor and related functions.

Option variable: ratprint

Default value: true

When ratprint is true, a message informing the user of the conversion of floating point numbers to rational numbers is displayed.

Function: ratsimp (expr)
Function: ratsimp (expr, x_1, …, x_n)

Simplifies the expression expr and all of its subexpressions, including the arguments to non-rational functions. The result is returned as the quotient of two polynomials in a recursive form, that is, the coefficients of the main variable are polynomials in the other variables. Variables may include non-rational functions (e.g., sin (x^2 + 1)) and the arguments to any such functions are also rationally simplified.

ratsimp (expr, x_1, ..., x_n) enables rational simplification with the specification of variable ordering as in ratvars.

When ratsimpexpons is true, ratsimp is applied to the exponents of expressions during simplification.

See also ratexpand. Note that ratsimp is affected by some of the flags which affect ratexpand.

Examples:

(%i1) sin (x/(x^2 + x)) = exp ((log(x) + 1)^2 - log(x)^2);
                                         2      2
                   x         (log(x) + 1)  - log (x)
(%o1)        sin(------) = %e
                  2
                 x  + x
(%i2) ratsimp (%);
                             1          2
(%o2)                  sin(-----) = %e x
                           x + 1
(%i3) ((x - 1)^(3/2) - (x + 1)*sqrt(x - 1))/sqrt((x - 1)*(x + 1));
                       3/2
                (x - 1)    - sqrt(x - 1) (x + 1)
(%o3)           --------------------------------
                     sqrt((x - 1) (x + 1))
(%i4) ratsimp (%);
                           2 sqrt(x - 1)
(%o4)                    - -------------
                                 2
                           sqrt(x  - 1)
(%i5) x^(a + 1/a), ratsimpexpons: true;
                               2
                              a  + 1
                              ------
                                a
(%o5)                        x

Option variable: ratsimpexpons

Default value: false

When ratsimpexpons is true, ratsimp is applied to the exponents of expressions during simplification.

Option variable: radsubstflag

Default value: false

radsubstflag, if true, permits ratsubst to make substitutions such as u for sqrt (x) in x.

Function: ratsubst (a, b, c)

Substitutes a for b in c and returns the resulting expression. b may be a sum, product, power, etc.

ratsubst knows something of the meaning of expressions whereas subst does a purely syntactic substitution. Thus subst (a, x + y, x + y + z) returns x + y + z whereas ratsubst returns z + a.

When radsubstflag is true, ratsubst makes substitutions for radicals in expressions which don’t explicitly contain them.

ratsubst ignores the value true of the option variable keepfloat.

Examples:

(%i1) ratsubst (a, x*y^2, x^4*y^3 + x^4*y^8);
                              3      4
(%o1)                      a x  y + a
(%i2) cos(x)^4 + cos(x)^3 + cos(x)^2 + cos(x) + 1;
               4         3         2
(%o2)       cos (x) + cos (x) + cos (x) + cos(x) + 1
(%i3) ratsubst (1 - sin(x)^2, cos(x)^2, %);
            4           2                     2
(%o3)    sin (x) - 3 sin (x) + cos(x) (2 - sin (x)) + 3
(%i4) ratsubst (1 - cos(x)^2, sin(x)^2, sin(x)^4);
                        4           2
(%o4)                cos (x) - 2 cos (x) + 1
(%i5) radsubstflag: false$
(%i6) ratsubst (u, sqrt(x), x);
(%o6)                           x
(%i7) radsubstflag: true$
(%i8) ratsubst (u, sqrt(x), x);
                                2
(%o8)                          u

Function: ratvars (x_1, …, x_n)
Function: ratvars ()
System variable: ratvars

Declares main variables x_1, …, x_n for rational expressions. x_n, if present in a rational expression, is considered the main variable. Otherwise, x_[n-1] is considered the main variable if present, and so on through the preceding variables to x_1, which is considered the main variable only if none of the succeeding variables are present.

If a variable in a rational expression is not present in the ratvars list, it is given a lower priority than x_1.

The arguments to ratvars can be either variables or non-rational functions such as sin(x).

The variable ratvars is a list of the arguments of the function ratvars when it was called most recently. Each call to the function ratvars resets the list. ratvars () clears the list.

Option variable: ratvarswitch

Default value: true

Maxima keeps an internal list in the Lisp variable VARLIST of the main variables for rational expressions. If ratvarswitch is true, every evaluation starts with a fresh list VARLIST. This is the default behavior. Otherwise, the main variables from previous evaluations are not removed from the internal list VARLIST.

The main variables, which are declared with the function ratvars are not affected by the option variable ratvarswitch.

Examples:

If ratvarswitch is true, every evaluation starts with a fresh list VARLIST.

(%i1) ratvarswitch:true$

(%i2) rat(2*x+y^2);
                             2
(%o2)/R/                    y  + 2 x
(%i3) :lisp varlist
($X $Y)

(%i3) rat(2*a+b^2);
                             2
(%o3)/R/                    b  + 2 a

(%i4) :lisp varlist
($A $B)

If ratvarswitch is false, the main variables from the last evaluation are still present.

(%i4) ratvarswitch:false$

(%i5) rat(2*x+y^2);
                             2
(%o5)/R/                    y  + 2 x
(%i6) :lisp varlist
($X $Y)

(%i6) rat(2*a+b^2);
                             2
(%o6)/R/                    b  + 2 a

(%i7) :lisp varlist
($A $B $X $Y)

Function: ratweight
    ratweight (x_1, w_1, …, x_n, w_n)
    ratweight ()

Assigns a weight w_i to the variable x_i. This causes a term to be replaced by 0 if its weight exceeds the value of the variable ratwtlvl (default yields no truncation). The weight of a term is the sum of the products of the weight of a variable in the term times its power. For example, the weight of 3 x_1^2 x_2 is 2 w_1 + w_2. Truncation according to ratwtlvl is carried out only when multiplying or exponentiating canonical rational expressions (CRE).

ratweight () returns the cumulative list of weight assignments.

Note: The ratfac and ratweight schemes are incompatible and may not both be used at the same time.

Examples:

(%i1) ratweight (a, 1, b, 1);
(%o1)                     [a, 1, b, 1]
(%i2) expr1: rat(a + b + 1)$
(%i3) expr1^2;
                  2                  2
(%o3)/R/         b  + (2 a + 2) b + a  + 2 a + 1
(%i4) ratwtlvl: 1$
(%i5) expr1^2;
(%o5)/R/                  2 b + 2 a + 1

System variable: ratweights

Default value: []

ratweights is the list of weights assigned by ratweight. The list is cumulative: each call to ratweight places additional items in the list.

kill (ratweights) and save (ratweights) both work as expected.

Option variable: ratwtlvl

Default value: false

ratwtlvl is used in combination with the ratweight function to control the truncation of canonical rational expressions (CRE). For the default value of false, no truncation occurs.

Function: remainder
    remainder (p_1, p_2)
    remainder (p_1, p_2, x_1, …, x_n)

Returns the remainder of the polynomial p_1 divided by the polynomial p_2. The arguments x_1, …, x_n are interpreted as in ratvars.

remainder returns the second element of the two-element list returned by divide.

Function: resultant (p_1, p_2, x)

The function resultant computes the resultant of the two polynomials p_1 and p_2, eliminating the variable x. The resultant is a determinant of the coefficients of x in p_1 and p_2, which equals zero if and only if p_1 and p_2 have a non-constant factor in common.

If p_1 or p_2 can be factored, it may be desirable to call factor before calling resultant.

The option variable resultant controls which algorithm will be used to compute the resultant. See the option variable resultant.

The function bezout takes the same arguments as resultant and returns a matrix. The determinant of the return value is the desired resultant.

Examples:

(%i1) resultant(2*x^2+3*x+1, 2*x^2+x+1, x);
(%o1)                           8
(%i2) resultant(x+1, x+1, x);
(%o2)                           0
(%i3) resultant((x+1)*x, (x+1), x);
(%o3)                           0
(%i4) resultant(a*x^2+b*x+1, c*x + 2, x);
                         2
(%o4)                   c  - 2 b c + 4 a

(%i5) bezout(a*x^2+b*x+1, c*x+2, x);
                        [ 2 a  2 b - c ]
(%o5)                   [              ]
                        [  c      2    ]
(%i6) determinant(%);
(%o6)                   4 a - (2 b - c) c

Option variable: resultant

Default value: subres

The option variable resultant controls which algorithm will be used to compute the resultant with the function resultant. The possible values are:

subres

for the subresultant polynomial remainder sequence (PRS) algorithm,

mod

(not enabled) for the modular resultant algorithm, and

red

for the reduced polynomial remainder sequence (PRS) algorithm.

On most problems the default value subres should be best.

Option variable: savefactors

Default value: false

When savefactors is true, causes the factors of an expression which is a product of factors to be saved by certain functions in order to speed up later factorizations of expressions containing some of the same factors.

Function: showratvars (expr)

Returns a list of the canonical rational expression (CRE) variables in expression expr.

See also ratvars.

Function: sqfr (expr)

is similar to factor except that the polynomial factors are "square-free." That is, they have factors only of degree one. This algorithm, which is also used by the first stage of factor, utilizes the fact that a polynomial has in common with its n’th derivative all its factors of degree greater than n. Thus by taking greatest common divisors with the polynomial of the derivatives with respect to each variable in the polynomial, all factors of degree greater than 1 can be found.

Example:

(%i1) sqfr (4*x^4 + 4*x^3 - 3*x^2 - 4*x - 1);
                                2   2
(%o1)                  (2 x + 1)  (x  - 1)

Function: tellrat
    tellrat (p_1, …, p_n)
    tellrat ()

Adds to the ring of algebraic integers known to Maxima the elements which are the solutions of the polynomials p_1, …, p_n. Each argument p_i is a polynomial with integer coefficients.

tellrat (x) effectively means substitute 0 for x in rational functions.

tellrat () returns a list of the current substitutions.

algebraic must be set to true in order for the simplification of algebraic integers to take effect.

Maxima initially knows about the imaginary unit %i and all roots of integers.

There is a command untellrat which takes kernels and removes tellrat properties.

When tellrat’ing a multivariate polynomial, e.g., tellrat (x^2 - y^2), there would be an ambiguity as to whether to substitute y^2 for x^2 or vice versa. Maxima picks a particular ordering, but if the user wants to specify which, e.g. tellrat (y^2 = x^2) provides a syntax which says replace y^2 by x^2.

Examples:

(%i1) 10*(%i + 1)/(%i + 3^(1/3));
                           10 (%i + 1)
(%o1)                      -----------
                                  1/3
                            %i + 3
(%i2) ev (ratdisrep (rat(%)), algebraic);
             2/3      1/3              2/3      1/3
(%o2)    (4 3    - 2 3    - 4) %i + 2 3    + 4 3    - 2
(%i3) tellrat (1 + a + a^2);
                            2
(%o3)                     [a  + a + 1]
(%i4) 1/(a*sqrt(2) - 1) + a/(sqrt(3) + sqrt(2));
                      1                 a
(%o4)           ------------- + -----------------
                sqrt(2) a - 1   sqrt(3) + sqrt(2)
(%i5) ev (ratdisrep (rat(%)), algebraic);
         (7 sqrt(3) - 10 sqrt(2) + 2) a - 2 sqrt(2) - 1
(%o5)    ----------------------------------------------
                               7
(%i6) tellrat (y^2 = x^2);
                        2    2   2
(%o6)                 [y  - x , a  + a + 1]

Function: totaldisrep (expr)

Converts every subexpression of expr from canonical rational expressions (CRE) to general form and returns the result. If expr is itself in CRE form then totaldisrep is identical to ratdisrep.

totaldisrep may be useful for ratdisrepping expressions such as equations, lists, matrices, etc., which have some subexpressions in CRE form.

Function: untellrat (x_1, …, x_n)

Removes tellrat properties from x_1, …, x_n.


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15 Special Functions


Next: , Previous: , Up: Special Functions   [Contents][Index]

15.1 Introduction to Special Functions

Special function notation follows:

bessel_j (index, expr)         Bessel function, 1st kind
bessel_y (index, expr)         Bessel function, 2nd kind
bessel_i (index, expr)         Modified Bessel function, 1st kind
bessel_k (index, expr)         Modified Bessel function, 2nd kind

hankel_1 (v,z)                 Hankel function of the 1st kind
hankel_2 (v,z)                 Hankel function of the 2nd kind
struve_h (v,z)                 Struve H function
struve_l (v,z)                 Struve L function

assoc_legendre_p[v,u] (z)      Legendre function of degree v and order u 
assoc_legendre_q[v,u] (z)      Legendre function, 2nd kind

%f[p,q] ([], [], expr)         Generalized Hypergeometric function
gamma (z)                      Gamma function
gamma_incomplete_lower (a,z)   Lower incomplete gamma function
gamma_incomplete (a,z)         Tail of incomplete gamma function
hypergeometric (l1, l2, z)     Hypergeometric function
slommel
%m[u,k] (z)                    Whittaker function, 1st kind
%w[u,k] (z)                    Whittaker function, 2nd kind
erfc (z)                       Complement of the erf function

expintegral_e (v,z)            Exponential integral E
expintegral_e1 (z)             Exponential integral E1
expintegral_ei (z)             Exponential integral Ei
expintegral_li (z)             Logarithmic integral Li
expintegral_si (z)             Exponential integral Si
expintegral_ci (z)             Exponential integral Ci
expintegral_shi (z)            Exponential integral Shi
expintegral_chi (z)            Exponential integral Chi

kelliptic (z)                  Complete elliptic integral of the first 
                               kind (K)
parabolic_cylinder_d (v,z)     Parabolic cylinder D function
·

Category: Bessel functions · Category: Airy functions · Category: Special functions


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15.2 Bessel Functions

Function: bessel_j (v, z)

The Bessel function of the first kind of order v and argument z.

bessel_j is defined as

                inf
                ====       k  - v - 2 k  v + 2 k
                \     (- 1)  2          z
                 >    --------------------------
                /        k! gamma(v + k + 1)
                ====
                k = 0

although the infinite series is not used for computations.

Function: bessel_y (v, z)

The Bessel function of the second kind of order v and argument z.

bessel_y is defined as

              cos(%pi v) bessel_j(v, z) - bessel_j(-v, z)
              -------------------------------------------
                             sin(%pi v)

when v is not an integer. When v is an integer n, the limit as v approaches n is taken.

Function: bessel_i (v, z)

The modified Bessel function of the first kind of order v and argument z.

bessel_i is defined as

                    inf
                    ====   - v - 2 k  v + 2 k
                    \     2          z
                     >    -------------------
                    /     k! gamma(v + k + 1)
                    ====
                    k = 0

although the infinite series is not used for computations.

Function: bessel_k (v, z)

The modified Bessel function of the second kind of order v and argument z.

bessel_k is defined as

           %pi csc(%pi v) (bessel_i(-v, z) - bessel_i(v, z))
           -------------------------------------------------
                                  2

when v is not an integer. If v is an integer n, then the limit as v approaches n is taken.

Function: hankel_1 (v, z)

The Hankel function of the first kind of order v and argument z (A&S 9.1.3). hankel_1 is defined as

   bessel_j(v,z) + %i * bessel_y(v,z)

Maxima evaluates hankel_1 numerically for a complex order v and complex argument z in float precision. The numerical evaluation in bigfloat precision is not supported.

When besselexpand is true, hankel_1 is expanded in terms of elementary functions when the order v is half of an odd integer. See besselexpand.

Maxima knows the derivative of hankel_1 wrt the argument z.

Examples:

Numerical evaluation:

(%i1) hankel_1(1,0.5);
(%o1)        0.24226845767487 - 1.471472392670243 %i
(%i2) hankel_1(1,0.5+%i);
(%o2)       - 0.25582879948621 %i - 0.23957560188301

Expansion of hankel_1 when besselexpand is true:

(%i1) hankel_1(1/2,z),besselexpand:true;
               sqrt(2) sin(z) - sqrt(2) %i cos(z)
(%o1)          ----------------------------------
                       sqrt(%pi) sqrt(z)

Derivative of hankel_1 wrt the argument z. The derivative wrt the order v is not supported. Maxima returns a noun form:

(%i1) diff(hankel_1(v,z),z);
             hankel_1(v - 1, z) - hankel_1(v + 1, z)
(%o1)        ---------------------------------------
                                2
(%i2) diff(hankel_1(v,z),v);
                       d
(%o2)                  -- (hankel_1(v, z))
                       dv

Function: hankel_2 (v, z)

The Hankel function of the second kind of order v and argument z (A&S 9.1.4). hankel_2 is defined as

   bessel_j(v,z) - %i * bessel_y(v,z)

Maxima evaluates hankel_2 numerically for a complex order v and complex argument z in float precision. The numerical evaluation in bigfloat precision is not supported.

When besselexpand is true, hankel_2 is expanded in terms of elementary functions when the order v is half of an odd integer. See besselexpand.

Maxima knows the derivative of hankel_2 wrt the argument z.

For examples see hankel_1.

Option variable: besselexpand

Default value: false

Controls expansion of the Bessel functions when the order is half of an odd integer. In this case, the Bessel functions can be expanded in terms of other elementary functions. When besselexpand is true, the Bessel function is expanded.

(%i1) besselexpand: false$
(%i2) bessel_j (3/2, z);
                                    3
(%o2)                      bessel_j(-, z)
                                    2
(%i3) besselexpand: true$
(%i4) bessel_j (3/2, z);
                                        sin(z)   cos(z)
                       sqrt(2) sqrt(z) (------ - ------)
                                           2       z
                                          z
(%o4)                  ---------------------------------
                                   sqrt(%pi)

Function: scaled_bessel_i (v, z)

The scaled modified Bessel function of the first kind of order v and argument z. That is, scaled_bessel_i(v,z) = exp(-abs(z))*bessel_i(v, z). This function is particularly useful for calculating bessel_i for large z, which is large. However, maxima does not otherwise know much about this function. For symbolic work, it is probably preferable to work with the expression exp(-abs(z))*bessel_i(v, z).

Function: scaled_bessel_i0 (z)

Identical to scaled_bessel_i(0,z).

Function: scaled_bessel_i1 (z)

Identical to scaled_bessel_i(1,z).

Function: %s [u,v] (z)

Lommel’s little s[u,v](z) function. Probably Gradshteyn & Ryzhik 8.570.1.


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15.3 Airy Functions

The Airy functions Ai(x) and Bi(x) are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Section 10.4.

y = Ai(x) and y = Bi(x) are two linearly independent solutions of the Airy differential equation diff (y(x), x, 2) - x y(x) = 0.

If the argument x is a real or complex floating point number, the numerical value of the function is returned.

Function: airy_ai (x)

The Airy function Ai(x). (A&S 10.4.2)

The derivative diff (airy_ai(x), x) is airy_dai(x).

See also airy_bi, airy_dai, airy_dbi.

Function: airy_dai (x)

The derivative of the Airy function Ai airy_ai(x).

See airy_ai.

Function: airy_bi (x)

The Airy function Bi(x). (A&S 10.4.3)

The derivative diff (airy_bi(x), x) is airy_dbi(x).

See airy_ai, airy_dbi.

Function: airy_dbi (x)

The derivative of the Airy Bi function airy_bi(x).

See airy_ai and airy_bi.


Next: , Previous: , Up: Special Functions   [Contents][Index]

15.4 Gamma and factorial Functions

The gamma function and the related beta, psi and incomplete gamma functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 6.

Function: bffac (expr, n)

Bigfloat version of the factorial (shifted gamma) function. The second argument is how many digits to retain and return, it’s a good idea to request a couple of extra.

Function: bfpsi (n, z, fpprec)
Function: bfpsi0 (z, fpprec)

bfpsi is the polygamma function of real argument z and integer order n. bfpsi0 is the digamma function. bfpsi0 (z, fpprec) is equivalent to bfpsi (0, z, fpprec).

These functions return bigfloat values. fpprec is the bigfloat precision of the return value.

Function: cbffac (z, fpprec)

Complex bigfloat factorial.

load ("bffac") loads this function.

Function: gamma (z)

The basic definition of the gamma function (A&S 6.1.1) is

                         inf
                        /
                        [     z - 1   - t
             gamma(z) = I    t      %e    dt
                        ]
                        /
                         0

Maxima simplifies gamma for positive integer and positive and negative rational numbers. For half integral values the result is a rational number times sqrt(%pi). The simplification for integer values is controlled by factlim. For integers greater than factlim the numerical result of the factorial function, which is used to calculate gamma, will overflow. The simplification for rational numbers is controlled by gammalim to avoid internal overflow. See factlim and gammalim.

For negative integers gamma is not defined.

Maxima can evalute gamma numerically for real and complex values in float and bigfloat precision.

gamma has mirror symmetry.

When gamma_expand is true, Maxima expands gamma for arguments z+n and z-n where n is an integer.

Maxima knows the derivate of gamma.

Examples:

Simplification for integer, half integral, and rational numbers:

(%i1) map('gamma,[1,2,3,4,5,6,7,8,9]);
(%o1)        [1, 1, 2, 6, 24, 120, 720, 5040, 40320]
(%i2) map('gamma,[1/2,3/2,5/2,7/2]);
                    sqrt(%pi)  3 sqrt(%pi)  15 sqrt(%pi)
(%o2)   [sqrt(%pi), ---------, -----------, ------------]
                        2           4            8
(%i3) map('gamma,[2/3,5/3,7/3]);
                                  2           1
                          2 gamma(-)  4 gamma(-)
                      2           3           3
(%o3)          [gamma(-), ----------, ----------]
                      3       3           9

Numerical evaluation for real and complex values:

(%i4) map('gamma,[2.5,2.5b0]);
(%o4)     [1.329340388179137, 1.3293403881791370205b0]
(%i5) map('gamma,[1.0+%i,1.0b0+%i]);
(%o5) [0.498015668118356 - .1549498283018107 %i, 
          4.9801566811835604272b-1 - 1.5494982830181068513b-1 %i]

gamma has mirror symmetry:

(%i6) declare(z,complex)$
(%i7) conjugate(gamma(z));
(%o7)                  gamma(conjugate(z))

Maxima expands gamma(z+n) and gamma(z-n), when gamma_expand is true:

(%i8) gamma_expand:true$

(%i9) [gamma(z+1),gamma(z-1),gamma(z+2)/gamma(z+1)];
                               gamma(z)
(%o9)             [z gamma(z), --------, z + 1]
                                z - 1

The deriviative of gamma:

(%i10) diff(gamma(z),z);
(%o10)                  psi (z) gamma(z)
                           0

See also makegamma.

The Euler-Mascheroni constant is %gamma.

Function: log_gamma (z)

The natural logarithm of the gamma function.

Function: gamma_incomplete_lower (a, z)

The lower incomplete gamma function (A&S 6.5.2):

                                    z
                                   /
                                   [  a - 1   - t
    gamma_incomplete_lower(a, z) = I t      %e    dt
                                   ]
                                   /
                                    0

See also gamma_incomplete (upper incomplete gamma function).

Function: gamma_incomplete (a, z)

The incomplete upper gamma function (A&S 6.5.3):

                              inf
                             /
                             [     a - 1   - t
    gamma_incomplete(a, z) = I    t      %e    dt
                             ]
                             /
                              z

See also gamma_expand for controlling how gamma_incomplete is expressed in terms of elementary functions and erfc.

Also see the related functions gamma_incomplete_regularized and gamma_incomplete_generalized.

Function: gamma_incomplete_regularized (a, z)

The regularized incomplete upper gamma function (A&S 6.5.1):

gamma_incomplete_regularized(a, z) = 
                                        gamma_incomplete(a, z)
                                        ----------------------
                                               gamma(a)

See also gamma_expand for controlling how gamma_incomplete is expressed in terms of elementary functions and erfc.

Also see gamma_incomplete.

Function: gamma_incomplete_generalized (a, z1, z1)

The generalized incomplete gamma function.

gamma_incomplete_generalized(a, z1, z2) = 
                                               z2
                                              /
                                              [    a - 1   - t
                                              I   t      %e    dt
                                              ]
                                              /
                                               z1

Also see gamma_incomplete and gamma_incomplete_regularized.

Option variable: gamma_expand

Default value: false

gamma_expand controls expansion of gamma_incomplete. When gamma_expand is true, gamma_incomplete(v,z) is expanded in terms of z, exp(z), and erfc(z) when possible.

(%i1) gamma_incomplete(2,z);
(%o1)                       gamma_incomplete(2, z)
(%i2) gamma_expand:true;
(%o2)                                true
(%i3) gamma_incomplete(2,z);
                                           - z
(%o3)                            (z + 1) %e
(%i4) gamma_incomplete(3/2,z);
                              - z   sqrt(%pi) erfc(sqrt(z))
(%o4)               sqrt(z) %e    + -----------------------
                                               2

Option variable: gammalim

Default value: 10000

gammalim controls simplification of the gamma function for integral and rational number arguments. If the absolute value of the argument is not greater than gammalim, then simplification will occur. Note that the factlim switch controls simplification of the result of gamma of an integer argument as well.

Function: makegamma (expr)

Transforms instances of binomial, factorial, and beta functions in expr into gamma functions.

See also makefact.

Function: beta (a, b)

The beta function is defined as gamma(a) gamma(b)/gamma(a+b) (A&S 6.2.1).

Maxima simplifies the beta function for positive integers and rational numbers, which sum to an integer. When beta_args_sum_to_integer is true, Maxima simplifies also general expressions which sum to an integer.

For a or b equal to zero the beta function is not defined.

In general the beta function is not defined for negative integers as an argument. The exception is for a=-n, n a positive integer and b a positive integer with b<=n, it is possible to define an analytic continuation. Maxima gives for this case a result.

When beta_expand is true, expressions like beta(a+n,b) and beta(a-n,b) or beta(a,b+n) and beta(a,b-n) with n an integer are simplified.

Maxima can evaluate the beta function for real and complex values in float and bigfloat precision. For numerical evaluation Maxima uses log_gamma:

           - log_gamma(b + a) + log_gamma(b) + log_gamma(a)
         %e

Maxima knows that the beta function is symmetric and has mirror symmetry.

Maxima knows the derivatives of the beta function with respect to a or b.

To express the beta function as a ratio of gamma functions see makegamma.

Examples:

Simplification, when one of the arguments is an integer:

(%i1) [beta(2,3),beta(2,1/3),beta(2,a)];
                               1   9      1
(%o1)                         [--, -, ---------]
                               12  4  a (a + 1)

Simplification for two rational numbers as arguments which sum to an integer:

(%i2) [beta(1/2,5/2),beta(1/3,2/3),beta(1/4,3/4)];
                          3 %pi   2 %pi
(%o2)                    [-----, -------, sqrt(2) %pi]
                            8    sqrt(3)

When setting beta_args_sum_to_integer to true more general expression are simplified, when the sum of the arguments is an integer:

(%i3) beta_args_sum_to_integer:true$
(%i4) beta(a+1,-a+2);
                                %pi (a - 1) a
(%o4)                         ------------------
                              2 sin(%pi (2 - a))

The possible results, when one of the arguments is a negative integer:

(%i5) [beta(-3,1),beta(-3,2),beta(-3,3)];
                                    1  1    1
(%o5)                            [- -, -, - -]
                                    3  6    3

beta(a+n,b) or beta(a-n) with n an integer simplifies when beta_expand is true:

(%i6) beta_expand:true$
(%i7) [beta(a+1,b),beta(a-1,b),beta(a+1,b)/beta(a,b+1)];
                    a beta(a, b)  beta(a, b) (b + a - 1)  a
(%o7)              [------------, ----------------------, -]
                       b + a              a - 1           b

Beta is not defined, when one of the arguments is zero:

(%i7) beta(0,b);
beta: expected nonzero arguments; found 0, b
 -- an error.  To debug this try debugmode(true);

Numercial evaluation for real and complex arguments in float or bigfloat precision:

(%i8) beta(2.5,2.3);
(%o8) .08694748611299981

(%i9) beta(2.5,1.4+%i);
(%o9) 0.0640144950796695 - .1502078053286415 %i

(%i10) beta(2.5b0,2.3b0);
(%o10) 8.694748611299969b-2

(%i11) beta(2.5b0,1.4b0+%i);
(%o11) 6.401449507966944b-2 - 1.502078053286415b-1 %i

Beta is symmetric and has mirror symmetry:

(%i14) beta(a,b)-beta(b,a);
(%o14)                                 0
(%i15) declare(a,complex,b,complex)$
(%i16) conjugate(beta(a,b));
(%o16)                 beta(conjugate(a), conjugate(b))

The derivative of the beta function wrt a:

(%i17) diff(beta(a,b),a);
(%o17)               - beta(a, b) (psi (b + a) - psi (a))
                                      0             0

Function: beta_incomplete (a, b, z)

The basic definition of the incomplete beta function (A&S 6.6.1) is

        z
       /
       [         b - 1  a - 1
       I  (1 - t)      t      dt
       ]
       /
        0

This definition is possible for realpart(a)>0 and realpart(b)>0 and abs(z)<1. For other values the incomplete beta function can be defined through a generalized hypergeometric function:

   gamma(a) hypergeometric_generalized([a, 1 - b], [a + 1], z) z

(See functions.wolfram.com for a complete definition of the incomplete beta function.)

For negative integers a = -n and positive integers b=m with m<=n the incomplete beta function is defined through

                            m - 1           k
                            ====  (1 - m)  z
                      n - 1 \            k
                     z       >    -----------
                            /     k! (n - k)
                            ====
                            k = 0

Maxima uses this definition to simplify beta_incomplete for a a negative integer.

For a a positive integer, beta_incomplete simplifies for any argument b and z and for b a positive integer for any argument a and z, with the exception of a a negative integer.

For z=0 and realpart(a)>0, beta_incomplete has the specific value zero. For z=1 and realpart(b)>0, beta_incomplete simplifies to the beta function beta(a,b).

Maxima evaluates beta_incomplete numerically for real and complex values in float or bigfloat precision. For the numerical evaluation an expansion of the incomplete beta function in continued fractions is used.

When the option variable beta_expand is true, Maxima expands expressions like beta_incomplete(a+n,b,z) and beta_incomplete(a-n,b,z) where n is a positive integer.

Maxima knows the derivatives of beta_incomplete with respect to the variables a, b and z and the integral with respect to the variable z.

Examples:

Simplification for a a positive integer:

(%i1) beta_incomplete(2,b,z);
                                       b
                            1 - (1 - z)  (b z + 1)
(%o1)                       ----------------------
                                  b (b + 1)

Simplification for b a positive integer:

(%i2) beta_incomplete(a,2,z);
                                               a
                              (a (1 - z) + 1) z
(%o2)                         ------------------
                                  a (a + 1)

Simplification for a and b a positive integer:

(%i3) beta_incomplete(3,2,z);
                                               3
                              (3 (1 - z) + 1) z
(%o3)                         ------------------
                                      12

a is a negative integer and b<=(-a), Maxima simplifies:

(%i4) beta_incomplete(-3,1,z);
                                       1
(%o4)                              - ----
                                        3
                                     3 z

For the specific values z=0 and z=1, Maxima simplifies:

(%i5) assume(a>0,b>0)$
(%i6) beta_incomplete(a,b,0);
(%o6)                                 0
(%i7) beta_incomplete(a,b,1);
(%o7)                            beta(a, b)

Numerical evaluation in float or bigfloat precision:

(%i8) beta_incomplete(0.25,0.50,0.9);
(%o8)                          4.594959440269333
(%i9)  fpprec:25$
(%i10) beta_incomplete(0.25,0.50,0.9b0);
(%o10)                    4.594959440269324086971203b0

For abs(z)>1 beta_incomplete returns a complex result:

(%i11) beta_incomplete(0.25,0.50,1.7);
(%o11)              5.244115108584249 - 1.45518047787844 %i

Results for more general complex arguments:

(%i14) beta_incomplete(0.25+%i,1.0+%i,1.7+%i);
(%o14)             2.726960675662536 - .3831175704269199 %i
(%i15) beta_incomplete(1/2,5/4*%i,2.8+%i);
(%o15)             13.04649635168716 %i - 5.802067956270001
(%i16) 

Expansion, when beta_expand is true:

(%i23) beta_incomplete(a+1,b,z),beta_expand:true;
                                                       b  a
                   a beta_incomplete(a, b, z)   (1 - z)  z
(%o23)             -------------------------- - -----------
                             b + a                 b + a

(%i24) beta_incomplete(a-1,b,z),beta_expand:true;
                                                           b  a - 1
           beta_incomplete(a, b, z) (- b - a + 1)   (1 - z)  z
(%o24)     -------------------------------------- - ---------------
                           1 - a                         1 - a

Derivative and integral for beta_incomplete:

(%i34) diff(beta_incomplete(a, b, z), z);
                              b - 1  a - 1
(%o34)                 (1 - z)      z
(%i35) integrate(beta_incomplete(a, b, z), z);
              b  a
       (1 - z)  z
(%o35) ----------- + beta_incomplete(a, b, z) z
          b + a
                                       a beta_incomplete(a, b, z)
                                     - --------------------------
                                                 b + a
(%i36) factor(diff(%, z));
(%o36)              beta_incomplete(a, b, z)

Function: beta_incomplete_regularized (a, b, z)

The regularized incomplete beta function (A&S 6.6.2), defined as

beta_incomplete_regularized(a, b, z) = 
                                      beta_incomplete(a, b, z)
                                      ------------------------
                                             beta(a, b)

As for beta_incomplete this definition is not complete. See functions.wolfram.com for a complete definition of beta_incomplete_regularized.

beta_incomplete_regularized simplifies a or b a positive integer.

For z=0 and realpart(a)>0, beta_incomplete_regularized has the specific value 0. For z=1 and realpart(b)>0, beta_incomplete_regularized simplifies to 1.

Maxima can evaluate beta_incomplete_regularized for real and complex arguments in float and bigfloat precision.

When beta_expand is true, Maxima expands beta_incomplete_regularized for arguments a+n or a-n, where n is an integer.

Maxima knows the derivatives of beta_incomplete_regularized with respect to the variables a, b, and z and the integral with respect to the variable z.

Examples:

Simplification for a or b a positive integer:

(%i1) beta_incomplete_regularized(2,b,z);
                                       b
(%o1)                       1 - (1 - z)  (b z + 1)

(%i2) beta_incomplete_regularized(a,2,z);
                                               a
(%o2)                         (a (1 - z) + 1) z

(%i3) beta_incomplete_regularized(3,2,z);
                                               3
(%o3)                         (3 (1 - z) + 1) z

For the specific values z=0 and z=1, Maxima simplifies:

(%i4) assume(a>0,b>0)$
(%i5) beta_incomplete_regularized(a,b,0);
(%o5)                                 0
(%i6) beta_incomplete_regularized(a,b,1);
(%o6)                                 1

Numerical evaluation for real and complex arguments in float and bigfloat precision:

(%i7) beta_incomplete_regularized(0.12,0.43,0.9);
(%o7)                         .9114011367359802
(%i8) fpprec:32$
(%i9) beta_incomplete_regularized(0.12,0.43,0.9b0);
(%o9)               9.1140113673598075519946998779975b-1
(%i10) beta_incomplete_regularized(1+%i,3/3,1.5*%i);
(%o10)             .2865367499935403 %i - 0.122995963334684
(%i11) fpprec:20$
(%i12) beta_incomplete_regularized(1+%i,3/3,1.5b0*%i);
(%o12)      2.8653674999354036142b-1 %i - 1.2299596333468400163b-1

Expansion, when beta_expand is true:

(%i13) beta_incomplete_regularized(a+1,b,z);
                                                     b  a
                                              (1 - z)  z
(%o13) beta_incomplete_regularized(a, b, z) - ------------
                                              a beta(a, b)
(%i14) beta_incomplete_regularized(a-1,b,z);
(%o14) beta_incomplete_regularized(a, b, z)
                                                     b  a - 1
                                              (1 - z)  z
                                         - ----------------------
                                           beta(a, b) (b + a - 1)

The derivative and the integral wrt z:

(%i15) diff(beta_incomplete_regularized(a,b,z),z);
                              b - 1  a - 1
                       (1 - z)      z
(%o15)                 -------------------
                           beta(a, b)
(%i16) integrate(beta_incomplete_regularized(a,b,z),z);
(%o16) beta_incomplete_regularized(a, b, z) z
                                                           b  a
                                                    (1 - z)  z
          a (beta_incomplete_regularized(a, b, z) - ------------)
                                                    a beta(a, b)
        - -------------------------------------------------------
                                   b + a

Function: beta_incomplete_generalized (a, b, z1, z2)

The basic definition of the generalized incomplete beta function is

             z2
           /
           [          b - 1  a - 1
           I   (1 - t)      t      dt
           ]
           /
            z1

Maxima simplifies beta_incomplete_regularized for a and b a positive integer.

For realpart(a)>0 and z1=0 or z2=0, Maxima simplifies beta_incomplete_generalized to beta_incomplete. For realpart(b)>0 and z1=1 or z2=1, Maxima simplifies to an expression with beta and beta_incomplete.

Maxima evaluates beta_incomplete_regularized for real and complex values in float and bigfloat precision.

When beta_expand is true, Maxima expands beta_incomplete_generalized for a+n and a-n, n a positive integer.

Maxima knows the derivative of beta_incomplete_generalized with respect to the variables a, b, z1, and z2 and the integrals with respect to the variables z1 and z2.

Examples:

Maxima simplifies beta_incomplete_generalized for a and b a positive integer:

(%i1) beta_incomplete_generalized(2,b,z1,z2);
                   b                      b
           (1 - z1)  (b z1 + 1) - (1 - z2)  (b z2 + 1)
(%o1)      -------------------------------------------
                            b (b + 1)
(%i2) beta_incomplete_generalized(a,2,z1,z2);
                              a                      a
           (a (1 - z2) + 1) z2  - (a (1 - z1) + 1) z1
(%o2)      -------------------------------------------
                            a (a + 1)
(%i3) beta_incomplete_generalized(3,2,z1,z2);
              2      2                       2      2
      (1 - z1)  (3 z1  + 2 z1 + 1) - (1 - z2)  (3 z2  + 2 z2 + 1)
(%o3) -----------------------------------------------------------
                                  12

Simplification for specific values z1=0, z2=0, z1=1, or z2=1:

(%i4) assume(a > 0, b > 0)$
(%i5) beta_incomplete_generalized(a,b,z1,0);
(%o5)                    - beta_incomplete(a, b, z1)

(%i6) beta_incomplete_generalized(a,b,0,z2);
(%o6)                    - beta_incomplete(a, b, z2)

(%i7) beta_incomplete_generalized(a,b,z1,1);
(%o7)              beta(a, b) - beta_incomplete(a, b, z1)

(%i8) beta_incomplete_generalized(a,b,1,z2);
(%o8)              beta_incomplete(a, b, z2) - beta(a, b)

Numerical evaluation for real arguments in float or bigfloat precision:

(%i9) beta_incomplete_generalized(1/2,3/2,0.25,0.31);
(%o9)                        .09638178086368676

(%i10) fpprec:32$
(%i10) beta_incomplete_generalized(1/2,3/2,0.25,0.31b0);
(%o10)               9.6381780863686935309170054689964b-2

Numerical evaluation for complex arguments in float or bigfloat precision:

(%i11) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31);
(%o11)           - .09625463003205376 %i - .003323847735353769
(%i12) fpprec:20$
(%i13) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31b0);
(%o13)     - 9.6254630032054178691b-2 %i - 3.3238477353543591914b-3

Expansion for a+n or a-n, n a positive integer, when beta_expand is true:

(%i14) beta_expand:true$

(%i15) beta_incomplete_generalized(a+1,b,z1,z2);

               b   a           b   a
       (1 - z1)  z1  - (1 - z2)  z2
(%o15) -----------------------------
                   b + a
                      a beta_incomplete_generalized(a, b, z1, z2)
                    + -------------------------------------------
                                         b + a
(%i16) beta_incomplete_generalized(a-1,b,z1,z2);

       beta_incomplete_generalized(a, b, z1, z2) (- b - a + 1)
(%o16) -------------------------------------------------------
                                1 - a
                                    b   a - 1           b   a - 1
                            (1 - z2)  z2      - (1 - z1)  z1
                          - -------------------------------------
                                            1 - a

Derivative wrt the variable z1 and integrals wrt z1 and z2:

(%i17) diff(beta_incomplete_generalized(a,b,z1,z2),z1);
                               b - 1   a - 1
(%o17)               - (1 - z1)      z1
(%i18) integrate(beta_incomplete_generalized(a,b,z1,z2),z1);
(%o18) beta_incomplete_generalized(a, b, z1, z2) z1
                                  + beta_incomplete(a + 1, b, z1)
(%i19) integrate(beta_incomplete_generalized(a,b,z1,z2),z2);
(%o19) beta_incomplete_generalized(a, b, z1, z2) z2
                                  - beta_incomplete(a + 1, b, z2)

Option variable: beta_expand

Default value: false

When beta_expand is true, beta(a,b) and related functions are expanded for arguments like a+n or a-n, where n is an integer.

Option variable: beta_args_sum_to_integer

Default value: false

When beta_args_sum_to_integer is true, Maxima simplifies beta(a,b), when the arguments a and b sum to an integer.

Function: psi [n](x)

The derivative of log (gamma (x)) of order n+1. Thus, psi[0](x) is the first derivative, psi[1](x) is the second derivative, etc.

Maxima does not know how, in general, to compute a numerical value of psi, but it can compute some exact values for rational args. Several variables control what range of rational args psi will return an exact value, if possible. See maxpsiposint, maxpsinegint, maxpsifracnum, and maxpsifracdenom. That is, x must lie between maxpsinegint and maxpsiposint. If the absolute value of the fractional part of x is rational and has a numerator less than maxpsifracnum and has a denominator less than maxpsifracdenom, psi will return an exact value.

The function bfpsi in the bffac package can compute numerical values.

Option variable: maxpsiposint

Default value: 20

maxpsiposint is the largest positive value for which psi[n](x) will try to compute an exact value.

Option variable: maxpsinegint

Default value: -10

maxpsinegint is the most negative value for which psi[n](x) will try to compute an exact value. That is if x is less than maxnegint, psi[n](x) will not return simplified answer, even if it could.

Option variable: maxpsifracnum

Default value: 6

Let x be a rational number less than one of the form p/q. If p is greater than maxpsifracnum, then psi[n](x) will not try to return a simplified value.

Option variable: maxpsifracdenom

Default value: 6

Let x be a rational number less than one of the form p/q. If q is greater than maxpsifracdenom, then psi[n](x) will not try to return a simplified value.

Function: makefact (expr)

Transforms instances of binomial, gamma, and beta functions in expr into factorials.

See also makegamma.

Function: numfactor (expr)

Returns the numerical factor multiplying the expression expr, which should be a single term.

content returns the greatest common divisor (gcd) of all terms in a sum.

(%i1) gamma (7/2);
                          15 sqrt(%pi)
(%o1)                     ------------
                               8
(%i2) numfactor (%);
                               15
(%o2)                          --
                               8


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15.5 Exponential Integrals

The Exponential Integral and related funtions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 5

Function: expintegral_e1 (z)

The Exponential Integral E1(z) (A&S 5.1.1) defined as

Function: expintegral_ei (z)

The Exponential Integral Ei(z) (A&S 5.1.2)

Function: expintegral_li (z)

The Exponential Integral Li(z) (A&S 5.1.3)

Function: expintegral_e (n,z)

The Exponential Integral En(z) (A&S 5.1.4) defined as

Function: expintegral_si (z)

The Exponential Integral Si(z) (A&S 5.2.1) defined as

Function: expintegral_ci (z)

The Exponential Integral Ci(z) (A&S 5.2.2) defined as

Function: expintegral_shi (z)

The Exponential Integral Shi(z) (A&S 5.2.3) defined as

Function: expintegral_chi (z)

The Exponential Integral Chi(z) (A&S 5.2.4) defined as

Option variable: expintrep

Default value: false

Change the representation of one of the exponential integrals, expintegral_e(m, z), expintegral_e1, or expintegral_ei to an equivalent form if possible.

Possible values for expintrep are false, gamma_incomplete, expintegral_e1, expintegral_ei, expintegral_li, expintegral_trig, or expintegral_hyp.

false means that the representation is not changed. Other values indicate the representation is to be changed to use the function specified where expintegral_trig means expintegral_si, expintegral_ci, and expintegral_hyp means expintegral_shi or expintegral_chi.

Option variable: expintexpand

Default value: false

Expand the Exponential Integral E[n](z) for half integral values in terms of Erfc or Erf and for positive integers in terms of Ei


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15.6 Error Function

The Error function and related funtions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 7

Function: erf (z)

The Error Function erf(z) (A&S 7.1.1)

See also flag erfflag.

Function: erfc (z)

The Complementary Error Function erfc(z) (A&S 7.1.2)

erfc(z) = 1-erf(z)

Function: erfi (z)

The Imaginary Error Function.

erfi(z) = -%i*erf(%i*z)

Function: erf_generalized (z1,z2)

Generalized Error function Erf(z1,z2)

Function: fresnel_c (z)

The Fresnel Integral C(z) = integrate(cos((%pi/2)*t^2),t,0,z). (A&S 7.3.1)

The simplification fresnel_c(-x) = -fresnel_c(x) is applied when flag trigsign is true.

The simplification fresnel_c(%i*x) = %i*fresnel_c(x) is applied when flag %iargs is true.

See flags erf_representation and hypergeometric_representation.

Function: fresnel_s (z)

The Fresnel Integral S(z) = integrate(sin((%pi/2)*t^2),t,0,z). (A&S 7.3.2)

The simplification fresnel_s(-x) = -fresnel_s(x) is applied when flag trigsign is true.

The simplification fresnel_s(%i*x) = -%i*fresnel_s(x) is applied when flag %iargs is true.

See flags erf_representation and hypergeometric_representation.

Option variable: erf_representation

Default value: false

When T erfc, erfi, erf_generalized, fresnel_s and fresnel_c are transformed to erf.

Option variable: hypergeometric_representation

Default value: false

Enables transformation to a Hypergeometric representation for fresnel_s and fresnel_c


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15.7 Struve Functions

The Struve functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 12.

Function: struve_h (v, z)

The Struve Function H of order v and argument z. (A&S 12.1.1)

Function: struve_l (v, z)

The Modified Struve Function L of order v and argument z. (A&S 12.2.1)


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15.8 Hypergeometric Functions

The Hypergeometric Functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapters 13 and 15.

Maxima has very limited knowledge of these functions. They can be returned from function hgfred.

Function: %m [k,u] (z)

Whittaker M function M[k,u](z) = exp(-z/2)*z^(1/2+u)*M(1/2+u-k,1+2*u,z). (A&S 13.1.32)

Function: %w [k,u] (z)

Whittaker W function. (A&S 13.1.33)

Function: %f [p,q] ([a],[b],z)

The pFq(a1,a2,..ap;b1,b2,..bq;z) hypergeometric function, where a a list of length p and b a list of length q.

Function: hypergeometric ([a1, ..., ap],[b1, ... ,bq], x)

The hypergeometric function. Unlike Maxima’s %f hypergeometric function, the function hypergeometric is a simplifying function; also, hypergeometric supports complex double and big floating point evaluation. For the Gauss hypergeometric function, that is p = 2 and q = 1, floating point evaluation outside the unit circle is supported, but in general, it is not supported.

When the option variable expand_hypergeometric is true (default is false) and one of the arguments a1 through ap is a negative integer (a polynomial case), hypergeometric returns an expanded polynomial.

Examples:

(%i1)  hypergeometric([],[],x);
(%o1) %e^x

Polynomial cases automatically expand when expand_hypergeometric is true:

(%i2) hypergeometric([-3],[7],x);
(%o2) hypergeometric([-3],[7],x)

(%i3) hypergeometric([-3],[7],x), expand_hypergeometric : true;
(%o3) -x^3/504+3*x^2/56-3*x/7+1

Both double float and big float evaluation is supported:

(%i4) hypergeometric([5.1],[7.1 + %i],0.42);
(%o4)       1.346250786375334 - 0.0559061414208204 %i
(%i5) hypergeometric([5,6],[8], 5.7 - %i);
(%o5)     .007375824009774946 - .001049813688578674 %i
(%i6) hypergeometric([5,6],[8], 5.7b0 - %i), fpprec : 30;
(%o6) 7.37582400977494674506442010824b-3
                          - 1.04981368857867315858055393376b-3 %i

Next: , Previous: , Up: Special Functions   [Contents][Index]

15.9 Parabolic Cylinder Functions

The Parabolic Cylinder Functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 19.

Maxima has very limited knowledge of these functions. They can be returned from function hgfred.

Function: parabolic_cylinder_d (v, z)

The parabolic cylinder function parabolic_cylinder_d(v,z). (A&S 19.3.1)


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15.10 Functions and Variables for Special Functions

Function: specint (exp(- s*t) * expr, t)

Compute the Laplace transform of expr with respect to the variable t. The integrand expr may contain special functions. The parameter s maybe be named something else; it is determined automatically, as the examples below show where p is used in some places.

The following special functions are handled by specint: incomplete gamma function, error functions (but not the error function erfi, it is easy to transform erfi e.g. to the error function erf), exponential integrals, bessel functions (including products of bessel functions), hankel functions, hermite and the laguerre polynomials.

Furthermore, specint can handle the hypergeometric function %f[p,q]([],[],z), the whittaker function of the first kind %m[u,k](z) and of the second kind %w[u,k](z).

The result may be in terms of special functions and can include unsimplified hypergeometric functions.

When laplace fails to find a Laplace transform, specint is called. Because laplace knows more general rules for Laplace transforms, it is preferable to use laplace and not specint.

demo("hypgeo") displays several examples of Laplace transforms computed by specint.

Examples:

(%i1) assume (p > 0, a > 0)$
(%i2) specint (t^(1/2) * exp(-a*t/4) * exp(-p*t), t);
                           sqrt(%pi)
(%o2)                     ------------
                                 a 3/2
                          2 (p + -)
                                 4
(%i3) specint (t^(1/2) * bessel_j(1, 2 * a^(1/2) * t^(1/2))
              * exp(-p*t), t);
                                   - a/p
                         sqrt(a) %e
(%o3)                    ---------------
                                2
                               p

Examples for exponential integrals:

(%i4) assume(s>0,a>0,s-a>0)$
(%i5) ratsimp(specint(%e^(a*t)
                      *(log(a)+expintegral_e1(a*t))*%e^(-s*t),t));
                             log(s)
(%o5)                        ------
                             s - a
(%i6) logarc:true$

(%i7) gamma_expand:true$

radcan(specint((cos(t)*expintegral_si(t)
                     -sin(t)*expintegral_ci(t))*%e^(-s*t),t));
                             log(s)
(%o8)                        ------
                              2
                             s  + 1
ratsimp(specint((2*t*log(a)+2/a*sin(a*t)
                      -2*t*expintegral_ci(a*t))*%e^(-s*t),t));
                               2    2
                          log(s  + a )
(%o9)                     ------------
                                2
                               s

Results when using the expansion of gamma_incomplete and when changing the representation to expintegral_e1:

(%i10) assume(s>0)$
(%i11) specint(1/sqrt(%pi*t)*unit_step(t-k)*%e^(-s*t),t);
                                            1
                            gamma_incomplete(-, k s)
                                            2
(%o11)                      ------------------------
                               sqrt(%pi) sqrt(s)

(%i12) gamma_expand:true$
(%i13) specint(1/sqrt(%pi*t)*unit_step(t-k)*%e^(-s*t),t);
                              erfc(sqrt(k) sqrt(s))
(%o13)                        ---------------------
                                     sqrt(s)

(%i14) expintrep:expintegral_e1$
(%i15) ratsimp(specint(1/(t+a)^2*%e^(-s*t),t));
                              a s
                        a s %e    expintegral_e1(a s) - 1
(%o15)                - ---------------------------------
                                        a

Function: hypergeometric_simp (e)

hypergeometric_simp simplifies hypergeometric functions by applying hgfred to the arguments of any hypergeometric functions in the expression e.

Only instances of hypergeometric are affected; any %f, %w, and %m in the expression e are not affected. Any unsimplified hypergeometric functions are returned unchanged (instead of changing to %f as hgfred would).

load(hypergeometric); loads this function.

See also hgfred.

Examples:

(%i1) load ("hypergeometric") $
(%i2) foo : [hypergeometric([1,1], [2], z), hypergeometric([1/2], [1], z)];
(%o2) [hypergeometric([1, 1], [2], z), 
                                                     1
                                     hypergeometric([-], [1], z)]
                                                     2
(%i3) hypergeometric_simp (foo);
                 log(1 - z)              z    z/2
(%o3)         [- ----------, bessel_i(0, -) %e   ]
                     z                   2
(%i4) bar : hypergeometric([n], [m], z + 1);
(%o4)            hypergeometric([n], [m], z + 1)
(%i5) hypergeometric_simp (bar);
(%o5)            hypergeometric([n], [m], z + 1)

Function: hgfred (a, b, t)

Simplify the generalized hypergeometric function in terms of other, simpler, forms. a is a list of numerator parameters and b is a list of the denominator parameters.

If hgfred cannot simplify the hypergeometric function, it returns an expression of the form %f[p,q]([a], [b], x) where p is the number of elements in a, and q is the number of elements in b. This is the usual pFq generalized hypergeometric function.

(%i1) assume(not(equal(z,0)));
(%o1)                          [notequal(z, 0)]
(%i2) hgfred([v+1/2],[2*v+1],2*%i*z);

                     v/2                               %i z
                    4    bessel_j(v, z) gamma(v + 1) %e
(%o2)               ---------------------------------------
                                       v
                                      z
(%i3) hgfred([1,1],[2],z);

                                   log(1 - z)
(%o3)                            - ----------
                                       z
(%i4) hgfred([a,a+1/2],[3/2],z^2);

                               1 - 2 a          1 - 2 a
                        (z + 1)        - (1 - z)
(%o4)                   -------------------------------
                                 2 (1 - 2 a) z

It can be beneficial to load orthopoly too as the following example shows. Note that L is the generalized Laguerre polynomial.

(%i5) load(orthopoly)$
(%i6) hgfred([-2],[a],z);

                                    (a - 1)
                                 2 L       (z)
                                    2
(%o6)                            -------------
                                   a (a + 1)
(%i7) ev(%);

                                  2
                                 z        2 z
(%o7)                         --------- - --- + 1
                              a (a + 1)    a

Function: lambert_w (z)

The principal branch of Lambert’s W function W(z), the solution of z = W(z) * exp(W(z)). (DLMF 4.13)

Function: generalized_lambert_w (k, z)

The k-th branch of Lambert’s W function W(z), the solution of z = W(z) * exp(W(z)). (DLMF 4.13)

The principal branch, denoted Wp(z) in DLMF, is lambert_w(z) = generalized_lambert_w(0,z).

The other branch with real values, denoted Wm(z) in DLMF, is generalized_lambert_w(-1,z).

Function: nzeta (z)

The Plasma Dispersion Function nzeta(z) = %i*sqrt(%pi)*exp(-z^2)*(1-erf(-%i*z))

Function: nzetar (z)

Returns realpart(nzeta(z)).

Function: nzetai (z)

Returns imagpart(nzeta(z)).


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16 Elliptic Functions


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16.1 Introduction to Elliptic Functions and Integrals

Maxima includes support for Jacobian elliptic functions and for complete and incomplete elliptic integrals. This includes symbolic manipulation of these functions and numerical evaluation as well. Definitions of these functions and many of their properties can by found in Abramowitz and Stegun, Chapter 16–17. As much as possible, we use the definitions and relationships given there.

In particular, all elliptic functions and integrals use the parameter m instead of the modulus k or the modular angle \alpha. This is one area where we differ from Abramowitz and Stegun who use the modular angle for the elliptic functions. The following relationships are true:

The elliptic functions and integrals are primarily intended to support symbolic computation. Therefore, most of derivatives of the functions and integrals are known. However, if floating-point values are given, a floating-point result is returned.

Support for most of the other properties of elliptic functions and integrals other than derivatives has not yet been written.

Some examples of elliptic functions:

(%i1) jacobi_sn (u, m);
(%o1)                    jacobi_sn(u, m)
(%i2) jacobi_sn (u, 1);
(%o2)                        tanh(u)
(%i3) jacobi_sn (u, 0);
(%o3)                        sin(u)
(%i4) diff (jacobi_sn (u, m), u);
(%o4)            jacobi_cn(u, m) jacobi_dn(u, m)
(%i5) diff (jacobi_sn (u, m), m);
(%o5) jacobi_cn(u, m) jacobi_dn(u, m)

      elliptic_e(asin(jacobi_sn(u, m)), m)
 (u - ------------------------------------)/(2 m)
                     1 - m

            2
   jacobi_cn (u, m) jacobi_sn(u, m)
 + --------------------------------
              2 (1 - m)

Some examples of elliptic integrals:

(%i1) elliptic_f (phi, m);
(%o1)                  elliptic_f(phi, m)
(%i2) elliptic_f (phi, 0);
(%o2)                          phi
(%i3) elliptic_f (phi, 1);
                               phi   %pi
(%o3)                  log(tan(--- + ---))
                                2     4
(%i4) elliptic_e (phi, 1);
(%o4)                       sin(phi)
(%i5) elliptic_e (phi, 0);
(%o5)                          phi
(%i6) elliptic_kc (1/2);
                                     1
(%o6)                    elliptic_kc(-)
                                     2
(%i7) makegamma (%);
                                 2 1
                            gamma (-)
                                   4
(%o7)                      -----------
                           4 sqrt(%pi)
(%i8) diff (elliptic_f (phi, m), phi);
                                1
(%o8)                 ---------------------
                                    2
                      sqrt(1 - m sin (phi))
(%i9) diff (elliptic_f (phi, m), m);
       elliptic_e(phi, m) - (1 - m) elliptic_f(phi, m)
(%o9) (-----------------------------------------------
                              m

                                 cos(phi) sin(phi)
                             - ---------------------)/(2 (1 - m))
                                             2
                               sqrt(1 - m sin (phi))

Support for elliptic functions and integrals was written by Raymond Toy. It is placed under the terms of the General Public License (GPL) that governs the distribution of Maxima.

·

Category: Elliptic functions


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16.2 Functions and Variables for Elliptic Functions

Function: jacobi_sn (u, m)

The Jacobian elliptic function sn(u,m).

Function: jacobi_cn (u, m)

The Jacobian elliptic function cn(u,m).

Function: jacobi_dn (u, m)

The Jacobian elliptic function dn(u,m).

Function: jacobi_ns (u, m)

The Jacobian elliptic function ns(u,m) = 1/sn(u,m).

Function: jacobi_sc (u, m)

The Jacobian elliptic function sc(u,m) = sn(u,m)/cn(u,m).

Function: jacobi_sd (u, m)

The Jacobian elliptic function sd(u,m) = sn(u,m)/dn(u,m).

Function: jacobi_nc (u, m)

The Jacobian elliptic function nc(u,m) = 1/cn(u,m).

Function: jacobi_cs (u, m)

The Jacobian elliptic function cs(u,m) = cn(u,m)/sn(u,m).

Function: jacobi_cd (u, m)

The Jacobian elliptic function cd(u,m) = cn(u,m)/dn(u,m).

Function: jacobi_nd (u, m)

The Jacobian elliptic function nd(u,m) = 1/dn(u,m).

Function: jacobi_ds (u, m)

The Jacobian elliptic function ds(u,m) = dn(u,m)/sn(u,m).

Function: jacobi_dc (u, m)

The Jacobian elliptic function dc(u,m) = dn(u,m)/cn(u,m).

Function: inverse_jacobi_sn (u, m)

The inverse of the Jacobian elliptic function sn(u,m).

Function: inverse_jacobi_cn (u, m)

The inverse of the Jacobian elliptic function cn(u,m).

Function: inverse_jacobi_dn (u, m)

The inverse of the Jacobian elliptic function dn(u,m).

Function: inverse_jacobi_ns (u, m)

The inverse of the Jacobian elliptic function ns(u,m).

Function: inverse_jacobi_sc (u, m)

The inverse of the Jacobian elliptic function sc(u,m).

Function: inverse_jacobi_sd (u, m)

The inverse of the Jacobian elliptic function sd(u,m).

Function: inverse_jacobi_nc (u, m)

The inverse of the Jacobian elliptic function nc(u,m).

Function: inverse_jacobi_cs (u, m)

The inverse of the Jacobian elliptic function cs(u,m).

Function: inverse_jacobi_cd (u, m)

The inverse of the Jacobian elliptic function cd(u,m).

Function: inverse_jacobi_nd (u, m)

The inverse of the Jacobian elliptic function nd(u,m).

Function: inverse_jacobi_ds (u, m)

The inverse of the Jacobian elliptic function ds(u,m).

Function: inverse_jacobi_dc (u, m)

The inverse of the Jacobian elliptic function dc(u,m).


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16.3 Functions and Variables for Elliptic Integrals

Function: elliptic_f (phi, m)

The incomplete elliptic integral of the first kind, defined as

integrate(1/sqrt(1 - m*sin(x)^2), x, 0, phi)

See also elliptic_e and elliptic_kc.

Function: elliptic_e (phi, m)

The incomplete elliptic integral of the second kind, defined as

elliptic_e(phi, m) = integrate(sqrt(1 - m*sin(x)^2), x, 0, phi)

See also elliptic_f and elliptic_ec.

Function: elliptic_eu (u, m)

The incomplete elliptic integral of the second kind, defined as

integrate(dn(v,m)^2,v,0,u) = integrate(sqrt(1-m*t^2)/sqrt(1-t^2), t, 0, tau)

where tau = sn(u,m).

This is related to elliptic_e by

elliptic_eu(u, m) = elliptic_e(asin(sn(u,m)),m)

See also elliptic_e.

Function: elliptic_pi (n, phi, m)

The incomplete elliptic integral of the third kind, defined as

integrate(1/(1-n*sin(x)^2)/sqrt(1 - m*sin(x)^2), x, 0, phi)

Function: elliptic_kc (m)

The complete elliptic integral of the first kind, defined as

integrate(1/sqrt(1 - m*sin(x)^2), x, 0, %pi/2)

For certain values of m, the value of the integral is known in terms of Gamma functions. Use makegamma to evaluate them.

Function: elliptic_ec (m)

The complete elliptic integral of the second kind, defined as

integrate(sqrt(1 - m*sin(x)^2), x, 0, %pi/2)

For certain values of m, the value of the integral is known in terms of Gamma functions. Use makegamma to evaluate them.


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17 Limits


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17.1 Functions and Variables for Limits

Option variable: lhospitallim

Default value: 4

lhospitallim is the maximum number of times L’Hospital’s rule is used in limit. This prevents infinite looping in cases like limit (cot(x)/csc(x), x, 0).

Function: limit
    limit (expr, x, val, dir)
    limit (expr, x, val)
    limit (expr)

Computes the limit of expr as the real variable x approaches the value val from the direction dir. dir may have the value plus for a limit from above, minus for a limit from below, or may be omitted (implying a two-sided limit is to be computed).

limit uses the following special symbols: inf (positive infinity) and minf (negative infinity). On output it may also use und (undefined), ind (indefinite but bounded) and infinity (complex infinity).

infinity (complex infinity) is returned when the limit of the absolute value of the expression is positive infinity, but the limit of the expression itself is not positive infinity or negative infinity. This includes cases where the limit of the complex argument is a constant, as in limit(log(x), x, minf), cases where the complex argument oscillates, as in limit((-2)^x, x, inf), and cases where the complex argument is different for either side of a two-sided limit, as in limit(1/x, x, 0) and limit(log(x), x, 0).

lhospitallim is the maximum number of times L’Hospital’s rule is used in limit. This prevents infinite looping in cases like limit (cot(x)/csc(x), x, 0).

tlimswitch when true will allow the limit command to use Taylor series expansion when necessary.

limsubst prevents limit from attempting substitutions on unknown forms. This is to avoid bugs like limit (f(n)/f(n+1), n, inf) giving 1. Setting limsubst to true will allow such substitutions.

limit with one argument is often called upon to simplify constant expressions, for example, limit (inf-1).

example (limit) displays some examples.

For the method see Wang, P., "Evaluation of Definite Integrals by Symbolic Manipulation", Ph.D. thesis, MAC TR-92, October 1971.

Option variable: limsubst

Default value: false

prevents limit from attempting substitutions on unknown forms. This is to avoid bugs like limit (f(n)/f(n+1), n, inf) giving 1. Setting limsubst to true will allow such substitutions.

Function: tlimit
    tlimit (expr, x, val, dir)
    tlimit (expr, x, val)
    tlimit (expr)

Take the limit of the Taylor series expansion of expr in x at val from direction dir.

Option variable: tlimswitch

Default value: true

When tlimswitch is true, the limit command will use a Taylor series expansion if the limit of the input expression cannot be computed directly. This allows evaluation of limits such as limit(x/(x-1)-1/log(x),x,1,plus). When tlimswitch is false and the limit of input expression cannot be computed directly, limit will return an unevaluated limit expression.


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18 Differentiation


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18.1 Functions and Variables for Differentiation

Function: antid (expr, x, u(x))

Returns a two-element list, such that an antiderivative of expr with respect to x can be constructed from the list. The expression expr may contain an unknown function u and its derivatives.

Let L, a list of two elements, be the return value of antid. Then L[1] + 'integrate (L[2], x) is an antiderivative of expr with respect to x.

When antid succeeds entirely, the second element of the return value is zero. Otherwise, the second element is nonzero, and the first element is nonzero or zero. If antid cannot make any progress, the first element is zero and the second nonzero.

load ("antid") loads this function. The antid package also defines the functions nonzeroandfreeof and linear.

antid is related to antidiff as follows. Let L, a list of two elements, be the return value of antid. Then the return value of antidiff is equal to L[1] + 'integrate (L[2], x) where x is the variable of integration.

Examples:

(%i1) load ("antid")$
(%i2) expr: exp (z(x)) * diff (z(x), x) * y(x);
                            z(x)  d
(%o2)                y(x) %e     (-- (z(x)))
                                  dx
(%i3) a1: antid (expr, x, z(x));
                       z(x)      z(x)  d
(%o3)          [y(x) %e    , - %e     (-- (y(x)))]
                                       dx
(%i4) a2: antidiff (expr, x, z(x));
                            /
                     z(x)   [   z(x)  d
(%o4)         y(x) %e     - I %e     (-- (y(x))) dx
                            ]         dx
                            /
(%i5) a2 - (first (a1) + 'integrate (second (a1), x));
(%o5)                           0
(%i6) antid (expr, x, y(x));
                             z(x)  d
(%o6)             [0, y(x) %e     (-- (z(x)))]
                                   dx
(%i7) antidiff (expr, x, y(x));
                  /
                  [        z(x)  d
(%o7)             I y(x) %e     (-- (z(x))) dx
                  ]              dx
                  /

Function: antidiff (expr, x, u(x))

Returns an antiderivative of expr with respect to x. The expression expr may contain an unknown function u and its derivatives.

When antidiff succeeds entirely, the resulting expression is free of integral signs (that is, free of the integrate noun). Otherwise, antidiff returns an expression which is partly or entirely within an integral sign. If antidiff cannot make any progress, the return value is entirely within an integral sign.

load ("antid") loads this function. The antid package also defines the functions nonzeroandfreeof and linear.

antidiff is related to antid as follows. Let L, a list of two elements, be the return value of antid. Then the return value of antidiff is equal to L[1] + 'integrate (L[2], x) where x is the variable of integration.

Examples:

(%i1) load ("antid")$
(%i2) expr: exp (z(x)) * diff (z(x), x) * y(x);
                            z(x)  d
(%o2)                y(x) %e     (-- (z(x)))
                                  dx
(%i3) a1: antid (expr, x, z(x));
                       z(x)      z(x)  d
(%o3)          [y(x) %e    , - %e     (-- (y(x)))]
                                       dx
(%i4) a2: antidiff (expr, x, z(x));
                            /
                     z(x)   [   z(x)  d
(%o4)         y(x) %e     - I %e     (-- (y(x))) dx
                            ]         dx
                            /
(%i5) a2 - (first (a1) + 'integrate (second (a1), x));
(%o5)                           0
(%i6) antid (expr, x, y(x));
                             z(x)  d
(%o6)             [0, y(x) %e     (-- (z(x)))]
                                   dx
(%i7) antidiff (expr, x, y(x));
                  /
                  [        z(x)  d
(%o7)             I y(x) %e     (-- (z(x))) dx
                  ]              dx
                  /

Function: at
    at (expr, [eqn_1, …, eqn_n])
    at (expr, eqn)

Evaluates the expression expr with the variables assuming the values as specified for them in the list of equations [eqn_1, ..., eqn_n] or the single equation eqn.

If a subexpression depends on any of the variables for which a value is specified but there is no atvalue specified and it can’t be otherwise evaluated, then a noun form of the at is returned which displays in a two-dimensional form.

at carries out multiple substitutions in parallel.

See also atvalue. For other functions which carry out substitutions, see also subst and ev.

Examples:

(%i1) atvalue (f(x,y), [x = 0, y = 1], a^2);
                                2
(%o1)                          a
(%i2) atvalue ('diff (f(x,y), x), x = 0, 1 + y);
(%o2)                        @2 + 1
(%i3) printprops (all, atvalue);
                                !
                  d             !
                 --- (f(@1, @2))!       = @2 + 1
                 d@1            !
                                !@1 = 0

                                     2
                          f(0, 1) = a

(%o3)                         done
(%i4) diff (4*f(x, y)^2 - u(x, y)^2, x);
                  d                          d
(%o4)  8 f(x, y) (-- (f(x, y))) - 2 u(x, y) (-- (u(x, y)))
                  dx                         dx
(%i5) at (%, [x = 0, y = 1]);
                                            !
                 2              d           !
(%o5)        16 a  - 2 u(0, 1) (-- (u(x, 1))!     )
                                dx          !
                                            !x = 0

Note that in the last line y is treated differently to x as y isn’t used as a differentiation variable.

The difference between subst, at and ev can be seen in the following example:

(%i1) e1:I(t)=C*diff(U(t),t)$
(%i2) e2:U(t)=L*diff(I(t),t)$
(%i3) at(e1,e2);
                               !
                      d        !
(%o3)       I(t) = C (-- (U(t))!                    )
                      dt       !          d
                               !U(t) = L (-- (I(t)))
                                          dt
(%i4) subst(e2,e1);
                            d      d
(%o4)             I(t) = C (-- (L (-- (I(t)))))
                            dt     dt
(%i5) ev(e1,e2,diff);
                                  2
                                 d
(%o5)                I(t) = C L (--- (I(t)))
                                   2
                                 dt

Property: atomgrad

atomgrad is the atomic gradient property of an expression. This property is assigned by gradef.

Function: atvalue
    atvalue (expr, [x_1 = a_1, …, x_m = a_m], c)
    atvalue (expr, x_1 = a_1, c)

Assigns the value c to expr at the point x = a. Typically boundary values are established by this mechanism.

expr is a function evaluation, f(x_1, ..., x_m), or a derivative, diff (f(x_1, ..., x_m), x_1, n_1, ..., x_n, n_m) in which the function arguments explicitly appear. n_i is the order of differentiation with respect to x_i.

The point at which the atvalue is established is given by the list of equations [x_1 = a_1, ..., x_m = a_m]. If there is a single variable x_1, the sole equation may be given without enclosing it in a list.

printprops ([f_1, f_2, ...], atvalue) displays the atvalues of the functions f_1, f_2, ... as specified by calls to atvalue. printprops (f, atvalue) displays the atvalues of one function f. printprops (all, atvalue) displays the atvalues of all functions for which atvalues are defined.

The symbols @1, @2, … represent the variables x_1, x_2, … when atvalues are displayed.

atvalue evaluates its arguments. atvalue returns c, the atvalue.

See also at.

Examples:

(%i1) atvalue (f(x,y), [x = 0, y = 1], a^2);
                                2
(%o1)                          a
(%i2) atvalue ('diff (f(x,y), x), x = 0, 1 + y);
(%o2)                        @2 + 1
(%i3) printprops (all, atvalue);
                                !
                  d             !
                 --- (f(@1, @2))!       = @2 + 1
                 d@1            !
                                !@1 = 0

                                     2
                          f(0, 1) = a

(%o3)                         done
(%i4) diff (4*f(x,y)^2 - u(x,y)^2, x);
                  d                          d
(%o4)  8 f(x, y) (-- (f(x, y))) - 2 u(x, y) (-- (u(x, y)))
                  dx                         dx
(%i5) at (%, [x = 0, y = 1]);
                                            !
                 2              d           !
(%o5)        16 a  - 2 u(0, 1) (-- (u(x, 1))!     )
                                dx          !
                                            !x = 0

Function: cartan

The exterior calculus of differential forms is a basic tool of differential geometry developed by Elie Cartan and has important applications in the theory of partial differential equations. The cartan package implements the functions ext_diff and lie_diff, along with the operators ~ (wedge product) and | (contraction of a form with a vector.) Type demo ("tensor") to see a brief description of these commands along with examples.

cartan was implemented by F.B. Estabrook and H.D. Wahlquist.

Function: del (x)

del (x) represents the differential of the variable x.

diff returns an expression containing del if an independent variable is not specified. In this case, the return value is the so-called "total differential".

Examples:

(%i1) diff (log (x));
                             del(x)
(%o1)                        ------
                               x
(%i2) diff (exp (x*y));
                     x y              x y
(%o2)            x %e    del(y) + y %e    del(x)
(%i3) diff (x*y*z);
(%o3)         x y del(z) + x z del(y) + y z del(x)

Function: delta (t)

The Dirac Delta function.

Currently only laplace knows about the delta function.

Example:

(%i1) laplace (delta (t - a) * sin(b*t), t, s);
Is  a  positive, negative, or zero?

p;
                                   - a s
(%o1)                   sin(a b) %e

System variable: dependencies
Function: dependencies (f_1, …, f_n)

The variable dependencies is the list of atoms which have functional dependencies, assigned by depends, the function dependencies, or gradef. The dependencies list is cumulative: each call to depends, dependencies, or gradef appends additional items. The default value of dependencies is [].

The function dependencies(f_1, …, f_n) appends f_1, …, f_n, to the dependencies list, where f_1, …, f_n are expressions of the form f(x_1, …, x_m), and x_1, …, x_m are any number of arguments.

dependencies(f(x_1, …, x_m)) is equivalent to depends(f, [x_1, …, x_m]).

See also depends and gradef.

(%i1) dependencies;
(%o1)                          []
(%i2) depends (foo, [bar, baz]);
(%o2)                    [foo(bar, baz)]
(%i3) depends ([g, h], [a, b, c]);
(%o3)               [g(a, b, c), h(a, b, c)]
(%i4) dependencies;
(%o4)        [foo(bar, baz), g(a, b, c), h(a, b, c)]
(%i5) dependencies (quux (x, y), mumble (u));
(%o5)                [quux(x, y), mumble(u)]
(%i6) dependencies;
(%o6) [foo(bar, baz), g(a, b, c), h(a, b, c), quux(x, y), 
                                                       mumble(u)]
(%i7) remove (quux, dependency);
(%o7)                         done
(%i8) dependencies;
(%o8)  [foo(bar, baz), g(a, b, c), h(a, b, c), mumble(u)]

Function: depends (f_1, x_1, …, f_n, x_n)

Declares functional dependencies among variables for the purpose of computing derivatives. In the absence of declared dependence, diff (f, x) yields zero. If depends (f, x) is declared, diff (f, x) yields a symbolic derivative (that is, a diff noun).

Each argument f_1, x_1, etc., can be the name of a variable or array, or a list of names. Every element of f_i (perhaps just a single element) is declared to depend on every element of x_i (perhaps just a single element). If some f_i is the name of an array or contains the name of an array, all elements of the array depend on x_i.

diff recognizes indirect dependencies established by depends and applies the chain rule in these cases.

remove (f, dependency) removes all dependencies declared for f.

depends returns a list of the dependencies established. The dependencies are appended to the global variable dependencies. depends evaluates its arguments.

diff is the only Maxima command which recognizes dependencies established by depends. Other functions (integrate, laplace, etc.) only recognize dependencies explicitly represented by their arguments. For example, integrate does not recognize the dependence of f on x unless explicitly represented as integrate (f(x), x).

depends(f, [x_1, …, x_n]) is equivalent to dependencies(f(x_1, …, x_n)).

(%i1) depends ([f, g], x);
(%o1)                     [f(x), g(x)]
(%i2) depends ([r, s], [u, v, w]);
(%o2)               [r(u, v, w), s(u, v, w)]
(%i3) depends (u, t);
(%o3)                        [u(t)]
(%i4) dependencies;
(%o4)      [f(x), g(x), r(u, v, w), s(u, v, w), u(t)]
(%i5) diff (r.s, u);
                         dr           ds
(%o5)                    -- . s + r . --
                         du           du
(%i6) diff (r.s, t);
                      dr du           ds du
(%o6)                 -- -- . s + r . -- --
                      du dt           du dt
(%i7) remove (r, dependency);
(%o7)                         done
(%i8) diff (r.s, t);
                                ds du
(%o8)                       r . -- --
                                du dt

Option variable: derivabbrev

Default value: false

When derivabbrev is true, symbolic derivatives (that is, diff nouns) are displayed as subscripts. Otherwise, derivatives are displayed in the Leibniz notation dy/dx.

Function: derivdegree (expr, y, x)

Returns the highest degree of the derivative of the dependent variable y with respect to the independent variable x occurring in expr.

Example:

(%i1) 'diff (y, x, 2) + 'diff (y, z, 3) + 'diff (y, x) * x^2;
                         3     2
                        d y   d y    2 dy
(%o1)                   --- + --- + x  --
                          3     2      dx
                        dz    dx
(%i2) derivdegree (%, y, x);
(%o2)                           2

Function: derivlist (var_1, …, var_k)

Causes only differentiations with respect to the indicated variables, within the ev command.

Option variable: derivsubst

Default value: false

When derivsubst is true, a non-syntactic substitution such as subst (x, 'diff (y, t), 'diff (y, t, 2)) yields 'diff (x, t).

Function: diff
    diff (expr, x_1, n_1, …, x_m, n_m)
    diff (expr, x, n)
    diff (expr, x)
    diff (expr)

Returns the derivative or differential of expr with respect to some or all variables in expr.

diff (expr, x, n) returns the n’th derivative of expr with respect to x.

diff (expr, x_1, n_1, ..., x_m, n_m) returns the mixed partial derivative of expr with respect to x_1, …, x_m. It is equivalent to diff (... (diff (expr, x_m, n_m) ...), x_1, n_1).

diff (expr, x) returns the first derivative of expr with respect to the variable x.

diff (expr) returns the total differential of expr, that is, the sum of the derivatives of expr with respect to each its variables times the differential del of each variable. No further simplification of del is offered.

The noun form of diff is required in some contexts, such as stating a differential equation. In these cases, diff may be quoted (as 'diff) to yield the noun form instead of carrying out the differentiation.

When derivabbrev is true, derivatives are displayed as subscripts. Otherwise, derivatives are displayed in the Leibniz notation, dy/dx.

Examples:

(%i1) diff (exp (f(x)), x, 2);
                     2
              f(x)  d               f(x)  d         2
(%o1)       %e     (--- (f(x))) + %e     (-- (f(x)))
                      2                   dx
                    dx
(%i2) derivabbrev: true$
(%i3) 'integrate (f(x, y), y, g(x), h(x));
                         h(x)
                        /
                        [
(%o3)                   I     f(x, y) dy
                        ]
                        /
                         g(x)
(%i4) diff (%, x);
       h(x)
      /
      [
(%o4) I     f(x, y)  dy + f(x, h(x)) h(x)  - f(x, g(x)) g(x)
      ]            x                     x                  x
      /
       g(x)

For the tensor package, the following modifications have been incorporated:

(1) The derivatives of any indexed objects in expr will have the variables x_i appended as additional arguments. Then all the derivative indices will be sorted.

(2) The x_i may be integers from 1 up to the value of the variable dimension [default value: 4]. This will cause the differentiation to be carried out with respect to the x_i’th member of the list coordinates which should be set to a list of the names of the coordinates, e.g., [x, y, z, t]. If coordinates is bound to an atomic variable, then that variable subscripted by x_i will be used for the variable of differentiation. This permits an array of coordinate names or subscripted names like X[1], X[2], … to be used. If coordinates has not been assigned a value, then the variables will be treated as in (1) above.

Special symbol: diff

When diff is present as an evflag in call to ev, all differentiations indicated in expr are carried out.

Function: express (expr)

Expands differential operator nouns into expressions in terms of partial derivatives. express recognizes the operators grad, div, curl, laplacian. express also expands the cross product ~.

Symbolic derivatives (that is, diff nouns) in the return value of express may be evaluated by including diff in the ev function call or command line. In this context, diff acts as an evfun.

load ("vect") loads this function.

Examples:

(%i1) load ("vect")$
(%i2) grad (x^2 + y^2 + z^2);
                              2    2    2
(%o2)                  grad (z  + y  + x )
(%i3) express (%);
       d    2    2    2   d    2    2    2   d    2    2    2
(%o3) [-- (z  + y  + x ), -- (z  + y  + x ), -- (z  + y  + x )]
       dx                 dy                 dz
(%i4) ev (%, diff);
(%o4)                    [2 x, 2 y, 2 z]
(%i5) div ([x^2, y^2, z^2]);
                              2   2   2
(%o5)                   div [x , y , z ]
(%i6) express (%);
                   d    2    d    2    d    2
(%o6)              -- (z ) + -- (y ) + -- (x )
                   dz        dy        dx
(%i7) ev (%, diff);
(%o7)                    2 z + 2 y + 2 x
(%i8) curl ([x^2, y^2, z^2]);
                               2   2   2
(%o8)                   curl [x , y , z ]
(%i9) express (%);
       d    2    d    2   d    2    d    2   d    2    d    2
(%o9) [-- (z ) - -- (y ), -- (x ) - -- (z ), -- (y ) - -- (x )]
       dy        dz       dz        dx       dx        dy
(%i10) ev (%, diff);
(%o10)                      [0, 0, 0]
(%i11) laplacian (x^2 * y^2 * z^2);
                                  2  2  2
(%o11)                laplacian (x  y  z )
(%i12) express (%);
         2                2                2
        d     2  2  2    d     2  2  2    d     2  2  2
(%o12)  --- (x  y  z ) + --- (x  y  z ) + --- (x  y  z )
          2                2                2
        dz               dy               dx
(%i13) ev (%, diff);
                      2  2      2  2      2  2
(%o13)             2 y  z  + 2 x  z  + 2 x  y
(%i14) [a, b, c] ~ [x, y, z];
(%o14)                [a, b, c] ~ [x, y, z]
(%i15) express (%);
(%o15)          [b z - c y, c x - a z, a y - b x]

Function: gradef
    gradef (f(x_1, …, x_n), g_1, …, g_m)
    gradef (a, x, expr)

Defines the partial derivatives (i.e., the components of the gradient) of the function f or variable a.

gradef (f(x_1, ..., x_n), g_1, ..., g_m) defines df/dx_i as g_i, where g_i is an expression; g_i may be a function call, but not the name of a function. The number of partial derivatives m may be less than the number of arguments n, in which case derivatives are defined with respect to x_1 through x_m only.

gradef (a, x, expr) defines the derivative of variable a with respect to x as expr. This also establishes the dependence of a on x (via depends (a, x)).

The first argument f(x_1, ..., x_n) or a is quoted, but the remaining arguments g_1, ..., g_m are evaluated. gradef returns the function or variable for which the partial derivatives are defined.

gradef can redefine the derivatives of Maxima’s built-in functions. For example, gradef (sin(x), sqrt (1 - sin(x)^2)) redefines the derivative of sin.

gradef cannot define partial derivatives for a subscripted function.

printprops ([f_1, ..., f_n], gradef) displays the partial derivatives of the functions f_1, ..., f_n, as defined by gradef.

printprops ([a_n, ..., a_n], atomgrad) displays the partial derivatives of the variables a_n, ..., a_n, as defined by gradef.

gradefs is the list of the functions for which partial derivatives have been defined by gradef. gradefs does not include any variables for which partial derivatives have been defined by gradef.

Gradients are needed when, for example, a function is not known explicitly but its first derivatives are and it is desired to obtain higher order derivatives.

System variable: gradefs

Default value: []

gradefs is the list of the functions for which partial derivatives have been defined by gradef. gradefs does not include any variables for which partial derivatives have been defined by gradef.

Function: laplace (expr, t, s)

Attempts to compute the Laplace transform of expr with respect to the variable t and transform parameter s.

laplace recognizes in expr the functions delta, exp, log, sin, cos, sinh, cosh, and erf, as well as derivative, integrate, sum, and ilt. If laplace fails to find a transform the function specint is called. specint can find the laplace transform for expressions with special functions like the bessel functions bessel_j, bessel_i, … and can handle the unit_step function. See also specint.

If specint cannot find a solution too, a noun laplace is returned.

expr may also be a linear, constant coefficient differential equation in which case atvalue of the dependent variable is used. The required atvalue may be supplied either before or after the transform is computed. Since the initial conditions must be specified at zero, if one has boundary conditions imposed elsewhere he can impose these on the general solution and eliminate the constants by solving the general solution for them and substituting their values back.

laplace recognizes convolution integrals of the form integrate (f(x) * g(t - x), x, 0, t); other kinds of convolutions are not recognized.

Functional relations must be explicitly represented in expr; implicit relations, established by depends, are not recognized. That is, if f depends on x and y, f (x, y) must appear in expr.

See also ilt, the inverse Laplace transform.

Examples:

(%i1) laplace (exp (2*t + a) * sin(t) * t, t, s);
                            a
                          %e  (2 s - 4)
(%o1)                    ---------------
                           2           2
                         (s  - 4 s + 5)
(%i2) laplace ('diff (f (x), x), x, s);
(%o2)             s laplace(f(x), x, s) - f(0)
(%i3) diff (diff (delta (t), t), t);
                          2
                         d
(%o3)                    --- (delta(t))
                           2
                         dt
(%i4) laplace (%, t, s);
                            !
               d            !         2
(%o4)        - -- (delta(t))!      + s  - delta(0) s
               dt           !
                            !t = 0
(%i5) assume(a>0)$
(%i6) laplace(gamma_incomplete(a,t),t,s),gamma_expand:true;
                                              - a - 1
                         gamma(a)   gamma(a) s
(%o6)                    -------- - -----------------
                            s            1     a
                                        (- + 1)
                                         s
(%i7) factor(laplace(gamma_incomplete(1/2,t),t,s));
                                              s + 1
                      sqrt(%pi) (sqrt(s) sqrt(-----) - 1)
                                                s
(%o7)                 -----------------------------------
                                3/2      s + 1
                               s    sqrt(-----)
                                           s
(%i8) assume(exp(%pi*s)>1)$
(%i9) laplace(sum((-1)^n*unit_step(t-n*%pi)*sin(t),n,0,inf),t,s),
         simpsum;
                         %i                         %i
              ------------------------ - ------------------------
                              - %pi s                    - %pi s
              (s + %i) (1 - %e       )   (s - %i) (1 - %e       )
(%o9)         ---------------------------------------------------
                                       2
(%i9) factor(%);
                                      %pi s
                                    %e
(%o9)                   -------------------------------
                                             %pi s
                        (s - %i) (s + %i) (%e      - 1)


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19 Integration


Next: , Previous: , Up: Integration   [Contents][Index]

19.1 Introduction to Integration

Maxima has several routines for handling integration. The integrate function makes use of most of them. There is also the antid package, which handles an unspecified function (and its derivatives, of course). For numerical uses, there is a set of adaptive integrators from QUADPACK, named quad_qag, quad_qags, etc., which are described under the heading QUADPACK. Hypergeometric functions are being worked on, see specint for details. Generally speaking, Maxima only handles integrals which are integrable in terms of the "elementary functions" (rational functions, trigonometrics, logs, exponentials, radicals, etc.) and a few extensions (error function, dilogarithm). It does not handle integrals in terms of unknown functions such as g(x) and h(x).


Next: , Previous: , Up: Integration   [Contents][Index]

19.2 Functions and Variables for Integration

Function: changevar (expr, f(x,y), y, x)

Makes the change of variable given by f(x,y) = 0 in all integrals occurring in expr with integration with respect to x. The new variable is y.

The change of variable can also be written f(x) = g(y).

(%i1) assume(a > 0)$
(%i2) 'integrate (%e**sqrt(a*y), y, 0, 4);
                      4
                     /
                     [    sqrt(a) sqrt(y)
(%o2)                I  %e                dy
                     ]
                     /
                      0
(%i3) changevar (%, y-z^2/a, z, y);
                      0
                     /
                     [                abs(z)
                   2 I            z %e       dz
                     ]
                     /
                      - 2 sqrt(a)
(%o3)            - ----------------------------
                                a

An expression containing a noun form, such as the instances of 'integrate above, may be evaluated by ev with the nouns flag. For example, the expression returned by changevar above may be evaluated by ev (%o3, nouns).

changevar may also be used to changes in the indices of a sum or product. However, it must be realized that when a change is made in a sum or product, this change must be a shift, i.e., i = j+ ..., not a higher degree function. E.g.,

(%i4) sum (a[i]*x^(i-2), i, 0, inf);
                         inf
                         ====
                         \         i - 2
(%o4)                     >    a  x
                         /      i
                         ====
                         i = 0
(%i5) changevar (%, i-2-n, n, i);
                        inf
                        ====
                        \               n
(%o5)                    >      a      x
                        /        n + 2
                        ====
                        n = - 2

Function: dblint (f, r, s, a, b)

A double-integral routine which was written in top-level Maxima and then translated and compiled to machine code. Use load ("dblint") to access this package. It uses the Simpson’s rule method in both the x and y directions to calculate

/b /s(x)
|  |
|  |    f(x,y) dy dx
|  |
/a /r(x)

The function f must be a translated or compiled function of two variables, and r and s must each be a translated or compiled function of one variable, while a and b must be floating point numbers. The routine has two global variables which determine the number of divisions of the x and y intervals: dblint_x and dblint_y, both of which are initially 10, and can be changed independently to other integer values (there are 2*dblint_x+1 points computed in the x direction, and 2*dblint_y+1 in the y direction). The routine subdivides the X axis and then for each value of X it first computes r(x) and s(x); then the Y axis between r(x) and s(x) is subdivided and the integral along the Y axis is performed using Simpson’s rule; then the integral along the X axis is done using Simpson’s rule with the function values being the Y-integrals. This procedure may be numerically unstable for a great variety of reasons, but is reasonably fast: avoid using it on highly oscillatory functions and functions with singularities (poles or branch points in the region). The Y integrals depend on how far apart r(x) and s(x) are, so if the distance s(x) - r(x) varies rapidly with X, there may be substantial errors arising from truncation with different step-sizes in the various Y integrals. One can increase dblint_x and dblint_y in an effort to improve the coverage of the region, at the expense of computation time. The function values are not saved, so if the function is very time-consuming, you will have to wait for re-computation if you change anything (sorry). It is required that the functions f, r, and s be either translated or compiled prior to calling dblint. This will result in orders of magnitude speed improvement over interpreted code in many cases!

demo ("dblint") executes a demonstration of dblint applied to an example problem.

Function: defint (expr, x, a, b)

Attempts to compute a definite integral. defint is called by integrate when limits of integration are specified, i.e., when integrate is called as integrate (expr, x, a, b). Thus from the user’s point of view, it is sufficient to call integrate.

defint returns a symbolic expression, either the computed integral or the noun form of the integral. See quad_qag and related functions for numerical approximation of definite integrals.

Option variable: erfflag

Default value: true

When erfflag is false, prevents risch from introducing the erf function in the answer if there were none in the integrand to begin with.

Function: ilt (expr, s, t)

Computes the inverse Laplace transform of expr with respect to s and parameter t. expr must be a ratio of polynomials whose denominator has only linear and quadratic factors. By using the functions laplace and ilt together with the solve or linsolve functions the user can solve a single differential or convolution integral equation or a set of them.

(%i1) 'integrate (sinh(a*x)*f(t-x), x, 0, t) + b*f(t) = t**2;
              t
             /
             [                                    2
(%o1)        I  f(t - x) sinh(a x) dx + b f(t) = t
             ]
             /
              0
(%i2) laplace (%, t, s);
                               a laplace(f(t), t, s)   2
(%o2)  b laplace(f(t), t, s) + --------------------- = --
                                       2    2           3
                                      s  - a           s
(%i3) linsolve ([%], ['laplace(f(t), t, s)]);
                                        2      2
                                     2 s  - 2 a
(%o3)     [laplace(f(t), t, s) = --------------------]
                                    5         2     3
                                 b s  + (a - a  b) s
(%i4) ilt (rhs (first (%)), s, t);
Is  a b (a b - 1)  positive, negative, or zero?

pos;
               sqrt(a b (a b - 1)) t
        2 cosh(---------------------)       2
                         b               a t
(%o4) - ----------------------------- + -------
              3  2      2               a b - 1
             a  b  - 2 a  b + a

                                                       2
                                             + ------------------
                                                3  2      2
                                               a  b  - 2 a  b + a

Option variable: intanalysis

Default value: true

When true, definite integration tries to find poles in the integrand in the interval of integration. If there are, then the integral is evaluated appropriately as a principal value integral. If intanalysis is false, this check is not performed and integration is done assuming there are no poles.

See also ldefint.

Examples:

Maxima can solve the following integrals, when intanalysis is set to false:

(%i1) integrate(1/(sqrt(x)+1),x,0,1);
                                1
                               /
                               [       1
(%o1)                          I  ----------- dx
                               ]  sqrt(x) + 1
                               /
                                0

(%i2) integrate(1/(sqrt(x)+1),x,0,1),intanalysis:false;
(%o2)                            2 - 2 log(2)

(%i3) integrate(cos(a)/sqrt((tan(a))^2 +1),a,-%pi/2,%pi/2);
The number 1 isn't in the domain of atanh
 -- an error. To debug this try: debugmode(true);

(%i4) intanalysis:false$
(%i5) integrate(cos(a)/sqrt((tan(a))^2+1),a,-%pi/2,%pi/2);
                                      %pi
(%o5)                                 ---
                                       2

Function: integrate
    integrate (expr, x)
    integrate (expr, x, a, b)

Attempts to symbolically compute the integral of expr with respect to x. integrate (expr, x) is an indefinite integral, while integrate (expr, x, a, b) is a definite integral, with limits of integration a and b. The limits should not contain x, although integrate does not enforce this restriction. a need not be less than b. If b is equal to a, integrate returns zero.

See quad_qag and related functions for numerical approximation of definite integrals. See residue for computation of residues (complex integration). See antid for an alternative means of computing indefinite integrals.

The integral (an expression free of integrate) is returned if integrate succeeds. Otherwise the return value is the noun form of the integral (the quoted operator 'integrate) or an expression containing one or more noun forms. The noun form of integrate is displayed with an integral sign.

In some circumstances it is useful to construct a noun form by hand, by quoting integrate with a single quote, e.g., 'integrate (expr, x). For example, the integral may depend on some parameters which are not yet computed. The noun may be applied to its arguments by ev (i, nouns) where i is the noun form of interest.

integrate handles definite integrals separately from indefinite, and employs a range of heuristics to handle each case. Special cases of definite integrals include limits of integration equal to zero or infinity (inf or minf), trigonometric functions with limits of integration equal to zero and %pi or 2 %pi, rational functions, integrals related to the definitions of the beta and psi functions, and some logarithmic and trigonometric integrals. Processing rational functions may include computation of residues. If an applicable special case is not found, an attempt will be made to compute the indefinite integral and evaluate it at the limits of integration. This may include taking a limit as a limit of integration goes to infinity or negative infinity; see also ldefint.

Special cases of indefinite integrals include trigonometric functions, exponential and logarithmic functions, and rational functions. integrate may also make use of a short table of elementary integrals.

integrate may carry out a change of variable if the integrand has the form f(g(x)) * diff(g(x), x). integrate attempts to find a subexpression g(x) such that the derivative of g(x) divides the integrand. This search may make use of derivatives defined by the gradef function. See also changevar and antid.

If none of the preceding heuristics find the indefinite integral, the Risch algorithm is executed. The flag risch may be set as an evflag, in a call to ev or on the command line, e.g., ev (integrate (expr, x), risch) or integrate (expr, x), risch. If risch is present, integrate calls the risch function without attempting heuristics first. See also risch.

integrate works only with functional relations represented explicitly with the f(x) notation. integrate does not respect implicit dependencies established by the depends function.

integrate may need to know some property of a parameter in the integrand. integrate will first consult the assume database, and, if the variable of interest is not there, integrate will ask the user. Depending on the question, suitable responses are yes; or no;, or pos;, zero;, or neg;.

integrate is not, by default, declared to be linear. See declare and linear.

integrate attempts integration by parts only in a few special cases.

Examples:

System variable: integration_constant

Default value: %c

When a constant of integration is introduced by indefinite integration of an equation, the name of the constant is constructed by concatenating integration_constant and integration_constant_counter.

integration_constant may be assigned any symbol.

Examples:

(%i1) integrate (x^2 = 1, x);
                           3
                          x
(%o1)                     -- = x + %c1
                          3
(%i2) integration_constant : 'k;
(%o2)                           k
(%i3) integrate (x^2 = 1, x);
                            3
                           x
(%o3)                      -- = x + k2
                           3

System variable: integration_constant_counter

Default value: 0

When a constant of integration is introduced by indefinite integration of an equation, the name of the constant is constructed by concatenating integration_constant and integration_constant_counter.

integration_constant_counter is incremented before constructing the next integration constant.

Examples:

(%i1) integrate (x^2 = 1, x);
                           3
                          x
(%o1)                     -- = x + %c1
                          3
(%i2) integrate (x^2 = 1, x);
                           3
                          x
(%o2)                     -- = x + %c2
                          3
(%i3) integrate (x^2 = 1, x);
                           3
                          x
(%o3)                     -- = x + %c3
                          3
(%i4) reset (integration_constant_counter);
(%o4)            [integration_constant_counter]
(%i5) integrate (x^2 = 1, x);
                           3
                          x
(%o5)                     -- = x + %c1
                          3

Option variable: integrate_use_rootsof

Default value: false

When integrate_use_rootsof is true and the denominator of a rational function cannot be factored, integrate returns the integral in a form which is a sum over the roots (not yet known) of the denominator.

For example, with integrate_use_rootsof set to false, integrate returns an unsolved integral of a rational function in noun form:

(%i1) integrate_use_rootsof: false$
(%i2) integrate (1/(1+x+x^5), x);
        /  2
        [ x  - 4 x + 5
        I ------------ dx                            2 x + 1
        ]  3    2                2            5 atan(-------)
        / x  - x  + 1       log(x  + x + 1)          sqrt(3)
(%o2)   ----------------- - --------------- + ---------------
                7                 14             7 sqrt(3)

Now we set the flag to be true and the unsolved part of the integral will be expressed as a summation over the roots of the denominator of the rational function:

(%i3) integrate_use_rootsof: true$
(%i4) integrate (1/(1+x+x^5), x);
      ====        2
      \       (%r4  - 4 %r4 + 5) log(x - %r4)
       >      -------------------------------
      /                    2
      ====            3 %r4  - 2 %r4
                        3      2
      %r4 in rootsof(%r4  - %r4  + 1, %r4)
(%o4) ----------------------------------------------------------
               7

                                                      2 x + 1
                                  2            5 atan(-------)
                             log(x  + x + 1)          sqrt(3)
                           - --------------- + ---------------
                                   14             7 sqrt(3)

Alternatively the user may compute the roots of the denominator separately, and then express the integrand in terms of these roots, e.g., 1/((x - a)*(x - b)*(x - c)) or 1/((x^2 - (a+b)*x + a*b)*(x - c)) if the denominator is a cubic polynomial. Sometimes this will help Maxima obtain a more useful result.

Function: ldefint (expr, x, a, b)

Attempts to compute the definite integral of expr by using limit to evaluate the indefinite integral of expr with respect to x at the upper limit b and at the lower limit a. If it fails to compute the definite integral, ldefint returns an expression containing limits as noun forms.

ldefint is not called from integrate, so executing ldefint (expr, x, a, b) may yield a different result than integrate (expr, x, a, b). ldefint always uses the same method to evaluate the definite integral, while integrate may employ various heuristics and may recognize some special cases.

Function: potential (givengradient)

The calculation makes use of the global variable potentialzeroloc[0] which must be nonlist or of the form

[indeterminatej=expressionj, indeterminatek=expressionk, ...]

the former being equivalent to the nonlist expression for all right-hand sides in the latter. The indicated right-hand sides are used as the lower limit of integration. The success of the integrations may depend upon their values and order. potentialzeroloc is initially set to 0.

Function: residue (expr, z, z_0)

Computes the residue in the complex plane of the expression expr when the variable z assumes the value z_0. The residue is the coefficient of (z - z_0)^(-1) in the Laurent series for expr.

(%i1) residue (s/(s**2+a**2), s, a*%i);
                                1
(%o1)                           -
                                2
(%i2) residue (sin(a*x)/x**4, x, 0);
                                 3
                                a
(%o2)                         - --
                                6

Function: risch (expr, x)

Integrates expr with respect to x using the transcendental case of the Risch algorithm. (The algebraic case of the Risch algorithm has not been implemented.) This currently handles the cases of nested exponentials and logarithms which the main part of integrate can’t do. integrate will automatically apply risch if given these cases.

erfflag, if false, prevents risch from introducing the erf function in the answer if there were none in the integrand to begin with.

(%i1) risch (x^2*erf(x), x);
                                                        2
             3                      2                - x
        %pi x  erf(x) + (sqrt(%pi) x  + sqrt(%pi)) %e
(%o1)   -------------------------------------------------
                              3 %pi
(%i2) diff(%, x), ratsimp;
                             2
(%o2)                       x  erf(x)

Function: tldefint (expr, x, a, b)

Equivalent to ldefint with tlimswitch set to true.


Next: , Previous: , Up: Integration   [Contents][Index]

19.3 Introduction to QUADPACK

QUADPACK is a collection of functions for the numerical computation of one-dimensional definite integrals. It originated from a joint project of R. Piessens 1, E. de Doncker 2, C. Ueberhuber 3, and D. Kahaner 4.

The QUADPACK library included in Maxima is an automatic translation (via the program f2cl) of the Fortran source code of QUADPACK as it appears in the SLATEC Common Mathematical Library, Version 4.1 5. The SLATEC library is dated July 1993, but the QUADPACK functions were written some years before. There is another version of QUADPACK at Netlib 6; it is not clear how that version differs from the SLATEC version.

The QUADPACK functions included in Maxima are all automatic, in the sense that these functions attempt to compute a result to a specified accuracy, requiring an unspecified number of function evaluations. Maxima’s Lisp translation of QUADPACK also includes some non-automatic functions, but they are not exposed at the Maxima level.

Further information about QUADPACK can be found in the QUADPACK book 7.

19.3.1 Overview

quad_qag

Integration of a general function over a finite interval. quad_qag implements a simple globally adaptive integrator using the strategy of Aind (Piessens, 1973). The caller may choose among 6 pairs of Gauss-Kronrod quadrature formulae for the rule evaluation component. The high-degree rules are suitable for strongly oscillating integrands.

quad_qags

Integration of a general function over a finite interval. quad_qags implements globally adaptive interval subdivision with extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn, 1956).

quad_qagi

Integration of a general function over an infinite or semi-infinite interval. The interval is mapped onto a finite interval and then the same strategy as in quad_qags is applied.

quad_qawo

Integration of cos(omega x) f(x) or sin(omega x) f(x) over a finite interval, where omega is a constant. The rule evaluation component is based on the modified Clenshaw-Curtis technique. quad_qawo applies adaptive subdivision with extrapolation, similar to quad_qags.

quad_qawf

Calculates a Fourier cosine or Fourier sine transform on a semi-infinite interval. The same approach as in quad_qawo is applied on successive finite intervals, and convergence acceleration by means of the Epsilon algorithm (Wynn, 1956) is applied to the series of the integral contributions.

quad_qaws

Integration of w(x) f(x) over a finite interval [a, b], where w is a function of the form (x - a)^alpha (b - x)^beta v(x) and v(x) is 1 or log(x - a) or log(b - x) or log(x - a) log(b - x), and alpha > -1 and beta > -1.

A globally adaptive subdivision strategy is applied, with modified Clenshaw-Curtis integration on the subintervals which contain a or b.

quad_qawc

Computes the Cauchy principal value of f(x)/(x - c) over a finite interval (a, b) and specified c. The strategy is globally adaptive, and modified Clenshaw-Curtis integration is used on the subranges which contain the point x = c.

quad_qagp

Basically the same as quad_qags but points of singularity or discontinuity of the integrand must be supplied. This makes it easier for the integrator to produce a good solution.

·

Category: Integral calculus · Category: Numerical methods · Category: Share packages · Category: Package quadpack


Previous: , Up: Integration   [Contents][Index]

19.4 Functions and Variables for QUADPACK

Function: quad_qag
    quad_qag (f(x), x, a, b, key, [epsrel, epsabs, limit])
    quad_qag (f, x, a, b, key, [epsrel, epsabs, limit])

Integration of a general function over a finite interval. quad_qag implements a simple globally adaptive integrator using the strategy of Aind (Piessens, 1973). The caller may choose among 6 pairs of Gauss-Kronrod quadrature formulae for the rule evaluation component. The high-degree rules are suitable for strongly oscillating integrands.

quad_qag computes the integral

integrate (f(x), x, a, b)

The function to be integrated is f(x), with dependent variable x, and the function is to be integrated between the limits a and b. key is the integrator to be used and should be an integer between 1 and 6, inclusive. The value of key selects the order of the Gauss-Kronrod integration rule. High-order rules are suitable for strongly oscillating integrands.

The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

The numerical integration is done adaptively by subdividing the integration region into sub-intervals until the desired accuracy is achieved.

The keyword arguments are optional and may be specified in any order. They all take the form key=val. The keyword arguments are:

epsrel

Desired relative error of approximation. Default is 1d-8.

epsabs

Desired absolute error of approximation. Default is 0.

limit

Size of internal work array. limit is the maximum number of subintervals to use. Default is 200.

quad_qag returns a list of four elements:

The error code (fourth element of the return value) can have the values:

0

if no problems were encountered;

1

if too many sub-intervals were done;

2

if excessive roundoff error is detected;

3

if extremely bad integrand behavior occurs;

6

if the input is invalid.

Examples:

(%i1) quad_qag (x^(1/2)*log(1/x), x, 0, 1, 3, 'epsrel=5d-8);
(%o1)    [.4444444444492108, 3.1700968502883E-9, 961, 0]
(%i2) integrate (x^(1/2)*log(1/x), x, 0, 1);
                                4
(%o2)                           -
                                9

Function: quad_qags
    quad_qags (f(x), x, a, b, [epsrel, epsabs, limit])
    quad_qags (f, x, a, b, [epsrel, epsabs, limit])

Integration of a general function over a finite interval. quad_qags implements globally adaptive interval subdivision with extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn, 1956).

quad_qags computes the integral

integrate (f(x), x, a, b)

The function to be integrated is f(x), with dependent variable x, and the function is to be integrated between the limits a and b.

The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

The keyword arguments are optional and may be specified in any order. They all take the form key=val. The keyword arguments are:

epsrel

Desired relative error of approximation. Default is 1d-8.

epsabs

Desired absolute error of approximation. Default is 0.

limit

Size of internal work array. limit is the maximum number of subintervals to use. Default is 200.

quad_qags returns a list of four elements:

The error code (fourth element of the return value) can have the values:

0

no problems were encountered;

1

too many sub-intervals were done;

2

excessive roundoff error is detected;

3

extremely bad integrand behavior occurs;

4

failed to converge

5

integral is probably divergent or slowly convergent

6

if the input is invalid.

Examples:

(%i1) quad_qags (x^(1/2)*log(1/x), x, 0, 1, 'epsrel=1d-10);
(%o1)   [.4444444444444448, 1.11022302462516E-15, 315, 0]

Note that quad_qags is more accurate and efficient than quad_qag for this integrand.

Function: quad_qagi
    quad_qagi (f(x), x, a, b, [epsrel, epsabs, limit])
    quad_qagi (f, x, a, b, [epsrel, epsabs, limit])

Integration of a general function over an infinite or semi-infinite interval. The interval is mapped onto a finite interval and then the same strategy as in quad_qags is applied.

quad_qagi evaluates one of the following integrals

integrate (f(x), x, a, inf)

integrate (f(x), x, minf, a)

integrate (f(x), x, minf, inf)

using the Quadpack QAGI routine. The function to be integrated is f(x), with dependent variable x, and the function is to be integrated over an infinite range.

The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

One of the limits of integration must be infinity. If not, then quad_qagi will just return the noun form.

The keyword arguments are optional and may be specified in any order. They all take the form key=val. The keyword arguments are:

epsrel

Desired relative error of approximation. Default is 1d-8.

epsabs

Desired absolute error of approximation. Default is 0.

limit

Size of internal work array. limit is the maximum number of subintervals to use. Default is 200.

quad_qagi returns a list of four elements:

The error code (fourth element of the return value) can have the values:

0

no problems were encountered;

1

too many sub-intervals were done;

2

excessive roundoff error is detected;

3

extremely bad integrand behavior occurs;

4

failed to converge

5

integral is probably divergent or slowly convergent

6

if the input is invalid.

Examples:

(%i1) quad_qagi (x^2*exp(-4*x), x, 0, inf, 'epsrel=1d-8);
(%o1)        [0.03125, 2.95916102995002E-11, 105, 0]
(%i2) integrate (x^2*exp(-4*x), x, 0, inf);
                               1
(%o2)                          --
                               32

Function: quad_qawc
    quad_qawc (f(x), x, c, a, b, [epsrel, epsabs, limit])
    quad_qawc (f, x, c, a, b, [epsrel, epsabs, limit])

Computes the Cauchy principal value of f(x)/(x - c) over a finite interval. The strategy is globally adaptive, and modified Clenshaw-Curtis integration is used on the subranges which contain the point x = c.

quad_qawc computes the Cauchy principal value of

integrate (f(x)/(x - c), x, a, b)

using the Quadpack QAWC routine. The function to be integrated is f(x)/(x - c), with dependent variable x, and the function is to be integrated over the interval a to b.

The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

The keyword arguments are optional and may be specified in any order. They all take the form key=val. The keyword arguments are:

epsrel

Desired relative error of approximation. Default is 1d-8.

epsabs

Desired absolute error of approximation. Default is 0.

limit

Size of internal work array. limit is the maximum number of subintervals to use. Default is 200.

quad_qawc returns a list of four elements:

The error code (fourth element of the return value) can have the values:

0

no problems were encountered;

1

too many sub-intervals were done;

2

excessive roundoff error is detected;

3

extremely bad integrand behavior occurs;

6

if the input is invalid.

Examples:

’epsrel=1d-7); x, 0, 5);

(%i1) quad_qawc (2^(-5)*((x-1)^2+4^(-5))^(-1), x, 2, 0, 5,
                 'epsrel=1d-7);
(%o1)    [- 3.130120337415925, 1.306830140249558E-8, 495, 0]
(%i2) integrate (2^(-alpha)*(((x-1)^2 + 4^(-alpha))*(x-2))^(-1),
      x, 0, 5);
Principal Value
                       alpha
        alpha       9 4                 9
       4      log(------------- + -------------)
                      alpha           alpha
                  64 4      + 4   64 4      + 4
(%o2) (-----------------------------------------
                        alpha
                     2 4      + 2

       3 alpha                       3 alpha
       -------                       -------
          2            alpha/2          2          alpha/2
    2 4        atan(4 4       )   2 4        atan(4       )   alpha
  - --------------------------- - -------------------------)/2
              alpha                        alpha
           2 4      + 2                 2 4      + 2
(%i3) ev (%, alpha=5, numer);
(%o3)                    - 3.130120337415917

Function: quad_qawf
    quad_qawf (f(x), x, a, omega, trig, [epsabs, limit, maxp1, limlst])
    quad_qawf (f, x, a, omega, trig, [epsabs, limit, maxp1, limlst])

Calculates a Fourier cosine or Fourier sine transform on a semi-infinite interval using the Quadpack QAWF function. The same approach as in quad_qawo is applied on successive finite intervals, and convergence acceleration by means of the Epsilon algorithm (Wynn, 1956) is applied to the series of the integral contributions.

quad_qawf computes the integral

integrate (f(x)*w(x), x, a, inf)

The weight function w is selected by trig:

cos

w(x) = cos (omega x)

sin

w(x) = sin (omega x)

The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

The keyword arguments are optional and may be specified in any order. They all take the form key=val. The keyword arguments are:

epsabs

Desired absolute error of approximation. Default is 1d-10.

limit

Size of internal work array. (limit - limlst)/2 is the maximum number of subintervals to use. Default is 200.

maxp1

Maximum number of Chebyshev moments. Must be greater than 0. Default is 100.

limlst

Upper bound on the number of cycles. Must be greater than or equal to 3. Default is 10.

quad_qawf returns a list of four elements:

The error code (fourth element of the return value) can have the values:

0

no problems were encountered;

1

too many sub-intervals were done;

2

excessive roundoff error is detected;

3

extremely bad integrand behavior occurs;

6

if the input is invalid.

Examples:

(%i1) quad_qawf (exp(-x^2), x, 0, 1, 'cos, 'epsabs=1d-9);
(%o1)   [.6901942235215714, 2.84846300257552E-11, 215, 0]
(%i2) integrate (exp(-x^2)*cos(x), x, 0, inf);
                          - 1/4
                        %e      sqrt(%pi)
(%o2)                   -----------------
                                2
(%i3) ev (%, numer);
(%o3)                   .6901942235215714

Function: quad_qawo
    quad_qawo (f(x), x, a, b, omega, trig, [epsrel, epsabs, limit, maxp1, limlst])
    quad_qawo (f, x, a, b, omega, trig, [epsrel, epsabs, limit, maxp1, limlst])

Integration of cos (omega x) f(x) or sin (omega x) f(x) over a finite interval, where omega is a constant. The rule evaluation component is based on the modified Clenshaw-Curtis technique. quad_qawo applies adaptive subdivision with extrapolation, similar to quad_qags.

quad_qawo computes the integral using the Quadpack QAWO routine:

integrate (f(x)*w(x), x, a, b)

The weight function w is selected by trig:

cos

w(x) = cos (omega x)

sin

w(x) = sin (omega x)

The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

The keyword arguments are optional and may be specified in any order. They all take the form key=val. The keyword arguments are:

epsrel

Desired relative error of approximation. Default is 1d-8.

epsabs

Desired absolute error of approximation. Default is 0.

limit

Size of internal work array. limit/2 is the maximum number of subintervals to use. Default is 200.

maxp1

Maximum number of Chebyshev moments. Must be greater than 0. Default is 100.

limlst

Upper bound on the number of cycles. Must be greater than or equal to 3. Default is 10.

quad_qawo returns a list of four elements:

The error code (fourth element of the return value) can have the values:

0

no problems were encountered;

1

too many sub-intervals were done;

2

excessive roundoff error is detected;

3

extremely bad integrand behavior occurs;

6

if the input is invalid.

Examples:

x, 0, inf));

(%i1) quad_qawo (x^(-1/2)*exp(-2^(-2)*x), x, 1d-8, 20*2^2, 1, cos);
(%o1)     [1.376043389877692, 4.72710759424899E-11, 765, 0]
(%i2) rectform (integrate (x^(-1/2)*exp(-2^(-alpha)*x) * cos(x),
      x, 0, inf));
                   alpha/2 - 1/2            2 alpha
        sqrt(%pi) 2              sqrt(sqrt(2        + 1) + 1)
(%o2)   -----------------------------------------------------
                               2 alpha
                         sqrt(2        + 1)
(%i3) ev (%, alpha=2, numer);
(%o3)                     1.376043390090716

Function: quad_qaws
    quad_qaws (f(x), x, a, b, alpha, beta, wfun, [epsrel, epsabs, limit])
    quad_qaws (f, x, a, b, alpha, beta, wfun, [epsrel, epsabs, limit])

Integration of w(x) f(x) over a finite interval, where w(x) is a certain algebraic or logarithmic function. A globally adaptive subdivision strategy is applied, with modified Clenshaw-Curtis integration on the subintervals which contain the endpoints of the interval of integration.

quad_qaws computes the integral using the Quadpack QAWS routine:

integrate (f(x)*w(x), x, a, b)

The weight function w is selected by wfun:

1

w(x) = (x - a)^alpha (b - x)^beta

2

w(x) = (x - a)^alpha (b - x)^beta log(x - a)

3

w(x) = (x - a)^alpha (b - x)^beta log(b - x)

4

w(x) = (x - a)^alpha (b - x)^beta log(x - a) log(b - x)

The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

The keyword arguments are optional and may be specified in any order. They all take the form key=val. The keyword arguments are:

epsrel

Desired relative error of approximation. Default is 1d-8.

epsabs

Desired absolute error of approximation. Default is 0.

limit

Size of internal work array. limitis the maximum number of subintervals to use. Default is 200.

quad_qaws returns a list of four elements:

The error code (fourth element of the return value) can have the values:

0

no problems were encountered;

1

too many sub-intervals were done;

2

excessive roundoff error is detected;

3

extremely bad integrand behavior occurs;

6

if the input is invalid.

Examples:

’epsabs=1d-9);

(%i1) quad_qaws (1/(x+1+2^(-4)), x, -1, 1, -0.5, -0.5, 1,
                 'epsabs=1d-9);
(%o1)     [8.750097361672832, 1.24321522715422E-10, 170, 0]
(%i2) integrate ((1-x*x)^(-1/2)/(x+1+2^(-alpha)), x, -1, 1);
       alpha
Is  4 2      - 1  positive, negative, or zero?

pos;
                          alpha         alpha
                   2 %pi 2      sqrt(2 2      + 1)
(%o2)              -------------------------------
                               alpha
                            4 2      + 2
(%i3) ev (%, alpha=4, numer);
(%o3)                     8.750097361672829

Function: quad_qagp
    quad_qagp (f(x), x, a, b, points, [epsrel, epsabs, limit])
    quad_qagp (f, x, a, b, points, [epsrel, epsabs, limit])

Integration of a general function over a finite interval. quad_qagp implements globally adaptive interval subdivision with extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn, 1956).

quad_qagp computes the integral

integrate (f(x), x, a, b)

The function to be integrated is f(x), with dependent variable x, and the function is to be integrated between the limits a and b.

The integrand may be specified as the name of a Maxima or Lisp function or operator, a Maxima lambda expression, or a general Maxima expression.

To help the integrator, the user must supply a list of points where the integrand is singular or discontinous.

The keyword arguments are optional and may be specified in any order. They all take the form key=val. The keyword arguments are:

epsrel

Desired relative error of approximation. Default is 1d-8.

epsabs

Desired absolute error of approximation. Default is 0.

limit

Size of internal work array. limit is the maximum number of subintervals to use. Default is 200.

quad_qagp returns a list of four elements:

The error code (fourth element of the return value) can have the values:

0

no problems were encountered;

1

too many sub-intervals were done;

2

excessive roundoff error is detected;

3

extremely bad integrand behavior occurs;

4

failed to converge

5

integral is probably divergent or slowly convergent

6

if the input is invalid.

Examples:

(%i1) quad_qagp(x^3*log(abs((x^2-1)*(x^2-2))),x,0,3,[1,sqrt(2)]);
(%o1)   [52.74074838347143, 2.6247632689546663e-7, 1029, 0]
(%i2) quad_qags(x^3*log(abs((x^2-1)*(x^2-2))), x, 0, 3);
(%o2)   [52.74074847951494, 4.088443219529836e-7, 1869, 0]

The integrand has singularities at 1 and sqrt(2) so we supply these points to quad_qagp. We also note that quad_qagp is more accurate and more efficient that quad_qags.

Function: quad_control (parameter, [value])

Control error handling for quadpack. The parameter should be one of the following symbols:

current_error

The current error number

control

Controls if messages are printed or not. If it is set to zero or less, messages are suppressed.

max_message

The maximum number of times any message is to be printed.

If value is not given, then the current value of the parameter is returned. If value is given, the value of parameter is set to the given value.


Next: , Previous: , Up: Top   [Contents][Index]

20 Equations


Previous: , Up: Equations   [Contents][Index]

20.1 Functions and Variables for Equations

System variable: %rnum

Default value: 0

%rnum is the counter for the %r variables introduced in solutions by solve and algsys.. The next %r variable is numbered %rnum+1.

See also %rnum_list.

System variable: %rnum_list

Default value: []

%rnum_list is the list of variables introduced in solutions by solve and algsys. %r variables are added to %rnum_list in the order they are created. This is convenient for doing substitutions into the solution later on.

See also %rnum.

It’s recommended to use this list rather than doing concat ('%r, j).

(%i1) solve ([x + y = 3], [x,y]);
(%o1)              [[x = 3 - %r1, y = %r1]]
(%i2) %rnum_list;
(%o2)                       [%r1]
(%i3) sol : solve ([x + 2*y + 3*z = 4], [x,y,z]);
(%o3)   [[x = - 2 %r3 - 3 %r2 + 4, y = %r3, z = %r2]]
(%i4) %rnum_list;
(%o4)                     [%r2, %r3]
(%i5) for i : 1 thru length (%rnum_list) do
        sol : subst (t[i], %rnum_list[i], sol)$
(%i6) sol;
(%o6)     [[x = - 2 t  - 3 t  + 4, y = t , z = t ]]
                     2      1           2       1

Option variable: algepsilon

Default value: 10^8

algepsilon is used by algsys.

Option variable: algexact

Default value: false

algexact affects the behavior of algsys as follows:

If algexact is true, algsys always calls solve and then uses realroots on solve’s failures.

If algexact is false, solve is called only if the eliminant was not univariate, or if it was a quadratic or biquadratic.

Thus algexact: true does not guarantee only exact solutions, just that algsys will first try as hard as it can to give exact solutions, and only yield approximations when all else fails.

Function: algsys
    algsys ([expr_1, …, expr_m], [x_1, …, x_n])
    algsys ([eqn_1, …, eqn_m], [x_1, …, x_n])

Solves the simultaneous polynomials expr_1, …, expr_m or polynomial equations eqn_1, …, eqn_m for the variables x_1, …, x_n. An expression expr is equivalent to an equation expr = 0. There may be more equations than variables or vice versa.

algsys returns a list of solutions, with each solution given as a list of equations stating values of the variables x_1, …, x_n which satisfy the system of equations. If algsys cannot find a solution, an empty list [] is returned.

The symbols %r1, %r2, …, are introduced as needed to represent arbitrary parameters in the solution; these variables are also appended to the list %rnum_list.

The method is as follows:

  1. First the equations are factored and split into subsystems.
  2. For each subsystem S_i, an equation E and a variable x are selected. The variable is chosen to have lowest nonzero degree. Then the resultant of E and E_j with respect to x is computed for each of the remaining equations E_j in the subsystem S_i. This yields a new subsystem S_i’ in one fewer variables, as x has been eliminated. The process now returns to (1).
  3. Eventually, a subsystem consisting of a single equation is obtained. If the equation is multivariate and no approximations in the form of floating point numbers have been introduced, then solve is called to find an exact solution.

    In some cases, solve is not be able to find a solution, or if it does the solution may be a very large expression.

    If the equation is univariate and is either linear, quadratic, or biquadratic, then again solve is called if no approximations have been introduced. If approximations have been introduced or the equation is not univariate and neither linear, quadratic, or biquadratic, then if the switch realonly is true, the function realroots is called to find the real-valued solutions. If realonly is false, then allroots is called which looks for real and complex-valued solutions.

    If algsys produces a solution which has fewer significant digits than required, the user can change the value of algepsilon to a higher value.

    If algexact is set to true, solve will always be called.

  4. Finally, the solutions obtained in step (3) are substituted into previous levels and the solution process returns to (1).

When algsys encounters a multivariate equation which contains floating point approximations (usually due to its failing to find exact solutions at an earlier stage), then it does not attempt to apply exact methods to such equations and instead prints the message: "algsys cannot solve - system too complicated."

Interactions with radcan can produce large or complicated expressions. In that case, it may be possible to isolate parts of the result with pickapart or reveal.

Occasionally, radcan may introduce an imaginary unit %i into a solution which is actually real-valued.

Examples:

(%i1) e1: 2*x*(1 - a1) - 2*(x - 1)*a2;
(%o1)              2 (1 - a1) x - 2 a2 (x - 1)
(%i2) e2: a2 - a1; 
(%o2)                        a2 - a1
(%i3) e3: a1*(-y - x^2 + 1); 
                                   2
(%o3)                   a1 (- y - x  + 1)
(%i4) e4: a2*(y - (x - 1)^2);
                                       2
(%o4)                   a2 (y - (x - 1) )
(%i5) algsys ([e1, e2, e3, e4], [x, y, a1, a2]);
(%o5) [[x = 0, y = %r1, a1 = 0, a2 = 0], 

                                  [x = 1, y = 0, a1 = 1, a2 = 1]]
(%i6) e1: x^2 - y^2;
                              2    2
(%o6)                        x  - y
(%i7) e2: -1 - y + 2*y^2 - x + x^2;
                         2        2
(%o7)                 2 y  - y + x  - x - 1
(%i8) algsys ([e1, e2], [x, y]);
                 1            1
(%o8) [[x = - -------, y = -------], 
              sqrt(3)      sqrt(3)

        1              1             1        1
[x = -------, y = - -------], [x = - -, y = - -], [x = 1, y = 1]]
     sqrt(3)        sqrt(3)          3        3

Function: allroots
    allroots (expr)
    allroots (eqn)

Computes numerical approximations of the real and complex roots of the polynomial expr or polynomial equation eqn of one variable.

The flag polyfactor when true causes allroots to factor the polynomial over the real numbers if the polynomial is real, or over the complex numbers, if the polynomial is complex.

allroots may give inaccurate results in case of multiple roots. If the polynomial is real, allroots (%i*p) may yield more accurate approximations than allroots (p), as allroots invokes a different algorithm in that case.

allroots rejects non-polynomials. It requires that the numerator after rat’ing should be a polynomial, and it requires that the denominator be at most a complex number. As a result of this allroots will always return an equivalent (but factored) expression, if polyfactor is true.

For complex polynomials an algorithm by Jenkins and Traub is used (Algorithm 419, Comm. ACM, vol. 15, (1972), p. 97). For real polynomials the algorithm used is due to Jenkins (Algorithm 493, ACM TOMS, vol. 1, (1975), p.178).

Examples:

(%i1) eqn: (1 + 2*x)^3 = 13.5*(1 + x^5);
                            3          5
(%o1)              (2 x + 1)  = 13.5 (x  + 1)
(%i2) soln: allroots (eqn);
(%o2) [x = .8296749902129361, x = - 1.015755543828121, 

x = .9659625152196369 %i - .4069597231924075, 

x = - .9659625152196369 %i - .4069597231924075, x = 1.0]
(%i3) for e in soln
        do (e2: subst (e, eqn), disp (expand (lhs(e2) - rhs(e2))));
                      - 3.5527136788005E-15

                     - 5.32907051820075E-15

         4.44089209850063E-15 %i - 4.88498130835069E-15

        - 4.44089209850063E-15 %i - 4.88498130835069E-15

                       3.5527136788005E-15

(%o3)                         done
(%i4) polyfactor: true$
(%i5) allroots (eqn);
(%o5) - 13.5 (x - 1.0) (x - .8296749902129361)

                           2
 (x + 1.015755543828121) (x  + .8139194463848151 x

 + 1.098699797110288)

Function: bfallroots
    bfallroots (expr)
    bfallroots (eqn)

Computes numerical approximations of the real and complex roots of the polynomial expr or polynomial equation eqn of one variable.

In all respects, bfallroots is identical to allroots except that bfallroots computes the roots using bigfloats. See allroots for more information.

Option variable: backsubst

Default value: true

When backsubst is false, prevents back substitution in linsolve after the equations have been triangularized. This may be helpful in very big problems where back substitution would cause the generation of extremely large expressions.

(%i1) eq1 : x + y + z = 6$
(%i2) eq2 : x - y + z = 2$
(%i3) eq3 : x + y - z = 0$
(%i4) backsubst : false$
(%i5) linsolve ([eq1, eq2, eq3], [x,y,z]);
(%o5)             [x = z - y, y = 2, z = 3]
(%i6) backsubst : true$
(%i7) linsolve ([eq1, eq2, eq3], [x,y,z]);
(%o7)               [x = 1, y = 2, z = 3]

Option variable: breakup

Default value: true

When breakup is true, solve expresses solutions of cubic and quartic equations in terms of common subexpressions, which are assigned to intermediate expression labels (%t1, %t2, etc.). Otherwise, common subexpressions are not identified.

breakup: true has an effect only when programmode is false.

Examples:

(%i1) programmode: false$
(%i2) breakup: true$
(%i3) solve (x^3 + x^2 - 1);

                        sqrt(23)    25 1/3
(%t3)                  (--------- + --)
                        6 sqrt(3)   54
Solution:

                                      sqrt(3) %i   1
                                      ---------- - -
                sqrt(3) %i   1            2        2   1
(%t4)    x = (- ---------- - -) %t3 + -------------- - -
                    2        2            9 %t3        3

                                      sqrt(3) %i   1
                                    - ---------- - -
              sqrt(3) %i   1              2        2   1
(%t5)    x = (---------- - -) %t3 + ---------------- - -
                  2        2             9 %t3         3

                                   1     1
(%t6)                  x = %t3 + ----- - -
                                 9 %t3   3
(%o6)                    [%t4, %t5, %t6]
(%i6) breakup: false$
(%i7) solve (x^3 + x^2 - 1);
Solution:

             sqrt(3) %i   1
             ---------- - -
                 2        2        sqrt(23)    25 1/3
(%t7) x = --------------------- + (--------- + --)
             sqrt(23)    25 1/3    6 sqrt(3)   54
          9 (--------- + --)
             6 sqrt(3)   54

                                              sqrt(3) %i   1    1
                                           (- ---------- - -) - -
                                                  2        2    3
           sqrt(23)    25 1/3  sqrt(3) %i   1
(%t8) x = (--------- + --)    (---------- - -)
           6 sqrt(3)   54          2        2

                                            sqrt(3) %i   1
                                          - ---------- - -
                                                2        2      1
                                      + --------------------- - -
                                           sqrt(23)    25 1/3   3
                                        9 (--------- + --)
                                           6 sqrt(3)   54
            sqrt(23)    25 1/3             1             1
(%t9)  x = (--------- + --)    + --------------------- - -
            6 sqrt(3)   54          sqrt(23)    25 1/3   3
                                 9 (--------- + --)
                                    6 sqrt(3)   54
(%o9)                    [%t7, %t8, %t9]

Function: dimension
    dimension (eqn)
    dimension (eqn_1, …, eqn_n)

dimen is a package for dimensional analysis. load ("dimen") loads this package. demo ("dimen") displays a short demonstration.

Option variable: dispflag

Default value: true

If set to false within a block will inhibit the display of output generated by the solve functions called from within the block. Termination of the block with a dollar sign, $, sets dispflag to false.

Function: funcsolve (eqn, g(t))

Returns [g(t) = ...] or [], depending on whether or not there exists a rational function g(t) satisfying eqn, which must be a first order, linear polynomial in (for this case) g(t) and g(t+1)

(%i1) eqn: (n + 1)*f(n) - (n + 3)*f(n + 1)/(n + 1) =
      (n - 1)/(n + 2);
                            (n + 3) f(n + 1)   n - 1
(%o1)        (n + 1) f(n) - ---------------- = -----
                                 n + 1         n + 2
(%i2) funcsolve (eqn, f(n));

Dependent equations eliminated:  (4 3)
                                   n
(%o2)                f(n) = ---------------
                            (n + 1) (n + 2)

Warning: this is a very rudimentary implementation – many safety checks and obvious generalizations are missing.

Option variable: globalsolve

Default value: false

When globalsolve is true, solved-for variables are assigned the solution values found by linsolve, and by solve when solving two or more linear equations.

When globalsolve is false, solutions found by linsolve and by solve when solving two or more linear equations are expressed as equations, and the solved-for variables are not assigned.

When solving anything other than two or more linear equations, solve ignores globalsolve. Other functions which solve equations (e.g., algsys) always ignore globalsolve.

Examples:

(%i1) globalsolve: true$
(%i2) solve ([x + 3*y = 2, 2*x - y = 5], [x, y]);
Solution

                                 17
(%t2)                        x : --
                                 7

                                   1
(%t3)                        y : - -
                                   7
(%o3)                     [[%t2, %t3]]
(%i3) x;
                               17
(%o3)                          --
                               7
(%i4) y;
                                 1
(%o4)                          - -
                                 7
(%i5) globalsolve: false$
(%i6) kill (x, y)$
(%i7) solve ([x + 3*y = 2, 2*x - y = 5], [x, y]);
Solution

                                 17
(%t7)                        x = --
                                 7

                                   1
(%t8)                        y = - -
                                   7
(%o8)                     [[%t7, %t8]]
(%i8) x;
(%o8)                           x
(%i9) y;
(%o9)                           y

Function: ieqn (ie, unk, tech, n, guess)

inteqn is a package for solving integral equations. load ("inteqn") loads this package.

ie is the integral equation; unk is the unknown function; tech is the technique to be tried from those given above (tech = first means: try the first technique which finds a solution; tech = all means: try all applicable techniques); n is the maximum number of terms to take for taylor, neumann, firstkindseries, or fredseries (it is also the maximum depth of recursion for the differentiation method); guess is the initial guess for neumann or firstkindseries.

Default values for the 2nd thru 5th parameters are:

unk: p(x), where p is the first function encountered in an integrand which is unknown to Maxima and x is the variable which occurs as an argument to the first occurrence of p found outside of an integral in the case of secondkind equations, or is the only other variable besides the variable of integration in firstkind equations. If the attempt to search for x fails, the user will be asked to supply the independent variable.

tech: first

n: 1

guess: none which will cause neumann and firstkindseries to use f(x) as an initial guess.

Option variable: ieqnprint

Default value: true

ieqnprint governs the behavior of the result returned by the ieqn command. When ieqnprint is false, the lists returned by the ieqn function are of the form

[solution, technique used, nterms, flag]

where flag is absent if the solution is exact.

Otherwise, it is the word approximate or incomplete corresponding to an inexact or non-closed form solution, respectively. If a series method was used, nterms gives the number of terms taken (which could be less than the n given to ieqn if an error prevented generation of further terms).

Function: lhs (expr)

Returns the left-hand side (that is, the first argument) of the expression expr, when the operator of expr is one of the relational operators < <= = # equal notequal >= >, one of the assignment operators := ::= : ::, or a user-defined binary infix operator, as declared by infix.

When expr is an atom or its operator is something other than the ones listed above, lhs returns expr.

See also rhs.

Examples:

(%i1) e: aa + bb = cc;
(%o1)                     bb + aa = cc
(%i2) lhs (e);
(%o2)                        bb + aa
(%i3) rhs (e);
(%o3)                          cc
(%i4) [lhs (aa < bb), lhs (aa <= bb), lhs (aa >= bb),
       lhs (aa > bb)];
(%o4)                   [aa, aa, aa, aa]
(%i5) [lhs (aa = bb), lhs (aa # bb), lhs (equal (aa, bb)),
       lhs (notequal (aa, bb))];
(%o5)                   [aa, aa, aa, aa]
(%i6) e1: '(foo(x) := 2*x);
(%o6)                     foo(x) := 2 x
(%i7) e2: '(bar(y) ::= 3*y);
(%o7)                    bar(y) ::= 3 y
(%i8) e3: '(x : y);
(%o8)                         x : y
(%i9) e4: '(x :: y);
(%o9)                        x :: y
(%i10) [lhs (e1), lhs (e2), lhs (e3), lhs (e4)];
(%o10)               [foo(x), bar(y), x, x]
(%i11) infix ("][");
(%o11)                         ][
(%i12) lhs (aa ][ bb);
(%o12)                         aa

Function: linsolve ([expr_1, …, expr_m], [x_1, …, x_n])

Solves the list of simultaneous linear equations for the list of variables. The expressions must each be polynomials in the variables and may be equations. If the length of the list of variables doesn’t match the number of linearly-independent equations to solve the result will be an empty list.

When globalsolve is true, each solved-for variable is bound to its value in the solution of the equations.

When backsubst is false, linsolve does not carry out back substitution after the equations have been triangularized. This may be necessary in very big problems where back substitution would cause the generation of extremely large expressions.

When linsolve_params is true, linsolve also generates the %r symbols used to represent arbitrary parameters described in the manual under algsys. Otherwise, linsolve solves an under-determined system of equations with some variables expressed in terms of others.

When programmode is false, linsolve displays the solution with intermediate expression (%t) labels, and returns the list of labels.

See also algsys, eliminate. and solve.

Examples:

(%i1) e1: x + z = y;
(%o1)                       z + x = y
(%i2) e2: 2*a*x - y = 2*a^2;
                                       2
(%o2)                   2 a x - y = 2 a
(%i3) e3: y - 2*z = 2;
(%o3)                      y - 2 z = 2
(%i4) [globalsolve: false, programmode: true];
(%o4)                     [false, true]
(%i5) linsolve ([e1, e2, e3], [x, y, z]);
(%o5)            [x = a + 1, y = 2 a, z = a - 1]
(%i6) [globalsolve: false, programmode: false];
(%o6)                    [false, false]
(%i7) linsolve ([e1, e2, e3], [x, y, z]);
Solution

(%t7)                       z = a - 1

(%t8)                        y = 2 a

(%t9)                       x = a + 1
(%o9)                    [%t7, %t8, %t9]
(%i9) ''%;
(%o9)            [z = a - 1, y = 2 a, x = a + 1]
(%i10) [globalsolve: true, programmode: false];
(%o10)                    [true, false]
(%i11) linsolve ([e1, e2, e3], [x, y, z]);
Solution

(%t11)                      z : a - 1

(%t12)                       y : 2 a

(%t13)                      x : a + 1
(%o13)                 [%t11, %t12, %t13]
(%i13) ''%;
(%o13)           [z : a - 1, y : 2 a, x : a + 1]
(%i14) [x, y, z];
(%o14)                 [a + 1, 2 a, a - 1]
(%i15) [globalsolve: true, programmode: true];
(%o15)                    [true, true]
(%i16) linsolve ([e1, e2, e3], '[x, y, z]);
(%o16)           [x : a + 1, y : 2 a, z : a - 1]
(%i17) [x, y, z];
(%o17)                 [a + 1, 2 a, a - 1]

Option variable: linsolvewarn

Default value: true

When linsolvewarn is true, linsolve prints a message "Dependent equations eliminated".

Option variable: linsolve_params

Default value: true

When linsolve_params is true, linsolve also generates the %r symbols used to represent arbitrary parameters described in the manual under algsys. Otherwise, linsolve solves an under-determined system of equations with some variables expressed in terms of others.

System variable: multiplicities

Default value: not_set_yet

multiplicities is set to a list of the multiplicities of the individual solutions returned by solve or realroots.

Function: nroots (p, low, high)

Returns the number of real roots of the real univariate polynomial p in the half-open interval (low, high]. The endpoints of the interval may be minf or inf.

nroots uses the method of Sturm sequences.

(%i1) p: x^10 - 2*x^4 + 1/2$
(%i2) nroots (p, -6, 9.1);
(%o2)                           4

Function: nthroot (p, n)

where p is a polynomial with integer coefficients and n is a positive integer returns q, a polynomial over the integers, such that q^n = p or prints an error message indicating that p is not a perfect nth power. This routine is much faster than factor or even sqfr.

Option variable: polyfactor

Default value: false

The option variable polyfactor when true causes allroots and bfallroots to factor the polynomial over the real numbers if the polynomial is real, or over the complex numbers, if the polynomial is complex.

See allroots for an example.

Option variable: programmode

Default value: true

When programmode is true, solve, realroots, allroots, and linsolve return solutions as elements in a list. (Except when backsubst is set to false, in which case programmode: false is assumed.)

When programmode is false, solve, etc. create intermediate expression labels %t1, %t2, etc., and assign the solutions to them.

Option variable: realonly

Default value: false

When realonly is true, algsys returns only those solutions which are free of %i.

Function: realroots
    realroots (expr, bound)
    realroots (eqn, bound)
    realroots (expr)
    realroots (eqn)

Computes rational approximations of the real roots of the polynomial expr or polynomial equation eqn of one variable, to within a tolerance of bound. Coefficients of expr or eqn must be literal numbers; symbol constants such as %pi are rejected.

realroots assigns the multiplicities of the roots it finds to the global variable multiplicities.

realroots constructs a Sturm sequence to bracket each root, and then applies bisection to refine the approximations. All coefficients are converted to rational equivalents before searching for roots, and computations are carried out by exact rational arithmetic. Even if some coefficients are floating-point numbers, the results are rational (unless coerced to floats by the float or numer flags).

When bound is less than 1, all integer roots are found exactly. When bound is unspecified, it is assumed equal to the global variable rootsepsilon.

When the global variable programmode is true, realroots returns a list of the form [x = x_1, x = x_2, ...]. When programmode is false, realroots creates intermediate expression labels %t1, %t2, …, assigns the results to them, and returns the list of labels.

Examples:

(%i1) realroots (-1 - x + x^5, 5e-6);
                               612003
(%o1)                     [x = ------]
                               524288
(%i2) ev (%[1], float);
(%o2)                 x = 1.167303085327148
(%i3) ev (-1 - x + x^5, %);
(%o3)                - 7.396496210176905E-6
(%i1) realroots (expand ((1 - x)^5 * (2 - x)^3 * (3 - x)), 1e-20);
(%o1)                 [x = 1, x = 2, x = 3]
(%i2) multiplicities;
(%o2)                       [5, 3, 1]

Function: rhs (expr)

Returns the right-hand side (that is, the second argument) of the expression expr, when the operator of expr is one of the relational operators < <= = # equal notequal >= >, one of the assignment operators := ::= : ::, or a user-defined binary infix operator, as declared by infix.

When expr is an atom or its operator is something other than the ones listed above, rhs returns 0.

See also lhs.

Examples:

(%i1) e: aa + bb = cc;
(%o1)                     bb + aa = cc
(%i2) lhs (e);
(%o2)                        bb + aa
(%i3) rhs (e);
(%o3)                          cc
(%i4) [rhs (aa < bb), rhs (aa <= bb), rhs (aa >= bb),
       rhs (aa > bb)];
(%o4)                   [bb, bb, bb, bb]
(%i5) [rhs (aa = bb), rhs (aa # bb), rhs (equal (aa, bb)),
       rhs (notequal (aa, bb))];
(%o5)                   [bb, bb, bb, bb]
(%i6) e1: '(foo(x) := 2*x);
(%o6)                     foo(x) := 2 x
(%i7) e2: '(bar(y) ::= 3*y);
(%o7)                    bar(y) ::= 3 y
(%i8) e3: '(x : y);
(%o8)                         x : y
(%i9) e4: '(x :: y);
(%o9)                        x :: y
(%i10) [rhs (e1), rhs (e2), rhs (e3), rhs (e4)];
(%o10)                  [2 x, 3 y, y, y]
(%i11) infix ("][");
(%o11)                         ][
(%i12) rhs (aa ][ bb);
(%o12)                         bb

Option variable: rootsconmode

Default value: true

rootsconmode governs the behavior of the rootscontract command. See rootscontract for details.

Function: rootscontract (expr)

Converts products of roots into roots of products. For example, rootscontract (sqrt(x)*y^(3/2)) yields sqrt(x*y^3).

When radexpand is true and domain is real, rootscontract converts abs into sqrt, e.g., rootscontract (abs(x)*sqrt(y)) yields sqrt(x^2*y).

There is an option rootsconmode affecting rootscontract as follows:

Problem            Value of        Result of applying
                  rootsconmode        rootscontract
      
x^(1/2)*y^(3/2)      false          (x*y^3)^(1/2)
x^(1/2)*y^(1/4)      false          x^(1/2)*y^(1/4)
x^(1/2)*y^(1/4)      true           (x*y^(1/2))^(1/2)
x^(1/2)*y^(1/3)      true           x^(1/2)*y^(1/3)
x^(1/2)*y^(1/4)      all            (x^2*y)^(1/4)
x^(1/2)*y^(1/3)      all            (x^3*y^2)^(1/6)

When rootsconmode is false, rootscontract contracts only with respect to rational number exponents whose denominators are the same. The key to the rootsconmode: true examples is simply that 2 divides into 4 but not into 3. rootsconmode: all involves taking the least common multiple of the denominators of the exponents.

rootscontract uses ratsimp in a manner similar to logcontract.

Examples:

(%i1) rootsconmode: false$
(%i2) rootscontract (x^(1/2)*y^(3/2));
                                   3
(%o2)                      sqrt(x y )
(%i3) rootscontract (x^(1/2)*y^(1/4));
                                   1/4
(%o3)                     sqrt(x) y
(%i4) rootsconmode: true$
(%i5) rootscontract (x^(1/2)*y^(1/4));
(%o5)                    sqrt(x sqrt(y))
(%i6) rootscontract (x^(1/2)*y^(1/3));
                                   1/3
(%o6)                     sqrt(x) y
(%i7) rootsconmode: all$
(%i8) rootscontract (x^(1/2)*y^(1/4));
                              2   1/4
(%o8)                       (x  y)
(%i9) rootscontract (x^(1/2)*y^(1/3));
                             3  2 1/6
(%o9)                      (x  y )
(%i10) rootsconmode: false$
(%i11) rootscontract (sqrt(sqrt(x) + sqrt(1 + x))
                    *sqrt(sqrt(1 + x) - sqrt(x)));
(%o11)                          1
(%i12) rootsconmode: true$
(%i13) rootscontract (sqrt(5+sqrt(5)) - 5^(1/4)*sqrt(1+sqrt(5)));
(%o13)                          0

Option variable: rootsepsilon

Default value: 1.0e-7

rootsepsilon is the tolerance which establishes the confidence interval for the roots found by the realroots function.

Function: solve
    solve (expr, x)
    solve (expr)
    solve ([eqn_1, …, eqn_n], [x_1, …, x_n])

Solves the algebraic equation expr for the variable x and returns a list of solution equations in x. If expr is not an equation, the equation expr = 0 is assumed in its place. x may be a function (e.g. f(x)), or other non-atomic expression except a sum or product. x may be omitted if expr contains only one variable. expr may be a rational expression, and may contain trigonometric functions, exponentials, etc.

The following method is used:

Let E be the expression and X be the variable. If E is linear in X then it is trivially solved for X. Otherwise if E is of the form A*X^N + B then the result is (-B/A)^1/N) times the N’th roots of unity.

If E is not linear in X then the gcd of the exponents of X in E (say N) is divided into the exponents and the multiplicity of the roots is multiplied by N. Then solve is called again on the result. If E factors then solve is called on each of the factors. Finally solve will use the quadratic, cubic, or quartic formulas where necessary.

In the case where E is a polynomial in some function of the variable to be solved for, say F(X), then it is first solved for F(X) (call the result C), then the equation F(X)=C can be solved for X provided the inverse of the function F is known.

breakup if false will cause solve to express the solutions of cubic or quartic equations as single expressions rather than as made up of several common subexpressions which is the default.

multiplicities - will be set to a list of the multiplicities of the individual solutions returned by solve, realroots, or allroots. Try apropos (solve) for the switches which affect solve. describe may then by used on the individual switch names if their purpose is not clear.

solve ([eqn_1, ..., eqn_n], [x_1, ..., x_n]) solves a system of simultaneous (linear or non-linear) polynomial equations by calling linsolve or algsys and returns a list of the solution lists in the variables. In the case of linsolve this list would contain a single list of solutions. It takes two lists as arguments. The first list represents the equations to be solved; the second list is a list of the unknowns to be determined. If the total number of variables in the equations is equal to the number of equations, the second argument-list may be omitted.

When programmode is false, solve displays solutions with intermediate expression (%t) labels, and returns the list of labels.

When globalsolve is true and the problem is to solve two or more linear equations, each solved-for variable is bound to its value in the solution of the equations.

Examples:

(%i1) solve (asin (cos (3*x))*(f(x) - 1), x);

solve: using arc-trig functions to get a solution.
Some solutions will be lost.
                            %pi
(%o1)                  [x = ---, f(x) = 1]
                             6
(%i2) ev (solve (5^f(x) = 125, f(x)), solveradcan);
                                log(125)
(%o2)                   [f(x) = --------]
                                 log(5)
(%i3) [4*x^2 - y^2 = 12, x*y - x = 2];
                      2    2
(%o3)             [4 x  - y  = 12, x y - x = 2]

(%i4) solve (%, [x, y]);
(%o4) [[x = 2, y = 2], [x = .5202594388652008 %i
 - .1331240357358706, y = .07678378523787788
 - 3.608003221870287 %i], [x = - .5202594388652008 %i
 - .1331240357358706, y = 3.608003221870287 %i
 + .07678378523787788], [x = - 1.733751846381093, 
y = - .1535675710019696]]

(%i5) solve (1 + a*x + x^3, x);

                                       3
              sqrt(3) %i   1   sqrt(4 a  + 27)   1 1/3
(%o5) [x = (- ---------- - -) (--------------- - -)
                  2        2      6 sqrt(3)      2

        sqrt(3) %i   1
       (---------- - -) a
            2        2
 - --------------------------, x = 
              3
      sqrt(4 a  + 27)   1 1/3
   3 (--------------- - -)
         6 sqrt(3)      2

                          3
 sqrt(3) %i   1   sqrt(4 a  + 27)   1 1/3
(---------- - -) (--------------- - -)
     2        2      6 sqrt(3)      2

         sqrt(3) %i   1
      (- ---------- - -) a
             2        2
 - --------------------------, x = 
              3
      sqrt(4 a  + 27)   1 1/3
   3 (--------------- - -)
         6 sqrt(3)      2

         3
 sqrt(4 a  + 27)   1 1/3               a
(--------------- - -)    - --------------------------]
    6 sqrt(3)      2                  3
                              sqrt(4 a  + 27)   1 1/3
                           3 (--------------- - -)
                                 6 sqrt(3)      2
(%i6) solve (x^3 - 1);
             sqrt(3) %i - 1        sqrt(3) %i + 1
(%o6)   [x = --------------, x = - --------------, x = 1]
                   2                     2
(%i7) solve (x^6 - 1);
           sqrt(3) %i + 1      sqrt(3) %i - 1
(%o7) [x = --------------, x = --------------, x = - 1, 
                 2                   2

                     sqrt(3) %i + 1        sqrt(3) %i - 1
               x = - --------------, x = - --------------, x = 1]
                           2                     2
(%i8) ev (x^6 - 1, %[1]);
                                      6
                      (sqrt(3) %i + 1)
(%o8)                 ----------------- - 1
                             64
(%i9) expand (%);
(%o9)                           0
(%i10) x^2 - 1;
                              2
(%o10)                       x  - 1
(%i11) solve (%, x);
(%o11)                  [x = - 1, x = 1]
(%i12) ev (%th(2), %[1]);
(%o12)                          0

The symbols %r are used to denote arbitrary constants in a solution.

(%i1) solve([x+y=1,2*x+2*y=2],[x,y]);

solve: dependent equations eliminated: (2)
(%o1)                      [[x = 1 - %r1, y = %r1]]

See algsys and %rnum_list for more information.

Option variable: solvedecomposes

Default value: true

When solvedecomposes is true, solve calls polydecomp if asked to solve polynomials.

Option variable: solveexplicit

Default value: false

When solveexplicit is true, inhibits solve from returning implicit solutions, that is, solutions of the form F(x) = 0 where F is some function.

Option variable: solvefactors

Default value: true

When solvefactors is false, solve does not try to factor the expression. The false setting may be desired in some cases where factoring is not necessary.

Option variable: solvenullwarn

Default value: true

When solvenullwarn is true, solve prints a warning message if called with either a null equation list or a null variable list. For example, solve ([], []) would print two warning messages and return [].

Option variable: solveradcan

Default value: false

When solveradcan is true, solve calls radcan which makes solve slower but will allow certain problems containing exponentials and logarithms to be solved.

Option variable: solvetrigwarn

Default value: true

When solvetrigwarn is true, solve may print a message saying that it is using inverse trigonometric functions to solve the equation, and thereby losing solutions.


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21 Differential Equations


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21.1 Introduction to Differential Equations

This section describes the functions available in Maxima to obtain analytic solutions for some specific types of first and second-order equations. To obtain a numerical solution for a system of differential equations, see the additional package dynamics. For graphical representations in phase space, see the additional package plotdf.


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21.2 Functions and Variables for Differential Equations

Function: bc2 (solution, xval1, yval1, xval2, yval2)

Solves a boundary value problem for a second order differential equation. Here: solution is a general solution to the equation, as found by ode2; xval1 specifies the value of the independent variable in a first point, in the form x = x1, and yval1 gives the value of the dependent variable in that point, in the form y = y1. The expressions xval2 and yval2 give the values for these variables at a second point, using the same form.

See ode2 for an example of its usage.

Function: desolve
    desolve (eqn, x)
    desolve ([eqn_1, ..., eqn_n], [x_1, ..., x_n])

The function desolve solves systems of linear ordinary differential equations using Laplace transform. Here the eqn’s are differential equations in the dependent variables x_1, ..., x_n. The functional dependence of x_1, ..., x_n on an independent variable, for instance x, must be explicitly indicated in the variables and its derivatives. For example, this would not be the correct way to define two equations:

eqn_1: 'diff(f,x,2) = sin(x) + 'diff(g,x);
eqn_2: 'diff(f,x) + x^2 - f = 2*'diff(g,x,2);

The correct way would be:

eqn_1: 'diff(f(x),x,2) = sin(x) + 'diff(g(x),x);
eqn_2: 'diff(f(x),x) + x^2 - f(x) = 2*'diff(g(x),x,2);

The call to the function desolve would then be

desolve([eqn_1, eqn_2], [f(x),g(x)]);

If initial conditions at x=0 are known, they can be supplied before calling desolve by using atvalue.

(%i1) 'diff(f(x),x)='diff(g(x),x)+sin(x);
                 d           d
(%o1)            -- (f(x)) = -- (g(x)) + sin(x)
                 dx          dx
(%i2) 'diff(g(x),x,2)='diff(f(x),x)-cos(x);
                  2
                 d            d
(%o2)            --- (g(x)) = -- (f(x)) - cos(x)
                   2          dx
                 dx
(%i3) atvalue('diff(g(x),x),x=0,a);
(%o3)                           a
(%i4) atvalue(f(x),x=0,1);
(%o4)                           1
(%i5) desolve([%o1,%o2],[f(x),g(x)]);
                  x
(%o5) [f(x) = a %e  - a + 1, g(x) = 

                                                x
                                   cos(x) + a %e  - a + g(0) - 1]
(%i6) [%o1,%o2],%o5,diff;
             x       x      x                x
(%o6)   [a %e  = a %e , a %e  - cos(x) = a %e  - cos(x)]

If desolve cannot obtain a solution, it returns false.

Function: ic1 (solution, xval, yval)

Solves initial value problems for first order differential equations. Here solution is a general solution to the equation, as found by ode2, xval gives an initial value for the independent variable in the form x = x0, and yval gives the initial value for the dependent variable in the form y = y0.

See ode2 for an example of its usage.

Function: ic2 (solution, xval, yval, dval)

Solves initial value problems for second-order differential equations. Here solution is a general solution to the equation, as found by ode2, xval gives the initial value for the independent variable in the form x = x0, yval gives the initial value of the dependent variable in the form y = y0, and dval gives the initial value for the first derivative of the dependent variable with respect to independent variable, in the form diff(y,x) = dy0 (diff does not have to be quoted).

See ode2 for an example of its usage.

Function: ode2 (eqn, dvar, ivar)

The function ode2 solves an ordinary differential equation (ODE) of first or second order. It takes three arguments: an ODE given by eqn, the dependent variable dvar, and the independent variable ivar. When successful, it returns either an explicit or implicit solution for the dependent variable. %c is used to represent the integration constant in the case of first-order equations, and %k1 and %k2 the constants for second-order equations. The dependence of the dependent variable on the independent variable does not have to be written explicitly, as in the case of desolve, but the independent variable must always be given as the third argument.

If ode2 cannot obtain a solution for whatever reason, it returns false, after perhaps printing out an error message. The methods implemented for first order equations in the order in which they are tested are: linear, separable, exact - perhaps requiring an integrating factor, homogeneous, Bernoulli’s equation, and a generalized homogeneous method. The types of second-order equations which can be solved are: constant coefficients, exact, linear homogeneous with non-constant coefficients which can be transformed to constant coefficients, the Euler or equi-dimensional equation, equations solvable by the method of variation of parameters, and equations which are free of either the independent or of the dependent variable so that they can be reduced to two first order linear equations to be solved sequentially.

In the course of solving ODE’s, several variables are set purely for informational purposes: method denotes the method of solution used (e.g., linear), intfactor denotes any integrating factor used, odeindex denotes the index for Bernoulli’s method or for the generalized homogeneous method, and yp denotes the particular solution for the variation of parameters technique.

In order to solve initial value problems (IVP) functions ic1 and ic2 are available for first and second order equations, and to solve second-order boundary value problems (BVP) the function bc2 can be used.

Example:

(%i1) x^2*'diff(y,x) + 3*y*x = sin(x)/x;
                      2 dy           sin(x)
(%o1)                x  -- + 3 x y = ------
                        dx             x
(%i2) ode2(%,y,x);
                             %c - cos(x)
(%o2)                    y = -----------
                                  3
                                 x
(%i3) ic1(%o2,x=%pi,y=0);
                              cos(x) + 1
(%o3)                   y = - ----------
                                   3
                                  x
(%i4) 'diff(y,x,2) + y*'diff(y,x)^3 = 0;
                         2
                        d y      dy 3
(%o4)                   --- + y (--)  = 0
                          2      dx
                        dx
(%i5) ode2(%,y,x);
                      3
                     y  + 6 %k1 y
(%o5)                ------------ = x + %k2
                          6
(%i6) ratsimp(ic2(%o5,x=0,y=0,'diff(y,x)=2));
                             3
                          2 y  - 3 y
(%o6)                   - ---------- = x
                              6
(%i7) bc2(%o5,x=0,y=1,x=1,y=3);
                         3
                        y  - 10 y       3
(%o7)                   --------- = x - -
                            6           2


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22 Numerical


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22.1 Introduction to fast Fourier transform

The fft package comprises functions for the numerical (not symbolic) computation of the fast Fourier transform.

·

Category: Fourier transform · Category: Numerical methods · Category: Share packages · Category: Package fft


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22.2 Functions and Variables for fast Fourier transform

Function: polartorect (r, t)

Translates complex values of the form r %e^(%i t) to the form a + b %i, where r is the magnitude and t is the phase. r and t are 1-dimensional arrays of the same size. The array size need not be a power of 2.

The original values of the input arrays are replaced by the real and imaginary parts, a and b, on return. The outputs are calculated as

a = r cos(t)
b = r sin(t)

polartorect is the inverse function of recttopolar.

load(fft) loads this function. See also fft.

Function: recttopolar (a, b)

Translates complex values of the form a + b %i to the form r %e^(%i t), where a is the real part and b is the imaginary part. a and b are 1-dimensional arrays of the same size. The array size need not be a power of 2.

The original values of the input arrays are replaced by the magnitude and angle, r and t, on return. The outputs are calculated as

r = sqrt(a^2 + b^2)
t = atan2(b, a)

The computed angle is in the range -%pi to %pi.

recttopolar is the inverse function of polartorect.

load(fft) loads this function. See also fft.

Function: inverse_fft (y)

Computes the inverse complex fast Fourier transform. y is a list or array (named or unnamed) which contains the data to transform. The number of elements must be a power of 2. The elements must be literal numbers (integers, rationals, floats, or bigfloats) or symbolic constants, or expressions a + b*%i where a and b are literal numbers or symbolic constants.

inverse_fft returns a new object of the same type as y, which is not modified. Results are always computed as floats or expressions a + b*%i where a and b are floats. If bigfloat precision is needed the function bf_inverse_fft can be used instead as a drop-in replacement of inverse_fft that is slower, but supports bfloats.

The inverse discrete Fourier transform is defined as follows. Let x be the output of the inverse transform. Then for j from 0 through n - 1,

x[j] = sum(y[k] exp(-2 %i %pi j k / n), k, 0, n - 1)

As there are various sign and normalization conventions possible, this definition of the transform may differ from that used by other mathematical software.

load(fft) loads this function.

See also fft (forward transform), recttopolar, and polartorect.

Examples:

Real data.

(%i1) load ("fft") $
(%i2) fpprintprec : 4 $
(%i3) L : [1, 2, 3, 4, -1, -2, -3, -4] $
(%i4) L1 : inverse_fft (L);
(%o4) [0.0, 14.49 %i - .8284, 0.0, 2.485 %i + 4.828, 0.0, 
                       4.828 - 2.485 %i, 0.0, - 14.49 %i - .8284]
(%i5) L2 : fft (L1);
(%o5) [1.0, 2.0 - 2.168L-19 %i, 3.0 - 7.525L-20 %i, 
4.0 - 4.256L-19 %i, - 1.0, 2.168L-19 %i - 2.0, 
7.525L-20 %i - 3.0, 4.256L-19 %i - 4.0]
(%i6) lmax (abs (L2 - L));
(%o6)                       3.545L-16

Complex data.

(%i1) load ("fft") $
(%i2) fpprintprec : 4 $                 
(%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $
(%i4) L1 : inverse_fft (L);
(%o4) [4.0, 2.711L-19 %i + 4.0, 2.0 %i - 2.0, 
- 2.828 %i - 2.828, 0.0, 5.421L-20 %i + 4.0, - 2.0 %i - 2.0, 
2.828 %i + 2.828]
(%i5) L2 : fft (L1);
(%o5) [4.066E-20 %i + 1.0, 1.0 %i + 1.0, 1.0 - 1.0 %i, 
1.55L-19 %i - 1.0, - 4.066E-20 %i - 1.0, 1.0 - 1.0 %i, 
1.0 %i + 1.0, 1.0 - 7.368L-20 %i]
(%i6) lmax (abs (L2 - L));                    
(%o6)                       6.841L-17

Function: fft (x)

Computes the complex fast Fourier transform. x is a list or array (named or unnamed) which contains the data to transform. The number of elements must be a power of 2. The elements must be literal numbers (integers, rationals, floats, or bigfloats) or symbolic constants, or expressions a + b*%i where a and b are literal numbers or symbolic constants.

fft returns a new object of the same type as x, which is not modified. Results are always computed as floats or expressions a + b*%i where a and b are floats. If bigfloat precision is needed the function bf_fft can be used instead as a drop-in replacement of fft that is slower, but supports bfloats. In addition if it is known that the input consists of only real values (no imaginary parts), real_fft can be used which is potentially faster.

The discrete Fourier transform is defined as follows. Let y be the output of the transform. Then for k from 0 through n - 1,

y[k] = (1/n) sum(x[j] exp(+2 %i %pi j k / n), j, 0, n - 1)

As there are various sign and normalization conventions possible, this definition of the transform may differ from that used by other mathematical software.

When the data x are real, real coefficients a and b can be computed such that

x[j] = sum(a[k]*cos(2*%pi*j*k/n)+b[k]*sin(2*%pi*j*k/n), k, 0, n/2)

with

a[0] = realpart (y[0])
b[0] = 0

and, for k from 1 through n/2 - 1,

a[k] = realpart (y[k] + y[n - k])
b[k] = imagpart (y[n - k] - y[k])

and

a[n/2] = realpart (y[n/2])
b[n/2] = 0

load(fft) loads this function.

See also inverse_fft (inverse transform), recttopolar, and polartorect.. See real_fft for FFTs of a real-valued input, and bf_fft and bf_real_fft for operations on bigfloat values.

Examples:

Real data.

(%i1) load ("fft") $
(%i2) fpprintprec : 4 $
(%i3) L : [1, 2, 3, 4, -1, -2, -3, -4] $
(%i4) L1 : fft (L);
(%o4) [0.0, - 1.811 %i - .1036, 0.0, .6036 - .3107 %i, 0.0, 
                         .3107 %i + .6036, 0.0, 1.811 %i - .1036]
(%i5) L2 : inverse_fft (L1);
(%o5) [1.0, 2.168L-19 %i + 2.0, 7.525L-20 %i + 3.0, 
4.256L-19 %i + 4.0, - 1.0, - 2.168L-19 %i - 2.0, 
- 7.525L-20 %i - 3.0, - 4.256L-19 %i - 4.0]
(%i6) lmax (abs (L2 - L));
(%o6)                       3.545L-16

Complex data.

(%i1) load ("fft") $
(%i2) fpprintprec : 4 $
(%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $
(%i4) L1 : fft (L);
(%o4) [0.5, .3536 %i + .3536, - 0.25 %i - 0.25, 
0.5 - 6.776L-21 %i, 0.0, - .3536 %i - .3536, 0.25 %i - 0.25, 
0.5 - 3.388L-20 %i]
(%i5) L2 : inverse_fft (L1);
(%o5) [1.0 - 4.066E-20 %i, 1.0 %i + 1.0, 1.0 - 1.0 %i, 
- 1.008L-19 %i - 1.0, 4.066E-20 %i - 1.0, 1.0 - 1.0 %i, 
1.0 %i + 1.0, 1.947L-20 %i + 1.0]
(%i6) lmax (abs (L2 - L));
(%o6)                       6.83L-17

Computation of sine and cosine coefficients.

(%i1) load ("fft") $
(%i2) fpprintprec : 4 $
(%i3) L : [1, 2, 3, 4, 5, 6, 7, 8] $
(%i4) n : length (L) $
(%i5) x : make_array (any, n) $
(%i6) fillarray (x, L) $
(%i7) y : fft (x) $
(%i8) a : make_array (any, n/2 + 1) $
(%i9) b : make_array (any, n/2 + 1) $
(%i10) a[0] : realpart (y[0]) $
(%i11) b[0] : 0 $
(%i12) for k : 1 thru n/2 - 1 do
   (a[k] : realpart (y[k] + y[n - k]),
    b[k] : imagpart (y[n - k] - y[k]));
(%o12)                        done
(%i13) a[n/2] : y[n/2] $
(%i14) b[n/2] : 0 $
(%i15) listarray (a);
(%o15)          [4.5, - 1.0, - 1.0, - 1.0, - 0.5]
(%i16) listarray (b);
(%o16)           [0, - 2.414, - 1.0, - .4142, 0]
(%i17) f(j) := sum (a[k]*cos(2*%pi*j*k/n) + b[k]*sin(2*%pi*j*k/n), 
                    k, 0, n/2) $
(%i18) makelist (float (f (j)), j, 0, n - 1);
(%o18)      [1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0]

Function: real_fft (x)

Computes the fast Fourier transform of a real-valued sequence x. This is equivalent to performing fft(x), except that only the first N/2+1 results are returned, where N is the length of x. N must be power of two.

No check is made that x contains only real values.

The symmetry properites of the Fourier transform of real sequences to reduce he complexity. In particular the first and last output values of real_fft are purely real. For larger sequencyes, real_fft may be computed more quickly than fft.

Since the output length is short, the normal inverse_fft cannot be directly used. Use inverse_real_fft to compute the inverse.

Function: inverse_real_fft (y)

Computes the inverse Fourier transfrom of y, which must have a length of N/2+1 where N is a power of two. That is, the input x is expected to be the output of real_fft.

No check is made to ensure that the input has the correct format. (The first and last elements must be purely real.)

Function: bf_inverse_fft (y)

Computes the inverse complex fast Fourier transform. This is the bigfloat version of inverse_fft that converts the input to bigfloats and returns a bigfloat result.

Function: bf_fft (y)

Computes the forward complex fast Fourier transform. This is the bigfloat version of fft that converts the input to bigfloats and returns a bigfloat result.

Function: bf_real_fft (x)

Computes the forward fast Fourier transform of a real-valued input returning a bigfloat result. This is the bigfloat version of real_fft.

Function: bf_inverse_real_fft (y)

Computes the inverse fast Fourier transfrom with a real-valued bigfloat output. This is the bigfloat version of inverse_real_fft.


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22.3 Functions for numerical solution of equations

Function: horner
    horner (expr, x)
    horner (expr)

Returns a rearranged representation of expr as in Horner’s rule, using x as the main variable if it is specified. x may be omitted in which case the main variable of the canonical rational expression form of expr is used.

horner sometimes improves stability if expr is to be numerically evaluated. It is also useful if Maxima is used to generate programs to be run in Fortran. See also stringout.

(%i1) expr: 1e-155*x^2 - 5.5*x + 5.2e155;
                           2
(%o1)             1.e-155 x  - 5.5 x + 5.2e+155
(%i2) expr2: horner (%, x), keepfloat: true;
(%o2)         1.0 ((1.e-155 x - 5.5) x + 5.2e+155)
(%i3) ev (expr, x=1e155);
Maxima encountered a Lisp error:

 arithmetic error FLOATING-POINT-OVERFLOW signalled

Automatically continuing.
To enable the Lisp debugger set *debugger-hook* to nil.
(%i4) ev (expr2, x=1e155);
(%o4)                 7.00000000000001e+154

Function: find_root (expr, x, a, b, [abserr, relerr])
Function: find_root (f, a, b, [abserr, relerr])
Function: bf_find_root (expr, x, a, b, [abserr, relerr])
Function: bf_find_root (f, a, b, [abserr, relerr])
Option variable: find_root_error
Option variable: find_root_abs
Option variable: find_root_rel

Finds a root of the expression expr or the function f over the closed interval [a, b]. The expression expr may be an equation, in which case find_root seeks a root of lhs(expr) - rhs(expr).

Given that Maxima can evaluate expr or f over [a, b] and that expr or f is continuous, find_root is guaranteed to find the root, or one of the roots if there is more than one.

find_root initially applies binary search. If the function in question appears to be smooth enough, find_root applies linear interpolation instead.

bf_find_root is a bigfloat version of find_root. The function is computed using bigfloat arithmetic and a bigfloat result is returned. Otherwise, bf_find_root is identical to find_root, and the following description is equally applicable to bf_find_root.

The accuracy of find_root is governed by abserr and relerr, which are optional keyword arguments to find_root. These keyword arguments take the form key=val. The keyword arguments are

abserr

Desired absolute error of function value at root. Default is find_root_abs.

relerr

Desired relative error of root. Default is find_root_rel.

find_root stops when the function in question evaluates to something less than or equal to abserr, or if successive approximants x_0, x_1 differ by no more than relerr * max(abs(x_0), abs(x_1)). The default values of find_root_abs and find_root_rel are both zero.

find_root expects the function in question to have a different sign at the endpoints of the search interval. When the function evaluates to a number at both endpoints and these numbers have the same sign, the behavior of find_root is governed by find_root_error. When find_root_error is true, find_root prints an error message. Otherwise find_root returns the value of find_root_error. The default value of find_root_error is true.

If f evaluates to something other than a number at any step in the search algorithm, find_root returns a partially-evaluated find_root expression.

The order of a and b is ignored; the region in which a root is sought is [min(a, b), max(a, b)].

Examples:

(%i1) f(x) := sin(x) - x/2;
                                        x
(%o1)                  f(x) := sin(x) - -
                                        2
(%i2) find_root (sin(x) - x/2, x, 0.1, %pi);
(%o2)                   1.895494267033981
(%i3) find_root (sin(x) = x/2, x, 0.1, %pi);
(%o3)                   1.895494267033981
(%i4) find_root (f(x), x, 0.1, %pi);
(%o4)                   1.895494267033981
(%i5) find_root (f, 0.1, %pi);
(%o5)                   1.895494267033981
(%i6) find_root (exp(x) = y, x, 0, 100);
                            x
(%o6)           find_root(%e  = y, x, 0.0, 100.0)
(%i7) find_root (exp(x) = y, x, 0, 100), y = 10;
(%o7)                   2.302585092994046
(%i8) log (10.0);
(%o8)                   2.302585092994046
(%i9) fpprec:32;
(%o9)                           32
(%i10) bf_find_root (exp(x) = y, x, 0, 100), y = 10;
(%o10)                  2.3025850929940456840179914546844b0
(%i11) log(10b0);
(%o11)                  2.3025850929940456840179914546844b0

Function: newton (expr, x, x_0, eps)

Returns an approximate solution of expr = 0 by Newton’s method, considering expr to be a function of one variable, x. The search begins with x = x_0 and proceeds until abs(expr) < eps (with expr evaluated at the current value of x).

newton allows undefined variables to appear in expr, so long as the termination test abs(expr) < eps evaluates to true or false. Thus it is not necessary that expr evaluate to a number.

load(newton1) loads this function.

See also realroots, allroots, find_root and mnewton-pkg

Examples:

(%i1) load ("newton1");
(%o1)  /maxima/share/numeric/newton1.mac
(%i2) newton (cos (u), u, 1, 1/100);
(%o2)                   1.570675277161251
(%i3) ev (cos (u), u = %);
(%o3)                 1.2104963335033529e-4
(%i4) assume (a > 0);
(%o4)                        [a > 0]
(%i5) newton (x^2 - a^2, x, a/2, a^2/100);
(%o5)                  1.00030487804878 a
(%i6) ev (x^2 - a^2, x = %);
                                           2
(%o6)                6.098490481853958e-4 a


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22.4 Introduction to numerical solution of differential equations

The Ordinary Differential Equations (ODE) solved by the functions in this section should have the form,

       dy
       -- = F(x,y)
       dx

which is a first-order ODE. Higher order differential equations of order n must be written as a system of n first-order equations of that kind. For instance, a second-order ODE should be written as a system of two equations

       dx               dy
       -- = G(x,y,t)    -- = F(x,y,t) 
       dt               dt

The first argument in the functions will be a list with the expressions on the right-side of the ODE’s. The variables whose derivatives are represented by those expressions should be given in a second list. In the case above those variables are x and y. The independent variable, t in the examples above, might be given in a separated option. If the expressions given do not depend on that independent variable, the system is called autonomous.

·

Category: Differential equations · Category: Numerical methods · Category: Plotting


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22.5 Functions for numerical solution of differential equations

Function: plotdf
    plotdf (dydx, options…)
    plotdf (dvdu, [u,v], options…)
    plotdf ([dxdt,cdydt], options…)
    plotdf ([dudt,cdvdt], [u,cv], options…)

The function plotdf creates a two-dimensional plot of the direction field (also called slope field) for a first-order Ordinary Differential Equation (ODE) or a system of two autonomous first-order ODE’s.

Plotdf requires Xmaxima, even if its run from a Maxima session in a console, since the plot will be created by the Tk scripts in Xmaxima. If Xmaxima is not installed plotdf will not work.

dydx, dxdt and dydt are expressions that depend on x and y. dvdu, dudt and dvdt are expressions that depend on u and v. In addition to those two variables, the expressions can also depend on a set of parameters, with numerical values given with the parameters option (the option syntax is given below), or with a range of allowed values specified by a sliders option.

Several other options can be given within the command, or selected in the menu. Integral curves can be obtained by clicking on the plot, or with the option trajectory_at. The direction of the integration can be controlled with the direction option, which can have values of forward, backward or both. The number of integration steps is given by nsteps; at each integration step the time increment will be adjusted automatically to produce displacements much smaller than the size of the plot window. The numerical method used is 4th order Runge-Kutta with variable time steps.

Plot window menu:

The menu bar of the plot window has the following seven icons:

An X. Can be used to close the plot window.

A wrench and a screwdriver. Opens the configuration menu with several fields that show the ODE(s) in use and various other settings. If a pair of coordinates are entered in the field Trajectory at and the enter key is pressed, a new integral curve will be shown, in addition to the ones already shown.

Two arrows following a circle. Replots the direction field with the new settings defined in the configuration menu and replots only the last integral curve that was previously plotted.

Hard disk drive with an arrow. Used to save a copy of the plot, in Postscript format, in the file specified in a field of the box that appears when that icon is clicked.

Magnifying glass with a plus sign. Zooms in the plot.

Magnifying glass with a minus sign. Zooms out the plot. The plot can be displaced by holding down the right mouse button while the mouse is moved.

Icon of a plot. Opens another window with a plot of the two variables in terms of time, for the last integral curve that was plotted.

Plot options:

Options can also be given within the plotdf itself, each one being a list of two or more elements. The first element in each option is the name of the option, and the remainder is the value or values assigned to the option.

The options which are recognized by plotdf are the following:

Examples:

Function: ploteq (exp, ...options...)

Plots equipotential curves for exp, which should be an expression depending on two variables. The curves are obtained by integrating the differential equation that define the orthogonal trajectories to the solutions of the autonomous system obtained from the gradient of the expression given. The plot can also show the integral curves for that gradient system (option fieldlines).

This program also requires Xmaxima, even if its run from a Maxima session in a console, since the plot will be created by the Tk scripts in Xmaxima. By default, the plot region will be empty until the user clicks in a point (or gives its coordinate with in the set-up menu or via the trajectory_at option).

Most options accepted by plotdf can also be used for ploteq and the plot interface is the same that was described in plotdf.

Example:

(%i1) V: 900/((x+1)^2+y^2)^(1/2)-900/((x-1)^2+y^2)^(1/2)$
(%i2) ploteq(V,[x,-2,2],[y,-2,2],[fieldlines,"blue"])$

Clicking on a point will plot the equipotential curve that passes by that point (in red) and the orthogonal trajectory (in blue).

Function: rk
    rk (ODE, var, initial, domain)
    rk ([ODE1, …, ODEm], [v1, …, vm], [init1, …, initm], domain)

The first form solves numerically one first-order ordinary differential equation, and the second form solves a system of m of those equations, using the 4th order Runge-Kutta method. var represents the dependent variable. ODE must be an expression that depends only on the independent and dependent variables and defines the derivative of the dependent variable with respect to the independent variable.

The independent variable is specified with domain, which must be a list of four elements as, for instance:

[t, 0, 10, 0.1]

the first element of the list identifies the independent variable, the second and third elements are the initial and final values for that variable, and the last element sets the increments that should be used within that interval.

If m equations are going to be solved, there should be m dependent variables v1, v2, ..., vm. The initial values for those variables will be init1, init2, ..., initm. There will still be just one independent variable defined by domain, as in the previous case. ODE1, ..., ODEm are the expressions that define the derivatives of each dependent variable in terms of the independent variable. The only variables that may appear in those expressions are the independent variable and any of the dependent variables. It is important to give the derivatives ODE1, ..., ODEm in the list in exactly the same order used for the dependent variables; for instance, the third element in the list will be interpreted as the derivative of the third dependent variable.

The program will try to integrate the equations from the initial value of the independent variable until its last value, using constant increments. If at some step one of the dependent variables takes an absolute value too large, the integration will be interrupted at that point. The result will be a list with as many elements as the number of iterations made. Each element in the results list is itself another list with m+1 elements: the value of the independent variable, followed by the values of the dependent variables corresponding to that point.

To solve numerically the differential equation

          dx/dt = t - x^2

With initial value x(t=0) = 1, in the interval of t from 0 to 8 and with increments of 0.1 for t, use:

(%i1) results: rk(t-x^2,x,1,[t,0,8,0.1])$
(%i2) plot2d ([discrete, results])$

the results will be saved in the list results and the plot will show the solution obtained, with t on the horizontal axis and x on the vertical axis.

To solve numerically the system:

        dx/dt = 4-x^2-4*y^2     dy/dt = y^2-x^2+1

for t between 0 and 4, and with values of -1.25 and 0.75 for x and y at t=0:

(%i1) sol: rk([4-x^2-4*y^2,y^2-x^2+1],[x,y],[-1.25,0.75],[t,0,4,0.02])$
(%i2) plot2d ([discrete,makelist([p[1],p[3]],p,sol)], [xlabel,"t"],[ylabel,"y"])$

The plot will show the solution for variable y as a function of t.


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23 Matrices and Linear Algebra


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23.1 Introduction to Matrices and Linear Algebra


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23.1.1 Dot

The operator . represents noncommutative multiplication and scalar product. When the operands are 1-column or 1-row matrices a and b, the expression a.b is equivalent to sum (a[i]*b[i], i, 1, length(a)). If a and b are not complex, this is the scalar product, also called the inner product or dot product, of a and b. The scalar product is defined as conjugate(a).b when a and b are complex; innerproduct in the eigen package provides the complex scalar product.

When the operands are more general matrices, the product is the matrix product a and b. The number of rows of b must equal the number of columns of a, and the result has number of rows equal to the number of rows of a and number of columns equal to the number of columns of b.

To distinguish . as an arithmetic operator from the decimal point in a floating point number, it may be necessary to leave spaces on either side. For example, 5.e3 is 5000.0 but 5 . e3 is 5 times e3.

There are several flags which govern the simplification of expressions involving ., namely dot0nscsimp, dot0simp, dot1simp, dotassoc, dotconstrules, dotdistrib, dotexptsimp, dotident, and dotscrules.


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23.1.2 Matrices

Matrices are handled with speed and memory-efficiency in mind. This means that assigning a matrix to a variable will create a reference to, not a copy of the matrix. If the matrix is modified all references to the matrix point to the modified object (See copymatrix for a way of avoiding this):

(%i1) M1:matrix([0,0],[0,0]);
                            [ 0  0 ]
(%o1)                       [      ]
                            [ 0  0 ]
(%i2) M2:M1;
                            [ 0  0 ]
(%o2)                       [      ]
                            [ 0  0 ]
(%i3) M1[1][1]:2;
(%o3)                           2
(%i4) M2;
                            [ 2  0 ]
(%o4)                       [      ]
                            [ 0  0 ]

Converting a matrix to nested lists and vice versa works the following way:

(%i1) l:[[1,2],[3,4]];
(%o1)                   [[1, 2], [3, 4]]
(%i2) M1:apply('matrix,l);
                            [ 1  2 ]
(%o2)                       [      ]
                            [ 3  4 ]
(%i3) M2:transpose(M1);
                            [ 1  3 ]
(%o3)                       [      ]
                            [ 2  4 ]
(%i4) args(M2);
(%o4)                   [[1, 3], [2, 4]]

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23.1.3 Vectors

vect is a package of functions for vector analysis. load ("vect") loads this package, and demo ("vect") displays a demonstration.

The vector analysis package can combine and simplify symbolic expressions including dot products and cross products, together with the gradient, divergence, curl, and Laplacian operators. The distribution of these operators over sums or products is governed by several flags, as are various other expansions, including expansion into components in any specific orthogonal coordinate systems. There are also functions for deriving the scalar or vector potential of a field.

The vect package contains these functions: vectorsimp, scalefactors, express, potential, and vectorpotential.

By default the vect package does not declare the dot operator to be a commutative operator. To get a commutative dot operator ., the command declare(".", commutative) must be executed.

·

Category: Vectors · Category: Share packages · Category: Package vect


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23.1.4 eigen

The package eigen contains several functions devoted to the symbolic computation of eigenvalues and eigenvectors. Maxima loads the package automatically if one of the functions eigenvalues or eigenvectors is invoked. The package may be loaded explicitly as load ("eigen").

demo ("eigen") displays a demonstration of the capabilities of this package. batch ("eigen") executes the same demonstration, but without the user prompt between successive computations.

The functions in the eigen package are:
innerproduct, unitvector, columnvector, gramschmidt, eigenvalues,
eigenvectors, uniteigenvectors, and similaritytransform.

·

Category: Vectors · Category: Matrices · Category: Share packages · Category: Package eigen


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23.2 Functions and Variables for Matrices and Linear Algebra

Function: addcol (M, list_1, …, list_n)

Appends the column(s) given by the one or more lists (or matrices) onto the matrix M.

See also addrow and append.

Function: addrow (M, list_1, …, list_n)

Appends the row(s) given by the one or more lists (or matrices) onto the matrix M.

See also addcol and append.

Function: adjoint (M)

Returns the adjoint of the matrix M. The adjoint matrix is the transpose of the matrix of cofactors of M.

Function: augcoefmatrix ([eqn_1, …, eqn_m], [x_1, …, x_n])

Returns the augmented coefficient matrix for the variables x_1, …, x_n of the system of linear equations eqn_1, …, eqn_m. This is the coefficient matrix with a column adjoined for the constant terms in each equation (i.e., those terms not dependent upon x_1, …, x_n).

(%i1) m: [2*x - (a - 1)*y = 5*b, c + b*y + a*x = 0]$
(%i2) augcoefmatrix (m, [x, y]);
                       [ 2  1 - a  - 5 b ]
(%o2)                  [                 ]
                       [ a    b      c   ]

Function: cauchy_matrix
    cauchy_matrix ([x_1, x_2, …, x_m], [y_1, y_2, …, y_n])
    cauchy_matrix ([x_1, x_2, …, x_n])

Returns a n by m Cauchy matrix with the elements a[i,j] = 1/(x_i+y_i). The second argument of cauchy_matrix is optional. For this case the elements of the Cauchy matrix are a[i,j] = 1/(x_i+x_j).

Remark: In the literature the Cauchy matrix can be found defined in two forms. A second definition is a[i,j] = 1/(x_i-y_i).

Examples:

(%i1) cauchy_matrix([x1,x2],[y1,y2]);
                      [    1        1    ]
                      [ -------  ------- ]
                      [ y1 + x1  y2 + x1 ]
(%o1)                 [                  ]
                      [    1        1    ]
                      [ -------  ------- ]
                      [ y1 + x2  y2 + x2 ]

(%i2) cauchy_matrix([x1,x2]);
                      [   1         1    ]
                      [  ----    ------- ]
                      [  2 x1    x2 + x1 ]
(%o2)                 [                  ]
                      [    1       1     ]
                      [ -------   ----   ]
                      [ x2 + x1   2 x2   ]

Function: charpoly (M, x)

Returns the characteristic polynomial for the matrix M with respect to variable x. That is, determinant (M - diagmatrix (length (M), x)).

(%i1) a: matrix ([3, 1], [2, 4]);
                            [ 3  1 ]
(%o1)                       [      ]
                            [ 2  4 ]
(%i2) expand (charpoly (a, lambda));
                           2
(%o2)                lambda  - 7 lambda + 10
(%i3) (programmode: true, solve (%));
(%o3)               [lambda = 5, lambda = 2]
(%i4) matrix ([x1], [x2]);
                             [ x1 ]
(%o4)                        [    ]
                             [ x2 ]
(%i5) ev (a . % - lambda*%, %th(2)[1]);
                          [ x2 - 2 x1 ]
(%o5)                     [           ]
                          [ 2 x1 - x2 ]
(%i6) %[1, 1] = 0;
(%o6)                     x2 - 2 x1 = 0
(%i7) x2^2 + x1^2 = 1;
                            2     2
(%o7)                     x2  + x1  = 1
(%i8) solve ([%th(2), %], [x1, x2]);
                  1               2
(%o8) [[x1 = - -------, x2 = - -------], 
               sqrt(5)         sqrt(5)

                                             1             2
                                    [x1 = -------, x2 = -------]]
                                          sqrt(5)       sqrt(5)

Function: coefmatrix ([eqn_1, …, eqn_m], [x_1, …, x_n])

Returns the coefficient matrix for the variables x_1, …, x_n of the system of linear equations eqn_1, …, eqn_m.

(%i1) coefmatrix([2*x-(a-1)*y+5*b = 0, b*y+a*x = 3], [x,y]);
                                 [ 2  1 - a ]
(%o1)                            [          ]
                                 [ a    b   ]

Function: col (M, i)

Returns the i’th column of the matrix M. The return value is a matrix.

Function: columnvector (L)
Function: covect (L)

Returns a matrix of one column and length (L) rows, containing the elements of the list L.

covect is a synonym for columnvector.

load ("eigen") loads this function.

This is useful if you want to use parts of the outputs of the functions in this package in matrix calculations.

Example:

(%i1) load ("eigen")$
Warning - you are redefining the Macsyma function eigenvalues
Warning - you are redefining the Macsyma function eigenvectors
(%i2) columnvector ([aa, bb, cc, dd]);
                             [ aa ]
                             [    ]
                             [ bb ]
(%o2)                        [    ]
                             [ cc ]
                             [    ]
                             [ dd ]

Function: copymatrix (M)

Returns a copy of the matrix M. This is the only way to make a copy aside from copying M element by element.

Note that an assignment of one matrix to another, as in m2: m1, does not copy m1. An assignment m2 [i,j]: x or setelmx(x, i, j, m2) also modifies m1 [i,j]. Creating a copy with copymatrix and then using assignment creates a separate, modified copy.

Function: determinant (M)

Computes the determinant of M by a method similar to Gaussian elimination.

The form of the result depends upon the setting of the switch ratmx.

There is a special routine for computing sparse determinants which is called when the switches ratmx and sparse are both true.

Option variable: detout

Default value: false

When detout is true, the determinant of a matrix whose inverse is computed is factored out of the inverse.

For this switch to have an effect doallmxops and doscmxops should be false (see their descriptions). Alternatively this switch can be given to ev which causes the other two to be set correctly.

Example:

(%i1) m: matrix ([a, b], [c, d]);
                            [ a  b ]
(%o1)                       [      ]
                            [ c  d ]
(%i2) detout: true$
(%i3) doallmxops: false$
(%i4) doscmxops: false$
(%i5) invert (m);
                          [  d   - b ]
                          [          ]
                          [ - c   a  ]
(%o5)                     ------------
                           a d - b c

Function: diagmatrix (n, x)

Returns a diagonal matrix of size n by n with the diagonal elements all equal to x. diagmatrix (n, 1) returns an identity matrix (same as ident (n)).

n must evaluate to an integer, otherwise diagmatrix complains with an error message.

x can be any kind of expression, including another matrix. If x is a matrix, it is not copied; all diagonal elements refer to the same instance, x.

Option variable: doallmxops

Default value: true

When doallmxops is true, all operations relating to matrices are carried out. When it is false then the setting of the individual dot switches govern which operations are performed.

Option variable: domxexpt

Default value: true

When domxexpt is true, a matrix exponential, exp (M) where M is a matrix, is interpreted as a matrix with element [i,j] equal to exp (m[i,j]). Otherwise exp (M) evaluates to exp (ev(M)).

domxexpt affects all expressions of the form base^power where base is an expression assumed scalar or constant, and power is a list or matrix.

Example:

(%i1) m: matrix ([1, %i], [a+b, %pi]);
                         [   1    %i  ]
(%o1)                    [            ]
                         [ b + a  %pi ]
(%i2) domxexpt: false$
(%i3) (1 - c)^m;
                             [   1    %i  ]
                             [            ]
                             [ b + a  %pi ]
(%o3)                 (1 - c)
(%i4) domxexpt: true$
(%i5) (1 - c)^m;
                  [                      %i  ]
                  [    1 - c      (1 - c)    ]
(%o5)             [                          ]
                  [        b + a         %pi ]
                  [ (1 - c)       (1 - c)    ]

Option variable: domxmxops

Default value: true

When domxmxops is true, all matrix-matrix or matrix-list operations are carried out (but not scalar-matrix operations); if this switch is false such operations are not carried out.

Option variable: domxnctimes

Default value: false

When domxnctimes is true, non-commutative products of matrices are carried out.

Option variable: dontfactor

Default value: []

dontfactor may be set to a list of variables with respect to which factoring is not to occur. (The list is initially empty.) Factoring also will not take place with respect to any variables which are less important, according the variable ordering assumed for canonical rational expression (CRE) form, than those on the dontfactor list.

Option variable: doscmxops

Default value: false

When doscmxops is true, scalar-matrix operations are carried out.

Option variable: doscmxplus

Default value: false

When doscmxplus is true, scalar-matrix operations yield a matrix result. This switch is not subsumed under doallmxops.

Option variable: dot0nscsimp

Default value: true

When dot0nscsimp is true, a non-commutative product of zero and a nonscalar term is simplified to a commutative product.

Option variable: dot0simp

Default value: true

When dot0simp is true, a non-commutative product of zero and a scalar term is simplified to a commutative product.

Option variable: dot1simp

Default value: true

When dot1simp is true, a non-commutative product of one and another term is simplified to a commutative product.

Option variable: dotassoc

Default value: true

When dotassoc is true, an expression (A.B).C simplifies to A.(B.C).

Option variable: dotconstrules

Default value: true

When dotconstrules is true, a non-commutative product of a constant and another term is simplified to a commutative product. Turning on this flag effectively turns on dot0simp, dot0nscsimp, and dot1simp as well.

Option variable: dotdistrib

Default value: false

When dotdistrib is true, an expression A.(B + C) simplifies to A.B + A.C.

Option variable: dotexptsimp

Default value: true

When dotexptsimp is true, an expression A.A simplifies to A^^2.

Option variable: dotident

Default value: 1

dotident is the value returned by X^^0.

Option variable: dotscrules

Default value: false

When dotscrules is true, an expression A.SC or SC.A simplifies to SC*A and A.(SC*B) simplifies to SC*(A.B).

Function: echelon (M)

Returns the echelon form of the matrix M, as produced by Gaussian elimination. The echelon form is computed from M by elementary row operations such that the first non-zero element in each row in the resulting matrix is one and the column elements under the first one in each row are all zero.

triangularize also carries out Gaussian elimination, but it does not normalize the leading non-zero element in each row.

lu_factor and cholesky are other functions which yield triangularized matrices.

(%i1) M: matrix ([3, 7, aa, bb], [-1, 8, 5, 2], [9, 2, 11, 4]);
                       [  3   7  aa  bb ]
                       [                ]
(%o1)                  [ - 1  8  5   2  ]
                       [                ]
                       [  9   2  11  4  ]
(%i2) echelon (M);
                  [ 1  - 8  - 5      - 2     ]
                  [                          ]
                  [         28       11      ]
                  [ 0   1   --       --      ]
(%o2)             [         37       37      ]
                  [                          ]
                  [              37 bb - 119 ]
                  [ 0   0    1   ----------- ]
                  [              37 aa - 313 ]

Function: eigenvalues (M)
Function: eivals (M)

Returns a list of two lists containing the eigenvalues of the matrix M. The first sublist of the return value is the list of eigenvalues of the matrix, and the second sublist is the list of the multiplicities of the eigenvalues in the corresponding order.

eivals is a synonym for eigenvalues.

eigenvalues calls the function solve to find the roots of the characteristic polynomial of the matrix. Sometimes solve may not be able to find the roots of the polynomial; in that case some other functions in this package (except innerproduct, unitvector, columnvector and gramschmidt) will not work. Sometimes solve may find only a subset of the roots of the polynomial. This may happen when the factoring of the polynomial contains polynomials of degree 5 or more. In such cases a warning message is displayed and the only the roots found and their corresponding multiplicities are returned.

In some cases the eigenvalues found by solve may be complicated expressions. (This may happen when solve returns a not-so-obviously real expression for an eigenvalue which is known to be real.) It may be possible to simplify the eigenvalues using some other functions.

The package eigen.mac is loaded automatically when eigenvalues or eigenvectors is referenced. If eigen.mac is not already loaded, load ("eigen") loads it. After loading, all functions and variables in the package are available.

Function: eigenvectors (M)
Function: eivects (M)

Computes eigenvectors of the matrix M. The return value is a list of two elements. The first is a list of the eigenvalues of M and a list of the multiplicities of the eigenvalues. The second is a list of lists of eigenvectors. There is one list of eigenvectors for each eigenvalue. There may be one or more eigenvectors in each list.

eivects is a synonym for eigenvectors.

The package eigen.mac is loaded automatically when eigenvalues or eigenvectors is referenced. If eigen.mac is not already loaded, load ("eigen") loads it. After loading, all functions and variables in the package are available.

Note that eigenvectors internally calls eigenvalues to obtain eigenvalues. So, when eigenvalues returns a subset of all the eigenvalues, the eigenvectors returns the corresponding subset of the all the eigenvectors, with the same warning displayed as eigenvalues.

The flags that affect this function are:

nondiagonalizable is set to true or false depending on whether the matrix is nondiagonalizable or diagonalizable after eigenvectors returns.

hermitianmatrix when true, causes the degenerate eigenvectors of the Hermitian matrix to be orthogonalized using the Gram-Schmidt algorithm.

knowneigvals when true causes the eigen package to assume the eigenvalues of the matrix are known to the user and stored under the global name listeigvals. listeigvals should be set to a list similar to the output eigenvalues.

The function algsys is used here to solve for the eigenvectors. Sometimes if the eigenvalues are messy, algsys may not be able to find a solution. In some cases, it may be possible to simplify the eigenvalues by first finding them using eigenvalues command and then using other functions to reduce them to something simpler. Following simplification, eigenvectors can be called again with the knowneigvals flag set to true.

See also eigenvalues.

Examples:

A matrix which has just one eigenvector per eigenvalue.

(%i1) M1 : matrix ([11, -1], [1, 7]);
                           [ 11  - 1 ]
(%o1)                      [         ]
                           [ 1    7  ]
(%i2) [vals, vecs] : eigenvectors (M1);
(%o2) [[[9 - sqrt(3), sqrt(3) + 9], [1, 1]], 
                        [[[1, sqrt(3) + 2]], [[1, 2 - sqrt(3)]]]]
(%i3) for i thru length (vals[1]) do disp (val[i] = vals[1][i],
  mult[i] = vals[2][i], vec[i] = vecs[i]);
                       val  = 9 - sqrt(3)
                          1

                            mult  = 1
                                1

                    vec  = [[1, sqrt(3) + 2]]
                       1

                       val  = sqrt(3) + 9
                          2

                            mult  = 1
                                2

                    vec  = [[1, 2 - sqrt(3)]]
                       2

(%o3)                         done

A matrix which has two eigenvectors for one eigenvalue (namely 2).

(%i1) M1 : matrix ([0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 2, 0], [0, 0, 0, 2]);
                         [ 0  1  0  0 ]
                         [            ]
                         [ 0  0  0  0 ]
(%o1)                    [            ]
                         [ 0  0  2  0 ]
                         [            ]
                         [ 0  0  0  2 ]
(%i2) [vals, vecs] : eigenvectors (M1);
(%o2) [[[0, 2], [2, 2]], [[[1, 0, 0, 0]], 
                                   [[0, 0, 1, 0], [0, 0, 0, 1]]]]
(%i3) for i thru length (vals[1]) do disp (val[i] = vals[1][i],
  mult[i] = vals[2][i], vec[i] = vecs[i]);
                            val  = 0
                               1

                            mult  = 2
                                1

                      vec  = [[1, 0, 0, 0]]
                         1

                            val  = 2
                               2

                            mult  = 2
                                2

               vec  = [[0, 0, 1, 0], [0, 0, 0, 1]]
                  2

(%o3)                         done

Function: ematrix (m, n, x, i, j)

Returns an m by n matrix, all elements of which are zero except for the [i, j] element which is x.

Function: entermatrix (m, n)

Returns an m by n matrix, reading the elements interactively.

If n is equal to m, Maxima prompts for the type of the matrix (diagonal, symmetric, antisymmetric, or general) and for each element. Each response is terminated by a semicolon ; or dollar sign $.

If n is not equal to m, Maxima prompts for each element.

The elements may be any expressions, which are evaluated. entermatrix evaluates its arguments.

(%i1) n: 3$
(%i2) m: entermatrix (n, n)$

Is the matrix  1. Diagonal  2. Symmetric  3. Antisymmetric 
4. General
Answer 1, 2, 3 or 4 : 
1$
Row 1 Column 1: 
(a+b)^n$
Row 2 Column 2: 
(a+b)^(n+1)$
Row 3 Column 3: 
(a+b)^(n+2)$

Matrix entered.
(%i3) m;
                [        3                     ]
                [ (b + a)      0         0     ]
                [                              ]
(%o3)           [                  4           ]
                [    0      (b + a)      0     ]
                [                              ]
                [                            5 ]
                [    0         0      (b + a)  ]

Function: genmatrix
    genmatrix (a, i_2, j_2, i_1, j_1)
    genmatrix (a, i_2, j_2, i_1)
    genmatrix (a, i_2, j_2)

Returns a matrix generated from a, taking element a[i_1, j_1] as the upper-left element and a[i_2, j_2] as the lower-right element of the matrix. Here a is a declared array (created by array but not by make_array) or a hashed array, or a memoizing function, or a lambda expression of two arguments. (A memoizing function is created like other functions with := or define, but arguments are enclosed in square brackets instead of parentheses.)

If j_1 is omitted, it is assumed equal to i_1. If both j_1 and i_1 are omitted, both are assumed equal to 1.

If a selected element i,j of the array is undefined, the matrix will contain a symbolic element a[i,j].

Examples:

(%i1) h [i, j] := 1 / (i + j - 1);
                                    1
(%o1)                  h     := ---------
                        i, j    i + j - 1
(%i2) genmatrix (h, 3, 3);
                           [    1  1 ]
                           [ 1  -  - ]
                           [    2  3 ]
                           [         ]
                           [ 1  1  1 ]
(%o2)                      [ -  -  - ]
                           [ 2  3  4 ]
                           [         ]
                           [ 1  1  1 ]
                           [ -  -  - ]
                           [ 3  4  5 ]
(%i3) array (a, fixnum, 2, 2);
(%o3)                           a
(%i4) a [1, 1] : %e;
(%o4)                          %e
(%i5) a [2, 2] : %pi;
(%o5)                          %pi
(%i6) genmatrix (a, 2, 2);
                           [ %e   0  ]
(%o6)                      [         ]
                           [ 0   %pi ]
(%i7) genmatrix (lambda ([i, j], j - i), 3, 3);
                         [  0    1   2 ]
                         [             ]
(%o7)                    [ - 1   0   1 ]
                         [             ]
                         [ - 2  - 1  0 ]
(%i8) genmatrix (B, 2, 2);
                        [ B      B     ]
                        [  1, 1   1, 2 ]
(%o8)                   [              ]
                        [ B      B     ]
                        [  2, 1   2, 2 ]

Function: gramschmidt
    gramschmidt (x)
    gramschmidt (x, F)

Carries out the Gram-Schmidt orthogonalization algorithm on x, which is either a matrix or a list of lists. x is not modified by gramschmidt. The inner product employed by gramschmidt is F, if present, otherwise the inner product is the function innerproduct.

If x is a matrix, the algorithm is applied to the rows of x. If x is a list of lists, the algorithm is applied to the sublists, which must have equal numbers of elements. In either case, the return value is a list of lists, the sublists of which are orthogonal and span the same space as x. If the dimension of the span of x is less than the number of rows or sublists, some sublists of the return value are zero.

factor is called at each stage of the algorithm to simplify intermediate results. As a consequence, the return value may contain factored integers.

load("eigen") loads this function.

Example:

Gram-Schmidt algorithm using default inner product function.

(%i1) load ("eigen")$
(%i2) x: matrix ([1, 2, 3], [9, 18, 30], [12, 48, 60]);
                         [ 1   2   3  ]
                         [            ]
(%o2)                    [ 9   18  30 ]
                         [            ]
                         [ 12  48  60 ]
(%i3) y: gramschmidt (x);
                       2      2            4     3
                      3      3   3 5      2  3  2  3
(%o3)  [[1, 2, 3], [- ---, - --, ---], [- ----, ----, 0]]
                      2 7    7   2 7       5     5
(%i4) map (innerproduct, [y[1], y[2], y[3]], [y[2], y[3], y[1]]);
(%o4)                       [0, 0, 0]

Gram-Schmidt algorithm using a specified inner product function.

(%i1) load ("eigen")$
(%i2) ip (f, g) := integrate (f * g, u, a, b);
(%o2)          ip(f, g) := integrate(f g, u, a, b)
(%i3) y : gramschmidt ([1, sin(u), cos(u)], ip), a= -%pi/2, b=%pi/2;
                               %pi cos(u) - 2
(%o3)              [1, sin(u), --------------]
                                    %pi
(%i4) map (ip, [y[1], y[2], y[3]], [y[2], y[3], y[1]]), a= -%pi/2, b=%pi/2;
(%o4)                       [0, 0, 0]

Function: ident (n)

Returns an n by n identity matrix.

Function: innerproduct (x, y)
Function: inprod (x, y)

Returns the inner product (also called the scalar product or dot product) of x and y, which are lists of equal length, or both 1-column or 1-row matrices of equal length. The return value is conjugate (x) . y, where . is the noncommutative multiplication operator.

load ("eigen") loads this function.

inprod is a synonym for innerproduct.

Function: invert_by_adjoint (M)

Returns the inverse of the matrix M. The inverse is computed by the adjoint method.

invert_by_adjoint honors the ratmx and detout flags, the same as invert.

Function: invert (M)

Returns the inverse of the matrix M. The inverse is computed via the LU decomposition.

When ratmx is true, elements of M are converted to canonical rational expressions (CRE), and the elements of the return value are also CRE.

When ratmx is false, elements of M are not converted to a common representation. In particular, float and bigfloat elements are not converted to rationals.

When detout is true, the determinant is factored out of the inverse. The global flags doallmxops and doscmxops must be false to prevent the determinant from being absorbed into the inverse. xthru can multiply the determinant into the inverse.

invert does not apply any simplifications to the elements of the inverse apart from the default arithmetic simplifications. ratsimp and expand can apply additional simplifications. In particular, when M has polynomial elements, expand(invert(M)) might be preferable.

invert(M) is equivalent to M^^-1.

Function: list_matrix_entries (M)

Returns a list containing the elements of the matrix M.

Example:

(%i1) list_matrix_entries(matrix([a,b],[c,d]));
(%o1)                     [a, b, c, d]

Option variable: lmxchar

Default value: [

lmxchar is the character displayed as the left delimiter of a matrix. See also rmxchar.

Example:

(%i1) lmxchar: "|"$
(%i2) matrix ([a, b, c], [d, e, f], [g, h, i]);
                           | a  b  c ]
                           |         ]
(%o2)                      | d  e  f ]
                           |         ]
                           | g  h  i ]

Function: matrix (row_1, …, row_n)

Returns a rectangular matrix which has the rows row_1, …, row_n. Each row is a list of expressions. All rows must be the same length.

The operations + (addition), - (subtraction), * (multiplication), and / (division), are carried out element by element when the operands are two matrices, a scalar and a matrix, or a matrix and a scalar. The operation ^ (exponentiation, equivalently **) is carried out element by element if the operands are a scalar and a matrix or a matrix and a scalar, but not if the operands are two matrices. All operations are normally carried out in full, including . (noncommutative multiplication).

Matrix multiplication is represented by the noncommutative multiplication operator .. The corresponding noncommutative exponentiation operator is ^^. For a matrix A, A.A = A^^2 and A^^-1 is the inverse of A, if it exists. A^^-1 is equivalent to invert(A).

There are switches for controlling simplification of expressions involving dot and matrix-list operations. These are doallmxops, domxexpt, domxmxops, doscmxops, and doscmxplus.

There are additional options which are related to matrices. These are: lmxchar, rmxchar, ratmx, listarith, detout, scalarmatrix and sparse.

There are a number of functions which take matrices as arguments or yield matrices as return values. See eigenvalues, eigenvectors, determinant, charpoly, genmatrix, addcol, addrow, copymatrix, transpose, echelon, and rank.

Examples:

(%i1) x: matrix ([17, 3], [-8, 11]);
                           [ 17   3  ]
(%o1)                      [         ]
                           [ - 8  11 ]
(%i2) y: matrix ([%pi, %e], [a, b]);
                           [ %pi  %e ]
(%o2)                      [         ]
                           [  a   b  ]
(%i3) x + y;
                      [ %pi + 17  %e + 3 ]
(%o3)                 [                  ]
                      [  a - 8    b + 11 ]
(%i4) x - y;
                      [ 17 - %pi  3 - %e ]
(%o4)                 [                  ]
                      [ - a - 8   11 - b ]
(%i5) x * y;
                        [ 17 %pi  3 %e ]
(%o5)                   [              ]
                        [ - 8 a   11 b ]
(%i6) x / y;
                        [ 17       - 1 ]
                        [ ---  3 %e    ]
                        [ %pi          ]
(%o6)                   [              ]
                        [   8    11    ]
                        [ - -    --    ]
                        [   a    b     ]
(%i7) x ^ 3;
                         [ 4913    27  ]
(%o7)                    [             ]
                         [ - 512  1331 ]
(%i8) exp(y); 
                         [   %pi    %e ]
                         [ %e     %e   ]
(%o8)                    [             ]
                         [    a     b  ]
                         [  %e    %e   ]
(%i9) x ^ y;
                                [ %pi  %e ]
                                [         ]
                                [  a   b  ]
                     [ 17   3  ]
(%o9)                [         ]
                     [ - 8  11 ]
(%i10) x . y;
                  [ 3 a + 17 %pi  3 b + 17 %e ]
(%o10)            [                           ]
                  [ 11 a - 8 %pi  11 b - 8 %e ]
(%i11) y . x;
                [ 17 %pi - 8 %e  3 %pi + 11 %e ]
(%o11)          [                              ]
                [  17 a - 8 b     11 b + 3 a   ]
(%i12) x ^^ 3;
                        [  3833   1719 ]
(%o12)                  [              ]
                        [ - 4584  395  ]
(%i13) %e ^^ y;
                         [   %pi    %e ]
                         [ %e     %e   ]
(%o13)                   [             ]
                         [    a     b  ]
                         [  %e    %e   ]
(%i14) x ^^ -1;
                         [ 11      3  ]
                         [ ---  - --- ]
                         [ 211    211 ]
(%o14)                   [            ]
                         [  8    17   ]
                         [ ---   ---  ]
                         [ 211   211  ]
(%i15) x . (x ^^ -1);
                            [ 1  0 ]
(%o15)                      [      ]
                            [ 0  1 ]

Function: matrixexp
    matrixexp (M)
    matrixexp (M, n)
    matrixexp (M, V)

Calculates the matrix exponential $$e^{M\cdot V}$$ . Instead of the vector V a number n can be specified as the second argument. If this argument is omitted matrixexp replaces it by 1.

The matrix exponential of a matrix M can be expressed as a power series: $$e^M=\sum_{k=0}^\infty{\left(\frac{M^k}{k!}\right)}$$

Function: matrixmap (f, M)

Returns a matrix with element i,j equal to f(M[i,j]).

See also map, fullmap, fullmapl, and apply.

Function: matrixp (expr)

Returns true if expr is a matrix, otherwise false.

Option variable: matrix_element_add

Default value: +

matrix_element_add is the operation invoked in place of addition in a matrix multiplication. matrix_element_add can be assigned any n-ary operator (that is, a function which handles any number of arguments). The assigned value may be the name of an operator enclosed in quote marks, the name of a function, or a lambda expression.

See also matrix_element_mult and matrix_element_transpose.

Example:

(%i1) matrix_element_add: "*"$
(%i2) matrix_element_mult: "^"$
(%i3) aa: matrix ([a, b, c], [d, e, f]);
                           [ a  b  c ]
(%o3)                      [         ]
                           [ d  e  f ]
(%i4) bb: matrix ([u, v, w], [x, y, z]);
                           [ u  v  w ]
(%o4)                      [         ]
                           [ x  y  z ]
(%i5) aa . transpose (bb);
                     [  u  v  w   x  y  z ]
                     [ a  b  c   a  b  c  ]
(%o5)                [                    ]
                     [  u  v  w   x  y  z ]
                     [ d  e  f   d  e  f  ]

Option variable: matrix_element_mult

Default value: *

matrix_element_mult is the operation invoked in place of multiplication in a matrix multiplication. matrix_element_mult can be assigned any binary operator. The assigned value may be the name of an operator enclosed in quote marks, the name of a function, or a lambda expression.

The dot operator . is a useful choice in some contexts.

See also matrix_element_add and matrix_element_transpose.

Example:

(%i1) matrix_element_add: lambda ([[x]], sqrt (apply ("+", x)))$
(%i2) matrix_element_mult: lambda ([x, y], (x - y)^2)$
(%i3) [a, b, c] . [x, y, z];
                          2          2          2
(%o3)         sqrt((c - z)  + (b - y)  + (a - x) )
(%i4) aa: matrix ([a, b, c], [d, e, f]);
                           [ a  b  c ]
(%o4)                      [         ]
                           [ d  e  f ]
(%i5) bb: matrix ([u, v, w], [x, y, z]);
                           [ u  v  w ]
(%o5)                      [         ]
                           [ x  y  z ]
(%i6) aa . transpose (bb);
               [             2          2          2  ]
               [ sqrt((c - w)  + (b - v)  + (a - u) ) ]
(%o6)  Col 1 = [                                      ]
               [             2          2          2  ]
               [ sqrt((f - w)  + (e - v)  + (d - u) ) ]

                         [             2          2          2  ]
                         [ sqrt((c - z)  + (b - y)  + (a - x) ) ]
                 Col 2 = [                                      ]
                         [             2          2          2  ]
                         [ sqrt((f - z)  + (e - y)  + (d - x) ) ]

Option variable: matrix_element_transpose

Default value: false

matrix_element_transpose is the operation applied to each element of a matrix when it is transposed. matrix_element_mult can be assigned any unary operator. The assigned value may be the name of an operator enclosed in quote marks, the name of a function, or a lambda expression.

When matrix_element_transpose equals transpose, the transpose function is applied to every element. When matrix_element_transpose equals nonscalars, the transpose function is applied to every nonscalar element. If some element is an atom, the nonscalars option applies transpose only if the atom is declared nonscalar, while the transpose option always applies transpose.

The default value, false, means no operation is applied.

See also matrix_element_add and matrix_element_mult.

Examples:

(%i1) declare (a, nonscalar)$
(%i2) transpose ([a, b]);
                        [ transpose(a) ]
(%o2)                   [              ]
                        [      b       ]
(%i3) matrix_element_transpose: nonscalars$
(%i4) transpose ([a, b]);
                        [ transpose(a) ]
(%o4)                   [              ]
                        [      b       ]
(%i5) matrix_element_transpose: transpose$
(%i6) transpose ([a, b]);
                        [ transpose(a) ]
(%o6)                   [              ]
                        [ transpose(b) ]
(%i7) matrix_element_transpose: lambda ([x], realpart(x)
      - %i*imagpart(x))$
(%i8) m: matrix ([1 + 5*%i, 3 - 2*%i], [7*%i, 11]);
                     [ 5 %i + 1  3 - 2 %i ]
(%o8)                [                    ]
                     [   7 %i       11    ]
(%i9) transpose (m);
                      [ 1 - 5 %i  - 7 %i ]
(%o9)                 [                  ]
                      [ 2 %i + 3    11   ]

Function: mattrace (M)

Returns the trace (that is, the sum of the elements on the main diagonal) of the square matrix M.

mattrace is called by ncharpoly, an alternative to Maxima’s charpoly.

load ("nchrpl") loads this function.

Function: minor (M, i, j)

Returns the i, j minor of the matrix M. That is, M with row i and column j removed.

Function: ncharpoly (M, x)

Returns the characteristic polynomial of the matrix M with respect to x. This is an alternative to Maxima’s charpoly.

ncharpoly works by computing traces of powers of the given matrix, which are known to be equal to sums of powers of the roots of the characteristic polynomial. From these quantities the symmetric functions of the roots can be calculated, which are nothing more than the coefficients of the characteristic polynomial. charpoly works by forming the determinant of x * ident [n] - a. Thus ncharpoly wins, for example, in the case of large dense matrices filled with integers, since it avoids polynomial arithmetic altogether.

load ("nchrpl") loads this file.

Function: newdet (M)

Computes the determinant of the matrix M by the Johnson-Gentleman tree minor algorithm. newdet returns the result in CRE form.

Function: permanent (M)

Computes the permanent of the matrix M by the Johnson-Gentleman tree minor algorithm. A permanent is like a determinant but with no sign changes. permanent returns the result in CRE form.

See also newdet.

Function: rank (M)

Computes the rank of the matrix M. That is, the order of the largest non-singular subdeterminant of M.

rank may return the wrong answer if it cannot determine that a matrix element that is equivalent to zero is indeed so.

Option variable: ratmx

Default value: false

When ratmx is false, determinant and matrix addition, subtraction, and multiplication are performed in the representation of the matrix elements and cause the result of matrix inversion to be left in general representation.

When ratmx is true, the 4 operations mentioned above are performed in CRE form and the result of matrix inverse is in CRE form. Note that this may cause the elements to be expanded (depending on the setting of ratfac) which might not always be desired.

Function: row (M, i)

Returns the i’th row of the matrix M. The return value is a matrix.

Option variable: rmxchar

Default value: ]

rmxchar is the character drawn on the right-hand side of a matrix.

See also lmxchar.

Option variable: scalarmatrixp

Default value: true

When scalarmatrixp is true, then whenever a 1 x 1 matrix is produced as a result of computing the dot product of matrices it is simplified to a scalar, namely the sole element of the matrix.

When scalarmatrixp is all, then all 1 x 1 matrices are simplified to scalars.

When scalarmatrixp is false, 1 x 1 matrices are not simplified to scalars.

Function: scalefactors (coordinatetransform)

Here the argument coordinatetransform evaluates to the form [[expression1, expression2, ...], indeterminate1, indeterminat2, ...], where the variables indeterminate1, indeterminate2, etc. are the curvilinear coordinate variables and where a set of rectangular Cartesian components is given in terms of the curvilinear coordinates by [expression1, expression2, ...]. coordinates is set to the vector [indeterminate1, indeterminate2,...], and dimension is set to the length of this vector. SF[1], SF[2], …, SF[DIMENSION] are set to the coordinate scale factors, and sfprod is set to the product of these scale factors. Initially, coordinates is [X, Y, Z], dimension is 3, and SF[1]=SF[2]=SF[3]=SFPROD=1, corresponding to 3-dimensional rectangular Cartesian coordinates. To expand an expression into physical components in the current coordinate system, there is a function with usage of the form

Function: setelmx (x, i, j, M)

Assigns x to the (i, j)’th element of the matrix M, and returns the altered matrix.

M [i, j]: x has the same effect, but returns x instead of M.

Function: similaritytransform (M)
Function: simtran (M)

similaritytransform computes a similarity transform of the matrix M. It returns a list which is the output of the uniteigenvectors command. In addition if the flag nondiagonalizable is false two global matrices leftmatrix and rightmatrix are computed. These matrices have the property that leftmatrix . M . rightmatrix is a diagonal matrix with the eigenvalues of M on the diagonal. If nondiagonalizable is true the left and right matrices are not computed.

If the flag hermitianmatrix is true then leftmatrix is the complex conjugate of the transpose of rightmatrix. Otherwise leftmatrix is the inverse of rightmatrix.

rightmatrix is the matrix the columns of which are the unit eigenvectors of M. The other flags (see eigenvalues and eigenvectors) have the same effects since similaritytransform calls the other functions in the package in order to be able to form rightmatrix.

load ("eigen") loads this function.

simtran is a synonym for similaritytransform.

Option variable: sparse

Default value: false

When sparse is true, and if ratmx is true, then determinant will use special routines for computing sparse determinants.

Function: submatrix
    submatrix (i_1, …, i_m, M, j_1, …, j_n)
    submatrix (i_1, …, i_m, M)
    submatrix (M, j_1, …, j_n)

Returns a new matrix composed of the matrix M with rows i_1, …, i_m deleted, and columns j_1, …, j_n deleted.

Function: transpose (M)

Returns the transpose of M.

If M is a matrix, the return value is another matrix N such that N[i,j] = M[j,i].

If M is a list, the return value is a matrix N of length (m) rows and 1 column, such that N[i,1] = M[i].

Otherwise M is a symbol, and the return value is a noun expression 'transpose (M).

Function: triangularize (M)

Returns the upper triangular form of the matrix M, as produced by Gaussian elimination. The return value is the same as echelon, except that the leading nonzero coefficient in each row is not normalized to 1.

lu_factor and cholesky are other functions which yield triangularized matrices.

(%i1) M: matrix ([3, 7, aa, bb], [-1, 8, 5, 2], [9, 2, 11, 4]);
                       [  3   7  aa  bb ]
                       [                ]
(%o1)                  [ - 1  8  5   2  ]
                       [                ]
                       [  9   2  11  4  ]
(%i2) triangularize (M);
             [ - 1   8         5            2      ]
             [                                     ]
(%o2)        [  0   - 74     - 56         - 22     ]
             [                                     ]
             [  0    0    626 - 74 aa  238 - 74 bb ]

Function: uniteigenvectors (M)
Function: ueivects (M)

Computes unit eigenvectors of the matrix M. The return value is a list of lists, the first sublist of which is the output of the eigenvalues command, and the other sublists of which are the unit eigenvectors of the matrix corresponding to those eigenvalues respectively.

The flags mentioned in the description of the eigenvectors command have the same effects in this one as well.

When knowneigvects is true, the eigen package assumes that the eigenvectors of the matrix are known to the user and are stored under the global name listeigvects. listeigvects should be set to a list similar to the output of the eigenvectors command.

If knowneigvects is set to true and the list of eigenvectors is given the setting of the flag nondiagonalizable may not be correct. If that is the case please set it to the correct value. The author assumes that the user knows what he is doing and will not try to diagonalize a matrix the eigenvectors of which do not span the vector space of the appropriate dimension.

load ("eigen") loads this function.

ueivects is a synonym for uniteigenvectors.

Function: unitvector (x)
Function: uvect (x)

Returns x/norm(x); this is a unit vector in the same direction as x.

load ("eigen") loads this function.

uvect is a synonym for unitvector.

Function: vectorpotential (givencurl)

Returns the vector potential of a given curl vector, in the current coordinate system. potentialzeroloc has a similar role as for potential, but the order of the left-hand sides of the equations must be a cyclic permutation of the coordinate variables.

Function: vectorsimp (expr)

Applies simplifications and expansions according to the following global flags:

expandall, expanddot, expanddotplus, expandcross, expandcrossplus, expandcrosscross, expandgrad, expandgradplus, expandgradprod, expanddiv, expanddivplus, expanddivprod, expandcurl, expandcurlplus, expandcurlcurl, expandlaplacian, expandlaplacianplus, and expandlaplacianprod.

All these flags have default value false. The plus suffix refers to employing additivity or distributivity. The prod suffix refers to the expansion for an operand that is any kind of product.

expandcrosscross

Simplifies p ~ (q ~ r) to (p . r)*q - (p . q)*r.

expandcurlcurl

Simplifies curl curl p to grad div p + div grad p.

expandlaplaciantodivgrad

Simplifies laplacian p to div grad p.

expandcross

Enables expandcrossplus and expandcrosscross.

expandplus

Enables expanddotplus, expandcrossplus, expandgradplus, expanddivplus, expandcurlplus, and expandlaplacianplus.

expandprod

Enables expandgradprod, expanddivprod, and expandlaplacianprod.

These flags have all been declared evflag.

Option variable: vect_cross

Default value: false

When vect_cross is true, it allows DIFF(X~Y,T) to work where ~ is defined in SHARE;VECT (where VECT_CROSS is set to true, anyway.)

Function: zeromatrix (m, n)

Returns an m by n matrix, all elements of which are zero.


Next: , Previous: , Up: Top   [Contents][Index]

24 Affine


Next: , Previous: , Up: Affine   [Contents][Index]

24.1 Introduction to Affine

affine is a package to work with groups of polynomials.

·

Category: Polynomials · Category: Groebner bases · Category: Share packages · Category: Package affine


Previous: , Up: Affine   [Contents][Index]

24.2 Functions and Variables for Affine

Function: fast_linsolve ([expr_1, …, expr_m], [x_1, …, x_n])

Solves the simultaneous linear equations expr_1, …, expr_m for the variables x_1, …, x_n. Each expr_i may be an equation or a general expression; if given as a general expression, it is treated as an equation of the form expr_i = 0.

The return value is a list of equations of the form [x_1 = a_1, …, x_n = a_n] where a_1, …, a_n are all free of x_1, …, x_n.

fast_linsolve is faster than linsolve for system of equations which are sparse.

load("affine") loads this function.

Function: grobner_basis ([expr_1, …, expr_m])

Returns a Groebner basis for the equations expr_1, …, expr_m. The function polysimp can then be used to simplify other functions relative to the equations.

grobner_basis ([3*x^2+1, y*x])$

polysimp (y^2*x + x^3*9 + 2) ==> -3*x + 2

polysimp(f) yields 0 if and only if f is in the ideal generated by expr_1, …, expr_m, that is, if and only if f is a polynomial combination of the elements of expr_1, …, expr_m.

load("affine") loads this function.

Function: set_up_dot_simplifications
    set_up_dot_simplifications (eqns, check_through_degree)
    set_up_dot_simplifications (eqns)

The eqns are polynomial equations in non commutative variables. The value of current_variables is the list of variables used for computing degrees. The equations must be homogeneous, in order for the procedure to terminate.

If you have checked overlapping simplifications in dot_simplifications above the degree of f, then the following is true: dotsimp (f) yields 0 if and only if f is in the ideal generated by the equations, i.e., if and only if f is a polynomial combination of the elements of the equations.

The degree is that returned by nc_degree. This in turn is influenced by the weights of individual variables.

load("affine") loads this function.

Function: declare_weights (x_1, w_1, …, x_n, w_n)

Assigns weights w_1, …, w_n to x_1, …, x_n, respectively. These are the weights used in computing nc_degree.

load("affine") loads this function.

Function: nc_degree (p)

Returns the degree of a noncommutative polynomial p. See declare_weights.

load("affine") loads this function.

Function: dotsimp (f)

Returns 0 if and only if f is in the ideal generated by the equations, i.e., if and only if f is a polynomial combination of the elements of the equations.

load("affine") loads this function.

Function: fast_central_elements ([x_1, …, x_n], n)

If set_up_dot_simplifications has been previously done, finds the central polynomials in the variables x_1, …, x_n in the given degree, n.

For example:

set_up_dot_simplifications ([y.x + x.y], 3);
fast_central_elements ([x, y], 2);
[y.y, x.x];

load("affine") loads this function.

Function: check_overlaps (n, add_to_simps)

Checks the overlaps thru degree n, making sure that you have sufficient simplification rules in each degree, for dotsimp to work correctly. This process can be speeded up if you know before hand what the dimension of the space of monomials is. If it is of finite global dimension, then hilbert should be used. If you don’t know the monomial dimensions, do not specify a rank_function. An optional third argument reset, false says don’t bother to query about resetting things.

load("affine") loads this function.

Function: mono ([x_1, …, x_n], n)

Returns the list of independent monomials relative to the current dot simplifications of degree n in the variables x_1, …, x_n.

load("affine") loads this function.

Function: monomial_dimensions (n)

Compute the Hilbert series through degree n for the current algebra.

load("affine") loads this function.

Function: extract_linear_equations ([p_1, …, p_n], [m_1, …, m_n])

Makes a list of the coefficients of the noncommutative polynomials p_1, …, p_n of the noncommutative monomials m_1, …, m_n. The coefficients should be scalars. Use list_nc_monomials to build the list of monomials.

load("affine") loads this function.

Function: list_nc_monomials
    list_nc_monomials ([p_1, …, p_n])
    list_nc_monomials (p)

Returns a list of the non commutative monomials occurring in a polynomial p or a list of polynomials p_1, …, p_n.

load("affine") loads this function.

Option variable: all_dotsimp_denoms

Default value: false

When all_dotsimp_denoms is a list, the denominators encountered by dotsimp are appended to the list. all_dotsimp_denoms may be initialized to an empty list [] before calling dotsimp.

By default, denominators are not collected by dotsimp.


Next: , Previous: , Up: Top   [Contents][Index]

25 itensor


Next: , Previous: , Up: itensor   [Contents][Index]

25.1 Introduction to itensor

Maxima implements symbolic tensor manipulation of two distinct types: component tensor manipulation (ctensor package) and indicial tensor manipulation (itensor package).

Nota bene: Please see the note on ’new tensor notation’ below.

Component tensor manipulation means that geometrical tensor objects are represented as arrays or matrices. Tensor operations such as contraction or covariant differentiation are carried out by actually summing over repeated (dummy) indices with do statements. That is, one explicitly performs operations on the appropriate tensor components stored in an array or matrix.

Indicial tensor manipulation is implemented by representing tensors as functions of their covariant, contravariant and derivative indices. Tensor operations such as contraction or covariant differentiation are performed by manipulating the indices themselves rather than the components to which they correspond.

These two approaches to the treatment of differential, algebraic and analytic processes in the context of Riemannian geometry have various advantages and disadvantages which reveal themselves only through the particular nature and difficulty of the user’s problem. However, one should keep in mind the following characteristics of the two implementations:

The representation of tensors and tensor operations explicitly in terms of their components makes ctensor easy to use. Specification of the metric and the computation of the induced tensors and invariants is straightforward. Although all of Maxima’s powerful simplification capacity is at hand, a complex metric with intricate functional and coordinate dependencies can easily lead to expressions whose size is excessive and whose structure is hidden. In addition, many calculations involve intermediate expressions which swell causing programs to terminate before completion. Through experience, a user can avoid many of these difficulties.

Because of the special way in which tensors and tensor operations are represented in terms of symbolic operations on their indices, expressions which in the component representation would be unmanageable can sometimes be greatly simplified by using the special routines for symmetrical objects in itensor. In this way the structure of a large expression may be more transparent. On the other hand, because of the special indicial representation in itensor, in some cases the user may find difficulty with the specification of the metric, function definition, and the evaluation of differentiated "indexed" objects.

The itensor package can carry out differentiation with respect to an indexed variable, which allows one to use the package when dealing with Lagrangian and Hamiltonian formalisms. As it is possible to differentiate a field Lagrangian with respect to an (indexed) field variable, one can use Maxima to derive the corresponding Euler-Lagrange equations in indicial form. These equations can be translated into component tensor (ctensor) programs using the ic_convert function, allowing us to solve the field equations in a particular coordinate representation, or to recast the equations of motion in Hamiltonian form. See einhil.dem and bradic.dem for two comprehensive examples. The first, einhil.dem, uses the Einstein-Hilbert action to derive the Einstein field tensor in the homogeneous and isotropic case (Friedmann equations) and the spherically symmetric, static case (Schwarzschild solution.) The second, bradic.dem, demonstrates how to compute the Friedmann equations from the action of Brans-Dicke gravity theory, and also derives the Hamiltonian associated with the theory’s scalar field.

·

Category: Tensors · Category: Share packages · Category: Package itensor

25.1.1 New tensor notation

Earlier versions of the itensor package in Maxima used a notation that sometimes led to incorrect index ordering. Consider the following, for instance:

(%i2) imetric(g);
(%o2)                                done
(%i3) ishow(g([],[j,k])*g([],[i,l])*a([i,j],[]))$
                                 i l  j k
(%t3)                           g    g    a
                                           i j
(%i4) ishow(contract(%))$
                                      k l
(%t4)                                a

This result is incorrect unless a happens to be a symmetric tensor. The reason why this happens is that although itensor correctly maintains the order within the set of covariant and contravariant indices, once an index is raised or lowered, its position relative to the other set of indices is lost.

To avoid this problem, a new notation has been developed that remains fully compatible with the existing notation and can be used interchangeably. In this notation, contravariant indices are inserted in the appropriate positions in the covariant index list, but with a minus sign prepended. Functions like contract_Itensor and ishow are now aware of this new index notation and can process tensors appropriately.

In this new notation, the previous example yields a correct result:

(%i5) ishow(g([-j,-k],[])*g([-i,-l],[])*a([i,j],[]))$
                                 i l       j k
(%t5)                           g    a    g
                                      i j
(%i6) ishow(contract(%))$
                                      l k
(%t6)                                a

Presently, the only code that makes use of this notation is the lc2kdt function. Through this notation, it achieves consistent results as it applies the metric tensor to resolve Levi-Civita symbols without resorting to numeric indices.

Since this code is brand new, it probably contains bugs. While it has been tested to make sure that it doesn’t break anything using the "old" tensor notation, there is a considerable chance that "new" tensors will fail to interoperate with certain functions or features. These bugs will be fixed as they are encountered... until then, caveat emptor!

25.1.2 Indicial tensor manipulation

The indicial tensor manipulation package may be loaded by load("itensor"). Demos are also available: try demo("tensor").

In itensor a tensor is represented as an "indexed object" . This is a function of 3 groups of indices which represent the covariant, contravariant and derivative indices. The covariant indices are specified by a list as the first argument to the indexed object, and the contravariant indices by a list as the second argument. If the indexed object lacks either of these groups of indices then the empty list [] is given as the corresponding argument. Thus, g([a,b],[c]) represents an indexed object called g which has two covariant indices (a,b), one contravariant index (c) and no derivative indices.

The derivative indices, if they are present, are appended as additional arguments to the symbolic function representing the tensor. They can be explicitly specified by the user or be created in the process of differentiation with respect to some coordinate variable. Since ordinary differentiation is commutative, the derivative indices are sorted alphanumerically, unless iframe_flag is set to true, indicating that a frame metric is being used. This canonical ordering makes it possible for Maxima to recognize that, for example, t([a],[b],i,j) is the same as t([a],[b],j,i). Differentiation of an indexed object with respect to some coordinate whose index does not appear as an argument to the indexed object would normally yield zero. This is because Maxima would not know that the tensor represented by the indexed object might depend implicitly on the corresponding coordinate. By modifying the existing Maxima function diff in itensor, Maxima now assumes that all indexed objects depend on any variable of differentiation unless otherwise stated. This makes it possible for the summation convention to be extended to derivative indices. It should be noted that itensor does not possess the capabilities of raising derivative indices, and so they are always treated as covariant.

The following functions are available in the tensor package for manipulating indexed objects. At present, with respect to the simplification routines, it is assumed that indexed objects do not by default possess symmetry properties. This can be overridden by setting the variable allsym[false] to true, which will result in treating all indexed objects completely symmetric in their lists of covariant indices and symmetric in their lists of contravariant indices.

The itensor package generally treats tensors as opaque objects. Tensorial equations are manipulated based on algebraic rules, specifically symmetry and contraction rules. In addition, the itensor package understands covariant differentiation, curvature, and torsion. Calculations can be performed relative to a metric of moving frame, depending on the setting of the iframe_flag variable.

A sample session below demonstrates how to load the itensor package, specify the name of the metric, and perform some simple calculations.

(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) imetric(g);
(%o2)                                done
(%i3) components(g([i,j],[]),p([i,j],[])*e([],[]))$
(%i4) ishow(g([k,l],[]))$
(%t4)                               e p
                                       k l
(%i5) ishow(diff(v([i],[]),t))$
(%t5)                                  0
(%i6) depends(v,t);
(%o6)                               [v(t)]
(%i7) ishow(diff(v([i],[]),t))$
                                    d
(%t7)                               -- (v )
                                    dt   i
(%i8) ishow(idiff(v([i],[]),j))$
(%t8)                                v
                                      i,j
(%i9) ishow(extdiff(v([i],[]),j))$
(%t9)                             v    - v
                                   j,i    i,j
                                  -----------
                                       2
(%i10) ishow(liediff(v,w([i],[])))$
                               %3          %3
(%t10)                        v   w     + v   w
                                   i,%3    ,i  %3
(%i11) ishow(covdiff(v([i],[]),j))$
                                              %4
(%t11)                        v    - v   ichr2
                               i,j    %4      i j
(%i12) ishow(ev(%,ichr2))$
                %4 %5
(%t12) v    - (g      v   (e p       + e   p     - e p       - e    p
        i,j            %4     j %5,i    ,i  j %5      i j,%5    ,%5  i j

                                         + e p       + e   p    ))/2
                                              i %5,j    ,j  i %5
(%i13) iframe_flag:true;
(%o13)                               true
(%i14) ishow(covdiff(v([i],[]),j))$
                                             %6
(%t14)                        v    - v   icc2
                               i,j    %6     i j
(%i15) ishow(ev(%,icc2))$
                                             %6
(%t15)                        v    - v   ifc2
                               i,j    %6     i j
(%i16) ishow(radcan(ev(%,ifc2,ifc1)))$
             %6 %7                    %6 %7
(%t16) - (ifg      v   ifb       + ifg      v   ifb       - 2 v
                    %6    j %7 i             %6    i j %7      i,j

                                             %6 %7
                                        - ifg      v   ifb      )/2
                                                    %6    %7 i j
(%i17) ishow(canform(s([i,j],[])-s([j,i])))$
(%t17)                            s    - s
                                   i j    j i
(%i18) decsym(s,2,0,[sym(all)],[]);
(%o18)                               done
(%i19) ishow(canform(s([i,j],[])-s([j,i])))$
(%t19)                                 0
(%i20) ishow(canform(a([i,j],[])+a([j,i])))$
(%t20)                            a    + a
                                   j i    i j
(%i21) decsym(a,2,0,[anti(all)],[]);
(%o21)                               done
(%i22) ishow(canform(a([i,j],[])+a([j,i])))$
(%t22)                                 0

Previous: , Up: itensor   [Contents][Index]

25.2 Functions and Variables for itensor

25.2.1 Managing indexed objects

Function: dispcon
    dispcon (tensor_1, tensor_2, …)
    dispcon (all)

Displays the contraction properties of its arguments as were given to defcon. dispcon (all) displays all the contraction properties which were defined.

Function: entertensor (name)

is a function which, by prompting, allows one to create an indexed object called name with any number of tensorial and derivative indices. Either a single index or a list of indices (which may be null) is acceptable input (see the example under covdiff).

Function: changename (old, new, expr)

will change the name of all indexed objects called old to new in expr. old may be either a symbol or a list of the form [name, m, n] in which case only those indexed objects called name with m covariant and n contravariant indices will be renamed to new.

Function: listoftens

Lists all tensors in a tensorial expression, complete with their indices. E.g.,


(%i6) ishow(a([i,j],[k])*b([u],[],v)+c([x,y],[])*d([],[])*e)$
                                         k
(%t6)                        d e c    + a    b
                                  x y    i j  u,v
(%i7) ishow(listoftens(%))$
                               k
(%t7)                        [a   , b   , c   , d]
                               i j   u,v   x y

Function: ishow (expr)

displays expr with the indexed objects in it shown having their covariant indices as subscripts and contravariant indices as superscripts. The derivative indices are displayed as subscripts, separated from the covariant indices by a comma (see the examples throughout this document).

Function: indices (expr)

Returns a list of two elements. The first is a list of the free indices in expr (those that occur only once). The second is the list of the dummy indices in expr (those that occur exactly twice) as the following example demonstrates.


(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) ishow(a([i,j],[k,l],m,n)*b([k,o],[j,m,p],q,r))$
                                k l      j m p
(%t2)                          a        b
                                i j,m n  k o,q r
(%i3) indices(%);
(%o3)                 [[l, p, i, n, o, q, r], [k, j, m]]

A tensor product containing the same index more than twice is syntactically illegal. indices attempts to deal with these expressions in a reasonable manner; however, when it is called to operate upon such an illegal expression, its behavior should be considered undefined.

Function: rename
    rename (expr)
    rename (expr, count)

Returns an expression equivalent to expr but with the dummy indices in each term chosen from the set [%1, %2,...], if the optional second argument is omitted. Otherwise, the dummy indices are indexed beginning at the value of count. Each dummy index in a product will be different. For a sum, rename will operate upon each term in the sum resetting the counter with each term. In this way rename can serve as a tensorial simplifier. In addition, the indices will be sorted alphanumerically (if allsym is true) with respect to covariant or contravariant indices depending upon the value of flipflag. If flipflag is false then the indices will be renamed according to the order of the contravariant indices. If flipflag is true the renaming will occur according to the order of the covariant indices. It often happens that the combined effect of the two renamings will reduce an expression more than either one by itself.


(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) allsym:true;
(%o2)                                true
(%i3) g([],[%4,%5])*g([],[%6,%7])*ichr2([%1,%4],[%3])*
ichr2([%2,%3],[u])*ichr2([%5,%6],[%1])*ichr2([%7,r],[%2])-
g([],[%4,%5])*g([],[%6,%7])*ichr2([%1,%2],[u])*
ichr2([%3,%5],[%1])*ichr2([%4,%6],[%3])*ichr2([%7,r],[%2]),noeval$
(%i4) expr:ishow(%)$
       %4 %5  %6 %7      %3         u          %1         %2
(%t4) g      g      ichr2      ichr2      ichr2      ichr2
                         %1 %4      %2 %3      %5 %6      %7 r

        %4 %5  %6 %7      u          %1         %3         %2
     - g      g      ichr2      ichr2      ichr2      ichr2
                          %1 %2      %3 %5      %4 %6      %7 r
(%i5) flipflag:true;
(%o5)                                true
(%i6) ishow(rename(expr))$
       %2 %5  %6 %7      %4         u          %1         %3
(%t6) g      g      ichr2      ichr2      ichr2      ichr2
                         %1 %2      %3 %4      %5 %6      %7 r

        %4 %5  %6 %7      u          %1         %3         %2
     - g      g      ichr2      ichr2      ichr2      ichr2
                          %1 %2      %3 %4      %5 %6      %7 r
(%i7) flipflag:false;
(%o7)                                false
(%i8) rename(%th(2));
(%o8)                                  0
(%i9) ishow(rename(expr))$
       %1 %2  %3 %4      %5         %6         %7        u
(%t9) g      g      ichr2      ichr2      ichr2     ichr2
                         %1 %6      %2 %3      %4 r      %5 %7

        %1 %2  %3 %4      %6         %5         %7        u
     - g      g      ichr2      ichr2      ichr2     ichr2
                          %1 %3      %2 %6      %4 r      %5 %7

Function: show (expr)

Displays expr with the indexed objects in it shown having covariant indices as subscripts, contravariant indices as superscripts. The derivative indices are displayed as subscripts, separated from the covariant indices by a comma.

Option variable: flipflag

Default value: false

If false then the indices will be renamed according to the order of the contravariant indices, otherwise according to the order of the covariant indices.

If flipflag is false then rename forms a list of the contravariant indices as they are encountered from left to right (if true then of the covariant indices). The first dummy index in the list is renamed to %1, the next to %2, etc. Then sorting occurs after the rename-ing (see the example under rename).

Function: defcon
    defcon (tensor_1)
    defcon (tensor_1, tensor_2, tensor_3)

gives tensor_1 the property that the contraction of a product of tensor_1 and tensor_2 results in tensor_3 with the appropriate indices. If only one argument, tensor_1, is given, then the contraction of the product of tensor_1 with any indexed object having the appropriate indices (say my_tensor) will yield an indexed object with that name, i.e. my_tensor, and with a new set of indices reflecting the contractions performed. For example, if imetric:g, then defcon(g) will implement the raising and lowering of indices through contraction with the metric tensor. More than one defcon can be given for the same indexed object; the latest one given which applies in a particular contraction will be used. contractions is a list of those indexed objects which have been given contraction properties with defcon.

Function: remcon
    remcon (tensor_1, ..., tensor_n)
    remcon (all)

Removes all the contraction properties from the (tensor_1, ..., tensor_n). remcon(all) removes all contraction properties from all indexed objects.

Function: contract (expr)

Carries out the tensorial contractions in expr which may be any combination of sums and products. This function uses the information given to the defcon function. For best results, expr should be fully expanded. ratexpand is the fastest way to expand products and powers of sums if there are no variables in the denominators of the terms. The gcd switch should be false if GCD cancellations are unnecessary.

Function: indexed_tensor (tensor)

Must be executed before assigning components to a tensor for which a built in value already exists as with ichr1, ichr2, icurvature. See the example under icurvature.

Function: components (tensor, expr)

permits one to assign an indicial value to an expression expr giving the values of the components of tensor. These are automatically substituted for the tensor whenever it occurs with all of its indices. The tensor must be of the form t([...],[...]) where either list may be empty. expr can be any indexed expression involving other objects with the same free indices as tensor. When used to assign values to the metric tensor wherein the components contain dummy indices one must be careful to define these indices to avoid the generation of multiple dummy indices. Removal of this assignment is given to the function remcomps.

It is important to keep in mind that components cares only about the valence of a tensor, not about any particular index ordering. Thus assigning components to, say, x([i,-j],[]), x([-j,i],[]), or x([i],[j]) all produce the same result, namely components being assigned to a tensor named x with valence (1,1).

Components can be assigned to an indexed expression in four ways, two of which involve the use of the components command:

1) As an indexed expression. For instance:

(%i2) components(g([],[i,j]),e([],[i])*p([],[j]))$
(%i3) ishow(g([],[i,j]))$
                                      i  j
(%t3)                                e  p

2) As a matrix:


(%i5) lg:-ident(4)$lg[1,1]:1$lg;
                            [ 1   0    0    0  ]
                            [                  ]
                            [ 0  - 1   0    0  ]
(%o5)                       [                  ]
                            [ 0   0   - 1   0  ]
                            [                  ]
                            [ 0   0    0   - 1 ]
(%i6) components(g([i,j],[]),lg);
(%o6)                                done
(%i7) ishow(g([i,j],[]))$
(%t7)                                g
                                      i j
(%i8) g([1,1],[]);
(%o8)                                  1
(%i9) g([4,4],[]);
(%o9)                                 - 1

3) As a function. You can use a Maxima function to specify the components of a tensor based on its indices. For instance, the following code assigns kdelta to h if h has the same number of covariant and contravariant indices and no derivative indices, and g otherwise:


(%i4) h(l1,l2,[l3]):=if length(l1)=length(l2) and length(l3)=0
  then kdelta(l1,l2) else apply(g,append([l1,l2], l3))$
(%i5) ishow(h([i],[j]))$
                                          j
(%t5)                               kdelta
                                          i
(%i6) ishow(h([i,j],[k],l))$
                                     k
(%t6)                               g
                                     i j,l

4) Using Maxima’s pattern matching capabilities, specifically the defrule and applyb1 commands:


(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) matchdeclare(l1,listp);
(%o2)                                done
(%i3) defrule(r1,m(l1,[]),(i1:idummy(),
      g([l1[1],l1[2]],[])*q([i1],[])*e([],[i1])))$

(%i4) defrule(r2,m([],l1),(i1:idummy(),
      w([],[l1[1],l1[2]])*e([i1],[])*q([],[i1])))$

(%i5) ishow(m([i,n],[])*m([],[i,m]))$
                                    i m
(%t5)                              m    m
                                         i n
(%i6) ishow(rename(applyb1(%,r1,r2)))$
                           %1  %2  %3 m
(%t6)                     e   q   w     q   e   g
                                         %1  %2  %3 n

Function: remcomps (tensor)

Unbinds all values from tensor which were assigned with the components function.

Function: showcomps (tensor)

Shows component assignments of a tensor, as made using the components command. This function can be particularly useful when a matrix is assigned to an indicial tensor using components, as demonstrated by the following example:


(%i1) load(ctensor);
(%o1)       /share/tensor/ctensor.mac
(%i2) load("itensor");
(%o2)      /share/tensor/itensor.lisp
(%i3) lg:matrix([sqrt(r/(r-2*m)),0,0,0],[0,r,0,0],
                [0,0,sin(theta)*r,0],[0,0,0,sqrt((r-2*m)/r)]);
               [         r                                     ]
               [ sqrt(-------)  0       0              0       ]
               [      r - 2 m                                  ]
               [                                               ]
               [       0        r       0              0       ]
(%o3)          [                                               ]
               [       0        0  r sin(theta)        0       ]
               [                                               ]
               [                                      r - 2 m  ]
               [       0        0       0        sqrt(-------) ]
               [                                         r     ]
(%i4) components(g([i,j],[]),lg);
(%o4)                                done
(%i5) showcomps(g([i,j],[]));
                  [         r                                     ]
                  [ sqrt(-------)  0       0              0       ]
                  [      r - 2 m                                  ]
                  [                                               ]
                  [       0        r       0              0       ]
(%t5)      g    = [                                               ]
            i j   [       0        0  r sin(theta)        0       ]
                  [                                               ]
                  [                                      r - 2 m  ]
                  [       0        0       0        sqrt(-------) ]
                  [                                         r     ]
(%o5)                                false

The showcomps command can also display components of a tensor of rank higher than 2.

Function: idummy ()

Increments icounter and returns as its value an index of the form %n where n is a positive integer. This guarantees that dummy indices which are needed in forming expressions will not conflict with indices already in use (see the example under indices).

Option variable: idummyx

Default value: %

Is the prefix for dummy indices (see the example under indices).

Option variable: icounter

Default value: 1

Determines the numerical suffix to be used in generating the next dummy index in the tensor package. The prefix is determined by the option idummy (default: %).

Function: kdelta (L1, L2)

is the generalized Kronecker delta function defined in the itensor package with L1 the list of covariant indices and L2 the list of contravariant indices. kdelta([i],[j]) returns the ordinary Kronecker delta. The command ev(expr,kdelta) causes the evaluation of an expression containing kdelta([],[]) to the dimension of the manifold.

In what amounts to an abuse of this notation, itensor also allows kdelta to have 2 covariant and no contravariant, or 2 contravariant and no covariant indices, in effect providing a co(ntra)variant "unit matrix" capability. This is strictly considered a programming aid and not meant to imply that kdelta([i,j],[]) is a valid tensorial object.

Function: kdels (L1, L2)

Symmetrized Kronecker delta, used in some calculations. For instance:


(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) kdelta([1,2],[2,1]);
(%o2)                                 - 1
(%i3) kdels([1,2],[2,1]);
(%o3)                                  1
(%i4) ishow(kdelta([a,b],[c,d]))$
                             c       d         d       c
(%t4)                  kdelta  kdelta  - kdelta  kdelta
                             a       b         a       b
(%i4) ishow(kdels([a,b],[c,d]))$
                             c       d         d       c
(%t4)                  kdelta  kdelta  + kdelta  kdelta
                             a       b         a       b

Function: levi_civita (L)

is the permutation (or Levi-Civita) tensor which yields 1 if the list L consists of an even permutation of integers, -1 if it consists of an odd permutation, and 0 if some indices in L are repeated.

Function: lc2kdt (expr)

Simplifies expressions containing the Levi-Civita symbol, converting these to Kronecker-delta expressions when possible. The main difference between this function and simply evaluating the Levi-Civita symbol is that direct evaluation often results in Kronecker expressions containing numerical indices. This is often undesirable as it prevents further simplification. The lc2kdt function avoids this problem, yielding expressions that are more easily simplified with rename or contract.


(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) expr:ishow('levi_civita([],[i,j])
                 *'levi_civita([k,l],[])*a([j],[k]))$
                                  i j  k
(%t2)                  levi_civita    a  levi_civita
                                       j            k l
(%i3) ishow(ev(expr,levi_civita))$
                                  i j  k       1 2
(%t3)                       kdelta    a  kdelta
                                  1 2  j       k l
(%i4) ishow(ev(%,kdelta))$
             i       j         j       i   k
(%t4) (kdelta  kdelta  - kdelta  kdelta ) a
             1       2         1       2   j

                               1       2         2       1
                        (kdelta  kdelta  - kdelta  kdelta )
                               k       l         k       l
(%i5) ishow(lc2kdt(expr))$
                     k       i       j    k       j       i
(%t5)               a  kdelta  kdelta  - a  kdelta  kdelta
                     j       k       l    j       k       l
(%i6) ishow(contract(expand(%)))$
                                 i           i
(%t6)                           a  - a kdelta
                                 l           l

The lc2kdt function sometimes makes use of the metric tensor. If the metric tensor was not defined previously with imetric, this results in an error.


(%i7) expr:ishow('levi_civita([],[i,j])
                 *'levi_civita([],[k,l])*a([j,k],[]))$
                                 i j            k l
(%t7)                 levi_civita    levi_civita    a
                                                     j k
(%i8) ishow(lc2kdt(expr))$
Maxima encountered a Lisp error:

 Error in $IMETRIC [or a callee]:
 $IMETRIC [or a callee] requires less than two arguments.

Automatically continuing.
To reenable the Lisp debugger set *debugger-hook* to nil.
(%i9) imetric(g);
(%o9)                                done
(%i10) ishow(lc2kdt(expr))$
         %3 i       k   %4 j       l     %3 i       l   %4 j
(%t10) (g     kdelta   g     kdelta   - g     kdelta   g    
                    %3             %4               %3
              k
        kdelta  ) a
              %4   j k
(%i11) ishow(contract(expand(%)))$
                                  l i    l i  j
(%t11)                           a    - g    a
                                              j

Function: lc_l

Simplification rule used for expressions containing the unevaluated Levi-Civita symbol (levi_civita). Along with `lc_u', it can be used to simplify many expressions more efficiently than the evaluation of levi_civita. For example:


(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) el1:ishow('levi_civita([i,j,k],[])*a([],[i])*a([],[j]))$
                             i  j
(%t2)                       a  a  levi_civita
                                             i j k
(%i3) el2:ishow('levi_civita([],[i,j,k])*a([i])*a([j]))$
                                       i j k
(%t3)                       levi_civita      a  a
                                              i  j
(%i4) canform(contract(expand(applyb1(el1,lc_l,lc_u))));
(%t4)                                  0
(%i5) canform(contract(expand(applyb1(el2,lc_l,lc_u))));
(%t5)                                  0

Function: lc_u

Simplification rule used for expressions containing the unevaluated Levi-Civita symbol (levi_civita). Along with lc_u, it can be used to simplify many expressions more efficiently than the evaluation of levi_civita. For details, see lc_l.

Function: canten (expr)

Simplifies expr by renaming (see rename) and permuting dummy indices. rename is restricted to sums of tensor products in which no derivatives are present. As such it is limited and should only be used if canform is not capable of carrying out the required simplification.

The canten function returns a mathematically correct result only if its argument is an expression that is fully symmetric in its indices. For this reason, canten returns an error if allsym is not set to true.

Function: concan (expr)

Similar to canten but also performs index contraction.

25.2.2 Tensor symmetries

Option variable: allsym

Default value: false

If true then all indexed objects are assumed symmetric in all of their covariant and contravariant indices. If false then no symmetries of any kind are assumed in these indices. Derivative indices are always taken to be symmetric unless iframe_flag is set to true.

Function: decsym (tensor, m, n, [cov_1, cov_2, ...], [contr_1, contr_2, ...])

Declares symmetry properties for tensor of m covariant and n contravariant indices. The cov_i and contr_i are pseudofunctions expressing symmetry relations among the covariant and contravariant indices respectively. These are of the form symoper(index_1, index_2,...) where symoper is one of sym, anti or cyc and the index_i are integers indicating the position of the index in the tensor. This will declare tensor to be symmetric, antisymmetric or cyclic respectively in the index_i. symoper(all) is also an allowable form which indicates all indices obey the symmetry condition. For example, given an object b with 5 covariant indices, decsym(b,5,3,[sym(1,2),anti(3,4)],[cyc(all)]) declares b symmetric in its first and second and antisymmetric in its third and fourth covariant indices, and cyclic in all of its contravariant indices. Either list of symmetry declarations may be null. The function which performs the simplifications is canform as the example below illustrates.


(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) expr:contract( expand( a([i1, j1, k1], [])
           *kdels([i, j, k], [i1, j1, k1])))$
(%i3) ishow(expr)$
(%t3)         a      + a      + a      + a      + a      + a
               k j i    k i j    j k i    j i k    i k j    i j k
(%i4) decsym(a,3,0,[sym(all)],[]);
(%o4)                                done
(%i5) ishow(canform(expr))$
(%t5)                              6 a
                                      i j k
(%i6) remsym(a,3,0);
(%o6)                                done
(%i7) decsym(a,3,0,[anti(all)],[]);
(%o7)                                done
(%i8) ishow(canform(expr))$
(%t8)                                  0
(%i9) remsym(a,3,0);
(%o9)                                done
(%i10) decsym(a,3,0,[cyc(all)],[]);
(%o10)                               done
(%i11) ishow(canform(expr))$
(%t11)                        3 a      + 3 a
                                 i k j      i j k
(%i12) dispsym(a,3,0);
(%o12)                     [[cyc, [[1, 2, 3]], []]]

Function: remsym (tensor, m, n)

Removes all symmetry properties from tensor which has m covariant indices and n contravariant indices.

Function: canform
    canform (expr)
    canform (expr, rename)

Simplifies expr by renaming dummy indices and reordering all indices as dictated by symmetry conditions imposed on them. If allsym is true then all indices are assumed symmetric, otherwise symmetry information provided by decsym declarations will be used. The dummy indices are renamed in the same manner as in the rename function. When canform is applied to a large expression the calculation may take a considerable amount of time. This time can be shortened by calling rename on the expression first. Also see the example under decsym. Note: canform may not be able to reduce an expression completely to its simplest form although it will always return a mathematically correct result.

The optional second parameter rename, if set to false, suppresses renaming.

25.2.3 Indicial tensor calculus

Function: diff (expr, v_1, [n_1, [v_2, n_2] ...])

is the usual Maxima differentiation function which has been expanded in its abilities for itensor. It takes the derivative of expr with respect to v_1 n_1 times, with respect to v_2 n_2 times, etc. For the tensor package, the function has been modified so that the v_i may be integers from 1 up to the value of the variable dim. This will cause the differentiation to be carried out with respect to the v_ith member of the list vect_coords. If vect_coords is bound to an atomic variable, then that variable subscripted by v_i will be used for the variable of differentiation. This permits an array of coordinate names or subscripted names like x[1], x[2], ... to be used.

A further extension adds the ability to diff to compute derivatives with respect to an indexed variable. In particular, the tensor package knows how to differentiate expressions containing combinations of the metric tensor and its derivatives with respect to the metric tensor and its first and second derivatives. This capability is particularly useful when considering Lagrangian formulations of a gravitational theory, allowing one to derive the Einstein tensor and field equations from the action principle.

Function: idiff (expr, v_1, [n_1, [v_2, n_2] ...])

Indicial differentiation. Unlike diff, which differentiates with respect to an independent variable, idiff) can be used to differentiate with respect to a coordinate. For an indexed object, this amounts to appending the v_i as derivative indices. Subsequently, derivative indices will be sorted, unless iframe_flag is set to true.

idiff can also differentiate the determinant of the metric tensor. Thus, if imetric has been bound to G then idiff(determinant(g),k) will return 2 * determinant(g) * ichr2([%i,k],[%i]) where the dummy index %i is chosen appropriately.

Function: liediff (v, ten)

Computes the Lie-derivative of the tensorial expression ten with respect to the vector field v. ten should be any indexed tensor expression; v should be the name (without indices) of a vector field. For example:


(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) ishow(liediff(v,a([i,j],[])*b([],[k],l)))$
       k    %2            %2          %2
(%t2) b   (v   a       + v   a     + v   a    )
       ,l       i j,%2    ,j  i %2    ,i  %2 j

                          %1  k        %1  k      %1  k
                      + (v   b      - b   v    + v   b   ) a
                              ,%1 l    ,l  ,%1    ,l  ,%1   i j

Function: rediff (ten)

Evaluates all occurrences of the idiff command in the tensorial expression ten.

Function: undiff (expr)

Returns an expression equivalent to expr but with all derivatives of indexed objects replaced by the noun form of the idiff function. Its arguments would yield that indexed object if the differentiation were carried out. This is useful when it is desired to replace a differentiated indexed object with some function definition resulting in expr and then carry out the differentiation by saying ev(expr, idiff).

Function: evundiff (expr)

Equivalent to the execution of undiff, followed by ev and rediff.

The point of this operation is to easily evalute expressions that cannot be directly evaluated in derivative form. For instance, the following causes an error:

(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) icurvature([i,j,k],[l],m);
Maxima encountered a Lisp error:

 Error in $ICURVATURE [or a callee]:
 $ICURVATURE [or a callee] requires less than three arguments.

Automatically continuing.
To reenable the Lisp debugger set *debugger-hook* to nil.

However, if icurvature is entered in noun form, it can be evaluated using evundiff:

(%i3) ishow('icurvature([i,j,k],[l],m))$
                                         l
(%t3)                          icurvature
                                         i j k,m
(%i4) ishow(evundiff(%))$
             l              l         %1           l           %1
(%t4) - ichr2        - ichr2     ichr2      - ichr2       ichr2
             i k,j m        %1 j      i k,m        %1 j,m      i k

             l              l         %1           l           %1
      + ichr2        + ichr2     ichr2      + ichr2       ichr2
             i j,k m        %1 k      i j,m        %1 k,m      i j

Note: In earlier versions of Maxima, derivative forms of the Christoffel-symbols also could not be evaluated. This has been fixed now, so evundiff is no longer necessary for expressions like this:

(%i5) imetric(g);
(%o5)                                done
(%i6) ishow(ichr2([i,j],[k],l))$
       k %3
      g     (g         - g         + g        )
              j %3,i l    i j,%3 l    i %3,j l
(%t6) -----------------------------------------
                          2

                         k %3
                        g     (g       - g       + g      )
                         ,l     j %3,i    i j,%3    i %3,j
                      + -----------------------------------
                                         2

Function: flush (expr, tensor_1, tensor_2, ...)

Set to zero, in expr, all occurrences of the tensor_i that have no derivative indices.

Function: flushd (expr, tensor_1, tensor_2, ...)

Set to zero, in expr, all occurrences of the tensor_i that have derivative indices.

Function: flushnd (expr, tensor, n)

Set to zero, in expr, all occurrences of the differentiated object tensor that have n or more derivative indices as the following example demonstrates.


(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) ishow(a([i],[J,r],k,r)+a([i],[j,r,s],k,r,s))$
                                J r      j r s
(%t2)                          a      + a
                                i,k r    i,k r s
(%i3) ishow(flushnd(%,a,3))$
                                     J r
(%t3)                               a
                                     i,k r

Function: coord (tensor_1, tensor_2, ...)

Gives tensor_i the coordinate differentiation property that the derivative of contravariant vector whose name is one of the tensor_i yields a Kronecker delta. For example, if coord(x) has been done then idiff(x([],[i]),j) gives kdelta([i],[j]). coord is a list of all indexed objects having this property.

Function: remcoord
    remcoord (tensor_1, tensor_2, ...)
    remcoord (all)

Removes the coordinate differentiation property from the tensor_i that was established by the function coord. remcoord(all) removes this property from all indexed objects.

Function: makebox (expr)

Display expr in the same manner as show; however, any tensor d’Alembertian occurring in expr will be indicated using the symbol []. For example, []p([m],[n]) represents g([],[i,j])*p([m],[n],i,j).

Function: conmetderiv (expr, tensor)

Simplifies expressions containing ordinary derivatives of both covariant and contravariant forms of the metric tensor (the current restriction). For example, conmetderiv can relate the derivative of the contravariant metric tensor with the Christoffel symbols as seen from the following:


(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) ishow(g([],[a,b],c))$
                                      a b
(%t2)                                g
                                      ,c
(%i3) ishow(conmetderiv(%,g))$
                         %1 b      a       %1 a      b
(%t3)                 - g     ichr2     - g     ichr2
                                   %1 c              %1 c

Function: simpmetderiv
    simpmetderiv (expr)
    simpmetderiv (expr[, stop])

Simplifies expressions containing products of the derivatives of the metric tensor. Specifically, simpmetderiv recognizes two identities:


   ab        ab           ab                 a
  g   g   + g   g     = (g   g  )   = (kdelta )   = 0
   ,d  bc        bc,d         bc ,d          c ,d

hence


   ab          ab
  g   g   = - g   g
   ,d  bc          bc,d

and


  ab          ab
 g   g     = g   g
  ,j  ab,i    ,i  ab,j

which follows from the symmetries of the Christoffel symbols.

The simpmetderiv function takes one optional parameter which, when present, causes the function to stop after the first successful substitution in a product expression. The simpmetderiv function also makes use of the global variable flipflag which determines how to apply a “canonical” ordering to the product indices.

Put together, these capabilities can be used to achieve powerful simplifications that are difficult or impossible to accomplish otherwise. This is demonstrated through the following example that explicitly uses the partial simplification features of simpmetderiv to obtain a contractible expression:


(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) imetric(g);
(%o2)                                done
(%i3) ishow(g([],[a,b])*g([],[b,c])*g([a,b],[],d)*g([b,c],[],e))$
                             a b  b c
(%t3)                       g    g    g      g
                                       a b,d  b c,e
(%i4) ishow(canform(%))$

errexp1 has improper indices
 -- an error.  Quitting.  To debug this try debugmode(true);
(%i5) ishow(simpmetderiv(%))$
                             a b  b c
(%t5)                       g    g    g      g
                                       a b,d  b c,e
(%i6) flipflag:not flipflag;
(%o6)                                true
(%i7) ishow(simpmetderiv(%th(2)))$
                               a b  b c
(%t7)                         g    g    g    g
                               ,d   ,e   a b  b c
(%i8) flipflag:not flipflag;
(%o8)                                false
(%i9) ishow(simpmetderiv(%th(2),stop))$
                               a b  b c
(%t9)                       - g    g    g      g
                                    ,e   a b,d  b c
(%i10) ishow(contract(%))$
                                    b c
(%t10)                           - g    g
                                    ,e   c b,d

See also weyl.dem for an example that uses simpmetderiv and conmetderiv together to simplify contractions of the Weyl tensor.

Function: flush1deriv (expr, tensor)

Set to zero, in expr, all occurrences of tensor that have exactly one derivative index.

25.2.4 Tensors in curved spaces

Function: imetric (g)
System variable: imetric

Specifies the metric by assigning the variable imetric:g in addition, the contraction properties of the metric g are set up by executing the commands defcon(g), defcon(g, g, kdelta). The variable imetric (unbound by default), is bound to the metric, assigned by the imetric(g) command.

Function: idim (n)

Sets the dimensions of the metric. Also initializes the antisymmetry properties of the Levi-Civita symbols for the given dimension.

Function: ichr1 ([i, j, k])

Yields the Christoffel symbol of the first kind via the definition

       (g      + g      - g     )/2 .
         ik,j     jk,i     ij,k

To evaluate the Christoffel symbols for a particular metric, the variable imetric must be assigned a name as in the example under chr2.

Function: ichr2 ([i, j], [k])

Yields the Christoffel symbol of the second kind defined by the relation

                       ks
   ichr2([i,j],[k]) = g    (g      + g      - g     )/2
                             is,j     js,i     ij,s

Function: icurvature ([i, j, k], [h])

Yields the Riemann curvature tensor in terms of the Christoffel symbols of the second kind (ichr2). The following notation is used:

            h             h            h         %1         h
  icurvature     = - ichr2      - ichr2     ichr2    + ichr2
            i j k         i k,j        %1 j      i k        i j,k
                            h          %1
                     + ichr2      ichr2
                            %1 k       i j

Function: covdiff (expr, v_1, v_2, ...)

Yields the covariant derivative of expr with respect to the variables v_i in terms of the Christoffel symbols of the second kind (ichr2). In order to evaluate these, one should use ev(expr,ichr2).


(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) entertensor()$
Enter tensor name: a;
Enter a list of the covariant indices: [i,j];
Enter a list of the contravariant indices: [k];
Enter a list of the derivative indices: [];
                                      k
(%t2)                                a
                                      i j
(%i3) ishow(covdiff(%,s))$
             k         %1     k         %1     k
(%t3)     - a     ichr2    - a     ichr2    + a
             i %1      j s    %1 j      i s    i j,s

             k     %1
      + ichr2     a
             %1 s  i j
(%i4) imetric:g;
(%o4)                                  g
(%i5) ishow(ev(%th(2),ichr2))$
         %1 %4  k
        g      a     (g       - g       + g      )
                i %1   s %4,j    j s,%4    j %4,s
(%t5) - ------------------------------------------
                            2
    %1 %3  k
   g      a     (g       - g       + g      )
           %1 j   s %3,i    i s,%3    i %3,s
 - ------------------------------------------
                       2
    k %2  %1
   g     a    (g        - g        + g       )
          i j   s %2,%1    %1 s,%2    %1 %2,s     k
 + ------------------------------------------- + a
                        2                         i j,s
(%i6)

Function: lorentz_gauge (expr)

Imposes the Lorentz condition by substituting 0 for all indexed objects in expr that have a derivative index identical to a contravariant index.

Function: igeodesic_coords (expr, name)

Causes undifferentiated Christoffel symbols and first derivatives of the metric tensor vanish in expr. The name in the `igeodesic_coords' function refers to the metric name (if it appears in expr) while the connection coefficients must be called with the names ichr1 and/or ichr2. The following example demonstrates the verification of the cyclic identity satisfied by the Riemann curvature tensor using the igeodesic_coords function.


(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) ishow(icurvature([r,s,t],[u]))$
             u            u         %1         u     
(%t2) - ichr2      - ichr2     ichr2    + ichr2      
             r t,s        %1 s      r t        r s,t 

                                              u         %1
                                       + ichr2     ichr2
                                              %1 t      r s
(%i3) ishow(igeodesic_coords(%,ichr2))$
                                 u            u
(%t3)                       ichr2      - ichr2
                                 r s,t        r t,s
(%i4) ishow(igeodesic_coords(icurvature([r,s,t],[u]),ichr2)+
            igeodesic_coords(icurvature([s,t,r],[u]),ichr2)+
            igeodesic_coords(icurvature([t,r,s],[u]),ichr2))$
             u            u            u            u
(%t4) - ichr2      + ichr2      + ichr2      - ichr2
             t s,r        t r,s        s t,r        s r,t

                                             u            u
                                      - ichr2      + ichr2
                                             r t,s        r s,t
(%i5) canform(%);
(%o5)                                  0

25.2.5 Moving frames

Maxima now has the ability to perform calculations using moving frames. These can be orthonormal frames (tetrads, vielbeins) or an arbitrary frame.

To use frames, you must first set iframe_flag to true. This causes the Christoffel-symbols, ichr1 and ichr2, to be replaced by the more general frame connection coefficients icc1 and icc2 in calculations. Speficially, the behavior of covdiff and icurvature is changed.

The frame is defined by two tensors: the inverse frame field (ifri, the dual basis tetrad), and the frame metric ifg. The frame metric is the identity matrix for orthonormal frames, or the Lorentz metric for orthonormal frames in Minkowski spacetime. The inverse frame field defines the frame base (unit vectors). Contraction properties are defined for the frame field and the frame metric.

When iframe_flag is true, many itensor expressions use the frame metric ifg instead of the metric defined by imetric for raising and lowerind indices.

IMPORTANT: Setting the variable iframe_flag to true does NOT undefine the contraction properties of a metric defined by a call to defcon or imetric. If a frame field is used, it is best to define the metric by assigning its name to the variable imetric and NOT invoke the imetric function.

Maxima uses these two tensors to define the frame coefficients (ifc1 and ifc2) which form part of the connection coefficients (icc1 and icc2), as the following example demonstrates:


(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) iframe_flag:true;
(%o2)                                true
(%i3) ishow(covdiff(v([],[i]),j))$
                               i        i     %1
(%t3)                         v   + icc2     v
                               ,j       %1 j
(%i4) ishow(ev(%,icc2))$
                               %1     i       i
(%t4)                         v   ifc2     + v
                                      %1 j    ,j
(%i5) ishow(ev(%,ifc2))$
                          %1    i %2                i
(%t5)                    v   ifg     ifc1        + v
                                         %1 j %2    ,j
(%i6) ishow(ev(%,ifc1))$
            %1    i %2
           v   ifg     (ifb        - ifb        + ifb       )
                           j %2 %1      %2 %1 j      %1 j %2     i
(%t6)      -------------------------------------------------- + v
                                   2                             ,j
(%i7) ishow(ifb([a,b,c]))$
                                                   %3    %4
(%t7)               (ifri        - ifri       ) ifr   ifr
                         a %3,%4       a %4,%3     b     c

An alternate method is used to compute the frame bracket (ifb) if the iframe_bracket_form flag is set to false:


(%i8) block([iframe_bracket_form:false],ishow(ifb([a,b,c])))$
                                %6    %5        %5      %6
(%t8)              ifri     (ifr   ifr     - ifr     ifr  )
                       a %5     b     c,%6      b,%6    c

Function: iframes ()

Since in this version of Maxima, contraction identities for ifr and ifri are always defined, as is the frame bracket (ifb), this function does nothing.

Variable: ifb

The frame bracket. The contribution of the frame metric to the connection coefficients is expressed using the frame bracket:


          - ifb      + ifb      + ifb
               c a b      b c a      a b c
ifc1    = --------------------------------
    abc                  2

The frame bracket itself is defined in terms of the frame field and frame metric. Two alternate methods of computation are used depending on the value of frame_bracket_form. If true (the default) or if the itorsion_flag is true:


          d      e                                      f
ifb =  ifr    ifr   (ifri      - ifri      - ifri    itr   )
   abc    b      c       a d,e       a e,d       a f    d e


Otherwise:


             e      d        d      e
ifb    = (ifr    ifr    - ifr    ifr   ) ifri
   abc       b      c,e      b,e    c        a d

Variable: icc1

Connection coefficients of the first kind. In itensor, defined as


icc1    = ichr1    - ikt1    - inmc1
    abc        abc       abc        abc

In this expression, if iframe_flag is true, the Christoffel-symbol ichr1 is replaced with the frame connection coefficient ifc1. If itorsion_flag is false, ikt1 will be omitted. It is also omitted if a frame base is used, as the torsion is already calculated as part of the frame bracket. Lastly, of inonmet_flag is false, inmc1 will not be present.

Variable: icc2

Connection coefficients of the second kind. In itensor, defined as


    c         c        c         c
icc2   = ichr2   - ikt2   - inmc2
    ab        ab       ab        ab

In this expression, if iframe_flag is true, the Christoffel-symbol ichr2 is replaced with the frame connection coefficient ifc2. If itorsion_flag is false, ikt2 will be omitted. It is also omitted if a frame base is used, as the torsion is already calculated as part of the frame bracket. Lastly, of inonmet_flag is false, inmc2 will not be present.

Variable: ifc1

Frame coefficient of the first kind (also known as Ricci-rotation coefficients.) This tensor represents the contribution of the frame metric to the connection coefficient of the first kind. Defined as:


          - ifb      + ifb      + ifb
               c a b      b c a      a b c
ifc1    = --------------------------------
    abc                   2


Variable: ifc2

Frame coefficient of the second kind. This tensor represents the contribution of the frame metric to the connection coefficient of the second kind. Defined as a permutation of the frame bracket (ifb) with the appropriate indices raised and lowered as necessary:


    c       cd
ifc2   = ifg   ifc1
    ab             abd

Variable: ifr

The frame field. Contracts with the inverse frame field (ifri) to form the frame metric (ifg).

Variable: ifri

The inverse frame field. Specifies the frame base (dual basis vectors). Along with the frame metric, it forms the basis of all calculations based on frames.

Variable: ifg

The frame metric. Defaults to kdelta, but can be changed using components.

Variable: ifgi

The inverse frame metric. Contracts with the frame metric (ifg) to kdelta.

Option variable: iframe_bracket_form

Default value: true

Specifies how the frame bracket (ifb) is computed.

25.2.6 Torsion and nonmetricity

Maxima can now take into account torsion and nonmetricity. When the flag itorsion_flag is set to true, the contribution of torsion is added to the connection coefficients. Similarly, when the flag inonmet_flag is true, nonmetricity components are included.

Variable: inm

The nonmetricity vector. Conformal nonmetricity is defined through the covariant derivative of the metric tensor. Normally zero, the metric tensor’s covariant derivative will evaluate to the following when inonmet_flag is set to true:


g     =- g  inm
 ij;k     ij   k

Variable: inmc1

Covariant permutation of the nonmetricity vector components. Defined as


           g   inm  - inm  g   - g   inm
            ab    c      a  bc    ac    b
inmc1    = ------------------------------
     abc                 2

(Substitute ifg in place of g if a frame metric is used.)

Variable: inmc2

Contravariant permutation of the nonmetricity vector components. Used in the connection coefficients if inonmet_flag is true. Defined as:


                      c         c         cd
          -inm  kdelta  - kdelta  inm  + g   inm  g
     c        a       b         a    b          d  ab
inmc2   = -------------------------------------------
     ab                        2

(Substitute ifg in place of g if a frame metric is used.)

Variable: ikt1

Covariant permutation of the torsion tensor (also known as contorsion). Defined as:


                  d           d       d
          -g   itr  - g    itr   - itr   g
            ad    cb    bd    ca      ab  cd
ikt1    = ----------------------------------
    abc                   2

(Substitute ifg in place of g if a frame metric is used.)

Variable: ikt2

Contravariant permutation of the torsion tensor (also known as contorsion). Defined as:


    c     cd
ikt2   = g   ikt1
    ab           abd

(Substitute ifg in place of g if a frame metric is used.)

Variable: itr

The torsion tensor. For a metric with torsion, repeated covariant differentiation on a scalar function will not commute, as demonstrated by the following example:


(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) imetric:g;
(%o2)                                  g
(%i3) covdiff( covdiff( f( [], []), i), j)
                      - covdiff( covdiff( f( [], []), j), i)$
(%i4) ishow(%)$
                                   %4              %2
(%t4)                    f    ichr2    - f    ichr2
                          ,%4      j i    ,%2      i j
(%i5) canform(%);
(%o5)                                  0
(%i6) itorsion_flag:true;
(%o6)                                true
(%i7) covdiff( covdiff( f( [], []), i), j)
                      - covdiff( covdiff( f( [], []), j), i)$
(%i8) ishow(%)$
                           %8             %6
(%t8)             f    icc2    - f    icc2    - f     + f
                   ,%8     j i    ,%6     i j    ,j i    ,i j
(%i9) ishow(canform(%))$
                                   %1             %1
(%t9)                     f    icc2    - f    icc2
                           ,%1     j i    ,%1     i j
(%i10) ishow(canform(ev(%,icc2)))$
                                   %1             %1
(%t10)                    f    ikt2    - f    ikt2
                           ,%1     i j    ,%1     j i
(%i11) ishow(canform(ev(%,ikt2)))$
                      %2 %1                    %2 %1
(%t11)          f    g      ikt1       - f    g      ikt1
                 ,%2            i j %1    ,%2            j i %1
(%i12) ishow(factor(canform(rename(expand(ev(%,ikt1))))))$
                           %3 %2            %1       %1
                     f    g      g      (itr    - itr   )
                      ,%3         %2 %1     j i      i j
(%t12)               ------------------------------------
                                      2
(%i13) decsym(itr,2,1,[anti(all)],[]);
(%o13)                               done
(%i14) defcon(g,g,kdelta);
(%o14)                               done
(%i15) subst(g,nounify(g),%th(3))$
(%i16) ishow(canform(contract(%)))$
                                           %1
(%t16)                           - f    itr
                                    ,%1    i j

25.2.7 Exterior algebra

The itensor package can perform operations on totally antisymmetric covariant tensor fields. A totally antisymmetric tensor field of rank (0,L) corresponds with a differential L-form. On these objects, a multiplication operation known as the exterior product, or wedge product, is defined.

Unfortunately, not all authors agree on the definition of the wedge product. Some authors prefer a definition that corresponds with the notion of antisymmetrization: in these works, the wedge product of two vector fields, for instance, would be defined as

            a a  - a a
             i j    j i
 a  /\ a  = -----------
  i     j        2

More generally, the product of a p-form and a q-form would be defined as

                       1     k1..kp l1..lq
A       /\ B       = ------ D              A       B
 i1..ip     j1..jq   (p+q)!  i1..ip j1..jq  k1..kp  l1..lq

where D stands for the Kronecker-delta.

Other authors, however, prefer a “geometric” definition that corresponds with the notion of the volume element:

a  /\ a  = a a  - a a
 i     j    i j    j i

and, in the general case

                       1    k1..kp l1..lq
A       /\ B       = ----- D              A       B
 i1..ip     j1..jq   p! q!  i1..ip j1..jq  k1..kp  l1..lq

Since itensor is a tensor algebra package, the first of these two definitions appears to be the more natural one. Many applications, however, utilize the second definition. To resolve this dilemma, a flag has been implemented that controls the behavior of the wedge product: if igeowedge_flag is false (the default), the first, "tensorial" definition is used, otherwise the second, "geometric" definition will be applied.

Operator: ~

The wedge product operator is denoted by the tilde ~. This is a binary operator. Its arguments should be expressions involving scalars, covariant tensors of rank one, or covariant tensors of rank l that have been declared antisymmetric in all covariant indices.

The behavior of the wedge product operator is controlled by the igeowedge_flag flag, as in the following example:

(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) ishow(a([i])~b([j]))$
                                 a  b  - b  a
                                  i  j    i  j
(%t2)                            -------------
                                       2
(%i3) decsym(a,2,0,[anti(all)],[]);
(%o3)                                done
(%i4) ishow(a([i,j])~b([k]))$
                          a    b  + b  a    - a    b
                           i j  k    i  j k    i k  j
(%t4)                     ---------------------------
                                       3
(%i5) igeowedge_flag:true;
(%o5)                                true
(%i6) ishow(a([i])~b([j]))$
(%t6)                            a  b  - b  a
                                  i  j    i  j
(%i7) ishow(a([i,j])~b([k]))$
(%t7)                     a    b  + b  a    - a    b
                           i j  k    i  j k    i k  j

Operator: |

The vertical bar | denotes the "contraction with a vector" binary operation. When a totally antisymmetric covariant tensor is contracted with a contravariant vector, the result is the same regardless which index was used for the contraction. Thus, it is possible to define the contraction operation in an index-free manner.

In the itensor package, contraction with a vector is always carried out with respect to the first index in the literal sorting order. This ensures better simplification of expressions involving the | operator. For instance:

(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) decsym(a,2,0,[anti(all)],[]);
(%o2)                                done
(%i3) ishow(a([i,j],[])|v)$
                                    %1
(%t3)                              v   a
                                        %1 j
(%i4) ishow(a([j,i],[])|v)$
                                     %1
(%t4)                             - v   a
                                         %1 j

Note that it is essential that the tensors used with the | operator be declared totally antisymmetric in their covariant indices. Otherwise, the results will be incorrect.

Function: extdiff (expr, i)

Computes the exterior derivative of expr with respect to the index i. The exterior derivative is formally defined as the wedge product of the partial derivative operator and a differential form. As such, this operation is also controlled by the setting of igeowedge_flag. For instance:

(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) ishow(extdiff(v([i]),j))$
                                  v    - v
                                   j,i    i,j
(%t2)                             -----------
                                       2
(%i3) decsym(a,2,0,[anti(all)],[]);
(%o3)                                done
(%i4) ishow(extdiff(a([i,j]),k))$
                           a      - a      + a
                            j k,i    i k,j    i j,k
(%t4)                      ------------------------
                                      3
(%i5) igeowedge_flag:true;
(%o5)                                true
(%i6) ishow(extdiff(v([i]),j))$
(%t6)                             v    - v
                                   j,i    i,j
(%i7) ishow(extdiff(a([i,j]),k))$
(%t7)                    - (a      - a      + a     )
                             k j,i    k i,j    j i,k

Function: hodge (expr)

Compute the Hodge-dual of expr. For instance:


(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) imetric(g);
(%o2)                            done
(%i3) idim(4);
(%o3)                            done
(%i4) icounter:100;
(%o4)                             100
(%i5) decsym(A,3,0,[anti(all)],[])$

(%i6) ishow(A([i,j,k],[]))$
(%t6)                           A
                                 i j k
(%i7) ishow(canform(hodge(%)))$
                          %1 %2 %3 %4
               levi_civita            g        A
                                       %1 %102  %2 %3 %4
(%t7)          -----------------------------------------
                                   6
(%i8) ishow(canform(hodge(%)))$
                 %1 %2 %3 %8            %4 %5 %6 %7
(%t8) levi_civita            levi_civita            g       
                                                     %1 %106
                             g        g        g      A         /6
                              %2 %107  %3 %108  %4 %8  %5 %6 %7
(%i9) lc2kdt(%)$

(%i10) %,kdelta$

(%i11) ishow(canform(contract(expand(%))))$
(%t11)                     - A
                              %106 %107 %108

Option variable: igeowedge_flag

Default value: false

Controls the behavior of the wedge product and exterior derivative. When set to false (the default), the notion of differential forms will correspond with that of a totally antisymmetric covariant tensor field. When set to true, differential forms will agree with the notion of the volume element.

25.2.8 Exporting TeX expressions

The itensor package provides limited support for exporting tensor expressions to TeX. Since itensor expressions appear as function calls, the regular Maxima tex command will not produce the expected output. You can try instead the tentex command, which attempts to translate tensor expressions into appropriately indexed TeX objects.

Function: tentex (expr)

To use the tentex function, you must first load tentex, as in the following example:

(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) load(tentex);
(%o2)       /share/tensor/tentex.lisp
(%i3) idummyx:m;
(%o3)                                  m
(%i4) ishow(icurvature([j,k,l],[i]))$
            m1       i           m1       i           i
(%t4)  ichr2    ichr2     - ichr2    ichr2     - ichr2
            j k      m1 l        j l      m1 k        j l,k

                                                      i
                                               + ichr2
                                                      j k,l
(%i5) tentex(%)$
$$\Gamma_{j\,k}^{m_1}\,\Gamma_{l\,m_1}^{i}-\Gamma_{j\,l}^{m_1}\,
 \Gamma_{k\,m_1}^{i}-\Gamma_{j\,l,k}^{i}+\Gamma_{j\,k,l}^{i}$$

Note the use of the idummyx assignment, to avoid the appearance of the percent sign in the TeX expression, which may lead to compile errors.

NB: This version of the tentex function is somewhat experimental.

25.2.9 Interfacing with ctensor

The itensor package has the ability to generate Maxima code that can then be executed in the context of the ctensor package. The function that performs this task is ic_convert.

Function: ic_convert (eqn)

Converts the itensor equation eqn to a ctensor assignment statement. Implied sums over dummy indices are made explicit while indexed objects are transformed into arrays (the array subscripts are in the order of covariant followed by contravariant indices of the indexed objects). The derivative of an indexed object will be replaced by the noun form of diff taken with respect to ct_coords subscripted by the derivative index. The Christoffel symbols ichr1 and ichr2 will be translated to lcs and mcs, respectively and if metricconvert is true then all occurrences of the metric with two covariant (contravariant) indices will be renamed to lg (ug). In addition, do loops will be introduced summing over all free indices so that the transformed assignment statement can be evaluated by just doing ev. The following examples demonstrate the features of this function.

(%i1) load("itensor");
(%o1)      /share/tensor/itensor.lisp
(%i2) eqn:ishow(t([i,j],[k])=f([],[])*g([l,m],[])*a([],[m],j)
      *b([i],[l,k]))$
                             k        m   l k
(%t2)                       t    = f a   b    g
                             i j      ,j  i    l m
(%i3) ic_convert(eqn);
(%o3) for i thru dim do (for j thru dim do (
       for k thru dim do
        t        : f sum(sum(diff(a , ct_coords ) b
         i, j, k                   m           j   i, l, k

 g    , l, 1, dim), m, 1, dim)))
  l, m
(%i4) imetric(g);
(%o4)                                done
(%i5) metricconvert:true;
(%o5)                                true
(%i6) ic_convert(eqn);
(%o6) for i thru dim do (for j thru dim do (
       for k thru dim do
        t        : f sum(sum(diff(a , ct_coords ) b
         i, j, k                   m           j   i, l, k

 lg    , l, 1, dim), m, 1, dim)))
   l, m

25.2.10 Reserved words

The following Maxima words are used by the itensor package internally and should not be redefined:

  Keyword    Comments
  ------------------------------------------
  indices2() Internal version of indices()
  conti      Lists contravariant indices
  covi       Lists covariant indices of a indexed object
  deri       Lists derivative indices of an indexed object
  name       Returns the name of an indexed object
  concan
  irpmon
  lc0
  _lc2kdt0
  _lcprod
  _extlc

Next: , Previous: , Up: Top   [Contents][Index]

26 ctensor


Next: , Previous: , Up: ctensor   [Contents][Index]

26.1 Introduction to ctensor

ctensor is a component tensor manipulation package. To use the ctensor package, type load("ctensor"). To begin an interactive session with ctensor, type csetup(). You are first asked to specify the dimension of the manifold. If the dimension is 2, 3 or 4 then the list of coordinates defaults to [x,y], [x,y,z] or [x,y,z,t] respectively. These names may be changed by assigning a new list of coordinates to the variable ct_coords (described below) and the user is queried about this. Care must be taken to avoid the coordinate names conflicting with other object definitions.

Next, the user enters the metric either directly or from a file by specifying its ordinal position. The metric is stored in the matrix lg. Finally, the metric inverse is computed and stored in the matrix ug. One has the option of carrying out all calculations in a power series.

A sample protocol is begun below for the static, spherically symmetric metric (standard coordinates) which will be applied to the problem of deriving Einstein’s vacuum equations (which lead to the Schwarzschild solution) as an example. Many of the functions in ctensor will be displayed for the standard metric as examples.

(%i1) load("ctensor");
(%o1)      /share/tensor/ctensor.mac
(%i2) csetup();
Enter the dimension of the coordinate system:
4;
Do you wish to change the coordinate names?
n;
Do you want to
1. Enter a new metric?

2. Enter a metric from a file?

3. Approximate a metric with a Taylor series?
1;

Is the matrix  1. Diagonal  2. Symmetric  3. Antisymmetric  4. General
Answer 1, 2, 3 or 4
1;
Row 1 Column 1:
a;
Row 2 Column 2:
x^2;
Row 3 Column 3:
x^2*sin(y)^2;
Row 4 Column 4:
-d;

Matrix entered.
Enter functional dependencies with the DEPENDS function or 'N' if none
depends([a,d],x);
Do you wish to see the metric?
y;
                          [ a  0       0        0  ]
                          [                        ]
                          [     2                  ]
                          [ 0  x       0        0  ]
                          [                        ]
                          [         2    2         ]
                          [ 0  0   x  sin (y)   0  ]
                          [                        ]
                          [ 0  0       0       - d ]
(%o2)                                done
(%i3) christof(mcs);
                                            a
                                             x
(%t3)                          mcs        = ---
                                  1, 1, 1   2 a

                                             1
(%t4)                           mcs        = -
                                   1, 2, 2   x

                                             1
(%t5)                           mcs        = -
                                   1, 3, 3   x

                                            d
                                             x
(%t6)                          mcs        = ---
                                  1, 4, 4   2 d

                                              x
(%t7)                          mcs        = - -
                                  2, 2, 1     a

                                           cos(y)
(%t8)                         mcs        = ------
                                 2, 3, 3   sin(y)

                                               2
                                          x sin (y)
(%t9)                      mcs        = - ---------
                              3, 3, 1         a

(%t10)                   mcs        = - cos(y) sin(y)
                            3, 3, 2

                                            d
                                             x
(%t11)                         mcs        = ---
                                  4, 4, 1   2 a
(%o11)                               done

·

Category: Tensors · Category: Share packages · Category: Package ctensor


Previous: , Up: ctensor   [Contents][Index]

26.2 Functions and Variables for ctensor

26.2.1 Initialization and setup

Function: csetup ()

A function in the ctensor (component tensor) package which initializes the package and allows the user to enter a metric interactively. See ctensor for more details.

Function: cmetric
    cmetric (dis)
    cmetric ()

A function in the ctensor (component tensor) package that computes the metric inverse and sets up the package for further calculations.

If cframe_flag is false, the function computes the inverse metric ug from the (user-defined) matrix lg. The metric determinant is also computed and stored in the variable gdet. Furthermore, the package determines if the metric is diagonal and sets the value of diagmetric accordingly. If the optional argument dis is present and not equal to false, the user is prompted to see the metric inverse.

If cframe_flag is true, the function expects that the values of fri (the inverse frame matrix) and lfg (the frame metric) are defined. From these, the frame matrix fr and the inverse frame metric ufg are computed.

Function: ct_coordsys
    ct_coordsys (coordinate_system, extra_arg)
    ct_coordsys (coordinate_system)

Sets up a predefined coordinate system and metric. The argument coordinate_system can be one of the following symbols:


 SYMBOL             Dim Coordinates     Description/comments
 ------------------------------------------------------------------
 cartesian2d           2  [x,y]             Cartesian 2D coordinate
                                            system
 polar                 2  [r,phi]           Polar coordinate system
 elliptic              2  [u,v]             Elliptic coord. system
 confocalelliptic      2  [u,v]             Confocal elliptic
                                            coordinates
 bipolar               2  [u,v]             Bipolar coord. system
 parabolic             2  [u,v]             Parabolic coord. system
 cartesian3d           3  [x,y,z]           Cartesian 3D coordinate
                                            system
 polarcylindrical      3  [r,theta,z]       Polar 2D with
                                            cylindrical z
 ellipticcylindrical   3  [u,v,z]           Elliptic 2D with
                                            cylindrical z
 confocalellipsoidal   3  [u,v,w]           Confocal ellipsoidal
 bipolarcylindrical    3  [u,v,z]           Bipolar 2D with
                                            cylindrical z
 paraboliccylindrical  3  [u,v,z]           Parabolic 2D with
                                            cylindrical z
 paraboloidal          3  [u,v,phi]         Paraboloidal coords.
 conical               3  [u,v,w]           Conical coordinates
 toroidal              3  [phi,u,v]         Toroidal coordinates
 spherical             3  [r,theta,phi]     Spherical coord. system
 oblatespheroidal      3  [u,v,phi]         Oblate spheroidal
                                            coordinates
 oblatespheroidalsqrt  3  [u,v,phi]
 prolatespheroidal     3  [u,v,phi]         Prolate spheroidal
                                            coordinates
 prolatespheroidalsqrt 3  [u,v,phi]
 ellipsoidal           3  [r,theta,phi]     Ellipsoidal coordinates
 cartesian4d           4  [x,y,z,t]         Cartesian 4D coordinate
                                            system
 spherical4d           4  [r,theta,eta,phi] Spherical 4D coordinate
                                            system
 exteriorschwarzschild 4  [t,r,theta,phi]   Schwarzschild metric
 interiorschwarzschild 4  [t,z,u,v]         Interior Schwarzschild
                                            metric
 kerr_newman           4  [t,r,theta,phi]   Charged axially
                                            symmetric metric

coordinate_system can also be a list of transformation functions, followed by a list containing the coordinate variables. For instance, you can specify a spherical metric as follows:


(%i1) load("ctensor");
(%o1)       /share/tensor/ctensor.mac
(%i2) ct_coordsys([r*cos(theta)*cos(phi),r*cos(theta)*sin(phi),
      r*sin(theta),[r,theta,phi]]);
(%o2)                                done
(%i3) lg:trigsimp(lg);
                           [ 1  0         0        ]
                           [                       ]
                           [     2                 ]
(%o3)                      [ 0  r         0        ]
                           [                       ]
                           [         2    2        ]
                           [ 0  0   r  cos (theta) ]
(%i4) ct_coords;
(%o4)                           [r, theta, phi]
(%i5) dim;
(%o5)                                  3

Transformation functions can also be used when cframe_flag is true:


(%i1) load("ctensor");
(%o1)       /share/tensor/ctensor.mac
(%i2) cframe_flag:true;
(%o2)                                true
(%i3) ct_coordsys([r*cos(theta)*cos(phi),r*cos(theta)*sin(phi),
      r*sin(theta),[r,theta,phi]]);
(%o3)                                done
(%i4) fri;
(%o4)
 [cos(phi)cos(theta) -cos(phi) r sin(theta) -sin(phi) r cos(theta)]
 [                                                                ]
 [sin(phi)cos(theta) -sin(phi) r sin(theta)  cos(phi) r cos(theta)]
 [                                                                ]
 [    sin(theta)           r cos(theta)                0          ]

(%i5) cmetric();
(%o5)                                false
(%i6) lg:trigsimp(lg);
                           [ 1  0         0        ]
                           [                       ]
                           [     2                 ]
(%o6)                      [ 0  r         0        ]
                           [                       ]
                           [         2    2        ]
                           [ 0  0   r  cos (theta) ]

The optional argument extra_arg can be any one of the following:

cylindrical tells ct_coordsys to attach an additional cylindrical coordinate.

minkowski tells ct_coordsys to attach an additional coordinate with negative metric signature.

all tells ct_coordsys to call cmetric and christof(false) after setting up the metric.

If the global variable verbose is set to true, ct_coordsys displays the values of dim, ct_coords, and either lg or lfg and fri, depending on the value of cframe_flag.

Function: init_ctensor ()

Initializes the ctensor package.

The init_ctensor function reinitializes the ctensor package. It removes all arrays and matrices used by ctensor, resets all flags, resets dim to 4, and resets the frame metric to the Lorentz-frame.

26.2.2 The tensors of curved space

The main purpose of the ctensor package is to compute the tensors of curved space(time), most notably the tensors used in general relativity.

When a metric base is used, ctensor can compute the following tensors:


 lg  -- ug
   \      \
    lcs -- mcs -- ric -- uric
              \      \       \
               \      tracer - ein -- lein
                \
                 riem -- lriem -- weyl
                     \
                      uriem


ctensor can also work using moving frames. When cframe_flag is set to true, the following tensors can be calculated:


 lfg -- ufg
     \
 fri -- fr -- lcs -- mcs -- lriem -- ric -- uric
      \                       |  \      \       \
       lg -- ug               |   weyl   tracer - ein -- lein
                              |\
                              | riem
                              |
                              \uriem

Function: christof (dis)

A function in the ctensor (component tensor) package. It computes the Christoffel symbols of both kinds. The argument dis determines which results are to be immediately displayed. The Christoffel symbols of the first and second kinds are stored in the arrays lcs[i,j,k] and mcs[i,j,k] respectively and defined to be symmetric in the first two indices. If the argument to christof is lcs or mcs then the unique non-zero values of lcs[i,j,k] or mcs[i,j,k], respectively, will be displayed. If the argument is all then the unique non-zero values of lcs[i,j,k] and mcs[i,j,k] will be displayed. If the argument is false then the display of the elements will not occur. The array elements mcs[i,j,k] are defined in such a manner that the final index is contravariant.

Function: ricci (dis)

A function in the ctensor (component tensor) package. ricci computes the covariant (symmetric) components ric[i,j] of the Ricci tensor. If the argument dis is true, then the non-zero components are displayed.

Function: uricci (dis)

This function first computes the covariant components ric[i,j] of the Ricci tensor. Then the mixed Ricci tensor is computed using the contravariant metric tensor. If the value of the argument dis is true, then these mixed components, uric[i,j] (the index i is covariant and the index j is contravariant), will be displayed directly. Otherwise, ricci(false) will simply compute the entries of the array uric[i,j] without displaying the results.

Function: scurvature ()

Returns the scalar curvature (obtained by contracting the Ricci tensor) of the Riemannian manifold with the given metric.

Function: einstein (dis)

A function in the ctensor (component tensor) package. einstein computes the mixed Einstein tensor after the Christoffel symbols and Ricci tensor have been obtained (with the functions christof and ricci). If the argument dis is true, then the non-zero values of the mixed Einstein tensor ein[i,j] will be displayed where j is the contravariant index. The variable rateinstein will cause the rational simplification on these components. If ratfac is true then the components will also be factored.

Function: leinstein (dis)

Covariant Einstein-tensor. leinstein stores the values of the covariant Einstein tensor in the array lein. The covariant Einstein-tensor is computed from the mixed Einstein tensor ein by multiplying it with the metric tensor. If the argument dis is true, then the non-zero values of the covariant Einstein tensor are displayed.

Function: riemann (dis)

A function in the ctensor (component tensor) package. riemann computes the Riemann curvature tensor from the given metric and the corresponding Christoffel symbols. The following index conventions are used:

                l      _l       _l       _l   _m    _l   _m
 R[i,j,k,l] =  R    = |      - |      + |    |   - |    |
                ijk     ij,k     ik,j     mk   ij    mj   ik

This notation is consistent with the notation used by the itensor package and its icurvature function. If the optional argument dis is true, the unique non-zero components riem[i,j,k,l] will be displayed. As with the Einstein tensor, various switches set by the user control the simplification of the components of the Riemann tensor. If ratriemann is true, then rational simplification will be done. If ratfac is true then each of the components will also be factored.

If the variable cframe_flag is false, the Riemann tensor is computed directly from the Christoffel-symbols. If cframe_flag is true, the covariant Riemann-tensor is computed first from the frame field coefficients.

Function: lriemann (dis)

Covariant Riemann-tensor (lriem[]).

Computes the covariant Riemann-tensor as the array lriem. If the argument dis is true, unique non-zero values are displayed.

If the variable cframe_flag is true, the covariant Riemann tensor is computed directly from the frame field coefficients. Otherwise, the (3,1) Riemann tensor is computed first.

For information on index ordering, see riemann.

Function: uriemann (dis)

Computes the contravariant components of the Riemann curvature tensor as array elements uriem[i,j,k,l]. These are displayed if dis is true.

Function: rinvariant ()

Forms the Kretchmann-invariant (kinvariant) obtained by contracting the tensors

lriem[i,j,k,l]*uriem[i,j,k,l].

This object is not automatically simplified since it can be very large.

Function: weyl (dis)

Computes the Weyl conformal tensor. If the argument dis is true, the non-zero components weyl[i,j,k,l] will be displayed to the user. Otherwise, these components will simply be computed and stored. If the switch ratweyl is set to true, then the components will be rationally simplified; if ratfac is true then the results will be factored as well.

26.2.3 Taylor series expansion

The ctensor package has the ability to truncate results by assuming that they are Taylor-series approximations. This behavior is controlled by the ctayswitch variable; when set to true, ctensor makes use internally of the function ctaylor when simplifying results.

The ctaylor function is invoked by the following ctensor functions:


    Function     Comments
    ---------------------------------
    christof()   For mcs only
    ricci()
    uricci()
    einstein()
    riemann()
    weyl()
    checkdiv()
Function: ctaylor ()

The ctaylor function truncates its argument by converting it to a Taylor-series using taylor, and then calling ratdisrep. This has the combined effect of dropping terms higher order in the expansion variable ctayvar. The order of terms that should be dropped is defined by ctaypov; the point around which the series expansion is carried out is specified in ctaypt.

As an example, consider a simple metric that is a perturbation of the Minkowski metric. Without further restrictions, even a diagonal metric produces expressions for the Einstein tensor that are far too complex:


(%i1) load("ctensor");
(%o1)       /share/tensor/ctensor.mac
(%i2) ratfac:true;
(%o2)                                true
(%i3) derivabbrev:true;
(%o3)                                true
(%i4) ct_coords:[t,r,theta,phi];
(%o4)                         [t, r, theta, phi]
(%i5) lg:matrix([-1,0,0,0],[0,1,0,0],[0,0,r^2,0],
                [0,0,0,r^2*sin(theta)^2]);
                        [ - 1  0  0         0        ]
                        [                            ]
                        [  0   1  0         0        ]
                        [                            ]
(%o5)                   [          2                 ]
                        [  0   0  r         0        ]
                        [                            ]
                        [              2    2        ]
                        [  0   0  0   r  sin (theta) ]
(%i6) h:matrix([h11,0,0,0],[0,h22,0,0],[0,0,h33,0],[0,0,0,h44]);
                            [ h11   0    0    0  ]
                            [                    ]
                            [  0   h22   0    0  ]
(%o6)                       [                    ]
                            [  0    0   h33   0  ]
                            [                    ]
                            [  0    0    0   h44 ]
(%i7) depends(l,r);
(%o7)                               [l(r)]
(%i8) lg:lg+l*h;
      [ h11 l - 1      0          0                 0            ]
      [                                                          ]
      [     0      h22 l + 1      0                 0            ]
      [                                                          ]
(%o8) [                        2                                 ]
      [     0          0      r  + h33 l            0            ]
      [                                                          ]
      [                                    2    2                ]
      [     0          0          0       r  sin (theta) + h44 l ]
(%i9) cmetric(false);
(%o9)                                done
(%i10) einstein(false);
(%o10)                               done
(%i11) ntermst(ein);
[[1, 1], 62]
[[1, 2], 0]
[[1, 3], 0]
[[1, 4], 0]
[[2, 1], 0]
[[2, 2], 24]
[[2, 3], 0]
[[2, 4], 0]
[[3, 1], 0]
[[3, 2], 0]
[[3, 3], 46]
[[3, 4], 0]
[[4, 1], 0]
[[4, 2], 0]
[[4, 3], 0]
[[4, 4], 46]
(%o12)                               done

However, if we recompute this example as an approximation that is linear in the variable l, we get much simpler expressions:


(%i14) ctayswitch:true;
(%o14)                               true
(%i15) ctayvar:l;
(%o15)                                 l
(%i16) ctaypov:1;
(%o16)                                 1
(%i17) ctaypt:0;
(%o17)                                 0
(%i18) christof(false);
(%o18)                               done
(%i19) ricci(false);
(%o19)                               done
(%i20) einstein(false);
(%o20)                               done
(%i21) ntermst(ein);
[[1, 1], 6]
[[1, 2], 0]
[[1, 3], 0]
[[1, 4], 0]
[[2, 1], 0]
[[2, 2], 13]
[[2, 3], 2]
[[2, 4], 0]
[[3, 1], 0]
[[3, 2], 2]
[[3, 3], 9]
[[3, 4], 0]
[[4, 1], 0]
[[4, 2], 0]
[[4, 3], 0]
[[4, 4], 9]
(%o21)                               done
(%i22) ratsimp(ein[1,1]);
                         2      2  4               2     2
(%o22) - (((h11 h22 - h11 ) (l )  r  - 2 h33 l    r ) sin (theta)
                              r               r r

                            2               2      4    2
              - 2 h44 l    r  - h33 h44 (l ) )/(4 r  sin (theta))
                       r r                r



This capability can be useful, for instance, when working in the weak field limit far from a gravitational source.

26.2.4 Frame fields

When the variable cframe_flag is set to true, the ctensor package performs its calculations using a moving frame.

Function: frame_bracket (fr, fri, diagframe)

The frame bracket (fb[]).

Computes the frame bracket according to the following definition:

   c          c         c        d     e
ifb   = ( ifri    - ifri    ) ifr   ifr
   ab         d,e       e,d      a     b

26.2.5 Algebraic classification

A new feature (as of November, 2004) of ctensor is its ability to compute the Petrov classification of a 4-dimensional spacetime metric. For a demonstration of this capability, see the file share/tensor/petrov.dem.

Function: nptetrad ()

Computes a Newman-Penrose null tetrad (np) and its raised-index counterpart (npi). See petrov for an example.

The null tetrad is constructed on the assumption that a four-dimensional orthonormal frame metric with metric signature (-,+,+,+) is being used. The components of the null tetrad are related to the inverse frame matrix as follows:


np  = (fri  + fri ) / sqrt(2)
  1       1      2

np  = (fri  - fri ) / sqrt(2)
  2       1      2

np  = (fri  + %i fri ) / sqrt(2)
  3       3         4

np  = (fri  - %i fri ) / sqrt(2)
  4       3         4

Function: psi (dis)

Computes the five Newman-Penrose coefficients psi[0]...psi[4]. If dis is set to true, the coefficients are displayed. See petrov for an example.

These coefficients are computed from the Weyl-tensor in a coordinate base. If a frame base is used, the Weyl-tensor is first converted to a coordinate base, which can be a computationally expensive procedure. For this reason, in some cases it may be more advantageous to use a coordinate base in the first place before the Weyl tensor is computed. Note however, that constructing a Newman-Penrose null tetrad requires a frame base. Therefore, a meaningful computation sequence may begin with a frame base, which is then used to compute lg (computed automatically by cmetric) and then ug. See petrov for an example. At this point, you can switch back to a coordinate base by setting cframe_flag to false before beginning to compute the Christoffel symbols. Changing to a frame base at a later stage could yield inconsistent results, as you may end up with a mixed bag of tensors, some computed in a frame base, some in a coordinate base, with no means to distinguish between the two.

Function: petrov ()

Computes the Petrov classification of the metric characterized by psi[0]...psi[4].

For example, the following demonstrates how to obtain the Petrov-classification of the Kerr metric:

(%i1) load("ctensor");
(%o1)       /share/tensor/ctensor.mac
(%i2) (cframe_flag:true,gcd:spmod,ctrgsimp:true,ratfac:true);
(%o2)                                true
(%i3) ct_coordsys(exteriorschwarzschild,all);
(%o3)                                done
(%i4) ug:invert(lg)$
(%i5) weyl(false);
(%o5)                                done
(%i6) nptetrad(true);
(%t6) np =

[ sqrt(r - 2 m)           sqrt(r)                                 ]
[---------------   ---------------------    0            0        ]
[sqrt(2) sqrt(r)   sqrt(2) sqrt(r - 2 m)                          ]
[                                                                 ]
[ sqrt(r - 2 m)            sqrt(r)                                ]
[---------------  - ---------------------   0            0        ]
[sqrt(2) sqrt(r)    sqrt(2) sqrt(r - 2 m)                         ]
[                                                                 ]
[                                          r      %i r sin(theta) ]
[       0                    0          -------   --------------- ]
[                                       sqrt(2)       sqrt(2)     ]
[                                                                 ]
[                                          r       %i r sin(theta)]
[       0                    0          -------  - ---------------]
[                                       sqrt(2)        sqrt(2)    ]

                             sqrt(r)         sqrt(r - 2 m)
(%t7) npi = matrix([- ---------------------,---------------, 0, 0],
                      sqrt(2) sqrt(r - 2 m) sqrt(2) sqrt(r)

          sqrt(r)            sqrt(r - 2 m)
[- ---------------------, - ---------------, 0, 0],
   sqrt(2) sqrt(r - 2 m)    sqrt(2) sqrt(r)

           1               %i
[0, 0, ---------, --------------------],
       sqrt(2) r  sqrt(2) r sin(theta)

           1                 %i
[0, 0, ---------, - --------------------])
       sqrt(2) r    sqrt(2) r sin(theta)

(%o7)                                done
(%i7) psi(true);
(%t8)                              psi  = 0
                                      0

(%t9)                              psi  = 0
                                      1

                                          m
(%t10)                             psi  = --
                                      2    3
                                          r

(%t11)                             psi  = 0
                                      3

(%t12)                             psi  = 0
                                      4
(%o12)                               done
(%i12) petrov();
(%o12)                                 D

The Petrov classification function is based on the algorithm published in "Classifying geometries in general relativity: III Classification in practice" by Pollney, Skea, and d’Inverno, Class. Quant. Grav. 17 2885-2902 (2000). Except for some simple test cases, the implementation is untested as of December 19, 2004, and is likely to contain errors.

26.2.6 Torsion and nonmetricity

ctensor has the ability to compute and include torsion and nonmetricity coefficients in the connection coefficients.

The torsion coefficients are calculated from a user-supplied tensor tr, which should be a rank (2,1) tensor. From this, the torsion coefficients kt are computed according to the following formulae:


              m          m      m
       - g  tr   - g   tr   - tr   g
          im  kj    jm   ki     ij  km
kt   = -------------------------------
  ijk                 2


  k     km
kt   = g   kt
  ij         ijm

Note that only the mixed-index tensor is calculated and stored in the array kt.

The nonmetricity coefficients are calculated from the user-supplied nonmetricity vector nm. From this, the nonmetricity coefficients nmc are computed as follows:


             k    k        km
       -nm  D  - D  nm  + g   nm  g
   k      i  j    i   j         m  ij
nmc  = ------------------------------
   ij                2

where D stands for the Kronecker-delta.

When ctorsion_flag is set to true, the values of kt are subtracted from the mixed-indexed connection coefficients computed by christof and stored in mcs. Similarly, if cnonmet_flag is set to true, the values of nmc are subtracted from the mixed-indexed connection coefficients.

If necessary, christof calls the functions contortion and nonmetricity in order to compute kt and nm.

Function: contortion (tr)

Computes the (2,1) contortion coefficients from the torsion tensor tr.

Function: nonmetricity (nm)

Computes the (2,1) nonmetricity coefficients from the nonmetricity vector nm.

26.2.7 Miscellaneous features

Function: ctransform (M)

A function in the ctensor (component tensor) package which will perform a coordinate transformation upon an arbitrary square symmetric matrix M. The user must input the functions which define the transformation. (Formerly called transform.)

Function: findde (A, n)

returns a list of the unique differential equations (expressions) corresponding to the elements of the n dimensional square array A. Presently, n may be 2 or 3. deindex is a global list containing the indices of A corresponding to these unique differential equations. For the Einstein tensor (ein), which is a two dimensional array, if computed for the metric in the example below, findde gives the following independent differential equations:

(%i1) load("ctensor");
(%o1)       /share/tensor/ctensor.mac
(%i2) derivabbrev:true;
(%o2)                                true
(%i3) dim:4;
(%o3)                                  4
(%i4) lg:matrix([a, 0, 0, 0], [ 0, x^2, 0, 0],
                              [0, 0, x^2*sin(y)^2, 0], [0,0,0,-d]);
                          [ a  0       0        0  ]
                          [                        ]
                          [     2                  ]
                          [ 0  x       0        0  ]
(%o4)                     [                        ]
                          [         2    2         ]
                          [ 0  0   x  sin (y)   0  ]
                          [                        ]
                          [ 0  0       0       - d ]
(%i5) depends([a,d],x);
(%o5)                            [a(x), d(x)]
(%i6) ct_coords:[x,y,z,t];
(%o6)                            [x, y, z, t]
(%i7) cmetric();
(%o7)                                done
(%i8) einstein(false);
(%o8)                                done
(%i9) findde(ein,2);
                                            2
(%o9) [d  x - a d + d, 2 a d d    x - a (d )  x - a  d d  x
        x                     x x         x        x    x

                                              2          2
                          + 2 a d d   - 2 a  d , a  x + a  - a]
                                   x       x      x
(%i10) deindex;
(%o10)                     [[1, 1], [2, 2], [4, 4]]

Function: cograd ()

Computes the covariant gradient of a scalar function allowing the user to choose the corresponding vector name as the example under contragrad illustrates.

Function: contragrad ()

Computes the contravariant gradient of a scalar function allowing the user to choose the corresponding vector name as the example below for the Schwarzschild metric illustrates:

(%i1) load("ctensor");
(%o1)       /share/tensor/ctensor.mac
(%i2) derivabbrev:true;
(%o2)                                true
(%i3) ct_coordsys(exteriorschwarzschild,all);
(%o3)                                done
(%i4) depends(f,r);
(%o4)                               [f(r)]
(%i5) cograd(f,g1);
(%o5)                                done
(%i6) listarray(g1);
(%o6)                            [0, f , 0, 0]
                                      r
(%i7) contragrad(f,g2);
(%o7)                                done
(%i8) listarray(g2);
                               f  r - 2 f  m
                                r        r
(%o8)                      [0, -------------, 0, 0]
                                     r

Function: dscalar ()

computes the tensor d’Alembertian of the scalar function once dependencies have been declared upon the function. For example:

(%i1) load("ctensor");
(%o1)       /share/tensor/ctensor.mac
(%i2) derivabbrev:true;
(%o2)                                true
(%i3) ct_coordsys(exteriorschwarzschild,all);
(%o3)                                done
(%i4) depends(p,r);
(%o4)                               [p(r)]
(%i5) factor(dscalar(p));
                          2
                    p    r  - 2 m p    r + 2 p  r - 2 m p
                     r r           r r        r          r
(%o5)               --------------------------------------
                                       2
                                      r

Function: checkdiv ()

computes the covariant divergence of the mixed second rank tensor (whose first index must be covariant) by printing the corresponding n components of the vector field (the divergence) where n = dim. If the argument to the function is g then the divergence of the Einstein tensor will be formed and must be zero. In addition, the divergence (vector) is given the array name div.

Function: cgeodesic (dis)

A function in the ctensor (component tensor) package. cgeodesic computes the geodesic equations of motion for a given metric. They are stored in the array geod[i]. If the argument dis is true then these equations are displayed.

Function: bdvac (f)

generates the covariant components of the vacuum field equations of the Brans- Dicke gravitational theory. The scalar field is specified by the argument f, which should be a (quoted) function name with functional dependencies, e.g., 'p(x).

The components of the second rank covariant field tensor are represented by the array bd.

Function: invariant1 ()

generates the mixed Euler- Lagrange tensor (field equations) for the invariant density of R^2. The field equations are the components of an array named inv1.

Function: invariant2 ()

*** NOT YET IMPLEMENTED ***

generates the mixed Euler- Lagrange tensor (field equations) for the invariant density of ric[i,j]*uriem[i,j]. The field equations are the components of an array named inv2.

Function: bimetric ()

*** NOT YET IMPLEMENTED ***

generates the field equations of Rosen’s bimetric theory. The field equations are the components of an array named rosen.

26.2.8 Utility functions

Function: diagmatrixp (M,n)

Returns true if the first n rows and n columns of M form a diagonal matrix or (2D) array.

Function: symmetricp (M, n)

Returns true if M is a n by n symmetric matrix or two-dimensional array, otherwise false.

If n is less than the size of M, symmetricp considers only the n by n submatrix (respectively, subarray) comprising rows 1 through n and columns 1 through n.

Function: ntermst (f)

gives the user a quick picture of the "size" of the doubly subscripted tensor (array) f. It prints two element lists where the second element corresponds to NTERMS of the components specified by the first elements. In this way, it is possible to quickly find the non-zero expressions and attempt simplification.

Function: cdisplay (ten)

displays all the elements of the tensor ten, as represented by a multidimensional array. Tensors of rank 0 and 1, as well as other types of variables, are displayed as with ldisplay. Tensors of rank 2 are displayed as 2-dimensional matrices, while tensors of higher rank are displayed as a list of 2-dimensional matrices. For instance, the Riemann-tensor of the Schwarzschild metric can be viewed as:

(%i1) load("ctensor");
(%o1)       /share/tensor/ctensor.mac
(%i2) ratfac:true;
(%o2)                                true
(%i3) ct_coordsys(exteriorschwarzschild,all);
(%o3)                                done
(%i4) riemann(false);
(%o4)                                done
(%i5) cdisplay(riem);
          [ 0               0                   0           0     ]
          [                                                       ]
          [                              2                        ]
          [      3 m (r - 2 m)   m    2 m                         ]
          [ 0  - ------------- + -- - ----      0           0     ]
          [            4          3     4                         ]
          [           r          r     r                          ]
          [                                                       ]
riem    = [                                m (r - 2 m)            ]
    1, 1  [ 0               0              -----------      0     ]
          [                                     4                 ]
          [                                    r                  ]
          [                                                       ]
          [                                           m (r - 2 m) ]
          [ 0               0                   0     ----------- ]
          [                                                4      ]
          [                                               r       ]

                                [    2 m (r - 2 m)       ]
                                [ 0  -------------  0  0 ]
                                [          4             ]
                                [         r              ]
                     riem     = [                        ]
                         1, 2   [ 0        0        0  0 ]
                                [                        ]
                                [ 0        0        0  0 ]
                                [                        ]
                                [ 0        0        0  0 ]

                                [         m (r - 2 m)    ]
                                [ 0  0  - -----------  0 ]
                                [              4         ]
                                [             r          ]
                     riem     = [                        ]
                         1, 3   [ 0  0        0        0 ]
                                [                        ]
                                [ 0  0        0        0 ]
                                [                        ]
                                [ 0  0        0        0 ]

                                [            m (r - 2 m) ]
                                [ 0  0  0  - ----------- ]
                                [                 4      ]
                                [                r       ]
                     riem     = [                        ]
                         1, 4   [ 0  0  0        0       ]
                                [                        ]
                                [ 0  0  0        0       ]
                                [                        ]
                                [ 0  0  0        0       ]

                               [       0         0  0  0 ]
                               [                         ]
                               [       2 m               ]
                               [ - ------------  0  0  0 ]
                    riem     = [    2                    ]
                        2, 1   [   r  (r - 2 m)          ]
                               [                         ]
                               [       0         0  0  0 ]
                               [                         ]
                               [       0         0  0  0 ]

             [     2 m                                         ]
             [ ------------  0        0               0        ]
             [  2                                              ]
             [ r  (r - 2 m)                                    ]
             [                                                 ]
             [      0        0        0               0        ]
             [                                                 ]
  riem     = [                         m                       ]
      2, 2   [      0        0  - ------------        0        ]
             [                     2                           ]
             [                    r  (r - 2 m)                 ]
             [                                                 ]
             [                                         m       ]
             [      0        0        0         - ------------ ]
             [                                     2           ]
             [                                    r  (r - 2 m) ]

                                [ 0  0       0        0 ]
                                [                       ]
                                [            m          ]
                                [ 0  0  ------------  0 ]
                     riem     = [        2              ]
                         2, 3   [       r  (r - 2 m)    ]
                                [                       ]
                                [ 0  0       0        0 ]
                                [                       ]
                                [ 0  0       0        0 ]

                                [ 0  0  0       0       ]
                                [                       ]
                                [               m       ]
                                [ 0  0  0  ------------ ]
                     riem     = [           2           ]
                         2, 4   [          r  (r - 2 m) ]
                                [                       ]
                                [ 0  0  0       0       ]
                                [                       ]
                                [ 0  0  0       0       ]

                                      [ 0  0  0  0 ]
                                      [            ]
                                      [ 0  0  0  0 ]
                                      [            ]
                           riem     = [ m          ]
                               3, 1   [ -  0  0  0 ]
                                      [ r          ]
                                      [            ]
                                      [ 0  0  0  0 ]

                                      [ 0  0  0  0 ]
                                      [            ]
                                      [ 0  0  0  0 ]
                                      [            ]
                           riem     = [    m       ]
                               3, 2   [ 0  -  0  0 ]
                                      [    r       ]
                                      [            ]
                                      [ 0  0  0  0 ]

                               [   m                      ]
                               [ - -   0   0       0      ]
                               [   r                      ]
                               [                          ]
                               [        m                 ]
                               [  0   - -  0       0      ]
                    riem     = [        r                 ]
                        3, 3   [                          ]
                               [  0    0   0       0      ]
                               [                          ]
                               [              2 m - r     ]
                               [  0    0   0  ------- + 1 ]
                               [                 r        ]

                                    [ 0  0  0    0   ]
                                    [                ]
                                    [ 0  0  0    0   ]
                                    [                ]
                         riem     = [            2 m ]
                             3, 4   [ 0  0  0  - --- ]
                                    [             r  ]
                                    [                ]
                                    [ 0  0  0    0   ]

                                [       0        0  0  0 ]
                                [                        ]
                                [       0        0  0  0 ]
                                [                        ]
                     riem     = [       0        0  0  0 ]
                         4, 1   [                        ]
                                [      2                 ]
                                [ m sin (theta)          ]
                                [ -------------  0  0  0 ]
                                [       r                ]

                                [ 0        0        0  0 ]
                                [                        ]
                                [ 0        0        0  0 ]
                                [                        ]
                     riem     = [ 0        0        0  0 ]
                         4, 2   [                        ]
                                [         2              ]
                                [    m sin (theta)       ]
                                [ 0  -------------  0  0 ]
                                [          r             ]

                              [ 0  0          0          0 ]
                              [                            ]
                              [ 0  0          0          0 ]
                              [                            ]
                   riem     = [ 0  0          0          0 ]
                       4, 3   [                            ]
                              [                2           ]
                              [         2 m sin (theta)    ]
                              [ 0  0  - ---------------  0 ]
                              [                r           ]

           [        2                                             ]
           [   m sin (theta)                                      ]
           [ - -------------         0                0         0 ]
           [         r                                            ]
           [                                                      ]
           [                         2                            ]
           [                    m sin (theta)                     ]
riem     = [        0         - -------------         0         0 ]
    4, 4   [                          r                           ]
           [                                                      ]
           [                                          2           ]
           [                                   2 m sin (theta)    ]
           [        0                0         ---------------  0 ]
           [                                          r           ]
           [                                                      ]
           [        0                0                0         0 ]

(%o5)                                done

Function: deleten (L, n)

Returns a new list consisting of L with the n’th element deleted.

26.2.9 Variables used by ctensor

Option variable: dim

Default value: 4

An option in the ctensor (component tensor) package. dim is the dimension of the manifold with the default 4. The command dim: n will reset the dimension to any other value n.

Option variable: diagmetric

Default value: false

An option in the ctensor (component tensor) package. If diagmetric is true special routines compute all geometrical objects (which contain the metric tensor explicitly) by taking into consideration the diagonality of the metric. Reduced run times will, of course, result. Note: this option is set automatically by csetup if a diagonal metric is specified.

Option variable: ctrgsimp

Causes trigonometric simplifications to be used when tensors are computed. Presently, ctrgsimp affects only computations involving a moving frame.

Option variable: cframe_flag

Causes computations to be performed relative to a moving frame as opposed to a holonomic metric. The frame is defined by the inverse frame array fri and the frame metric lfg. For computations using a Cartesian frame, lfg should be the unit matrix of the appropriate dimension; for computations in a Lorentz frame, lfg should have the appropriate signature.

Option variable: ctorsion_flag

Causes the contortion tensor to be included in the computation of the connection coefficients. The contortion tensor itself is computed by contortion from the user-supplied tensor tr.

Option variable: cnonmet_flag

Causes the nonmetricity coefficients to be included in the computation of the connection coefficients. The nonmetricity coefficients are computed from the user-supplied nonmetricity vector nm by the function nonmetricity.

Option variable: ctayswitch

If set to true, causes some ctensor computations to be carried out using Taylor-series expansions. Presently, christof, ricci, uricci, einstein, and weyl take into account this setting.

Option variable: ctayvar

Variable used for Taylor-series expansion if ctayswitch is set to true.

Option variable: ctaypov

Maximum power used in Taylor-series expansion when ctayswitch is set to true.

Option variable: ctaypt

Point around which Taylor-series expansion is carried out when ctayswitch is set to true.

System variable: gdet

The determinant of the metric tensor lg. Computed by cmetric when cframe_flag is set to false.

Option variable: ratchristof

Causes rational simplification to be applied by christof.

Option variable: rateinstein

Default value: true

If true rational simplification will be performed on the non-zero components of Einstein tensors; if ratfac is true then the components will also be factored.

Option variable: ratriemann

Default value: true

One of the switches which controls simplification of Riemann tensors; if true, then rational simplification will be done; if ratfac is true then each of the components will also be factored.

Option variable: ratweyl

Default value: true

If true, this switch causes the weyl function to apply rational simplification to the values of the Weyl tensor. If ratfac is true, then the components will also be factored.

Variable: lfg

The covariant frame metric. By default, it is initialized to the 4-dimensional Lorentz frame with signature (+,+,+,-). Used when cframe_flag is true.

Variable: ufg

The inverse frame metric. Computed from lfg when cmetric is called while cframe_flag is set to true.

Variable: riem

The (3,1) Riemann tensor. Computed when the function riemann is invoked. For information about index ordering, see the description of riemann.

If cframe_flag is true, riem is computed from the covariant Riemann-tensor lriem.

Variable: lriem

The covariant Riemann tensor. Computed by lriemann.

Variable: uriem

The contravariant Riemann tensor. Computed by uriemann.

Variable: ric

The covariant Ricci-tensor. Computed by ricci.

Variable: uric

The mixed-index Ricci-tensor. Computed by uricci.

Variable: lg

The metric tensor. This tensor must be specified (as a dim by dim matrix) before other computations can be performed.

Variable: ug

The inverse of the metric tensor. Computed by cmetric.

Variable: weyl

The Weyl tensor. Computed by weyl.

Variable: fb

Frame bracket coefficients, as computed by frame_bracket.

Variable: kinvariant

The Kretchmann invariant. Computed by rinvariant.

Variable: np

A Newman-Penrose null tetrad. Computed by nptetrad.

Variable: npi

The raised-index Newman-Penrose null tetrad. Computed by nptetrad. Defined as ug.np. The product np.transpose(npi) is constant:

(%i39) trigsimp(np.transpose(npi));
                              [  0   - 1  0  0 ]
                              [                ]
                              [ - 1   0   0  0 ]
(%o39)                        [                ]
                              [  0    0   0  1 ]
                              [                ]
                              [  0    0   1  0 ]

Variable: tr

User-supplied rank-3 tensor representing torsion. Used by contortion.

Variable: kt

The contortion tensor, computed from tr by contortion.

Variable: nm

User-supplied nonmetricity vector. Used by nonmetricity.

Variable: nmc

The nonmetricity coefficients, computed from nm by nonmetricity.

System variable: tensorkill

Variable indicating if the tensor package has been initialized. Set and used by csetup, reset by init_ctensor.

Option variable: ct_coords

Default value: []

An option in the ctensor (component tensor) package. ct_coords contains a list of coordinates. While normally defined when the function csetup is called, one may redefine the coordinates with the assignment ct_coords: [j1, j2, ..., jn] where the j’s are the new coordinate names. See also csetup.

26.2.10 Reserved names

The following names are used internally by the ctensor package and should not be redefined:

  Name         Description
  ---------------------------------------------------------------------
  _lg()        Evaluates to lfg if frame metric used, lg otherwise
  _ug()        Evaluates to ufg if frame metric used, ug otherwise
  cleanup()    Removes items drom the deindex list
  contract4()  Used by psi()
  filemet()    Used by csetup() when reading the metric from a file
  findde1()    Used by findde()
  findde2()    Used by findde()
  findde3()    Used by findde()
  kdelt()      Kronecker-delta (not generalized)
  newmet()     Used by csetup() for setting up a metric interactively
  setflags()   Used by init_ctensor()
  readvalue()
  resimp()
  sermet()     Used by csetup() for entering a metric as Taylor-series
  txyzsum()
  tmetric()    Frame metric, used by cmetric() when cframe_flag:true
  triemann()   Riemann-tensor in frame base, used when cframe_flag:true
  tricci()     Ricci-tensor in frame base, used when cframe_flag:true
  trrc()       Ricci rotation coefficients, used by christof()
  yesp()

26.2.11 Changes

In November, 2004, the ctensor package was extensively rewritten. Many functions and variables have been renamed in order to make the package compatible with the commercial version of Macsyma.

  New Name     Old Name        Description
  ---------------------------------------------------------------------
  ctaylor()    DLGTAYLOR()     Taylor-series expansion of an expression
  lgeod[]      EM              Geodesic equations
  ein[]        G[]             Mixed Einstein-tensor
  ric[]        LR[]            Mixed Ricci-tensor
  ricci()      LRICCICOM()     Compute the mixed Ricci-tensor
  ctaypov      MINP            Maximum power in Taylor-series expansion
  cgeodesic()  MOTION          Compute geodesic equations
  ct_coords    OMEGA           Metric coordinates
  ctayvar      PARAM           Taylor-series expansion variable
  lriem[]      R[]             Covariant Riemann-tensor
  uriemann()   RAISERIEMANN()  Compute the contravariant Riemann-tensor
  ratriemann   RATRIEMAN       Rational simplif. of the Riemann-tensor
  uric[]       RICCI[]         Contravariant Ricci-tensor
  uricci()     RICCICOM()      Compute the contravariant Ricci-tensor
  cmetric()    SETMETRIC()     Set up the metric
  ctaypt       TAYPT           Point for Taylor-series expansion
  ctayswitch   TAYSWITCH       Taylor-series setting switch
  csetup()     TSETUP()        Start interactive setup session
  ctransform() TTRANSFORM()    Interactive coordinate transformation
  uriem[]      UR[]            Contravariant Riemann-tensor
  weyl[]       W[]             (3,1) Weyl-tensor


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27 atensor


Next: , Previous: , Up: atensor   [Contents][Index]

27.1 Introduction to atensor

atensor is an algebraic tensor manipulation package. To use atensor, type load("atensor"), followed by a call to the init_atensor function.

The essence of atensor is a set of simplification rules for the noncommutative (dot) product operator ("."). atensor recognizes several algebra types; the corresponding simplification rules are put into effect when the init_atensor function is called.

The capabilities of atensor can be demonstrated by defining the algebra of quaternions as a Clifford-algebra Cl(0,2) with two basis vectors. The three quaternionic imaginary units are then the two basis vectors and their product, i.e.:

    i = v     j = v     k = v  . v
         1         2         1    2

Although the atensor package has a built-in definition for the quaternion algebra, it is not used in this example, in which we endeavour to build the quaternion multiplication table as a matrix:

(%i1) load("atensor");
(%o1)       /share/tensor/atensor.mac
(%i2) init_atensor(clifford,0,0,2);
(%o2)                                done
(%i3) atensimp(v[1].v[1]);
(%o3)                                 - 1
(%i4) atensimp((v[1].v[2]).(v[1].v[2]));
(%o4)                                 - 1
(%i5) q:zeromatrix(4,4);
                                [ 0  0  0  0 ]
                                [            ]
                                [ 0  0  0  0 ]
(%o5)                           [            ]
                                [ 0  0  0  0 ]
                                [            ]
                                [ 0  0  0  0 ]
(%i6) q[1,1]:1;
(%o6)                                  1
(%i7) for i thru adim do q[1,i+1]:q[i+1,1]:v[i];
(%o7)                                done
(%i8) q[1,4]:q[4,1]:v[1].v[2];
(%o8)                               v  . v
                                     1    2
(%i9) for i from 2 thru 4 do for j from 2 thru 4 do
      q[i,j]:atensimp(q[i,1].q[1,j]);
(%o9)                                done
(%i10) q;
                   [    1        v         v      v  . v  ]
                   [              1         2      1    2 ]
                   [                                      ]
                   [   v         - 1     v  . v    - v    ]
                   [    1                 1    2      2   ]
(%o10)             [                                      ]
                   [   v      - v  . v     - 1      v     ]
                   [    2        1    2              1    ]
                   [                                      ]
                   [ v  . v      v        - v       - 1   ]
                   [  1    2      2          1            ]

atensor recognizes as base vectors indexed symbols, where the symbol is that stored in asymbol and the index runs between 1 and adim. For indexed symbols, and indexed symbols only, the bilinear forms sf, af, and av are evaluated. The evaluation substitutes the value of aform[i,j] in place of fun(v[i],v[j]) where v represents the value of asymbol and fun is either af or sf; or, it substitutes v[aform[i,j]] in place of av(v[i],v[j]).

Needless to say, the functions sf, af and av can be redefined.

When the atensor package is loaded, the following flags are set:

dotscrules:true;
dotdistrib:true;
dotexptsimp:false;

If you wish to experiment with a nonassociative algebra, you may also consider setting dotassoc to false. In this case, however, atensimp will not always be able to obtain the desired simplifications.

·

Category: Tensors · Category: Share packages · Category: Package atensor


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27.2 Functions and Variables for atensor

Function: init_atensor
    init_atensor (alg_type, opt_dims)
    init_atensor (alg_type)

Initializes the atensor package with the specified algebra type. alg_type can be one of the following:

universal: The universal algebra has no commutation rules.

grassmann: The Grassman algebra is defined by the commutation relation u.v+v.u=0.

clifford: The Clifford algebra is defined by the commutation relation u.v+v.u=-2*sf(u,v) where sf is a symmetric scalar-valued function. For this algebra, opt_dims can be up to three nonnegative integers, representing the number of positive, degenerate, and negative dimensions of the algebra, respectively. If any opt_dims values are supplied, atensor will configure the values of adim and aform appropriately. Otherwise, adim will default to 0 and aform will not be defined.

symmetric: The symmetric algebra is defined by the commutation relation u.v-v.u=0.

symplectic: The symplectic algebra is defined by the commutation relation u.v-v.u=2*af(u,v) where af is an antisymmetric scalar-valued function. For the symplectic algebra, opt_dims can be up to two nonnegative integers, representing the nondegenerate and degenerate dimensions, respectively. If any opt_dims values are supplied, atensor will configure the values of adim and aform appropriately. Otherwise, adim will default to 0 and aform will not be defined.

lie_envelop: The algebra of the Lie envelope is defined by the commutation relation u.v-v.u=2*av(u,v) where av is an antisymmetric function.

The init_atensor function also recognizes several predefined algebra types:

complex implements the algebra of complex numbers as the Clifford algebra Cl(0,1). The call init_atensor(complex) is equivalent to init_atensor(clifford,0,0,1).

quaternion implements the algebra of quaternions. The call init_atensor (quaternion) is equivalent to init_atensor (clifford,0,0,2).

pauli implements the algebra of Pauli-spinors as the Clifford-algebra Cl(3,0). A call to init_atensor(pauli) is equivalent to init_atensor(clifford,3).

dirac implements the algebra of Dirac-spinors as the Clifford-algebra Cl(3,1). A call to init_atensor(dirac) is equivalent to init_atensor(clifford,3,0,1).

Function: atensimp (expr)

Simplifies an algebraic tensor expression expr according to the rules configured by a call to init_atensor. Simplification includes recursive application of commutation relations and resolving calls to sf, af, and av where applicable. A safeguard is used to ensure that the function always terminates, even for complex expressions.

Function: alg_type

The algebra type. Valid values are universal, grassmann, clifford, symmetric, symplectic and lie_envelop.

Variable: adim

Default value: 0

The dimensionality of the algebra. atensor uses the value of adim to determine if an indexed object is a valid base vector. See abasep.

Variable: aform

Default value: ident(3)

Default values for the bilinear forms sf, af, and av. The default is the identity matrix ident(3).

Variable: asymbol

Default value: v

The symbol for base vectors.

Function: sf (u, v)

A symmetric scalar function that is used in commutation relations. The default implementation checks if both arguments are base vectors using abasep and if that is the case, substitutes the corresponding value from the matrix aform.

Function: af (u, v)

An antisymmetric scalar function that is used in commutation relations. The default implementation checks if both arguments are base vectors using abasep and if that is the case, substitutes the corresponding value from the matrix aform.

Function: av (u, v)

An antisymmetric function that is used in commutation relations. The default implementation checks if both arguments are base vectors using abasep and if that is the case, substitutes the corresponding value from the matrix aform.

For instance:

(%i1) load("atensor");
(%o1)       /share/tensor/atensor.mac
(%i2) adim:3;
(%o2)                                  3
(%i3) aform:matrix([0,3,-2],[-3,0,1],[2,-1,0]);
                               [  0    3   - 2 ]
                               [               ]
(%o3)                          [ - 3   0    1  ]
                               [               ]
                               [  2   - 1   0  ]
(%i4) asymbol:x;
(%o4)                                  x
(%i5) av(x[1],x[2]);
(%o5)                                 x
                                       3

Function: abasep (v)

Checks if its argument is an atensor base vector. That is, if it is an indexed symbol, with the symbol being the same as the value of asymbol, and the index having a numeric value between 1 and adim.


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28 Sums, Products, and Series


Next: , Previous: , Up: Sums Products and Series   [Contents][Index]

28.1 Functions and Variables for Sums and Products

Function: bashindices (expr)

Transforms the expression expr by giving each summation and product a unique index. This gives changevar greater precision when it is working with summations or products. The form of the unique index is jnumber. The quantity number is determined by referring to gensumnum, which can be changed by the user. For example, gensumnum:0$ resets it.

Function: lsum (expr, x, L)

Represents the sum of expr for each element x in L. A noun form 'lsum is returned if the argument L does not evaluate to a list.

Examples:

(%i1) lsum (x^i, i, [1, 2, 7]);
                            7    2
(%o1)                      x  + x  + x
(%i2) lsum (i^2, i, rootsof (x^3 - 1, x));
                     ====
                     \      2
(%o2)                 >    i
                     /
                     ====
                                   3
                     i in rootsof(x  - 1, x)

Function: intosum (expr)

Moves multiplicative factors outside a summation to inside. If the index is used in the outside expression, then the function tries to find a reasonable index, the same as it does for sumcontract. This is essentially the reverse idea of the outative property of summations, but note that it does not remove this property, it only bypasses it.

In some cases, a scanmap (multthru, expr) may be necessary before the intosum.

Option variable: simpproduct

Default value: false

When simpproduct is true, the result of a product is simplified. This simplification may sometimes be able to produce a closed form. If simpproduct is false or if the quoted form 'product is used, the value is a product noun form which is a representation of the pi notation used in mathematics.

Function: product (expr, i, i_0, i_1)

Represents a product of the values of expr as the index i varies from i_0 to i_1. The noun form 'product is displayed as an uppercase letter pi.

product evaluates expr and lower and upper limits i_0 and i_1, product quotes (does not evaluate) the index i.

If the upper and lower limits differ by an integer, expr is evaluated for each value of the index i, and the result is an explicit product.

Otherwise, the range of the index is indefinite. Some rules are applied to simplify the product. When the global variable simpproduct is true, additional rules are applied. In some cases, simplification yields a result which is not a product; otherwise, the result is a noun form 'product.

See also nouns and evflag.

Examples:

(%i1) product (x + i*(i+1)/2, i, 1, 4);
(%o1)           (x + 1) (x + 3) (x + 6) (x + 10)
(%i2) product (i^2, i, 1, 7);
(%o2)                       25401600
(%i3) product (a[i], i, 1, 7);
(%o3)                 a  a  a  a  a  a  a
                       1  2  3  4  5  6  7
(%i4) product (a(i), i, 1, 7);
(%o4)          a(1) a(2) a(3) a(4) a(5) a(6) a(7)
(%i5) product (a(i), i, 1, n);
                             n
                           /===\
                            ! !
(%o5)                       ! !  a(i)
                            ! !
                           i = 1
(%i6) product (k, k, 1, n);
                               n
                             /===\
                              ! !
(%o6)                         ! !  k
                              ! !
                             k = 1
(%i7) product (k, k, 1, n), simpproduct;
(%o7)                          n!
(%i8) product (integrate (x^k, x, 0, 1), k, 1, n);
                             n
                           /===\
                            ! !    1
(%o8)                       ! !  -----
                            ! !  k + 1
                           k = 1
(%i9) product (if k <= 5 then a^k else b^k, k, 1, 10);
                              15  40
(%o9)                        a   b

Option variable: simpsum

Default value: false

When simpsum is true, the result of a sum is simplified. This simplification may sometimes be able to produce a closed form. If simpsum is false or if the quoted form 'sum is used, the value is a sum noun form which is a representation of the sigma notation used in mathematics.

Function: sum (expr, i, i_0, i_1)

Represents a summation of the values of expr as the index i varies from i_0 to i_1. The noun form 'sum is displayed as an uppercase letter sigma.

sum evaluates its summand expr and lower and upper limits i_0 and i_1, sum quotes (does not evaluate) the index i.

If the upper and lower limits differ by an integer, the summand expr is evaluated for each value of the summation index i, and the result is an explicit sum.

Otherwise, the range of the index is indefinite. Some rules are applied to simplify the summation. When the global variable simpsum is true, additional rules are applied. In some cases, simplification yields a result which is not a summation; otherwise, the result is a noun form 'sum.

When the evflag (evaluation flag) cauchysum is true, a product of summations is expressed as a Cauchy product, in which the index of the inner summation is a function of the index of the outer one, rather than varying independently.

The global variable genindex is the alphabetic prefix used to generate the next index of summation, when an automatically generated index is needed.

gensumnum is the numeric suffix used to generate the next index of summation, when an automatically generated index is needed. When gensumnum is false, an automatically-generated index is only genindex with no numeric suffix.

See also lsum, sumcontract, intosum, bashindices, niceindices, nouns, evflag, and zeilberger-pkg

Examples:

(%i1) sum (i^2, i, 1, 7);
(%o1)                          140
(%i2) sum (a[i], i, 1, 7);
(%o2)           a  + a  + a  + a  + a  + a  + a
                 7    6    5    4    3    2    1
(%i3) sum (a(i), i, 1, 7);
(%o3)    a(7) + a(6) + a(5) + a(4) + a(3) + a(2) + a(1)
(%i4) sum (a(i), i, 1, n);
                            n
                           ====
                           \
(%o4)                       >    a(i)
                           /
                           ====
                           i = 1
(%i5) sum (2^i + i^2, i, 0, n);
                          n
                         ====
                         \       i    2
(%o5)                     >    (2  + i )
                         /
                         ====
                         i = 0
(%i6) sum (2^i + i^2, i, 0, n), simpsum;
                              3      2
                   n + 1   2 n  + 3 n  + n
(%o6)             2      + --------------- - 1
                                  6
(%i7) sum (1/3^i, i, 1, inf);
                            inf
                            ====
                            \     1
(%o7)                        >    --
                            /      i
                            ====  3
                            i = 1
(%i8) sum (1/3^i, i, 1, inf), simpsum;
                                1
(%o8)                           -
                                2
(%i9) sum (i^2, i, 1, 4) * sum (1/i^2, i, 1, inf);
                              inf
                              ====
                              \     1
(%o9)                      30  >    --
                              /      2
                              ====  i
                              i = 1
(%i10) sum (i^2, i, 1, 4) * sum (1/i^2, i, 1, inf), simpsum;
                                  2
(%o10)                       5 %pi
(%i11) sum (integrate (x^k, x, 0, 1), k, 1, n);
                            n
                           ====
                           \       1
(%o11)                      >    -----
                           /     k + 1
                           ====
                           k = 1
(%i12) sum (if k <= 5 then a^k else b^k, k, 1, 10);
          10    9    8    7    6    5    4    3    2
(%o12)   b   + b  + b  + b  + b  + a  + a  + a  + a  + a

Function: sumcontract (expr)

Combines all sums of an addition that have upper and lower bounds that differ by constants. The result is an expression containing one summation for each set of such summations added to all appropriate extra terms that had to be extracted to form this sum. sumcontract combines all compatible sums and uses one of the indices from one of the sums if it can, and then try to form a reasonable index if it cannot use any supplied.

It may be necessary to do an intosum (expr) before the sumcontract.

Option variable: sumexpand

Default value: false

When sumexpand is true, products of sums and exponentiated sums simplify to nested sums.

See also cauchysum.

Examples:

(%i1) sumexpand: true$
(%i2) sum (f (i), i, 0, m) * sum (g (j), j, 0, n);
                     m      n
                    ====   ====
                    \      \
(%o2)                >      >     f(i1) g(i2)
                    /      /
                    ====   ====
                    i1 = 0 i2 = 0
(%i3) sum (f (i), i, 0, m)^2;
                     m      m
                    ====   ====
                    \      \
(%o3)                >      >     f(i3) f(i4)
                    /      /
                    ====   ====
                    i3 = 0 i4 = 0


Next: , Previous: , Up: Sums Products and Series   [Contents][Index]

28.2 Introduction to Series

Maxima contains functions taylor and powerseries for finding the series of differentiable functions. It also has tools such as nusum capable of finding the closed form of some series. Operations such as addition and multiplication work as usual on series. This section presents the global variables which control the expansion.


Next: , Previous: , Up: Sums Products and Series   [Contents][Index]

28.3 Functions and Variables for Series

Option variable: cauchysum

Default value: false

When multiplying together sums with inf as their upper limit, if sumexpand is true and cauchysum is true then the Cauchy product will be used rather than the usual product. In the Cauchy product the index of the inner summation is a function of the index of the outer one rather than varying independently.

Example:

(%i1) sumexpand: false$
(%i2) cauchysum: false$
(%i3) s: sum (f(i), i, 0, inf) * sum (g(j), j, 0, inf);
                      inf         inf
                      ====        ====
                      \           \
(%o3)                ( >    f(i))  >    g(j)
                      /           /
                      ====        ====
                      i = 0       j = 0
(%i4) sumexpand: true$
(%i5) cauchysum: true$
(%i6) expand(s,0,0);
                 inf     i1
                 ====   ====
                 \      \
(%o6)             >      >     g(i1 - i2) f(i2)
                 /      /
                 ====   ====
                 i1 = 0 i2 = 0

Function: deftaylor (f_1(x_1), expr_1, …, f_n(x_n), expr_n)

For each function f_i of one variable x_i, deftaylor defines expr_i as the Taylor series about zero. expr_i is typically a polynomial in x_i or a summation; more general expressions are accepted by deftaylor without complaint.

powerseries (f_i(x_i), x_i, 0) returns the series defined by deftaylor.

deftaylor returns a list of the functions f_1, …, f_n. deftaylor evaluates its arguments.

Example:

(%i1) deftaylor (f(x), x^2 + sum(x^i/(2^i*i!^2), i, 4, inf));
(%o1)                          [f]
(%i2) powerseries (f(x), x, 0);
                      inf
                      ====      i1
                      \        x         2
(%o2)                  >     -------- + x
                      /       i1    2
                      ====   2   i1!
                      i1 = 4
(%i3) taylor (exp (sqrt (f(x))), x, 0, 4);
                      2         3          4
                     x    3073 x    12817 x
(%o3)/T/     1 + x + -- + ------- + -------- + . . .
                     2     18432     307200

Option variable: maxtayorder

Default value: true

When maxtayorder is true, then during algebraic manipulation of (truncated) Taylor series, taylor tries to retain as many terms as are known to be correct.

Function: niceindices (expr)

Renames the indices of sums and products in expr. niceindices attempts to rename each index to the value of niceindicespref[1], unless that name appears in the summand or multiplicand, in which case niceindices tries the succeeding elements of niceindicespref in turn, until an unused variable is found. If the entire list is exhausted, additional indices are constructed by appending integers to the value of niceindicespref[1], e.g., i0, i1, i2, …

niceindices returns an expression. niceindices evaluates its argument.

Example:

(%i1) niceindicespref;
(%o1)                  [i, j, k, l, m, n]
(%i2) product (sum (f (foo + i*j*bar), foo, 1, inf), bar, 1, inf);
                 inf    inf
                /===\   ====
                 ! !    \
(%o2)            ! !     >      f(bar i j + foo)
                 ! !    /
                bar = 1 ====
                        foo = 1
(%i3) niceindices (%);
                     inf  inf
                    /===\ ====
                     ! !  \
(%o3)                ! !   >    f(i j l + k)
                     ! !  /
                    l = 1 ====
                          k = 1

Option variable: niceindicespref

Default value: [i, j, k, l, m, n]

niceindicespref is the list from which niceindices takes the names of indices for sums and products.

The elements of niceindicespref are typically names of variables, although that is not enforced by niceindices.

Example:

(%i1) niceindicespref: [p, q, r, s, t, u]$
(%i2) product (sum (f (foo + i*j*bar), foo, 1, inf), bar, 1, inf);
                 inf    inf
                /===\   ====
                 ! !    \
(%o2)            ! !     >      f(bar i j + foo)
                 ! !    /
                bar = 1 ====
                        foo = 1
(%i3) niceindices (%);
                     inf  inf
                    /===\ ====
                     ! !  \
(%o3)                ! !   >    f(i j q + p)
                     ! !  /
                    q = 1 ====
                          p = 1

Function: nusum (expr, x, i_0, i_1)

Carries out indefinite hypergeometric summation of expr with respect to x using a decision procedure due to R.W. Gosper. expr and the result must be expressible as products of integer powers, factorials, binomials, and rational functions.

The terms "definite" and "indefinite summation" are used analogously to "definite" and "indefinite integration". To sum indefinitely means to give a symbolic result for the sum over intervals of variable length, not just e.g. 0 to inf. Thus, since there is no formula for the general partial sum of the binomial series, nusum can’t do it.

nusum and unsum know a little about sums and differences of finite products. See also unsum.

Examples:

(%i1) nusum (n*n!, n, 0, n);

Dependent equations eliminated:  (1)
(%o1)                     (n + 1)! - 1
(%i2) nusum (n^4*4^n/binomial(2*n,n), n, 0, n);
                     4        3       2              n
      2 (n + 1) (63 n  + 112 n  + 18 n  - 22 n + 3) 4      2
(%o2) ------------------------------------------------ - ------
                    693 binomial(2 n, n)                 3 11 7
(%i3) unsum (%, n);
                              4  n
                             n  4
(%o3)                   ----------------
                        binomial(2 n, n)
(%i4) unsum (prod (i^2, i, 1, n), n);
                    n - 1
                    /===\
                     ! !   2
(%o4)              ( ! !  i ) (n - 1) (n + 1)
                     ! !
                    i = 1
(%i5) nusum (%, n, 1, n);

Dependent equations eliminated:  (2 3)
                            n
                          /===\
                           ! !   2
(%o5)                      ! !  i  - 1
                           ! !
                          i = 1

Function: pade (taylor_series, numer_deg_bound, denom_deg_bound)

Returns a list of all rational functions which have the given Taylor series expansion where the sum of the degrees of the numerator and the denominator is less than or equal to the truncation level of the power series, i.e. are "best" approximants, and which additionally satisfy the specified degree bounds.

taylor_series is a univariate Taylor series. numer_deg_bound and denom_deg_bound are positive integers specifying degree bounds on the numerator and denominator.

taylor_series can also be a Laurent series, and the degree bounds can be inf which causes all rational functions whose total degree is less than or equal to the length of the power series to be returned. Total degree is defined as numer_deg_bound + denom_deg_bound. Length of a power series is defined as "truncation level" + 1 - min(0, "order of series").

(%i1) taylor (1 + x + x^2 + x^3, x, 0, 3);
                              2    3
(%o1)/T/             1 + x + x  + x  + . . .
(%i2) pade (%, 1, 1);
                                 1
(%o2)                       [- -----]
                               x - 1
(%i3) t: taylor(-(83787*x^10 - 45552*x^9 - 187296*x^8
                   + 387072*x^7 + 86016*x^6 - 1507328*x^5
                   + 1966080*x^4 + 4194304*x^3 - 25165824*x^2
                   + 67108864*x - 134217728)
       /134217728, x, 0, 10);
                    2    3       4       5       6        7
             x   3 x    x    15 x    23 x    21 x    189 x
(%o3)/T/ 1 - - + ---- - -- - ----- + ----- - ----- - ------
             2    16    32   1024    2048    32768   65536

                                  8         9          10
                            5853 x    2847 x    83787 x
                          + ------- + ------- - --------- + . . .
                            4194304   8388608   134217728
(%i4) pade (t, 4, 4);
(%o4)                          []

There is no rational function of degree 4 numerator/denominator, with this power series expansion. You must in general have degree of the numerator and degree of the denominator adding up to at least the degree of the power series, in order to have enough unknown coefficients to solve.

(%i5) pade (t, 5, 5);
                     5                4                 3
(%o5) [- (520256329 x  - 96719020632 x  - 489651410240 x

                  2
 - 1619100813312 x  - 2176885157888 x - 2386516803584)

               5                 4                  3
/(47041365435 x  + 381702613848 x  + 1360678489152 x

                  2
 + 2856700692480 x  + 3370143559680 x + 2386516803584)]

Function: powerseries (expr, x, a)

Returns the general form of the power series expansion for expr in the variable x about the point a (which may be inf for infinity):

           inf
           ====
           \               n
            >    b  (x - a)
           /      n
           ====
           n = 0

If powerseries is unable to expand expr, taylor may give the first several terms of the series.

When verbose is true, powerseries prints progress messages.

(%i1) verbose: true$
(%i2) powerseries (log(sin(x)/x), x, 0);
can't expand 
                                 log(sin(x))
so we'll try again after applying the rule:
                                        d
                                      / -- (sin(x))
                                      [ dx
                        log(sin(x)) = i ----------- dx
                                      ]   sin(x)
                                      /
in the first simplification we have returned:
                             /
                             [
                             i cot(x) dx - log(x)
                             ]
                             /
                    inf
                    ====        i1  2 i1             2 i1
                    \      (- 1)   2     bern(2 i1) x
                     >     ------------------------------
                    /                i1 (2 i1)!
                    ====
                    i1 = 1
(%o2)                -------------------------------------
                                      2

Option variable: psexpand

Default value: false

When psexpand is true, an extended rational function expression is displayed fully expanded. The switch ratexpand has the same effect.

When psexpand is false, a multivariate expression is displayed just as in the rational function package.

When psexpand is multi, then terms with the same total degree in the variables are grouped together.

Function: revert (expr, x)
Function: revert2 (expr, x, n)

These functions return the reversion of expr, a Taylor series about zero in the variable x. revert returns a polynomial of degree equal to the highest power in expr. revert2 returns a polynomial of degree n, which may be greater than, equal to, or less than the degree of expr.

load ("revert") loads these functions.

Examples:

(%i1) load ("revert")$
(%i2) t: taylor (exp(x) - 1, x, 0, 6);
                   2    3    4    5     6
                  x    x    x    x     x
(%o2)/T/      x + -- + -- + -- + --- + --- + . . .
                  2    6    24   120   720
(%i3) revert (t, x);
               6       5       4       3       2
           10 x  - 12 x  + 15 x  - 20 x  + 30 x  - 60 x
(%o3)/R/ - --------------------------------------------
                                60
(%i4) ratexpand (%);
                     6    5    4    3    2
                    x    x    x    x    x
(%o4)             - -- + -- - -- + -- - -- + x
                    6    5    4    3    2
(%i5) taylor (log(x+1), x, 0, 6);
                    2    3    4    5    6
                   x    x    x    x    x
(%o5)/T/       x - -- + -- - -- + -- - -- + . . .
                   2    3    4    5    6
(%i6) ratsimp (revert (t, x) - taylor (log(x+1), x, 0, 6));
(%o6)                           0
(%i7) revert2 (t, x, 4);
                          4    3    2
                         x    x    x
(%o7)                  - -- + -- - -- + x
                         4    3    2

Function: taylor
    taylor (expr, x, a, n)
    taylor (expr, [x_1, x_2, …], a, n)
    taylor (expr, [x, a, n, 'asymp])
    taylor (expr, [x_1, x_2, …], [a_1, a_2, …], [n_1, n_2, …])
    taylor (expr, [x_1, a_1, n_1], [x_2, a_2, n_2], …)

taylor (expr, x, a, n) expands the expression expr in a truncated Taylor or Laurent series in the variable x around the point a, containing terms through (x - a)^n.

If expr is of the form f(x)/g(x) and g(x) has no terms up to degree n then taylor attempts to expand g(x) up to degree 2 n. If there are still no nonzero terms, taylor doubles the degree of the expansion of g(x) so long as the degree of the expansion is less than or equal to n 2^taylordepth.

taylor (expr, [x_1, x_2, ...], a, n) returns a truncated power series of degree n in all variables x_1, x_2, … about the point (a, a, ...).

taylor (expr, [x_1, a_1, n_1], [x_2, a_2, n_2], ...) returns a truncated power series in the variables x_1, x_2, … about the point (a_1, a_2, ...), truncated at n_1, n_2, …

taylor (expr, [x_1, x_2, ...], [a_1, a_2, ...], [n_1, n_2, ...]) returns a truncated power series in the variables x_1, x_2, … about the point (a_1, a_2, ...), truncated at n_1, n_2, …

taylor (expr, [x, a, n, 'asymp]) returns an expansion of expr in negative powers of x - a. The highest order term is (x - a)^-n.

When maxtayorder is true, then during algebraic manipulation of (truncated) Taylor series, taylor tries to retain as many terms as are known to be correct.

When psexpand is true, an extended rational function expression is displayed fully expanded. The switch ratexpand has the same effect. When psexpand is false, a multivariate expression is displayed just as in the rational function package. When psexpand is multi, then terms with the same total degree in the variables are grouped together.

See also the taylor_logexpand switch for controlling expansion.

Examples:

(%i1) taylor (sqrt (sin(x) + a*x + 1), x, 0, 3);
                           2             2
             (a + 1) x   (a  + 2 a + 1) x
(%o1)/T/ 1 + --------- - -----------------
                 2               8

                                   3      2             3
                               (3 a  + 9 a  + 9 a - 1) x
                             + -------------------------- + . . .
                                           48
(%i2) %^2;
                                    3
                                   x
(%o2)/T/           1 + (a + 1) x - -- + . . .
                                   6
(%i3) taylor (sqrt (x + 1), x, 0, 5);
                       2    3      4      5
                  x   x    x    5 x    7 x
(%o3)/T/      1 + - - -- + -- - ---- + ---- + . . .
                  2   8    16   128    256
(%i4) %^2;
(%o4)/T/                  1 + x + . . .
(%i5) product ((1 + x^i)^2.5, i, 1, inf)/(1 + x^2);
                         inf
                        /===\
                         ! !    i     2.5
                         ! !  (x  + 1)
                         ! !
                        i = 1
(%o5)                   -----------------
                              2
                             x  + 1
(%i6) ev (taylor(%, x,  0, 3), keepfloat);
                               2           3
(%o6)/T/    1 + 2.5 x + 3.375 x  + 6.5625 x  + . . .
(%i7) taylor (1/log (x + 1), x, 0, 3);
                               2       3
                 1   1   x    x    19 x
(%o7)/T/         - + - - -- + -- - ----- + . . .
                 x   2   12   24    720
(%i8) taylor (cos(x) - sec(x), x, 0, 5);
                                4
                           2   x
(%o8)/T/                - x  - -- + . . .
                               6
(%i9) taylor ((cos(x) - sec(x))^3, x, 0, 5);
(%o9)/T/                    0 + . . .
(%i10) taylor (1/(cos(x) - sec(x))^3, x, 0, 5);
                                               2          4
            1     1       11      347    6767 x    15377 x
(%o10)/T/ - -- + ---- + ------ - ----- - ------- - --------
             6      4        2   15120   604800    7983360
            x    2 x    120 x

                                                          + . . .
(%i11) taylor (sqrt (1 - k^2*sin(x)^2), x, 0, 6);
               2  2       4      2   4
              k  x    (3 k  - 4 k ) x
(%o11)/T/ 1 - ----- - ----------------
                2            24

                                    6       4       2   6
                               (45 k  - 60 k  + 16 k ) x
                             - -------------------------- + . . .
                                          720
(%i12) taylor ((x + 1)^n, x, 0, 4);
                      2       2     3      2         3
                    (n  - n) x    (n  - 3 n  + 2 n) x
(%o12)/T/ 1 + n x + ----------- + --------------------
                         2                 6

                               4      3       2         4
                             (n  - 6 n  + 11 n  - 6 n) x
                           + ---------------------------- + . . .
                                          24
(%i13) taylor (sin (y + x), x, 0, 3, y, 0, 3);
               3                 2
              y                 y
(%o13)/T/ y - -- + . . . + (1 - -- + . . .) x
              6                 2

                    3                       2
               y   y            2      1   y            3
          + (- - + -- + . . .) x  + (- - + -- + . . .) x  + . . .
               2   12                  6   12
(%i14) taylor (sin (y + x), [x, y], 0, 3);
                     3        2      2      3
                    x  + 3 y x  + 3 y  x + y
(%o14)/T/   y + x - ------------------------- + . . .
                                6
(%i15) taylor (1/sin (y + x), x, 0, 3, y, 0, 3);
          1   y              1    1               1            2
(%o15)/T/ - + - + . . . + (- -- + - + . . .) x + (-- + . . .) x
          y   6               2   6                3
                             y                    y

                                           1            3
                                      + (- -- + . . .) x  + . . .
                                            4
                                           y
(%i16) taylor (1/sin (y + x), [x, y], 0, 3);
                             3         2       2        3
            1     x + y   7 x  + 21 y x  + 21 y  x + 7 y
(%o16)/T/ ----- + ----- + ------------------------------- + . . .
          x + y     6                   360

Option variable: taylordepth

Default value: 3

If there are still no nonzero terms, taylor doubles the degree of the expansion of g(x) so long as the degree of the expansion is less than or equal to n 2^taylordepth.

Function: taylorinfo (expr)

Returns information about the Taylor series expr. The return value is a list of lists. Each list comprises the name of a variable, the point of expansion, and the degree of the expansion.

taylorinfo returns false if expr is not a Taylor series.

Example:

(%i1) taylor ((1 - y^2)/(1 - x), x, 0, 3, [y, a, inf]);
                  2                       2
(%o1)/T/ - (y - a)  - 2 a (y - a) + (1 - a )

         2                        2
 + (1 - a  - 2 a (y - a) - (y - a) ) x

         2                        2   2
 + (1 - a  - 2 a (y - a) - (y - a) ) x

         2                        2   3
 + (1 - a  - 2 a (y - a) - (y - a) ) x  + . . .
(%i2) taylorinfo(%);
(%o2)               [[y, a, inf], [x, 0, 3]]

Function: taylorp (expr)

Returns true if expr is a Taylor series, and false otherwise.

Option variable: taylor_logexpand

Default value: true

taylor_logexpand controls expansions of logarithms in taylor series.

When taylor_logexpand is true, all logarithms are expanded fully so that zero-recognition problems involving logarithmic identities do not disturb the expansion process. However, this scheme is not always mathematically correct since it ignores branch information.

When taylor_logexpand is set to false, then the only expansion of logarithms that occur is that necessary to obtain a formal power series.

Option variable: taylor_order_coefficients

Default value: true

taylor_order_coefficients controls the ordering of coefficients in a Taylor series.

When taylor_order_coefficients is true, coefficients of taylor series are ordered canonically.

Function: taylor_simplifier (expr)

Simplifies coefficients of the power series expr. taylor calls this function.

Option variable: taylor_truncate_polynomials

Default value: true

When taylor_truncate_polynomials is true, polynomials are truncated based upon the input truncation levels.

Otherwise, polynomials input to taylor are considered to have infinite precison.

Function: taytorat (expr)

Converts expr from taylor form to canonical rational expression (CRE) form. The effect is the same as rat (ratdisrep (expr)), but faster.

Function: trunc (expr)

Annotates the internal representation of the general expression expr so that it is displayed as if its sums were truncated Taylor series. expr is not otherwise modified.

Example:

(%i1) expr: x^2 + x + 1;
                            2
(%o1)                      x  + x + 1
(%i2) trunc (expr);
                                2
(%o2)                  1 + x + x  + . . .
(%i3) is (expr = trunc (expr));
(%o3)                         true

Function: unsum (f, n)

Returns the first backward difference f(n) - f(n - 1). Thus unsum in a sense is the inverse of sum.

See also nusum.

Examples:

(%i1) g(p) := p*4^n/binomial(2*n,n);
                                     n
                                  p 4
(%o1)               g(p) := ----------------
                            binomial(2 n, n)
(%i2) g(n^4);
                              4  n
                             n  4
(%o2)                   ----------------
                        binomial(2 n, n)
(%i3) nusum (%, n, 0, n);
                     4        3       2              n
      2 (n + 1) (63 n  + 112 n  + 18 n  - 22 n + 3) 4      2
(%o3) ------------------------------------------------ - ------
                    693 binomial(2 n, n)                 3 11 7
(%i4) unsum (%, n);
                              4  n
                             n  4
(%o4)                   ----------------
                        binomial(2 n, n)

Option variable: verbose

Default value: false

When verbose is true, powerseries prints progress messages.


Next: , Previous: , Up: Sums Products and Series   [Contents][Index]

28.4 Introduction to Fourier series

The fourie package comprises functions for the symbolic computation of Fourier series. There are functions in the fourie package to calculate Fourier integral coefficients and some functions for manipulation of expressions.

·

Category: Fourier transform · Category: Share packages · Category: Package fourie


Next: , Previous: , Up: Sums Products and Series   [Contents][Index]

28.5 Functions and Variables for Fourier series

Function: equalp (x, y)

Returns true if equal (x, y) otherwise false (doesn’t give an error message like equal (x, y) would do in this case).

Function: remfun
    remfun (f, expr)
    remfun (f, expr, x)

remfun (f, expr) replaces all occurrences of f (arg) by arg in expr.

remfun (f, expr, x) replaces all occurrences of f (arg) by arg in expr only if arg contains the variable x.

Function: funp
    funp (f, expr)
    funp (f, expr, x)

funp (f, expr) returns true if expr contains the function f.

funp (f, expr, x) returns true if expr contains the function f and the variable x is somewhere in the argument of one of the instances of f.

Function: absint
    absint (f, x, halfplane)
    absint (f, x)
    absint (f, x, a, b)

absint (f, x, halfplane) returns the indefinite integral of f with respect to x in the given halfplane (pos, neg, or both). f may contain expressions of the form abs (x), abs (sin (x)), abs (a) * exp (-abs (b) * abs (x)).

absint (f, x) is equivalent to absint (f, x, pos).

absint (f, x, a, b) returns the definite integral of f with respect to x from a to b. f may include absolute values.

Function: fourier (f, x, p)

Returns a list of the Fourier coefficients of f(x) defined on the interval [-p, p].

Function: foursimp (l)

Simplifies sin (n %pi) to 0 if sinnpiflag is true and cos (n %pi) to (-1)^n if cosnpiflag is true.

Option variable: sinnpiflag

Default value: true

See foursimp.

Option variable: cosnpiflag

Default value: true

See foursimp.

Function: fourexpand (l, x, p, limit)

Constructs and returns the Fourier series from the list of Fourier coefficients l up through limit terms (limit may be inf). x and p have same meaning as in fourier.

Function: fourcos (f, x, p)

Returns the Fourier cosine coefficients for f(x) defined on [0, p].

Function: foursin (f, x, p)

Returns the Fourier sine coefficients for f(x) defined on [0, p].

Function: totalfourier (f, x, p)

Returns fourexpand (foursimp (fourier (f, x, p)), x, p, 'inf).

Function: fourint (f, x)

Constructs and returns a list of the Fourier integral coefficients of f(x) defined on [minf, inf].

Function: fourintcos (f, x)

Returns the Fourier cosine integral coefficients for f(x) on [0, inf].

Function: fourintsin (f, x)

Returns the Fourier sine integral coefficients for f(x) on [0, inf].


Previous: , Up: Sums Products and Series   [Contents][Index]

28.6 Functions and Variables for Poisson series

Function: intopois (a)

Converts a into a Poisson encoding.

Function: outofpois (a)

Converts a from Poisson encoding to general representation. If a is not in Poisson form, outofpois carries out the conversion, i.e., the return value is outofpois (intopois (a)). This function is thus a canonical simplifier for sums of powers of sine and cosine terms of a particular type.

Function: poisdiff (a, b)

Differentiates a with respect to b. b must occur only in the trig arguments or only in the coefficients.

Function: poisexpt (a, b)

Functionally identical to intopois (a^b). b must be a positive integer.

Function: poisint (a, b)

Integrates in a similarly restricted sense (to poisdiff). Non-periodic terms in b are dropped if b is in the trig arguments.

Option variable: poislim

Default value: 5

poislim determines the domain of the coefficients in the arguments of the trig functions. The initial value of 5 corresponds to the interval [-2^(5-1)+1,2^(5-1)], or [-15,16], but it can be set to [-2^(n-1)+1, 2^(n-1)].

Function: poismap (series, sinfn, cosfn)

will map the functions sinfn on the sine terms and cosfn on the cosine terms of the Poisson series given. sinfn and cosfn are functions of two arguments which are a coefficient and a trigonometric part of a term in series respectively.

Function: poisplus (a, b)

Is functionally identical to intopois (a + b).

Function: poissimp (a)

Converts a into a Poisson series for a in general representation.

Special symbol: poisson

The symbol /P/ follows the line label of Poisson series expressions.

Function: poissubst (a, b, c)

Substitutes a for b in c. c is a Poisson series.

(1) Where B is a variable u, v, w, x, y, or z, then a must be an expression linear in those variables (e.g., 6*u + 4*v).

(2) Where b is other than those variables, then a must also be free of those variables, and furthermore, free of sines or cosines.

poissubst (a, b, c, d, n) is a special type of substitution which operates on a and b as in type (1) above, but where d is a Poisson series, expands cos(d) and sin(d) to order n so as to provide the result of substituting a + d for b in c. The idea is that d is an expansion in terms of a small parameter. For example, poissubst (u, v, cos(v), %e, 3) yields cos(u)*(1 - %e^2/2) - sin(u)*(%e - %e^3/6).

Function: poistimes (a, b)

Is functionally identical to intopois (a*b).

Function: poistrim ()

is a reserved function name which (if the user has defined it) gets applied during Poisson multiplication. It is a predicate function of 6 arguments which are the coefficients of the u, v, ..., z in a term. Terms for which poistrim is true (for the coefficients of that term) are eliminated during multiplication.

Function: printpois (a)

Prints a Poisson series in a readable format. In common with outofpois, it will convert a into a Poisson encoding first, if necessary.


Next: , Previous: , Up: Top   [Contents][Index]

29 Number Theory


Previous: , Up: Number Theory   [Contents][Index]

29.1 Functions and Variables for Number Theory

Function: bern (n)

Returns the n’th Bernoulli number for integer n. Bernoulli numbers equal to zero are suppressed if zerobern is false.

See also burn.

(%i1) zerobern: true$
(%i2) map (bern, [0, 1, 2, 3, 4, 5, 6, 7, 8]);
                      1  1       1      1        1
(%o2)           [1, - -, -, 0, - --, 0, --, 0, - --]
                      2  6       30     42       30
(%i3) zerobern: false$
(%i4) map (bern, [0, 1, 2, 3, 4, 5, 6, 7, 8]);
                      1  1    1   1     1   5     691   7
(%o4)           [1, - -, -, - --, --, - --, --, - ----, -]
                      2  6    30  42    30  66    2730  6

Function: bernpoly (x, n)

Returns the n’th Bernoulli polynomial in the variable x.

Function: bfzeta (s, n)

Returns the Riemann zeta function for the argument s. The return value is a big float (bfloat); n is the number of digits in the return value.

Function: bfhzeta (s, h, n)

Returns the Hurwitz zeta function for the arguments s and h. The return value is a big float (bfloat); n is the number of digits in the return value.

The Hurwitz zeta function is defined as

                        inf
                        ====
                        \        1
         zeta (s,h)  =   >    --------
                        /            s
                        ====  (k + h)
                        k = 0

load ("bffac") loads this function.

Function: burn (n)

Returns a rational number, which is an approximation of the n’th Bernoulli number for integer n. burn exploits the observation that (rational) Bernoulli numbers can be approximated by (transcendental) zetas with tolerable efficiency:

                   n - 1  1 - 2 n
              (- 1)      2        zeta(2 n) (2 n)!
     B(2 n) = ------------------------------------
                                2 n
                             %pi

burn may be more efficient than bern for large, isolated n as bern computes all the Bernoulli numbers up to index n before returning. burn invokes the approximation for even integers n > 255. For odd integers and n <= 255 the function bern is called.

load ("bffac") loads this function. See also bern.

Function: chinese ([r_1, …, r_n], [m_1, …, m_n])

Solves the system of congruences x = r_1 mod m_1, …, x = r_n mod m_n. The remainders r_n may be arbitrary integers while the moduli m_n have to be positive and pairwise coprime integers.

(%i1) mods : [1000, 1001, 1003, 1007];
(%o1)                   [1000, 1001, 1003, 1007]
(%i2) lreduce('gcd, mods);
(%o2)                               1
(%i3) x : random(apply("*", mods));
(%o3)                         685124877004
(%i4) rems : map(lambda([z], mod(x, z)), mods);
(%o4)                       [4, 568, 54, 624]
(%i5) chinese(rems, mods);
(%o5)                         685124877004
(%i6) chinese([1, 2], [3, n]);
(%o6)                    chinese([1, 2], [3, n])
(%i7) %, n = 4;
(%o7)                              10

Function: cf (expr)

Computes a continued fraction approximation. expr is an expression comprising continued fractions, square roots of integers, and literal real numbers (integers, rational numbers, ordinary floats, and bigfloats). cf computes exact expansions for rational numbers, but expansions are truncated at ratepsilon for ordinary floats and 10^(-fpprec) for bigfloats.

Operands in the expression may be combined with arithmetic operators. Maxima does not know about operations on continued fractions outside of cf.

cf evaluates its arguments after binding listarith to false. cf returns a continued fraction, represented as a list.

A continued fraction a + 1/(b + 1/(c + ...)) is represented by the list [a, b, c, ...]. The list elements a, b, c, … must evaluate to integers. expr may also contain sqrt (n) where n is an integer. In this case cf will give as many terms of the continued fraction as the value of the variable cflength times the period.

A continued fraction can be evaluated to a number by evaluating the arithmetic representation returned by cfdisrep. See also cfexpand for another way to evaluate a continued fraction.

See also cfdisrep, cfexpand, and cflength.

Examples:

Function: cfdisrep (list)

Constructs and returns an ordinary arithmetic expression of the form a + 1/(b + 1/(c + ...)) from the list representation of a continued fraction [a, b, c, ...].

(%i1) cf ([1, 2, -3] + [1, -2, 1]);
(%o1)                     [1, 1, 1, 2]
(%i2) cfdisrep (%);
                                  1
(%o2)                     1 + ---------
                                    1
                              1 + -----
                                      1
                                  1 + -
                                      2

Function: cfexpand (x)

Returns a matrix of the numerators and denominators of the last (column 1) and next-to-last (column 2) convergents of the continued fraction x.

(%i1) cf (rat (ev (%pi, numer)));

`rat' replaced 3.141592653589793 by 103993/33102 =3.141592653011902
(%o1)                  [3, 7, 15, 1, 292]
(%i2) cfexpand (%); 
                         [ 103993  355 ]
(%o2)                    [             ]
                         [ 33102   113 ]
(%i3) %[1,1]/%[2,1], numer;
(%o3)                   3.141592653011902

Option variable: cflength

Default value: 1

cflength controls the number of terms of the continued fraction the function cf will give, as the value cflength times the period. Thus the default is to give one period.

(%i1) cflength: 1$
(%i2) cf ((1 + sqrt(5))/2);
(%o2)                    [1, 1, 1, 1, 2]
(%i3) cflength: 2$
(%i4) cf ((1 + sqrt(5))/2);
(%o4)               [1, 1, 1, 1, 1, 1, 1, 2]
(%i5) cflength: 3$
(%i6) cf ((1 + sqrt(5))/2);
(%o6)           [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]

Function: divsum
    divsum (n, k)
    divsum (n)

divsum (n, k) returns the sum of the divisors of n raised to the k’th power.

divsum (n) returns the sum of the divisors of n.

(%i1) divsum (12);
(%o1)                          28
(%i2) 1 + 2 + 3 + 4 + 6 + 12;
(%o2)                          28
(%i3) divsum (12, 2);
(%o3)                          210
(%i4) 1^2 + 2^2 + 3^2 + 4^2 + 6^2 + 12^2;
(%o4)                          210

Function: euler (n)

Returns the n’th Euler number for nonnegative integer n. Euler numbers equal to zero are suppressed if zerobern is false.

For the Euler-Mascheroni constant, see %gamma.

(%i1) zerobern: true$
(%i2) map (euler, [0, 1, 2, 3, 4, 5, 6]);
(%o2)               [1, 0, - 1, 0, 5, 0, - 61]
(%i3) zerobern: false$
(%i4) map (euler, [0, 1, 2, 3, 4, 5, 6]);
(%o4)               [1, - 1, 5, - 61, 1385, - 50521, 2702765]

Option variable: factors_only

Default value: false

Controls the value returned by ifactors. The default false causes ifactors to provide information about multiplicities of the computed prime factors. If factors_only is set to true, ifactors returns nothing more than a list of prime factors.

Example: See ifactors.

Function: fib (n)

Returns the n’th Fibonacci number. fib(0) is equal to 0 and fib(1) equal to 1, and fib (-n) equal to (-1)^(n + 1) * fib(n).

After calling fib, prevfib is equal to fib(n - 1), the Fibonacci number preceding the last one computed.

(%i1) map (fib, [-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8]);
(%o1)           [- 3, 2, - 1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21]

Function: fibtophi (expr)

Expresses Fibonacci numbers in expr in terms of the constant %phi, which is (1 + sqrt(5))/2, approximately 1.61803399.

Examples:

(%i1) fibtophi (fib (n));
                           n             n
                       %phi  - (1 - %phi)
(%o1)                  -------------------
                           2 %phi - 1
(%i2) fib (n-1) + fib (n) - fib (n+1);
(%o2)          - fib(n + 1) + fib(n) + fib(n - 1)
(%i3) fibtophi (%);
            n + 1             n + 1       n             n
        %phi      - (1 - %phi)        %phi  - (1 - %phi)
(%o3) - --------------------------- + -------------------
                2 %phi - 1                2 %phi - 1
                                          n - 1             n - 1
                                      %phi      - (1 - %phi)
                                    + ---------------------------
                                              2 %phi - 1
(%i4) ratsimp (%);
(%o4)                           0

Function: ifactors (n)

For a positive integer n returns the factorization of n. If n=p1^e1..pk^nk is the decomposition of n into prime factors, ifactors returns [[p1, e1], ... , [pk, ek]].

Factorization methods used are trial divisions by primes up to 9973, Pollard’s rho and p-1 method and elliptic curves.

If the variable ifactor_verbose is set to true ifactor produces detailed output about what it is doing including immediate feedback as soon as a factor has been found.

The value returned by ifactors is controlled by the option variable factors_only. The default false causes ifactors to provide information about the multiplicities of the computed prime factors. If factors_only is set to true, ifactors simply returns the list of prime factors.

(%i1) ifactors(51575319651600);
(%o1)     [[2, 4], [3, 2], [5, 2], [1583, 1], [9050207, 1]]
(%i2) apply("*", map(lambda([u], u[1]^u[2]), %));
(%o2)                        51575319651600
(%i3) ifactors(51575319651600), factors_only : true;
(%o3)                   [2, 3, 5, 1583, 9050207]

Function: igcdex (n, k)

Returns a list [a, b, u] where u is the greatest common divisor of n and k, and u is equal to a n + b k. The arguments n and k must be integers.

igcdex implements the Euclidean algorithm. See also gcdex.

The command load("gcdex") loads the function.

Examples:

(%i1) load("gcdex")$

(%i2) igcdex(30,18);
(%o2)                      [- 1, 2, 6]
(%i3) igcdex(1526757668, 7835626735736);
(%o3)            [845922341123, - 164826435, 4]
(%i4) igcdex(fib(20), fib(21));
(%o4)                   [4181, - 2584, 1]

Function: inrt (x, n)

Returns the integer n’th root of the absolute value of x.

(%i1) l: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]$
(%i2) map (lambda ([a], inrt (10^a, 3)), l);
(%o2) [2, 4, 10, 21, 46, 100, 215, 464, 1000, 2154, 4641, 10000]

Function: inv_mod (n, m)

Computes the inverse of n modulo m. inv_mod (n,m) returns false, if n is a zero divisor modulo m.

(%i1) inv_mod(3, 41);
(%o1)                           14
(%i2) ratsimp(3^-1), modulus = 41;
(%o2)                           14
(%i3) inv_mod(3, 42);
(%o3)                          false

Function: isqrt (x)

Returns the "integer square root" of the absolute value of x, which is an integer.

Function: jacobi (p, q)

Returns the Jacobi symbol of p and q.

(%i1) l: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]$
(%i2) map (lambda ([a], jacobi (a, 9)), l);
(%o2)         [1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0]

Function: lcm (expr_1, …, expr_n)

Returns the least common multiple of its arguments. The arguments may be general expressions as well as integers.

load ("functs") loads this function.

Function: lucas (n)

Returns the n’th Lucas number. lucas(0) is equal to 2 and lucas(1) equal to 1, and lucas(-n) equal to (-1)^(-n) * lucas(n).

(%i1) map (lucas, [-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8]);
(%o1)             [7, - 4, 3, - 1, 2, 1, 3, 4, 7, 11, 18, 29, 47]

After calling lucas, the global variable next_lucas is equal to lucas (n + 1), the Lucas number following the last returned. The example shows how Fibonacci numbers can be computed via lucas and next_lucas.

(%i1) fib_via_lucas(n) := 
         block([lucas : lucas(n)],
         signum(n) * (2*next_lucas - lucas)/5 )$
(%i2) map (fib_via_lucas, [-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8]);
(%o2)             [- 3, 2, - 1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21]

Function: mod (x, y)

If x and y are real numbers and y is nonzero, return x - y * floor(x / y). Further for all real x, we have mod (x, 0) = x. For a discussion of the definition mod (x, 0) = x, see Section 3.4, of "Concrete Mathematics," by Graham, Knuth, and Patashnik. The function mod (x, 1) is a sawtooth function with period 1 with mod (1, 1) = 0 and mod (0, 1) = 0.

To find the principal argument (a number in the interval (-%pi, %pi]) of a complex number, use the function x |-> %pi - mod (%pi - x, 2*%pi), where x is an argument.

When x and y are constant expressions (10 * %pi, for example), mod uses the same big float evaluation scheme that floor and ceiling uses. Again, it’s possible, although unlikely, that mod could return an erroneous value in such cases.

For nonnumerical arguments x or y, mod knows several simplification rules:

(%i1) mod (x, 0);
(%o1)                           x
(%i2) mod (a*x, a*y);
(%o2)                      a mod(x, y)
(%i3) mod (0, x);
(%o3)                           0

Function: next_prime (n)

Returns the smallest prime bigger than n.

(%i1) next_prime(27);
(%o1)                       29

Function: partfrac (expr, var)

Expands the expression expr in partial fractions with respect to the main variable var. partfrac does a complete partial fraction decomposition. The algorithm employed is based on the fact that the denominators of the partial fraction expansion (the factors of the original denominator) are relatively prime. The numerators can be written as linear combinations of denominators, and the expansion falls out.

partfrac ignores the value true of the option variable keepfloat.

(%i1) 1/(1+x)^2 - 2/(1+x) + 2/(2+x);
                      2       2        1
(%o1)               ----- - ----- + --------
                    x + 2   x + 1          2
                                    (x + 1)
(%i2) ratsimp (%);
                                 x
(%o2)                 - -------------------
                         3      2
                        x  + 4 x  + 5 x + 2
(%i3) partfrac (%, x);
                      2       2        1
(%o3)               ----- - ----- + --------
                    x + 2   x + 1          2
                                    (x + 1)
Function: power_mod (a, n, m)

Uses a modular algorithm to compute a^n mod m where a and n are integers and m is a positive integer. If n is negative, inv_mod is used to find the modular inverse.

(%i1) power_mod(3, 15, 5);
(%o1)                          2
(%i2) mod(3^15,5);
(%o2)                          2
(%i3) power_mod(2, -1, 5);
(%o3)                          3
(%i4) inv_mod(2,5);
(%o4)                          3

Function: primep (n)

Primality test. If primep (n) returns false, n is a composite number and if it returns true, n is a prime number with very high probability.

For n less than 341550071728321 a deterministic version of Miller-Rabin’s test is used. If primep (n) returns true, then n is a prime number.

For n bigger than 341550071728321 primep uses primep_number_of_tests Miller-Rabin’s pseudo-primality tests and one Lucas pseudo-primality test. The probability that a non-prime n will pass one Miller-Rabin test is less than 1/4. Using the default value 25 for primep_number_of_tests, the probability of n being composite is much smaller that 10^-15.

Option variable: primep_number_of_tests

Default value: 25

Number of Miller-Rabin’s tests used in primep.

Function: primes (start, end)

Returns the list of all primes from start to end.

(%i1) primes(3, 7);
(%o1)                     [3, 5, 7]

Function: prev_prime (n)

Returns the greatest prime smaller than n.

(%i1) prev_prime(27);
(%o1)                       23

Function: qunit (n)

Returns the principal unit of the real quadratic number field sqrt (n) where n is an integer, i.e., the element whose norm is unity. This amounts to solving Pell’s equation a^2 - n b^2 = 1.

(%i1) qunit (17);
(%o1)                     sqrt(17) + 4
(%i2) expand (% * (sqrt(17) - 4));
(%o2)                           1

Function: totient (n)

Returns the number of integers less than or equal to n which are relatively prime to n.

Option variable: zerobern

Default value: true

When zerobern is false, bern excludes the Bernoulli numbers and euler excludes the Euler numbers which are equal to zero. See bern and euler.

Function: zeta (n)

Returns the Riemann zeta function. If n is a negative integer, 0, or a positive even integer, the Riemann zeta function simplifies to an exact value. For a positive even integer the option variable zeta%pi has to be true in addition (See zeta%pi). For a floating point or bigfloat number the Riemann zeta function is evaluated numerically. Maxima returns a noun form zeta (n) for all other arguments, including rational noninteger, and complex arguments, or for even integers, if zeta%pi has the value false.

zeta(1) is undefined, but Maxima knows the limit limit(zeta(x), x, 1) from above and below.

The Riemann zeta function distributes over lists, matrices, and equations.

See also bfzeta and zeta%pi.

Examples:

(%i1) zeta([-2, -1, 0, 0.5, 2, 3, 1+%i]);
                                             2
            1     1                       %pi
(%o1) [0, - --, - -, - 1.460354508809586, ----, zeta(3), 
            12    2                        6
                                                    zeta(%i + 1)]
(%i2) limit(zeta(x),x,1,plus);
(%o2)                          inf
(%i3) limit(zeta(x),x,1,minus);
(%o3)                         minf

Option variable: zeta%pi

Default value: true

When zeta%pi is true, zeta returns an expression proportional to %pi^n for even integer n. Otherwise, zeta returns a noun form zeta (n) for even integer n.

Examples:

(%i1) zeta%pi: true$
(%i2) zeta (4);
                                 4
                              %pi
(%o2)                         ----
                               90
(%i3) zeta%pi: false$
(%i4) zeta (4);
(%o4)                        zeta(4)

Function: zn_add_table (n)

Shows an addition table of all elements in (Z/nZ).

See also zn_mult_table, zn_power_table.

Function: zn_characteristic_factors (n)

Returns a list containing the characteristic factors of the totient of n.

Using the characteristic factors a multiplication group modulo n can be expressed as a group direct product of cyclic subgroups.

In case the group itself is cyclic the list only contains the totient and using zn_primroot a generator can be computed. If the totient splits into more than one characteristic factors zn_factor_generators finds generators of the corresponding subgroups.

Each of the r factors in the list divides the right following factors. For the last factor f_r therefore holds a^f_r = 1 (mod n) for all a coprime to n. This factor is also known as Carmichael function or Carmichael lambda.

If n > 2, then totient(n)/2^r is the number of quadratic residues, and each of these has 2^r square roots.

See also totient, zn_primroot, zn_factor_generators.

Examples:

The multiplication group modulo 14 is cyclic and its 6 elements can be generated by a primitive root.

(%i1) [zn_characteristic_factors(14), phi: totient(14)];
(%o1)                              [[6], 6]
(%i2) [zn_factor_generators(14), g: zn_primroot(14)];
(%o2)                              [[3], 3]
(%i3) M14: makelist(power_mod(g,i,14), i,0,phi-1);
(%o3)                         [1, 3, 9, 13, 11, 5]

The multiplication group modulo 15 is not cyclic and its 8 elements can be generated by two factor generators.

(%i1) [[f1,f2]: zn_characteristic_factors(15), totient(15)];
(%o1)                             [[2, 4], 8]
(%i2) [[g1,g2]: zn_factor_generators(15), zn_primroot(15)];
(%o2)                           [[11, 7], false]
(%i3) UG1: makelist(power_mod(g1,i,15), i,0,f1-1);
(%o3)                               [1, 11]
(%i4) UG2: makelist(power_mod(g2,i,15), i,0,f2-1);
(%o4)                            [1, 7, 4, 13]
(%i5) M15: create_list(mod(i*j,15), i,UG1, j,UG2);
(%o5)                      [1, 7, 4, 13, 11, 2, 14, 8]

For the last characteristic factor 4 it holds that a^4 = 1 (mod 15) for all a in M15.

M15 has two characteristic factors and therefore 8/2^2 quadratic residues, and each of these has 2^2 square roots.

(%i6) zn_power_table(15);
                               [ 1   1  1   1 ]
                               [              ]
                               [ 2   4  8   1 ]
                               [              ]
                               [ 4   1  4   1 ]
                               [              ]
                               [ 7   4  13  1 ]
(%o6)                          [              ]
                               [ 8   4  2   1 ]
                               [              ]
                               [ 11  1  11  1 ]
                               [              ]
                               [ 13  4  7   1 ]
                               [              ]
                               [ 14  1  14  1 ]
(%i7) map(lambda([i], zn_nth_root(i,2,15)), [1,4]);
(%o7)                   [[1, 4, 11, 14], [2, 7, 8, 13]]

Function: zn_carmichael_lambda (n)

Returns 1 if n is 1 and otherwise the greatest characteristic factor of the totient of n.

For remarks and examples see zn_characteristic_factors.

Function: zn_determinant (matrix, p)

Uses the technique of LU-decomposition to compute the determinant of matrix over (Z/pZ). p must be a prime.

However if the determinant is equal to zero the LU-decomposition might fail. In that case zn_determinant computes the determinant non-modular and reduces thereafter.

See also zn_invert_by_lu.

Examples:

(%i1) m : matrix([1,3],[2,4]);
                                [ 1  3 ]
(%o1)                           [      ]
                                [ 2  4 ]
(%i2) zn_determinant(m, 5);
(%o2)                               3
(%i3) m : matrix([2,4,1],[3,1,4],[4,3,2]);
                               [ 2  4  1 ]
                               [         ]
(%o3)                          [ 3  1  4 ]
                               [         ]
                               [ 4  3  2 ]
(%i4) zn_determinant(m, 5);
(%o4)                               0

Function: zn_factor_generators (n)

Returns a list containing factor generators corresponding to the characteristic factors of the totient of n.

For remarks and examples see zn_characteristic_factors.

Function: zn_invert_by_lu (matrix, p)

Uses the technique of LU-decomposition to compute the modular inverse of matrix over (Z/pZ). p must be a prime and matrix invertible. zn_invert_by_lu returns false if matrix is not invertible.

See also zn_determinant.

Example:

(%i1) m : matrix([1,3],[2,4]);
                                [ 1  3 ]
(%o1)                           [      ]
                                [ 2  4 ]
(%i2) zn_determinant(m, 5);
(%o2)                               3
(%i3) mi : zn_invert_by_lu(m, 5);
                                [ 3  4 ]
(%o3)                           [      ]
                                [ 1  2 ]
(%i4) matrixmap(lambda([a], mod(a, 5)), m . mi);
                                [ 1  0 ]
(%o4)                           [      ]
                                [ 0  1 ]

Function: zn_log
    zn_log (a, g, n)
    zn_log (a, g, n, [[p1, e1], …, [pk, ek]])

Computes the discrete logarithm. Let (Z/nZ)* be a cyclic group, g a primitive root modulo n and let a be a member of this group. zn_log (a, g, n) then solves the congruence g^x = a mod n.

The applied algorithm needs a prime factorization of totient(n). This factorization might be time consuming as well and in some cases it can be useful to factor first and then to pass the list of factors to zn_log as the fourth argument. The list must be of the same form as the list returned by ifactors(totient(n)) using the default option factors_only : false.

The algorithm uses a Pohlig-Hellman-reduction and Pollard’s Rho-method for discrete logarithms. The run time of zn_log primarily depends on the bitlength of the totient’s greatest prime factor.

See also zn_primroot, zn_order, ifactors, totient.

Examples:

zn_log (a, g, n) solves the congruence g^x = a mod n.

(%i1) n : 22$
(%i2) g : zn_primroot(n);
(%o2)                               7
(%i3) ord_7 : zn_order(7, n);
(%o3)                              10
(%i4) powers_7 : makelist(power_mod(g, x, n), x, 0, ord_7 - 1);
(%o4)              [1, 7, 5, 13, 3, 21, 15, 17, 9, 19]
(%i5) zn_log(21, g, n);
(%o5)                               5
(%i6) map(lambda([x], zn_log(x, g, n)), powers_7);
(%o6)                [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

The optional fourth argument must be of the same form as the list returned by ifactors(totient(n)). The run time primarily depends on the bitlength of the totient’s greatest prime factor.

(%i1) (p : 2^127-1, primep(p));
(%o1)                             true
(%i2) ifs : ifactors(p - 1)$
(%i3) g : zn_primroot(p, ifs);
(%o3)                              43
(%i4) a : power_mod(g, 1234567890, p)$
(%i5) zn_log(a, g, p, ifs);
(%o5)                          1234567890
(%i6) time(%o5);  
(%o6)                            [1.204]
(%i7) f_max : last(ifs);
(%o7)                       [77158673929, 1]
(%i8) slength( printf(false, "~b", f_max[1]) );
(%o8)                              37

Function: zn_mult_table
    zn_mult_table (n)
    zn_mult_table (n, gcd)

Without the optional argument gcd zn_mult_table(n) shows a multiplication table of all elements in (Z/nZ)* which are all elements coprime to n.

The optional second argument gcd allows to select a specific subset of (Z/nZ). If gcd is an integer, a multiplication table of all residues x with gcd(x,n) = gcd are returned. Additionally row and column headings are added for better readability. If necessary, these can be easily removed by submatrix(1, table, 1).

If gcd is set to all, the table is printed for all non-zero elements in (Z/nZ).

The second example shows an alternative way to create a multiplication table for subgroups.

See also zn_add_table, zn_power_table.

Examples:

The default table shows all elements in (Z/nZ)* and allows to demonstrate and study basic properties of modular multiplication groups. E.g. the principal diagonal contains all quadratic residues, each row and column contains every element, the tables are symmetric, etc..

If gcd is set to all, the table is printed for all non-zero elements in (Z/nZ).

(%i1) zn_mult_table(8);
                                [ 1  3  5  7 ]
                                [            ]
                                [ 3  1  7  5 ]
(%o1)                           [            ]
                                [ 5  7  1  3 ]
                                [            ]
                                [ 7  5  3  1 ]
(%i2) zn_mult_table(8, all);
                            [ 1  2  3  4  5  6  7 ]
                            [                     ]
                            [ 2  4  6  0  2  4  6 ]
                            [                     ]
                            [ 3  6  1  4  7  2  5 ]
                            [                     ]
(%o2)                       [ 4  0  4  0  4  0  4 ]
                            [                     ]
                            [ 5  2  7  4  1  6  3 ]
                            [                     ]
                            [ 6  4  2  0  6  4  2 ]
                            [                     ]
                            [ 7  6  5  4  3  2  1 ]

If gcd is an integer, row and column headings are added for better readability.

If the subset chosen by gcd is a group there is another way to create a multiplication table. An isomorphic mapping from a group with 1 as identity builds a table which is easy to read. The mapping is accomplished via CRT.

In the second version of T36_4 the identity, here 28, is placed in the top left corner, just like in table T9.

(%i1) T36_4: zn_mult_table(36,4);
                        [ *   4   8   16  20  28  32 ]
                        [                            ]
                        [ 4   16  32  28  8   4   20 ]
                        [                            ]
                        [ 8   32  28  20  16  8   4  ]
                        [                            ]
(%o1)                   [ 16  28  20  4   32  16  8  ]
                        [                            ]
                        [ 20  8   16  32  4   20  28 ]
                        [                            ]
                        [ 28  4   8   16  20  28  32 ]
                        [                            ]
                        [ 32  20  4   8   28  32  16 ]
(%i2) T9: zn_mult_table(36/4);
                             [ 1  2  4  5  7  8 ]
                             [                  ]
                             [ 2  4  8  1  5  7 ]
                             [                  ]
                             [ 4  8  7  2  1  5 ]
(%o2)                        [                  ]
                             [ 5  1  2  7  8  4 ]
                             [                  ]
                             [ 7  5  1  8  4  2 ]
                             [                  ]
                             [ 8  7  5  4  2  1 ]
(%i3) T36_4: matrixmap(lambda([x], chinese([0,x],[4,9])), T9);
                          [ 28  20  4   32  16  8  ]
                          [                        ]
                          [ 20  4   8   28  32  16 ]
                          [                        ]
                          [ 4   8   16  20  28  32 ]
(%o3)                     [                        ]
                          [ 32  28  20  16  8   4  ]
                          [                        ]
                          [ 16  32  28  8   4   20 ]
                          [                        ]
                          [ 8   16  32  4   20  28 ]

Function: zn_nth_root
    zn_nth_root (x, n, m)
    zn_nth_root (x, n, m, [[p1, e1], …, [pk, ek]])

Returns a list with all n-th roots of x from the multiplication subgroup of (Z/mZ) which contains x, or false, if x is no n-th power modulo m or not contained in any multiplication subgroup of (Z/mZ).

x is an element of a multiplication subgroup modulo m, if the greatest common divisor g = gcd(x,m) is coprime to m/g.

zn_nth_root is based on an algorithm by Adleman, Manders and Miller and on theorems about modulo multiplication groups by Daniel Shanks.

The algorithm needs a prime factorization of the modulus m. So in case the factorization of m is known, the list of factors can be passed as the fourth argument. This optional argument must be of the same form as the list returned by ifactors(m) using the default option factors_only: false.

Examples:

A power table of the multiplication group modulo 14 followed by a list of lists containing all n-th roots of 1 with n from 1 to 6.

(%i1) zn_power_table(14);
                         [ 1   1   1   1   1   1 ]
                         [                       ]
                         [ 3   9   13  11  5   1 ]
                         [                       ]
                         [ 5   11  13  9   3   1 ]
(%o1)                    [                       ]
                         [ 9   11  1   9   11  1 ]
                         [                       ]
                         [ 11  9   1   11  9   1 ]
                         [                       ]
                         [ 13  1   13  1   13  1 ]
(%i2) makelist(zn_nth_root(1,n,14), n,1,6);
(%o2)  [[1], [1, 13], [1, 9, 11], [1, 13], [1], [1, 3, 5, 9, 11, 13]]

In the following example x is not coprime to m, but is a member of a multiplication subgroup of (Z/mZ) and any n-th root is a member of the same subgroup.

The residue class 3 is no member of any multiplication subgroup of (Z/63Z) and is therefore not returned as a third root of 27.

Here zn_power_table shows all residues x in (Z/63Z) with gcd(x,63) = 9. This subgroup is isomorphic to (Z/7Z)* and its identity 36 is computed via CRT.

(%i1) m: 7*9$

(%i2) zn_power_table(m,9);
                         [ 9   18  36  9   18  36 ]
                         [                        ]
                         [ 18  9   36  18  9   36 ]
                         [                        ]
                         [ 27  36  27  36  27  36 ]
(%o2)                    [                        ]
                         [ 36  36  36  36  36  36 ]
                         [                        ]
                         [ 45  9   27  18  54  36 ]
                         [                        ]
                         [ 54  18  27  9   45  36 ]
(%i3) zn_nth_root(27,3,m);
(%o3)                           [27, 45, 54]
(%i4) id7:1$  id63_9: chinese([id7,0],[7,9]);
(%o5)                                36

In the following RSA-like example, where the modulus N is squarefree, i.e. it splits into exclusively first power factors, every x from 0 to N-1 is contained in a multiplication subgroup.

The process of decryption needs the e-th root. e is coprime to totient(N) and therefore the e-th root is unique. In this case zn_nth_root effectively performs CRT-RSA. (Please note that flatten removes braces but no solutions.)

(%i1) [p,q,e]: [5,7,17]$  N: p*q$

(%i3) xs: makelist(x,x,0,N-1)$

(%i4) ys: map(lambda([x],power_mod(x,e,N)),xs)$

(%i5) zs: flatten(map(lambda([y], zn_nth_root(y,e,N)), ys))$

(%i6) is(zs = xs);
(%o6)                             true

In the following example the factorization of the modulus is known and passed as the fourth argument.

(%i1) p: 2^107-1$  q: 2^127-1$  N: p*q$

(%i4) ibase: obase: 16$

(%i5) msg: 11223344556677889900aabbccddeeff$

(%i6) enc: power_mod(msg, 10001, N);
(%o6)    1a8db7892ae588bdc2be25dd5107a425001fe9c82161abc673241c8b383
(%i7) zn_nth_root(enc, 10001, N, [[p,1],[q,1]]);
(%o7)               [11223344556677889900aabbccddeeff]

Function: zn_order
    zn_order (x, n)
    zn_order (x, n, [[p1, e1], …, [pk, ek]])

Returns the order of x if it is a unit of the finite group (Z/nZ)* or returns false. x is a unit modulo n if it is coprime to n.

The applied algorithm needs a prime factorization of totient(n). This factorization might be time consuming in some cases and it can be useful to factor first and then to pass the list of factors to zn_log as the third argument. The list must be of the same form as the list returned by ifactors(totient(n)) using the default option factors_only : false.

See also zn_primroot, ifactors, totient.

Examples:

zn_order computes the order of the unit x in (Z/nZ)*.

(%i1) n : 22$
(%i2) g : zn_primroot(n);
(%o2)                               7
(%i3) units_22 : sublist(makelist(i,i,1,21), lambda([x], gcd(x, n) = 1));
(%o3)              [1, 3, 5, 7, 9, 13, 15, 17, 19, 21]
(%i4) (ord_7 : zn_order(7, n)) = totient(n);
(%o4)                            10 = 10
(%i5) powers_7 : makelist(power_mod(g,i,n), i,0,ord_7 - 1);
(%o5)              [1, 7, 5, 13, 3, 21, 15, 17, 9, 19]
(%i6) map(lambda([x], zn_order(x, n)), powers_7);
(%o6)              [1, 10, 5, 10, 5, 2, 5, 10, 5, 10]
(%i7) map(lambda([x], ord_7/gcd(x, ord_7)), makelist(i, i,0,ord_7 - 1));
(%o7)              [1, 10, 5, 10, 5, 2, 5, 10, 5, 10]
(%i8) totient(totient(n));
(%o8)                               4

The optional third argument must be of the same form as the list returned by ifactors(totient(n)).

(%i1) (p : 2^142 + 217, primep(p));
(%o1)                             true
(%i2) ifs : ifactors( totient(p) )$
(%i3) g : zn_primroot(p, ifs);
(%o3)                               3
(%i4) is( (ord_3 : zn_order(g, p, ifs)) = totient(p) );
(%o4)                             true
(%i5) map(lambda([x], ord_3/zn_order(x, p, ifs)), makelist(i,i,2,15));
(%o5)        [22, 1, 44, 10, 5, 2, 22, 2, 8, 2, 1, 1, 20, 1]

Function: zn_power_table
    zn_power_table (n)
    zn_power_table (n, gcd)
    zn_power_table (n, gcd, max_exp)

Without any optional argument zn_power_table(n) shows a power table of all elements in (Z/nZ)* which are all residue classes coprime to n. The exponent loops from 1 to the greatest characteristic factor of totient(n) (also known as Carmichael function or Carmichael lambda) and the table ends with a column of ones on the right side.

The optional second argument gcd allows to select powers of a specific subset of (Z/nZ). If gcd is an integer, powers of all residue classes x with gcd(x,n) = gcd are returned, i.e. the default value for gcd is 1. If gcd is set to all, the table contains powers of all elements in (Z/nZ).

If the optional third argument max_exp is given, the exponent loops from 1 to max_exp.

See also zn_add_table, zn_mult_table.

Examples:

The default which is gcd = 1 allows to demonstrate and study basic theorems of e.g. Fermat and Euler.

The argument gcd allows to select subsets of (Z/nZ) and to study multiplication subgroups and isomorphisms. E.g. the groups G10 and G10_2 are under multiplication both isomorphic to G5. 1 is the identity in G5. So are 1 resp. 6 the identities in G10 resp. G10_2. There are corresponding mappings for primitive roots, n-th roots, etc..

(%i1) zn_power_table(10);
                              [ 1  1  1  1 ]
                              [            ]
                              [ 3  9  7  1 ]
(%o1)                         [            ]
                              [ 7  9  3  1 ]
                              [            ]
                              [ 9  1  9  1 ]
(%i2) zn_power_table(10,2);
                              [ 2  4  8  6 ]
                              [            ]
                              [ 4  6  4  6 ]
(%o2)                         [            ]
                              [ 6  6  6  6 ]
                              [            ]
                              [ 8  4  2  6 ]
(%i3) zn_power_table(10,5);
(%o3)                         [ 5  5  5  5 ]
(%i4) zn_power_table(10,10);
(%o4)                         [ 0  0  0  0 ]
(%i5) G5: [1,2,3,4];
(%o6)                          [1, 2, 3, 4]
(%i6) G10_2: map(lambda([x], chinese([0,x],[2,5])), G5);
(%o6)                          [6, 2, 8, 4]
(%i7) G10: map(lambda([x], power_mod(3, zn_log(x,2,5), 10)), G5);
(%o7)                          [1, 3, 7, 9]

If gcd is set to all, the table contains powers of all elements in (Z/nZ).

The third argument max_exp allows to set the highest exponent. The following table shows a very small example of RSA.

(%i1) N:2*5$ phi:totient(N)$ e:7$ d:inv_mod(e,phi)$

(%i5) zn_power_table(N, all, e*d);
       [ 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ]
       [                                                               ]
       [ 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 ]
       [                                                               ]
       [ 2  4  8  6  2  4  8  6  2  4  8  6  2  4  8  6  2  4  8  6  2 ]
       [                                                               ]
       [ 3  9  7  1  3  9  7  1  3  9  7  1  3  9  7  1  3  9  7  1  3 ]
       [                                                               ]
       [ 4  6  4  6  4  6  4  6  4  6  4  6  4  6  4  6  4  6  4  6  4 ]
(%o5)  [                                                               ]
       [ 5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5 ]
       [                                                               ]
       [ 6  6  6  6  6  6  6  6  6  6  6  6  6  6  6  6  6  6  6  6  6 ]
       [                                                               ]
       [ 7  9  3  1  7  9  3  1  7  9  3  1  7  9  3  1  7  9  3  1  7 ]
       [                                                               ]
       [ 8  4  2  6  8  4  2  6  8  4  2  6  8  4  2  6  8  4  2  6  8 ]
       [                                                               ]
       [ 9  1  9  1  9  1  9  1  9  1  9  1  9  1  9  1  9  1  9  1  9 ]

Function: zn_primroot
    zn_primroot (n)
    zn_primroot (n, [[p1, e1], …, [pk, ek]])

If the multiplicative group (Z/nZ)* is cyclic, zn_primroot computes the smallest primitive root modulo n. (Z/nZ)* is cyclic if n is equal to 2, 4, p^k or 2*p^k, where p is prime and greater than 2 and k is a natural number. zn_primroot performs an according pretest if the option variable zn_primroot_pretest (default: false) is set to true. In any case the computation is limited by the upper bound zn_primroot_limit.

If (Z/nZ)* is not cyclic or if there is no primitive root up to zn_primroot_limit, zn_primroot returns false.

The applied algorithm needs a prime factorization of totient(n). This factorization might be time consuming in some cases and it can be useful to factor first and then to pass the list of factors to zn_log as an additional argument. The list must be of the same form as the list returned by ifactors(totient(n)) using the default option factors_only : false.

See also zn_primroot_p, zn_order, ifactors, totient.

Examples:

zn_primroot computes the smallest primitive root modulo n or returns false.

(%i1) n : 14$
(%i2) g : zn_primroot(n);
(%o2)                               3
(%i3) zn_order(g, n) = totient(n);
(%o3)                             6 = 6
(%i4) n : 15$
(%i5) zn_primroot(n);
(%o5)                             false

The optional second argument must be of the same form as the list returned by ifactors(totient(n)).

(%i1) (p : 2^142 + 217, primep(p));
(%o1)                             true
(%i2) ifs : ifactors( totient(p) )$
(%i3) g : zn_primroot(p, ifs);
(%o3)                               3
(%i4) [time(%o2), time(%o3)];
(%o4)                    [[15.556972], [0.004]]
(%i5) is(zn_order(g, p, ifs) = p - 1);
(%o5)                             true
(%i6) n : 2^142 + 216$
(%i7) ifs : ifactors(totient(n))$
(%i8) zn_primroot(n, ifs), 
      zn_primroot_limit : 200, zn_primroot_verbose : true;
`zn_primroot' stopped at zn_primroot_limit = 200
(%o8)                             false

Option variable: zn_primroot_limit

Default value: 1000

If zn_primroot cannot find a primitve root, it stops at this upper bound. If the option variable zn_primroot_verbose (default: false) is set to true, a message will be printed when zn_primroot_limit is reached.

Function: zn_primroot_p
    zn_primroot_p (x, n)
    zn_primroot_p (x, n, [[p1, e1], …, [pk, ek]])

Checks whether x is a primitive root in the multiplicative group (Z/nZ)*.

The applied algorithm needs a prime factorization of totient(n). This factorization might be time consuming and in case zn_primroot_p will be consecutively applied to a list of candidates it can be useful to factor first and then to pass the list of factors to zn_log as a third argument. The list must be of the same form as the list returned by ifactors(totient(n)) using the default option factors_only : false.

See also zn_primroot, zn_order, ifactors, totient.

Examples:

zn_primroot_p as a predicate function.

(%i1) n : 14$
(%i2) units_14 : sublist(makelist(i,i,1,13), lambda([i], gcd(i, n) = 1));
(%o2)                     [1, 3, 5, 9, 11, 13]
(%i3) zn_primroot_p(13, n);
(%o3)                            false
(%i4) sublist(units_14, lambda([x], zn_primroot_p(x, n)));
(%o4)                            [3, 5]
(%i5) map(lambda([x], zn_order(x, n)), units_14);
(%o5)                      [1, 6, 6, 3, 3, 2]

The optional third argument must be of the same form as the list returned by ifactors(totient(n)).

(%i1) (p : 2^142 + 217, primep(p));
(%o1)                             true
(%i2) ifs : ifactors( totient(p) )$
(%i3) sublist(makelist(i,i,1,50), lambda([x], zn_primroot_p(x, p, ifs)));
(%o3)      [3, 12, 13, 15, 21, 24, 26, 27, 29, 33, 38, 42, 48]
(%i4) [time(%o2), time(%o3)];
(%o4)                   [[7.748484], [0.036002]]

Option variable: zn_primroot_pretest

Default value: false

The multiplicative group (Z/nZ)* is cyclic if n is equal to 2, 4, p^k or 2*p^k, where p is prime and greater than 2 and k is a natural number.

zn_primroot_pretest controls whether zn_primroot will check if one of these cases occur before it computes the smallest primitive root. Only if zn_primroot_pretest is set to true this pretest will be performed.

Option variable: zn_primroot_verbose

Default value: false

Controls whether zn_primroot prints a message when reaching zn_primroot_limit.


Next: , Previous: , Up: Top   [Contents][Index]

30 Symmetries


Next: , Previous: , Up: Symmetries   [Contents][Index]

30.1 Introduction to Symmetries

sym is a package for working with symmetric groups of polynomials.

It was written for Macsyma-Symbolics by Annick Valibouze (http://www-calfor.lip6.fr/~avb/). The algorithms are described in the following papers:

  1. Fonctions symétriques et changements de bases. Annick Valibouze. EUROCAL’87 (Leipzig, 1987), 323–332, Lecture Notes in Comput. Sci 378. Springer, Berlin, 1989.
    http://www.stix.polytechnique.fr/publications/1984-1994.html
  2. Résolvantes et fonctions symétriques. Annick Valibouze. Proceedings of the ACM-SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation, ISSAC’89 (Portland, Oregon). ACM Press, 390-399, 1989.
    http://www-calfor.lip6.fr/~avb/DonneesTelechargeables/MesArticles/issac89ACMValibouze.pdf
  3. Symbolic computation with symmetric polynomials, an extension to Macsyma. Annick Valibouze. Computers and Mathematics (MIT, USA, June 13-17, 1989), Springer-Verlag, New York Berlin, 308-320, 1989.
    http://www.stix.polytechnique.fr/publications/1984-1994.html
  4. Théorie de Galois Constructive. Annick Valibouze. Mémoire d’habilitation à diriger les recherches (HDR), Université P. et M. Curie (Paris VI), 1994.
·

Category: Group theory · Category: Polynomials · Category: Share packages · Category: Package sym


Previous: , Up: Symmetries   [Contents][Index]

30.2 Functions and Variables for Symmetries

30.2.1 Changing bases

Function: comp2pui (n, L)

implements passing from the complete symmetric functions given in the list L to the elementary symmetric functions from 0 to n. If the list L contains fewer than n+1 elements, it will be completed with formal values of the type h1, h2, etc. If the first element of the list L exists, it specifies the size of the alphabet, otherwise the size is set to n.

(%i1) comp2pui (3, [4, g]);
                        2                    2
(%o1)    [4, g, 2 h2 - g , 3 h3 - g h2 + g (g  - 2 h2)]

Function: ele2pui (m, L)

goes from the elementary symmetric functions to the complete functions. Similar to comp2ele and comp2pui.

Other functions for changing bases: comp2ele.

Function: ele2comp (m, L)

Goes from the elementary symmetric functions to the compete functions. Similar to comp2ele and comp2pui.

Other functions for changing bases: comp2ele.

Function: elem (ele, sym, lvar)

decomposes the symmetric polynomial sym, in the variables contained in the list lvar, in terms of the elementary symmetric functions given in the list ele. If the first element of ele is given, it will be the size of the alphabet, otherwise the size will be the degree of the polynomial sym. If values are missing in the list ele, formal values of the type e1, e2, etc. will be added. The polynomial sym may be given in three different forms: contracted (elem should then be 1, its default value), partitioned (elem should be 3), or extended (i.e. the entire polynomial, and elem should then be 2). The function pui is used in the same way.

On an alphabet of size 3 with e1, the first elementary symmetric function, with value 7, the symmetric polynomial in 3 variables whose contracted form (which here depends on only two of its variables) is x^4-2*x*y decomposes as follows in elementary symmetric functions:

(%i1) elem ([3, 7], x^4 - 2*x*y, [x, y]);
(%o1) 7 (e3 - 7 e2 + 7 (49 - e2)) + 21 e3

                                         + (- 2 (49 - e2) - 2) e2
(%i2) ratsimp (%);
                              2
(%o2)             28 e3 + 2 e2  - 198 e2 + 2401

Other functions for changing bases: comp2ele.

Function: mon2schur (L)

The list L represents the Schur function S_L: we have L = [i_1, i_2, ..., i_q], with i_1 <= i_2 <= ... <= i_q. The Schur function S_[i_1, i_2, ..., i_q] is the minor of the infinite matrix h_[i-j], i <= 1, j <= 1, consisting of the q first rows and the columns 1 + i_1, 2 + i_2, ..., q + i_q.

This Schur function can be written in terms of monomials by using treinat and kostka. The form returned is a symmetric polynomial in a contracted representation in the variables x_1,x_2,...

(%i1) mon2schur ([1, 1, 1]);
(%o1)                       x1 x2 x3
(%i2) mon2schur ([3]);
                                  2        3
(%o2)                x1 x2 x3 + x1  x2 + x1
(%i3) mon2schur ([1, 2]);
                                      2
(%o3)                  2 x1 x2 x3 + x1  x2

which means that for 3 variables this gives:

   2 x1 x2 x3 + x1^2 x2 + x2^2 x1 + x1^2 x3 + x3^2 x1
    + x2^2 x3 + x3^2 x2

Other functions for changing bases: comp2ele.

Function: multi_elem (l_elem, multi_pc, l_var)

decomposes a multi-symmetric polynomial in the multi-contracted form multi_pc in the groups of variables contained in the list of lists l_var in terms of the elementary symmetric functions contained in l_elem.

(%i1) multi_elem ([[2, e1, e2], [2, f1, f2]], a*x + a^2 + x^3,
      [[x, y], [a, b]]);
                                                  3
(%o1)         - 2 f2 + f1 (f1 + e1) - 3 e1 e2 + e1
(%i2) ratsimp (%);
                         2                       3
(%o2)         - 2 f2 + f1  + e1 f1 - 3 e1 e2 + e1

Other functions for changing bases: comp2ele.

Function: multi_pui

is to the function pui what the function multi_elem is to the function elem.

(%i1) multi_pui ([[2, p1, p2], [2, t1, t2]], a*x + a^2 + x^3,
      [[x, y], [a, b]]);
                                            3
                                3 p1 p2   p1
(%o1)              t2 + p1 t1 + ------- - ---
                                   2       2

Function: pui (L, sym, lvar)

decomposes the symmetric polynomial sym, in the variables in the list lvar, in terms of the power functions in the list L. If the first element of L is given, it will be the size of the alphabet, otherwise the size will be the degree of the polynomial sym. If values are missing in the list L, formal values of the type p1, p2 , etc. will be added. The polynomial sym may be given in three different forms: contracted (elem should then be 1, its default value), partitioned (elem should be 3), or extended (i.e. the entire polynomial, and elem should then be 2). The function pui is used in the same way.

(%i1) pui;
(%o1)                           1
(%i2) pui ([3, a, b], u*x*y*z, [x, y, z]);
                       2
                   a (a  - b) u   (a b - p3) u
(%o2)              ------------ - ------------
                        6              3
(%i3) ratsimp (%);
                                       3
                      (2 p3 - 3 a b + a ) u
(%o3)                 ---------------------
                                6

Other functions for changing bases: comp2ele.

Function: pui2comp (n, lpui)

renders the list of the first n complete functions (with the length first) in terms of the power functions given in the list lpui. If the list lpui is empty, the cardinal is n, otherwise it is its first element (as in comp2ele and comp2pui).

(%i1) pui2comp (2, []);
                                       2
                                p2 + p1
(%o1)                   [2, p1, --------]
                                   2
(%i2) pui2comp (3, [2, a1]);
                                            2
                                 a1 (p2 + a1 )
                         2  p3 + ------------- + a1 p2
                  p2 + a1              2
(%o2)     [2, a1, --------, --------------------------]
                     2                  3
(%i3) ratsimp (%);
                            2                     3
                     p2 + a1   2 p3 + 3 a1 p2 + a1
(%o3)        [2, a1, --------, --------------------]
                        2               6

Other functions for changing bases: comp2ele.

Function: pui2ele (n, lpui)

effects the passage from power functions to the elementary symmetric functions. If the flag pui2ele is girard, it will return the list of elementary symmetric functions from 1 to n, and if the flag is close, it will return the n-th elementary symmetric function.

Other functions for changing bases: comp2ele.

Function: puireduc (n, lpui)

lpui is a list whose first element is an integer m. puireduc gives the first n power functions in terms of the first m.

(%i1) puireduc (3, [2]);
                                         2
                                   p1 (p1  - p2)
(%o1)          [2, p1, p2, p1 p2 - -------------]
                                         2
(%i2) ratsimp (%);
                                           3
                               3 p1 p2 - p1
(%o2)              [2, p1, p2, -------------]
                                     2

Function: schur2comp (P, l_var)

P is a polynomial in the variables of the list l_var. Each of these variables represents a complete symmetric function. In l_var the i-th complete symmetric function is represented by the concatenation of the letter h and the integer i: hi. This function expresses P in terms of Schur functions.

(%i1) schur2comp (h1*h2 - h3, [h1, h2, h3]);
(%o1)                         s
                               1, 2
(%i2) schur2comp (a*h3, [h3]);
(%o2)                         s  a
                               3

30.2.2 Changing representations

Function: cont2part (pc, lvar)

returns the partitioned polynomial associated to the contracted form pc whose variables are in lvar.

(%i1) pc: 2*a^3*b*x^4*y + x^5;
                           3    4      5
(%o1)                   2 a  b x  y + x
(%i2) cont2part (pc, [x, y]);
                                   3
(%o2)              [[1, 5, 0], [2 a  b, 4, 1]]

Function: contract (psym, lvar)

returns a contracted form (i.e. a monomial orbit under the action of the symmetric group) of the polynomial psym in the variables contained in the list lvar. The function explose performs the inverse operation. The function tcontract tests the symmetry of the polynomial.

(%i1) psym: explose (2*a^3*b*x^4*y, [x, y, z]);
         3      4      3      4      3    4        3    4
(%o1) 2 a  b y z  + 2 a  b x z  + 2 a  b y  z + 2 a  b x  z

                                           3      4      3    4
                                      + 2 a  b x y  + 2 a  b x  y
(%i2) contract (psym, [x, y, z]);
                              3    4
(%o2)                      2 a  b x  y

Function: explose (pc, lvar)

returns the symmetric polynomial associated with the contracted form pc. The list lvar contains the variables.

(%i1) explose (a*x + 1, [x, y, z]);
(%o1)                  a z + a y + a x + 1

Function: part2cont (ppart, lvar)

goes from the partitioned form to the contracted form of a symmetric polynomial. The contracted form is rendered with the variables in lvar.

(%i1) part2cont ([[2*a^3*b, 4, 1]], [x, y]);
                              3    4
(%o1)                      2 a  b x  y

Function: partpol (psym, lvar)

psym is a symmetric polynomial in the variables of the list lvar. This function retturns its partitioned representation.

(%i1) partpol (-a*(x + y) + 3*x*y, [x, y]);
(%o1)               [[3, 1, 1], [- a, 1, 0]]

Function: tcontract (pol, lvar)

tests if the polynomial pol is symmetric in the variables of the list lvar. If so, it returns a contracted representation like the function contract.

Function: tpartpol (pol, lvar)

tests if the polynomial pol is symmetric in the variables of the list lvar. If so, it returns its partitioned representation like the function partpol.

30.2.3 Groups and orbits

Function: direct ([p_1, ..., p_n], y, f, [lvar_1, ..., lvar_n])

calculates the direct image (see M. Giusti, D. Lazard et A. Valibouze, ISSAC 1988, Rome) associated to the function f, in the lists of variables lvar_1, ..., lvar_n, and in the polynomials p_1, ..., p_n in a variable y. The arity of the function f is important for the calulation. Thus, if the expression for f does not depend on some variable, it is useless to include this variable, and not including it will also considerably reduce the amount of computation.

(%i1) direct ([z^2  - e1* z + e2, z^2  - f1* z + f2],
              z, b*v + a*u, [[u, v], [a, b]]);
       2
(%o1) y  - e1 f1 y

                                 2            2             2   2
                  - 4 e2 f2 - (e1  - 2 e2) (f1  - 2 f2) + e1  f1
                + -----------------------------------------------
                                         2
(%i2) ratsimp (%);
              2                2                   2
(%o2)        y  - e1 f1 y + (e1  - 4 e2) f2 + e2 f1
(%i3) ratsimp (direct ([z^3-e1*z^2+e2*z-e3,z^2  - f1* z + f2],
              z, b*v + a*u, [[u, v], [a, b]]));
       6            5         2                        2    2   4
(%o3) y  - 2 e1 f1 y  + ((2 e1  - 6 e2) f2 + (2 e2 + e1 ) f1 ) y

                          3                               3   3
 + ((9 e3 + 5 e1 e2 - 2 e1 ) f1 f2 + (- 2 e3 - 2 e1 e2) f1 ) y

         2       2        4    2
 + ((9 e2  - 6 e1  e2 + e1 ) f2

                    2       2       2                   2    4
 + (- 9 e1 e3 - 6 e2  + 3 e1  e2) f1  f2 + (2 e1 e3 + e2 ) f1 )

  2          2                      2     3          2
 y  + (((9 e1  - 27 e2) e3 + 3 e1 e2  - e1  e2) f1 f2

                 2            2    3                5
 + ((15 e2 - 2 e1 ) e3 - e1 e2 ) f1  f2 - 2 e2 e3 f1 ) y

           2                   3           3     2   2    3
 + (- 27 e3  + (18 e1 e2 - 4 e1 ) e3 - 4 e2  + e1  e2 ) f2

         2      3                   3    2   2
 + (27 e3  + (e1  - 9 e1 e2) e3 + e2 ) f1  f2

                   2    4        2   6
 + (e1 e2 e3 - 9 e3 ) f1  f2 + e3  f1

Finding the polynomial whose roots are the sums a+u where a is a root of z^2 - e_1 z + e_2 and u is a root of z^2 - f_1 z + f_2.

(%i1) ratsimp (direct ([z^2 - e1* z + e2, z^2 - f1* z + f2],
                          z, a + u, [[u], [a]]));
       4                    3             2
(%o1) y  + (- 2 f1 - 2 e1) y  + (2 f2 + f1  + 3 e1 f1 + 2 e2

     2   2                              2               2
 + e1 ) y  + ((- 2 f1 - 2 e1) f2 - e1 f1  + (- 2 e2 - e1 ) f1

                  2                     2            2
 - 2 e1 e2) y + f2  + (e1 f1 - 2 e2 + e1 ) f2 + e2 f1  + e1 e2 f1

     2
 + e2

direct accepts two flags: elementaires and puissances (default) which allow decomposing the symmetric polynomials appearing in the calculation into elementary symmetric functions, or power functions, respectively.

Functions of sym used in this function:

multi_orbit (so orbit), pui_direct, multi_elem (so elem), multi_pui (so pui), pui2ele, ele2pui (if the flag direct is in puissances).

Function: multi_orbit (P, [lvar_1, lvar_2,..., lvar_p])

P is a polynomial in the set of variables contained in the lists lvar_1, lvar_2, ..., lvar_p. This function returns the orbit of the polynomial P under the action of the product of the symmetric groups of the sets of variables represented in these p lists.

(%i1) multi_orbit (a*x + b*y, [[x, y], [a, b]]);
(%o1)                [b y + a x, a y + b x]
(%i2) multi_orbit (x + y + 2*a, [[x, y], [a, b, c]]);
(%o2)        [y + x + 2 c, y + x + 2 b, y + x + 2 a]

Also see: orbit for the action of a single symmetric group.

Function: multsym (ppart_1, ppart_2, n)

returns the product of the two symmetric polynomials in n variables by working only modulo the action of the symmetric group of order n. The polynomials are in their partitioned form.

Given the 2 symmetric polynomials in x, y: 3*(x + y) + 2*x*y and 5*(x^2 + y^2) whose partitioned forms are [[3, 1], [2, 1, 1]] and [[5, 2]], their product will be

(%i1) multsym ([[3, 1], [2, 1, 1]], [[5, 2]], 2);
(%o1)         [[10, 3, 1], [15, 3, 0], [15, 2, 1]]

that is 10*(x^3*y + y^3*x) + 15*(x^2*y + y^2*x) + 15*(x^3 + y^3).

Functions for changing the representations of a symmetric polynomial:

contract, cont2part, explose, part2cont, partpol, tcontract, tpartpol.

Function: orbit (P, lvar)

computes the orbit of the polynomial P in the variables in the list lvar under the action of the symmetric group of the set of variables in the list lvar.

(%i1) orbit (a*x + b*y, [x, y]);
(%o1)                [a y + b x, b y + a x]
(%i2) orbit (2*x + x^2, [x, y]);
                        2         2
(%o2)                 [y  + 2 y, x  + 2 x]

See also multi_orbit for the action of a product of symmetric groups on a polynomial.

Function: pui_direct (orbite, [lvar_1, ..., lvar_n], [d_1, d_2, ..., d_n])

Let f be a polynomial in n blocks of variables lvar_1, ..., lvar_n. Let c_i be the number of variables in lvar_i, and SC be the product of n symmetric groups of degree c_1, ..., c_n. This group acts naturally on f. The list orbite is the orbit, denoted SC(f), of the function f under the action of SC. (This list may be obtained by the function multi_orbit.) The di are integers s.t. c_1 <= d_1, c_2 <= d_2, ..., c_n <= d_n.

Let SD be the product of the symmetric groups S_[d_1] x S_[d_2] x ... x S_[d_n]. The function pui_direct returns the first n power functions of SD(f) deduced from the power functions of SC(f), where n is the size of SD(f).

The result is in multi-contracted form w.r.t. SD, i.e. only one element is kept per orbit, under the action of SD.

(%i1) l: [[x, y], [a, b]];
(%o1)                   [[x, y], [a, b]]
(%i2) pui_direct (multi_orbit (a*x + b*y, l), l, [2, 2]);
                                       2  2
(%o2)               [a x, 4 a b x y + a  x ]
(%i3) pui_direct (multi_orbit (a*x + b*y, l), l, [3, 2]);
                             2  2     2    2        3  3
(%o3) [2 a x, 4 a b x y + 2 a  x , 3 a  b x  y + 2 a  x , 

    2  2  2  2      3    3        4  4
12 a  b  x  y  + 4 a  b x  y + 2 a  x , 

    3  2  3  2      4    4        5  5
10 a  b  x  y  + 5 a  b x  y + 2 a  x , 

    3  3  3  3       4  2  4  2      5    5        6  6
40 a  b  x  y  + 15 a  b  x  y  + 6 a  b x  y + 2 a  x ]
(%i4) pui_direct ([y + x + 2*c, y + x + 2*b, y + x + 2*a],
      [[x, y], [a, b, c]], [2, 3]);
                             2              2
(%o4) [3 x + 2 a, 6 x y + 3 x  + 4 a x + 4 a , 

                 2                   3        2       2        3
              9 x  y + 12 a x y + 3 x  + 6 a x  + 12 a  x + 8 a ]

30.2.4 Partitions

Function: kostka (part_1, part_2)

written by P. Esperet, calculates the Kostka number of the partition part_1 and part_2.

(%i1) kostka ([3, 3, 3], [2, 2, 2, 1, 1, 1]);
(%o1)                           6

Function: lgtreillis (n, m)

returns the list of partitions of weight n and length m.

(%i1) lgtreillis (4, 2);
(%o1)                   [[3, 1], [2, 2]]

Also see: ltreillis, treillis and treinat.

Function: ltreillis (n, m)

returns the list of partitions of weight n and length less than or equal to m.

(%i1) ltreillis (4, 2);
(%o1)               [[4, 0], [3, 1], [2, 2]]

Also see: lgtreillis, treillis and treinat.

Function: treillis (n)

returns all partitions of weight n.

(%i1) treillis (4);
(%o1)    [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]

See also: lgtreillis, ltreillis and treinat.

Function: treinat (part)

retruns the list of partitions inferior to the partition part w.r.t. the natural order.

(%i1) treinat ([5]);
(%o1)                         [[5]]
(%i2) treinat ([1, 1, 1, 1, 1]);
(%o2) [[5], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], 

                                                 [1, 1, 1, 1, 1]]
(%i3) treinat ([3, 2]);
(%o3)                 [[5], [4, 1], [3, 2]]

See also: lgtreillis, ltreillis and treillis.

30.2.5 Polynomials and their roots

Function: ele2polynome (L, z)

returns the polynomial in z s.t. the elementary symmetric functions of its roots are in the list L = [n, e_1, ..., e_n], where n is the degree of the polynomial and e_i the i-th elementary symmetric function.

(%i1) ele2polynome ([2, e1, e2], z);
                          2
(%o1)                    z  - e1 z + e2
(%i2) polynome2ele (x^7 - 14*x^5 + 56*x^3  - 56*x + 22, x);
(%o2)          [7, 0, - 14, 0, 56, 0, - 56, - 22]
(%i3) ele2polynome ([7, 0, -14, 0, 56, 0, -56, -22], x);
                  7       5       3
(%o3)            x  - 14 x  + 56 x  - 56 x + 22

The inverse: polynome2ele (P, z).

Also see: polynome2ele, pui2polynome.

Function: polynome2ele (P, x)

gives the list l = [n, e_1, ..., e_n] where n is the degree of the polynomial P in the variable x and e_i is the i-the elementary symmetric function of the roots of P.

(%i1) polynome2ele (x^7 - 14*x^5 + 56*x^3 - 56*x + 22, x);
(%o1)          [7, 0, - 14, 0, 56, 0, - 56, - 22]
(%i2) ele2polynome ([7, 0, -14, 0, 56, 0, -56, -22], x);
                  7       5       3
(%o2)            x  - 14 x  + 56 x  - 56 x + 22

The inverse: ele2polynome (l, x)

Function: prodrac (L, k)

L is a list containing the elementary symmetric functions on a set A. prodrac returns the polynomial whose roots are the k by k products of the elements of A.

Also see somrac.

Function: pui2polynome (x, lpui)

calculates the polynomial in x whose power functions of the roots are given in the list lpui.

(%i1) pui;
(%o1)                           1
(%i2) kill(labels);
(%o0)                         done
(%i1) polynome2ele (x^3 - 4*x^2 + 5*x - 1, x);
(%o1)                     [3, 4, 5, 1]
(%i2) ele2pui (3, %);
(%o2)                     [3, 4, 6, 7]
(%i3) pui2polynome (x, %);
                        3      2
(%o3)                  x  - 4 x  + 5 x - 1

See also: polynome2ele, ele2polynome.

Function: somrac (L, k)

The list L contains elementary symmetric functions of a polynomial P . The function computes the polynomial whose roots are the k by k distinct sums of the roots of P.

Also see prodrac.

30.2.6 Resolvents

Function: resolvante (P, x, f, [x_1,..., x_d])

calculates the resolvent of the polynomial P in x of degree n >= d by the function f expressed in the variables x_1, ..., x_d. For efficiency of computation it is important to not include in the list [x_1, ..., x_d] variables which do not appear in the transformation function f.

To increase the efficiency of the computation one may set flags in resolvante so as to use appropriate algorithms:

If the function f is unitary:

the flag of resolvante may be, respectively:

(%i1) resolvante: unitaire$
(%i2) resolvante (x^7 - 14*x^5 + 56*x^3 - 56*x + 22, x, x^3 - 1,
      [x]);

" resolvante unitaire " [7, 0, 28, 0, 168, 0, 1120, - 154, 7840,
                         - 2772, 56448, - 33880, 

413952, - 352352, 3076668, - 3363360, 23114112, - 30494464, 

175230832, - 267412992, 1338886528, - 2292126760] 
  3       6      3       9      6      3
[x  - 1, x  - 2 x  + 1, x  - 3 x  + 3 x  - 1, 

 12      9      6      3       15      12       9       6      3
x   - 4 x  + 6 x  - 4 x  + 1, x   - 5 x   + 10 x  - 10 x  + 5 x

       18      15       12       9       6      3
 - 1, x   - 6 x   + 15 x   - 20 x  + 15 x  - 6 x  + 1, 

 21      18       15       12       9       6      3
x   - 7 x   + 21 x   - 35 x   + 35 x  - 21 x  + 7 x  - 1] 
[- 7, 1127, - 6139, 431767, - 5472047, 201692519, - 3603982011] 
       7      6        5         4          3           2
(%o2) y  + 7 y  - 539 y  - 1841 y  + 51443 y  + 315133 y

                                              + 376999 y + 125253
(%i3) resolvante: lineaire$
(%i4) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3]);

" resolvante lineaire " 
       24       20         16            12             8
(%o4) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y

                                                    4
                                       + 344489984 y  + 655360000
(%i5) resolvante: general$
(%i6) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3]);

" resolvante generale " 
       24       20         16            12             8
(%o6) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y

                                                    4
                                       + 344489984 y  + 655360000
(%i7) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3, x4]);

" resolvante generale " 
       24       20         16            12             8
(%o7) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y

                                                    4
                                       + 344489984 y  + 655360000
(%i8) direct ([x^4 - 1], x, x1 + 2*x2 + 3*x3, [[x1, x2, x3]]);
       24       20         16            12             8
(%o8) y   + 80 y   + 7520 y   + 1107200 y   + 49475840 y

                                                    4
                                       + 344489984 y  + 655360000
(%i9) resolvante :lineaire$
(%i10) resolvante (x^4 - 1, x, x1 + x2 + x3, [x1, x2, x3]);

" resolvante lineaire " 
                              4
(%o10)                       y  - 1
(%i11) resolvante: symetrique$
(%i12) resolvante (x^4 - 1, x, x1 + x2 + x3, [x1, x2, x3]);

" resolvante symetrique " 
                              4
(%o12)                       y  - 1
(%i13) resolvante (x^4 + x + 1, x, x1 - x2, [x1, x2]);

" resolvante symetrique " 
                           6      2
(%o13)                    y  - 4 y  - 1
(%i14) resolvante: alternee$
(%i15) resolvante (x^4 + x + 1, x, x1 - x2, [x1, x2]);

" resolvante alternee " 
            12      8       6        4        2
(%o15)     y   + 8 y  + 26 y  - 112 y  + 216 y  + 229
(%i16) resolvante: produit$
(%i17) resolvante (x^7 - 7*x + 3, x, x1*x2*x3, [x1, x2, x3]);

" resolvante produit "
        35      33         29        28         27        26
(%o17) y   - 7 y   - 1029 y   + 135 y   + 7203 y   - 756 y

         24           23          22            21           20
 + 1323 y   + 352947 y   - 46305 y   - 2463339 y   + 324135 y

          19           18             17              15
 - 30618 y   - 453789 y   - 40246444 y   + 282225202 y

             14              12             11            10
 - 44274492 y   + 155098503 y   + 12252303 y   + 2893401 y

              9            8            7             6
 - 171532242 y  + 6751269 y  + 2657205 y  - 94517766 y

            5             3
 - 3720087 y  + 26040609 y  + 14348907
(%i18) resolvante: symetrique$
(%i19) resolvante (x^7 - 7*x + 3, x, x1*x2*x3, [x1, x2, x3]);

" resolvante symetrique " 
        35      33         29        28         27        26
(%o19) y   - 7 y   - 1029 y   + 135 y   + 7203 y   - 756 y

         24           23          22            21           20
 + 1323 y   + 352947 y   - 46305 y   - 2463339 y   + 324135 y

          19           18             17              15
 - 30618 y   - 453789 y   - 40246444 y   + 282225202 y

             14              12             11            10
 - 44274492 y   + 155098503 y   + 12252303 y   + 2893401 y

              9            8            7             6
 - 171532242 y  + 6751269 y  + 2657205 y  - 94517766 y

            5             3
 - 3720087 y  + 26040609 y  + 14348907
(%i20) resolvante: cayley$
(%i21) resolvante (x^5 - 4*x^2 + x + 1, x, a, []);

" resolvante de Cayley "
        6       5         4          3            2
(%o21) x  - 40 x  + 4080 x  - 92928 x  + 3772160 x  + 37880832 x

                                                       + 93392896

For the Cayley resolvent, the 2 last arguments are neutral and the input polynomial must necessarily be of degree 5.

See also:

resolvante_bipartite, resolvante_produit_sym, resolvante_unitaire, resolvante_alternee1, resolvante_klein, resolvante_klein3, resolvante_vierer, resolvante_diedrale.

Function: resolvante_alternee1 (P, x)

calculates the transformation P(x) of degree n by the function product(x_i - x_j, 1 <= i < j <= n - 1).

See also:

resolvante_produit_sym, resolvante_unitaire, resolvante, resolvante_klein, resolvante_klein3, resolvante_vierer, resolvante_diedrale, resolvante_bipartite.

Function: resolvante_bipartite (P, x)

calculates the transformation of P(x) of even degree n by the function x_1 x_2 ... x_[n/2] + x_[n/2 + 1] ... x_n.

(%i1) resolvante_bipartite (x^6 + 108, x);
              10        8           6             4
(%o1)        y   - 972 y  + 314928 y  - 34012224 y

See also:

resolvante_produit_sym, resolvante_unitaire, resolvante, resolvante_klein, resolvante_klein3, resolvante_vierer, resolvante_diedrale, resolvante_alternee1.

Function: resolvante_diedrale (P, x)

calculates the transformation of P(x) by the function x_1 x_2 + x_3 x_4.

(%i1) resolvante_diedrale (x^5 - 3*x^4 + 1, x);
       15       12       11       10        9         8         7
(%o1) x   - 21 x   - 81 x   - 21 x   + 207 x  + 1134 x  + 2331 x

        6         5          4          3          2
 - 945 x  - 4970 x  - 18333 x  - 29079 x  - 20745 x  - 25326 x

 - 697

See also:

resolvante_produit_sym, resolvante_unitaire, resolvante_alternee1, resolvante_klein, resolvante_klein3, resolvante_vierer, resolvante.

Function: resolvante_klein (P, x)

calculates the transformation of P(x) by the function x_1 x_2 x_4 + x_4.

See also:

resolvante_produit_sym, resolvante_unitaire, resolvante_alternee1, resolvante, resolvante_klein3, resolvante_vierer, resolvante_diedrale.

Function: resolvante_klein3 (P, x)

calculates the transformation of P(x) by the function x_1 x_2 x_4 + x_4.

See also:

resolvante_produit_sym, resolvante_unitaire, resolvante_alternee1, resolvante_klein, resolvante, resolvante_vierer, resolvante_diedrale.

Function: resolvante_produit_sym (P, x)

calculates the list of all product resolvents of the polynomial P(x).

(%i1) resolvante_produit_sym (x^5 + 3*x^4 + 2*x - 1, x);
        5      4             10      8       7       6       5
(%o1) [y  + 3 y  + 2 y - 1, y   - 2 y  - 21 y  - 31 y  - 14 y

    4       3      2       10      8       7    6       5       4
 - y  + 14 y  + 3 y  + 1, y   + 3 y  + 14 y  - y  - 14 y  - 31 y

       3      2       5      4
 - 21 y  - 2 y  + 1, y  - 2 y  - 3 y - 1, y - 1]
(%i2) resolvante: produit$
(%i3) resolvante (x^5 + 3*x^4 + 2*x - 1, x, a*b*c, [a, b, c]);

" resolvante produit "
       10      8       7    6        5       4       3     2
(%o3) y   + 3 y  + 14 y  - y  - 14 y  - 31 y  - 21 y  - 2 y  + 1

See also:

resolvante, resolvante_unitaire, resolvante_alternee1, resolvante_klein, resolvante_klein3, resolvante_vierer, resolvante_diedrale.

Function: resolvante_unitaire (P, Q, x)

computes the resolvent of the polynomial P(x) by the polynomial Q(x).

See also:

resolvante_produit_sym, resolvante, resolvante_alternee1, resolvante_klein, resolvante_klein3, resolvante_vierer, resolvante_diedrale.

Function: resolvante_vierer (P, x)

computes the transformation of P(x) by the function x_1 x_2 - x_3 x_4.

See also:

resolvante_produit_sym, resolvante_unitaire, resolvante_alternee1, resolvante_klein, resolvante_klein3, resolvante, resolvante_diedrale.

30.2.7 Miscellaneous

Function: multinomial (r, part)

where r is the weight of the partition part. This function returns the associate multinomial coefficient: if the parts of part are i_1, i_2, ..., i_k, the result is r!/(i_1! i_2! ... i_k!).

Function: permut (L)

returns the list of permutations of the list L.


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31 Groups


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31.1 Functions and Variables for Groups

Function: todd_coxeter
    todd_coxeter (relations, subgroup)
    todd_coxeter (relations)

Find the order of G/H where G is the Free Group modulo relations, and H is the subgroup of G generated by subgroup. subgroup is an optional argument, defaulting to []. In doing this it produces a multiplication table for the right action of G on G/H, where the cosets are enumerated [H,Hg2,Hg3,...]. This can be seen internally in the variable todd_coxeter_state.

Example:

(%i1) symet(n):=create_list(
        if (j - i) = 1 then (p(i,j))^^3 else
            if (not i = j) then (p(i,j))^^2 else
                p(i,i) , j, 1, n-1, i, 1, j);
                                                       <3>
(%o1) symet(n) := create_list(if j - i = 1 then p(i, j)

                                <2>
 else (if not i = j then p(i, j)    else p(i, i)), j, 1, n - 1,

i, 1, j)
(%i2) p(i,j) := concat(x,i).concat(x,j);
(%o2)        p(i, j) := concat(x, i) . concat(x, j)
(%i3) symet(5);
         <2>           <3>    <2>           <2>           <3>
(%o3) [x1   , (x1 . x2)   , x2   , (x1 . x3)   , (x2 . x3)   ,

            <2>           <2>           <2>           <3>    <2>
          x3   , (x1 . x4)   , (x2 . x4)   , (x3 . x4)   , x4   ]
(%i4) todd_coxeter(%o3);

Rows tried 426
(%o4)                          120
(%i5) todd_coxeter(%o3,[x1]);

Rows tried 213
(%o5)                          60
(%i6) todd_coxeter(%o3,[x1,x2]);

Rows tried 71
(%o6)                          20


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32 Runtime Environment


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32.1 Introduction for Runtime Environment

maxima-init.mac is a file which is loaded automatically when Maxima starts. You can use maxima-init.mac to customize your Maxima environment. maxima-init.mac, if it exists, is typically placed in the directory named by maxima_userdir, although it can be in any directory searched by the function file_search.

Here is an example maxima-init.mac file:

setup_autoload ("specfun.mac", ultraspherical, assoc_legendre_p);
showtime:all;

In this example, setup_autoload tells Maxima to load the specified file (specfun.mac) if any of the functions (ultraspherical, assoc_legendre_p) are called but not yet defined. Thus you needn’t remember to load the file before calling the functions.

The statement showtime: all tells Maxima to set the showtime variable. The maxima-init.mac file can contain any other assignments or other Maxima statements.

·

Category: Session management


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32.2 Interrupts

The user can stop a time-consuming computation with the ^C (control-C) character. The default action is to stop the computation and print another user prompt. In this case, it is not possible to restart a stopped computation.

If the Lisp variable *debugger-hook* is set to nil, by executing

:lisp (setq *debugger-hook* nil)

then upon receiving ^C, Maxima will enter the Lisp debugger, and the user may use the debugger to inspect the Lisp environment. The stopped computation can be restarted by entering continue in the Lisp debugger. The means of returning to Maxima from the Lisp debugger (other than running the computation to completion) is different for each version of Lisp.

On Unix systems, the character ^Z (control-Z) causes Maxima to stop altogether, and control is returned to the shell prompt. The fg command causes Maxima to resume from the point at which it was stopped.

·

Category: Console interaction


Previous: , Up: Runtime Environment   [Contents][Index]

32.3 Functions and Variables for Runtime Environment

System variable: maxima_tempdir

maxima_tempdir names the directory in which Maxima creates some temporary files. In particular, temporary files for plotting are created in maxima_tempdir.

The initial value of maxima_tempdir is the user’s home directory, if Maxima can locate it; otherwise Maxima makes a guess about a suitable directory.

maxima_tempdir may be assigned a string which names a directory.

System variable: maxima_userdir

maxima_userdir names a directory which Maxima searches to find Maxima and Lisp files. (Maxima searches some other directories as well; file_search_maxima and file_search_lisp are the complete lists.)

The initial value of maxima_userdir is a subdirectory of the user’s home directory, if Maxima can locate it; otherwise Maxima makes a guess about a suitable directory.

maxima_userdir may be assigned a string which names a directory. However, assigning to maxima_userdir does not automatically change file_search_maxima and file_search_lisp; those variables must be changed separately.

Function: room
    room ()
    room (true)
    room (false)

Prints out a description of the state of storage and stack management in Maxima. room calls the Lisp function of the same name.

Function: sstatus (keyword, item)

When keyword is the symbol feature, item is put on the list of system features. After sstatus (keyword, item) is executed, status (feature, item) returns true. If keyword is the symbol nofeature, item is deleted from the list of system features. This can be useful for package writers, to keep track of what features they have loaded in.

See also status.

Function: status
    status (feature)
    status (feature, item)

Returns information about the presence or absence of certain system-dependent features.

See also sstatus.

The variable features contains a list of features which apply to mathematical expressions. See features and featurep for more information.

Function: system (command)

Executes command as a separate process. The command is passed to the default shell for execution. system is not supported by all operating systems, but generally exists in Unix and Unix-like environments.

Supposing _hist.out is a list of frequencies which you wish to plot as a bar graph using xgraph.

(%i1) (with_stdout("_hist.out",
           for i:1 thru length(hist) do (
             print(i,hist[i]))),
       system("xgraph -bar -brw .7 -nl < _hist.out"));

In order to make the plot be done in the background (returning control to Maxima) and remove the temporary file after it is done do:

system("(xgraph -bar -brw .7 -nl < _hist.out;  rm -f _hist.out)&")
Function: time (%o1, %o2, %o3, …)

Returns a list of the times, in seconds, taken to compute the output lines %o1, %o2, %o3, … The time returned is Maxima’s estimate of the internal computation time, not the elapsed time. time can only be applied to output line variables; for any other variables, time returns unknown.

Set showtime: true to make Maxima print out the computation time and elapsed time with each output line.

Function: timedate
    timedate (T, tz_offset)
    timedate (T)
    timedate ()

timedate(T, tz_offset) returns a string representing the time T in the time zone tz_offset. The string format is YYYY-MM-DD HH:MM:SS.NNN[+|-]ZZ:ZZ (using as many digits as necessary to represent the fractional part) if T has a nonzero fractional part, or YYYY-MM-DD HH:MM:SS[+|-]ZZ:ZZ if its fractional part is zero.

T measures time, in seconds, since midnight, January 1, 1900, in the GMT time zone.

tz_offset measures the offset of the time zone, in hours, east (positive) or west (negative) of GMT. tz_offset must be an integer, rational, or float between -24 and 24, inclusive. If tz_offset is not a multiple of 1/60, it is rounded to the nearest multiple of 1/60.

timedate(T) is equivalent to timedate(T, tz_offset) with tz_offset equal to the offset of the local time zone.

timedate() is equivalent to timedate(absolute_real_time()). That is, it returns the current time in the local time zone.

Example:

timedate with no argument returns a string representing the current time and date.

(%i1) d : timedate ();
(%o1)                      2010-06-08 04:08:09+01:00
(%i2) print ("timedate reports current time", d) $
timedate reports current time 2010-06-08 04:08:09+01:00

timedate with an argument returns a string representing the argument.

(%i1) timedate (0);
(%o1)                      1900-01-01 01:00:00+01:00
(%i2) timedate (absolute_real_time () - 7*24*3600);
(%o2)                      2010-06-01 04:19:51+01:00

timedate with optional timezone offset.

(%i1) timedate (1000000000, -9.5);
(%o1)               1931-09-09 16:16:40-09:30

Function: parse_timedate
    parse_timedate (S)

Parses a string S representing a date or date and time of day and returns the number of seconds since midnight, January 1, 1900 GMT. If there is a nonzero fractional part, the value returned is a rational number, otherwise, it is an integer. parse_timedate returns false if it cannot parse S according to any of the allowed formats.

The string S must have one of the following formats, optionally followed by a timezone designation:

where the fields are year, month, day, hours, minutes, seconds, and fraction of a second, and square brackets indicate acceptable alternatives. The fraction may contain one or more digits.

Except for the fraction of a second, each field must have exactly the number of digits indicated: four digits for the year, and two for the month, day of the month, hours, minutes, and seconds.

A timezone designation must have one of the following forms:

where hh and mm indicate hours and minutes east (+) or west (-) of GMT. The timezone may be from +24 hours (inclusive) to -24 hours (inclusive).

A literal character Z is equivalent to +00:00 and its variants, indicating GMT.

If no timezone is indicated, the time is assumed to be in the local time zone.

Any leading or trailing whitespace (space, tab, newline, and carriage return) is ignored, but any other leading or trailing characters cause parse_timedate to fail and return false.

See also timedate and absolute_real_time.

Examples:

Midnight, January 1, 1900, in the local time zone, in each acceptable format. The result is the number of seconds the local time zone is ahead (negative result) or behind (positive result) GMT. In this example, the local time zone is 8 hours behind GMT.

(%i1) parse_timedate ("1900-01-01 00:00:00,000");
(%o1)                         28800
(%i2) parse_timedate ("1900-01-01 00:00:00.000");
(%o2)                         28800
(%i3) parse_timedate ("1900-01-01T00:00:00,000");
(%o3)                         28800
(%i4) parse_timedate ("1900-01-01T00:00:00.000");
(%o4)                         28800
(%i5) parse_timedate ("1900-01-01 00:00:00");
(%o5)                         28800
(%i6) parse_timedate ("1900-01-01T00:00:00");
(%o6)                         28800
(%i7) parse_timedate ("1900-01-01");
(%o7)                         28800

Midnight, January 1, 1900, GMT, in different indicated time zones.

(%i1) parse_timedate ("1900-01-01 19:00:00+19:00");
(%o1)                           0
(%i2) parse_timedate ("1900-01-01 07:00:00+07:00");
(%o2)                           0
(%i3) parse_timedate ("1900-01-01 01:00:00+01:00");
(%o3)                           0
(%i4) parse_timedate ("1900-01-01Z");
(%o4)                           0
(%i5) parse_timedate ("1899-12-31 21:00:00-03:00");
(%o5)                           0
(%i6) parse_timedate ("1899-12-31 13:00:00-11:00");
(%o6)                           0
(%i7) parse_timedate ("1899-12-31 08:00:00-16:00");
(%o7)                           0

Function: encode_time
    encode_time (year, month, day, hours, minutes, seconds, tz_offset)
    encode_time (year, month, day, hours, minutes, seconds)

Given a time and date specified by year, month, day, hours, minutes, and seconds, encode_time returns the number of seconds (possibly including a fractional part) since midnight, January 1, 1900 GMT.

year must be an integer greater than or equal to 1899. However, 1899 is allowed only if the resulting encoded time is greater than or equal to 0.

month must be an integer from 1 to 12, inclusive.

day must be an integer from 1 to n, inclusive, where n is the number of days in the month specified by month.

hours must be an integer from 0 to 23, inclusive.

minutes must be an integer from 0 to 59, inclusive.

seconds must be an integer, rational, or float greater than or equal to 0 and less than 60. When seconds is not an integer, encode_time returns a rational, such that the fractional part of the return value is equal to the fractional part of seconds. Otherwise, seconds is an integer, and the return value is likewise an integer.

tz_offset measures the offset of the time zone, in hours, east (positive) or west (negative) of GMT. tz_offset must be an integer, rational, or float between -24 and 24, inclusive. If tz_offset is not a multiple of 1/3600, it is rounded to the nearest multiple of 1/3600.

If tz_offset is not present, the offset of the local time zone is assumed.

See also decode_time.

Examples:

(%i1) encode_time (1900, 1, 1, 0, 0, 0, 0);
(%o1)                           0
(%i2) encode_time (1970, 1, 1, 0, 0, 0, 0);
(%o2)                      2208988800
(%i3) encode_time (1970, 1, 1, 8, 30, 0, 8.5);
(%o3)                      2208988800
(%i4) encode_time (1969, 12, 31, 16, 0, 0, -8);
(%o4)                      2208988800
(%i5) encode_time (1969, 12, 31, 16, 0, 1/1000, -8);
                          2208988800001
(%o5)                     -------------
                              1000
(%i6) % - 2208988800;
                               1
(%o6)                         ----
                              1000

Function: decode_time
    decode_time (T, tz_offset)
    decode_time (T)

Given the number of seconds (possibly including a fractional part) since midnight, January 1, 1900 GMT, returns the date and time as represented by a list comprising the year, month, day of the month, hours, minutes, seconds, and time zone offset.

tz_offset measures the offset of the time zone, in hours, east (positive) or west (negative) of GMT. tz_offset must be an integer, rational, or float between -24 and 24, inclusive. If tz_offset is not a multiple of 1/3600, it is rounded to the nearest multiple of 1/3600.

If tz_offset is not present, the offset of the local time zone is assumed.

See also encode_time.

Examples:

(%i1) decode_time (0, 0);
(%o1)               [1900, 1, 1, 0, 0, 0, 0]
(%i2) decode_time (0);
(%o2)             [1899, 12, 31, 16, 0, 0, - 8]
(%i3) decode_time (2208988800, 9.25);
                                          37
(%o3)              [1970, 1, 1, 9, 15, 0, --]
                                          4
(%i4) decode_time (2208988800);
(%o4)             [1969, 12, 31, 16, 0, 0, - 8]
(%i5) decode_time (2208988800 + 1729/1000, -6);
                                      1729
(%o5)           [1969, 12, 31, 18, 0, ----, - 6]
                                      1000
(%i6) decode_time (2208988800 + 1729/1000);
                                      1729
(%o6)           [1969, 12, 31, 16, 0, ----, - 8]
                                      1000

Function: absolute_real_time ()

Returns the number of seconds since midnight, January 1, 1900 GMT. The return value is an integer.

See also elapsed_real_time and elapsed_run_time.

Example:

(%i1) absolute_real_time ();
(%o1)                      3385045277
(%i2) 1900 + absolute_real_time () / (365.25 * 24 * 3600);
(%o2)                   2007.265612087104

Function: elapsed_real_time ()

Returns the number of seconds (including fractions of a second) since Maxima was most recently started or restarted. The return value is a floating-point number.

See also absolute_real_time and elapsed_run_time.

Example:

(%i1) elapsed_real_time ();
(%o1)                       2.559324
(%i2) expand ((a + b)^500)$
(%i3) elapsed_real_time ();
(%o3)                       7.552087

Function: elapsed_run_time ()

Returns an estimate of the number of seconds (including fractions of a second) which Maxima has spent in computations since Maxima was most recently started or restarted. The return value is a floating-point number.

See also absolute_real_time and elapsed_real_time.

Example:

(%i1) elapsed_run_time ();
(%o1)                         0.04
(%i2) expand ((a + b)^500)$
(%i3) elapsed_run_time ();
(%o3)                         1.26


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33 Miscellaneous Options


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33.1 Introduction to Miscellaneous Options

In this section various options are discussed which have a global effect on the operation of Maxima. Also various lists such as the list of all user defined functions, are discussed.


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33.2 Share

The Maxima "share" directory contains programs and other files of interest to Maxima users, but not part of the core implementation of Maxima. These programs are typically loaded via load or setup_autoload.

:lisp *maxima-sharedir* displays the location of the share directory within the user’s file system.

printfile ("share.usg") prints an out-of-date list of share packages. Users may find it more informative to browse the share directory using a file system browser.


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33.3 Functions and Variables for Miscellaneous Options

System variable: askexp

When asksign is called, askexp is the expression asksign is testing.

At one time, it was possible for a user to inspect askexp by entering a Maxima break with control-A.

Option variable: genindex

Default value: i

genindex is the alphabetic prefix used to generate the next variable of summation when necessary.

Option variable: gensumnum

Default value: 0

gensumnum is the numeric suffix used to generate the next variable of summation. If it is set to false then the index will consist only of genindex with no numeric suffix.

Function: gensym
    gensym ()
    gensym (x)

gensym() creates and returns a fresh symbol.

The name of the new symbol is the concatenation of a prefix, which defaults to "g", and a suffix, which is an integer that defaults to the value of an internal counter.

If x is supplied, and is a string, then that string is used as a prefix instead of "g" for this call to gensym only.

If x is supplied, and is a nonnegative integer, then that integer, instead of the value of the internal counter, is used as the suffix for this call to gensym only.

If and only if no explicit suffix is supplied, the internal counter is incremented after it is used.

Examples:

(%i1) gensym();
(%o1)                         g887
(%i2) gensym("new");
(%o2)                        new888
(%i3) gensym(123);
(%o3)                         g123
Option variable: packagefile

Default value: false

Package designers who use save or translate to create packages (files) for others to use may want to set packagefile: true to prevent information from being added to Maxima’s information-lists (e.g. values, functions) except where necessary when the file is loaded in. In this way, the contents of the package will not get in the user’s way when he adds his own data. Note that this will not solve the problem of possible name conflicts. Also note that the flag simply affects what is output to the package file. Setting the flag to true is also useful for creating Maxima init files.

Function: remvalue
    remvalue (name_1, …, name_n)
    remvalue remvalue (all)

Removes the values of user variables name_1, …, name_n (which can be subscripted) from the system.

remvalue (all) removes the values of all variables in values, the list of all variables given names by the user (as opposed to those which are automatically assigned by Maxima).

See also values.

Function: rncombine (expr)

Transforms expr by combining all terms of expr that have identical denominators or denominators that differ from each other by numerical factors only. This is slightly different from the behavior of combine, which collects terms that have identical denominators.

Setting pfeformat: true and using combine yields results similar to those that can be obtained with rncombine, but rncombine takes the additional step of cross-multiplying numerical denominator factors. This results in neater forms, and the possibility of recognizing some cancellations.

load(rncomb) loads this function.

Function: setup_autoload (filename, function_1, …, function_n)

Specifies that if any of function_1, …, function_n are referenced and not yet defined, filename is loaded via load. filename usually contains definitions for the functions specified, although that is not enforced.

setup_autoload does not work for memoizing functions.

setup_autoload quotes its arguments.

Example:

(%i1) legendre_p (1, %pi);
(%o1)                  legendre_p(1, %pi)
(%i2) setup_autoload ("specfun.mac", legendre_p, ultraspherical);
(%o2)                         done
(%i3) ultraspherical (2, 1/2, %pi);
Warning - you are redefining the Macsyma function ultraspherical
Warning - you are redefining the Macsyma function legendre_p
                            2
                 3 (%pi - 1)
(%o3)            ------------ + 3 (%pi - 1) + 1
                      2
(%i4) legendre_p (1, %pi);
(%o4)                          %pi
(%i5) legendre_q (1, %pi);
                              %pi + 1
                      %pi log(-------)
                              1 - %pi
(%o5)                 ---------------- - 1
                             2

Function: tcl_output
    tcl_output (list, i0, skip)
    tcl_output (list, i0)
    tcl_output ([list_1, …, list_n], i)

Prints elements of a list enclosed by curly braces { }, suitable as part of a program in the Tcl/Tk language.

tcl_output (list, i0, skip) prints list, beginning with element i0 and printing elements i0 + skip, i0 + 2 skip, etc.

tcl_output (list, i0) is equivalent to tcl_output (list, i0, 2).

tcl_output ([list_1, ..., list_n], i) prints the i’th elements of list_1, …, list_n.

Examples:

(%i1) tcl_output ([1, 2, 3, 4, 5, 6], 1, 3)$

 {1.000000000     4.000000000     
 }
(%i2) tcl_output ([1, 2, 3, 4, 5, 6], 2, 3)$

 {2.000000000     5.000000000     
 }
(%i3) tcl_output ([3/7, 5/9, 11/13, 13/17], 1)$

 {((RAT SIMP) 3 7) ((RAT SIMP) 11 13) 
 }
(%i4) tcl_output ([x1, y1, x2, y2, x3, y3], 2)$

 {$Y1 $Y2 $Y3 
 }
(%i5) tcl_output ([[1, 2, 3], [11, 22, 33]], 1)$

 {SIMP 1.000000000     11.00000000     
 }

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34 Rules and Patterns


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34.1 Introduction to Rules and Patterns

This section describes user-defined pattern matching and simplification rules. There are two groups of functions which implement somewhat different pattern matching schemes. In one group are tellsimp, tellsimpafter, defmatch, defrule, apply1, applyb1, and apply2. In the other group are let and letsimp. Both schemes define patterns in terms of pattern variables declared by matchdeclare.

Pattern-matching rules defined by tellsimp and tellsimpafter are applied automatically by the Maxima simplifier. Rules defined by defmatch, defrule, and let are applied by an explicit function call.

There are additional mechanisms for rules applied to polynomials by tellrat, and for commutative and noncommutative algebra in affine package.

·

Category: Simplification · Category: Rules and patterns


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34.2 Functions and Variables for Rules and Patterns

Function: apply1 (expr, rule_1, …, rule_n)

Repeatedly applies rule_1 to expr until it fails, then repeatedly applies the same rule to all subexpressions of expr, left to right, until rule_1 has failed on all subexpressions. Call the result of transforming expr in this manner expr_2. Then rule_2 is applied in the same fashion starting at the top of expr_2. When rule_n fails on the final subexpression, the result is returned.

maxapplydepth is the depth of the deepest subexpressions processed by apply1 and apply2.

See also applyb1, apply2 and let.

Function: apply2 (expr, rule_1, …, rule_n)

If rule_1 fails on a given subexpression, then rule_2 is repeatedly applied, etc. Only if all rules fail on a given subexpression is the whole set of rules repeatedly applied to the next subexpression. If one of the rules succeeds, then the same subexpression is reprocessed, starting with the first rule.

maxapplydepth is the depth of the deepest subexpressions processed by apply1 and apply2.

See also apply1 and let.

Function: applyb1 (expr, rule_1, …, rule_n)

Repeatedly applies rule_1 to the deepest subexpression of expr until it fails, then repeatedly applies the same rule one level higher (i.e., larger subexpressions), until rule_1 has failed on the top-level expression. Then rule_2 is applied in the same fashion to the result of rule_1. After rule_n has been applied to the top-level expression, the result is returned.

applyb1 is similar to apply1 but works from the bottom up instead of from the top down.

maxapplyheight is the maximum height which applyb1 reaches before giving up.

See also apply1, apply2 and let.

Option variable: current_let_rule_package

Default value: default_let_rule_package

current_let_rule_package is the name of the rule package that is used by functions in the let package (letsimp, etc.) if no other rule package is specified. This variable may be assigned the name of any rule package defined via the let command.

If a call such as letsimp (expr, rule_pkg_name) is made, the rule package rule_pkg_name is used for that function call only, and the value of current_let_rule_package is not changed.

Option variable: default_let_rule_package

Default value: default_let_rule_package

default_let_rule_package is the name of the rule package used when one is not explicitly set by the user with let or by changing the value of current_let_rule_package.

Function: defmatch
    defmatch (progname, pattern, x_1, …, x_n)
    defmatch (progname, pattern)

Defines a function progname(expr, x_1, ..., x_n) which tests expr to see if it matches pattern.

pattern is an expression containing the pattern arguments x_1, …, x_n (if any) and some pattern variables (if any). The pattern arguments are given explicitly as arguments to defmatch while the pattern variables are declared by the matchdeclare function. Any variable not declared as a pattern variable in matchdeclare or as a pattern argument in defmatch matches only itself.

The first argument to the created function progname is an expression to be matched against the pattern and the other arguments are the actual arguments which correspond to the dummy variables x_1, …, x_n in the pattern.

If the match is successful, progname returns a list of equations whose left sides are the pattern arguments and pattern variables, and whose right sides are the subexpressions which the pattern arguments and variables matched. The pattern variables, but not the pattern arguments, are assigned the subexpressions they match. If the match fails, progname returns false.

A literal pattern (that is, a pattern which contains neither pattern arguments nor pattern variables) returns true if the match succeeds.

See also matchdeclare, defrule, tellsimp and tellsimpafter.

Examples:

Define a function linearp(expr, x) which tests expr to see if it is of the form a*x + b such that a and b do not contain x and a is nonzero. This match function matches expressions which are linear in any variable, because the pattern argument x is given to defmatch.

(%i1) matchdeclare (a, lambda ([e], e#0 and freeof(x, e)), b,
                    freeof(x));
(%o1)                         done
(%i2) defmatch (linearp, a*x + b, x);
(%o2)                        linearp
(%i3) linearp (3*z + (y + 1)*z + y^2, z);
                         2
(%o3)              [b = y , a = y + 4, x = z]
(%i4) a;
(%o4)                         y + 4
(%i5) b;
                                2
(%o5)                          y
(%i6) x;
(%o6)                           x

Define a function linearp(expr) which tests expr to see if it is of the form a*x + b such that a and b do not contain x and a is nonzero. This match function only matches expressions linear in x, not any other variable, because no pattern argument is given to defmatch.

(%i1) matchdeclare (a, lambda ([e], e#0 and freeof(x, e)), b,
                    freeof(x));
(%o1)                         done
(%i2) defmatch (linearp, a*x + b);
(%o2)                        linearp
(%i3) linearp (3*z + (y + 1)*z + y^2);
(%o3)                         false
(%i4) linearp (3*x + (y + 1)*x + y^2);
                             2
(%o4)                  [b = y , a = y + 4]

Define a function checklimits(expr) which tests expr to see if it is a definite integral.

(%i1) matchdeclare ([a, f], true);
(%o1)                         done
(%i2) constinterval (l, h) := constantp (h - l);
(%o2)        constinterval(l, h) := constantp(h - l)
(%i3) matchdeclare (b, constinterval (a));
(%o3)                         done
(%i4) matchdeclare (x, atom);
(%o4)                         done
(%i5) simp : false;
(%o5)                         false
(%i6) defmatch (checklimits, 'integrate (f, x, a, b));
(%o6)                      checklimits
(%i7) simp : true;
(%o7)                         true
(%i8) 'integrate (sin(t), t, %pi + x, 2*%pi + x);
                       x + 2 %pi
                      /
                      [
(%o8)                 I          sin(t) dt
                      ]
                      /
                       x + %pi
(%i9) checklimits (%);
(%o9)    [b = x + 2 %pi, a = x + %pi, x = t, f = sin(t)]

Function: defrule (rulename, pattern, replacement)

Defines and names a replacement rule for the given pattern. If the rule named rulename is applied to an expression (by apply1, applyb1, or apply2), every subexpression matching the pattern will be replaced by the replacement. All variables in the replacement which have been assigned values by the pattern match are assigned those values in the replacement which is then simplified.

The rules themselves can be treated as functions which transform an expression by one operation of the pattern match and replacement. If the match fails, the rule function returns false.

Function: disprule
    disprule (rulename_1, …, rulename_2)
    disprule (all)

Display rules with the names rulename_1, …, rulename_n, as returned by defrule, tellsimp, or tellsimpafter, or a pattern defined by defmatch. Each rule is displayed with an intermediate expression label (%t).

disprule (all) displays all rules.

disprule quotes its arguments. disprule returns the list of intermediate expression labels corresponding to the displayed rules.

See also letrules, which displays rules defined by let.

Examples:

(%i1) tellsimpafter (foo (x, y), bar (x) + baz (y));
(%o1)                   [foorule1, false]
(%i2) tellsimpafter (x + y, special_add (x, y));
(%o2)                   [+rule1, simplus]
(%i3) defmatch (quux, mumble (x));
(%o3)                         quux
(%i4) disprule (foorule1, ?\+rule1, quux);
(%t4)        foorule1 : foo(x, y) -> baz(y) + bar(x)

(%t5)          +rule1 : y + x -> special_add(x, y)

(%t6)                quux : mumble(x) -> []

(%o6)                    [%t4, %t5, %t6]
(%i7) ev(%);
(%o7) [foorule1 : foo(x, y) -> baz(y) + bar(x), 
     +rule1 : y + x -> special_add(x, y), quux : mumble(x) -> []]

Function: let
    let (prod, repl, predname, arg_1, …, arg_n)
    let ([prod, repl, predname, arg_1, …, arg_n], package_name)

Defines a substitution rule for letsimp such that prod is replaced by repl. prod is a product of positive or negative powers of the following terms:

A term to a positive power will only match a term having at least that power. A term to a negative power on the other hand will only match a term with a power at least as negative. In the case of negative powers in prod the switch letrat must be set to true. See also letrat.

If a predicate is included in the let function followed by a list of arguments, a tentative match (i.e. one that would be accepted if the predicate were omitted) is accepted only if predname (arg_1', ..., arg_n') evaluates to true where arg_i’ is the value matched to arg_i. The arg_i may be the name of any atom or the argument of any kernel appearing in prod. repl may be any rational expression. If any of the atoms or arguments from prod appear in repl the appropriate substitutions are made.

The global flag letrat controls the simplification of quotients by letsimp. When letrat is false, letsimp simplifies the numerator and denominator of expr separately, and does not simplify the quotient. Substitutions such as n!/n goes to (n-1)! then fail. When letrat is true, then the numerator, denominator, and the quotient are simplified in that order.

These substitution functions allow you to work with several rule packages at once. Each rule package can contain any number of let rules and is referenced by a user-defined name. The command let ([prod, repl, predname, arg_1, ..., arg_n], package_name) adds the rule predname to the rule package package_name. The command letsimp (expr, package_name) applies the rules in package_name. letsimp (expr, package_name1, package_name2, ...) is equivalent to letsimp (expr, package_name1) followed by letsimp (%, package_name2), …

current_let_rule_package is the name of the rule package that is presently being used. This variable may be assigned the name of any rule package defined via the let command. Whenever any of the functions comprising the let package are called with no package name, the package named by current_let_rule_package is used. If a call such as letsimp (expr, rule_pkg_name) is made, the rule package rule_pkg_name is used for that letsimp command only, and current_let_rule_package is not changed. If not otherwise specified, current_let_rule_package defaults to default_let_rule_package.

(%i1) matchdeclare ([a, a1, a2], true)$
(%i2) oneless (x, y) := is (x = y-1)$
(%i3) let (a1*a2!, a1!, oneless, a2, a1);
(%o3)         a1 a2! --> a1! where oneless(a2, a1)
(%i4) letrat: true$
(%i5) let (a1!/a1, (a1-1)!);
                        a1!
(%o5)                   --- --> (a1 - 1)!
                        a1
(%i6) letsimp (n*m!*(n-1)!/m);
(%o6)                      (m - 1)! n!
(%i7) let (sin(a)^2, 1 - cos(a)^2);
                        2               2
(%o7)                sin (a) --> 1 - cos (a)
(%i8) letsimp (sin(x)^4);
                        4           2
(%o8)                cos (x) - 2 cos (x) + 1

Option variable: letrat

Default value: false

When letrat is false, letsimp simplifies the numerator and denominator of a ratio separately, and does not simplify the quotient.

When letrat is true, the numerator, denominator, and their quotient are simplified in that order.

(%i1) matchdeclare (n, true)$
(%i2) let (n!/n, (n-1)!);
                         n!
(%o2)                    -- --> (n - 1)!
                         n
(%i3) letrat: false$
(%i4) letsimp (a!/a);
                               a!
(%o4)                          --
                               a
(%i5) letrat: true$
(%i6) letsimp (a!/a);
(%o6)                       (a - 1)!

Function: letrules
    letrules ()
    letrules (package_name)

Displays the rules in a rule package. letrules () displays the rules in the current rule package. letrules (package_name) displays the rules in package_name.

The current rule package is named by current_let_rule_package. If not otherwise specified, current_let_rule_package defaults to default_let_rule_package.

See also disprule, which displays rules defined by tellsimp and tellsimpafter.

Function: letsimp
    letsimp (expr)
    letsimp (expr, package_name)
    letsimp (expr, package_name_1, …, package_name_n)

Repeatedly applies the substitution rules defined by let until no further change is made to expr.

letsimp (expr) uses the rules from current_let_rule_package.

letsimp (expr, package_name) uses the rules from package_name without changing current_let_rule_package.

letsimp (expr, package_name_1, ..., package_name_n) is equivalent to letsimp (expr, package_name_1), followed by letsimp (%, package_name_2), and so on.

See also let. For other ways to do substitutions see also subst, psubst, at and ratsubst.

(%i1) e0:e(k)=-(9*y(k))/(5*z)-u(k-1)/(5*z)+(4*y(k))/(5*z^2)+(3*u(k-1))/(5*z^2)+y(k)-(2*u(k-1))/5;
                9 y(k)    u(k - 1)   4 y(k)   3 u(k - 1)
(%o1) e(k) = (- ------) - -------- + ------ + ---------- + y(k)
                 5 z        5 z          2          2
                                      5 z        5 z
                                                       2 u(k - 1)
                                                     - ----------
                                                           5
(%i2) matchdeclare(h,any)$
(%i3) let(u(h)/z,u(h-1));
                        u(h)
(%o3)                   ---- --> u(h - 1)
                         z
(%i4) let(y(h)/z, y(h-1));
                        y(h)
(%o4)                   ---- --> y(h - 1)
                         z
(%i5) e1:letsimp(e0);
                2 u(k - 1)           3 u(k - 3)   4 y(k - 2)
(%o5) e(k) = (- ----------) + y(k) + ---------- + ----------
                    5                    5            5
                                       u(k - 2)       9 y(k - 1)
                                  + (- --------) + (- ----------)
                                          5               5

Option variable: let_rule_packages

Default value: [default_let_rule_package]

let_rule_packages is a list of all user-defined let rule packages plus the default package default_let_rule_package.

Function: matchdeclare (a_1, pred_1, …, a_n, pred_n)

Associates a predicate pred_k with a variable or list of variables a_k so that a_k matches expressions for which the predicate returns anything other than false.

A predicate is the name of a function, or a lambda expression, or a function call or lambda call missing the last argument, or true or all. Any expression matches true or all. If the predicate is specified as a function call or lambda call, the expression to be tested is appended to the list of arguments; the arguments are evaluated at the time the match is evaluated. Otherwise, the predicate is specified as a function name or lambda expression, and the expression to be tested is the sole argument. A predicate function need not be defined when matchdeclare is called; the predicate is not evaluated until a match is attempted.

A predicate may return a Boolean expression as well as true or false. Boolean expressions are evaluated by is within the constructed rule function, so it is not necessary to call is within the predicate.

If an expression satisfies a match predicate, the match variable is assigned the expression, except for match variables which are operands of addition + or multiplication *. Only addition and multiplication are handled specially; other n-ary operators (both built-in and user-defined) are treated like ordinary functions.

In the case of addition and multiplication, the match variable may be assigned a single expression which satisfies the match predicate, or a sum or product (respectively) of such expressions. Such multiple-term matching is greedy: predicates are evaluated in the order in which their associated variables appear in the match pattern, and a term which satisfies more than one predicate is taken by the first predicate which it satisfies. Each predicate is tested against all operands of the sum or product before the next predicate is evaluated. In addition, if 0 or 1 (respectively) satisfies a match predicate, and there are no other terms which satisfy the predicate, 0 or 1 is assigned to the match variable associated with the predicate.

The algorithm for processing addition and multiplication patterns makes some match results (for example, a pattern in which a "match anything" variable appears) dependent on the ordering of terms in the match pattern and in the expression to be matched. However, if all match predicates are mutually exclusive, the match result is insensitive to ordering, as one match predicate cannot accept terms matched by another.

Calling matchdeclare with a variable a as an argument changes the matchdeclare property for a, if one was already declared; only the most recent matchdeclare is in effect when a rule is defined. Later changes to the matchdeclare property (via matchdeclare or remove) do not affect existing rules.

propvars (matchdeclare) returns the list of all variables for which there is a matchdeclare property. printprops (a, matchdeclare) returns the predicate for variable a. printprops (all, matchdeclare) returns the list of predicates for all matchdeclare variables. remove (a, matchdeclare) removes the matchdeclare property from a.

The functions defmatch, defrule, tellsimp, tellsimpafter, and let construct rules which test expressions against patterns.

matchdeclare quotes its arguments. matchdeclare always returns done.

Examples:

A predicate is the name of a function, or a lambda expression, or a function call or lambda call missing the last argument, or true or all.

(%i1) matchdeclare (aa, integerp);
(%o1)                         done
(%i2) matchdeclare (bb, lambda ([x], x > 0));
(%o2)                         done
(%i3) matchdeclare (cc, freeof (%e, %pi, %i));
(%o3)                         done
(%i4) matchdeclare (dd, lambda ([x, y], gcd (x, y) = 1) (1728));
(%o4)                         done
(%i5) matchdeclare (ee, true);
(%o5)                         done
(%i6) matchdeclare (ff, all);
(%o6)                         done

If an expression satisfies a match predicate, the match variable is assigned the expression.

(%i1) matchdeclare (aa, integerp, bb, atom);
(%o1)                         done
(%i2) defrule (r1, bb^aa, ["integer" = aa, "atom" = bb]);
                    aa
(%o2)        r1 : bb   -> [integer = aa, atom = bb]
(%i3) r1 (%pi^8);
(%o3)               [integer = 8, atom = %pi]

In the case of addition and multiplication, the match variable may be assigned a single expression which satisfies the match predicate, or a sum or product (respectively) of such expressions.

(%i1) matchdeclare (aa, atom, bb, lambda ([x], not atom(x)));
(%o1)                         done
(%i2) defrule (r1, aa + bb, ["all atoms" = aa, "all nonatoms" =
               bb]);
(%o2)  r1 : bb + aa -> [all atoms = aa, all nonatoms = bb]
(%i3) r1 (8 + a*b + sin(x));
(%o3)     [all atoms = 8, all nonatoms = sin(x) + a b]
(%i4) defrule (r2, aa * bb, ["all atoms" = aa, "all nonatoms" =
               bb]);
(%o4)   r2 : aa bb -> [all atoms = aa, all nonatoms = bb]
(%i5) r2 (8 * (a + b) * sin(x));
(%o5)    [all atoms = 8, all nonatoms = (b + a) sin(x)]

When matching arguments of + and *, if all match predicates are mutually exclusive, the match result is insensitive to ordering, as one match predicate cannot accept terms matched by another.

(%i1) matchdeclare (aa, atom, bb, lambda ([x], not atom(x)));
(%o1)                         done
(%i2) defrule (r1, aa + bb, ["all atoms" = aa, "all nonatoms" =
               bb]);
(%o2)  r1 : bb + aa -> [all atoms = aa, all nonatoms = bb]
(%i3) r1 (8 + a*b + %pi + sin(x) - c + 2^n);
                                                     n
(%o3) [all atoms = %pi + 8, all nonatoms = sin(x) + 2  - c + a b]
(%i4) defrule (r2, aa * bb, ["all atoms" = aa, "all nonatoms" =
               bb]);
(%o4)   r2 : aa bb -> [all atoms = aa, all nonatoms = bb]
(%i5) r2 (8 * (a + b) * %pi * sin(x) / c * 2^n);
                                                n + 3
                                       (b + a) 2      sin(x)
(%o5) [all atoms = %pi, all nonatoms = ---------------------]
                                                 c

The functions propvars and printprops return information about match variables.

(%i1) matchdeclare ([aa, bb, cc], atom, [dd, ee], integerp);
(%o1)                         done
(%i2) matchdeclare (ff, floatnump, gg, lambda ([x], x > 100));
(%o2)                         done
(%i3) propvars (matchdeclare);
(%o3)             [aa, bb, cc, dd, ee, ff, gg]
(%i4) printprops (ee, matchdeclare);
(%o4)                    [integerp(ee)]
(%i5) printprops (gg, matchdeclare);
(%o5)              [lambda([x], x > 100, gg)]
(%i6) printprops (all, matchdeclare);
(%o6) [lambda([x], x > 100, gg), floatnump(ff), integerp(ee), 
                      integerp(dd), atom(cc), atom(bb), atom(aa)]

Option variable: maxapplydepth

Default value: 10000

maxapplydepth is the maximum depth to which apply1 and apply2 will delve.

Option variable: maxapplyheight

Default value: 10000

maxapplyheight is the maximum height to which applyb1 will reach before giving up.

Function: remlet
    remlet (prod, name)
    remlet ()
    remlet (all)
    remlet (all, name)

Deletes the substitution rule, prod –> repl, most recently defined by the let function. If name is supplied the rule is deleted from the rule package name.

remlet() and remlet(all) delete all substitution rules from the current rule package. If the name of a rule package is supplied, e.g. remlet (all, name), the rule package name is also deleted.

If a substitution is to be changed using the same product, remlet need not be called, just redefine the substitution using the same product (literally) with the let function and the new replacement and/or predicate name. Should remlet (prod) now be called the original substitution rule is revived.

See also remrule, which removes a rule defined by tellsimp or tellsimpafter.

Function: remrule
    remrule (op, rulename)
    remrule (op, all)

Removes rules defined by tellsimp or tellsimpafter.

remrule (op, rulename) removes the rule with the name rulename from the operator op. When op is a built-in or user-defined operator (as defined by infix, prefix, etc.), op and rulename must be enclosed in double quote marks.

remrule (op, all) removes all rules for the operator op.

See also remlet, which removes a rule defined by let.

Examples:

(%i1) tellsimp (foo (aa, bb), bb - aa);
(%o1)                   [foorule1, false]
(%i2) tellsimpafter (aa + bb, special_add (aa, bb));
(%o2)                   [+rule1, simplus]
(%i3) infix ("@@");
(%o3)                          @@
(%i4) tellsimp (aa @@ bb, bb/aa);
(%o4)                   [@@rule1, false]
(%i5) tellsimpafter (quux (%pi, %e), %pi - %e);
(%o5)                  [quuxrule1, false]
(%i6) tellsimpafter (quux (%e, %pi), %pi + %e);
(%o6)             [quuxrule2, quuxrule1, false]
(%i7) [foo (aa, bb), aa + bb, aa @@ bb, quux (%pi, %e),
       quux (%e, %pi)];
                                     bb
(%o7) [bb - aa, special_add(aa, bb), --, %pi - %e, %pi + %e]
                                     aa
(%i8) remrule (foo, foorule1);
(%o8)                          foo
(%i9) remrule ("+", ?\+rule1);
(%o9)                           +
(%i10) remrule ("@@", ?\@\@rule1);
(%o10)                         @@
(%i11) remrule (quux, all);
(%o11)                        quux
(%i12) [foo (aa, bb), aa + bb, aa @@ bb, quux (%pi, %e),
        quux (%e, %pi)];
(%o12) [foo(aa, bb), bb + aa, aa @@ bb, quux(%pi, %e), 
                                                   quux(%e, %pi)]

Function: tellsimp (pattern, replacement)

is similar to tellsimpafter but places new information before old so that it is applied before the built-in simplification rules.

tellsimp is used when it is important to modify the expression before the simplifier works on it, for instance if the simplifier "knows" something about the expression, but what it returns is not to your liking. If the simplifier "knows" something about the main operator of the expression, but is simply not doing enough for you, you probably want to use tellsimpafter.

The pattern may not be a sum, product, single variable, or number.

The system variable rules is the list of rules defined by defrule, defmatch, tellsimp, and tellsimpafter.

Examples:

(%i1) matchdeclare (x, freeof (%i));
(%o1)                         done
(%i2) %iargs: false$
(%i3) tellsimp (sin(%i*x), %i*sinh(x));
(%o3)                 [sinrule1, simp-%sin]
(%i4) trigexpand (sin (%i*y + x));
(%o4)         sin(x) cos(%i y) + %i cos(x) sinh(y)
(%i5) %iargs:true$
(%i6) errcatch(0^0);
 0
0  has been generated
(%o6)                          []
(%i7) ev (tellsimp (0^0, 1), simp: false);
(%o7)                  [^rule1, simpexpt]
(%i8) 0^0;
(%o8)                           1
(%i9) remrule ("^", %th(2)[1]);
(%o9)                           ^
(%i10) tellsimp (sin(x)^2, 1 - cos(x)^2);
(%o10)                 [^rule2, simpexpt]
(%i11) (1 + sin(x))^2;
                                      2
(%o11)                    (sin(x) + 1)
(%i12) expand (%);
                                   2
(%o12)               2 sin(x) - cos (x) + 2
(%i13) sin(x)^2;
                                  2
(%o13)                     1 - cos (x)
(%i14) kill (rules);
(%o14)                        done
(%i15) matchdeclare (a, true);
(%o15)                        done
(%i16) tellsimp (sin(a)^2, 1 - cos(a)^2);
(%o16)                 [^rule3, simpexpt]
(%i17) sin(y)^2;
                                  2
(%o17)                     1 - cos (y)

Function: tellsimpafter (pattern, replacement)

Defines a simplification rule which the Maxima simplifier applies after built-in simplification rules. pattern is an expression, comprising pattern variables (declared by matchdeclare) and other atoms and operators, considered literals for the purpose of pattern matching. replacement is substituted for an actual expression which matches pattern; pattern variables in replacement are assigned the values matched in the actual expression.

pattern may be any nonatomic expression in which the main operator is not a pattern variable; the simplification rule is associated with the main operator. The names of functions (with one exception, described below), lists, and arrays may appear in pattern as the main operator only as literals (not pattern variables); this rules out expressions such as aa(x) and bb[y] as patterns, if aa and bb are pattern variables. Names of functions, lists, and arrays which are pattern variables may appear as operators other than the main operator in pattern.

There is one exception to the above rule concerning names of functions. The name of a subscripted function in an expression such as aa[x](y) may be a pattern variable, because the main operator is not aa but rather the Lisp atom mqapply. This is a consequence of the representation of expressions involving subscripted functions.

Simplification rules are applied after evaluation (if not suppressed through quotation or the flag noeval). Rules established by tellsimpafter are applied in the order they were defined, and after any built-in rules. Rules are applied bottom-up, that is, applied first to subexpressions before application to the whole expression. It may be necessary to repeatedly simplify a result (for example, via the quote-quote operator '' or the flag infeval) to ensure that all rules are applied.

Pattern variables are treated as local variables in simplification rules. Once a rule is defined, the value of a pattern variable does not affect the rule, and is not affected by the rule. An assignment to a pattern variable which results from a successful rule match does not affect the current assignment (or lack of it) of the pattern variable. However, as with all atoms in Maxima, the properties of pattern variables (as declared by put and related functions) are global.

The rule constructed by tellsimpafter is named after the main operator of pattern. Rules for built-in operators, and user-defined operators defined by infix, prefix, postfix, matchfix, and nofix, have names which are Lisp identifiers. Rules for other functions have names which are Maxima identifiers.

The treatment of noun and verb forms is slightly confused. If a rule is defined for a noun (or verb) form and a rule for the corresponding verb (or noun) form already exists, the newly-defined rule applies to both forms (noun and verb). If a rule for the corresponding verb (or noun) form does not exist, the newly-defined rule applies only to the noun (or verb) form.

The rule constructed by tellsimpafter is an ordinary Lisp function. If the name of the rule is $foorule1, the construct :lisp (trace $foorule1) traces the function, and :lisp (symbol-function '$foorule1) displays its definition.

tellsimpafter quotes its arguments. tellsimpafter returns the list of rules for the main operator of pattern, including the newly established rule.

See also matchdeclare, defmatch, defrule, tellsimp, let, kill, remrule and clear_rules.

Examples:

pattern may be any nonatomic expression in which the main operator is not a pattern variable.

(%i1) matchdeclare (aa, atom, [ll, mm], listp, xx, true)$
(%i2) tellsimpafter (sin (ll), map (sin, ll));
(%o2)                 [sinrule1, simp-%sin]
(%i3) sin ([1/6, 1/4, 1/3, 1/2, 1]*%pi);
                    1     1     sqrt(3)
(%o3)              [-, -------, -------, 1, 0]
                    2  sqrt(2)     2
(%i4) tellsimpafter (ll^mm, map ("^", ll, mm));
(%o4)                  [^rule1, simpexpt]
(%i5) [a, b, c]^[1, 2, 3];
                                2   3
(%o5)                      [a, b , c ]
(%i6) tellsimpafter (foo (aa (xx)), aa (foo (xx)));
(%o6)                   [foorule1, false]
(%i7) foo (bar (u - v));
(%o7)                    bar(foo(u - v))

Rules are applied in the order they were defined. If two rules can match an expression, the rule which was defined first is applied.

(%i1) matchdeclare (aa, integerp);
(%o1)                         done
(%i2) tellsimpafter (foo (aa), bar_1 (aa));
(%o2)                   [foorule1, false]
(%i3) tellsimpafter (foo (aa), bar_2 (aa));
(%o3)              [foorule2, foorule1, false]
(%i4) foo (42);
(%o4)                       bar_1(42)

Pattern variables are treated as local variables in simplification rules. (Compare to defmatch, which treats pattern variables as global variables.)

(%i1) matchdeclare (aa, integerp, bb, atom);
(%o1)                         done
(%i2) tellsimpafter (foo(aa, bb), bar('aa=aa, 'bb=bb));
(%o2)                   [foorule1, false]
(%i3) bb: 12345;
(%o3)                         12345
(%i4) foo (42, %e);
(%o4)                 bar(aa = 42, bb = %e)
(%i5) bb;
(%o5)                         12345

As with all atoms, properties of pattern variables are global even though values are local. In this example, an assignment property is declared via define_variable. This is a property of the atom bb throughout Maxima.

(%i1) matchdeclare (aa, integerp, bb, atom);
(%o1)                         done
(%i2) tellsimpafter (foo(aa, bb), bar('aa=aa, 'bb=bb));
(%o2)                   [foorule1, false]
(%i3) foo (42, %e);
(%o3)                 bar(aa = 42, bb = %e)
(%i4) define_variable (bb, true, boolean);
(%o4)                         true
(%i5) foo (42, %e);
translator: bb was declared with mode boolean, but it has value: 
                                                               %e
 -- an error. To debug this try: debugmode(true);

Rules are named after main operators. Names of rules for built-in and user-defined operators are Lisp identifiers, while names for other functions are Maxima identifiers.

(%i1) tellsimpafter (foo (%pi + %e), 3*%pi);
(%o1)                   [foorule1, false]
(%i2) tellsimpafter (foo (%pi * %e), 17*%e);
(%o2)              [foorule2, foorule1, false]
(%i3) tellsimpafter (foo (%i ^ %e), -42*%i);
(%o3)         [foorule3, foorule2, foorule1, false]
(%i4) tellsimpafter (foo (9) + foo (13), quux (22));
(%o4)                   [+rule1, simplus]
(%i5) tellsimpafter (foo (9) * foo (13), blurf (22));
(%o5)                  [*rule1, simptimes]
(%i6) tellsimpafter (foo (9) ^ foo (13), mumble (22));
(%o6)                  [^rule1, simpexpt]
(%i7) rules;
(%o7) [foorule1, foorule2, foorule3, +rule1, *rule1, ^rule1]
(%i8) foorule_name: first (%o1);
(%o8)                       foorule1
(%i9) plusrule_name: first (%o4);
(%o9)                        +rule1
(%i10) remrule (foo, foorule1);
(%o10)                         foo
(%i11) remrule ("^", ?\^rule1);
(%o11)                          ^
(%i12) rules;
(%o12)        [foorule2, foorule3, +rule1, *rule1]

A worked example: anticommutative multiplication.

(%i1) gt (i, j) := integerp(j) and i < j;
(%o1)          gt(i, j) := integerp(j) and (i < j)
(%i2) matchdeclare (i, integerp, j, gt(i));
(%o2)                         done
(%i3) tellsimpafter (s[i]^^2, 1);
(%o3)                 [^^rule1, simpncexpt]
(%i4) tellsimpafter (s[i] . s[j], -s[j] . s[i]);
(%o4)                   [.rule1, simpnct]
(%i5) s[1] . (s[1] + s[2]);
(%o5)                    s  . (s  + s )
                          1     2    1
(%i6) expand (%);
(%o6)                      1 - s  . s
                                2    1
(%i7) factor (expand (sum (s[i], i, 0, 9)^^5));
(%o7) 100 (s  + s  + s  + s  + s  + s  + s  + s  + s  + s )
            9    8    7    6    5    4    3    2    1    0

Function: clear_rules ()

Executes kill (rules) and then resets the next rule number to 1 for addition +, multiplication *, and exponentiation ^.


Next: , Previous: , Up: Top   [Contents][Index]

35 Sets


Next: , Previous: , Up: Sets   [Contents][Index]

35.1 Introduction to Sets

Maxima provides set functions, such as intersection and union, for finite sets that are defined by explicit enumeration. Maxima treats lists and sets as distinct objects. This feature makes it possible to work with sets that have members that are either lists or sets.

In addition to functions for finite sets, Maxima provides some functions related to combinatorics; these include the Stirling numbers of the first and second kind, the Bell numbers, multinomial coefficients, partitions of nonnegative integers, and a few others. Maxima also defines a Kronecker delta function.

35.1.1 Usage

To construct a set with members a_1, ..., a_n, write set(a_1, ..., a_n) or {a_1, ..., a_n}; to construct the empty set, write set() or {}. In input, set(...) and { ... } are equivalent. Sets are always displayed with curly braces.

If a member is listed more than once, simplification eliminates the redundant member.

(%i1) set();
(%o1)                          {}
(%i2) set(a, b, a);
(%o2)                        {a, b}
(%i3) set(a, set(b));
(%o3)                       {a, {b}}
(%i4) set(a, [b]);
(%o4)                       {a, [b]}
(%i5) {};
(%o5)                          {}
(%i6) {a, b, a};
(%o6)                        {a, b}
(%i7) {a, {b}};
(%o7)                       {a, {b}}
(%i8) {a, [b]};
(%o8)                       {a, [b]}

Two would-be elements x and y are redundant (i.e., considered the same for the purpose of set construction) if and only if is(x = y) yields true. Note that is(equal(x, y)) can yield true while is(x = y) yields false; in that case the elements x and y are considered distinct.

(%i1) x: a/c + b/c;
                              b   a
(%o1)                         - + -
                              c   c
(%i2) y: a/c + b/c;
                              b   a
(%o2)                         - + -
                              c   c
(%i3) z: (a + b)/c;
                              b + a
(%o3)                         -----
                                c
(%i4) is (x = y);
(%o4)                         true
(%i5) is (y = z);
(%o5)                         false
(%i6) is (equal (y, z));
(%o6)                         true
(%i7) y - z;
                           b + a   b   a
(%o7)                    - ----- + - + -
                             c     c   c
(%i8) ratsimp (%);
(%o8)                           0
(%i9) {x, y, z};
                          b + a  b   a
(%o9)                    {-----, - + -}
                            c    c   c

To construct a set from the elements of a list, use setify.

(%i1) setify ([b, a]);
(%o1)                        {a, b}

Set members x and y are equal provided is(x = y) evaluates to true. Thus rat(x) and x are equal as set members; consequently,

(%i1) {x, rat(x)};
(%o1)                          {x}

Further, since is((x - 1)*(x + 1) = x^2 - 1) evaluates to false, (x - 1)*(x + 1) and x^2 - 1 are distinct set members; thus

(%i1) {(x - 1)*(x + 1), x^2 - 1};
                                       2
(%o1)               {(x - 1) (x + 1), x  - 1}

To reduce this set to a singleton set, apply rat to each set member:

(%i1) {(x - 1)*(x + 1), x^2 - 1};
                                       2
(%o1)               {(x - 1) (x + 1), x  - 1}
(%i2) map (rat, %);
                              2
(%o2)/R/                    {x  - 1}

To remove redundancies from other sets, you may need to use other simplification functions. Here is an example that uses trigsimp:

(%i1) {1, cos(x)^2 + sin(x)^2};
                            2         2
(%o1)                {1, sin (x) + cos (x)}
(%i2) map (trigsimp, %);
(%o2)                          {1}

A set is simplified when its members are non-redundant and sorted. The current version of the set functions uses the Maxima function orderlessp to order sets; however, future versions of the set functions might use a different ordering function.

Some operations on sets, such as substitution, automatically force a re-simplification; for example,

(%i1) s: {a, b, c}$
(%i2) subst (c=a, s);
(%o2)                        {a, b}
(%i3) subst ([a=x, b=x, c=x], s);
(%o3)                          {x}
(%i4) map (lambda ([x], x^2), set (-1, 0, 1));
(%o4)                        {0, 1}

Maxima treats lists and sets as distinct objects; functions such as union and intersection complain if any argument is not a set. If you need to apply a set function to a list, use the setify function to convert it to a set. Thus

(%i1) union ([1, 2], {a, b});
Function union expects a set, instead found [1,2]
 -- an error.  Quitting.  To debug this try debugmode(true);
(%i2) union (setify ([1, 2]), {a, b});
(%o2)                     {1, 2, a, b}

To extract all set elements of a set s that satisfy a predicate f, use subset(s, f). (A predicate is a boolean-valued function.) For example, to find the equations in a given set that do not depend on a variable z, use

(%i1) subset ({x + y + z, x - y + 4, x + y - 5},
                                    lambda ([e], freeof (z, e)));
(%o1)               {- y + x + 4, y + x - 5}

The section Functions and Variables for Sets has a complete list of the set functions in Maxima.

·

Category: Sets

35.1.2 Set Member Iteration

There two ways to to iterate over set members. One way is the use map; for example:

(%i1) map (f, {a, b, c});
(%o1)                  {f(a), f(b), f(c)}

The other way is to use for x in s do

(%i1) s: {a, b, c};
(%o1)                       {a, b, c}
(%i2) for si in s do print (concat (si, 1));
a1 
b1 
c1 
(%o2)                         done

The Maxima functions first and rest work correctly on sets. Applied to a set, first returns the first displayed element of a set; which element that is may be implementation-dependent. If s is a set, then rest(s) is equivalent to disjoin(first(s), s). Currently, there are other Maxima functions that work correctly on sets. In future versions of the set functions, first and rest may function differently or not at all.

Maxima’s orderless and ordergreat mechanisms are incompatible with the set functions. If you need to use either orderless or ordergreat, call those functions before constructing any sets, and do not call unorder.

35.1.3 Authors

Stavros Macrakis of Cambridge, Massachusetts and Barton Willis of the University of Nebraska at Kearney (UNK) wrote the Maxima set functions and their documentation.


Previous: , Up: Sets   [Contents][Index]

35.2 Functions and Variables for Sets

Function: adjoin (x, a)

Returns the union of the set a with {x}.

adjoin complains if a is not a literal set.

adjoin(x, a) and union(set(x), a) are equivalent; however, adjoin may be somewhat faster than union.

See also disjoin.

Examples:

(%i1) adjoin (c, {a, b});
(%o1)                       {a, b, c}
(%i2) adjoin (a, {a, b});
(%o2)                        {a, b}

Function: belln (n)

Represents the n-th Bell number. belln(n) is the number of partitions of a set with n members.

For nonnegative integers n, belln(n) simplifies to the n-th Bell number. belln does not simplify for any other arguments.

belln distributes over equations, lists, matrices, and sets.

Examples:

belln applied to nonnegative integers.

(%i1) makelist (belln (i), i, 0, 6);
(%o1)               [1, 1, 2, 5, 15, 52, 203]
(%i2) is (cardinality (set_partitions ({})) = belln (0));
(%o2)                         true
(%i3) is (cardinality (set_partitions ({1, 2, 3, 4, 5, 6})) =
                       belln (6));
(%o3)                         true

belln applied to arguments which are not nonnegative integers.

(%i1) [belln (x), belln (sqrt(3)), belln (-9)];
(%o1)        [belln(x), belln(sqrt(3)), belln(- 9)]

Function: cardinality (a)

Returns the number of distinct elements of the set a.

cardinality ignores redundant elements even when simplification is disabled.

Examples:

(%i1) cardinality ({});
(%o1)                           0
(%i2) cardinality ({a, a, b, c});
(%o2)                           3
(%i3) simp : false;
(%o3)                         false
(%i4) cardinality ({a, a, b, c});
(%o4)                           3

Function: cartesian_product (b_1, ... , b_n)

Returns a set of lists of the form [x_1, ..., x_n], where x_1, ..., x_n are elements of the sets b_1, ... , b_n, respectively.

cartesian_product complains if any argument is not a literal set.

Examples:

(%i1) cartesian_product ({0, 1});
(%o1)                      {[0], [1]}
(%i2) cartesian_product ({0, 1}, {0, 1});
(%o2)           {[0, 0], [0, 1], [1, 0], [1, 1]}
(%i3) cartesian_product ({x}, {y}, {z});
(%o3)                      {[x, y, z]}
(%i4) cartesian_product ({x}, {-1, 0, 1});
(%o4)              {[x, - 1], [x, 0], [x, 1]}

Function: disjoin (x, a)

Returns the set a without the member x. If x is not a member of a, return a unchanged.

disjoin complains if a is not a literal set.

disjoin(x, a), delete(x, a), and setdifference(a, set(x)) are all equivalent. Of these, disjoin is generally faster than the others.

Examples:

(%i1) disjoin (a, {a, b, c, d});
(%o1)                       {b, c, d}
(%i2) disjoin (a + b, {5, z, a + b, %pi});
(%o2)                      {5, %pi, z}
(%i3) disjoin (a - b, {5, z, a + b, %pi});
(%o3)                  {5, %pi, b + a, z}

Function: disjointp (a, b)

Returns true if and only if the sets a and b are disjoint.

disjointp complains if either a or b is not a literal set.

Examples:

(%i1) disjointp ({a, b, c}, {1, 2, 3});
(%o1)                         true
(%i2) disjointp ({a, b, 3}, {1, 2, 3});
(%o2)                         false

Function: divisors (n)

Represents the set of divisors of n.

divisors(n) simplifies to a set of integers when n is a nonzero integer. The set of divisors includes the members 1 and n. The divisors of a negative integer are the divisors of its absolute value.

divisors distributes over equations, lists, matrices, and sets.

Examples:

We can verify that 28 is a perfect number: the sum of its divisors (except for itself) is 28.

(%i1) s: divisors(28);
(%o1)                 {1, 2, 4, 7, 14, 28}
(%i2) lreduce ("+", args(s)) - 28;
(%o2)                          28

divisors is a simplifying function. Substituting 8 for a in divisors(a) yields the divisors without reevaluating divisors(8).

(%i1) divisors (a);
(%o1)                      divisors(a)
(%i2) subst (8, a, %);
(%o2)                     {1, 2, 4, 8}

divisors distributes over equations, lists, matrices, and sets.

(%i1) divisors (a = b);
(%o1)               divisors(a) = divisors(b)
(%i2) divisors ([a, b, c]);
(%o2)        [divisors(a), divisors(b), divisors(c)]
(%i3) divisors (matrix ([a, b], [c, d]));
                  [ divisors(a)  divisors(b) ]
(%o3)             [                          ]
                  [ divisors(c)  divisors(d) ]
(%i4) divisors ({a, b, c});
(%o4)        {divisors(a), divisors(b), divisors(c)}

Function: elementp (x, a)

Returns true if and only if x is a member of the set a.

elementp complains if a is not a literal set.

Examples:

(%i1) elementp (sin(1), {sin(1), sin(2), sin(3)});
(%o1)                         true
(%i2) elementp (sin(1), {cos(1), cos(2), cos(3)});
(%o2)                         false

Function: emptyp (a)

Return true if and only if a is the empty set or the empty list.

Examples:

(%i1) map (emptyp, [{}, []]);
(%o1)                     [true, true]
(%i2) map (emptyp, [a + b, {{}}, %pi]);
(%o2)                 [false, false, false]

Function: equiv_classes (s, F)

Returns a set of the equivalence classes of the set s with respect to the equivalence relation F.

F is a function of two variables defined on the Cartesian product of s with s. The return value of F is either true or false, or an expression expr such that is(expr) is either true or false.

When F is not an equivalence relation, equiv_classes accepts it without complaint, but the result is generally incorrect in that case.

Examples:

The equivalence relation is a lambda expression which returns true or false.

(%i1) equiv_classes ({1, 1.0, 2, 2.0, 3, 3.0},
                        lambda ([x, y], is (equal (x, y))));
(%o1)            {{1, 1.0}, {2, 2.0}, {3, 3.0}}

The equivalence relation is the name of a relational function which is evaluates to true or false.

(%i1) equiv_classes ({1, 1.0, 2, 2.0, 3, 3.0}, equal);
(%o1)            {{1, 1.0}, {2, 2.0}, {3, 3.0}}

The equivalence classes are numbers which differ by a multiple of 3.

(%i1) equiv_classes ({1, 2, 3, 4, 5, 6, 7},
                     lambda ([x, y], remainder (x - y, 3) = 0));
(%o1)              {{1, 4, 7}, {2, 5}, {3, 6}}

Function: every
    every (f, s)
    every (f, L_1, ..., L_n)

Returns true if the predicate f is true for all given arguments.

Given one set as the second argument, every(f, s) returns true if is(f(a_i)) returns true for all a_i in s. every may or may not evaluate f for all a_i in s. Since sets are unordered, every may evaluate f(a_i) in any order.

Given one or more lists as arguments, every(f, L_1, ..., L_n) returns true if is(f(x_1, ..., x_n)) returns true for all x_1, ..., x_n in L_1, ..., L_n, respectively. every may or may not evaluate f for every combination x_1, ..., x_n. every evaluates lists in the order of increasing index.

Given an empty set {} or empty lists [] as arguments, every returns true.

When the global flag maperror is true, all lists L_1, ..., L_n must have equal lengths. When maperror is false, list arguments are effectively truncated to the length of the shortest list.

Return values of the predicate f which evaluate (via is) to something other than true or false are governed by the global flag prederror. When prederror is true, such values are treated as false, and the return value from every is false. When prederror is false, such values are treated as unknown, and the return value from every is unknown.

Examples:

every applied to a single set. The predicate is a function of one argument.

(%i1) every (integerp, {1, 2, 3, 4, 5, 6});
(%o1)                         true
(%i2) every (atom, {1, 2, sin(3), 4, 5 + y, 6});
(%o2)                         false

every applied to two lists. The predicate is a function of two arguments.

(%i1) every ("=", [a, b, c], [a, b, c]);
(%o1)                         true
(%i2) every ("#", [a, b, c], [a, b, c]);
(%o2)                         false

Return values of the predicate f which evaluate to something other than true or false are governed by the global flag prederror.

(%i1) prederror : false;
(%o1)                         false
(%i2) map (lambda ([a, b], is (a < b)), [x, y, z],
                   [x^2, y^2, z^2]);
(%o2)              [unknown, unknown, unknown]
(%i3) every ("<", [x, y, z], [x^2, y^2, z^2]);
(%o3)                        unknown
(%i4) prederror : true;
(%o4)                         true
(%i5) every ("<", [x, y, z], [x^2, y^2, z^2]);
(%o5)                         false

Function: extremal_subset
    extremal_subset (s, f, max)
    extremal_subset (s, f, min)

Returns the subset of s for which the function f takes on maximum or minimum values.

extremal_subset(s, f, max) returns the subset of the set or list s for which the real-valued function f takes on its maximum value.

extremal_subset(s, f, min) returns the subset of the set or list s for which the real-valued function f takes on its minimum value.

Examples:

(%i1) extremal_subset ({-2, -1, 0, 1, 2}, abs, max);
(%o1)                       {- 2, 2}
(%i2) extremal_subset ({sqrt(2), 1.57, %pi/2}, sin, min);
(%o2)                       {sqrt(2)}

Function: flatten (expr)

Collects arguments of subexpressions which have the same operator as expr and constructs an expression from these collected arguments.

Subexpressions in which the operator is different from the main operator of expr are copied without modification, even if they, in turn, contain some subexpressions in which the operator is the same as for expr.

It may be possible for flatten to construct expressions in which the number of arguments differs from the declared arguments for an operator; this may provoke an error message from the simplifier or evaluator. flatten does not try to detect such situations.

Expressions with special representations, for example, canonical rational expressions (CRE), cannot be flattened; in such cases, flatten returns its argument unchanged.

Examples:

Applied to a list, flatten gathers all list elements that are lists.

(%i1) flatten ([a, b, [c, [d, e], f], [[g, h]], i, j]);
(%o1)            [a, b, c, d, e, f, g, h, i, j]

Applied to a set, flatten gathers all members of set elements that are sets.

(%i1) flatten ({a, {b}, {{c}}});
(%o1)                       {a, b, c}
(%i2) flatten ({a, {[a], {a}}});
(%o2)                       {a, [a]}

flatten is similar to the effect of declaring the main operator n-ary. However, flatten has no effect on subexpressions which have an operator different from the main operator, while an n-ary declaration affects those.

(%i1) expr: flatten (f (g (f (f (x)))));
(%o1)                     f(g(f(f(x))))
(%i2) declare (f, nary);
(%o2)                         done
(%i3) ev (expr);
(%o3)                      f(g(f(x)))

flatten treats subscripted functions the same as any other operator.

(%i1) flatten (f[5] (f[5] (x, y), z));
(%o1)                      f (x, y, z)
                            5

It may be possible for flatten to construct expressions in which the number of arguments differs from the declared arguments for an operator;

(%i1) 'mod (5, 'mod (7, 4));
(%o1)                   mod(5, mod(7, 4))
(%i2) flatten (%);
(%o2)                     mod(5, 7, 4)
(%i3) ''%, nouns;
Wrong number of arguments to mod
 -- an error.  Quitting.  To debug this try debugmode(true);

Function: full_listify (a)

Replaces every set operator in a by a list operator, and returns the result. full_listify replaces set operators in nested subexpressions, even if the main operator is not set.

listify replaces only the main operator.

Examples:

(%i1) full_listify ({a, b, {c, {d, e, f}, g}});
(%o1)               [a, b, [c, [d, e, f], g]]
(%i2) full_listify (F (G ({a, b, H({c, d, e})})));
(%o2)              F(G([a, b, H([c, d, e])]))

Function: fullsetify (a)

When a is a list, replaces the list operator with a set operator, and applies fullsetify to each member which is a set. When a is not a list, it is returned unchanged.

setify replaces only the main operator.

Examples:

In line (%o2), the argument of f isn’t converted to a set because the main operator of f([b]) isn’t a list.

(%i1) fullsetify ([a, [a]]);
(%o1)                       {a, {a}}
(%i2) fullsetify ([a, f([b])]);
(%o2)                      {a, f([b])}

Function: identity (x)

Returns x for any argument x.

Examples:

identity may be used as a predicate when the arguments are already Boolean values.

(%i1) every (identity, [true, true]);
(%o1)                         true
Function: integer_partitions
    integer_partitions (n)
    integer_partitions (n, len)

Returns integer partitions of n, that is, lists of integers which sum to n.

integer_partitions(n) returns the set of all partitions of the integer n. Each partition is a list sorted from greatest to least.

integer_partitions(n, len) returns all partitions that have length len or less; in this case, zeros are appended to each partition with fewer than len terms to make each partition have exactly len terms. Each partition is a list sorted from greatest to least.

A list [a_1, ..., a_m] is a partition of a nonnegative integer n when (1) each a_i is a nonzero integer, and (2) a_1 + ... + a_m = n. Thus 0 has no partitions.

Examples:

(%i1) integer_partitions (3);
(%o1)               {[1, 1, 1], [2, 1], [3]}
(%i2) s: integer_partitions (25)$
(%i3) cardinality (s);
(%o3)                         1958
(%i4) map (lambda ([x], apply ("+", x)), s);
(%o4)                         {25}
(%i5) integer_partitions (5, 3);
(%o5) {[2, 2, 1], [3, 1, 1], [3, 2, 0], [4, 1, 0], [5, 0, 0]}
(%i6) integer_partitions (5, 2);
(%o6)               {[3, 2], [4, 1], [5, 0]}

To find all partitions that satisfy a condition, use the function subset; here is an example that finds all partitions of 10 that consist of prime numbers.

(%i1) s: integer_partitions (10)$
(%i2) cardinality (s);
(%o2)                          42
(%i3) xprimep(x) := integerp(x) and (x > 1) and primep(x)$
(%i4) subset (s, lambda ([x], every (xprimep, x)));
(%o4) {[2, 2, 2, 2, 2], [3, 3, 2, 2], [5, 3, 2], [5, 5], [7, 3]}

Function: intersect (a_1, ..., a_n)

intersect is the same as intersection, which see.

Function: intersection (a_1, ..., a_n)

Returns a set containing the elements that are common to the sets a_1 through a_n.

intersection complains if any argument is not a literal set.

Examples:

(%i1) S_1 : {a, b, c, d};
(%o1)                     {a, b, c, d}
(%i2) S_2 : {d, e, f, g};
(%o2)                     {d, e, f, g}
(%i3) S_3 : {c, d, e, f};
(%o3)                     {c, d, e, f}
(%i4) S_4 : {u, v, w};
(%o4)                       {u, v, w}
(%i5) intersection (S_1, S_2);
(%o5)                          {d}
(%i6) intersection (S_2, S_3);
(%o6)                       {d, e, f}
(%i7) intersection (S_1, S_2, S_3);
(%o7)                          {d}
(%i8) intersection (S_1, S_2, S_3, S_4);
(%o8)                          {}

Function: kron_delta (x1, x2, …, xp)

Represents the Kronecker delta function.

kron_delta simplifies to 1 when xi and yj are equal for all pairs of arguments, and it simplifies to 0 when xi and yj are not equal for some pair of arguments. Equality is determined using is(equal(xi,xj)) and inequality by is(notequal(xi,xj)). For exactly one argument, kron_delta signals an error.

Examples:

(%i1) kron_delta(a,a);
(%o1)                                  1
(%i2) kron_delta(a,b,a,b);
(%o2)                          kron_delta(a, b)
(%i3) kron_delta(a,a,b,a+1);
(%o3)                                  0
(%i4) assume(equal(x,y));
(%o4)                            [equal(x, y)]
(%i5) kron_delta(x,y);
(%o5)                                  1
Function: listify (a)

Returns a list containing the members of a when a is a set. Otherwise, listify returns a.

full_listify replaces all set operators in a by list operators.

Examples:

(%i1) listify ({a, b, c, d});
(%o1)                     [a, b, c, d]
(%i2) listify (F ({a, b, c, d}));
(%o2)                    F({a, b, c, d})

Function: makeset (expr, x, s)

Returns a set with members generated from the expression expr, where x is a list of variables in expr, and s is a set or list of lists. To generate each set member, expr is evaluated with the variables x bound in parallel to a member of s.

Each member of s must have the same length as x. The list of variables x must be a list of symbols, without subscripts. Even if there is only one symbol, x must be a list of one element, and each member of s must be a list of one element.

See also makelist.

Examples:

(%i1) makeset (i/j, [i, j], [[1, a], [2, b], [3, c], [4, d]]);
                           1  2  3  4
(%o1)                     {-, -, -, -}
                           a  b  c  d
(%i2) S : {x, y, z}$
(%i3) S3 : cartesian_product (S, S, S);
(%o3) {[x, x, x], [x, x, y], [x, x, z], [x, y, x], [x, y, y], 
[x, y, z], [x, z, x], [x, z, y], [x, z, z], [y, x, x], 
[y, x, y], [y, x, z], [y, y, x], [y, y, y], [y, y, z], 
[y, z, x], [y, z, y], [y, z, z], [z, x, x], [z, x, y], 
[z, x, z], [z, y, x], [z, y, y], [z, y, z], [z, z, x], 
[z, z, y], [z, z, z]}
(%i4) makeset (i + j + k, [i, j, k], S3);
(%o4) {3 x, 3 y, y + 2 x, 2 y + x, 3 z, z + 2 x, z + y + x, 
                                       z + 2 y, 2 z + x, 2 z + y}
(%i5) makeset (sin(x), [x], {[1], [2], [3]});
(%o5)               {sin(1), sin(2), sin(3)}

Function: moebius (n)

Represents the Moebius function.

When n is product of k distinct primes, moebius(n) simplifies to (-1)^k; when n = 1, it simplifies to 1; and it simplifies to 0 for all other positive integers.

moebius distributes over equations, lists, matrices, and sets.

Examples:

(%i1) moebius (1);
(%o1)                           1
(%i2) moebius (2 * 3 * 5);
(%o2)                          - 1
(%i3) moebius (11 * 17 * 29 * 31);
(%o3)                           1
(%i4) moebius (2^32);
(%o4)                           0
(%i5) moebius (n);
(%o5)                      moebius(n)
(%i6) moebius (n = 12);
(%o6)                    moebius(n) = 0
(%i7) moebius ([11, 11 * 13, 11 * 13 * 15]);
(%o7)                      [- 1, 1, 1]
(%i8) moebius (matrix ([11, 12], [13, 14]));
                           [ - 1  0 ]
(%o8)                      [        ]
                           [ - 1  1 ]
(%i9) moebius ({21, 22, 23, 24});
(%o9)                      {- 1, 0, 1}

Function: multinomial_coeff
    multinomial_coeff (a_1, ..., a_n)
    multinomial_coeff ()

Returns the multinomial coefficient.

When each a_k is a nonnegative integer, the multinomial coefficient gives the number of ways of placing a_1 + ... + a_n distinct objects into n boxes with a_k elements in the k’th box. In general, multinomial_coeff (a_1, ..., a_n) evaluates to (a_1 + ... + a_n)!/(a_1! ... a_n!).

multinomial_coeff() (with no arguments) evaluates to 1.

minfactorial may be able to simplify the value returned by multinomial_coeff.

Examples:

(%i1) multinomial_coeff (1, 2, x);
                            (x + 3)!
(%o1)                       --------
                              2 x!
(%i2) minfactorial (%);
                     (x + 1) (x + 2) (x + 3)
(%o2)                -----------------------
                                2
(%i3) multinomial_coeff (-6, 2);
                             (- 4)!
(%o3)                       --------
                            2 (- 6)!
(%i4) minfactorial (%);
(%o4)                          10

Function: num_distinct_partitions
    num_distinct_partitions (n)
    num_distinct_partitions (n, list)

Returns the number of distinct integer partitions of n when n is a nonnegative integer. Otherwise, num_distinct_partitions returns a noun expression.

num_distinct_partitions(n, list) returns a list of the number of distinct partitions of 1, 2, 3, ..., n.

A distinct partition of n is a list of distinct positive integers k_1, ..., k_m such that n = k_1 + ... + k_m.

Examples:

(%i1) num_distinct_partitions (12);
(%o1)                          15
(%i2) num_distinct_partitions (12, list);
(%o2)      [1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15]
(%i3) num_distinct_partitions (n);
(%o3)              num_distinct_partitions(n)

Function: num_partitions
    num_partitions (n)
    num_partitions (n, list)

Returns the number of integer partitions of n when n is a nonnegative integer. Otherwise, num_partitions returns a noun expression.

num_partitions(n, list) returns a list of the number of integer partitions of 1, 2, 3, ..., n.

For a nonnegative integer n, num_partitions(n) is equal to cardinality(integer_partitions(n)); however, num_partitions does not actually construct the set of partitions, so it is much faster.

Examples:

(%i1) num_partitions (5) = cardinality (integer_partitions (5));
(%o1)                         7 = 7
(%i2) num_partitions (8, list);
(%o2)            [1, 1, 2, 3, 5, 7, 11, 15, 22]
(%i3) num_partitions (n);
(%o3)                   num_partitions(n)

Function: partition_set (a, f)

Partitions the set a according to the predicate f.

partition_set returns a list of two sets. The first set comprises the elements of a for which f evaluates to false, and the second comprises any other elements of a. partition_set does not apply is to the return value of f.

partition_set complains if a is not a literal set.

See also subset.

Examples:

(%i1) partition_set ({2, 7, 1, 8, 2, 8}, evenp);
(%o1)                   [{1, 7}, {2, 8}]
(%i2) partition_set ({x, rat(y), rat(y) + z, 1},
                     lambda ([x], ratp(x)));
(%o2)/R/              [{1, x}, {y, y + z}]

Function: permutations (a)

Returns a set of all distinct permutations of the members of the list or set a. Each permutation is a list, not a set.

When a is a list, duplicate members of a are included in the permutations.

permutations complains if a is not a literal list or set.

See also random_permutation.

Examples:

(%i1) permutations ([a, a]);
(%o1)                       {[a, a]}
(%i2) permutations ([a, a, b]);
(%o2)           {[a, a, b], [a, b, a], [b, a, a]}

Function: powerset
    powerset (a)
    powerset (a, n)

Returns the set of all subsets of a, or a subset of that set.

powerset(a) returns the set of all subsets of the set a. powerset(a) has 2^cardinality(a) members.

powerset(a, n) returns the set of all subsets of a that have cardinality n.

powerset complains if a is not a literal set, or if n is not a nonnegative integer.

Examples:

(%i1) powerset ({a, b, c});
(%o1) {{}, {a}, {a, b}, {a, b, c}, {a, c}, {b}, {b, c}, {c}}
(%i2) powerset ({w, x, y, z}, 4);
(%o2)                    {{w, x, y, z}}
(%i3) powerset ({w, x, y, z}, 3);
(%o3)     {{w, x, y}, {w, x, z}, {w, y, z}, {x, y, z}}
(%i4) powerset ({w, x, y, z}, 2);
(%o4)   {{w, x}, {w, y}, {w, z}, {x, y}, {x, z}, {y, z}}
(%i5) powerset ({w, x, y, z}, 1);
(%o5)                 {{w}, {x}, {y}, {z}}
(%i6) powerset ({w, x, y, z}, 0);
(%o6)                         {{}}

Function: random_permutation (a)

Returns a random permutation of the set or list a, as constructed by the Knuth shuffle algorithm.

The return value is a new list, which is distinct from the argument even if all elements happen to be the same. However, the elements of the argument are not copied.

Examples:

(%i1) random_permutation ([a, b, c, 1, 2, 3]);
(%o1)                  [c, 1, 2, 3, a, b]
(%i2) random_permutation ([a, b, c, 1, 2, 3]);
(%o2)                  [b, 3, 1, c, a, 2]
(%i3) random_permutation ({x + 1, y + 2, z + 3});
(%o3)                 [y + 2, z + 3, x + 1]
(%i4) random_permutation ({x + 1, y + 2, z + 3});
(%o4)                 [x + 1, y + 2, z + 3]

Function: setdifference (a, b)

Returns a set containing the elements in the set a that are not in the set b.

setdifference complains if either a or b is not a literal set.

Examples:

(%i1) S_1 : {a, b, c, x, y, z};
(%o1)                  {a, b, c, x, y, z}
(%i2) S_2 : {aa, bb, c, x, y, zz};
(%o2)                 {aa, bb, c, x, y, zz}
(%i3) setdifference (S_1, S_2);
(%o3)                       {a, b, z}
(%i4) setdifference (S_2, S_1);
(%o4)                     {aa, bb, zz}
(%i5) setdifference (S_1, S_1);
(%o5)                          {}
(%i6) setdifference (S_1, {});
(%o6)                  {a, b, c, x, y, z}
(%i7) setdifference ({}, S_1);
(%o7)                          {}

Function: setequalp (a, b)

Returns true if sets a and b have the same number of elements and is(x = y) is true for x in the elements of a and y in the elements of b, considered in the order determined by listify. Otherwise, setequalp returns false.

Examples:

(%i1) setequalp ({1, 2, 3}, {1, 2, 3});
(%o1)                         true
(%i2) setequalp ({a, b, c}, {1, 2, 3});
(%o2)                         false
(%i3) setequalp ({x^2 - y^2}, {(x + y) * (x - y)});
(%o3)                         false

Function: setify (a)

Constructs a set from the elements of the list a. Duplicate elements of the list a are deleted and the elements are sorted according to the predicate orderlessp.

setify complains if a is not a literal list.

Examples:

(%i1) setify ([1, 2, 3, a, b, c]);
(%o1)                  {1, 2, 3, a, b, c}
(%i2) setify ([a, b, c, a, b, c]);
(%o2)                       {a, b, c}
(%i3) setify ([7, 13, 11, 1, 3, 9, 5]);
(%o3)                {1, 3, 5, 7, 9, 11, 13}

Function: setp (a)

Returns true if and only if a is a Maxima set.

setp returns true for unsimplified sets (that is, sets with redundant members) as well as simplified sets.

setp is equivalent to the Maxima function setp(a) := not atom(a) and op(a) = 'set.

Examples:

(%i1) simp : false;
(%o1)                         false
(%i2) {a, a, a};
(%o2)                       {a, a, a}
(%i3) setp (%);
(%o3)                         true

Function: set_partitions
    set_partitions (a)
    set_partitions (a, n)

Returns the set of all partitions of a, or a subset of that set.

set_partitions(a, n) returns a set of all decompositions of a into n nonempty disjoint subsets.

set_partitions(a) returns the set of all partitions.

stirling2 returns the cardinality of the set of partitions of a set.

A set of sets P is a partition of a set S when

  1. each member of P is a nonempty set,
  2. distinct members of P are disjoint,
  3. the union of the members of P equals S.

Examples:

The empty set is a partition of itself, the conditions 1 and 2 being vacuously true.

(%i1) set_partitions ({});
(%o1)                         {{}}

The cardinality of the set of partitions of a set can be found using stirling2.

(%i1) s: {0, 1, 2, 3, 4, 5}$
(%i2) p: set_partitions (s, 3)$ 
(%i3) cardinality(p) = stirling2 (6, 3);
(%o3)                        90 = 90

Each member of p should have n = 3 members; let’s check.

(%i1) s: {0, 1, 2, 3, 4, 5}$
(%i2) p: set_partitions (s, 3)$ 
(%i3) map (cardinality, p);
(%o3)                          {3}

Finally, for each member of p, the union of its members should equal s; again let’s check.

(%i1) s: {0, 1, 2, 3, 4, 5}$
(%i2) p: set_partitions (s, 3)$ 
(%i3) map (lambda ([x], apply (union, listify (x))), p);
(%o3)                 {{0, 1, 2, 3, 4, 5}}

Function: some
    some (f, a)
    some (f, L_1, ..., L_n)

Returns true if the predicate f is true for one or more given arguments.

Given one set as the second argument, some(f, s) returns true if is(f(a_i)) returns true for one or more a_i in s. some may or may not evaluate f for all a_i in s. Since sets are unordered, some may evaluate f(a_i) in any order.

Given one or more lists as arguments, some(f, L_1, ..., L_n) returns true if is(f(x_1, ..., x_n)) returns true for one or more x_1, ..., x_n in L_1, ..., L_n, respectively. some may or may not evaluate f for some combinations x_1, ..., x_n. some evaluates lists in the order of increasing index.

Given an empty set {} or empty lists [] as arguments, some returns false.

When the global flag maperror is true, all lists L_1, ..., L_n must have equal lengths. When maperror is false, list arguments are effectively truncated to the length of the shortest list.

Return values of the predicate f which evaluate (via is) to something other than true or false are governed by the global flag prederror. When prederror is true, such values are treated as false. When prederror is false, such values are treated as unknown.

Examples:

some applied to a single set. The predicate is a function of one argument.

(%i1) some (integerp, {1, 2, 3, 4, 5, 6});
(%o1)                         true
(%i2) some (atom, {1, 2, sin(3), 4, 5 + y, 6});
(%o2)                         true

some applied to two lists. The predicate is a function of two arguments.

(%i1) some ("=", [a, b, c], [a, b, c]);
(%o1)                         true
(%i2) some ("#", [a, b, c], [a, b, c]);
(%o2)                         false

Return values of the predicate f which evaluate to something other than true or false are governed by the global flag prederror.

(%i1) prederror : false;
(%o1)                         false
(%i2) map (lambda ([a, b], is (a < b)), [x, y, z],
           [x^2, y^2, z^2]);
(%o2)              [unknown, unknown, unknown]
(%i3) some ("<", [x, y, z], [x^2, y^2, z^2]);
(%o3)                        unknown
(%i4) some ("<", [x, y, z], [x^2, y^2, z + 1]);
(%o4)                         true
(%i5) prederror : true;
(%o5)                         true
(%i6) some ("<", [x, y, z], [x^2, y^2, z^2]);
(%o6)                         false
(%i7) some ("<", [x, y, z], [x^2, y^2, z + 1]);
(%o7)                         true

Function: stirling1 (n, m)

Represents the Stirling number of the first kind.

When n and m are nonnegative integers, the magnitude of stirling1 (n, m) is the number of permutations of a set with n members that have m cycles.

stirling1 is a simplifying function. Maxima knows the following identities:

  1. stirling1(1,k) = kron_delta(1,k), k >= 0,(see http://dlmf.nist.gov/26.8.E2)
  2. stirling1(n,n) = 1, n >= 0 (see http://dlmf.nist.gov/26.8.E1)
  3. stirling1(n,n-1) = -binomial(n,2), n >= 1, (see http://dlmf.nist.gov/26.8.E16)
  4. stirling1(n,0) = kron_delta(n,0), n >=0 (see http://dlmf.nist.gov/26.8.E14 and http://dlmf.nist.gov/26.8.E1)
  5. stirling1(n,1) =(-1)^(n-1) (n-1)!, n >= 1 (see http://dlmf.nist.gov/26.8.E14)
  6. stirling1(n,k) = 0, n >= 0 and k > n.

These identities are applied when the arguments are literal integers or symbols declared as integers, and the first argument is nonnegative. stirling1 does not simplify for non-integer arguments.

Examples:

(%i1) declare (n, integer)$
(%i2) assume (n >= 0)$
(%i3) stirling1 (n, n);
(%o3)                           1

Function: stirling2 (n, m)

Represents the Stirling number of the second kind.

When n and m are nonnegative integers, stirling2 (n, m) is the number of ways a set with cardinality n can be partitioned into m disjoint subsets.

stirling2 is a simplifying function. Maxima knows the following identities.

  1. stirling2(n,0) = 1, n >= 1 (see http://dlmf.nist.gov/26.8.E17 and stirling2(0,0) = 1)
  2. stirling2(n,n) = 1, n >= 0, (see http://dlmf.nist.gov/26.8.E4)
  3. stirling2(n,1) = 1, n >= 1, (see http://dlmf.nist.gov/26.8.E17 and stirling2(0,1) = 0)
  4. stirling2(n,2) = 2^(n-1) -1, n >= 1, (see http://dlmf.nist.gov/26.8.E17)
  5. stirling2(n,n-1) = binomial(n,2), n>= 1 (see http://dlmf.nist.gov/26.8.E16)
  6. stirling2(n,k) = 0, n >= 0 and k > n.

These identities are applied when the arguments are literal integers or symbols declared as integers, and the first argument is nonnegative. stirling2 does not simplify for non-integer arguments.

Examples:

(%i1) declare (n, integer)$
(%i2) assume (n >= 0)$
(%i3) stirling2 (n, n);
(%o3)                           1

stirling2 does not simplify for non-integer arguments.

(%i1) stirling2 (%pi, %pi);
(%o1)                  stirling2(%pi, %pi)

Function: subset (a, f)

Returns the subset of the set a that satisfies the predicate f.

subset returns a set which comprises the elements of a for which f returns anything other than false. subset does not apply is to the return value of f.

subset complains if a is not a literal set.

See also partition_set.

Examples:

(%i1) subset ({1, 2, x, x + y, z, x + y + z}, atom);
(%o1)                     {1, 2, x, z}
(%i2) subset ({1, 2, 7, 8, 9, 14}, evenp);
(%o2)                      {2, 8, 14}

Function: subsetp (a, b)

Returns true if and only if the set a is a subset of b.

subsetp complains if either a or b is not a literal set.

Examples:

(%i1) subsetp ({1, 2, 3}, {a, 1, b, 2, c, 3});
(%o1)                         true
(%i2) subsetp ({a, 1, b, 2, c, 3}, {1, 2, 3});
(%o2)                         false

Function: symmdifference (a_1, …, a_n)

Returns the symmetric difference of sets a_1, …, a_n.

Given two arguments, symmdifference (a, b) is the same as union (setdifference (a, b), setdifference (b, a)).

symmdifference complains if any argument is not a literal set.

Examples:

(%i1) S_1 : {a, b, c};
(%o1)                       {a, b, c}
(%i2) S_2 : {1, b, c};
(%o2)                       {1, b, c}
(%i3) S_3 : {a, b, z};
(%o3)                       {a, b, z}
(%i4) symmdifference ();
(%o4)                          {}
(%i5) symmdifference (S_1);
(%o5)                       {a, b, c}
(%i6) symmdifference (S_1, S_2);
(%o6)                        {1, a}
(%i7) symmdifference (S_1, S_2, S_3);
(%o7)                        {1, b, z}
(%i8) symmdifference ({}, S_1, S_2, S_3);
(%o8)                        {1,b, z}

Function: union (a_1, ..., a_n)

Returns the union of the sets a_1 through a_n.

union() (with no arguments) returns the empty set.

union complains if any argument is not a literal set.

Examples:

(%i1) S_1 : {a, b, c + d, %e};
(%o1)                   {%e, a, b, d + c}
(%i2) S_2 : {%pi, %i, %e, c + d};
(%o2)                 {%e, %i, %pi, d + c}
(%i3) S_3 : {17, 29, 1729, %pi, %i};
(%o3)                {17, 29, 1729, %i, %pi}
(%i4) union ();
(%o4)                          {}
(%i5) union (S_1);
(%o5)                   {%e, a, b, d + c}
(%i6) union (S_1, S_2);
(%o6)              {%e, %i, %pi, a, b, d + c}
(%i7) union (S_1, S_2, S_3);
(%o7)       {17, 29, 1729, %e, %i, %pi, a, b, d + c}
(%i8) union ({}, S_1, S_2, S_3);
(%o8)       {17, 29, 1729, %e, %i, %pi, a, b, d + c}


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36 Function Definition


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36.1 Introduction to Function Definition


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36.2 Function

·

Category: Function definition · Category: Programming

36.2.1 Ordinary functions

To define a function in Maxima you use the := operator. E.g.

f(x) := sin(x)

defines a function f. Anonymous functions may also be created using lambda. For example

lambda ([i, j], ...)

can be used instead of f where

f(i,j) := block ([], ...);
map (lambda ([i], i+1), l)

would return a list with 1 added to each term.

You may also define a function with a variable number of arguments, by having a final argument which is assigned to a list of the extra arguments:

(%i1) f ([u]) := u;
(%o1)                      f([u]) := u
(%i2) f (1, 2, 3, 4);
(%o2)                     [1, 2, 3, 4]
(%i3) f (a, b, [u]) := [a, b, u];
(%o3)               f(a, b, [u]) := [a, b, u]
(%i4) f (1, 2, 3, 4, 5, 6);
(%o4)                 [1, 2, [3, 4, 5, 6]]

The right hand side of a function is an expression. Thus if you want a sequence of expressions, you do

f(x) := (expr1, expr2, ...., exprn);

and the value of exprn is what is returned by the function.

If you wish to make a return from some expression inside the function then you must use block and return.

block ([], expr1, ..., if (a > 10) then return(a), ..., exprn)

is itself an expression, and so could take the place of the right hand side of a function definition. Here it may happen that the return happens earlier than the last expression.

The first [] in the block, may contain a list of variables and variable assignments, such as [a: 3, b, c: []], which would cause the three variables a,b,and c to not refer to their global values, but rather have these special values for as long as the code executes inside the block, or inside functions called from inside the block. This is called dynamic binding, since the variables last from the start of the block to the time it exits. Once you return from the block, or throw out of it, the old values (if any) of the variables will be restored. It is certainly a good idea to protect your variables in this way. Note that the assignments in the block variables, are done in parallel. This means, that if you had used c: a in the above, the value of c would have been the value of a at the time you just entered the block, but before a was bound. Thus doing something like

block ([a: a], expr1, ... a: a+3, ..., exprn)

will protect the external value of a from being altered, but would let you access what that value was. Thus the right hand side of the assignments, is evaluated in the entering context, before any binding occurs. Using just block ([x], ...) would cause the x to have itself as value, just as if it would have if you entered a fresh Maxima session.

The actual arguments to a function are treated in exactly same way as the variables in a block. Thus in

f(x) := (expr1, ..., exprn);

and

f(1);

we would have a similar context for evaluation of the expressions as if we had done

block ([x: 1], expr1, ..., exprn)

Inside functions, when the right hand side of a definition, may be computed at runtime, it is useful to use define and possibly buildq.

36.2.2 Memoizing Functions

A memoizing function caches the result the first time it is called with a given argument, and returns the stored value, without recomputing it, when that same argument is given. Memoizing functions are often called array function and are in fact handled like arrays in many ways:

The names of memoizing functions are appended to the global list arrays (not the global list functions). arrayinfo returns the list of arguments for which there are stored values, and listarray returns the stored values. dispfun and fundef return the array function definition.

arraymake constructs an array function call, analogous to funmake for ordinary functions. arrayapply applies an array function to its arguments, analogous to apply for ordinary functions. There is nothing exactly analogous to map for array functions, although map(lambda([x], a[x]), L) or makelist(a[x], x, L), where L is a list, are not too far off the mark.

remarray removes an array function definition (including any stored function values), analogous to remfunction for ordinary functions.

kill(a[x]) removes the value of the array function a stored for the argument x; the next time a is called with argument x, the function value is recomputed. However, there is no way to remove all of the stored values at once, except for kill(a) or remarray(a), which also remove the function definition.

Examples

If evaluating the function needs much time and only a limited number of points is ever evaluated (which means not much time is spent looking up results in a long list of cached results) Memoizing functions can speed up calculations considerably.

(%i1) showtime:true$
Evaluation took 0.0000 seconds (0.0000 elapsed) using 0 bytes.
(%i2) a[x]:=float(sum(sin(x*t),t,1,10000));
Evaluation took 0.0000 seconds (0.0000 elapsed) using 0 bytes.
(%o2)        a  := float(sum(sin(x t), t, 1, 10000))
              x
(%i3) a[1];
Evaluation took 5.1250 seconds (5.1260 elapsed) using 775.250 MB.
(%o3)                   1.633891021792447
(%i4) a[1];
Evaluation took 0.0000 seconds (0.0000 elapsed) using 0 bytes.
(%o4)                   1.633891021792447

As the memoizing function is only evaluated once for each input value changes in variables the memoizing function uses are not considered for values that are already cached:

(%i1) a[x]:=b*x;
(%o1)                       a  := b x
                             x
(%i2) b:1;
(%o2)                           1
(%i3) a[2];
(%o3)                           2
(%i4) b:2;
(%o4)                           2
(%i5) a[1];
(%o5)                           2
(%i6) a[2];
(%o6)                           2

Next: , Previous: , Up: Function Definition   [Contents][Index]

36.3 Macros

Function: buildq (L, expr)

Substitutes variables named by the list L into the expression expr, in parallel, without evaluating expr. The resulting expression is simplified, but not evaluated, after buildq carries out the substitution.

The elements of L are symbols or assignment expressions symbol: value, evaluated in parallel. That is, the binding of a variable on the right-hand side of an assignment is the binding of that variable in the context from which buildq was called, not the binding of that variable in the variable list L. If some variable in L is not given an explicit assignment, its binding in buildq is the same as in the context from which buildq was called.

Then the variables named by L are substituted into expr in parallel. That is, the substitution for every variable is determined before any substitution is made, so the substitution for one variable has no effect on any other.

If any variable x appears as splice (x) in expr, then x must be bound to a list, and the list is spliced (interpolated) into expr instead of substituted.

Any variables in expr not appearing in L are carried into the result verbatim, even if they have bindings in the context from which buildq was called.

Examples

a is explicitly bound to x, while b has the same binding (namely 29) as in the calling context, and c is carried through verbatim. The resulting expression is not evaluated until the explicit evaluation ''%.

(%i1) (a: 17, b: 29, c: 1729)$
(%i2) buildq ([a: x, b], a + b + c);
(%o2)                      x + c + 29
(%i3) ''%;
(%o3)                       x + 1758

e is bound to a list, which appears as such in the arguments of foo, and interpolated into the arguments of bar.

(%i1) buildq ([e: [a, b, c]], foo (x, e, y));
(%o1)                 foo(x, [a, b, c], y)
(%i2) buildq ([e: [a, b, c]], bar (x, splice (e), y));
(%o2)                  bar(x, a, b, c, y)

The result is simplified after substitution. If simplification were applied before substitution, these two results would be the same.

(%i1) buildq ([e: [a, b, c]], splice (e) + splice (e));
(%o1)                    2 c + 2 b + 2 a
(%i2) buildq ([e: [a, b, c]], 2 * splice (e));
(%o2)                        2 a b c

The variables in L are bound in parallel; if bound sequentially, the first result would be foo (b, b). Substitutions are carried out in parallel; compare the second result with the result of subst, which carries out substitutions sequentially.

(%i1) buildq ([a: b, b: a], foo (a, b));
(%o1)                       foo(b, a)
(%i2) buildq ([u: v, v: w, w: x, x: y, y: z, z: u],
              bar (u, v, w, x, y, z));
(%o2)                 bar(v, w, x, y, z, u)
(%i3) subst ([u=v, v=w, w=x, x=y, y=z, z=u],
             bar (u, v, w, x, y, z));
(%o3)                 bar(u, u, u, u, u, u)

Construct a list of equations with some variables or expressions on the left-hand side and their values on the right-hand side. macroexpand shows the expression returned by show_values.

(%i1) show_values ([L]) ::= buildq ([L], map ("=", 'L, L));
(%o1)   show_values([L]) ::= buildq([L], map("=", 'L, L))
(%i2) (a: 17, b: 29, c: 1729)$
(%i3) show_values (a, b, c - a - b);
(%o3)          [a = 17, b = 29, c - b - a = 1683]
(%i4) macroexpand (show_values (a, b, c - a - b));
(%o4)    map(=, '([a, b, c - b - a]), [a, b, c - b - a])

Given a function of several arguments, create another function for which some of the arguments are fixed.

(%i1) curry (f, [a]) :=
        buildq ([f, a], lambda ([[x]], apply (f, append (a, x))))$
(%i2) by3 : curry ("*", 3);
(%o2)        lambda([[x]], apply(*, append([3], x)))
(%i3) by3 (a + b);
(%o3)                       3 (b + a)

Function: macroexpand (expr)

Returns the macro expansion of expr without evaluating it, when expr is a macro function call. Otherwise, macroexpand returns expr.

If the expansion of expr yields another macro function call, that macro function call is also expanded.

macroexpand quotes its argument. However, if the expansion of a macro function call has side effects, those side effects are executed.

See also ::=, macros, and macroexpand1..

Examples

(%i1) g (x) ::= x / 99;
                                    x
(%o1)                      g(x) ::= --
                                    99
(%i2) h (x) ::= buildq ([x], g (x - a));
(%o2)            h(x) ::= buildq([x], g(x - a))
(%i3) a: 1234;
(%o3)                         1234
(%i4) macroexpand (h (y));
                              y - a
(%o4)                         -----
                               99
(%i5) h (y);
                            y - 1234
(%o5)                       --------
                               99

Function: macroexpand1 (expr)

Returns the macro expansion of expr without evaluating it, when expr is a macro function call. Otherwise, macroexpand1 returns expr.

macroexpand1 quotes its argument. However, if the expansion of a macro function call has side effects, those side effects are executed.

If the expansion of expr yields another macro function call, that macro function call is not expanded.

See also ::=, macros, and macroexpand.

Examples

(%i1) g (x) ::= x / 99;
                                    x
(%o1)                      g(x) ::= --
                                    99
(%i2) h (x) ::= buildq ([x], g (x - a));
(%o2)            h(x) ::= buildq([x], g(x - a))
(%i3) a: 1234;
(%o3)                         1234
(%i4) macroexpand1 (h (y));
(%o4)                       g(y - a)
(%i5) h (y);
                            y - 1234
(%o5)                       --------
                               99

Global variable: macros

Default value: []

macros is the list of user-defined macro functions. The macro function definition operator ::= puts a new macro function onto this list, and kill, remove, and remfunction remove macro functions from the list.

See also infolists.

Function: splice (a)

Splices (interpolates) the list named by the atom a into an expression, but only if splice appears within buildq; otherwise, splice is treated as an undefined function. If appearing within buildq as a alone (without splice), a is substituted (not interpolated) as a list into the result. The argument of splice can only be an atom; it cannot be a literal list or an expression which yields a list.

Typically splice supplies the arguments for a function or operator. For a function f, the expression f (splice (a)) within buildq expands to f (a[1], a[2], a[3], ...). For an operator o, the expression "o" (splice (a)) within buildq expands to "o" (a[1], a[2], a[3], ...), where o may be any type of operator (typically one which takes multiple arguments). Note that the operator must be enclosed in double quotes ".

Examples

(%i1) buildq ([x: [1, %pi, z - y]], foo (splice (x)) / length (x));
                       foo(1, %pi, z - y)
(%o1)                -----------------------
                     length([1, %pi, z - y])
(%i2) buildq ([x: [1, %pi]], "/" (splice (x)));
                                1
(%o2)                          ---
                               %pi
(%i3) matchfix ("<>", "<>");
(%o3)                          <>
(%i4) buildq ([x: [1, %pi, z - y]], "<>" (splice (x)));
(%o4)                   <>1, %pi, z - y<>


Previous: , Up: Function Definition   [Contents][Index]

36.4 Functions and Variables for Function Definition

Function: apply (F, [x_1, …, x_n])

Constructs and evaluates an expression F(arg_1, ..., arg_n).

apply does not attempt to distinguish a memoizing function from an ordinary function; when F is the name of a memoizing function, apply evaluates F(...) (that is, a function call with parentheses instead of square brackets). arrayapply evaluates a function call with square brackets in this case.

See also funmake and args.

Examples:

apply evaluates its arguments. In this example, min is applied to the value of L.

(%i1) L : [1, 5, -10.2, 4, 3];
(%o1)                 [1, 5, - 10.2, 4, 3]
(%i2) apply (min, L);
(%o2)                        - 10.2

apply evaluates arguments, even if the function F quotes them.

(%i1) F (x) := x / 1729;
                                   x
(%o1)                     F(x) := ----
                                  1729
(%i2) fname : F;
(%o2)                           F
(%i3) dispfun (F);
                                   x
(%t3)                     F(x) := ----
                                  1729

(%o3)                         [%t3]
(%i4) dispfun (fname);
fundef: no such function: fname
 -- an error. To debug this try: debugmode(true);
(%i5) apply (dispfun, [fname]);
                                   x
(%t5)                     F(x) := ----
                                  1729

(%o5)                         [%t5]

apply evaluates the function name F. Single quote ' defeats evaluation. demoivre is the name of a global variable and also a function.

(%i1) demoivre;
(%o1)                         false
(%i2) demoivre (exp (%i * x));
(%o2)                  %i sin(x) + cos(x)
(%i3) apply (demoivre, [exp (%i * x)]);
apply: found false where a function was expected.
 -- an error. To debug this try: debugmode(true);
(%i4) apply ('demoivre, [exp (%i * x)]);
(%o4)                  %i sin(x) + cos(x)

How to convert a nested list into a matrix:

(%i1) a:[[1,2],[3,4]];
(%o1)                   [[1, 2], [3, 4]]
(%i2) apply(matrix,a);
                            [ 1  2 ]
(%o2)                       [      ]
                            [ 3  4 ]

Function: block
    block ([v_1, …, v_m], expr_1, …, expr_n)
    block (expr_1, …, expr_n)

The function block allows to make the variables v_1, …, v_m to be local for a sequence of commands. If these variables are already bound block saves the current values of the variables v_1, …, v_m (if any) upon entry to the block, then unbinds the variables so that they evaluate to themselves; The local variables may be bound to arbitrary values within the block but when the block is exited the saved values are restored, and the values assigned within the block are lost.

If there is no need to define local variables then the list at the beginning of the block command may be omitted. In this case if neither return nor go are used block behaves similar to the following construct:

( expr_1, expr_2,... , expr_n );

expr_1, …, expr_n will be evaluated in sequence and the value of the last expression will be returned. The sequence can be modified by the go, throw, and return functions. The last expression is expr_n unless return or an expression containing throw is evaluated.

The declaration local(v_1, ..., v_m) within block saves the properties associated with the symbols v_1, …, v_m, removes any properties before evaluating other expressions, and restores any saved properties on exit from the block. Some declarations are implemented as properties of a symbol, including :=, array, dependencies, atvalue, matchdeclare, atomgrad, constant, nonscalar, assume, and some others. The effect of local is to make such declarations effective only within the block; otherwise declarations within a block are actually global declarations.

block may appear within another block. Local variables are established each time a new block is evaluated. Local variables appear to be global to any enclosed blocks. If a variable is non-local in a block, its value is the value most recently assigned by an enclosing block, if any, otherwise, it is the value of the variable in the global environment. This policy may coincide with the usual understanding of "dynamic scope".

The value of the block is the value of the last statement or the value of the argument to the function return which may be used to exit explicitly from the block. The function go may be used to transfer control to the statement of the block that is tagged with the argument to go. To tag a statement, precede it by an atomic argument as another statement in the block. For example: block ([x], x:1, loop, x: x+1, ..., go(loop), ...). The argument to go must be the name of a tag appearing within the block. One cannot use go to transfer to a tag in a block other than the one containing the go.

Blocks typically appear on the right side of a function definition but can be used in other places as well.

See also return and go.

Function: break (expr_1, …, expr_n)

Evaluates and prints expr_1, …, expr_n and then causes a Maxima break at which point the user can examine and change his environment. Upon typing exit; the computation resumes.

Function: catch (expr_1, …, expr_n)

Evaluates expr_1, …, expr_n one by one; if any leads to the evaluation of an expression of the form throw (arg), then the value of the catch is the value of throw (arg), and no further expressions are evaluated. This "non-local return" thus goes through any depth of nesting to the nearest enclosing catch. If there is no catch enclosing a throw, an error message is printed.

If the evaluation of the arguments does not lead to the evaluation of any throw then the value of catch is the value of expr_n.

(%i1) lambda ([x], if x < 0 then throw(x) else f(x))$
(%i2) g(l) := catch (map (''%, l))$
(%i3) g ([1, 2, 3, 7]);
(%o3)               [f(1), f(2), f(3), f(7)]
(%i4) g ([1, 2, -3, 7]);
(%o4)                          - 3

The function g returns a list of f of each element of l if l consists only of non-negative numbers; otherwise, g "catches" the first negative element of l and "throws" it up.

Function: compfile
    compfile (filename, f_1, …, f_n)
    compfile (filename, functions)
    compfile (filename, all)

Translates Maxima functions into Lisp and writes the translated code into the file filename.

compfile(filename, f_1, ..., f_n) translates the specified functions. compfile (filename, functions) and compfile (filename, all) translate all user-defined functions.

The Lisp translations are not evaluated, nor is the output file processed by the Lisp compiler. translate creates and evaluates Lisp translations. compile_file translates Maxima into Lisp, and then executes the Lisp compiler.

See also translate, translate_file, and compile_file.

Function: compile
    compile (f_1, …, f_n)
    compile (functions)
    compile (all)

Translates Maxima functions f_1, …, f_n into Lisp, evaluates the Lisp translations, and calls the Lisp function COMPILE on each translated function. compile returns a list of the names of the compiled functions.

compile (all) or compile (functions) compiles all user-defined functions.

compile quotes its arguments; the quote-quote operator '' defeats quotation.

Compiling a function to native code can mean a big increase in speed and might cause the memory footprint to reduce drastically. Code tends to be especially effective when the flexibility it needs to provide is limited. If compilation doesn’t provide the speed that is needed a few ways to limit the code’s functionality are the following:

Function: define
    define (f(x_1, …, x_n), expr)
    define (f[x_1, …, x_n], expr)
    define (f[x_1, …, x_n](y_1, …, y_m), expr)
    define (funmake (f, [x_1, …, x_n]), expr)
    define (arraymake (f, [x_1, …, x_n]), expr)
    define (ev (expr_1), expr_2)

Defines a function named f with arguments x_1, …, x_n and function body expr. define always evaluates its second argument (unless explicitly quoted). The function so defined may be an ordinary Maxima function (with arguments enclosed in parentheses) or a memoizing function (with arguments enclosed in square brackets).

When the last or only function argument x_n is a list of one element, the function defined by define accepts a variable number of arguments. Actual arguments are assigned one-to-one to formal arguments x_1, …, x_(n - 1), and any further actual arguments, if present, are assigned to x_n as a list.

When the first argument of define is an expression of the form f(x_1, ..., x_n) or f[x_1, ..., x_n], the function arguments are evaluated but f is not evaluated, even if there is already a function or variable by that name.

When the first argument is an expression with operator funmake, arraymake, or ev, the first argument is evaluated; this allows for the function name to be computed, as well as the body.

All function definitions appear in the same namespace; defining a function f within another function g does not automatically limit the scope of f to g. However, local(f) makes the definition of function f effective only within the block or other compound expression in which local appears.

If some formal argument x_k is a quoted symbol (after evaluation), the function defined by define does not evaluate the corresponding actual argument. Otherwise all actual arguments are evaluated.

See also := and ::=.

Examples:

define always evaluates its second argument (unless explicitly quoted).

(%i1) expr : cos(y) - sin(x);
(%o1)                    cos(y) - sin(x)
(%i2) define (F1 (x, y), expr);
(%o2)              F1(x, y) := cos(y) - sin(x)
(%i3) F1 (a, b);
(%o3)                    cos(b) - sin(a)
(%i4) F2 (x, y) := expr;
(%o4)                   F2(x, y) := expr
(%i5) F2 (a, b);
(%o5)                    cos(y) - sin(x)

The function defined by define may be an ordinary Maxima function or a memoizing function.

(%i1) define (G1 (x, y), x.y - y.x);
(%o1)               G1(x, y) := x . y - y . x
(%i2) define (G2 [x, y], x.y - y.x);
(%o2)                G2     := x . y - y . x
                       x, y

When the last or only function argument x_n is a list of one element, the function defined by define accepts a variable number of arguments.

(%i1) define (H ([L]), '(apply ("+", L)));
(%o1)                H([L]) := apply("+", L)
(%i2) H (a, b, c);
(%o2)                       c + b + a

When the first argument is an expression with operator funmake, arraymake, or ev, the first argument is evaluated.

(%i1) [F : I, u : x];
(%o1)                        [I, x]
(%i2) funmake (F, [u]);
(%o2)                         I(x)
(%i3) define (funmake (F, [u]), cos(u) + 1);
(%o3)                  I(x) := cos(x) + 1
(%i4) define (arraymake (F, [u]), cos(u) + 1);
(%o4)                   I  := cos(x) + 1
                         x
(%i5) define (foo (x, y), bar (y, x));
(%o5)                foo(x, y) := bar(y, x)
(%i6) define (ev (foo (x, y)), sin(x) - cos(y));
(%o6)             bar(y, x) := sin(x) - cos(y)

Function: define_variable (name, default_value, mode)

Introduces a global variable into the Maxima environment. define_variable is useful in user-written packages, which are often translated or compiled as it gives the compiler hints of the type (“mode”) of a variable and therefore avoids requiring it to generate generic code that can deal with every variable being an integer, float, maxima object, array etc.

define_variable carries out the following steps:

  1. mode_declare (name, mode) declares the mode (“type”) of name to the translator which can considerably speed up compiled code as it allows having to create generic code. See mode_declare for a list of the possible modes.
  2. If the variable is unbound, default_value is assigned to name.
  3. Associates name with a test function to ensure that name is only assigned values of the declared mode.

The value_check property can be assigned to any variable which has been defined via define_variable with a mode other than any. The value_check property is a lambda expression or the name of a function of one variable, which is called when an attempt is made to assign a value to the variable. The argument of the value_check function is the would-be assigned value.

define_variable evaluates default_value, and quotes name and mode. define_variable returns the current value of name, which is default_value if name was unbound before, and otherwise it is the previous value of name.

Examples:

foo is a Boolean variable, with the initial value true.

(%i1) define_variable (foo, true, boolean);
(%o1)                         true
(%i2) foo;
(%o2)                         true
(%i3) foo: false;
(%o3)                         false
(%i4) foo: %pi;
translator: foo was declared with mode boolean
                                          , but it has value: %pi
 -- an error. To debug this try: debugmode(true);
(%i5) foo;
(%o5)                         false

bar is an integer variable, which must be prime.

(%i1) define_variable (bar, 2, integer);
(%o1)                           2
(%i2) qput (bar, prime_test, value_check);
(%o2)                      prime_test
(%i3) prime_test (y) := if not primep(y) then
                           error (y, "is not prime.");
(%o3) prime_test(y) := if not primep(y)
                                   then error(y, "is not prime.")
(%i4) bar: 1439;
(%o4)                         1439
(%i5) bar: 1440;
1440 is not prime.
#0: prime_test(y=1440)
 -- an error. To debug this try: debugmode(true);
(%i6) bar;
(%o6)                         1439

baz_quux is a variable which cannot be assigned a value. The mode any_check is like any, but any_check enables the value_check mechanism, and any does not.

(%i1) define_variable (baz_quux, 'baz_quux, any_check);
(%o1)                       baz_quux
(%i2) F: lambda ([y], if y # 'baz_quux then
                 error ("Cannot assign to `baz_quux'."));
(%o2) lambda([y], if y # 'baz_quux
                        then error(Cannot assign to `baz_quux'.))
(%i3) qput (baz_quux, ''F, value_check);
(%o3) lambda([y], if y # 'baz_quux
                        then error(Cannot assign to `baz_quux'.))
(%i4) baz_quux: 'baz_quux;
(%o4)                       baz_quux
(%i5) baz_quux: sqrt(2);
Cannot assign to `baz_quux'.
#0: lambda([y],if y # 'baz_quux then error("Cannot assign to `baz_quux'."))(y=sqrt(2))
 -- an error. To debug this try: debugmode(true);
(%i6) baz_quux;
(%o6)                       baz_quux

Function: dispfun
    dispfun (f_1, …, f_n)
    dispfun (all)

Displays the definition of the user-defined functions f_1, …, f_n. Each argument may be the name of a macro (defined with ::=), an ordinary function (defined with := or define), an array function (defined with := or define, but enclosing arguments in square brackets [ ]), a subscripted function (defined with := or define, but enclosing some arguments in square brackets and others in parentheses ( )), one of a family of subscripted functions selected by a particular subscript value, or a subscripted function defined with a constant subscript.

dispfun (all) displays all user-defined functions as given by the functions, arrays, and macros lists, omitting subscripted functions defined with constant subscripts.

dispfun creates an intermediate expression label (%t1, %t2, etc.) for each displayed function, and assigns the function definition to the label. In contrast, fundef returns the function definition.

dispfun quotes its arguments; the quote-quote operator '' defeats quotation. dispfun returns the list of intermediate expression labels corresponding to the displayed functions.

Examples:

(%i1) m(x, y) ::= x^(-y);
                                     - y
(%o1)                   m(x, y) ::= x
(%i2) f(x, y) :=  x^(-y);
                                     - y
(%o2)                    f(x, y) := x
(%i3) g[x, y] :=  x^(-y);
                                    - y
(%o3)                     g     := x
                           x, y
(%i4) h[x](y) :=  x^(-y);
                                    - y
(%o4)                     h (y) := x
                           x
(%i5) i[8](y) :=  8^(-y);
                                    - y
(%o5)                     i (y) := 8
                           8
(%i6) dispfun (m, f, g, h, h[5], h[10], i[8]);
                                     - y
(%t6)                   m(x, y) ::= x

                                     - y
(%t7)                    f(x, y) := x

                                    - y
(%t8)                     g     := x
                           x, y

                                    - y
(%t9)                     h (y) := x
                           x

                                    1
(%t10)                     h (y) := --
                            5        y
                                    5

                                     1
(%t11)                    h  (y) := ---
                           10         y
                                    10

                                    - y
(%t12)                    i (y) := 8
                           8

(%o12)       [%t6, %t7, %t8, %t9, %t10, %t11, %t12]
(%i13) ''%;
                     - y              - y            - y
(%o13) [m(x, y) ::= x   , f(x, y) := x   , g     := x   , 
                                            x, y
                  - y           1              1             - y
        h (y) := x   , h (y) := --, h  (y) := ---, i (y) := 8   ]
         x              5        y   10         y   8
                                5             10

Function: fullmap (f, expr_1, …)

Similar to map, but fullmap keeps mapping down all subexpressions until the main operators are no longer the same.

fullmap is used by the Maxima simplifier for certain matrix manipulations; thus, Maxima sometimes generates an error message concerning fullmap even though fullmap was not explicitly called by the user.

Examples:

(%i1) a + b * c;
(%o1)                        b c + a
(%i2) fullmap (g, %);
(%o2)                   g(b) g(c) + g(a)
(%i3) map (g, %th(2));
(%o3)                     g(b c) + g(a)

Function: fullmapl (f, list_1, …)

Similar to fullmap, but fullmapl only maps onto lists and matrices.

Example:

(%i1) fullmapl ("+", [3, [4, 5]], [[a, 1], [0, -1.5]]);
(%o1)                [[a + 3, 4], [4, 3.5]]

System variable: functions

Default value: []

functions is the list of ordinary Maxima functions in the current session. An ordinary function is a function constructed by define or := and called with parentheses (). A function may be defined at the Maxima prompt or in a Maxima file loaded by load or batch.

Memoizing functions (called with square brackets, e.g., F[x]) and subscripted functions (called with square brackets and parentheses, e.g., F[x](y)) are listed by the global variable arrays, and not by functions.

Lisp functions are not kept on any list.

Examples:

(%i1) F_1 (x) := x - 100;
(%o1)                   F_1(x) := x - 100
(%i2) F_2 (x, y) := x / y;
                                      x
(%o2)                    F_2(x, y) := -
                                      y
(%i3) define (F_3 (x), sqrt (x));
(%o3)                   F_3(x) := sqrt(x)
(%i4) G_1 [x] := x - 100;
(%o4)                    G_1  := x - 100
                            x
(%i5) G_2 [x, y] := x / y;
                                     x
(%o5)                     G_2     := -
                             x, y    y
(%i6) define (G_3 [x], sqrt (x));
(%o6)                    G_3  := sqrt(x)
                            x
(%i7) H_1 [x] (y) := x^y;
                                      y
(%o7)                     H_1 (y) := x
                             x
(%i8) functions;
(%o8)              [F_1(x), F_2(x, y), F_3(x)]
(%i9) arrays;
(%o9)                 [G_1, G_2, G_3, H_1]

Function: fundef (f)

Returns the definition of the function f.

The argument may be

fundef quotes its argument; the quote-quote operator '' defeats quotation.

fundef (f) returns the definition of f. In contrast, dispfun (f) creates an intermediate expression label and assigns the definition to the label.

Function: funmake (F, [arg_1, …, arg_n])

Returns an expression F(arg_1, ..., arg_n). The return value is simplified, but not evaluated, so the function F is not called, even if it exists.

funmake does not attempt to distinguish memoizing functions from ordinary functions; when F is the name of a memoizing function, funmake returns F(...) (that is, a function call with parentheses instead of square brackets). arraymake returns a function call with square brackets in this case.

funmake evaluates its arguments.

See also apply and args.

Examples:

funmake applied to an ordinary Maxima function.

(%i1) F (x, y) := y^2 - x^2;
                                   2    2
(%o1)                  F(x, y) := y  - x
(%i2) funmake (F, [a + 1, b + 1]);
(%o2)                    F(a + 1, b + 1)
(%i3) ''%;
                              2          2
(%o3)                  (b + 1)  - (a + 1)

funmake applied to a macro.

(%i1) G (x) ::= (x - 1)/2;
                                  x - 1
(%o1)                    G(x) ::= -----
                                    2
(%i2) funmake (G, [u]);
(%o2)                         G(u)
(%i3) ''%;
                              u - 1
(%o3)                         -----
                                2

funmake applied to a subscripted function.

(%i1) H [a] (x) := (x - 1)^a;
                                        a
(%o1)                   H (x) := (x - 1)
                         a
(%i2) funmake (H [n], [%e]);
                                       n
(%o2)               lambda([x], (x - 1) )(%e)
(%i3) ''%;
                                    n
(%o3)                       (%e - 1)
(%i4) funmake ('(H [n]), [%e]);
(%o4)                        H (%e)
                              n
(%i5) ''%;
                                    n
(%o5)                       (%e - 1)

funmake applied to a symbol which is not a defined function of any kind.

(%i1) funmake (A, [u]);
(%o1)                         A(u)
(%i2) ''%;
(%o2)                         A(u)

funmake evaluates its arguments, but not the return value.

(%i1) det(a,b,c) := b^2 -4*a*c;
                                    2
(%o1)              det(a, b, c) := b  - 4 a c
(%i2) (x : 8, y : 10, z : 12);
(%o2)                          12
(%i3) f : det;
(%o3)                          det
(%i4) funmake (f, [x, y, z]);
(%o4)                    det(8, 10, 12)
(%i5) ''%;
(%o5)                         - 284

Maxima simplifies funmake’s return value.

(%i1) funmake (sin, [%pi / 2]);
(%o1)                           1

Function: lambda
    lambda ([x_1, …, x_m], expr_1, …, expr_n)
    lambda ([[L]], expr_1, …, expr_n)
    lambda ([x_1, …, x_m, [L]], expr_1, …, expr_n)

Defines and returns a lambda expression (that is, an anonymous function). The function may have required arguments x_1, …, x_m and/or optional arguments L, which appear within the function body as a list. The return value of the function is expr_n. A lambda expression can be assigned to a variable and evaluated like an ordinary function. A lambda expression may appear in some contexts in which a function name is expected.

When the function is evaluated, unbound local variables x_1, …, x_m are created. lambda may appear within block or another lambda; local variables are established each time another block or lambda is evaluated. Local variables appear to be global to any enclosed block or lambda. If a variable is not local, its value is the value most recently assigned in an enclosing block or lambda, if any, otherwise, it is the value of the variable in the global environment. This policy may coincide with the usual understanding of "dynamic scope".

After local variables are established, expr_1 through expr_n are evaluated in turn. The special variable %%, representing the value of the preceding expression, is recognized. throw and catch may also appear in the list of expressions.

return cannot appear in a lambda expression unless enclosed by block, in which case return defines the return value of the block and not of the lambda expression, unless the block happens to be expr_n. Likewise, go cannot appear in a lambda expression unless enclosed by block.

lambda quotes its arguments; the quote-quote operator '' defeats quotation.

Examples:

(%i1) f: lambda ([x], x^2);
                                      2
(%o1)                    lambda([x], x )
(%i2) f(a);
                                2
(%o2)                          a
(%i1) lambda ([x], x^2) (a);
                                2
(%o1)                          a
(%i2) apply (lambda ([x], x^2), [a]);
                                2
(%o2)                          a
(%i3) map (lambda ([x], x^2), [a, b, c, d, e]);
                        2   2   2   2   2
(%o3)                 [a , b , c , d , e ]
(%i1) a: %pi$
(%i2) b: %e$
(%i3) g: lambda ([a], a*b);
(%o3)                   lambda([a], a b)
(%i4) b: %gamma$
(%i5) g(1/2);
                             %gamma
(%o5)                        ------
                               2
(%i6) g2: lambda ([a], a*''b);
(%o6)                 lambda([a], a %gamma)
(%i7) b: %e$
(%i8) g2(1/2);
                             %gamma
(%o8)                        ------
                               2
(%i1) h: lambda ([a, b], h2: lambda ([a], a*b), h2(1/2));
                                                   1
(%o1)     lambda([a, b], h2 : lambda([a], a b), h2(-))
                                                   2
(%i2) h(%pi, %gamma);
                             %gamma
(%o2)                        ------
                               2
(%i1) i: lambda ([a], lambda ([x], a*x));
(%o1)             lambda([a], lambda([x], a x))
(%i2) i(1/2);
(%o2)                   lambda([x], a x)
(%i3) i2: lambda([a], buildq([a: a], lambda([x], a*x)));
(%o3)    lambda([a], buildq([a : a], lambda([x], a x)))
(%i4) i2(1/2);
                                    1
(%o4)                  lambda([x], (-) x)
                                    2
(%i5) i2(1/2)(%pi);
                               %pi
(%o5)                          ---
                                2
(%i1) f : lambda ([aa, bb, [cc]], aa * cc + bb);
(%o1)          lambda([aa, bb, [cc]], aa cc + bb)
(%i2) f (foo, %i, 17, 29, 256);
(%o2)       [17 foo + %i, 29 foo + %i, 256 foo + %i]
(%i3) g : lambda ([[aa]], apply ("+", aa));
(%o3)             lambda([[aa]], apply(+, aa))
(%i4) g (17, 29, x, y, z, %e);
(%o4)                  z + y + x + %e + 46

Function: local (v_1, …, v_n)

Saves the properties associated with the symbols v_1, …, v_n, removes any properties before evaluating other expressions, and restores any saved properties on exit from the block or other compound expression in which local appears.

Some declarations are implemented as properties of a symbol, including :=, array, dependencies, atvalue, matchdeclare, atomgrad, constant, nonscalar, assume, and some others. The effect of local is to make such declarations effective only within the block or other compound expression in which local appears; otherwise such declarations are global declarations.

local can only appear in block or in the body of a function definition or lambda expression, and only one occurrence is permitted in each.

local quotes its arguments. local returns done.

Example:

A local function definition.

(%i1) foo (x) := 1 - x;
(%o1)                    foo(x) := 1 - x
(%i2) foo (100);
(%o2)                         - 99
(%i3) block (local (foo), foo (x) := 2 * x, foo (100));
(%o3)                          200
(%i4) foo (100);
(%o4)                         - 99

Option variable: macroexpansion

Default value: false

macroexpansion controls whether the expansion (that is, the return value) of a macro function is substituted for the macro function call. A substitution may speed up subsequent expression evaluations, at the cost of storing the expansion.

false

The expansion of a macro function is not substituted for the macro function call.

expand

The first time a macro function call is evaluated, the expansion is stored. The expansion is not recomputed on subsequent calls; any side effects (such as print or assignment to global variables) happen only when the macro function call is first evaluated. Expansion in an expression does not affect other expressions which have the same macro function call.

displace

The first time a macro function call is evaluated, the expansion is substituted for the call, thus modifying the expression from which the macro function was called. The expansion is not recomputed on subsequent calls; any side effects happen only when the macro function call is first evaluated. Expansion in an expression does not affect other expressions which have the same macro function call.

Examples

When macroexpansion is false, a macro function is called every time the calling expression is evaluated, and the calling expression is not modified.

(%i1) f (x) := h (x) / g (x);
                                  h(x)
(%o1)                     f(x) := ----
                                  g(x)
(%i2) g (x) ::= block (print ("x + 99 is equal to", x),
                       return (x + 99));
(%o2) g(x) ::= block(print("x + 99 is equal to", x), 
                                                  return(x + 99))
(%i3) h (x) ::= block (print ("x - 99 is equal to", x),
                       return (x - 99));
(%o3) h(x) ::= block(print("x - 99 is equal to", x), 
                                                  return(x - 99))
(%i4) macroexpansion: false;
(%o4)                         false
(%i5) f (a * b);
x - 99 is equal to x 
x + 99 is equal to x 
                            a b - 99
(%o5)                       --------
                            a b + 99
(%i6) dispfun (f);
                                  h(x)
(%t6)                     f(x) := ----
                                  g(x)

(%o6)                         [%t6]
(%i7) f (a * b);
x - 99 is equal to x 
x + 99 is equal to x 
                            a b - 99
(%o7)                       --------
                            a b + 99

When macroexpansion is expand, a macro function is called once, and the calling expression is not modified.

(%i1) f (x) := h (x) / g (x);
                                  h(x)
(%o1)                     f(x) := ----
                                  g(x)
(%i2) g (x) ::= block (print ("x + 99 is equal to", x),
                       return (x + 99));
(%o2) g(x) ::= block(print("x + 99 is equal to", x), 
                                                  return(x + 99))
(%i3) h (x) ::= block (print ("x - 99 is equal to", x),
                       return (x - 99));
(%o3) h(x) ::= block(print("x - 99 is equal to", x), 
                                                  return(x - 99))
(%i4) macroexpansion: expand;
(%o4)                        expand
(%i5) f (a * b);
x - 99 is equal to x 
x + 99 is equal to x 
                            a b - 99
(%o5)                       --------
                            a b + 99
(%i6) dispfun (f);
                      mmacroexpanded(x - 99, h(x))
(%t6)         f(x) := ----------------------------
                      mmacroexpanded(x + 99, g(x))

(%o6)                         [%t6]
(%i7) f (a * b);
                            a b - 99
(%o7)                       --------
                            a b + 99

When macroexpansion is displace, a macro function is called once, and the calling expression is modified.

(%i1) f (x) := h (x) / g (x);
                                  h(x)
(%o1)                     f(x) := ----
                                  g(x)
(%i2) g (x) ::= block (print ("x + 99 is equal to", x),
                       return (x + 99));
(%o2) g(x) ::= block(print("x + 99 is equal to", x), 
                                                  return(x + 99))
(%i3) h (x) ::= block (print ("x - 99 is equal to", x),
                       return (x - 99));
(%o3) h(x) ::= block(print("x - 99 is equal to", x), 
                                                  return(x - 99))
(%i4) macroexpansion: displace;
(%o4)                       displace
(%i5) f (a * b);
x - 99 is equal to x 
x + 99 is equal to x 
                            a b - 99
(%o5)                       --------
                            a b + 99
(%i6) dispfun (f);
                                 x - 99
(%t6)                    f(x) := ------
                                 x + 99

(%o6)                         [%t6]
(%i7) f (a * b);
                            a b - 99
(%o7)                       --------
                            a b + 99

Function: mode_declare (y_1, mode_1, …, y_n, mode_n)

A mode_declare informs the compiler which type (lisp programmers name the type: “mode”) a function parameter or its return value will be of. This can greatly boost the efficiency of the code the compiler generates: Without knowing the type of all variables and knowing the return value of all functions a function uses in advance very generic (and thus potentially slow) code needs to be generated.

The arguments of mode_declare are pairs consisting of a variable (or a list of variables all having the same mode) and a mode. Available modes (“types”) are:

array            an declared array (see the detailed description below)
string           a string
boolean          true or false
integer          integers (including arbitrary-size integers)
fixnum           integers (excluding arbitrary-size integers)
float            machine-size floating-point numbers
real             machine-size floating-point or integer
number           Numbers
even             Even integers
odd              Odd integers
any              any kind of object (useful for arrays of any)

A function parameter named a can be declared as an array filled with elements of the type t the following way:

mode_declare (a, array(t, dim1, dim2, ...))

If none of the elements of the array a needs to be checked if it still doesn’t contain a value additional code can be omitted by declaring this fact, too:

mode_declare (a, array (t, complete, dim1, dim2, ...))

The complete has no effect if all array elements are of the type fixnum or float: Machine-sized numbers inevitably contain a value (and will automatically be initialized to 0 in most lisp implementations).

Another way to tell that all entries of the array a are of the type (“mode”) m and have been assigned a value to would be:

mode_declare (completearray (a), m))

Numeric code using arrays might run faster still if the size of the array is known at compile time, as well, as in:

mode_declare (completearray (a [10, 10]), float)

for a floating point number array named a which is 10 x 10.

mode_declare also can be used in order to declare the type of the result of a function. In this case the function compilation needs to be preceded by another mode_declare statement. For example the expression,

mode_declare ([function (f_1, f_2, ...)], fixnum)

declares that the values returned by f_1, f_2, … are single-word integers.

modedeclare is a synonym for mode_declare.

If the type of function parameters and results doesn’t match the declaration by mode_declare the function may misbehave or a warning or an error might occur, see mode_checkp, mode_check_errorp and mode_check_warnp.

See mode_identity for declaring the type of lists and define_variable for declaring the type of all global variables compiled code uses, as well.

Example:

(%i1) square_float(f):=(
     mode_declare(f,float),
     f*f
 );
(%o1)   square_float(f) := (mode_declare(f, float), f f)
(%i2) mode_declare([function(f)],float);
(%o2)                    [[function(f)]]
(%i3) compile(square_float);
(%o3)                    [square_float]
(%i4) square_float(100.0);
(%o4)                        10000.0

Option variable: mode_checkp

Default value: true

When mode_checkp is true, mode_declare does not only define which type a variable will be of so the compiler can generate more efficient code, but will also create a runtime warning if the variable isn’t of the variable type the code was compiled to deal with.

(%i1) mode_checkp:true;
(%o1)                         true
(%i2) square(f):=(
    mode_declare(f,float),
    f^2);
                                                   2
(%o2)       square(f) := (mode_declare(f, float), f )
(%i3) compile(square);
(%o3)                       [square]
(%i4) square(2.3);
(%o4)                   5.289999999999999
(%i5) square(4);
Maxima encountered a Lisp error:

 The value
   4
 is not of type
   DOUBLE-FLOAT
 when binding $F

Automatically continuing.
To enable the Lisp debugger set *debugger-hook* to nil.

Option variable: mode_check_errorp

Default value: false

When mode_check_errorp is true, mode_declare calls error.

Option variable: mode_check_warnp

Default value: true

When mode_check_warnp is true, mode errors are described.

Function: mode_identity (arg_1, arg_2)

mode_identity works similar to mode_declare, but is used for informing the compiler that a thing like a macro or a list operation will only return a specific type of object. The purpose of doing so is that maxima supports many objects: Machine integers, arbitrary length integers, equations, machine floats, big floats, which means that for everything that deals with return values of operations that can result in any object the compiler needs to output generic (and therefore potentially slow) code.

The first argument to mode_identity is the type of return value something will return (for possible types see mode_declare). (i.e., one of float, fixnum, number, The second argument is the expression that will return an object of this type.

If the the return value of this expression is of a type the code was not compiled for error or warning is signalled.

If you knew that first (l) returned a number then you could write

mode_identity (number, first (l)).

However, if you need this construct more often it would be more efficient to define a function that returns a number fist:

firstnumb (x) ::= buildq ([x], mode_identity (number, first(x)));
compile(firstnumb)

firstnumb now can be used every time you need the first element of a list that is guaranteed to be filled with numbers.

Function: remfunction
    remfunction (f_1, …, f_n)
    remfunction (all)

Unbinds the function definitions of the symbols f_1, …, f_n. The arguments may be the names of ordinary functions (created by := or define) or macro functions (created by ::=).

remfunction (all) unbinds all function definitions.

remfunction quotes its arguments.

remfunction returns a list of the symbols for which the function definition was unbound. false is returned in place of any symbol for which there is no function definition.

remfunction does not apply to memoizing functions or subscripted functions. remarray applies to those types of functions.

Option variable: savedef

Default value: true

When savedef is true, the Maxima version of a user function is preserved when the function is translated. This permits the definition to be displayed by dispfun and allows the function to be edited.

When savedef is false, the names of translated functions are removed from the functions list.

Option variable: transcompile

Default value: true

When transcompile is true, translate and translate_file generate declarations to make the translated code more suitable for compilation.

compfile sets transcompile: true for the duration.

Function: translate
    translate (f_1, …, f_n)
    translate (functions)
    translate (all)

Translates the user-defined functions f_1, …, f_n from the Maxima language into Lisp and evaluates the Lisp translations. Typically the translated functions run faster than the originals.

translate (all) or translate (functions) translates all user-defined functions.

Functions to be translated should include a call to mode_declare at the beginning when possible in order to produce more efficient code. For example:

f (x_1, x_2, ...) := block ([v_1, v_2, ...],
    mode_declare (v_1, mode_1, v_2, mode_2, ...), ...)

where the x_1, x_2, … are the parameters to the function and the v_1, v_2, … are the local variables.

The names of translated functions are removed from the functions list if savedef is false (see below) and are added to the props lists.

Functions should not be translated unless they are fully debugged.

Expressions are assumed simplified; if they are not, correct but non-optimal code gets generated. Thus, the user should not set the simp switch to false which inhibits simplification of the expressions to be translated.

The switch translate, if true, causes automatic translation of a user’s function to Lisp.

Note that translated functions may not run identically to the way they did before translation as certain incompatibilities may exist between the Lisp and Maxima versions. Principally, the rat function with more than one argument and the ratvars function should not be used if any variables are mode_declare’d canonical rational expressions (CRE). Also the prederror: false setting will not translate.

savedef - if true will cause the Maxima version of a user function to remain when the function is translate’d. This permits the definition to be displayed by dispfun and allows the function to be edited.

transrun - if false will cause the interpreted version of all functions to be run (provided they are still around) rather than the translated version.

The result returned by translate is a list of the names of the functions translated.

Function: translate_file
    translate_file (maxima_filename)
    translate_file (maxima_filename, lisp_filename)

Translates a file of Maxima code into a file of Lisp code. translate_file returns a list of three filenames: the name of the Maxima file, the name of the Lisp file, and the name of file containing additional information about the translation. translate_file evaluates its arguments.

translate_file ("foo.mac"); load("foo.LISP") is the same as the command batch ("foo.mac") except for certain restrictions, the use of '' and %, for example.

translate_file (maxima_filename) translates a Maxima file maxima_filename into a similarly-named Lisp file. For example, foo.mac is translated into foo.LISP. The Maxima filename may include a directory name or names, in which case the Lisp output file is written to the same directory from which the Maxima input comes.

translate_file (maxima_filename, lisp_filename) translates a Maxima file maxima_filename into a Lisp file lisp_filename. translate_file ignores the filename extension, if any, of lisp_filename; the filename extension of the Lisp output file is always LISP. The Lisp filename may include a directory name or names, in which case the Lisp output file is written to the specified directory.

translate_file also writes a file of translator warning messages of various degrees of severity. The filename extension of this file is UNLISP. This file may contain valuable information, though possibly obscure, for tracking down bugs in translated code. The UNLISP file is always written to the same directory from which the Maxima input comes.

translate_file emits Lisp code which causes some declarations and definitions to take effect as soon as the Lisp code is compiled. See compile_file for more on this topic.

See also

tr_array_as_ref tr_bound_function_applyp, tr_exponent tr_file_tty_messagesp, tr_float_can_branch_complex, tr_function_call_default, tr_numer, tr_optimize_max_loop, tr_semicompile, tr_state_vars, tr_warnings_get, tr_warn_bad_function_calls tr_warn_fexpr, tr_warn_meval, tr_warn_mode, tr_warn_undeclared, and tr_warn_undefined_variable.

Option variable: transrun

Default value: true

When transrun is false will cause the interpreted version of all functions to be run (provided they are still around) rather than the translated version.

Option variable: tr_array_as_ref

Default value: true

If translate_fast_arrays is false, array references in Lisp code emitted by translate_file are affected by tr_array_as_ref. When tr_array_as_ref is true, array names are evaluated, otherwise array names appear as literal symbols in translated code.

tr_array_as_ref has no effect if translate_fast_arrays is true.

Option variable: tr_bound_function_applyp

Default value: true

When tr_bound_function_applyp is true, Maxima gives a warning if a bound variable (such as a function argument) is found being used as a function. tr_bound_function_applyp does not affect the code generated in such cases.

For example, an expression such as g (f, x) := f (x+1) will trigger the warning message.

Option variable: tr_file_tty_messagesp

Default value: false

When tr_file_tty_messagesp is true, messages generated by translate_file during translation of a file are displayed on the console and inserted into the UNLISP file. When false, messages about translation of the file are only inserted into the UNLISP file.

Option variable: tr_float_can_branch_complex

Default value: true

Tells the Maxima-to-Lisp translator to assume that the functions acos, asin, asec, and acsc can return complex results.

The ostensible effect of tr_float_can_branch_complex is the following. However, it appears that this flag has no effect on the translator output.

When it is true then acos(x) is of mode any even if x is of mode float (as set by mode_declare). When false then acos(x) is of mode float if and only if x is of mode float.

Option variable: tr_function_call_default

Default value: general

false means give up and call meval, expr means assume Lisp fixed arg function. general, the default gives code good for mexprs and mlexprs but not macros. general assures variable bindings are correct in compiled code. In general mode, when translating F(X), if F is a bound variable, then it assumes that apply (f, [x]) is meant, and translates a such, with appropriate warning. There is no need to turn this off. With the default settings, no warning messages implies full compatibility of translated and compiled code with the Maxima interpreter.

Option variable: tr_numer

Default value: false

When tr_numer is true, numer properties are used for atoms which have them, e.g. %pi.

Option variable: tr_optimize_max_loop

Default value: 100

tr_optimize_max_loop is the maximum number of times the macro-expansion and optimization pass of the translator will loop in considering a form. This is to catch macro expansion errors, and non-terminating optimization properties.

Option variable: tr_semicompile

Default value: false

When tr_semicompile is true, translate_file and compfile output forms which will be macroexpanded but not compiled into machine code by the Lisp compiler.

System variable: tr_state_vars

Default value:

[transcompile, tr_semicompile, tr_warn_undeclared, tr_warn_meval,
tr_warn_fexpr, tr_warn_mode, tr_warn_undefined_variable,
tr_function_call_default, tr_array_as_ref,tr_numer]

The list of the switches that affect the form of the translated output. This information is useful to system people when trying to debug the translator. By comparing the translated product to what should have been produced for a given state, it is possible to track down bugs.

Function: tr_warnings_get ()

Prints a list of warnings which have been given by the translator during the current translation.

Option variable: tr_warn_bad_function_calls

Default value: true

- Gives a warning when when function calls are being made which may not be correct due to improper declarations that were made at translate time.

Option variable: tr_warn_fexpr

Default value: compfile

- Gives a warning if any FEXPRs are encountered. FEXPRs should not normally be output in translated code, all legitimate special program forms are translated.

Option variable: tr_warn_meval

Default value: compfile

- Gives a warning if the function meval gets called. If meval is called that indicates problems in the translation.

Option variable: tr_warn_mode

Default value: all

- Gives a warning when variables are assigned values inappropriate for their mode.

Option variable: tr_warn_undeclared

Default value: compile

- Determines when to send warnings about undeclared variables to the TTY.

Option variable: tr_warn_undefined_variable

Default value: all

- Gives a warning when undefined global variables are seen.

Function: compile_file
    compile_file (filename)
    compile_file (filename, compiled_filename)
    compile_file (filename, compiled_filename, lisp_filename)

Translates the Maxima file filename into Lisp, executes the Lisp compiler, and, if the translation and compilation succeed, loads the compiled code into Maxima.

compile_file returns a list of the names of four files: the original Maxima file, the Lisp translation, notes on translation, and the compiled code. If the compilation fails, the fourth item is false.

Some declarations and definitions take effect as soon as the Lisp code is compiled (without loading the compiled code). These include functions defined with the := operator, macros define with the ::= operator, alias, declare, define_variable, mode_declare, and infix, matchfix, nofix, postfix, prefix, and compfile.

Assignments and function calls are not evaluated until the compiled code is loaded. In particular, within the Maxima file, assignments to the translation flags (tr_numer, etc.) have no effect on the translation.

filename may not contain :lisp statements.

compile_file evaluates its arguments.

Function: declare_translated (f_1, f_2, …)

When translating a file of Maxima code to Lisp, it is important for the translator to know which functions it sees in the file are to be called as translated or compiled functions, and which ones are just Maxima functions or undefined. Putting this declaration at the top of the file, lets it know that although a symbol does which does not yet have a Lisp function value, will have one at call time. (MFUNCTION-CALL fn arg1 arg2 ...) is generated when the translator does not know fn is going to be a Lisp function.


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37 Program Flow


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37.1 Lisp and Maxima

Maxima is a fairly complete programming language. But since it is written in Lisp, it additionally can provide easy access to Lisp functions and variables from Maxima and vice versa. Lisp and Maxima symbols are distinguished by a naming convention. A Lisp symbol which begins with a dollar sign $ corresponds to a Maxima symbol without the dollar sign.

A Maxima symbol which begins with a question mark ? corresponds to a Lisp symbol without the question mark. For example, the Maxima symbol foo corresponds to the Lisp symbol $FOO, while the Maxima symbol ?foo corresponds to the Lisp symbol FOO. Note that ?foo is written without a space between ? and foo; otherwise it might be mistaken for describe ("foo").

Hyphen -, asterisk *, or other special characters in Lisp symbols must be escaped by backslash \ where they appear in Maxima code. For example, the Lisp identifier *foo-bar* is written ?\*foo\-bar\* in Maxima.

Lisp code may be executed from within a Maxima session. A single line of Lisp (containing one or more forms) may be executed by the special command :lisp. For example,

(%i1) :lisp (foo $x $y)

calls the Lisp function foo with Maxima variables x and y as arguments. The :lisp construct can appear at the interactive prompt or in a file processed by batch or demo, but not in a file processed by load, batchload, translate_file, or compile_file.

The function to_lisp opens an interactive Lisp session. Entering (to-maxima) closes the Lisp session and returns to Maxima.

Lisp functions and variables which are to be visible in Maxima as functions and variables with ordinary names (no special punctuation) must have Lisp names beginning with the dollar sign $.

Maxima is case-sensitive, distinguishing between lowercase and uppercase letters in identifiers. There are some rules governing the translation of names between Lisp and Maxima.

  1. A Lisp identifier not enclosed in vertical bars corresponds to a Maxima identifier in lowercase. Whether the Lisp identifier is uppercase, lowercase, or mixed case, is ignored. E.g., Lisp $foo, $FOO, and $Foo all correspond to Maxima foo. But this is because $foo, $FOO and $Foo are converted by the Lisp reader by default to the Lisp symbol $FOO.
  2. A Lisp identifier which is all uppercase or all lowercase and enclosed in vertical bars corresponds to a Maxima identifier with case reversed. That is, uppercase is changed to lowercase and lowercase to uppercase. E.g., Lisp |$FOO| and |$foo| correspond to Maxima foo and FOO, respectively.
  3. A Lisp identifier which is mixed uppercase and lowercase and enclosed in vertical bars corresponds to a Maxima identifier with the same case. E.g., Lisp |$Foo| corresponds to Maxima Foo.

The #$ Lisp macro allows the use of Maxima expressions in Lisp code. #$expr$ expands to a Lisp expression equivalent to the Maxima expression expr.

(msetq $foo #$[x, y]$)

This has the same effect as entering

(%i1) foo: [x, y];

The Lisp function displa prints an expression in Maxima format.

(%i1) :lisp #$[x, y, z]$ 
((MLIST SIMP) $X $Y $Z)
(%i1) :lisp (displa '((MLIST SIMP) $X $Y $Z))
[x, y, z]
NIL

Functions defined in Maxima are not ordinary Lisp functions. The Lisp function mfuncall calls a Maxima function. For example:

(%i1) foo(x,y) := x*y$
(%i2) :lisp (mfuncall '$foo 'a 'b)
((MTIMES SIMP) A B)

Some Lisp functions are shadowed in the Maxima package, namely the following.

   complement     continue      //
   float          functionp     array
   exp            listen        signum
   atan           asin          acos
   asinh          acosh         atanh
   tanh           cosh          sinh
   tan            break         gcd
·

Category: Programming


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37.2 Garbage Collection

One of the advantages of using lisp is that it uses “Garbage Collection”. In other words it automatically takes care of freeing memory occupied for example of intermediate results that were used during symbolic computation.

Garbage Collection avoids many errors frequently found in C programs (memory being freed too early, multiple times or not at all).

Function: garbage_collect ()

Tries to manually trigger the lisp’s garbage collection. This rarely is necessary as the lisp will employ an excellent algorithm for determining when to start garbage collection.

If maxima knows how to do manually trigger the garbage collection for the current lisp garbage_collect returns true, else false.


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37.3 Introduction to Program Flow

Maxima provides a do loop for iteration, as well as more primitive constructs such as go.


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37.4 Functions and Variables for Program Flow

Function: backtrace
    backtrace ()
    backtrace (n)

Prints the call stack, that is, the list of functions which called the currently active function.

backtrace () prints the entire call stack.

backtrace (n) prints the n most recent functions, including the currently active function.

backtrace can be called from a script, a function, or the interactive prompt (not only in a debugging context).

Examples:

Special operator: do
Special operator: while
Special operator: unless
Special operator: for
Special operator: from
Special operator: thru
Special operator: step
Special operator: next
Special operator: in

The do statement is used for performing iteration. The general form of the do statements maxima supports is:

If the loop is expected to generate a list as output the command makelist may be the appropriate command to use instead, See Performance considerations for Lists.

initial_value, increment, limit, and body can be any expression. list is a list. If the increment is 1 then "step 1" may be omitted; As always, if body needs to contain more than one command these commands can be specified as a comma-separated list surrounded by parenthesis or as a block. Due to its great generality the do statement will be described in two parts. The first form of the do statement (which is shown in the first three items above) is analogous to that used in several other programming languages (Fortran, Algol, PL/I, etc.); then the other features will be mentioned.

The execution of the do statement proceeds by first assigning the initial_value to the variable (henceforth called the control-variable). Then: (1) If the control-variable has exceeded the limit of a thru specification, or if the condition of the unless is true, or if the condition of the while is false then the do terminates. (2) The body is evaluated. (3) The increment is added to the control-variable. The process from (1) to (3) is performed repeatedly until the termination condition is satisfied. One may also give several termination conditions in which case the do terminates when any of them is satisfied.

In general the thru test is satisfied when the control-variable is greater than the limit if the increment was non-negative, or when the control-variable is less than the limit if the increment was negative. The increment and limit may be non-numeric expressions as long as this inequality can be determined. However, unless the increment is syntactically negative (e.g. is a negative number) at the time the do statement is input, Maxima assumes it will be positive when the do is executed. If it is not positive, then the do may not terminate properly.

Note that the limit, increment, and termination condition are evaluated each time through the loop. Thus if any of these involve much computation, and yield a result that does not change during all the executions of the body, then it is more efficient to set a variable to their value prior to the do and use this variable in the do form.

The value normally returned by a do statement is the atom done. However, the function return may be used inside the body to exit the do prematurely and give it any desired value. Note however that a return within a do that occurs in a block will exit only the do and not the block. Note also that the go function may not be used to exit from a do into a surrounding block.

The control-variable is always local to the do and thus any variable may be used without affecting the value of a variable with the same name outside of the do. The control-variable is unbound after the do terminates.

(%i1) for a:-3 thru 26 step 7 do display(a)$
                             a = - 3

                              a = 4

                             a = 11

                             a = 18

                             a = 25
(%i1) s: 0$
(%i2) for i: 1 while i <= 10 do s: s+i;
(%o2)                         done
(%i3) s;
(%o3)                          55

Note that the condition while i <= 10 is equivalent to unless i > 10 and also thru 10.

(%i1) series: 1$
(%i2) term: exp (sin (x))$
(%i3) for p: 1 unless p > 7 do
          (term: diff (term, x)/p, 
           series: series + subst (x=0, term)*x^p)$
(%i4) series;
                  7    6     5    4    2
                 x    x     x    x    x
(%o4)            -- - --- - -- - -- + -- + x + 1
                 90   240   15   8    2

which gives 8 terms of the Taylor series for e^sin(x).

(%i1) poly: 0$
(%i2) for i: 1 thru 5 do
          for j: i step -1 thru 1 do
              poly: poly + i*x^j$
(%i3) poly;
                  5      4       3       2
(%o3)          5 x  + 9 x  + 12 x  + 14 x  + 15 x
(%i4) guess: -3.0$
(%i5) for i: 1 thru 10 do
          (guess: subst (guess, x, 0.5*(x + 10/x)),
           if abs (guess^2 - 10) < 0.00005 then return (guess));
(%o5)                  - 3.162280701754386

This example computes the negative square root of 10 using the Newton- Raphson iteration a maximum of 10 times. Had the convergence criterion not been met the value returned would have been done.

Instead of always adding a quantity to the control-variable one may sometimes wish to change it in some other way for each iteration. In this case one may use next expression instead of step increment. This will cause the control-variable to be set to the result of evaluating expression each time through the loop.

(%i6) for count: 2 next 3*count thru 20 do display (count)$
                            count = 2

                            count = 6

                           count = 18

As an alternative to for variable: value ...do... the syntax for variable from value ...do... may be used. This permits the from value to be placed after the step or next value or after the termination condition. If from value is omitted then 1 is used as the initial value.

Sometimes one may be interested in performing an iteration where the control-variable is never actually used. It is thus permissible to give only the termination conditions omitting the initialization and updating information as in the following example to compute the square-root of 5 using a poor initial guess.

(%i1) x: 1000$
(%i2) thru 20 do x: 0.5*(x + 5.0/x)$
(%i3) x;
(%o3)                   2.23606797749979
(%i4) sqrt(5), numer;
(%o4)                   2.23606797749979

If it is desired one may even omit the termination conditions entirely and just give do body which will continue to evaluate the body indefinitely. In this case the function return should be used to terminate execution of the do.

(%i1) newton (f, x):= ([y, df, dfx], df: diff (f ('x), 'x),
          do (y: ev(df), x: x - f(x)/y, 
              if abs (f (x)) < 5e-6 then return (x)))$
(%i2) sqr (x) := x^2 - 5.0$
(%i3) newton (sqr, 1000);
(%o3)                   2.236068027062195

(Note that return, when executed, causes the current value of x to be returned as the value of the do. The block is exited and this value of the do is returned as the value of the block because the do is the last statement in the block.)

One other form of the do is available in Maxima. The syntax is:

for variable in list end_tests do body

The elements of list are any expressions which will successively be assigned to the variable on each iteration of the body. The optional termination tests end_tests can be used to terminate execution of the do; otherwise it will terminate when the list is exhausted or when a return is executed in the body. (In fact, list may be any non-atomic expression, and successive parts are taken.)

(%i1)  for f in [log, rho, atan] do ldisp(f(1))$
(%t1)                                  0
(%t2)                                rho(1)
                                     %pi
(%t3)                                 ---
                                      4
(%i4) ev(%t3,numer);
(%o4)                             0.78539816

Function: errcatch (expr_1, …, expr_n)

Evaluates expr_1, …, expr_n one by one and returns [expr_n] (a list) if no error occurs. If an error occurs in the evaluation of any argument, errcatch prevents the error from propagating and returns the empty list [] without evaluating any more arguments.

errcatch is useful in batch files where one suspects an error might occur which would terminate the batch if the error weren’t caught.

See also errormsg.

Function: error (expr_1, …, expr_n)
System variable: error

Evaluates and prints expr_1, …, expr_n, and then causes an error return to top level Maxima or to the nearest enclosing errcatch.

The variable error is set to a list describing the error. The first element of error is a format string, which merges all the strings among the arguments expr_1, …, expr_n, and the remaining elements are the values of any non-string arguments.

errormsg() formats and prints error. This is effectively reprinting the most recent error message.

Function: warning (expr_1, …, expr_n)

Evaluates and prints expr_1, …, expr_n, as a warning message that is formatted in a standard way so a maxima front-end may be able to recognize the warning and to format it accordingly.

The function warning always returns false.

Option variable: error_size

Default value: 10

error_size modifies error messages according to the size of expressions which appear in them. If the size of an expression (as determined by the Lisp function ERROR-SIZE) is greater than error_size, the expression is replaced in the message by a symbol, and the symbol is assigned the expression. The symbols are taken from the list error_syms.

Otherwise, the expression is smaller than error_size, and the expression is displayed in the message.

See also error and error_syms.

Example:

The size of U, as determined by ERROR-SIZE, is 24.

(%i1) U: (C^D^E + B + A)/(cos(X-1) + 1)$

(%i2) error_size: 20$

(%i3) error ("Example expression is", U);

Example expression is errexp1
 -- an error.  Quitting.  To debug this try debugmode(true);
(%i4) errexp1;
                            E
                           D
                          C   + B + A
(%o4)                    --------------
                         cos(X - 1) + 1
(%i5) error_size: 30$

(%i6) error ("Example expression is", U);

                         E
                        D
                       C   + B + A
Example expression is --------------
                      cos(X - 1) + 1
 -- an error.  Quitting.  To debug this try debugmode(true);

Option variable: error_syms

Default value: [errexp1, errexp2, errexp3]

In error messages, expressions larger than error_size are replaced by symbols, and the symbols are set to the expressions. The symbols are taken from the list error_syms. The first too-large expression is replaced by error_syms[1], the second by error_syms[2], and so on.

If there are more too-large expressions than there are elements of error_syms, symbols are constructed automatically, with the n-th symbol equivalent to concat ('errexp, n).

See also error and error_size.

Function: errormsg ()

Reprints the most recent error message. The variable error holds the message, and errormsg formats and prints it.

Option variable: errormsg

Default value: true

When false the output of error messages is suppressed.

The option variable errormsg can not be set in a block to a local value. The global value of errormsg is always present.

(%i1) errormsg;
(%o1)                         true
(%i2) sin(a,b);
sin: wrong number of arguments.
 -- an error. To debug this try: debugmode(true);
(%i3) errormsg:false;
(%o3)                         false
(%i4) sin(a,b);
 -- an error. To debug this try: debugmode(true);

The option variable errormsg can not be set in a block to a local value.

(%i1) f(bool):=block([errormsg:bool], print ("value of errormsg is",errormsg))$
(%i2) errormsg:true;
(%o2)                         true
(%i3) f(false);
value of errormsg is true 
(%o3)                         true
(%i4) errormsg:false;
(%o4)                         false
(%i5) f(true);
value of errormsg is false 
(%o5)                         false

Function: go (tag)

is used within a block to transfer control to the statement of the block which is tagged with the argument to go. To tag a statement, precede it by an atomic argument as another statement in the block. For example:

block ([x], x:1, loop, x+1, ..., go(loop), ...)

The argument to go must be the name of a tag appearing in the same block. One cannot use go to transfer to tag in a block other than the one containing the go.

Special operator: if

Represents conditional evaluation. Various forms of if expressions are recognized.

if cond_1 then expr_1 else expr_0 evaluates to expr_1 if cond_1 evaluates to true, otherwise the expression evaluates to expr_0.

The command if cond_1 then expr_1 elseif cond_2 then expr_2 elseif ... else expr_0 evaluates to expr_k if cond_k is true and all preceding conditions are false. If none of the conditions are true, the expression evaluates to expr_0.

A trailing else false is assumed if else is missing. That is, the command if cond_1 then expr_1 is equivalent to if cond_1 then expr_1 else false, and the command if cond_1 then expr_1 elseif ... elseif cond_n then expr_n is equivalent to if cond_1 then expr_1 elseif ... elseif cond_n then expr_n else false.

The alternatives expr_0, …, expr_n may be any Maxima expressions, including nested if expressions. The alternatives are neither simplified nor evaluated unless the corresponding condition is true.

The conditions cond_1, …, cond_n are expressions which potentially or actually evaluate to true or false. When a condition does not actually evaluate to true or false, the behavior of if is governed by the global flag prederror. When prederror is true, it is an error if any evaluated condition does not evaluate to true or false. Otherwise, conditions which do not evaluate to true or false are accepted, and the result is a conditional expression.

Among other elements, conditions may comprise relational and logical operators as follows.

Operation            Symbol      Type
 
less than            <           relational infix
less than            <=
  or equal to                    relational infix
equality (syntactic) =           relational infix
negation of =        #           relational infix
equality (value)     equal       relational function
negation of equal    notequal    relational function
greater than         >=
  or equal to                    relational infix
greater than         >           relational infix
and                  and         logical infix
or                   or          logical infix
not                  not         logical prefix

Function: map (f, expr_1, …, expr_n)

Returns an expression whose leading operator is the same as that of the expressions expr_1, …, expr_n but whose subparts are the results of applying f to the corresponding subparts of the expressions. f is either the name of a function of n arguments or is a lambda form of n arguments.

maperror - if false will cause all of the mapping functions to (1) stop when they finish going down the shortest expr_i if not all of the expr_i are of the same length and (2) apply f to [expr_1, expr_2, …] if the expr_i are not all the same type of object. If maperror is true then an error message will be given in the above two instances.

One of the uses of this function is to map a function (e.g. partfrac) onto each term of a very large expression where it ordinarily wouldn’t be possible to use the function on the entire expression due to an exhaustion of list storage space in the course of the computation.

See also scanmap, maplist, outermap, matrixmap and apply.

(%i1) map(f,x+a*y+b*z);
(%o1)                        f(b z) + f(a y) + f(x)
(%i2) map(lambda([u],partfrac(u,x)),x+1/(x^3+4*x^2+5*x+2));
                           1       1        1
(%o2)                     ----- - ----- + -------- + x
                         x + 2   x + 1          2
                                         (x + 1)
(%i3) map(ratsimp, x/(x^2+x)+(y^2+y)/y);
                                      1
(%o3)                            y + ----- + 1
                                    x + 1
(%i4) map("=",[a,b],[-0.5,3]);
(%o4)                          [a = - 0.5, b = 3]


Function: mapatom (expr)

Returns true if and only if expr is treated by the mapping routines as an atom. "Mapatoms" are atoms, numbers (including rational numbers), and subscripted variables.

Option variable: maperror

Default value: true

When maperror is false, causes all of the mapping functions, for example

map (f, expr_1, expr_2, ...)

to (1) stop when they finish going down the shortest expr_i if not all of the expr_i are of the same length and (2) apply f to [expr_1, expr_2, …] if the expr_i are not all the same type of object.

If maperror is true then an error message is displayed in the above two instances.

Option variable: mapprint

Default value: true

When mapprint is true, various information messages from map, maplist, and fullmap are produced in certain situations. These include situations where map would use apply, or map is truncating on the shortest list.

If mapprint is false, these messages are suppressed.

Function: maplist (f, expr_1, …, expr_n)

Returns a list of the applications of f to the parts of the expressions expr_1, …, expr_n. f is the name of a function, or a lambda expression.

maplist differs from map(f, expr_1, ..., expr_n) which returns an expression with the same main operator as expr_i has (except for simplifications and the case where map does an apply).

Option variable: prederror

Default value: false

When prederror is true, an error message is displayed whenever the predicate of an if statement or an is function fails to evaluate to either true or false.

If false, unknown is returned instead in this case. The prederror: false mode is not supported in translated code; however, maybe is supported in translated code.

See also is and maybe.

Function: return (value)

May be used to exit explicitly from the current block, while, for or do loop bringing its argument. It therefore can be compared with the return statement found in other programming languages but it yields one difference: In maxima only returns from the current block, not from the entire function it was called in. In this aspect it more closely resembles the break statement from C.

(%i1) for i:1 thru 10 do o:i;
(%o1)                         done
(%i2) for i:1 thru 10 do if i=3 then return(i);
(%o2)                           3
(%i3) for i:1 thru 10 do
    (
        block([i],
            i:3,
            return(i)
        ),
        return(8)
    );
(%o3)                           8
(%i4) block([i],
    i:4,
    block([o],
        o:5,
        return(o)
    ),
    return(i),
    return(10)
 );
(%o4)                           4

See also for, while, do and block.

Function: scanmap
    scanmap (f, expr)
    scanmap (f, expr, bottomup)

Recursively applies f to expr, in a top down manner. This is most useful when complete factorization is desired, for example:

(%i1) exp:(a^2+2*a+1)*y + x^2$
(%i2) scanmap(factor,exp);
                                    2      2
(%o2)                         (a + 1)  y + x

Note the way in which scanmap applies the given function factor to the constituent subexpressions of expr; if another form of expr is presented to scanmap then the result may be different. Thus, %o2 is not recovered when scanmap is applied to the expanded form of exp:

(%i3) scanmap(factor,expand(exp));
                           2                  2
(%o3)                      a  y + 2 a y + y + x

Here is another example of the way in which scanmap recursively applies a given function to all subexpressions, including exponents:

(%i4) expr : u*v^(a*x+b) + c$
(%i5) scanmap('f, expr);
                    f(f(f(a) f(x)) + f(b))
(%o5) f(f(f(u) f(f(v)                      )) + f(c))

scanmap (f, expr, bottomup) applies f to expr in a bottom-up manner. E.g., for undefined f,

scanmap(f,a*x+b) ->
   f(a*x+b) -> f(f(a*x)+f(b)) -> f(f(f(a)*f(x))+f(b))
scanmap(f,a*x+b,bottomup) -> f(a)*f(x)+f(b)
    -> f(f(a)*f(x))+f(b) ->
     f(f(f(a)*f(x))+f(b))

In this case, you get the same answer both ways.

Function: throw (expr)

Evaluates expr and throws the value back to the most recent catch. throw is used with catch as a nonlocal return mechanism.

Function: outermap (f, a_1, …, a_n)

Applies the function f to each one of the elements of the outer product a_1 cross a_2 … cross a_n.

f is the name of a function of n arguments or a lambda expression of n arguments. Each argument a_k may be a list or nested list, or a matrix, or any other kind of expression.

The outermap return value is a nested structure. Let x be the return value. Then x has the same structure as the first list, nested list, or matrix argument, x[i_1]...[i_m] has the same structure as the second list, nested list, or matrix argument, x[i_1]...[i_m][j_1]...[j_n] has the same structure as the third list, nested list, or matrix argument, and so on, where m, n, … are the numbers of indices required to access the elements of each argument (one for a list, two for a matrix, one or more for a nested list). Arguments which are not lists or matrices have no effect on the structure of the return value.

Note that the effect of outermap is different from that of applying f to each one of the elements of the outer product returned by cartesian_product. outermap preserves the structure of the arguments in the return value, while cartesian_product does not.

outermap evaluates its arguments.

See also map, maplist, and apply.

Examples:

Elementary examples of outermap. To show the argument combinations more clearly, F is left undefined.

(%i1) outermap (F, [a, b, c], [1, 2, 3]);
(%o1) [[F(a, 1), F(a, 2), F(a, 3)], [F(b, 1), F(b, 2), F(b, 3)], 
                                     [F(c, 1), F(c, 2), F(c, 3)]]
(%i2) outermap (F, matrix ([a, b], [c, d]), matrix ([1, 2], [3, 4]));
         [ [ F(a, 1)  F(a, 2) ]  [ F(b, 1)  F(b, 2) ] ]
         [ [                  ]  [                  ] ]
         [ [ F(a, 3)  F(a, 4) ]  [ F(b, 3)  F(b, 4) ] ]
(%o2)    [                                            ]
         [ [ F(c, 1)  F(c, 2) ]  [ F(d, 1)  F(d, 2) ] ]
         [ [                  ]  [                  ] ]
         [ [ F(c, 3)  F(c, 4) ]  [ F(d, 3)  F(d, 4) ] ]
(%i3) outermap (F, [a, b], x, matrix ([1, 2], [3, 4]));
       [ F(a, x, 1)  F(a, x, 2) ]  [ F(b, x, 1)  F(b, x, 2) ]
(%o3) [[                        ], [                        ]]
       [ F(a, x, 3)  F(a, x, 4) ]  [ F(b, x, 3)  F(b, x, 4) ]
(%i4) outermap (F, [a, b], matrix ([1, 2]), matrix ([x], [y]));
       [ [ F(a, 1, x) ]  [ F(a, 2, x) ] ]
(%o4) [[ [            ]  [            ] ], 
       [ [ F(a, 1, y) ]  [ F(a, 2, y) ] ]
                              [ [ F(b, 1, x) ]  [ F(b, 2, x) ] ]
                              [ [            ]  [            ] ]]
                              [ [ F(b, 1, y) ]  [ F(b, 2, y) ] ]
(%i5) outermap ("+", [a, b, c], [1, 2, 3]);
(%o5) [[a + 1, a + 2, a + 3], [b + 1, b + 2, b + 3], 
                                           [c + 1, c + 2, c + 3]]

A closer examination of the outermap return value. The first, second, and third arguments are a matrix, a list, and a matrix, respectively. The return value is a matrix. Each element of that matrix is a list, and each element of each list is a matrix.

(%i1) arg_1 :  matrix ([a, b], [c, d]);
                            [ a  b ]
(%o1)                       [      ]
                            [ c  d ]
(%i2) arg_2 : [11, 22];
(%o2)                       [11, 22]
(%i3) arg_3 : matrix ([xx, yy]);
(%o3)                      [ xx  yy ]
(%i4) xx_0 : outermap (lambda ([x, y, z], x / y + z), arg_1,
                                                   arg_2, arg_3);
               [  [      a        a  ]  [      a        a  ]  ]
               [ [[ xx + --  yy + -- ], [ xx + --  yy + -- ]] ]
               [  [      11       11 ]  [      22       22 ]  ]
(%o4)  Col 1 = [                                              ]
               [  [      c        c  ]  [      c        c  ]  ]
               [ [[ xx + --  yy + -- ], [ xx + --  yy + -- ]] ]
               [  [      11       11 ]  [      22       22 ]  ]
                 [  [      b        b  ]  [      b        b  ]  ]
                 [ [[ xx + --  yy + -- ], [ xx + --  yy + -- ]] ]
                 [  [      11       11 ]  [      22       22 ]  ]
         Col 2 = [                                              ]
                 [  [      d        d  ]  [      d        d  ]  ]
                 [ [[ xx + --  yy + -- ], [ xx + --  yy + -- ]] ]
                 [  [      11       11 ]  [      22       22 ]  ]
(%i5) xx_1 : xx_0 [1][1];
           [      a        a  ]  [      a        a  ]
(%o5)     [[ xx + --  yy + -- ], [ xx + --  yy + -- ]]
           [      11       11 ]  [      22       22 ]
(%i6) xx_2 : xx_0 [1][1] [1];
                      [      a        a  ]
(%o6)                 [ xx + --  yy + -- ]
                      [      11       11 ]
(%i7) xx_3 : xx_0 [1][1] [1] [1][1];
                                  a
(%o7)                        xx + --
                                  11
(%i8) [op (arg_1), op (arg_2), op (arg_3)];
(%o8)                  [matrix, [, matrix]
(%i9) [op (xx_0), op (xx_1), op (xx_2)];
(%o9)                  [matrix, [, matrix]

outermap preserves the structure of the arguments in the return value, while cartesian_product does not.

(%i1) outermap (F, [a, b, c], [1, 2, 3]);
(%o1) [[F(a, 1), F(a, 2), F(a, 3)], [F(b, 1), F(b, 2), F(b, 3)], 
                                     [F(c, 1), F(c, 2), F(c, 3)]]
(%i2) setify (flatten (%));
(%o2) {F(a, 1), F(a, 2), F(a, 3), F(b, 1), F(b, 2), F(b, 3), 
                                       F(c, 1), F(c, 2), F(c, 3)}
(%i3) map (lambda ([L], apply (F, L)),
                     cartesian_product ({a, b, c}, {1, 2, 3}));
(%o3) {F(a, 1), F(a, 2), F(a, 3), F(b, 1), F(b, 2), F(b, 3), 
                                       F(c, 1), F(c, 2), F(c, 3)}
(%i4) is (equal (%, %th (2)));
(%o4)                         true


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38 Debugging


Next: , Up: Debugging   [Contents][Index]

38.1 Source Level Debugging

Maxima has a built-in source level debugger. The user can set a breakpoint at a function, and then step line by line from there. The call stack may be examined, together with the variables bound at that level.

The command :help or :h shows the list of debugger commands. (In general, commands may be abbreviated if the abbreviation is unique. If not unique, the alternatives will be listed.) Within the debugger, the user can also use any ordinary Maxima functions to examine, define, and manipulate variables and expressions.

A breakpoint is set by the :br command at the Maxima prompt. Within the debugger, the user can advance one line at a time using the :n (“next”) command. The :bt (“backtrace”) command shows a list of stack frames. The :r (“resume”) command exits the debugger and continues with execution. These commands are demonstrated in the example below.

(%i1) load ("/tmp/foobar.mac");

(%o1)                           /tmp/foobar.mac

(%i2) :br foo
Turning on debugging debugmode(true)
Bkpt 0 for foo (in /tmp/foobar.mac line 1) 

(%i2) bar (2,3);
Bkpt 0:(foobar.mac 1)
/tmp/foobar.mac:1::

(dbm:1) :bt                        <-- :bt typed here gives a backtrace
#0: foo(y=5)(foobar.mac line 1)
#1: bar(x=2,y=3)(foobar.mac line 9)

(dbm:1) :n                         <-- Here type :n to advance line
(foobar.mac 2)
/tmp/foobar.mac:2::

(dbm:1) :n                         <-- Here type :n to advance line
(foobar.mac 3)
/tmp/foobar.mac:3::

(dbm:1) u;                         <-- Investigate value of u
28

(dbm:1) u: 33;                     <-- Change u to be 33
33

(dbm:1) :r                         <-- Type :r to resume the computation

(%o2)                                1094

The file /tmp/foobar.mac is the following:

foo(y) := block ([u:y^2],
  u: u+3,
  u: u^2,
  u);
 
bar(x,y) := (
  x: x+2,
  y: y+2,
  x: foo(y),
  x+y);

USE OF THE DEBUGGER THROUGH EMACS

If the user is running the code under GNU emacs in a shell window (dbl shell), or is running the graphical interface version, Xmaxima, then if he stops at a break point, he will see his current position in the source file which will be displayed in the other half of the window, either highlighted in red, or with a little arrow pointing at the right line. He can advance single lines at a time by typing M-n (Alt-n).

Under Emacs you should run in a dbl shell, which requires the dbl.el file in the elisp directory. Make sure you install the elisp files or add the Maxima elisp directory to your path: e.g., add the following to your .emacs file or the site-init.el

(setq load-path (cons "/usr/share/maxima/5.9.1/emacs" load-path))
(autoload 'dbl "dbl")

then in emacs

M-x dbl

should start a shell window in which you can run programs, for example Maxima, gcl, gdb etc. This shell window also knows about source level debugging, and display of source code in the other window.

The user may set a break point at a certain line of the file by typing C-x space. This figures out which function the cursor is in, and then it sees which line of that function the cursor is on. If the cursor is on, say, line 2 of foo, then it will insert in the other window the command, “:br foo 2”, to break foo at its second line. To have this enabled, the user must have maxima-mode.el turned on in the window in which the file foobar.mac is visiting. There are additional commands available in that file window, such as evaluating the function into the Maxima, by typing Alt-Control-x.

·

Category: Debugging


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38.2 Keyword Commands

Keyword commands are special keywords which are not interpreted as Maxima expressions. A keyword command can be entered at the Maxima prompt or the debugger prompt, although not at the break prompt. Keyword commands start with a colon, ’:’. For example, to evaluate a Lisp form you may type :lisp followed by the form to be evaluated.

(%i1) :lisp (+ 2 3) 
5

The number of arguments taken depends on the particular command. Also, you need not type the whole command, just enough to be unique among the break keywords. Thus :br would suffice for :break.

The keyword commands are listed below.

:break F n

Set a breakpoint in function F at line offset n from the beginning of the function. If F is given as a string, then it is assumed to be a file, and n is the offset from the beginning of the file. The offset is optional. If not given, it is assumed to be zero (first line of the function or file).

:bt

Print a backtrace of the stack frames

:continue

Continue the computation

:delete

Delete the specified breakpoints, or all if none are specified

:disable

Disable the specified breakpoints, or all if none are specified

:enable

Enable the specified breakpoints, or all if none are specified

:frame n

Print stack frame n, or the current frame if none is specified

:help

Print help on a debugger command, or all commands if none is specified

:info

Print information about item

:lisp some-form

Evaluate some-form as a Lisp form

:lisp-quiet some-form

Evaluate Lisp form some-form without any output

:next

Like :step, except :next steps over function calls

:quit

Quit the current debugger level without completing the computation

:resume

Continue the computation

:step

Continue the computation until it reaches a new source line

:top

Return to the Maxima prompt (from any debugger level) without completing the computation

·

Category: Debugging


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38.3 Functions and Variables for Debugging

Option variable: debugmode

Default value: false

When a Maxima error occurs, Maxima will start the debugger if debugmode is true. The user may enter commands to examine the call stack, set breakpoints, step through Maxima code, and so on. See debugging for a list of debugger commands.

Enabling debugmode will not catch Lisp errors.

Option variable: refcheck

Default value: false

When refcheck is true, Maxima prints a message each time a bound variable is used for the first time in a computation.

Option variable: setcheck

Default value: false

If setcheck is set to a list of variables (which can be subscripted), Maxima prints a message whenever the variables, or subscripted occurrences of them, are bound with the ordinary assignment operator :, the :: assignment operator, or function argument binding, but not the function assignment := nor the macro assignment ::= operators. The message comprises the name of the variable and the value it is bound to.

setcheck may be set to all or true thereby including all variables.

Each new assignment of setcheck establishes a new list of variables to check, and any variables previously assigned to setcheck are forgotten.

The names assigned to setcheck must be quoted if they would otherwise evaluate to something other than themselves. For example, if x, y, and z are already bound, then enter

setcheck: ['x, 'y, 'z]$

to put them on the list of variables to check.

No printout is generated when a variable on the setcheck list is assigned to itself, e.g., X: 'X.

Option variable: setcheckbreak

Default value: false

When setcheckbreak is true, Maxima will present a break prompt whenever a variable on the setcheck list is assigned a new value. The break occurs before the assignment is carried out. At this point, setval holds the value to which the variable is about to be assigned. Hence, one may assign a different value by assigning to setval.

See also setcheck and setval.

System variable: setval

Holds the value to which a variable is about to be set when a setcheckbreak occurs. Hence, one may assign a different value by assigning to setval.

See also setcheck and setcheckbreak.

Function: timer (f_1, …, f_n)
    timer (all)
    timer ()

Given functions f_1, …, f_n, timer puts each one on the list of functions for which timing statistics are collected. timer(f)$ timer(g)$ puts f and then g onto the list; the list accumulates from one call to the next.

timer(all) puts all user-defined functions (as named by the global variable functions) on the list of timed functions.

With no arguments, timer returns the list of timed functions.

Maxima records how much time is spent executing each function on the list of timed functions. timer_info returns the timing statistics, including the average time elapsed per function call, the number of calls, and the total time elapsed. untimer removes functions from the list of timed functions.

timer quotes its arguments. f(x) := x^2$ g:f$ timer(g)$ does not put f on the timer list.

If trace(f) is in effect, then timer(f) has no effect; trace and timer cannot both be in effect at the same time.

See also timer_devalue.

Function: untimer (f_1, …, f_n)
    untimer ()

Given functions f_1, …, f_n, untimer removes each function from the timer list.

With no arguments, untimer removes all functions currently on the timer list.

After untimer (f) is executed, timer_info (f) still returns previously collected timing statistics, although timer_info() (with no arguments) does not return information about any function not currently on the timer list. timer (f) resets all timing statistics to zero and puts f on the timer list again.

Option variable: timer_devalue

Default value: false

When timer_devalue is true, Maxima subtracts from each timed function the time spent in other timed functions. Otherwise, the time reported for each function includes the time spent in other functions. Note that time spent in untimed functions is not subtracted from the total time.

See also timer and timer_info.

Function: timer_info (f_1, ..., f_n)
    timer_info ()

Given functions f_1, ..., f_n, timer_info returns a matrix containing timing information for each function. With no arguments, timer_info returns timing information for all functions currently on the timer list.

The matrix returned by timer_info contains the function name, time per function call, number of function calls, total time, and gctime, which meant "garbage collection time" in the original Macsyma but is now always zero.

The data from which timer_info constructs its return value can also be obtained by the get function:

get(f, 'calls);  get(f, 'runtime);  get(f, 'gctime);

See also timer.

Function: trace (f_1, …, f_n)
    trace (all)
    trace ()

Given functions f_1, …, f_n, trace instructs Maxima to print out debugging information whenever those functions are called. trace(f)$ trace(g)$ puts f and then g onto the list of functions to be traced; the list accumulates from one call to the next.

trace(all) puts all user-defined functions (as named by the global variable functions) on the list of functions to be traced.

With no arguments, trace returns a list of all the functions currently being traced.

The untrace function disables tracing. See also trace_options.

trace quotes its arguments. Thus, f(x) := x^2$ g:f$ trace(g)$ does not put f on the trace list.

When a function is redefined, it is removed from the timer list. Thus after timer(f)$ f(x) := x^2$, function f is no longer on the timer list.

If timer (f) is in effect, then trace (f) has no effect; trace and timer can’t both be in effect for the same function.

Function: trace_options (f, option_1, …, option_n)
    trace_options (f)

Sets the trace options for function f. Any previous options are superseded. trace_options (f, ...) has no effect unless trace (f) is also called (either before or after trace_options).

trace_options (f) resets all options to their default values.

The option keywords are:

Trace options are specified in two forms. The presence of the option keyword alone puts the option into effect unconditionally. (Note that option foo is not put into effect by specifying foo: true or a similar form; note also that keywords need not be quoted.) Specifying the option keyword with a predicate function makes the option conditional on the predicate.

The argument list to the predicate function is always [level, direction, function, item] where level is the recursion level for the function, direction is either enter or exit, function is the name of the function, and item is the argument list (on entering) or the return value (on exiting).

Here is an example of unconditional trace options:

(%i1) ff(n) := if equal(n, 0) then 1 else n * ff(n - 1)$

(%i2) trace (ff)$

(%i3) trace_options (ff, lisp_print, break)$

(%i4) ff(3);

Here is the same function, with the break option conditional on a predicate:

(%i5) trace_options (ff, break(pp))$

(%i6) pp (level, direction, function, item) := block (print (item),
    return (function = 'ff and level = 3 and direction = exit))$

(%i7) ff(6);

Function: untrace
    untrace (f_1, …, f_n)
    untrace ()

Given functions f_1, …, f_n, untrace disables tracing enabled by the trace function. With no arguments, untrace disables tracing for all functions.

untrace returns a list of the functions for which it disabled tracing.


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39 alt-display


Next: , Previous: , Up: alt-display-pkg   [Contents][Index]

39.1 Introduction to alt-display

The alt-display package provides a means to change the way that Maxima displays its output. The *alt-display1d* and *alt-display2d* Lisp hooks were introduced to Maxima in 2002, but were not easily accessible from the Maxima REPL until the introduction of this package.

The package provides a general purpose function to define alternative display functions, and a separate function to set the display function. The package also provides customized display functions to produce output in TeX, Texinfo, XML and all three output formats within Texinfo.

Here is a sample session:

(%i1) load("alt-display.mac")$
(%i2) set_alt_display(2,tex_display)$

(%i3) x/(x^2+y^2) = 1;
\mbox{\tt\red({\it \%o_3}) \black}$${{x}\over{y^2+x^2}}=1$$

(%i4) set_alt_display(2,mathml_display)$

(%i5) x/(x^2+y^2) = 1;
<math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>mlabel</mi> 
<mfenced separators=""><msub><mi>%o</mi> <mn>5</mn></msub> 
<mo>,</mo><mfrac><mrow><mi>x</mi> </mrow> <mrow><msup><mrow>
<mi>y</mi> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup><mrow>
<mi>x</mi> </mrow> <mn>2</mn> </msup> </mrow></mfrac> <mo>=</mo> 
<mn>1</mn> </mfenced> </math>

(%i6) set_alt_display(2,multi_display_for_texinfo)$

(%i7) x/(x^2+y^2) = 1;

@iftex
@tex
\mbox{\tt\red({\it \%o_7}) \black}$${{x}\over{y^2+x^2}}=1$$
@end tex
@end iftex
@ifhtml
@html

<math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>mlabel</mi> 
<mfenced separators=""><msub><mi>%o</mi> <mn>7</mn></msub> 
<mo>,</mo><mfrac><mrow><mi>x</mi> </mrow> <mrow><msup><mrow>
<mi>y</mi> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup><mrow>
<mi>x</mi> </mrow> <mn>2</mn> </msup> </mrow></mfrac> <mo>=</mo> 
<mn>1</mn> </mfenced> </math>
@end html
@end ifhtml
@ifinfo
@example
(%o7) x/(y^2+x^2) = 1
@end example
@end ifinfo

If the alternative display function causes an error, the error is trapped and the display function is reset to the default display. In the following example, the error function is set to display the output. This throws an error, which is handled by resetting the 2d-display to the default.

(%i8) set_alt_display(2,?error)$

(%i9) x;

Error in *alt-display2d*.
Messge: Condition designator ((MLABEL) $%O9 $X) is not of type (OR SYMBOL STRING
                                                             FUNCTION).
*alt-display2d* reset to nil.
 -- an error. To debug this try: debugmode(true);

(%i10) x;
(%o10)                                 x

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39.2 Functions and Variables for alt-display

Function: define_alt_display (function(input), expr)

This function is similar to define: it evaluates its arguments and expands into a function definition. The function is a function of a single input input. For convenience, a substitution is applied to expr after evaluation, to provide easy access to Lisp variable names.

Set a time-stamp on each prompt:

(%i1) load("alt-display.mac")$

(%i2) display2d:false$

(%i3) define_alt_display(time_stamp(x),
                   block([alt_display1d:false,alt_display2d:false],
                         prompt_prefix:printf(false,"~a~%",timedate()),
                         displa(x)));

(%o3) time_stamp(x):=block([\*alt\-display1d\*:false,
                            \*alt\-display2d\*:false],
                 \*prompt\-prefix\*:printf(false,"~a~%",timedate()),displa(x))
(%i4) set_alt_display(1,time_stamp);

(%o4) done
2017-11-27 16:15:58-06:00
(%i5) 

The input line %i3 defines time_stamp using define_alt_display. The output line %o3 shows that the Maxima variable names alt_display1d, alt_display2d and prompt_prefix have been replaced by their Lisp translations, as has displa been replaced by ?displa (the display function).

The display variables alt_display1d and alt_display2d are both bound to false in the body of time_stamp to prevent an infinite recursion in displa.

Function: info_display (form)

This is an alias for the default 1-d display function. It may be used as an alternative 1-d or 2-d display function.

(%i1) load("alt-display.mac")$

(%i2) set_alt_display(2,info_display);

(%o2) done
(%i3) x/y;

(%o3) x/y

Function: mathml_display (form)

Produces MathML output.

(%i1) load("alt-display.mac")$

(%i2) set_alt_display(2,mathml_display);
<math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>mlabel</mi> 
 <mfenced separators=""><msub><mi>%o</mi> <mn>2</mn></msub> 
 <mo>,</mo><mi>done</mi> </mfenced> </math>

Function: tex_display (form)

Produces TeX output.

(%i2) set_alt_display(2,tex_display);
\mbox{\tt\red({\it \%o_2}) \black}$$\mathbf{done}$$
(%i3) x/(x^2+y^2);
\mbox{\tt\red({\it \%o_3}) \black}$${{x}\over{y^2+x^2}}$$

Function: multi_display_for_texinfo (form)

Produces Texinfo output using all three display functions.

(%i2) set_alt_display(2,multi_display_for_texinfo)$

(%i3) x/(x^2+y^2);

@iftex
@tex
\mbox{\tt\red({\it \%o_3}) \black}$${{x}\over{y^2+x^2}}$$
@end tex
@end iftex
@ifhtml
@html

   <math xmlns="http://www.w3.org/1998/Math/MathML"> <mi>mlabel</mi> 
   <mfenced separators=""><msub><mi>%o</mi> <mn>3</mn></msub> 
   <mo>,</mo><mfrac><mrow><mi>x</mi> </mrow> <mrow><msup><mrow>
   <mi>y</mi> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup><mrow>
   <mi>x</mi> </mrow> <mn>2</mn> </msup> </mrow></mfrac> </mfenced> </math>
@end html
@end ifhtml
@ifinfo
@example
(%o3) x/(y^2+x^2)
@end example
@end ifinfo

Functions: reset_displays ()

Resets the prompt prefix and suffix to the empty string, and sets both 1-d and 2-d display functions to the default.

Function: set_alt_display (num, display-function)

The input num is the display to set; it may be either 1 or 2. The second input display-function is the display function to use. The display function may be either a Maxima function or a lambda expression.

Here is an example where the display function is a lambda expression; it just displays the result as TeX.

(%i1) load("alt-display.mac")$

(%i2) set_alt_display(2, lambda([form], tex(?caddr(form))))$

(%i3) integrate(exp(-t^2),t,0,inf);
$${{\sqrt{\pi}}\over{2}}$$

A user-defined display function should take care that it prints its output. A display function that returns a string will appear to display nothing, nor cause any errors.

Function: set_prompt (fix, expr)

Set the prompt prefix or suffix to expr. The input fix must evaluate to one of prefix, suffix, general, prolog or epilog. The input expr must evaluate to either a string or false; if false, the fix is reset to the default value.

(%i1) load("alt-display.mac")$
(%i2) set_prompt('prefix,printf(false,"It is now: ~a~%",timedate()))$

It is now: 2014-01-07 15:23:23-05:00
(%i3) 

The following example shows the effect of each option, except prolog. Note that the epilog prompt is printed as Maxima closes down. The general is printed between the end of input and the output, unless the input line ends in $.

Here is an example to show where the prompt strings are placed.

(%i1) load("alt-display.mac")$

(%i2) set_prompt(prefix,"<<prefix>> ",suffix,"<<suffix>> ",general,
           printf(false,"<<general>>~%"),epilog,printf(false,"<<epilog>>~%"));

(%o2)                                done
<<prefix>> (%i3) <<suffix>> x/y;
<<general>>
                                       x
(%o3)                                  -
                                       y
<<prefix>> (%i4) <<suffix>> quit();
<<general>>
<<epilog>>

Here is an example that shows how to colorize the input and output when Maxima is running in a terminal or terminal emulator like Emacs8.

figures/color_terminal

Each prompt string starts with the ASCII escape character (27) followed by an open square bracket (91); each string ends with a lower-case m (109). The webpages http://misc.flogisoft.com/bash/tip_colors_and_formatting and http://www.tldp.org/HOWTO/Bash-Prompt-HOWTO/x329.html provide information on how to use control strings to set the terminal colors.


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40 asympa


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40.1 Introduction to asympa

Function: asympa

asympa is a package for asymptotic analysis. The package contains simplification functions for asymptotic analysis, including the “big O” and “little o” functions that are widely used in complexity analysis and numerical analysis.

load ("asympa") loads this package.


Previous: , Up: asympa-pkg   [Contents][Index]

40.2 Functions and variables for asympa


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41 augmented_lagrangian


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41.1 Functions and Variables for augmented_lagrangian

Function: augmented_lagrangian_method
    augmented_lagrangian_method (FOM, xx, C, yy)
    augmented_lagrangian_method (FOM, xx, C, yy, optional_args)
    augmented_lagrangian_method ([FOM, grad], xx, C, yy)
    augmented_lagrangian_method ([FOM, grad], xx, C, yy, optional_args)

Returns an approximate minimum of the expression FOM with respect to the variables xx, holding the constraints C equal to zero. yy is a list of initial guesses for xx. The method employed is the augmented Lagrangian method (see Refs [1] and [2]).

grad, if present, is the gradient of FOM with respect to xx, represented as a list of expressions, one for each variable in xx. If not present, the gradient is constructed automatically.

FOM and each element of grad, if present, must be ordinary expressions, not names of functions or lambda expressions.

optional_args represents additional arguments, specified as symbol = value. The optional arguments recognized are:

niter

Number of iterations of the augmented Lagrangian algorithm

lbfgs_tolerance

Tolerance supplied to LBFGS

iprint

IPRINT parameter (a list of two integers which controls verbosity) supplied to LBFGS

%lambda

Initial value of %lambda to be used for calculating the augmented Lagrangian

This implementation minimizes the augmented Lagrangian by applying the limited-memory BFGS (LBFGS) algorithm, which is a quasi-Newton algorithm.

load("augmented_lagrangian") loads this function.

See also lbfgs-pkg

References:

[1] http://www-fp.mcs.anl.gov/otc/Guide/OptWeb/continuous/constrained/nonlinearcon/auglag.html

[2] http://www.cs.ubc.ca/spider/ascher/542/chap10.pdf

Examples:

(%i1) load ("lbfgs");
(%o1)  /home/gunter/src/maxima-code/share/lbfgs/lbfgs.mac
(%i2) load ("augmented_lagrangian");
(%o2) /home/gunter/src/maxima-code/share/contrib/augmented_lagra\
ngian.mac
(%i3) FOM: x^2 + 2*y^2;
                               2    2
(%o3)                       2 y  + x
(%i4) xx: [x, y];
(%o4)                        [x, y]
(%i5) C: [x + y - 1];
(%o5)                      [y + x - 1]
(%i6) yy: [1, 1];
(%o6)                        [1, 1]
(%i7) augmented_lagrangian_method(FOM, xx, C, yy, iprint=[-1,0]);
(%o7) [[x = 0.666659841080023, y = 0.333340272455448], 
                                 %lambda = [- 1.333337940892518]]

Same example as before, but this time the gradient is supplied as an argument.

(%i1) load ("lbfgs")$
(%i2) load ("augmented_lagrangian")$
(%i3) FOM: x^2 + 2*y^2;
                               2    2
(%o3)                       2 y  + x
(%i4) xx: [x, y];
(%o4)                        [x, y]
(%i5) grad : [2*x, 4*y];
(%o5)                      [2 x, 4 y]
(%i6) C: [x + y - 1];
(%o6)                      [y + x - 1]
(%i7) yy: [1, 1];
(%o7)                        [1, 1]
(%i8) augmented_lagrangian_method ([FOM, grad], xx, C, yy,
                             iprint = [-1, 0]);
(%o8) [[x = 0.6666598410800247, y = 0.3333402724554464], 
                                 %lambda = [- 1.333337940892525]]


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42 Bernstein


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42.1 Functions and Variables for Bernstein

Function: bernstein_poly (k, n, x)

Provided k is not a negative integer, the Bernstein polynomials are defined by bernstein_poly(k,n,x) = binomial(n,k) x^k (1-x)^(n-k); for a negative integer k, the Bernstein polynomial bernstein_poly(k,n,x) vanishes. When either k or n are non integers, the option variable bernstein_explicit controls the expansion of the Bernstein polynomials into its explicit form; example:

(%i1) load("bernstein")$

(%i2) bernstein_poly(k,n,x);
(%o2)                bernstein_poly(k, n, x)
(%i3) bernstein_poly(k,n,x), bernstein_explicit : true;
                                       n - k  k
(%o3)            binomial(n, k) (1 - x)      x

The Bernstein polynomials have both a gradef property and an integrate property:

(%i4) diff(bernstein_poly(k,n,x),x);
(%o4) (bernstein_poly(k - 1, n - 1, x)
                                 - bernstein_poly(k, n - 1, x)) n
(%i5) integrate(bernstein_poly(k,n,x),x);
(%o5) 
                                                            k + 1
 hypergeometric([k + 1, k - n], [k + 2], x) binomial(n, k) x
 ----------------------------------------------------------------
                              k + 1

For numeric inputs, both real and complex, the Bernstein polynomials evaluate to a numeric result:

(%i6) bernstein_poly(5,9, 1/2 + %i);
                        39375 %i   39375
(%o6)                   -------- + -----
                          128       256
(%i7) bernstein_poly(5,9, 0.5b0 + %i);
(%o7)           3.076171875b2 %i + 1.5380859375b2

To use bernstein_poly, first load("bernstein").

Variable: bernstein_explicit

Default value: false

When either k or n are non integers, the option variable bernstein_explicit controls the expansion of bernstein(k,n,x) into its explicit form; example:

(%i1) bernstein_poly(k,n,x);
(%o1)                bernstein_poly(k, n, x)
(%i2) bernstein_poly(k,n,x), bernstein_explicit : true;
                                       n - k  k
(%o2)            binomial(n, k) (1 - x)      x

When both k and n are explicitly integers, bernstein(k,n,x) always expands to its explicit form.

Function: multibernstein_poly ([k1,k2,…, kp], [n1,n2,…, np], [x1,x2,…, xp])

The multibernstein polynomial multibernstein_poly ([k1, k2, ..., kp], [n1, n2, ..., np], [x1, x2, ..., xp]) is the product of bernstein polynomials bernstein_poly(k1, n1, x1) bernstein_poly(k2, n2, x2) ... bernstein_poly(kp, np, xp).

To use multibernstein_poly, first load("bernstein").

Function: bernstein_approx (f, [x1, x1, …, xn], n)

Return the n-th order uniform Bernstein polynomial approximation for the function (x1, x2, ..., xn) |--> f. Examples

(%i1) bernstein_approx(f(x),[x], 2);
                 2       1                          2
(%o1)      f(1) x  + 2 f(-) (1 - x) x + f(0) (1 - x)
                         2
(%i2) bernstein_approx(f(x,y),[x,y], 2);
               2  2       1                2
(%o2) f(1, 1) x  y  + 2 f(-, 1) (1 - x) x y
                          2
                  2  2          1   2
 + f(0, 1) (1 - x)  y  + 2 f(1, -) x  (1 - y) y
                                2
       1  1                               1         2
 + 4 f(-, -) (1 - x) x (1 - y) y + 2 f(0, -) (1 - x)  (1 - y) y
       2  2                               2
            2        2       1                      2
 + f(1, 0) x  (1 - y)  + 2 f(-, 0) (1 - x) x (1 - y)
                             2
                  2        2
 + f(0, 0) (1 - x)  (1 - y)

To use bernstein_approx, first load("bernstein").

Function: bernstein_expand (e, [x1, x1, …, xn])

Express the polynomial e exactly as a linear combination of multi-variable Bernstein polynomials.

(%i1) bernstein_expand(x*y+1,[x,y]);
(%o1)    2 x y + (1 - x) y + x (1 - y) + (1 - x) (1 - y)
(%i2) expand(%);
(%o2)                        x y + 1

Maxima signals an error when the first argument isn’t a polynomial.

To use bernstein_expand, first load("bernstein").


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43 bitwise

The package bitwise provides functions that allow to manipulate bits of integer constants. As always maxima attempts to simplify the result of the operation if the actual value of a constant isn’t known considering attributes that might be known for the variables, see the declare mechanism.


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43.1 Functions and Variables for bitwise

Function: bit_not (int)

Inverts all bits of a signed integer. The result of this action reads -int - 1.

(%i1) load("bitwise")$
(%i2) bit_not(i);
(%o2)                      bit_not(i)
(%i3) bit_not(bit_not(i));
(%o3)                           i
(%i4) bit_not(3);
(%o4)                          - 4
(%i5) bit_not(100);
(%o5)                         - 101
(%i6) bit_not(-101);
(%o6)                          100

Function: bit_and (int1, ...)

This function calculates a bitwise and of two or more signed integers.

(%i1) load("bitwise")$
(%i2) bit_and(i,i);
(%o2)                           i
(%i3) bit_and(i,i,i);
(%o3)                           i
(%i4) bit_and(1,3);
(%o4)                           1
(%i5) bit_and(-7,7);
(%o5)                           1

If it is known if one of the parameters to bit_and is even this information is taken into consideration by the function.

(%i1) load("bitwise")$
(%i2) declare(e,even,o,odd);
(%o2)                         done
(%i3) bit_and(1,e);
(%o3)                           0
(%i4) bit_and(1,o);
(%o4)                           1

Function: bit_or (int1, ...)

This function calculates a bitwise or of two or more signed integers.

(%i1) load("bitwise")$
(%i2) bit_or(i,i);
(%o2)                           i
(%i3) bit_or(i,i,i);
(%o3)                           i
(%i4) bit_or(1,3);
(%o4)                           3
(%i5) bit_or(-7,7);
(%o5)                          - 1

If it is known if one of the parameters to bit_or is even this information is taken into consideration by the function.

(%i1) load("bitwise")$
(%i2) declare(e,even,o,odd);
(%o2)                         done
(%i3) bit_or(1,e);
(%o3)                         e + 1
(%i4) bit_or(1,o);
(%o4)                           o

Function: bit_xor (int1, ...)

This function calculates a bitwise or of two or more signed integers.

(%i1) load("bitwise")$
(%i2) bit_xor(i,i);
(%o2)                           0
(%i3) bit_xor(i,i,i);
(%o3)                           i
(%i4) bit_xor(1,3);
(%o4)                           2
(%i5) bit_xor(-7,7);
(%o5)                          - 2

If it is known if one of the parameters to bit_xor is even this information is taken into consideration by the function.

(%i1) load("bitwise")$
(%i2) declare(e,even,o,odd);
(%o2)                         done
(%i3) bit_xor(1,e);
(%o3)                         e + 1
(%i4) bit_xor(1,o);
(%o4)                         o - 1

Function: bit_lsh (int, nBits)

This function shifts all bits of the signed integer int to the left by nBits bits. The width of the integer is extended by nBits for this process. The result of bit_lsh therefore is int * 2.

(%i1) load("bitwise")$
(%i2) bit_lsh(0,1);
(%o2)                           0
(%i3) bit_lsh(1,0);
(%o3)                           1
(%i4) bit_lsh(1,1);
(%o4)                           2
(%i5) bit_lsh(1,i);
(%o5)                     bit_lsh(1, i)
(%i6) bit_lsh(-3,1);
(%o6)                          - 6
(%i7) bit_lsh(-2,1);
(%o7)                          - 4

Function: bit_rsh (int, nBits)

This function shifts all bits of the signed integer int to the right by nBits bits. The width of the integer is reduced by nBits for this process.

(%i1) load("bitwise")$
(%i2) bit_rsh(0,1);
(%o2)                           0
(%i3) bit_rsh(2,0);
(%o3)                           2
(%i4) bit_rsh(2,1);
(%o4)                           1
(%i5) bit_rsh(2,2);
(%o5)                           0
(%i6) bit_rsh(-3,1);
(%o6)                          - 2
(%i7) bit_rsh(-2,1);
(%o7)                          - 1
(%i8) bit_rsh(-2,2);
(%o8)                          - 1

Function: bit_length (int)

determines how many bits a variable needs to be long in order to store the number int. This function only operates on positive numbers.

(%i1) load("bitwise")$
(%i2) bit_length(0);
(%o2)                           0
(%i3) bit_length(1);
(%o3)                           1
(%i4) bit_length(7);
(%o4)                           3
(%i5) bit_length(8);
(%o5)                           4

Function: bit_onep (int, nBit)

determines if bits nBit is set in the signed integer int.

(%i1) load("bitwise")$
(%i2) bit_onep(85,0);
(%o2)                         true
(%i3) bit_onep(85,1);
(%o3)                         false
(%i4) bit_onep(85,2);
(%o4)                         true
(%i5) bit_onep(85,3);
(%o5)                         false
(%i6) bit_onep(85,100);
(%o6)                         false
(%i7) bit_onep(i,100);
(%o7)                   bit_onep(i, 100)

For signed numbers the sign bit is interpreted to be more than nBit to the left of the leftmost bit of int that reads 1.

(%i1) load("bitwise")$
(%i2) bit_onep(-2,0);
(%o2)                         false
(%i3) bit_onep(-2,1);
(%o3)                         true
(%i4) bit_onep(-2,2);
(%o4)                         true
(%i5) bit_onep(-2,3);
(%o5)                         true
(%i6) bit_onep(-2,4);
(%o6)                         true

If it is known if the number to be tested is even this information is taken into consideration by the function.

(%i1) load("bitwise")$
(%i2) declare(e,even,o,odd);
(%o2)                         done
(%i3) bit_onep(e,0);
(%o3)                         false
(%i4) bit_onep(o,0);
(%o4)                         true


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44 bode


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44.1 Functions and Variables for bode

Function: bode_gain (H, range, ...plot_opts...)

Function to draw Bode gain plots.

Examples (1 through 7 from

8 from Ron Crummett):

(%i1) load("bode")$

(%i2) H1 (s) := 100 * (1 + s) / ((s + 10) * (s + 100))$

(%i3) bode_gain (H1 (s), [w, 1/1000, 1000])$

(%i4) H2 (s) := 1 / (1 + s/omega0)$

(%i5) bode_gain (H2 (s), [w, 1/1000, 1000]), omega0 = 10$

(%i6) H3 (s) := 1 / (1 + s/omega0)^2$

(%i7) bode_gain (H3 (s), [w, 1/1000, 1000]), omega0 = 10$

(%i8) H4 (s) := 1 + s/omega0$

(%i9) bode_gain (H4 (s), [w, 1/1000, 1000]), omega0 = 10$

(%i10) H5 (s) := 1/s$

(%i11) bode_gain (H5 (s), [w, 1/1000, 1000])$

(%i12) H6 (s) := 1/((s/omega0)^2 + 2 * zeta * (s/omega0) + 1)$

(%i13) bode_gain (H6 (s), [w, 1/1000, 1000]), 
                  omega0 = 10, zeta = 1/10$

(%i14) H7 (s) := (s/omega0)^2 + 2 * zeta * (s/omega0) + 1$

(%i15) bode_gain (H7 (s), [w, 1/1000, 1000]),
                  omega0 = 10, zeta = 1/10$

(%i16) H8 (s) := 0.5 / (0.0001 * s^3 + 0.002 * s^2 + 0.01 * s)$

(%i17) bode_gain (H8 (s), [w, 1/1000, 1000])$

To use this function write first load("bode"). See also bode_phase.

Function: bode_phase (H, range, ...plot_opts...)

Function to draw Bode phase plots.

Examples (1 through 7 from

8 from Ron Crummett):

(%i1) load("bode")$

(%i2) H1 (s) := 100 * (1 + s) / ((s + 10) * (s + 100))$

(%i3) bode_phase (H1 (s), [w, 1/1000, 1000])$

(%i4) H2 (s) := 1 / (1 + s/omega0)$

(%i5) bode_phase (H2 (s), [w, 1/1000, 1000]), omega0 = 10$

(%i6) H3 (s) := 1 / (1 + s/omega0)^2$

(%i7) bode_phase (H3 (s), [w, 1/1000, 1000]), omega0 = 10$

(%i8) H4 (s) := 1 + s/omega0$

(%i9) bode_phase (H4 (s), [w, 1/1000, 1000]), omega0 = 10$

(%i10) H5 (s) := 1/s$

(%i11) bode_phase (H5 (s), [w, 1/1000, 1000])$

(%i12) H6 (s) := 1/((s/omega0)^2 + 2 * zeta * (s/omega0) + 1)$

(%i13) bode_phase (H6 (s), [w, 1/1000, 1000]), 
                   omega0 = 10, zeta = 1/10$

(%i14) H7 (s) := (s/omega0)^2 + 2 * zeta * (s/omega0) + 1$

(%i15) bode_phase (H7 (s), [w, 1/1000, 1000]), 
                   omega0 = 10, zeta = 1/10$

(%i16) H8 (s) := 0.5 / (0.0001 * s^3 + 0.002 * s^2 + 0.01 * s)$

(%i17) bode_phase (H8 (s), [w, 1/1000, 1000])$

(%i18) block ([bode_phase_unwrap : false],
              bode_phase (H8 (s), [w, 1/1000, 1000]));

(%i19) block ([bode_phase_unwrap : true], 
              bode_phase (H8 (s), [w, 1/1000, 1000]));

To use this function write first load("bode"). See also bode_gain.


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45 celine


Up: celine-pkg   [Contents][Index]

45.1 Introduction to celine

Maxima implementation of Sister Celine’s method. Barton Willis wrote this code. It is released under the Creative Commons CC0 license.

Celine’s method is described in Sections 4.1–4.4 of the book "A=B", by Marko Petkovsek, Herbert S. Wilf, and Doron Zeilberger. This book is available at http://www.math.rutgers.edu/~zeilberg/AeqB.pdf

Let f = F(n,k). The function celine returns a set of recursion relations for F of the form

p_0(n) * fff(n,k) + p_1(n) * fff(n+1,k) + ... + p_p(n) * fff(n+p,k+q),

where p_0 through p_p are polynomials. If Maxima is unable to determine that sum(sum(a(i,j) * F(n+i,k+j),i,0,p),j,0,q) / F(n,k) is a rational function of n and k, celine returns the empty set. When f involves parameters (variables other than n or k), celine might make assumptions about these parameters. Using ’put’ with a key of ’proviso,’ Maxima saves these assumptions on the input label.

To use this function, first load the package integer_sequence, opsubst, and to_poly_solve.

Examples:

(%i1) load("integer_sequence")$
(%i2) load("opsubst")$
(%i3) load("to_poly_solve")$
(%i4) load("celine")$
(%i5) celine(n!,n,k,1,0);
(%o5)       {fff(n + 1, k) - n fff(n, k) - fff(n, k)}

Verification that this result is correct:

(%i1) load("integer_sequence")$
(%i2) load("opsubst")$
(%i3) load("to_poly_solve")$
(%i4) load("celine")$
(%i5) g1:{fff(n+1,k)-n*fff(n,k)-fff(n,k)};
(%o5)       {fff(n + 1, k) - n fff(n, k) - fff(n, k)}
(%i6) ratsimp(minfactorial(first(g1))),fff(n,k) := n!;
(%o6)                           0

An example with parameters including the test that the result of the example is correct:

(%i1) load("integer_sequence")$
(%i2) load("opsubst")$
(%i3) load("to_poly_solve")$
(%i4) load("celine")$
(%i5) e : pochhammer(a,k) * pochhammer(-k,n) / (pochhammer(b,k));
                           (a)  (- k)
                              k      n
(%o5)                      -----------
                              (b)
                                 k
(%i6) recur : celine(e,n,k,2,1);
(%o6) {fff(n + 2, k + 1) - fff(n + 2, k) - b fff(n + 1, k + 1)
 + n ((- fff(n + 1, k + 1)) + 2 fff(n + 1, k) - a fff(n, k)
 - fff(n, k)) + a (fff(n + 1, k) - fff(n, k)) + 2 fff(n + 1, k)
    2
 - n  fff(n, k)}
(%i7) /* Test this result for correctness */
(%i8) first(%), fff(n,k) := ''(e)$
(%i9) makefact(makegamma(%))$
(%o9)                           0
(%i10) minfactorial(factor(minfactorial(factor(%))));

The proviso data suggests that setting a = b may result in a lower order recursion which is shown by the following example:

(%i1) load("integer_sequence")$
(%i2) load("opsubst")$
(%i3) load("to_poly_solve")$
(%i4) load("celine")$
(%i5) e : pochhammer(a,k) * pochhammer(-k,n) / (pochhammer(b,k));
                           (a)  (- k)
                              k      n
(%o5)                      -----------
                              (b)
                                 k
(%i6) recur : celine(e,n,k,2,1);
(%o6) {fff(n + 2, k + 1) - fff(n + 2, k) - b fff(n + 1, k + 1)
 + n ((- fff(n + 1, k + 1)) + 2 fff(n + 1, k) - a fff(n, k)
 - fff(n, k)) + a (fff(n + 1, k) - fff(n, k)) + 2 fff(n + 1, k)
    2
 - n  fff(n, k)}
(%i7) get('%,'proviso);
(%o7)                         false
(%i8) celine(subst(b=a,e),n,k,1,1);
(%o8) {fff(n + 1, k + 1) - fff(n + 1, k) + n fff(n, k)
                                                     + fff(n, k)}

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46 clebsch_gordan


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46.1 Functions and Variables for clebsch_gordan

Function: clebsch_gordan (j1, j2, m1, m2, j, m)

Compute the Clebsch-Gordan coefficient <j1, j2, m1, m2 | j, m>.

Function: racah_v (a, b, c, a1, b1, c1)

Compute Racah’s V coefficient (computed in terms of a related Clebsch-Gordan coefficient).

Function: racah_w (j1, j2, j5, j4, j3, j6)

Compute Racah’s W coefficient (computed in terms of a Wigner 6j symbol)

Function: wigner_3j (j1, j2, j3, m1, m2, m3)

Compute Wigner’s 3j symbol (computed in terms of a related Clebsch-Gordan coefficient).

Function: wigner_6j (j1, j2, j3, j4, j5, j6)

Compute Wigner’s 6j symbol.

Function: wigner_9j (a, b, c, d, e, f, g, h, i, j,)

Compute Wigner’s 9j symbol.


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47 cobyla


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47.1 Introduction to cobyla

fmin_cobyla is a Common Lisp translation (via f2cl) of the Fortran constrained optimization routine COBYLA by Powell[1][2][3].

COBYLA minimizes an objective function F(X) subject to M inequality constraints of the form g(X) >= 0 on X, where X is a vector of variables that has N components.

Equality constraints g(X)=0 can often be implemented by a pair of inequality constraints g(X)>=0 and -g(X)>= 0. Maxima’s interface to COBYLA allows equality constraints and internally converts the equality constraints to a pair of inequality constraints.

The algorithm employs linear approximations to the objective and constraint functions, the approximations being formed by linear interpolation at N+1 points in the space of the variables. The interpolation points are regarded as vertices of a simplex. The parameter RHO controls the size of the simplex and it is reduced automatically from RHOBEG to RHOEND. For each RHO the subroutine tries to achieve a good vector of variables for the current size, and then RHO is reduced until the value RHOEND is reached. Therefore RHOBEG and RHOEND should be set to reasonable initial changes to and the required accuracy in the variables respectively, but this accuracy should be viewed as a subject for experimentation because it is not guaranteed. The routine treats each constraint individually when calculating a change to the variables, rather than lumping the constraints together into a single penalty function. The name of the subroutine is derived from the phrase Constrained Optimization BY Linear Approximations.

References:

[1] Fortran Code is from http://plato.asu.edu/sub/nlores.html#general

[2] M. J. D. Powell, "A direct search optimization method that models the objective and constraint functions by linear interpolation," in Advances in Optimization and Numerical Analysis, eds. S. Gomez and J.-P. Hennart (Kluwer Academic: Dordrecht, 1994), p. 51-67.

[3] M. J. D. Powell, "Direct search algorithms for optimization calculations," Acta Numerica 7, 287-336 (1998). Also available as University of Cambridge, Department of Applied Mathematics and Theoretical Physics, Numerical Analysis Group, Report NA1998/04 from http://www.damtp.cam.ac.uk/user/na/reports.html

·

Category: Numerical methods · Category: Optimization · Category: Share packages · Category: Package cobyla


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47.2 Functions and Variables for cobyla

Function: fmin_cobyla
    fmin_cobyla (F, X, Y)
    fmin_cobyla (F, X, Y, optional_args)

Returns an approximate minimum of the expression F with respect to the variables X, subject to an optional set of constraints. Y is a list of initial guesses for X.

F must be an ordinary expressions, not names of functions or lambda expressions.

optional_args represents additional arguments, specified as symbol = value. The optional arguments recognized are:

constraints

List of inequality and equality constraints that must be satisfied by X. The inequality constraints must be actual inequalities of the form g(X) >= h(X) or g(X) <= h(X). The equality constraints must be of the form g(X) = h(X).

rhobeg

Initial value of the internal RHO variable which controls the size of simplex. (Defaults to 1.0)

rhoend

The desired final value rho parameter. It is approximately the accuracy in the variables. (Defaults to 1d-6.)

iprint

Verbose output level. (Defaults to 0)

  • 0 - No output
  • 1 - Summary at the end of the calculation
  • 2 - Each new value of RHO and SIGMA is printed, including the vector of variables, some function information when RHO is reduced.
  • 3 - Like 2, but information is printed when F(X) is computed.
maxfun

The maximum number of function evaluations. (Defaults to 1000).

On return, a vector is given:

  1. The value of the variables giving the minimum. This is a list of elements of the form var = value for each of the variables listed in X.
  2. The minimized function value
  3. The number of function evaluations.
  4. Return code with the following meanings
    1. 0 - No errors.
    2. 1 - Limit on maximum number of function evaluations reached.
    3. 2 - Rounding errors inhibiting progress.

load("fmin_cobyla") loads this function.

Function: bf_fmin_cobyla
    bf_fmin_cobyla (F, X, Y)
    bf_fmin_cobyla (F, X, Y, optional_args)

This function is identical to fmin_cobyla, except that bigfloat operations are used, and the default value for rhoend is 10^(fpprec/2).

See fmin_cobyla for more information.

load("bf_fmin_cobyla") loads this function.


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47.3 Examples for cobyla

Minimize x1*x2 with 1-x1^2-x2^2 >= 0. The theoretical solution is x1 = 1/sqrt(2), x2 = -1/sqrt(2).

(%i1) load("fmin_cobyla")$
(%i2) fmin_cobyla(x1*x2, [x1, x2], [1,1], 
                  constraints = [x1^2+x2^2<=1], iprint=1);
   Normal return from subroutine COBYLA

   NFVALS =   66   F =-5.000000E-01    MAXCV = 1.999845E-12
   X = 7.071058E-01  -7.071077E-01
(%o2) [[x1 = 0.70710584934848, x2 = - 0.7071077130248], 
       - 0.49999999999926, [[-1.999955756559757e-12],[]], 66]

There are additional examples in the share/cobyla/ex directory.


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48 combinatorics


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48.1 Package combinatorics

The combinatorics package provides several functions to work with permutations and to permute elements of a list. The permutations of degree n are all the n! possible orderings of the first n positive integers, 1, 2, …, n. The functions in this packages expect a permutation to be represented by a list of those integers.

Cycles are represented as a list of two or more integers i_1, i_2, …, i_m, all different. Such a list represents a permutation where the integer i_2 appears in the i_1th position, the integer i_3 appears in the i_2th position and so on, until the integer i_1, which appears in the i_mth position.

For instance, [4, 2, 1, 3] is one of the 24 permutations of degree four, which can also be represented by the cycle [1, 4, 3]. The functions where cycles are used to represent permutations also require the order of the permutation to avoid ambiguity. For instance, the same cycle [1, 4, 3] could refer to the permutation of order 6: [4, 2, 1, 3, 5, 6]. A product of cycles must be represented by a list of cycles; the cycles at the end of the list are applied first. For example, [[2, 4], [1, 3, 6, 5]] is equivalent to the permutation [3, 4, 6, 2, 1, 5].

A cycle can be written in several ways. for instance, [1, 3, 6, 5], [3, 6, 5, 1] and [6, 5, 1, 3] are all equivalent. The canonical form used in the package is the one that places the lowest index in the first place. A cycle with only two indices is also called a transposition and if the two indices are consecutive, it is called an adjacent transposition.

To run an interactive tutorial, use the command demo (combinatorics). Since this is an additional package, it must be loaded with the command load("combinatorics").

·

Category: Share packages · Category: Package combinatorics


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48.2 Functions and Variables for Combinatorics

Function: apply_cycles (cl,l)

Permutes the list or set l applying to it the list of cycles cl. The cycles at the end of the list are applied first and the first cycle in the list cl is the last one to be applied.

See also permute.

Example:

(%i1) load("combinatorics")$
(%i2) lis1:[a,b*c^2,4,z,x/y,1/2,ff23(x),0];
                        2        x  1
(%o2)            [a, b c , 4, z, -, -, ff23(x), 0]
                                 y  2
(%i3) apply_cycles ([[1, 6], [2, 6, 5, 7]], lis1);
                  x  1                       2
(%o3)            [-, -, 4, z, ff23(x), a, b c , 0]
                  y  2

Function: cyclep (c, n)

Returns true if c is a valid cycle of order n namely, a list of non-repeated positive integers less or equal to n. Otherwise, it returns false.

See also permp.

Examples:

(%i1) load("combinatorics")$
(%i2) cyclep ([-2,3,4], 5);
(%o2)                          false
(%i3) cyclep ([2,3,4,2], 5);
(%o3)                          false
(%i4) cyclep ([6,3,4], 5);
(%o4)                          false
(%i5) cyclep ([6,3,4], 6);
(%o5)                          true

Function: perm_cycles (p)

Returns permutation p as a product of cycles. The cycles are written in a canonical form, in which the lowest index in the cycle is placed in the first position.

See also perm_decomp.

Example:

(%i1) load("combinatorics")$
(%i2) perm_cycles ([4, 6, 3, 1, 7, 5, 2, 8]);
(%o2)                 [[1, 4], [2, 6, 5, 7]]

Function: perm_decomp (p)

Returns the minimum set of adjacent transpositions whose product equals the given permutation p.

See also perm_cycles.

Example:

(%i1) load("combinatorics")$
(%i2) perm_decomp ([4, 6, 3, 1, 7, 5, 2, 8]);
(%o2) [[6, 7], [5, 6], [6, 7], [3, 4], [4, 5], [2, 3], [3, 4], 
                            [4, 5], [5, 6], [1, 2], [2, 3], [3, 4]]

Function: perm_inverse (p)

Returns the inverse of a permutation of p, namely, a permutation q such that the products pq and qp are equal to the identity permutation: [1, 2, 3, …, n], where n is the length of p.

See also permult.

Example:

(%i1) load("combinatorics")$
(%i2) perm_inverse ([4, 6, 3, 1, 7, 5, 2, 8]);
(%o2)                [4, 7, 3, 1, 6, 2, 5, 8]

Function: perm_length (p)

Determines the minimum number of adjacent transpositions necessary to write permutation p as a product of adjacent transpositions. An adjacent transposition is a cycle with only two numbers, which are consecutive integers.

See also perm_decomp.

Example:

(%i1) load("combinatorics")$
(%i2) perm_length ([4, 6, 3, 1, 7, 5, 2, 8]);
(%o2)                           12

Function: perm_lex_next (p)

Returns the permutation that comes after the given permutation p, in the sequence of permutations in lexicographic order.

Example:

(%i1) load("combinatorics")$
(%i2) perm_lex_next ([4, 6, 3, 1, 7, 5, 2, 8]);
(%o2)                [4, 6, 3, 1, 7, 5, 8, 2]

Function: perm_lex_rank (p)

Finds the position of permutation p, an integer from 1 to the degree n of the permutation, in the sequence of permutations in lexicographic order.

See also perm_lex_unrank and perms_lex.

Example:

(%i1) load("combinatorics")$
(%i2) perm_lex_rank ([4, 6, 3, 1, 7, 5, 2, 8]);
(%o2)                          18255

Function: perm_lex_unrank (n, i)

Returns the n-degree permutation at position i (from 1 to n!) in the lexicographic ordering of permutations.

See also perm_lex_rank and perms_lex.

Example:

(%i1) load("combinatorics")$
(%i2) perm_lex_unrank (8, 18255);
(%o2)                [4, 6, 3, 1, 7, 5, 2, 8]

Function: perm_next (p)

Returns the permutation that comes after the given permutation p, in the sequence of permutations in Trotter-Johnson order.

See also perms.

Example:

(%i1) load("combinatorics")$
(%i2) perm_next ([4, 6, 3, 1, 7, 5, 2, 8]);
(%o2)                [4, 6, 3, 1, 7, 5, 8, 2]

Function: perm_parity (p)

Finds the parity of permutation p: 0 if the minimum number of adjacent transpositions necessary to write permutation p as a product of adjacent transpositions is even, or 1 if that number is odd.

See also perm_decomp.

Example:

(%i1) load("combinatorics")$
(%i2) perm_parity ([4, 6, 3, 1, 7, 5, 2, 8]);
(%o2)                            0

Function: perm_rank (p)

Finds the position of permutation p, an integer from 1 to the degree n of the permutation, in the sequence of permutations in Trotter-Johnson order.

See also perm_unrank and perms.

Example:

(%i1) load("combinatorics")$
(%i2) perm_rank ([4, 6, 3, 1, 7, 5, 2, 8]);
(%o2)                          19729

Function: perm_undecomp (cl, n)

Converts the list of cycles cl of degree n into an n degree permutation, equal to their product.

See also perm_decomp.

Example:

(%i1) load("combinatorics")$
(%i2) perm_undecomp ([[1,6],[2,6,5,7]], 8);
(%o2)                [5, 6, 3, 4, 7, 1, 2, 8]

Function: perm_unrank (n, i)

Returns the n-degree permutation at position i (from 1 to n!) in the Trotter-Johnson ordering of permutations.

See also perm_rank and perms.

Example:

(%i1) load("combinatorics")$
(%i2) perm_unrank (8, 19729);
(%o2)                [4, 6, 3, 1, 7, 5, 2, 8]

Function: permp (p)

Returns true if p is a valid permutation namely, a list of length n, whose elements are all the positive integers from 1 to n, without repetitions. Otherwise, it returns false.

Examples:

(%i1) load("combinatorics")$
(%i2) permp ([2,0,3,1]);
(%o2)                          false
(%i3) permp ([2,1,4,3]);
(%o3)                          true

Function: perms
    perms (n)
    perms (n, i)
    perms (n, i, j)

perms(n) returns a list of all n-degree permutations in the so-called Trotter-Johnson order.

perms(n, i) returns the n-degree permutation which is at the ith position (from 1 to n!) in the Trotter-Johnson ordering of the permutations.

perms(n, i, j) returns a list of the n-degree permutations between positions i and j in the Trotter-Johnson ordering of the permutations.

The sequence of permutations in Trotter-Johnson order starts with the identity permutation and each consecutive permutation can be obtained from the previous one a by single adjacent transposition.

See also perm_next, perm_rank and perm_unrank.

Examples:

(%i1) load("combinatorics")$
(%i2) perms (4);
(%o2) [[1, 2, 3, 4], [1, 2, 4, 3], [1, 4, 2, 3], [4, 1, 2, 3], 
[4, 1, 3, 2], [1, 4, 3, 2], [1, 3, 4, 2], [1, 3, 2, 4], 
[3, 1, 2, 4], [3, 1, 4, 2], [3, 4, 1, 2], [4, 3, 1, 2], 
[4, 3, 2, 1], [3, 4, 2, 1], [3, 2, 4, 1], [3, 2, 1, 4], 
[2, 3, 1, 4], [2, 3, 4, 1], [2, 4, 3, 1], [4, 2, 3, 1], 
[4, 2, 1, 3], [2, 4, 1, 3], [2, 1, 4, 3], [2, 1, 3, 4]]
(%i3) perms (4, 12);
(%o3)                     [[4, 3, 1, 2]]
(%i4) perms (4, 12, 14);
(%o4)       [[4, 3, 1, 2], [4, 3, 2, 1], [3, 4, 2, 1]]

Function: perms_lex
    perms_lex (n)
    perms_lex (n, i)
    perms_lex (n, i, j)

perms_lex(n) returns a list of all n-degree permutations in the so-called lexicographic order.

perms_lex(n, i) returns the n-degree permutation which is at the ith position (from 1 to n!) in the lexicographic ordering of the permutations.

perms_lex(n, i, j) returns a list of the n-degree permutations between positions i and j in the lexicographic ordering of the permutations.

The sequence of permutations in lexicographic order starts with all the permutations with the lowest index, 1, followed by all permutations starting with the following index, 2, and so on. The permutations starting by an index i are the permutations of the first n integers different from i and they are also placed in lexicographic order, where the permutations with the lowest of those integers are placed first and so on.

See also perm_lex_next, perm_lex_rank and perm_lex_unrank.

Examples:

(%i1) load("combinatorics")$
(%i2) perms_lex (4);
(%o2) [[1, 2, 3, 4], [1, 2, 4, 3], [1, 3, 2, 4], [1, 3, 4, 2], 
[1, 4, 2, 3], [1, 4, 3, 2], [2, 1, 3, 4], [2, 1, 4, 3], 
[2, 3, 1, 4], [2, 3, 4, 1], [2, 4, 1, 3], [2, 4, 3, 1], 
[3, 1, 2, 4], [3, 1, 4, 2], [3, 2, 1, 4], [3, 2, 4, 1], 
[3, 4, 1, 2], [3, 4, 2, 1], [4, 1, 2, 3], [4, 1, 3, 2], 
[4, 2, 1, 3], [4, 2, 3, 1], [4, 3, 1, 2], [4, 3, 2, 1]]
(%i3) perms_lex (4, 12);
(%o3)                     [[2, 4, 3, 1]]
(%i4) perms_lex (4, 12, 14);
(%o4)       [[2, 4, 3, 1], [3, 1, 2, 4], [3, 1, 4, 2]]

Function: permult (p_1, …, p_m)

Returns the product of two or more permutations p_1, …, p_m.

Example:

(%i1) load("combinatorics")$
(%i2) permult ([2,3,1], [3,1,2], [2,1,3]);
(%o2)                        [2, 1, 3]

Function: permute (p, l)

Applies the permutation p to the elements of the list (or set) l.

Example:

(%i1) load("combinatorics")$
(%i2) lis1: [a,b*c^2,4,z,x/y,1/2,ff23(x),0];
                        2        x  1
(%o2)            [a, b c , 4, z, -, -, ff23(x), 0]
                                 y  2
(%i3) permute ([4, 6, 3, 1, 7, 5, 2, 8], lis1);
                     1                 x     2
(%o3)            [z, -, 4, a, ff23(x), -, b c , 0]
                     2                 y

Function: random_perm (n)

Returns a random permutation of degree n.

See also random_permutation.

Example:

(%i1) load("combinatorics")$
(%i2) random_perm (7);
(%o2)                  [6, 3, 4, 7, 5, 1, 2]


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49 contrib_ode


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49.1 Introduction to contrib_ode

Maxima’s ordinary differential equation (ODE) solver ode2 solves elementary linear ODEs of first and second order. The function contrib_ode extends ode2 with additional methods for linear and non-linear first order ODEs and linear homogeneous second order ODEs. The code is still under development and the calling sequence may change in future releases. Once the code has stabilized it may be moved from the contrib directory and integrated into Maxima.

This package must be loaded with the command load('contrib_ode) before use.

The calling convention for contrib_ode is identical to ode2. It takes three arguments: an ODE (only the left hand side need be given if the right hand side is 0), the dependent variable, and the independent variable. When successful, it returns a list of solutions.

The form of the solution differs from ode2. As non-linear equations can have multiple solutions, contrib_ode returns a list of solutions. Each solution can have a number of forms:

%c is used to represent the constant of integration for first order equations. %k1 and %k2 are the constants for second order equations. If contrib_ode cannot obtain a solution for whatever reason, it returns false, after perhaps printing out an error message.

It is necessary to return a list of solutions, as even first order non-linear ODEs can have multiple solutions. For example:

(%i1) load('contrib_ode)$
(%i2) eqn:x*'diff(y,x)^2-(1+x*y)*'diff(y,x)+y=0;
                    dy 2             dy
(%o2)            x (--)  - (1 + x y) -- + y = 0
                    dx               dx
(%i3) contrib_ode(eqn,y,x);
                    dy 2             dy
(%t3)            x (--)  - (1 + x y) -- + y = 0
                    dx               dx

              first order equation not linear in y'

                                             x
(%o3)             [y = log(x) + %c, y = %c %e ]
(%i4) method;
(%o4)                        factor

Nonlinear ODEs can have singular solutions without constants of integration, as in the second solution of the following example:

(%i1) load('contrib_ode)$
(%i2) eqn:'diff(y,x)^2+x*'diff(y,x)-y=0;
                       dy 2     dy
(%o2)                 (--)  + x -- - y = 0
                       dx       dx
(%i3) contrib_ode(eqn,y,x);
                       dy 2     dy
(%t3)                 (--)  + x -- - y = 0
                       dx       dx

              first order equation not linear in y'

                                           2
                                 2        x
(%o3)              [y = %c x + %c , y = - --]
                                          4
(%i4) method;
(%o4)                       clairault

The following ODE has two parametric solutions in terms of the dummy variable %t. In this case the parametric solutions can be manipulated to give explicit solutions.

(%i1) load('contrib_ode)$
(%i2) eqn:'diff(y,x)=(x+y)^2;
                          dy          2
(%o2)                     -- = (x + y)
                          dx
(%i3) contrib_ode(eqn,y,x);
(%o3) [[x = %c - atan(sqrt(%t)), y = (- x) - sqrt(%t)], 
                     [x = atan(sqrt(%t)) + %c, y = sqrt(%t) - x]]
(%i4) method;
(%o4)                       lagrange

The following example (Kamke 1.112) demonstrates an implicit solution.

(%i1) load('contrib_ode)$
(%i2) assume(x>0,y>0);
(%o2)                    [x > 0, y > 0]
(%i3) eqn:x*'diff(y,x)-x*sqrt(y^2+x^2)-y;
                     dy           2    2
(%o3)              x -- - x sqrt(y  + x ) - y
                     dx
(%i4) contrib_ode(eqn,y,x);
                                  y
(%o4)                  [x - asinh(-) = %c]
                                  x
(%i5) method;
(%o5)                          lie

The following Riccati equation is transformed into a linear second order ODE in the variable %u. Maxima is unable to solve the new ODE, so it is returned unevaluated.

(%i1) load('contrib_ode)$
(%i2) eqn:x^2*'diff(y,x)=a+b*x^n+c*x^2*y^2;
                    2 dy      2  2      n
(%o2)              x  -- = c x  y  + b x  + a
                      dx
(%i3) contrib_ode(eqn,y,x);
               d%u
               ---                            2
               dx        2  a       n - 2    d %u
(%o3)  [[y = - ----, %u c  (-- + b x     ) + ---- c = 0]]
               %u c          2                 2
                            x                dx
(%i4) method;
(%o4)                        riccati

For first order ODEs contrib_ode calls ode2. It then tries the following methods: factorization, Clairault, Lagrange, Riccati, Abel and Lie symmetry methods. The Lie method is not attempted on Abel equations if the Abel method fails, but it is tried if the Riccati method returns an unsolved second order ODE.

For second order ODEs contrib_ode calls ode2 then odelin.

Extensive debugging traces and messages are displayed if the command put('contrib_ode,true,'verbose) is executed.

·

Category: Differential equations · Category: Share packages · Category: Package contrib_ode


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49.2 Functions and Variables for contrib_ode

Function: contrib_ode (eqn, y, x)

Returns a list of solutions of the ODE eqn with independent variable x and dependent variable y.

Function: odelin (eqn, y, x)

odelin solves linear homogeneous ODEs of first and second order with independent variable x and dependent variable y. It returns a fundamental solution set of the ODE.

For second order ODEs, odelin uses a method, due to Bronstein and Lafaille, that searches for solutions in terms of given special functions.

(%i1) load('contrib_ode)$
(%i2) odelin(x*(x+1)*'diff(y,x,2)+(x+5)*'diff(y,x,1)+(-4)*y,y,x);
       gauss_a(- 6, - 2, - 3, - x)  gauss_b(- 6, - 2, - 3, - x)
(%o2) {---------------------------, ---------------------------}
                    4                            4
                   x                            x

Function: ode_check (eqn, soln)

Returns the value of ODE eqn after substituting a possible solution soln. The value is equivalent to zero if soln is a solution of eqn.

(%i1) load('contrib_ode)$
(%i2) eqn:'diff(y,x,2)+(a*x+b)*y;
                         2
                        d y
(%o2)                   --- + (b + a x) y
                          2
                        dx
(%i3) ans:[y = bessel_y(1/3,2*(a*x+b)^(3/2)/(3*a))*%k2*sqrt(a*x+b)
         +bessel_j(1/3,2*(a*x+b)^(3/2)/(3*a))*%k1*sqrt(a*x+b)];
                                  3/2
                    1  2 (b + a x)
(%o3) [y = bessel_y(-, --------------) %k2 sqrt(a x + b)
                    3       3 a
                                          3/2
                            1  2 (b + a x)
                 + bessel_j(-, --------------) %k1 sqrt(a x + b)]
                            3       3 a
(%i4) ode_check(eqn,ans[1]);
(%o4)                           0

System variable: method

The variable method is set to the successful solution method.

Variable: %c

%c is the integration constant for first order ODEs.

Variable: %k1

%k1 is the first integration constant for second order ODEs.

Variable: %k2

%k2 is the second integration constant for second order ODEs.

Function: gauss_a (a, b, c, x)

gauss_a(a,b,c,x) and gauss_b(a,b,c,x) are 2F1 geometric functions. They represent any two independent solutions of the hypergeometric differential equation x(1-x) diff(y,x,2) + [c-(a+b+1)x] diff(y,x) - aby = 0 (A&S 15.5.1).

The only use of these functions is in solutions of ODEs returned by odelin and contrib_ode. The definition and use of these functions may change in future releases of Maxima.

See also gauss_b, dgauss_a and gauss_b.

Function: gauss_b (a, b, c, x)

See gauss_a.

Function: dgauss_a (a, b, c, x)

The derivative with respect to x of gauss_a(a, b, c, x).

Function: dgauss_b (a, b, c, x)

The derivative with respect to x of gauss_b(a, b, c, x).

Function: kummer_m (a, b, x)

Kummer’s M function, as defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Section 13.1.2.

The only use of this function is in solutions of ODEs returned by odelin and contrib_ode. The definition and use of this function may change in future releases of Maxima.

See also kummer_u, dkummer_m, and dkummer_u.

Function: kummer_u (a, b, x)

Kummer’s U function, as defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Section 13.1.3.

See kummer_m.

Function: dkummer_m (a, b, x)

The derivative with respect to x of kummer_m(a, b, x).

Function: dkummer_u (a, b, x)

The derivative with respect to x of kummer_u(a, b, x).

Function: bessel_simplify (expr)

Simplifies expressions containing Bessel functions bessel_j, bessel_y, bessel_i, bessel_k, hankel_1, hankel_2, strauve_h and strauve_l. Recurrence relations (given in Abramowitz and Stegun, Handbook of Mathematical Functions, Section 9.1.27) are used to replace functions of highest order n by functions of order n-1 and n-2.

This process repeated until all the orders differ by less than 2.

(%i1) load('contrib_ode)$
(%i2) bessel_simplify(4*bessel_j(n,x^2)*(x^2-n^2/x^2)
  +x*((bessel_j(n-2,x^2)-bessel_j(n,x^2))*x
  -(bessel_j(n,x^2)-bessel_j(n+2,x^2))*x)
  -2*bessel_j(n+1,x^2)+2*bessel_j(n-1,x^2));
(%o2)                           0
(%i3) bessel_simplify(-2*bessel_j(1,z)*z^3-10*bessel_j(2,z)*z^2
 +15*%pi*bessel_j(1,z)*struve_h(3,z)*z-15*%pi*struve_h(1,z)*bessel_j(3,z)*z
 -15*%pi*bessel_j(0,z)*struve_h(2,z)*z+15*%pi*struve_h(0,z)*bessel_j(2,z)*z
 -30*%pi*bessel_j(1,z)*struve_h(2,z)+30*%pi*struve_h(1,z)*bessel_j(2,z));
(%o3)                           0

Function: expintegral_e_simplify (expr)

Simplify expressions containing exponential integral expintegral_e using the recurrence (A&S 5.1.14).

expintegral_e(n+1,z) = (1/n) * (exp(-z)-z*expintegral_e(n,z)) n = 1,2,3 ....


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49.3 Possible improvements to contrib_ode

These routines are work in progress. I still need to:


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49.4 Test cases for contrib_ode

The routines have been tested on a approximately one thousand test cases from Murphy, Kamke, Zwillinger and elsewhere. These are included in the tests subdirectory.


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49.5 References for contrib_ode

  1. E. Kamke, Differentialgleichungen Losungsmethoden und Losungen, Vol 1, Geest & Portig, Leipzig, 1961
  2. G. M. Murphy, Ordinary Differential Equations and Their Solutions, Van Nostrand, New York, 1960
  3. D. Zwillinger, Handbook of Differential Equations, 3rd edition, Academic Press, 1998
  4. F. Schwarz, Symmetry Analysis of Abel’s Equation, Studies in Applied Mathematics, 100:269-294 (1998)
  5. F. Schwarz, Algorithmic Solution of Abel’s Equation, Computing 61, 39-49 (1998)
  6. E. S. Cheb-Terrab, A. D. Roche, Symmetries and First Order ODE Patterns, Computer Physics Communications 113 (1998), p 239. (http://lie.uwaterloo.ca/papers/ode_vii.pdf)
  7. E. S. Cheb-Terrab, T. Kolokolnikov, First Order ODEs, Symmetries and Linear Transformations, European Journal of Applied Mathematics, Vol. 14, No. 2, pp. 231-246 (2003). (http://arxiv.org/abs/math-ph/0007023,
    http://lie.uwaterloo.ca/papers/ode_iv.pdf)
  8. G. W. Bluman, S. C. Anco, Symmetry and Integration Methods for Differential Equations, Springer, (2002)
  9. M. Bronstein, S. Lafaille, Solutions of linear ordinary differential equations in terms of special functions, Proceedings of ISSAC 2002, Lille, ACM Press, 23-28. (http://www-sop.inria.fr/cafe/Manuel.Bronstein/publications/issac2002.pdf)

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50 descriptive


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50.1 Introduction to descriptive

Package descriptive contains a set of functions for making descriptive statistical computations and graphing. Together with the source code there are three data sets in your Maxima tree: pidigits.data, wind.data and biomed.data.

Any statistics manual can be used as a reference to the functions in package descriptive.

For comments, bugs or suggestions, please contact me at ’riotorto AT yahoo DOT com’.

Here is a simple example on how the descriptive functions in descriptive do they work, depending on the nature of their arguments, lists or matrices,

(%i1) load ("descriptive")$
(%i2) /* univariate sample */   mean ([a, b, c]);
                            c + b + a
(%o2)                       ---------
                                3
(%i3) matrix ([a, b], [c, d], [e, f]);
                            [ a  b ]
                            [      ]
(%o3)                       [ c  d ]
                            [      ]
                            [ e  f ]
(%i4) /* multivariate sample */ mean (%);
                      e + c + a  f + d + b
(%o4)                [---------, ---------]
                          3          3

Note that in multivariate samples the mean is calculated for each column.

In case of several samples with possible different sizes, the Maxima function map can be used to get the desired results for each sample,

(%i1) load ("descriptive")$
(%i2) map (mean, [[a, b, c], [d, e]]);
                        c + b + a  e + d
(%o2)                  [---------, -----]
                            3        2

In this case, two samples of sizes 3 and 2 were stored into a list.

Univariate samples must be stored in lists like

(%i1) s1 : [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5];
(%o1)           [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5]

and multivariate samples in matrices as in

(%i1) s2 : matrix ([13.17, 9.29], [14.71, 16.88], [18.50, 16.88],
             [10.58, 6.63], [13.33, 13.25], [13.21,  8.12]);
                        [ 13.17  9.29  ]
                        [              ]
                        [ 14.71  16.88 ]
                        [              ]
                        [ 18.5   16.88 ]
(%o1)                   [              ]
                        [ 10.58  6.63  ]
                        [              ]
                        [ 13.33  13.25 ]
                        [              ]
                        [ 13.21  8.12  ]

In this case, the number of columns equals the random variable dimension and the number of rows is the sample size.

Data can be introduced by hand, but big samples are usually stored in plain text files. For example, file pidigits.data contains the first 100 digits of number %pi:

      3
      1
      4
      1
      5
      9
      2
      6
      5
      3 ...

In order to load these digits in Maxima,

(%i1) s1 : read_list (file_search ("pidigits.data"))$
(%i2) length (s1);
(%o2)                          100

On the other hand, file wind.data contains daily average wind speeds at 5 meteorological stations in the Republic of Ireland (This is part of a data set taken at 12 meteorological stations. The original file is freely downloadable from the StatLib Data Repository and its analysis is discused in Haslett, J., Raftery, A. E. (1989) Space-time Modelling with Long-memory Dependence: Assessing Ireland’s Wind Power Resource, with Discussion. Applied Statistics 38, 1-50). This loads the data:

(%i1) s2 : read_matrix (file_search ("wind.data"))$
(%i2) length (s2);
(%o2)                          100
(%i3) s2 [%]; /* last record */
(%o3)            [3.58, 6.0, 4.58, 7.62, 11.25]

Some samples contain non numeric data. As an example, file biomed.data (which is part of another bigger one downloaded from the StatLib Data Repository) contains four blood measures taken from two groups of patients, A and B, of different ages,

(%i1) s3 : read_matrix (file_search ("biomed.data"))$
(%i2) length (s3);
(%o2)                          100
(%i3) s3 [1]; /* first record */
(%o3)            [A, 30, 167.0, 89.0, 25.6, 364]

The first individual belongs to group A, is 30 years old and his/her blood measures were 167.0, 89.0, 25.6 and 364.

One must take care when working with categorical data. In the next example, symbol a is assigned a value in some previous moment and then a sample with categorical value a is taken,

(%i1) a : 1$
(%i2) matrix ([a, 3], [b, 5]);
                            [ 1  3 ]
(%o2)                       [      ]
                            [ b  5 ]
·

Category: Descriptive statistics · Category: Share packages · Category: Package descriptive


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50.2 Functions and Variables for data manipulation

Function: build_sample
    build_sample (list)
    build_sample (matrix)

Builds a sample from a table of absolute frequencies. The input table can be a matrix or a list of lists, all of them of equal size. The number of columns or the length of the lists must be greater than 1. The last element of each row or list is interpreted as the absolute frequency. The output is always a sample in matrix form.

Examples:

Univariate frequency table.

(%i1) load ("descriptive")$
(%i2) sam1: build_sample([[6,1], [j,2], [2,1]]);
                       [ 6 ]
                       [   ]
                       [ j ]
(%o2)                  [   ]
                       [ j ]
                       [   ]
                       [ 2 ]
(%i3) mean(sam1);
                      2 j + 8
(%o3)                [-------]
                         4
(%i4) barsplot(sam1) $

Multivariate frequency table.

(%i1) load ("descriptive")$
(%i2) sam2: build_sample([[6,3,1], [5,6,2], [u,2,1],[6,8,2]]) ;
                           [ 6  3 ]
                           [      ]
                           [ 5  6 ]
                           [      ]
                           [ 5  6 ]
(%o2)                      [      ]
                           [ u  2 ]
                           [      ]
                           [ 6  8 ]
                           [      ]
                           [ 6  8 ]
(%i3) cov(sam2);
       [   2                 2                            ]
       [  u  + 158   (u + 28)     2 u + 174   11 (u + 28) ]
       [  -------- - ---------    --------- - ----------- ]
(%o3)  [     6          36            6           12      ]
       [                                                  ]
       [ 2 u + 174   11 (u + 28)            21            ]
       [ --------- - -----------            --            ]
       [     6           12                 4             ]
(%i4) barsplot(sam2, grouping=stacked) $

Function: continuous_freq
    continuous_freq (data)
    continuous_freq (data, m)

The first argument of continuous_freq must be a list or 1-dimensional array (as created by make_array) of numbers. Divides the range in intervals and counts how many values are inside them. The second argument is optional and either equals the number of classes we want, 10 by default, or equals a list containing the class limits and the number of classes we want, or a list containing only the limits.

If sample values are all equal, this function returns only one class of amplitude 2.

Examples:

Optional argument indicates the number of classes we want. The first list in the output contains the interval limits, and the second the corresponding counts: there are 16 digits inside the interval [0, 1.8], 24 digits in (1.8, 3.6], and so on.

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) continuous_freq (s1, 5);
(%o3) [[0, 1.8, 3.6, 5.4, 7.2, 9.0], [16, 24, 18, 17, 25]]

Optional argument indicates we want 7 classes with limits -2 and 12:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) continuous_freq (s1, [-2,12,7]);
(%o3) [[- 2, 0, 2, 4, 6, 8, 10, 12], [8, 20, 22, 17, 20, 13, 0]]

Optional argument indicates we want the default number of classes with limits -2 and 12:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) continuous_freq (s1, [-2,12]);
                3  4  11  18     32  39  46  53
(%o3)  [[- 2, - -, -, --, --, 5, --, --, --, --, 12], 
                5  5  5   5      5   5   5   5
               [0, 8, 20, 12, 18, 9, 8, 25, 0, 0]]

The first argument may be an array.

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) a1 : make_array (fixnum, length (s1)) $
(%i4) fillarray (a1, s1);
(%o4) {Lisp Array: 
#(3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 \
5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 0 5 8 2 0 9 7 4 9 4 4 5 9
  2 3 0 7 8 1 6 4 0 6 2 8 6 2 0 8 9 9 8 6 2 8 0 3 4 8 2 5 3 4 2 \
1 1 7 0 6 7)}
(%i5) continuous_freq (a1);
           9   9  27  18  9  27  63  36  81
(%o5) [[0, --, -, --, --, -, --, --, --, --, 9], 
           10  5  10  5   2  5   10  5   10
                             [8, 8, 12, 12, 10, 8, 9, 8, 12, 13]]

Function: discrete_freq (data)

Counts absolute frequencies in discrete samples, both numeric and categorical. Its unique argument is a list, or 1-dimensional array (as created by make_array).

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) discrete_freq (s1);
(%o3) [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 
                             [8, 8, 12, 12, 10, 8, 9, 8, 12, 13]]

The first list gives the sample values and the second their absolute frequencies. Commands ? col and ? transpose should help you to understand the last input.

The argument may be an array.

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) a1 : make_array (fixnum, length (s1)) $
(%i4) fillarray (a1, s1);
(%o4) {Lisp Array: 
#(3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 \
5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 0 5 8 2 0 9 7 4 9 4 4 5 9
  2 3 0 7 8 1 6 4 0 6 2 8 6 2 0 8 9 9 8 6 2 8 0 3 4 8 2 5 3 4 2 \
1 1 7 0 6 7)}
(%i5) discrete_freq (a1);
(%o5) [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], 
                             [8, 8, 12, 12, 10, 8, 9, 8, 12, 13]]

Function: standardize
    standardize (list)
    standardize (matrix)

Subtracts to each element of the list the sample mean and divides the result by the standard deviation. When the input is a matrix, standardize subtracts to each row the multivariate mean, and then divides each component by the corresponding standard deviation.

Function: subsample
    subsample (data_matrix, predicate_function)
    subsample (data_matrix, predicate_function, col_num1, col_num2, ...)

This is a sort of variant of the Maxima submatrix function. The first argument is the data matrix, the second is a predicate function and optional additional arguments are the numbers of the columns to be taken. Its behaviour is better understood with examples.

These are multivariate records in which the wind speed in the first meteorological station were greater than 18. See that in the lambda expression the i-th component is referred to as v[i].

(%i1) load ("descriptive")$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) subsample (s2, lambda([v], v[1] > 18));
              [ 19.38  15.37  15.12  23.09  25.25 ]
              [                                   ]
              [ 18.29  18.66  19.08  26.08  27.63 ]
(%o3)         [                                   ]
              [ 20.25  21.46  19.95  27.71  23.38 ]
              [                                   ]
              [ 18.79  18.96  14.46  26.38  21.84 ]

In the following example, we request only the first, second and fifth components of those records with wind speeds greater or equal than 16 in station number 1 and less than 25 knots in station number 4. The sample contains only data from stations 1, 2 and 5. In this case, the predicate function is defined as an ordinary Maxima function.

(%i1) load ("descriptive")$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) g(x):= x[1] >= 16 and x[4] < 25$
(%i4) subsample (s2, g, 1, 2, 5);
                     [ 19.38  15.37  25.25 ]
                     [                     ]
                     [ 17.33  14.67  19.58 ]
(%o4)                [                     ]
                     [ 16.92  13.21  21.21 ]
                     [                     ]
                     [ 17.25  18.46  23.87 ]

Here is an example with the categorical variables of biomed.data. We want the records corresponding to those patients in group B who are older than 38 years.

(%i1) load ("descriptive")$
(%i2) s3 : read_matrix (file_search ("biomed.data"))$
(%i3) h(u):= u[1] = B and u[2] > 38 $
(%i4) subsample (s3, h);
                [ B  39  28.0  102.3  17.1  146 ]
                [                               ]
                [ B  39  21.0  92.4   10.3  197 ]
                [                               ]
                [ B  39  23.0  111.5  10.0  133 ]
                [                               ]
                [ B  39  26.0  92.6   12.3  196 ]
(%o4)           [                               ]
                [ B  39  25.0  98.7   10.0  174 ]
                [                               ]
                [ B  39  21.0  93.2   5.9   181 ]
                [                               ]
                [ B  39  18.0  95.0   11.3  66  ]
                [                               ]
                [ B  39  39.0  88.5   7.6   168 ]

Probably, the statistical analysis will involve only the blood measures,

(%i1) load ("descriptive")$
(%i2) s3 : read_matrix (file_search ("biomed.data"))$
(%i3) subsample (s3, lambda([v], v[1] = B and v[2] > 38),
                 3, 4, 5, 6);
                   [ 28.0  102.3  17.1  146 ]
                   [                        ]
                   [ 21.0  92.4   10.3  197 ]
                   [                        ]
                   [ 23.0  111.5  10.0  133 ]
                   [                        ]
                   [ 26.0  92.6   12.3  196 ]
(%o3)              [                        ]
                   [ 25.0  98.7   10.0  174 ]
                   [                        ]
                   [ 21.0  93.2   5.9   181 ]
                   [                        ]
                   [ 18.0  95.0   11.3  66  ]
                   [                        ]
                   [ 39.0  88.5   7.6   168 ]

This is the multivariate mean of s3,

(%i1) load ("descriptive")$
(%i2) s3 : read_matrix (file_search ("biomed.data"))$
(%i3) mean (s3);
       65 B + 35 A  317          6 NA + 8144.999999999999
(%o3) [-----------, ---, 87.178, ------------------------, 
           100      10                     100
                                                    3 NA + 19587
                                            18.123, ------------]
                                                        100

Here, the first component is meaningless, since A and B are categorical, the second component is the mean age of individuals in rational form, and the fourth and last values exhibit some strange behaviour. This is because symbol NA is used here to indicate non available data, and the two means are nonsense. A possible solution would be to take out from the matrix those rows with NA symbols, although this deserves some loss of information.

(%i1) load ("descriptive")$
(%i2) s3 : read_matrix (file_search ("biomed.data"))$
(%i3) g(v):= v[4] # NA and v[6] # NA $
(%i4) mean (subsample (s3, g, 3, 4, 5, 6));
(%o4) [79.4923076923077, 86.2032967032967, 16.93186813186813, 
                                                            2514
                                                            ----]
                                                             13

Function: transform_sample (matrix, varlist, exprlist)

Transforms the sample matrix, where each column is called according to varlist, following expressions in exprlist.

Examples:

The second argument assigns names to the three columns. With these names, a list of expressions define the transformation of the sample.

(%i1) load ("descriptive")$
(%i2) data: matrix([3,2,7],[3,7,2],[8,2,4],[5,2,4]) $
(%i3) transform_sample(data, [a,b,c], [c, a*b, log(a)]);
                               [ 7  6   log(3) ]
                               [               ]
                               [ 2  21  log(3) ]
(%o3)                          [               ]
                               [ 4  16  log(8) ]
                               [               ]
                               [ 4  10  log(5) ]

Add a constant column and remove the third variable.

(%i1) load ("descriptive")$
(%i2) data: matrix([3,2,7],[3,7,2],[8,2,4],[5,2,4]) $
(%i3) transform_sample(data, [a,b,c], [makelist(1,k,length(data)),a,b]);
                                  [ 1  3  2 ]
                                  [         ]
                                  [ 1  3  7 ]
(%o3)                             [         ]
                                  [ 1  8  2 ]
                                  [         ]
                                  [ 1  5  2 ]


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50.3 Functions and Variables for descriptive statistics

Function: mean
    mean (list)
    mean (matrix)

This is the sample mean, defined as

                       n
                     ====
             _   1   \
             x = -    >    x
                 n   /      i
                     ====
                     i = 1

Example:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) mean (s1);
                               471
(%o3)                          ---
                               100
(%i4) %, numer;
(%o4)                         4.71
(%i5) s2 : read_matrix (file_search ("wind.data"))$
(%i6) mean (s2);
(%o6)     [9.9485, 10.1607, 10.8685, 15.7166, 14.8441]

Function: var
    var (list)
    var (matrix)

This is the sample variance, defined as

                     n
                   ====
           2   1   \          _ 2
          s  = -    >    (x - x)
               n   /       i
                   ====
                   i = 1

Example:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) var (s1), numer;
(%o3)                   8.425899999999999

See also function var1.

Function: var1
    var1 (list)
    var1 (matrix)

This is the sample variance, defined as

                     n
                   ====
               1   \          _ 2
              ---   >    (x - x)
              n-1  /       i
                   ====
                   i = 1

Example:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) var1 (s1), numer;
(%o3)                    8.5110101010101
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) var1 (s2);
(%o5) [17.39586540404041, 15.13912778787879, 15.63204924242424, 
                            32.50152569696971, 24.66977392929294]

See also function var.

Function: std
    std (list)
    std (matrix)

This is the square root of the function var, the variance with denominator n.

Example:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) std (s1), numer;
(%o3)                   2.902740084816414
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) std (s2);
(%o5) [4.149928523480858, 3.871399812729241, 3.933920277534866, 
                            5.672434260526957, 4.941970881136392]

See also functions var and std1.

Function: std1
    std1 (list)
    std1 (matrix)

This is the square root of the function var1, the variance with denominator n-1.

Example:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) std1 (s1), numer;
(%o3)                   2.917363553109228
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) std1 (s2);
(%o5) [4.170835096721089, 3.89090320978032, 3.953738641137555, 
                            5.701010936401517, 4.966867617451963]

See also functions var1 and std.

Function: noncentral_moment
    noncentral_moment (list, k)
    noncentral_moment (matrix, k)

The non central moment of order k, defined as

                       n
                     ====
                 1   \      k
                 -    >    x
                 n   /      i
                     ====
                     i = 1

Example:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) noncentral_moment (s1, 1), numer; /* the mean */
(%o3)                         4.71
(%i5) s2 : read_matrix (file_search ("wind.data"))$
(%i6) noncentral_moment (s2, 5);
(%o6) [319793.8724761505, 320532.1923892463,
      391249.5621381556, 2502278.205988911, 1691881.797742255]

See also function central_moment.

Function: central_moment
    central_moment (list, k)
    central_moment (matrix, k)

The central moment of order k, defined as

                    n
                  ====
              1   \          _ k
              -    >    (x - x)
              n   /       i
                  ====
                  i = 1

Example:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) central_moment (s1, 2), numer; /* the variance */
(%o3)                   8.425899999999999
(%i5) s2 : read_matrix (file_search ("wind.data"))$
(%i6) central_moment (s2, 3);
(%o6) [11.29584771375004, 16.97988248298583, 5.626661952750102,
                             37.5986572057918, 25.85981904394192]

See also functions central_moment and mean.

Function: cv
    cv (list)
    cv (matrix)

The variation coefficient is the quotient between the sample standard deviation (std) and the mean,

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) cv (s1), numer;
(%o3)                   .6193977819764815
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) cv (s2);
(%o5) [.4192426091090204, .3829365309260502, 0.363779605385983, 
                            .3627381836021478, .3346021393989506]

See also functions std and mean.

Function: smin
    smin (list)
    smin (matrix)

This is the minimum value of the sample list. When the argument is a matrix, smin returns a list containing the minimum values of the columns, which are associated to statistical variables.

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) smin (s1);
(%o3)                           0
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) smin (s2);
(%o5)             [0.58, 0.5, 2.67, 5.25, 5.17]

See also function smax.

Function: smax
    smax (list)
    smax (matrix)

This is the maximum value of the sample list. When the argument is a matrix, smax returns a list containing the maximum values of the columns, which are associated to statistical variables.

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) smax (s1);
(%o3)                           9
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) smax (s2);
(%o5)          [20.25, 21.46, 20.04, 29.63, 27.63]

See also function smin.

Function: range
    range (list)
    range (matrix)

The range is the difference between the extreme values.

Example:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) range (s1);
(%o3)                           9
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) range (s2);
(%o5)          [19.67, 20.96, 17.37, 24.38, 22.46]

Function: quantile
    quantile (list, p)
    quantile (matrix, p)

This is the p-quantile, with p a number in [0, 1], of the sample list. Although there are several definitions for the sample quantile (Hyndman, R. J., Fan, Y. (1996) Sample quantiles in statistical packages. American Statistician, 50, 361-365), the one based on linear interpolation is implemented in package descriptive-pkg

Example:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) /* 1st and 3rd quartiles */
         [quantile (s1, 1/4), quantile (s1, 3/4)], numer;
(%o3)                      [2.0, 7.25]
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) quantile (s2, 1/4);
(%o5)    [7.2575, 7.477500000000001, 7.82, 11.28, 11.48]

Function: median
    median (list)
    median (matrix)

Once the sample is ordered, if the sample size is odd the median is the central value, otherwise it is the mean of the two central values.

Example:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) median (s1);
                                9
(%o3)                           -
                                2
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) median (s2);
(%o5)         [10.06, 9.855, 10.73, 15.48, 14.105]

The median is the 1/2-quantile.

See also function quantile.

Function: qrange
    qrange (list)
    qrange (matrix)

The interquartilic range is the difference between the third and first quartiles, quantile(list,3/4) - quantile(list,1/4),

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) qrange (s1);
                               21
(%o3)                          --
                               4
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) qrange (s2);
(%o5) [5.385, 5.572499999999998, 6.022500000000001, 
                            8.729999999999999, 6.649999999999999]

See also function quantile.

Function: mean_deviation
    mean_deviation (list)
    mean_deviation (matrix)

The mean deviation, defined as

                     n
                   ====
               1   \          _
               -    >    |x - x|
               n   /       i
                   ====
                   i = 1

Example:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) mean_deviation (s1);
                               51
(%o3)                          --
                               20
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) mean_deviation (s2);
(%o5) [3.287959999999999, 3.075342, 3.23907, 4.715664000000001, 
                                               4.028546000000002]

See also function mean.

Function: median_deviation
    median_deviation (list)
    median_deviation (matrix)

The median deviation, defined as

                 n
               ====
           1   \
           -    >    |x - med|
           n   /       i
               ====
               i = 1

where med is the median of list.

Example:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) median_deviation (s1);
                                5
(%o3)                           -
                                2
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) median_deviation (s2);
(%o5)           [2.75, 2.755, 3.08, 4.315, 3.31]

See also function mean.

Function: harmonic_mean
    harmonic_mean (list)
    harmonic_mean (matrix)

The harmonic mean, defined as

                  n
               --------
                n
               ====
               \     1
                >    --
               /     x
               ====   i
               i = 1

Example:

(%i1) load ("descriptive")$
(%i2) y : [5, 7, 2, 5, 9, 5, 6, 4, 9, 2, 4, 2, 5]$
(%i3) harmonic_mean (y), numer;
(%o3)                   3.901858027632205
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) harmonic_mean (s2);
(%o5) [6.948015590052786, 7.391967752360356, 9.055658197151745, 
                            13.44199028193692, 13.01439145898509]

See also functions mean and geometric_mean.

Function: geometric_mean
    geometric_mean (list)
    geometric_mean (matrix)

The geometric mean, defined as

                 /  n      \ 1/n
                 | /===\   |
                 |  ! !    |
                 |  ! !  x |
                 |  ! !   i|
                 | i = 1   |
                 \         /

Example:

(%i1) load ("descriptive")$
(%i2) y : [5, 7, 2, 5, 9, 5, 6, 4, 9, 2, 4, 2, 5]$
(%i3) geometric_mean (y), numer;
(%o3)                   4.454845412337012
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) geometric_mean (s2);
(%o5) [8.82476274347979, 9.22652604739361, 10.0442675714889, 
                            14.61274126349021, 13.96184163444275]

See also functions mean and harmonic_mean.

Function: kurtosis
    kurtosis (list)
    kurtosis (matrix)

The kurtosis coefficient, defined as

                    n
                  ====
            1     \          _ 4
           ----    >    (x - x)  - 3
              4   /       i
           n s    ====
                  i = 1

Example:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) kurtosis (s1), numer;
(%o3)                  - 1.273247946514421
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) kurtosis (s2);
(%o5) [- .2715445622195385, 0.119998784429451, 
     - .4275233490482861, - .6405361979019522, - .4952382132352935]

See also functions mean, var and skewness.

Function: skewness
    skewness (list)
    skewness (matrix)

The skewness coefficient, defined as

                    n
                  ====
            1     \          _ 3
           ----    >    (x - x)
              3   /       i
           n s    ====
                  i = 1

Example:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) skewness (s1), numer;
(%o3)                  .009196180476450424
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) skewness (s2);
(%o5) [.1580509020000978, .2926379232061854, .09242174416107717, 
                            .2059984348148687, .2142520248890831]

See also functions mean,, var and kurtosis.

Function: pearson_skewness
    pearson_skewness (list)
    pearson_skewness (matrix)

Pearson’s skewness coefficient, defined as

                _
             3 (x - med)
             -----------
                  s

where med is the median of list.

Example:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) pearson_skewness (s1), numer;
(%o3)                   .2159484029093895
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) pearson_skewness (s2);
(%o5) [- .08019976629211892, .2357036272952649, 
         .1050904062491204, .1245042340592368, .4464181795804519]

See also functions mean, var and median.

Function: quartile_skewness
    quartile_skewness (list)
    quartile_skewness (matrix)

The quartile skewness coefficient, defined as

               c    - 2 c    + c
                3/4      1/2    1/4
               --------------------
                   c    - c
                    3/4    1/4

where c_p is the p-quantile of sample list.

Example:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) quartile_skewness (s1), numer;
(%o3)                  .04761904761904762
(%i4) s2 : read_matrix (file_search ("wind.data"))$
(%i5) quartile_skewness (s2);
(%o5) [- 0.0408542246982353, .1467025572005382, 
       0.0336239103362392, .03780068728522298, .2105263157894735]

See also function quantile.

Function: km
    km (list, option ...)
    km (matrix, option ...)

Kaplan Meier estimator of the survival, or reliability, function S(x)=1-F(x).

Data can be introduced as a list of pairs, or as a two column matrix. The first component is the observed time, and the second component a censoring index (1 = non censored, 0 = right censored).

The optional argument is the name of the variable in the returned expression, which is x by default.

Examples:

Sample as a list of pairs.

(%i1) load ("descriptive")$
(%i2) S: km([[2,1], [3,1], [5,0], [8,1]]);
                       charfun((3 <= x) and (x < 8))
(%o2) charfun(x < 0) + -----------------------------
                                     2
                3 charfun((2 <= x) and (x < 3))
              + ------------------------------- 
                               4
              + charfun((0 <= x) and (x < 2))
(%i3) load ("draw")$
(%i4) draw2d(
        line_width = 3, grid = true,
        explicit(S, x, -0.1, 10))$

Estimate survival probabilities.

(%i1) load ("descriptive")$
(%i2) S(t):= ''(km([[2,1], [3,1], [5,0], [8,1]], t)) $
(%i3) S(6);
                            1
(%o3)                       -
                            2

Function: cdf_empirical
    cdf_empirical (list, option ...)
    cdf_empirical (matrix, option ...)

Empirical distribution function F(x).

Data can be introduced as a list of numbers, or as a one column matrix.

The optional argument is the name of the variable in the returned expression, which is x by default.

Example:

Empirical distribution function.

(%i1) load ("descriptive")$
(%i2) F(x):= ''(cdf_empirical([1,3,3,5,7,7,7,8,9]));
(%o2) F(x) := (charfun(x >= 9) + charfun(x >= 8)
               + 3 charfun(x >= 7) + charfun(x >= 5)
               + 2 charfun(x >= 3) + charfun(x >= 1))/9
(%i3) F(6);
                           4
(%o3)                      -
                           9
(%i4) load(draw)$
(%i5) draw2d(
        line_width = 3,
        grid       = true,
        explicit(F(z), z, -2, 12)) $

Function: cov (matrix)

The covariance matrix of the multivariate sample, defined as

              n
             ====
          1  \           _        _
      S = -   >    (X  - X) (X  - X)'
          n  /       j        j
             ====
             j = 1

where X_j is the j-th row of the sample matrix.

Example:

(%i1) load ("descriptive")$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) fpprintprec : 7$  /* change precision for pretty output */
(%i4) cov (s2);
      [ 17.22191  13.61811  14.37217  19.39624  15.42162 ]
      [                                                  ]
      [ 13.61811  14.98774  13.30448  15.15834  14.9711  ]
      [                                                  ]
(%o4) [ 14.37217  13.30448  15.47573  17.32544  16.18171 ]
      [                                                  ]
      [ 19.39624  15.15834  17.32544  32.17651  20.44685 ]
      [                                                  ]
      [ 15.42162  14.9711   16.18171  20.44685  24.42308 ]

See also function cov1.

Function: cov1 (matrix)

The covariance matrix of the multivariate sample, defined as

              n
             ====
         1   \           _        _
   S  = ---   >    (X  - X) (X  - X)'
    1   n-1  /       j        j
             ====
             j = 1

where X_j is the j-th row of the sample matrix.

Example:

(%i1) load ("descriptive")$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) fpprintprec : 7$ /* change precision for pretty output */
(%i4) cov1 (s2);
      [ 17.39587  13.75567  14.51734  19.59216  15.5774  ]
      [                                                  ]
      [ 13.75567  15.13913  13.43887  15.31145  15.12232 ]
      [                                                  ]
(%o4) [ 14.51734  13.43887  15.63205  17.50044  16.34516 ]
      [                                                  ]
      [ 19.59216  15.31145  17.50044  32.50153  20.65338 ]
      [                                                  ]
      [ 15.5774   15.12232  16.34516  20.65338  24.66977 ]

See also function cov.

Function: global_variances
    global_variances (matrix)
    global_variances (matrix, options ...)

Function global_variances returns a list of global variance measures:

where p is the dimension of the multivariate random variable and S_1 the covariance matrix returned by cov1.

Option:

Example:

(%i1) load ("descriptive")$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) global_variances (s2);
(%o3) [105.338342060606, 21.06766841212119, 12874.34690469686, 
         113.4651792608501, 6.636590811800795, 2.576158149609762]

Calculate the global_variances from the covariance matrix.

(%i1) load ("descriptive")$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) s : cov1 (s2)$
(%i4) global_variances (s, data=false);
(%o4) [105.338342060606, 21.06766841212119, 12874.34690469686, 
         113.4651792608501, 6.636590811800795, 2.576158149609762]

See also cov and cov1.

Function: cor
    cor (matrix)
    cor (matrix, logical_value)

The correlation matrix of the multivariate sample.

Option:

Example:

(%i1) load ("descriptive")$
(%i2) fpprintprec : 7 $
(%i3) s2 : read_matrix (file_search ("wind.data"))$
(%i4) cor (s2);
      [   1.0     .8476339  .8803515  .8239624  .7519506 ]
      [                                                  ]
      [ .8476339    1.0     .8735834  .6902622  0.782502 ]
      [                                                  ]
(%o4) [ .8803515  .8735834    1.0     .7764065  .8323358 ]
      [                                                  ]
      [ .8239624  .6902622  .7764065    1.0     .7293848 ]
      [                                                  ]
      [ .7519506  0.782502  .8323358  .7293848    1.0    ]

Calculate de correlation matrix from the covariance matrix.

(%i1) load ("descriptive")$
(%i2) fpprintprec : 7 $
(%i3) s2 : read_matrix (file_search ("wind.data"))$
(%i4) s : cov1 (s2)$
(%i5) cor (s, data=false); /* this is faster */
      [   1.0     .8476339  .8803515  .8239624  .7519506 ]
      [                                                  ]
      [ .8476339    1.0     .8735834  .6902622  0.782502 ]
      [                                                  ]
(%o5) [ .8803515  .8735834    1.0     .7764065  .8323358 ]
      [                                                  ]
      [ .8239624  .6902622  .7764065    1.0     .7293848 ]
      [                                                  ]
      [ .7519506  0.782502  .8323358  .7293848    1.0    ]

See also cov and cov1.

Function: list_correlations
    list_correlations (matrix)
    list_correlations (matrix, options ...)

Function list_correlations returns a list of correlation measures:

Option:

Example:

(%i1) load ("descriptive")$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) z : list_correlations (s2)$
(%i4) fpprintprec : 5$ /* for pretty output */
(%i5) z[1];  /* precision matrix */
      [  .38486   - .13856   - .15626   - .10239    .031179  ]
      [                                                      ]
      [ - .13856   .34107    - .15233    .038447   - .052842 ]
      [                                                      ]
(%o5) [ - .15626  - .15233    .47296    - .024816  - .10054  ]
      [                                                      ]
      [ - .10239   .038447   - .024816   .10937    - .034033 ]
      [                                                      ]
      [ .031179   - .052842  - .10054   - .034033   .14834   ]
(%i6) z[2];  /* multiple correlation vector */
(%o6)      [.85063, .80634, .86474, .71867, .72675]
(%i7) z[3];  /* partial correlation matrix */
      [  - 1.0     .38244   .36627   .49908   - .13049 ]
      [                                                ]
      [  .38244    - 1.0    .37927  - .19907   .23492  ]
      [                                                ]
(%o7) [  .36627    .37927   - 1.0    .10911    .37956  ]
      [                                                ]
      [  .49908   - .19907  .10911   - 1.0     .26719  ]
      [                                                ]
      [ - .13049   .23492   .37956   .26719    - 1.0   ]

See also cov and cov1.

Function: principal_components
    principal_components (matrix)
    principal_components (matrix, options ...)

Calculates the principal componentes of a multivariate sample. Principal components are used in multivariate statistical analysis to reduce the dimensionality of the sample.

Option:

The output of function principal_components is a list with the following results:

Examples:

In this sample, the first component explains 83.13 per cent of total variance.

(%i1) load ("descriptive")$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) fpprintprec:4 $
(%i4) res: principal_components(s2);
0 errors, 0 warnings
(%o4) [[87.57, 8.753, 5.515, 1.889, 1.613], 
[83.13, 8.31, 5.235, 1.793, 1.531], 
[ .4149  .03379   - .4757  - 0.581   - .5126 ]
[                                            ]
[ 0.369  - .3657  - .4298   .7237    - .1469 ]
[                                            ]
[ .3959  - .2178  - .2181  - .2749    .8201  ]]
[                                            ]
[ .5548   .7744    .1857    .2319    .06498  ]
[                                            ]
[ .4765  - .4669   0.712   - .09605  - .1969 ]
(%i5) /* accumulated percentages  */
    block([ap: copy(res[2])],
      for k:2 thru length(ap) do ap[k]: ap[k]+ap[k-1],
      ap);
(%o5)                 [83.13, 91.44, 96.68, 98.47, 100.0]
(%i6) /* sample dimension */
      p: length(first(res));
(%o6)                                  5
(%i7) /* plot percentages to select number of
         principal components for further work */
     draw2d(
        fill_density = 0.2,
        apply(bars, makelist([k, res[2][k], 1/2], k, p)),
        points_joined = true,
        point_type    = filled_circle,
        point_size    = 3,
        points(makelist([k, res[2][k]], k, p)),
        xlabel = "Variances",
        ylabel = "Percentages",
        xtics  = setify(makelist([concat("PC",k),k], k, p))) $

In case de covariance matrix is known, it can be passed to the function, but option data=false must be used.

(%i1) load ("descriptive")$
(%i2) S: matrix([1,-2,0],[-2,5,0],[0,0,2]);
                                [  1   - 2  0 ]
                                [             ]
(%o2)                           [ - 2   5   0 ]
                                [             ]
                                [  0    0   2 ]
(%i3) fpprintprec:4 $
(%i4) /* the argumment is a covariance matrix */
      res: principal_components(S, data=false);
0 errors, 0 warnings
                                                  [ - .3827  0.0  .9239 ]
                                                  [                     ]
(%o4) [[5.828, 2.0, .1716], [72.86, 25.0, 2.145], [  .9239   0.0  .3827 ]]
                                                  [                     ]
                                                  [   0.0    1.0   0.0  ]
(%i5) /* transformation to get the principal components
         from original records */
      matrix([a1,b2,c3],[a2,b2,c2]).last(res);
             [ .9239 b2 - .3827 a1  1.0 c3  .3827 b2 + .9239 a1 ]
(%o5)        [                                                  ]
             [ .9239 b2 - .3827 a2  1.0 c2  .3827 b2 + .9239 a2 ]


Previous: , Up: descriptive-pkg   [Contents][Index]

50.4 Functions and Variables for statistical graphs

Function: barsplot (data1, data2, …, option_1, option_2, …)

Plots bars diagrams for discrete statistical variables, both for one or multiple samples.

data can be a list of outcomes representing one sample, or a matrix of m rows and n columns, representing n samples of size m each.

Available options are:

There is also a function wxbarsplot for creating embedded histograms in interfaces wxMaxima and iMaxima. barsplot in a multiplot context.

Examples:

Univariate sample in matrix form. Absolute frequencies.

(%i1) load ("descriptive")$
(%i2) m : read_matrix (file_search ("biomed.data"))$
(%i3) barsplot(
        col(m,2),
        title        = "Ages",
        xlabel       = "years",
        box_width    = 1/2,
        fill_density = 3/4)$

Two samples of different sizes, with relative frequencies and user declared colors.

(%i1) load ("descriptive")$
(%i2) l1:makelist(random(10),k,1,50)$
(%i3) l2:makelist(random(10),k,1,100)$
(%i4) barsplot(
        l1,l2,
        box_width    = 1,
        fill_density = 1,
        bars_colors  = [black, grey],
        frequency = relative,
        sample_keys = ["A", "B"])$

Four non numeric samples of equal size.

(%i1) load ("descriptive")$
(%i2) barsplot(
        makelist([Yes, No, Maybe][random(3)+1],k,1,50),
        makelist([Yes, No, Maybe][random(3)+1],k,1,50),
        makelist([Yes, No, Maybe][random(3)+1],k,1,50),
        makelist([Yes, No, Maybe][random(3)+1],k,1,50),
        title  = "Asking for something to four groups",
        ylabel = "# of individuals",
        groups_gap   = 3,
        fill_density = 0.5,
        ordering     = ordergreatp)$

Stacked bars.

(%i1) load ("descriptive")$
(%i2) barsplot(
        makelist([Yes, No, Maybe][random(3)+1],k,1,50),
        makelist([Yes, No, Maybe][random(3)+1],k,1,50),
        makelist([Yes, No, Maybe][random(3)+1],k,1,50),
        makelist([Yes, No, Maybe][random(3)+1],k,1,50),
        title  = "Asking for something to four groups",
        ylabel = "# of individuals",
        grouping     = stacked,
        fill_density = 0.5,
        ordering     = ordergreatp)$

For bars diagrams related options, see barsplot of package draw-pkg See also functions histogram and piechart.

Function: barsplot_description (…)

Function barsplot_description creates a graphic object suitable for creating complex scenes, together with other graphic objects.

Example: barsplot in a multiplot context.

(%i1) load ("descriptive")$
(%i2) l1:makelist(random(10),k,1,50)$
(%i3) l2:makelist(random(10),k,1,100)$
(%i4) bp1 : 
        barsplot_description(
         l1,
         box_width = 1,
         fill_density = 0.5,
         bars_colors = [blue],
         frequency = relative)$
(%i5) bp2 : 
        barsplot_description(
         l2,
         box_width = 1,
         fill_density = 0.5,
         bars_colors = [red],
         frequency = relative)$
(%i6) draw(gr2d(bp1), gr2d(bp2))$

Function: boxplot (data)
    boxplot (data, option_1, option_2, …)

This function plots box-and-whisker diagrams. Argument data can be a list, which is not of great interest, since these diagrams are mainly used for comparing different samples, or a matrix, so it is possible to compare two or more components of a multivariate statistical variable. But it is also allowed data to be a list of samples with possible different sample sizes, in fact this is the only function in package descriptive that admits this type of data structure.

The box is plotted from the first quartile to the third, with an horizontal segment situated at the second quartile or median. By default, lower and upper whiskers are plotted at the minimum and maximum values, respectively. Option range can be used to indicate that values greater than quantile(x,3/4)+range*(quantile(x,3/4)-quantile(x,1/4)) or less than quantile(x,1/4)-range*(quantile(x,3/4)-quantile(x,1/4)) must be considered as outliers, in which case they are plotted as isolated points, and the whiskers are located at the extremes of the rest of the sample.

Available options are:

There is also a function wxboxplot for creating embedded histograms in interfaces wxMaxima and iMaxima.

Examples:

Box-and-whisker diagram from a multivariate sample.

(%i1) load ("descriptive")$
(%i2) s2 : read_matrix(file_search("wind.data"))$
(%i3) boxplot(s2,
        box_width  = 0.2,
        title      = "Windspeed in knots",
        xlabel     = "Stations",
        color      = red,
        line_width = 2)$

Box-and-whisker diagram from three samples of different sizes.

(%i1) load ("descriptive")$
(%i2) A :
       [[6, 4, 6, 2, 4, 8, 6, 4, 6, 4, 3, 2],
        [8, 10, 7, 9, 12, 8, 10],
        [16, 13, 17, 12, 11, 18, 13, 18, 14, 12]]$
(%i3) boxplot (A, box_orientation = horizontal)$

Option range can be used to handle outliers.

(%i1) load ("descriptive")$
(%i2) B: [[7, 15, 5, 8, 6, 5, 7, 3, 1],
          [10, 8, 12, 8, 11, 9, 20],
          [23, 17, 19, 7, 22, 19]] $
(%i3) boxplot (B, range=1)$
(%i4) boxplot (B, range=1.5, box_orientation = horizontal)$
(%i5) draw2d(
        boxplot_description(
          B,
          range            = 1.5,
          line_width       = 3,
          outliers_size    = 2,
          color            = red,
          background_color = light_gray),
        xtics = {["Low",1],["Medium",2],["High",3]}) $

Function: boxplot_description (…)

Function boxplot_description creates a graphic object suitable for creating complex scenes, together with other graphic objects.

Function: histogram
    histogram (list)
    histogram (list, option_1, option_2, …)
    histogram (one_column_matrix)
    histogram (one_column_matrix, option_1, option_2, …)
    histogram (one_row_matrix)
    histogram (one_row_matrix, option_1, option_2, …)

This function plots an histogram from a continuous sample. Sample data must be stored in a list of numbers or a one dimensional matrix.

Available options are:

There is also a function wxhistogram for creating embedded histograms in interfaces wxMaxima and iMaxima.

Examples:

A simple with eight classes:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) histogram (
           s1,
           nclasses     = 8,
           title        = "pi digits",
           xlabel       = "digits",
           ylabel       = "Absolute frequency",
           fill_color   = grey,
           fill_density = 0.6)$

Setting the limits of the histogram to -2 and 12, with 3 classes. Also, we introduce predefined tics:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) histogram (
           s1,
           nclasses     = [-2,12,3],
           htics        = ["A", "B", "C"],
           terminal     = png,
           fill_color   = "#23afa0",
           fill_density = 0.6)$

Bounds for varying bin widths.

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) histogram (s1, nclasses = {0,3,6,7,11})$

Freedmann - Diakonis robust method for optimal search of the number of classes.

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) histogram(s1, nclasses=fd) $

Function: histogram_description (…)

Function histogram_description creates a graphic object suitable for creating complex scenes, together with other graphic objects. We make use of histogram_description for setting the xrange and adding an explicit curve into the scene:

(%i1) load ("descriptive")$
(%i2) ( load("distrib"),
        m: 14, s: 2,
        s2: random_normal(m, s, 1000) ) $
(%i3) draw2d(
        grid   = true,
        xrange = [5, 25],
        histogram_description(
          s2,
          nclasses     = 9,
          frequency    = density,
          fill_density = 0.5),
        explicit(pdf_normal(x,m,s), x, m - 3*s, m + 3* s))$

Function: piechart
    piechart (list)
    piechart (list, option_1, option_2, …)
    piechart (one_column_matrix)
    piechart (one_column_matrix, option_1, option_2, …)
    piechart (one_row_matrix)
    piechart (one_row_matrix, option_1, option_2, …)

Similar to barsplot, but plots sectors instead of rectangles.

Available options are:

There is also a function wxpiechart for creating embedded histograms in interfaces wxMaxima and iMaxima.

Example:

(%i1) load ("descriptive")$
(%i2) s1 : read_list (file_search ("pidigits.data"))$
(%i3) piechart(
        s1,
        xrange  = [-1.1, 1.3],
        yrange  = [-1.1, 1.1],
        title   = "Digit frequencies in pi")$

See also function barsplot.

Function: piechart_description (…)

Function piechart_description creates a graphic object suitable for creating complex scenes, together with other graphic objects.

Function: scatterplot
    scatterplot (list)
    scatterplot (list, option_1, option_2, …)
    scatterplot (matrix)
    scatterplot (matrix, option_1, option_2, …)

Plots scatter diagrams both for univariate (list) and multivariate (matrix) samples.

Available options are the same admitted by histogram.

There is also a function wxscatterplot for creating embedded histograms in interfaces wxMaxima and iMaxima.

Examples:

Univariate scatter diagram from a simulated Gaussian sample.

(%i1) load ("descriptive")$
(%i2) load ("distrib")$
(%i3) scatterplot(
        random_normal(0,1,200),
        xaxis      = true,
        point_size = 2,
        dimensions = [600,150])$

Two dimensional scatter plot.

(%i1) load ("descriptive")$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) scatterplot(
       submatrix(s2, 1,2,3),
       title      = "Data from stations #4 and #5",
       point_type = diamant,
       point_size = 2,
       color      = blue)$

Three dimensional scatter plot.

(%i1) load ("descriptive")$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) scatterplot(submatrix (s2, 1,2), nclasses=4)$

Five dimensional scatter plot, with five classes histograms.

(%i1) load ("descriptive")$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) scatterplot(
        s2,
        nclasses     = 5,
        frequency    = relative,
        fill_color   = blue,
        fill_density = 0.3,
        xtics        = 5)$

For plotting isolated or line-joined points in two and three dimensions, see points. See also histogram.

Function: scatterplot_description (…)

Function scatterplot_description creates a graphic object suitable for creating complex scenes, together with other graphic objects.

Function: starplot (data1, data2, …, option_1, option_2, …)

Plots star diagrams for discrete statistical variables, both for one or multiple samples.

data can be a list of outcomes representing one sample, or a matrix of m rows and n columns, representing n samples of size m each.

Available options are:

There is also a function wxstarplot for creating embedded histograms in interfaces wxMaxima and iMaxima.

Example:

Plot based on absolute frequencies. Location and radius defined by the user.

(%i1) load ("descriptive")$
(%i2) l1: makelist(random(10),k,1,50)$
(%i3) l2: makelist(random(10),k,1,200)$
(%i4) starplot(
        l1, l2,
        stars_colors = [blue,red],
        sample_keys = ["1st sample", "2nd sample"],
        star_center = [1,2],
        star_radius = 4,
        proportional_axes = xy,
        line_width = 2 ) $ 

Function: starplot_description (…)

Function starplot_description creates a graphic object suitable for creating complex scenes, together with other graphic objects.

Function: stemplot
    stemplot (data)
    stemplot (data, option)

Plots stem and leaf diagrams.

Unique available option is:

Example:

(%i1) load ("descriptive")$
(%i2) load(distrib)$
(%i3) stemplot(
        random_normal(15, 6, 100),
        leaf_unit = 0.1);
-5|4
 0|37
 1|7
 3|6
 4|4
 5|4
 6|57
 7|0149
 8|3
 9|1334588
10|07888
11|01144467789
12|12566889
13|24778
14|047
15|223458
16|4
17|11557
18|000247
19|4467799
20|00
21|1
22|2335
23|01457
24|12356
25|455
27|79
key: 6|3 =  6.3
(%o3)                  done


Next: , Previous: , Up: Top   [Contents][Index]

51 diag


Previous: , Up: diag-pkg   [Contents][Index]

51.1 Functions and Variables for diag

Function: diag (lm)

Constructs a matrix that is the block sum of the elements of lm. The elements of lm are assumed to be matrices; if an element is scalar, it treated as a 1 by 1 matrix.

The resulting matrix will be square if each of the elements of lm is square.

Example:

(%i1) load("diag")$

(%i2) a1:matrix([1,2,3],[0,4,5],[0,0,6])$

(%i3) a2:matrix([1,1],[1,0])$

(%i4) diag([a1,x,a2]);
                   [ 1  2  3  0  0  0 ]
                   [                  ]
                   [ 0  4  5  0  0  0 ]
                   [                  ]
                   [ 0  0  6  0  0  0 ]
(%o4)              [                  ]
                   [ 0  0  0  x  0  0 ]
                   [                  ]
                   [ 0  0  0  0  1  1 ]
                   [                  ]
                   [ 0  0  0  0  1  0 ]
(%i5) diag ([matrix([1,2]), 3]);
                        [ 1  2  0 ]
(%o5)                   [         ]
                        [ 0  0  3 ]

To use this function write first load("diag").

Function: JF (lambda,n)

Returns the Jordan cell of order n with eigenvalue lambda.

Example:

(%i1) load("diag")$

(%i2) JF(2,5);
                    [ 2  1  0  0  0 ]
                    [               ]
                    [ 0  2  1  0  0 ]
                    [               ]
(%o2)               [ 0  0  2  1  0 ]
                    [               ]
                    [ 0  0  0  2  1 ]
                    [               ]
                    [ 0  0  0  0  2 ]
(%i3) JF(3,2);
                         [ 3  1 ]
(%o3)                    [      ]
                         [ 0  3 ]

To use this function write first load("diag").

Function: jordan (mat)

Returns the Jordan form of matrix mat, encoded as a list in a particular format. To get the corresponding matrix, call the function dispJordan using the output of jordan as the argument.

The elements of the returned list are themselves lists. The first element of each is an eigenvalue of mat. The remaining elements are positive integers which are the lengths of the Jordan blocks for this eigenvalue. These integers are listed in decreasing order. Eigenvalues are not repeated.

The functions dispJordan, minimalPoly and ModeMatrix expect the output of a call to jordan as an argument. If you construct this argument by hand, rather than by calling jordan, you must ensure that each eigenvalue only appears once and that the block sizes are listed in decreasing order, otherwise the functions might give incorrect answers.

Example:

(%i1) load("diag")$
(%i2) A: matrix([2,0,0,0,0,0,0,0],
                [1,2,0,0,0,0,0,0],
                [-4,1,2,0,0,0,0,0],
                [2,0,0,2,0,0,0,0],
                [-7,2,0,0,2,0,0,0],
                [9,0,-2,0,1,2,0,0],
                [-34,7,1,-2,-1,1,2,0],
                [145,-17,-16,3,9,-2,0,3])$
(%i3) jordan (A);
(%o3)                [[2, 3, 3, 1], [3, 1]]
(%i4) dispJordan (%);
                   [ 2  1  0  0  0  0  0  0 ]
                   [                        ]
                   [ 0  2  1  0  0  0  0  0 ]
                   [                        ]
                   [ 0  0  2  0  0  0  0  0 ]
                   [                        ]
                   [ 0  0  0  2  1  0  0  0 ]
(%o4)              [                        ]
                   [ 0  0  0  0  2  1  0  0 ]
                   [                        ]
                   [ 0  0  0  0  0  2  0  0 ]
                   [                        ]
                   [ 0  0  0  0  0  0  2  0 ]
                   [                        ]
                   [ 0  0  0  0  0  0  0  3 ]

To use this function write first load("diag"). See also dispJordan and minimalPoly.

Function: dispJordan (l)

Returns a matrix in Jordan canonical form (JCF) corresponding to the list of eigenvalues and multiplicities given by l. This list should be in the format given by the jordan function. See jordan for details of this format.

Example:

(%i1) load("diag")$

(%i2) b1:matrix([0,0,1,1,1],
                [0,0,0,1,1],
                [0,0,0,0,1],
                [0,0,0,0,0],
                [0,0,0,0,0])$

(%i3) jordan(b1);
(%o3)                  [[0, 3, 2]]
(%i4) dispJordan(%);
                    [ 0  1  0  0  0 ]
                    [               ]
                    [ 0  0  1  0  0 ]
                    [               ]
(%o4)               [ 0  0  0  0  0 ]
                    [               ]
                    [ 0  0  0  0  1 ]
                    [               ]
                    [ 0  0  0  0  0 ]

To use this function write first load("diag"). See also jordan and minimalPoly.

Function: minimalPoly (l)

Returns the minimal polynomial of the matrix whose Jordan form is described by the list l. This list should be in the format given by the jordan function. See jordan for details of this format.

Example:

(%i1) load("diag")$

(%i2) a:matrix([2,1,2,0],
               [-2,2,1,2],
               [-2,-1,-1,1],
               [3,1,2,-1])$

(%i3) jordan(a);
(%o3)               [[- 1, 1], [1, 3]]
(%i4) minimalPoly(%);
                            3
(%o4)                (x - 1)  (x + 1)

To use this function write first load("diag"). See also jordan and dispJordan.

Function: ModeMatrix (A, [jordan_info])

Returns an invertible matrix M such that (M^^-1).A.M is the Jordan form of A.

To calculate this, Maxima must find the Jordan form of A, which might be quite computationally expensive. If that has already been calculated by a previous call to jordan, pass it as a second argument, jordan_info. See jordan for details of the required format.

Example:

(%i1) load("diag")$
(%i2) A: matrix([2,1,2,0], [-2,2,1,2], [-2,-1,-1,1], [3,1,2,-1])$
(%i3) M: ModeMatrix (A);
                      [  1    - 1   1   1 ]
                      [                   ]
                      [   1               ]
                      [ - -   - 1   0   0 ]
                      [   9               ]
                      [                   ]
(%o3)                 [   13              ]
                      [ - --   1   - 1  0 ]
                      [   9               ]
                      [                   ]
                      [  17               ]
                      [  --   - 1   1   1 ]
                      [  9                ]
(%i4) is ((M^^-1) . A . M = dispJordan (jordan (A)));
(%o4)                         true

Note that, in this example, the Jordan form of A is computed twice. To avoid this, we could have stored the output of jordan(A) in a variable and passed that to both ModeMatrix and dispJordan.

To use this function write first load("diag"). See also jordan and dispJordan.

Function: mat_function (f,A)

Returns f(A), where f is an analytic function and A a matrix. This computation is based on the Taylor expansion of f. It is not efficient for numerical evaluation, but can give symbolic answers for small matrices.

Example 1:

The exponential of a matrix. We only give the first row of the answer, since the output is rather large.

(%i1) load("diag")$
(%i2) A: matrix ([0,1,0], [0,0,1], [-1,-3,-3])$
(%i3) ratsimp (mat_function (exp, t*A)[1]);
           2              - t                   2   - t
         (t  + 2 t + 2) %e       2        - t  t  %e
(%o3)   [--------------------, (t  + t) %e   , --------]
                  2                               2

Example 2:

Comparison with the Taylor series for the exponential and also comparing exp(%i*A) with sine and cosine.

(%i1) load("diag")$
(%i2) A: matrix ([0,1,1,1],
                 [0,0,0,1],
                 [0,0,0,1],
                 [0,0,0,0])$
(%i3) ratsimp (mat_function (exp, t*A));
                       [           2     ]
                       [ 1  t  t  t  + t ]
                       [                 ]
(%o3)                  [ 0  1  0    t    ]
                       [                 ]
                       [ 0  0  1    t    ]
                       [                 ]
                       [ 0  0  0    1    ]
(%i4) minimalPoly (jordan (A));
                                3
(%o4)                          x
(%i5) ratsimp (ident(4) + t*A + 1/2*(t^2)*A^^2);
                       [           2     ]
                       [ 1  t  t  t  + t ]
                       [                 ]
(%o5)                  [ 0  1  0    t    ]
                       [                 ]
                       [ 0  0  1    t    ]
                       [                 ]
                       [ 0  0  0    1    ]
(%i6) ratsimp (mat_function (exp, %i*t*A));
                  [                        2 ]
                  [ 1  %i t  %i t  %i t - t  ]
                  [                          ]
(%o6)             [ 0   1     0      %i t    ]
                  [                          ]
                  [ 0   0     1      %i t    ]
                  [                          ]
                  [ 0   0     0        1     ]
(%i7) ratsimp (mat_function (cos, t*A) + %i*mat_function (sin, t*A));
                  [                        2 ]
                  [ 1  %i t  %i t  %i t - t  ]
                  [                          ]
(%o7)             [ 0   1     0      %i t    ]
                  [                          ]
                  [ 0   0     1      %i t    ]
                  [                          ]
                  [ 0   0     0        1     ]

Example 3:

Power operations.

(%i1) load("diag")$
(%i2) A: matrix([1,2,0], [0,1,0], [1,0,1])$
(%i3) integer_pow(x) := block ([k], declare (k, integer), x^k)$
(%i4) mat_function (integer_pow, A);
                       [ 1     2 k     0 ]
                       [                 ]
(%o4)                  [ 0      1      0 ]
                       [                 ]
                       [ k  (k - 1) k  1 ]
(%i5) A^^20;
                         [ 1   40   0 ]
                         [            ]
(%o5)                    [ 0    1   0 ]
                         [            ]
                         [ 20  380  1 ]

To use this function write first load("diag").


Next: , Previous: , Up: Top   [Contents][Index]

52 distrib


Next: , Previous: , Up: distrib-pkg   [Contents][Index]

52.1 Introduction to distrib

Package distrib contains a set of functions for making probability computations on both discrete and continuous univariate models.

What follows is a short reminder of basic probabilistic related definitions.

Let f(x) be the density function of an absolute continuous random variable X. The distribution function is defined as

                       x
                      /
                      [
               F(x) = I     f(u) du
                      ]
                      /
                       minf

which equals the probability Pr(X <= x).

The mean value is a localization parameter and is defined as

                     inf
                    /
                    [
           E[X]  =  I   x f(x) dx
                    ]
                    /
                     minf

The variance is a measure of variation,

                 inf
                /
                [                    2
         V[X] = I     f(x) (x - E[X])  dx
                ]
                /
                 minf

which is a positive real number. The square root of the variance is the standard deviation, D[X]=sqrt(V[X]), and it is another measure of variation.

The skewness coefficient is a measure of non-symmetry,

                 inf
                /
            1   [                    3
  SK[X] = ----- I     f(x) (x - E[X])  dx
              3 ]
          D[X]  /
                 minf

And the kurtosis coefficient measures the peakedness of the distribution,

                 inf
                /
            1   [                    4
  KU[X] = ----- I     f(x) (x - E[X])  dx - 3
              4 ]
          D[X]  /
                 minf

If X is gaussian, KU[X]=0. In fact, both skewness and kurtosis are shape parameters used to measure the non–gaussianity of a distribution.

If the random variable X is discrete, the density, or probability, function f(x) takes positive values within certain countable set of numbers x_i, and zero elsewhere. In this case, the distribution function is

                       ====
                       \
                F(x) =  >    f(x )
                       /        i
                       ====
                      x <= x
                       i

The mean, variance, standard deviation, skewness coefficient and kurtosis coefficient take the form

                       ====
                       \
                E[X] =  >  x  f(x ) ,
                       /    i    i
                       ====
                        x 
                         i
                ====
                \                     2
        V[X] =   >    f(x ) (x - E[X])  ,
                /        i    i
                ====
                 x
                  i
               D[X] = sqrt(V[X]),
                     ====
              1      \                     3
  SK[X] =  -------    >    f(x ) (x - E[X])  
           D[X]^3    /        i    i
                     ====
                      x
                       i

and

                     ====
              1      \                     4
  KU[X] =  -------    >    f(x ) (x - E[X])   - 3 ,
           D[X]^4    /        i    i
                     ====
                      x
                       i

respectively.

There is a naming convention in package distrib. Every function name has two parts, the first one makes reference to the function or parameter we want to calculate,

Functions:
   Density function            (pdf_*)
   Distribution function       (cdf_*)
   Quantile                    (quantile_*)
   Mean                        (mean_*)
   Variance                    (var_*)
   Standard deviation          (std_*)
   Skewness coefficient        (skewness_*)
   Kurtosis coefficient        (kurtosis_*)
   Random variate              (random_*)

The second part is an explicit reference to the probabilistic model,

Continuous distributions:
   Normal              (*normal)
   Student             (*student_t)
   Chi^2               (*chi2)
   Noncentral Chi^2    (*noncentral_chi2)
   F                   (*f)
   Exponential         (*exp)
   Lognormal           (*lognormal)
   Gamma               (*gamma)
   Beta                (*beta)
   Continuous uniform  (*continuous_uniform)
   Logistic            (*logistic)
   Pareto              (*pareto)
   Weibull             (*weibull)
   Rayleigh            (*rayleigh)
   Laplace             (*laplace)
   Cauchy              (*cauchy)
   Gumbel              (*gumbel)

Discrete distributions:
   Binomial             (*binomial)
   Poisson              (*poisson)
   Bernoulli            (*bernoulli)
   Geometric            (*geometric)
   Discrete uniform     (*discrete_uniform)
   hypergeometric       (*hypergeometric)
   Negative binomial    (*negative_binomial)
   Finite discrete      (*general_finite_discrete)

For example, pdf_student_t(x,n) is the density function of the Student distribution with n degrees of freedom, std_pareto(a,b) is the standard deviation of the Pareto distribution with parameters a and b and kurtosis_poisson(m) is the kurtosis coefficient of the Poisson distribution with mean m.

In order to make use of package distrib you need first to load it by typing

(%i1) load("distrib")$

For comments, bugs or suggestions, please contact the author at ’riotorto AT yahoo DOT com’.

·

Category: Statistical functions · Category: Share packages · Category: Package distrib


Next: , Previous: , Up: distrib-pkg   [Contents][Index]

52.2 Functions and Variables for continuous distributions

Function: pdf_normal (x,m,s)

Returns the value at x of the density function of a Normal(m,s) random variable, with s>0. To make use of this function, write first load("distrib").

Function: cdf_normal (x,m,s)

Returns the value at x of the distribution function of a Normal(m,s) random variable, with s>0. This function is defined in terms of Maxima’s built-in error function erf.

(%i1) load ("distrib")$
(%i2) cdf_normal(x,m,s);
                                    x - m
                              erf(---------)
                                  sqrt(2) s    1
(%o2)                         -------------- + -
                                    2          2

See also erf.

Function: quantile_normal (q,m,s)

Returns the q-quantile of a Normal(m,s) random variable, with s>0; in other words, this is the inverse of cdf_normal. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").

(%i1) load ("distrib")$
(%i2) quantile_normal(95/100,0,1);
                                      9
(%o2)             sqrt(2) inverse_erf(--)
                                      10
(%i3) float(%);
(%o3)               1.644853626951472

Function: mean_normal (m,s)

Returns the mean of a Normal(m,s) random variable, with s>0, namely m. To make use of this function, write first load("distrib").

Function: var_normal (m,s)

Returns the variance of a Normal(m,s) random variable, with s>0, namely s^2. To make use of this function, write first load("distrib").

Function: std_normal (m,s)

Returns the standard deviation of a Normal(m,s) random variable, with s>0, namely s. To make use of this function, write first load("distrib").

Function: skewness_normal (m,s)

Returns the skewness coefficient of a Normal(m,s) random variable, with s>0, which is always equal to 0. To make use of this function, write first load("distrib").

Function: kurtosis_normal (m,s)

Returns the kurtosis coefficient of a Normal(m,s) random variable, with s>0, which is always equal to 0. To make use of this function, write first load("distrib").

Function: random_normal (m,s)
    random_normal (m,s,n)

Returns a Normal(m,s) random variate, with s>0. Calling random_normal with a third argument n, a random sample of size n will be simulated.

This is an implementation of the Box-Mueller algorithm, as described in Knuth, D.E. (1981) Seminumerical Algorithms. The Art of Computer Programming. Addison-Wesley.

To make use of this function, write first load("distrib").

Function: pdf_student_t (x,n)

Returns the value at x of the density function of a Student random variable t(n), with n>0 degrees of freedom. To make use of this function, write first load("distrib").

Function: cdf_student_t (x,n)

Returns the value at x of the distribution function of a Student random variable t(n), with n>0 degrees of freedom.

(%i1) load ("distrib")$
(%i2) cdf_student_t(1/2, 7/3);
                                         7  1  28
             beta_incomplete_regularized(-, -, --)
                                         6  2  31
(%o2)    1 - -------------------------------------
                               2
(%i3) float(%);
(%o3)                .6698450596140415

Function: quantile_student_t (q,n)

Returns the q-quantile of a Student random variable t(n), with n>0; in other words, this is the inverse of cdf_student_t. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").

Function: mean_student_t (n)

Returns the mean of a Student random variable t(n), with n>0, which is always equal to 0. To make use of this function, write first load("distrib").

Function: var_student_t (n)

Returns the variance of a Student random variable t(n), with n>2.

(%i1) load ("distrib")$
(%i2) var_student_t(n);
                                n
(%o2)                         -----
                              n - 2

Function: std_student_t (n)

Returns the standard deviation of a Student random variable t(n), with n>2. To make use of this function, write first load("distrib").

Function: skewness_student_t (n)

Returns the skewness coefficient of a Student random variable t(n), with n>3, which is always equal to 0. To make use of this function, write first load("distrib").

Function: kurtosis_student_t (n)

Returns the kurtosis coefficient of a Student random variable t(n), with n>4. To make use of this function, write first load("distrib").

Function: random_student_t (n)
    random_student_t (n,m)

Returns a Student random variate t(n), with n>0. Calling random_student_t with a second argument m, a random sample of size m will be simulated.

The implemented algorithm is based on the fact that if Z is a normal random variable N(0,1) and S^2 is a chi square random variable with n degrees of freedom, Chi^2(n), then

                           Z
                 X = -------------
                     /   2  \ 1/2
                     |  S   |
                     | ---  |
                     \  n   /

is a Student random variable with n degrees of freedom, t(n).

To make use of this function, write first load("distrib").

Function: pdf_noncentral_student_t (x,n,ncp)

Returns the value at x of the density function of a noncentral Student random variable nc_t(n,ncp), with n>0 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first load("distrib").

Sometimes an extra work is necessary to get the final result.

(%i1) load ("distrib")$
(%i2) expand(pdf_noncentral_student_t(3,5,0.1));
                           7/2                         7/2
      0.04296414417400905 5      1.323650307289301e-6 5
(%o2) ------------------------ + -------------------------
         3/2   5/2                       sqrt(%pi)
        2    14    sqrt(%pi)
                                                        7/2
                                   1.94793720435093e-4 5
                                 + ------------------------
                                             %pi
(%i3) float(%);
(%o3)          .02080593159405669

Function: cdf_noncentral_student_t (x,n,ncp)

Returns the value at x of the distribution function of a noncentral Student random variable nc_t(n,ncp), with n>0 degrees of freedom and noncentrality parameter ncp. This function has no closed form and it is numerically computed.

(%i1) load ("distrib")$
(%i2) cdf_noncentral_student_t(-2,5,-5);
(%o2)          .9952030093319743

Function: quantile_noncentral_student_t (q,n,ncp)

Returns the q-quantile of a noncentral Student random variable nc_t(n,ncp), with n>0 degrees of freedom and noncentrality parameter ncp; in other words, this is the inverse of cdf_noncentral_student_t. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").

Function: mean_noncentral_student_t (n,ncp)

Returns the mean of a noncentral Student random variable nc_t(n,ncp), with n>1 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first load("distrib").

(%i1) load ("distrib")$
(%i2) mean_noncentral_student_t(df,k);
                   df - 1
             gamma(------) sqrt(df) k
                     2
(%o2)        ------------------------
                              df
                sqrt(2) gamma(--)
                              2

Function: var_noncentral_student_t (n,ncp)

Returns the variance of a noncentral Student random variable nc_t(n,ncp), with n>2 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first load("distrib").

Function: std_noncentral_student_t (n,ncp)

Returns the standard deviation of a noncentral Student random variable nc_t(n,ncp), with n>2 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first load("distrib").

Function: skewness_noncentral_student_t (n,ncp)

Returns the skewness coefficient of a noncentral Student random variable nc_t(n,ncp), with n>3 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first load("distrib").

Function: kurtosis_noncentral_student_t (n,ncp)

Returns the kurtosis coefficient of a noncentral Student random variable nc_t(n,ncp), with n>4 degrees of freedom and noncentrality parameter ncp. To make use of this function, write first load("distrib").

Function: random_noncentral_student_t (n,ncp)
    random_noncentral_student_t (n,ncp,m)

Returns a noncentral Student random variate nc_t(n,ncp), with n>0. Calling random_noncentral_student_t with a third argument m, a random sample of size m will be simulated.

The implemented algorithm is based on the fact that if X is a normal random variable N(ncp,1) and S^2 is a chi square random variable with n degrees of freedom, Chi^2(n), then

                           X
                 U = -------------
                     /   2  \ 1/2
                     |  S   |
                     | ---  |
                     \  n   /

is a noncentral Student random variable with n degrees of freedom and noncentrality parameter ncp, nc_t(n,ncp).

To make use of this function, write first load("distrib").

Function: pdf_chi2 (x,n)

Returns the value at x of the density function of a Chi-square random variable Chi^2(n), with n>0. The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

(%i1) load ("distrib")$
(%i2) pdf_chi2(x,n);
                         n/2 - 1   - x/2
                        x        %e
(%o2)                   ----------------
                          n/2       n
                         2    gamma(-)
                                    2

Function: cdf_chi2 (x,n)

Returns the value at x of the distribution function of a Chi-square random variable Chi^2(n), with n>0.

(%i1) load ("distrib")$
(%i2) cdf_chi2(3,4);
                                               3
(%o2)      1 - gamma_incomplete_regularized(2, -)
                                               2
(%i3) float(%);
(%o3)               .4421745996289256

Function: quantile_chi2 (q,n)

Returns the q-quantile of a Chi-square random variable Chi^2(n), with n>0; in other words, this is the inverse of cdf_chi2. Argument q must be an element of [0,1].

This function has no closed form and it is numerically computed.

(%i1) load ("distrib")$
(%i2) quantile_chi2(0.99,9);
(%o2)                   21.66599433346194

Function: mean_chi2 (n)

Returns the mean of a Chi-square random variable Chi^2(n), with n>0.

The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

(%i1) load ("distrib")$
(%i2) mean_chi2(n);
(%o2)                           n

Function: var_chi2 (n)

Returns the variance of a Chi-square random variable Chi^2(n), with n>0.

The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

(%i1) load ("distrib")$
(%i2) var_chi2(n);
(%o2)                          2 n

Function: std_chi2 (n)

Returns the standard deviation of a Chi-square random variable Chi^2(n), with n>0.

The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

(%i1) load ("distrib")$
(%i2) std_chi2(n);
(%o2)                    sqrt(2) sqrt(n)

Function: skewness_chi2 (n)

Returns the skewness coefficient of a Chi-square random variable Chi^2(n), with n>0.

The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

(%i1) load ("distrib")$
(%i2) skewness_chi2(n);
                                     3/2
                                    2
(%o2)                              -------
                                   sqrt(n)

Function: kurtosis_chi2 (n)

Returns the kurtosis coefficient of a Chi-square random variable Chi^2(n), with n>0.

The Chi^2(n) random variable is equivalent to the Gamma(n/2,2).

(%i1) load ("distrib")$
(%i2) kurtosis_chi2(n);
                               12
(%o2)                          --
                               n

Function: random_chi2 (n)
    random_chi2 (n,m)

Returns a Chi-square random variate Chi^2(n), with n>0. Calling random_chi2 with a second argument m, a random sample of size m will be simulated.

The simulation is based on the Ahrens-Cheng algorithm. See random_gamma for details.

To make use of this function, write first load("distrib").

Function: pdf_noncentral_chi2 (x,n,ncp)

Returns the value at x of the density function of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0. To make use of this function, write first load("distrib").

Function: cdf_noncentral_chi2 (x,n,ncp)

Returns the value at x of the distribution function of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0. To make use of this function, write first load("distrib").

Function: quantile_noncentral_chi2 (q,n,ncp)

Returns the q-quantile of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0; in other words, this is the inverse of cdf_noncentral_chi2. Argument q must be an element of [0,1].

This function has no closed form and it is numerically computed.

Function: mean_noncentral_chi2 (n,ncp)

Returns the mean of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

Function: var_noncentral_chi2 (n,ncp)

Returns the variance of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

Function: std_noncentral_chi2 (n,ncp)

Returns the standard deviation of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

Function: skewness_noncentral_chi2 (n,ncp)

Returns the skewness coefficient of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

Function: kurtosis_noncentral_chi2 (n,ncp)

Returns the kurtosis coefficient of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.

Function: random_noncentral_chi2 (n,ncp)
    random_noncentral_chi2 (n,ncp,m)

Returns a noncentral Chi-square random variate nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0. Calling random_noncentral_chi2 with a third argument m, a random sample of size m will be simulated.

To make use of this function, write first load("distrib").

Function: pdf_f (x,m,n)

Returns the value at x of the density function of a F random variable F(m,n), with m,n>0. To make use of this function, write first load("distrib").

Function: cdf_f (x,m,n)

Returns the value at x of the distribution function of a F random variable F(m,n), with m,n>0.

(%i1) load ("distrib")$
(%i2) cdf_f(2,3,9/4);
                                         9  3  3
(%o2)    1 - beta_incomplete_regularized(-, -, --)
                                         8  2  11
(%i3) float(%);
(%o3)                 0.66756728179008

Function: quantile_f (q,m,n)

Returns the q-quantile of a F random variable F(m,n), with m,n>0; in other words, this is the inverse of cdf_f. Argument q must be an element of [0,1].

(%i1) load ("distrib")$
(%i2) quantile_f(2/5,sqrt(3),5);
(%o2)                   0.518947838573693

Function: mean_f (m,n)

Returns the mean of a F random variable F(m,n), with m>0, n>2. To make use of this function, write first load("distrib").

Function: var_f (m,n)

Returns the variance of a F random variable F(m,n), with m>0, n>4. To make use of this function, write first load("distrib").

Function: std_f (m,n)

Returns the standard deviation of a F random variable F(m,n), with m>0, n>4. To make use of this function, write first load("distrib").

Function: skewness_f (m,n)

Returns the skewness coefficient of a F random variable F(m,n), with m>0, n>6. To make use of this function, write first load("distrib").

Function: kurtosis_f (m,n)

Returns the kurtosis coefficient of a F random variable F(m,n), with m>0, n>8. To make use of this function, write first load("distrib").

Function: random_f (m,n)
    random_f (m,n,k)

Returns a F random variate F(m,n), with m,n>0. Calling random_f with a third argument k, a random sample of size k will be simulated.

The simulation algorithm is based on the fact that if X is a Chi^2(m) random variable and Y is a Chi^2(n) random variable, then

                        n X
                    F = ---
                        m Y

is a F random variable with m and n degrees of freedom, F(m,n).

To make use of this function, write first load("distrib").

Function: pdf_exp (x,m)

Returns the value at x of the density function of an Exponential(m) random variable, with m>0.

The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

(%i1) load ("distrib")$
(%i2) pdf_exp(x,m);
                                - m x
(%o2)                       m %e

Function: cdf_exp (x,m)

Returns the value at x of the distribution function of an Exponential(m) random variable, with m>0.

The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

(%i1) load ("distrib")$
(%i2) cdf_exp(x,m);
                                 - m x
(%o2)                      1 - %e

Function: quantile_exp (q,m)

Returns the q-quantile of an Exponential(m) random variable, with m>0; in other words, this is the inverse of cdf_exp. Argument q must be an element of [0,1].

The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

(%i1) load ("distrib")$
(%i2) quantile_exp(0.56,5);
(%o2)                   .1641961104139661
(%i3) quantile_exp(0.56,m);
                             0.8209805520698303
(%o3)                        ------------------
                                     m

Function: mean_exp (m)

Returns the mean of an Exponential(m) random variable, with m>0.

The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

(%i1) load ("distrib")$
(%i2) mean_exp(m);
                                1
(%o2)                           -
                                m

Function: var_exp (m)

Returns the variance of an Exponential(m) random variable, with m>0.

The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

(%i1) load ("distrib")$
(%i2) var_exp(m);
                               1
(%o2)                          --
                                2
                               m

Function: std_exp (m)

Returns the standard deviation of an Exponential(m) random variable, with m>0.

The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

(%i1) load ("distrib")$
(%i2) std_exp(m);
                                1
(%o2)                           -
                                m

Function: skewness_exp (m)

Returns the skewness coefficient of an Exponential(m) random variable, with m>0.

The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

(%i1) load ("distrib")$
(%i2) skewness_exp(m);
(%o2)                           2

Function: kurtosis_exp (m)

Returns the kurtosis coefficient of an Exponential(m) random variable, with m>0.

The Exponential(m) random variable is equivalent to the Weibull(1,1/m).

(%i1) load ("distrib")$
(%i2) kurtosis_exp(m);
(%o3)                           6

Function: random_exp (m)
    random_exp (m,k)

Returns an Exponential(m) random variate, with m>0. Calling random_exp with a second argument k, a random sample of size k will be simulated.

The simulation algorithm is based on the general inverse method.

To make use of this function, write first load("distrib").

Function: pdf_lognormal (x,m,s)

Returns the value at x of the density function of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load("distrib").

Function: cdf_lognormal (x,m,s)

Returns the value at x of the distribution function of a Lognormal(m,s) random variable, with s>0. This function is defined in terms of Maxima’s built-in error function erf.

(%i1) load ("distrib")$
(%i2) cdf_lognormal(x,m,s);
                           log(x) - m
                       erf(----------)
                           sqrt(2) s     1
(%o2)                  --------------- + -
                              2          2

See also erf.

Function: quantile_lognormal (q,m,s)

Returns the q-quantile of a Lognormal(m,s) random variable, with s>0; in other words, this is the inverse of cdf_lognormal. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").

(%i1) load ("distrib")$
(%i2) quantile_lognormal(95/100,0,1);
                  sqrt(2) inverse_erf(9/10)
(%o2)           %e
(%i3) float(%);
(%o3)               5.180251602233015

Function: mean_lognormal (m,s)

Returns the mean of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load("distrib").

Function: var_lognormal (m,s)

Returns the variance of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load("distrib").

Function: std_lognormal (m,s)

Returns the standard deviation of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load("distrib").

Function: skewness_lognormal (m,s)

Returns the skewness coefficient of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load("distrib").

Function: kurtosis_lognormal (m,s)

Returns the kurtosis coefficient of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load("distrib").

Function: random_lognormal (m,s)
    random_lognormal (m,s,n)

Returns a Lognormal(m,s) random variate, with s>0. Calling random_lognormal with a third argument n, a random sample of size n will be simulated.

Log-normal variates are simulated by means of random normal variates. See random_normal for details.

To make use of this function, write first load("distrib").

Function: pdf_gamma (x,a,b)

Returns the value at x of the density function of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").

Function: cdf_gamma (x,a,b)

Returns the value at x of the distribution function of a Gamma(a,b) random variable, with a,b>0.

(%i1) load ("distrib")$
(%i2) cdf_gamma(3,5,21);
                                              1
(%o2)     1 - gamma_incomplete_regularized(5, -)
                                              7
(%i3) float(%);
(%o3)              4.402663157376807E-7

Function: quantile_gamma (q,a,b)

Returns the q-quantile of a Gamma(a,b) random variable, with a,b>0; in other words, this is the inverse of cdf_gamma. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").

Function: mean_gamma (a,b)

Returns the mean of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").

Function: var_gamma (a,b)

Returns the variance of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").

Function: std_gamma (a,b)

Returns the standard deviation of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").

Function: skewness_gamma (a,b)

Returns the skewness coefficient of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").

Function: kurtosis_gamma (a,b)

Returns the kurtosis coefficient of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").

Function: random_gamma (a,b)
    random_gamma (a,b,n)

Returns a Gamma(a,b) random variate, with a,b>0. Calling random_gamma with a third argument n, a random sample of size n will be simulated.

The implemented algorithm is a combinantion of two procedures, depending on the value of parameter a:

For a>=1, Cheng, R.C.H. and Feast, G.M. (1979). Some simple gamma variate generators. Appl. Stat., 28, 3, 290-295.

For 0<a<1, Ahrens, J.H. and Dieter, U. (1974). Computer methods for sampling from gamma, beta, poisson and binomial cdf_tributions. Computing, 12, 223-246.

To make use of this function, write first load("distrib").

Function: pdf_beta (x,a,b)

Returns the value at x of the density function of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").

Function: cdf_beta (x,a,b)

Returns the value at x of the distribution function of a Beta(a,b) random variable, with a,b>0.

(%i1) load ("distrib")$
(%i2) cdf_beta(1/3,15,2);
                             11
(%o2)                     --------
                          14348907
(%i3) float(%);
(%o3)              7.666089131388195E-7

Function: quantile_beta (q,a,b)

Returns the q-quantile of a Beta(a,b) random variable, with a,b>0; in other words, this is the inverse of cdf_beta. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").

Function: mean_beta (a,b)

Returns the mean of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").

Function: var_beta (a,b)

Returns the variance of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").

Function: std_beta (a,b)

Returns the standard deviation of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").

Function: skewness_beta (a,b)

Returns the skewness coefficient of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").

Function: kurtosis_beta (a,b)

Returns the kurtosis coefficient of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").

Function: random_beta (a,b)
    random_beta (a,b,n)

Returns a Beta(a,b) random variate, with a,b>0. Calling random_beta with a third argument n, a random sample of size n will be simulated.

The implemented algorithm is defined in Cheng, R.C.H. (1978). Generating Beta Variates with Nonintegral Shape Parameters. Communications of the ACM, 21:317-322

To make use of this function, write first load("distrib").

Function: pdf_continuous_uniform (x,a,b)

Returns the value at x of the density function of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first load("distrib").

Function: cdf_continuous_uniform (x,a,b)

Returns the value at x of the distribution function of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first load("distrib").

Function: quantile_continuous_uniform (q,a,b)

Returns the q-quantile of a Continuous Uniform(a,b) random variable, with a<b; in other words, this is the inverse of cdf_continuous_uniform. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").

Function: mean_continuous_uniform (a,b)

Returns the mean of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first load("distrib").

Function: var_continuous_uniform (a,b)

Returns the variance of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first load("distrib").

Function: std_continuous_uniform (a,b)

Returns the standard deviation of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first load("distrib").

Function: skewness_continuous_uniform (a,b)

Returns the skewness coefficient of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first load("distrib").

Function: kurtosis_continuous_uniform (a,b)

Returns the kurtosis coefficient of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first load("distrib").

Function: random_continuous_uniform (a,b)
    random_continuous_uniform (a,b,n)

Returns a Continuous Uniform(a,b) random variate, with a<b. Calling random_continuous_uniform with a third argument n, a random sample of size n will be simulated.

This is a direct application of the random built-in Maxima function.

See also random. To make use of this function, write first load("distrib").

Function: pdf_logistic (x,a,b)

Returns the value at x of the density function of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib").

Function: cdf_logistic (x,a,b)

Returns the value at x of the distribution function of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib").

Function: quantile_logistic (q,a,b)

Returns the q-quantile of a Logistic(a,b) random variable , with b>0; in other words, this is the inverse of cdf_logistic. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").

Function: mean_logistic (a,b)

Returns the mean of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib").

Function: var_logistic (a,b)

Returns the variance of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib").

Function: std_logistic (a,b)

Returns the standard deviation of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib").

Function: skewness_logistic (a,b)

Returns the skewness coefficient of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib").

Function: kurtosis_logistic (a,b)

Returns the kurtosis coefficient of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib").

Function: random_logistic (a,b)
    random_logistic (a,b,n)

Returns a Logistic(a,b) random variate, with b>0. Calling random_logistic with a third argument n, a random sample of size n will be simulated.

The implemented algorithm is based on the general inverse method.

To make use of this function, write first load("distrib").

Function: pdf_pareto (x,a,b)

Returns the value at x of the density function of a Pareto(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").

Function: cdf_pareto (x,a,b)

Returns the value at x of the distribution function of a Pareto(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").

Function: quantile_pareto (q,a,b)

Returns the q-quantile of a Pareto(a,b) random variable, with a,b>0; in other words, this is the inverse of cdf_pareto. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").

Function: mean_pareto (a,b)

Returns the mean of a Pareto(a,b) random variable, with a>1,b>0. To make use of this function, write first load("distrib").

Function: var_pareto (a,b)

Returns the variance of a Pareto(a,b) random variable, with a>2,b>0. To make use of this function, write first load("distrib").

Function: std_pareto (a,b)

Returns the standard deviation of a Pareto(a,b) random variable, with a>2,b>0. To make use of this function, write first load("distrib").

Function: skewness_pareto (a,b)

Returns the skewness coefficient of a Pareto(a,b) random variable, with a>3,b>0. To make use of this function, write first load("distrib").

Function: kurtosis_pareto (a,b)

Returns the kurtosis coefficient of a Pareto(a,b) random variable, with a>4,b>0. To make use of this function, write first load("distrib").

Function: random_pareto (a,b)
    random_pareto (a,b,n)

Returns a Pareto(a,b) random variate, with a>0,b>0. Calling random_pareto with a third argument n, a random sample of size n will be simulated.

The implemented algorithm is based on the general inverse method.

To make use of this function, write first load("distrib").

Function: pdf_weibull (x,a,b)

Returns the value at x of the density function of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").

Function: cdf_weibull (x,a,b)

Returns the value at x of the distribution function of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").

Function: quantile_weibull (q,a,b)

Returns the q-quantile of a Weibull(a,b) random variable, with a,b>0; in other words, this is the inverse of cdf_weibull. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").

Function: mean_weibull (a,b)

Returns the mean of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").

Function: var_weibull (a,b)

Returns the variance of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").

Function: std_weibull (a,b)

Returns the standard deviation of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").

Function: skewness_weibull (a,b)

Returns the skewness coefficient of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").

Function: kurtosis_weibull (a,b)

Returns the kurtosis coefficient of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib").

Function: random_weibull (a,b)
    random_weibull (a,b,n)

Returns a Weibull(a,b) random variate, with a,b>0. Calling random_weibull with a third argument n, a random sample of size n will be simulated.

The implemented algorithm is based on the general inverse method.

To make use of this function, write first load("distrib").

Function: pdf_rayleigh (x,b)

Returns the value at x of the density function of a Rayleigh(b) random variable, with b>0.

The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

(%i1) load ("distrib")$
(%i2) pdf_rayleigh(x,b);
                                    2  2
                           2     - b  x
(%o2)                   2 b  x %e

Function: cdf_rayleigh (x,b)

Returns the value at x of the distribution function of a Rayleigh(b) random variable, with b>0.

The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

(%i1) load ("distrib")$
(%i2) cdf_rayleigh(x,b);
                                   2  2
                                - b  x
(%o2)                     1 - %e

Function: quantile_rayleigh (q,b)

Returns the q-quantile of a Rayleigh(b) random variable, with b>0; in other words, this is the inverse of cdf_rayleigh. Argument q must be an element of [0,1].

The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

(%i1) load ("distrib")$
(%i2) quantile_rayleigh(0.99,b);
                        2.145966026289347
(%o2)                   -----------------
                                b

Function: mean_rayleigh (b)

Returns the mean of a Rayleigh(b) random variable, with b>0.

The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

(%i1) load ("distrib")$
(%i2) mean_rayleigh(b);
                            sqrt(%pi)
(%o2)                       ---------
                               2 b

Function: var_rayleigh (b)

Returns the variance of a Rayleigh(b) random variable, with b>0.

The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

(%i1) load ("distrib")$
(%i2) var_rayleigh(b);
                                 %pi
                             1 - ---
                                  4
(%o2)                        -------
                                2
                               b

Function: std_rayleigh (b)

Returns the standard deviation of a Rayleigh(b) random variable, with b>0.

The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

(%i1) load ("distrib")$
(%i2) std_rayleigh(b);
                                   %pi
                          sqrt(1 - ---)
                                    4
(%o2)                     -------------
                                b

Function: skewness_rayleigh (b)

Returns the skewness coefficient of a Rayleigh(b) random variable, with b>0.

The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

(%i1) load ("distrib")$
(%i2) skewness_rayleigh(b);
                         3/2
                      %pi      3 sqrt(%pi)
                      ------ - -----------
                        4           4
(%o2)                 --------------------
                               %pi 3/2
                          (1 - ---)
                                4

Function: kurtosis_rayleigh (b)

Returns the kurtosis coefficient of a Rayleigh(b) random variable, with b>0.

The Rayleigh(b) random variable is equivalent to the Weibull(2,1/b).

(%i1) load ("distrib")$
(%i2) kurtosis_rayleigh(b);
                                  2
                             3 %pi
                         2 - ------
                               16
(%o2)                    ---------- - 3
                              %pi 2
                         (1 - ---)
                               4

Function: random_rayleigh (b)
    random_rayleigh (b,n)

Returns a Rayleigh(b) random variate, with b>0. Calling random_rayleigh with a second argument n, a random sample of size n will be simulated.

The implemented algorithm is based on the general inverse method.

To make use of this function, write first load("distrib").

Function: pdf_laplace (x,a,b)

Returns the value at x of the density function of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib").

Function: cdf_laplace (x,a,b)

Returns the value at x of the distribution function of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib").

Function: quantile_laplace (q,a,b)

Returns the q-quantile of a Laplace(a,b) random variable, with b>0; in other words, this is the inverse of cdf_laplace. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").

Function: mean_laplace (a,b)

Returns the mean of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib").

Function: var_laplace (a,b)

Returns the variance of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib").

Function: std_laplace (a,b)

Returns the standard deviation of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib").

Function: skewness_laplace (a,b)

Returns the skewness coefficient of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib").

Function: kurtosis_laplace (a,b)

Returns the kurtosis coefficient of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib").

Function: random_laplace (a,b)
    random_laplace (a,b,n)

Returns a Laplace(a,b) random variate, with b>0. Calling random_laplace with a third argument n, a random sample of size n will be simulated.

The implemented algorithm is based on the general inverse method.

To make use of this function, write first load("distrib").

Function: pdf_cauchy (x,a,b)

Returns the value at x of the density function of a Cauchy(a,b) random variable, with b>0. To make use of this function, write first load("distrib").

Function: cdf_cauchy (x,a,b)

Returns the value at x of the distribution function of a Cauchy(a,b) random variable, with b>0. To make use of this function, write first load("distrib").

Function: quantile_cauchy (q,a,b)

Returns the q-quantile of a Cauchy(a,b) random variable, with b>0; in other words, this is the inverse of cdf_cauchy. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").

Function: random_cauchy (a,b)
    random_cauchy (a,b,n)

Returns a Cauchy(a,b) random variate, with b>0. Calling random_cauchy with a third argument n, a random sample of size n will be simulated.

The implemented algorithm is based on the general inverse method.

To make use of this function, write first load("distrib").

Function: pdf_gumbel (x,a,b)

Returns the value at x of the density function of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first load("distrib").

Function: cdf_gumbel (x,a,b)

Returns the value at x of the distribution function of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first load("distrib").

Function: quantile_gumbel (q,a,b)

Returns the q-quantile of a Gumbel(a,b) random variable, with b>0; in other words, this is the inverse of cdf_gumbel. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").

Function: mean_gumbel (a,b)

Returns the mean of a Gumbel(a,b) random variable, with b>0.

(%i1) load ("distrib")$
(%i2) mean_gumbel(a,b);
(%o2)                     %gamma b + a

where symbol %gamma stands for the Euler-Mascheroni constant. See also %gamma.

Function: var_gumbel (a,b)

Returns the variance of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first load("distrib").

Function: std_gumbel (a,b)

Returns the standard deviation of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first load("distrib").

Function: skewness_gumbel (a,b)

Returns the skewness coefficient of a Gumbel(a,b) random variable, with b>0.

(%i1) load ("distrib")$
(%i2) skewness_gumbel(a,b);
                                  3/2
                               2 6    zeta(3)
(%o2)                          --------------
                                       3
                                    %pi

where zeta stands for the Riemann’s zeta function.

Function: kurtosis_gumbel (a,b)

Returns the kurtosis coefficient of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first load("distrib").

Function: random_gumbel (a,b)
    random_gumbel (a,b,n)

Returns a Gumbel(a,b) random variate, with b>0. Calling random_gumbel with a third argument n, a random sample of size n will be simulated.

The implemented algorithm is based on the general inverse method.

To make use of this function, write first load("distrib").


Previous: , Up: distrib-pkg   [Contents][Index]

52.3 Functions and Variables for discrete distributions

Function: pdf_general_finite_discrete (x,v)

Returns the value at x of the probability function of a general finite discrete random variable, with vector probabilities v, such that Pr(X=i) = v_i. Vector v can be a list of nonnegative expressions, whose components will be normalized to get a vector of probabilities. To make use of this function, write first load("distrib").

(%i1) load ("distrib")$
(%i2) pdf_general_finite_discrete(2, [1/7, 4/7, 2/7]);
                                4
(%o2)                           -
                                7
(%i3) pdf_general_finite_discrete(2, [1, 4, 2]);
                                4
(%o3)                           -
                                7

Function: cdf_general_finite_discrete (x,v)

Returns the value at x of the distribution function of a general finite discrete random variable, with vector probabilities v.

See pdf_general_finite_discrete for more details.

(%i1) load ("distrib")$
(%i2) cdf_general_finite_discrete(2, [1/7, 4/7, 2/7]);
                                5
(%o2)                           -
                                7
(%i3) cdf_general_finite_discrete(2, [1, 4, 2]);
                                5
(%o3)                           -
                                7
(%i4) cdf_general_finite_discrete(2+1/2, [1, 4, 2]);
                                5
(%o4)                           -
                                7

Function: quantile_general_finite_discrete (q,v)

Returns the q-quantile of a general finite discrete random variable, with vector probabilities v.

See pdf_general_finite_discrete for more details.

Function: mean_general_finite_discrete (v)

Returns the mean of a general finite discrete random variable, with vector probabilities v.

See pdf_general_finite_discrete for more details.

Function: var_general_finite_discrete (v)

Returns the variance of a general finite discrete random variable, with vector probabilities v.

See pdf_general_finite_discrete for more details.

Function: std_general_finite_discrete (v)

Returns the standard deviation of a general finite discrete random variable, with vector probabilities v.

See pdf_general_finite_discrete for more details.

Function: skewness_general_finite_discrete (v)

Returns the skewness coefficient of a general finite discrete random variable, with vector probabilities v.

See pdf_general_finite_discrete for more details.

Function: kurtosis_general_finite_discrete (v)

Returns the kurtosis coefficient of a general finite discrete random variable, with vector probabilities v.

See pdf_general_finite_discrete for more details.

Function: random_general_finite_discrete (v)
    random_general_finite_discrete (v,m)

Returns a general finite discrete random variate, with vector probabilities v. Calling random_general_finite_discrete with a second argument m, a random sample of size m will be simulated.

See pdf_general_finite_discrete for more details.

(%i1) load ("distrib")$
(%i2) random_general_finite_discrete([1,3,1,5]);
(%o2)                          4
(%i3) random_general_finite_discrete([1,3,1,5], 10);
(%o3)           [4, 2, 2, 3, 2, 4, 4, 1, 2, 2]

Function: pdf_binomial (x,n,p)

Returns the value at x of the probability function of a Binomial(n,p) random variable, with 0 \leq p \leq 1 and n a positive integer. To make use of this function, write first load("distrib").

Function: cdf_binomial (x,n,p)

Returns the value at x of the distribution function of a Binomial(n,p) random variable, with 0 \leq p \leq 1 and n a positive integer.

(%i1) load ("distrib")$
(%i2) cdf_binomial(5,7,1/6);
                            7775
(%o2)                       ----
                            7776
(%i3) float(%);
(%o3)               .9998713991769548

Function: quantile_binomial (q,n,p)

Returns the q-quantile of a Binomial(n,p) random variable, with 0 \leq p \leq 1 and n a positive integer; in other words, this is the inverse of cdf_binomial. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").

Function: mean_binomial (n,p)

Returns the mean of a Binomial(n,p) random variable, with 0 \leq p \leq 1 and n a positive integer. To make use of this function, write first load("distrib").

Function: var_binomial (n,p)

Returns the variance of a Binomial(n,p) random variable, with 0 \leq p \leq 1 and n a positive integer. To make use of this function, write first load("distrib").

Function: std_binomial (n,p)

Returns the standard deviation of a Binomial(n,p) random variable, with 0 \leq p \leq 1 and n a positive integer. To make use of this function, write first load("distrib").

Function: skewness_binomial (n,p)

Returns the skewness coefficient of a Binomial(n,p) random variable, with 0 \leq p \leq 1 and n a positive integer. To make use of this function, write first load("distrib").

Function: kurtosis_binomial (n,p)

Returns the kurtosis coefficient of a Binomial(n,p) random variable, with 0 \leq p \leq 1 and n a positive integer. To make use of this function, write first load("distrib").

Function: random_binomial (n,p)
    random_binomial (n,p,m)

Returns a Binomial(n,p) random variate, with 0 \leq p \leq 1 and n a positive integer. Calling random_binomial with a third argument m, a random sample of size m will be simulated.

The implemented algorithm is based on the one described in Kachitvichyanukul, V. and Schmeiser, B.W. (1988) Binomial Random Variate Generation. Communications of the ACM, 31, Feb., 216.

To make use of this function, write first load("distrib").

Function: pdf_poisson (x,m)

Returns the value at x of the probability function of a Poisson(m) random variable, with m>0. To make use of this function, write first load("distrib").

Function: cdf_poisson (x,m)

Returns the value at x of the distribution function of a Poisson(m) random variable, with m>0.

(%i1) load ("distrib")$
(%i2) cdf_poisson(3,5);
(%o2)       gamma_incomplete_regularized(4, 5)
(%i3) float(%);
(%o3)               .2650259152973623

Function: quantile_poisson (q,m)

Returns the q-quantile of a Poisson(m) random variable, with m>0; in other words, this is the inverse of cdf_poisson. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").

Function: mean_poisson (m)

Returns the mean of a Poisson(m) random variable, with m>0. To make use of this function, write first load("distrib").

Function: var_poisson (m)

Returns the variance of a Poisson(m) random variable, with m>0. To make use of this function, write first load("distrib").

Function: std_poisson (m)

Returns the standard deviation of a Poisson(m) random variable, with m>0. To make use of this function, write first load("distrib").

Function: skewness_poisson (m)

Returns the skewness coefficient of a Poisson(m) random variable, with m>0. To make use of this function, write first load("distrib").

Function: kurtosis_poisson (m)

Returns the kurtosis coefficient of a Poisson random variable Poi(m), with m>0. To make use of this function, write first load("distrib").

Function: random_poisson (m)
    random_poisson (m,n)

Returns a Poisson(m) random variate, with m>0. Calling random_poisson with a second argument n, a random sample of size n will be simulated.

The implemented algorithm is the one described in Ahrens, J.H. and Dieter, U. (1982) Computer Generation of Poisson Deviates From Modified Normal Distributions. ACM Trans. Math. Software, 8, 2, June,163-179.

To make use of this function, write first load("distrib").

Function: pdf_bernoulli (x,p)

Returns the value at x of the probability function of a Bernoulli(p) random variable, with 0 \leq p \leq 1.

The Bernoulli(p) random variable is equivalent to the Binomial(1,p).

(%i1) load ("distrib")$
(%i2) pdf_bernoulli(1,p);
(%o2)                           p

Function: cdf_bernoulli (x,p)

Returns the value at x of the distribution function of a Bernoulli(p) random variable, with 0 \leq p \leq 1. To make use of this function, write first load("distrib").

Function: quantile_bernoulli (q,p)

Returns the q-quantile of a Bernoulli(p) random variable, with 0 \leq p \leq 1; in other words, this is the inverse of cdf_bernoulli. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").

Function: mean_bernoulli (p)

Returns the mean of a Bernoulli(p) random variable, with 0 \leq p \leq 1.

The Bernoulli(p) random variable is equivalent to the Binomial(1,p).

(%i1) load ("distrib")$
(%i2) mean_bernoulli(p);
(%o2)                           p

Function: var_bernoulli (p)

Returns the variance of a Bernoulli(p) random variable, with 0 \leq p \leq 1.

The Bernoulli(p) random variable is equivalent to the Binomial(1,p).

(%i1) load ("distrib")$
(%i2) var_bernoulli(p);
(%o2)                       (1 - p) p

Function: std_bernoulli (p)

Returns the standard deviation of a Bernoulli(p) random variable, with 0 \leq p \leq 1.

The Bernoulli(p) random variable is equivalent to the Binomial(1,p).

(%i1) load ("distrib")$
(%i2) std_bernoulli(p);
(%o2)                           sqrt((1 - p) p)

Function: skewness_bernoulli (p)

Returns the skewness coefficient of a Bernoulli(p) random variable, with 0 \leq p \leq 1.

The Bernoulli(p) random variable is equivalent to the Binomial(1,p).

(%i1) load ("distrib")$
(%i2) skewness_bernoulli(p);
                                    1 - 2 p
(%o2)                           ---------------
                                sqrt((1 - p) p)

Function: kurtosis_bernoulli (p)

Returns the kurtosis coefficient of a Bernoulli(p) random variable, with 0 \leq p \leq 1.

The Bernoulli(p) random variable is equivalent to the Binomial(1,p).

(%i1) load ("distrib")$
(%i2) kurtosis_bernoulli(p);
                         1 - 6 (1 - p) p
(%o2)                    ---------------
                            (1 - p) p

Function: random_bernoulli (p)
    random_bernoulli (p,n)

Returns a Bernoulli(p) random variate, with 0 \leq p \leq 1. Calling random_bernoulli with a second argument n, a random sample of size n will be simulated.

This is a direct application of the random built-in Maxima function.

See also random. To make use of this function, write first load("distrib").

Function: pdf_geometric (x,p)

Returns the value at x of the probability function of a Geometric(p) random variable, with 0 < p <= 1.

The probability function is defined as p (1 - p)^x. This is interpreted as the probability of x failures before the first success.

load("distrib") loads this function.

Function: cdf_geometric (x,p)

Returns the value at x of the distribution function of a Geometric(p) random variable, with 0 < p <= 1.

The probability from which the distribution function is derived is defined as p (1 - p)^x. This is interpreted as the probability of x failures before the first success.

load("distrib") loads this function.

Function: quantile_geometric (q,p)

Returns the q-quantile of a Geometric(p) random variable, with 0 < p <= 1; in other words, this is the inverse of cdf_geometric. Argument q must be an element of [0,1].

The probability from which the quantile is derived is defined as p (1 - p)^x. This is interpreted as the probability of x failures before the first success.

load("distrib") loads this function.

Function: mean_geometric (p)

Returns the mean of a Geometric(p) random variable, with 0 < p <= 1.

The probability from which the mean is derived is defined as p (1 - p)^x. This is interpreted as the probability of x failures before the first success.

load("distrib") loads this function.

Function: var_geometric (p)

Returns the variance of a Geometric(p) random variable, with 0 < p <= 1.

The probability from which the variance is derived is defined as p (1 - p)^x. This is interpreted as the probability of x failures before the first success.

load("distrib") loads this function.

Function: std_geometric (p)

Returns the standard deviation of a Geometric(p) random variable, with 0 < p <= 1.

The probability from which the standard deviation is derived is defined as p (1 - p)^x. This is interpreted as the probability of x failures before the first success.

load("distrib") loads this function.

Function: skewness_geometric (p)

Returns the skewness coefficient of a Geometric(p) random variable, with 0 < p <= 1.

The probability from which the skewness is derived is defined as p (1 - p)^x. This is interpreted as the probability of x failures before the first success.

load("distrib") loads this function.

Function: kurtosis_geometric (p)

Returns the kurtosis coefficient of a geometric random variable Geometric(p), with 0 < p <= 1.

The probability from which the kurtosis is derived is defined as p (1 - p)^x. This is interpreted as the probability of x failures before the first success.

load("distrib") loads this function.

Function: random_geometric (p)
    random_geometric (p,n)

random_geometric(p) returns one random sample from a Geometric(p) distribution, with 0 < p <= 1.

random_geometric(p, n) returns a list of n random samples.

The algorithm is based on simulation of Bernoulli trials.

The probability from which the random sample is derived is defined as p (1 - p)^x. This is interpreted as the probability of x failures before the first success.

load("distrib") loads this function.

Function: pdf_discrete_uniform (x,n)

Returns the value at x of the probability function of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first load("distrib").

Function: cdf_discrete_uniform (x,n)

Returns the value at x of the distribution function of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first load("distrib").

Function: quantile_discrete_uniform (q,n)

Returns the q-quantile of a Discrete Uniform(n) random variable, with n a strictly positive integer; in other words, this is the inverse of cdf_discrete_uniform. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").

Function: mean_discrete_uniform (n)

Returns the mean of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first load("distrib").

Function: var_discrete_uniform (n)

Returns the variance of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first load("distrib").

Function: std_discrete_uniform (n)

Returns the standard deviation of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first load("distrib").

Function: skewness_discrete_uniform (n)

Returns the skewness coefficient of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first load("distrib").

Function: kurtosis_discrete_uniform (n)

Returns the kurtosis coefficient of a Discrete Uniform(n) random variable, with n a strictly positive integer. To make use of this function, write first load("distrib").

Function: random_discrete_uniform (n)
    random_discrete_uniform (n,m)

Returns a Discrete Uniform(n) random variate, with n a strictly positive integer. Calling random_discrete_uniform with a second argument m, a random sample of size m will be simulated.

This is a direct application of the random built-in Maxima function.

See also random. To make use of this function, write first load("distrib").

Function: pdf_hypergeometric (x,n1,n2,n)

Returns the value at x of the probability function of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2. Being n1 the number of objects of class A, n2 the number of objects of class B, and n the size of the sample without replacement, this function returns the probability of event "exactly x objects are of class A".

To make use of this function, write first load("distrib").

Function: cdf_hypergeometric (x,n1,n2,n)

Returns the value at x of the distribution function of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2. See pdf_hypergeometric for a more complete description.

To make use of this function, write first load("distrib").

Function: quantile_hypergeometric (q,n1,n2,n)

Returns the q-quantile of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2; in other words, this is the inverse of cdf_hypergeometric. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").

Function: mean_hypergeometric (n1,n2,n)

Returns the mean of a discrete uniform random variable Hyp(n1,n2,n), with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first load("distrib").

Function: var_hypergeometric (n1,n2,n)

Returns the variance of a hypergeometric random variable Hyp(n1,n2,n), with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first load("distrib").

Function: std_hypergeometric (n1,n2,n)

Returns the standard deviation of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first load("distrib").

Function: skewness_hypergeometric (n1,n2,n)

Returns the skewness coefficient of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first load("distrib").

Function: kurtosis_hypergeometric (n1,n2,n)

Returns the kurtosis coefficient of a Hypergeometric(n1,n2,n) random variable, with n1, n2 and n non negative integers and n<=n1+n2. To make use of this function, write first load("distrib").

Function: random_hypergeometric (n1,n2,n)
    random_hypergeometric (n1,n2,n,m)

Returns a Hypergeometric(n1,n2,n) random variate, with n1, n2 and n non negative integers and n<=n1+n2. Calling random_hypergeometric with a fourth argument m, a random sample of size m will be simulated.

Algorithm described in Kachitvichyanukul, V., Schmeiser, B.W. (1985) Computer generation of hypergeometric random variates. Journal of Statistical Computation and Simulation 22, 127-145.

To make use of this function, write first load("distrib").

Function: pdf_negative_binomial (x,n,p)

Returns the value at x of the probability function of a Negative Binomial(n,p) random variable, with 0 < p \leq 1 and n a positive number. To make use of this function, write first load("distrib").

Function: cdf_negative_binomial (x,n,p)

Returns the value at x of the distribution function of a Negative Binomial(n,p) random variable, with 0 < p \leq 1 and n a positive number.

(%i1) load ("distrib")$
(%i2) cdf_negative_binomial(3,4,1/8);
                            3271
(%o2)                      ------
                           524288

Function: quantile_negative_binomial (q,n,p)

Returns the q-quantile of a Negative Binomial(n,p) random variable, with 0 < p \leq 1 and n a positive number; in other words, this is the inverse of cdf_negative_binomial. Argument q must be an element of [0,1]. To make use of this function, write first load("distrib").

Function: mean_negative_binomial (n,p)

Returns the mean of a Negative Binomial(n,p) random variable, with 0 < p \leq 1 and n a positive number. To make use of this function, write first load("distrib").

Function: var_negative_binomial (n,p)

Returns the variance of a Negative Binomial(n,p) random variable, with 0 < p \leq 1 and n a positive number. To make use of this function, write first load("distrib").

Function: std_negative_binomial (n,p)

Returns the standard deviation of a Negative Binomial(n,p) random variable, with 0 < p \leq 1 and n a positive number. To make use of this function, write first load("distrib").

Function: skewness_negative_binomial (n,p)

Returns the skewness coefficient of a Negative Binomial(n,p) random variable, with 0 < p \leq 1 and n a positive number. To make use of this function, write first load("distrib").

Function: kurtosis_negative_binomial (n,p)

Returns the kurtosis coefficient of a Negative Binomial(n,p) random variable, with 0 < p \leq 1 and n a positive number. To make use of this function, write first load("distrib").

Function: random_negative_binomial (n,p)
    random_negative_binomial (n,p,m)

Returns a Negative Binomial(n,p) random variate, with 0 < p \leq 1 and n a positive number. Calling random_negative_binomial with a third argument m, a random sample of size m will be simulated.

Algorithm described in Devroye, L. (1986) Non-Uniform Random Variate Generation. Springer Verlag, p. 480.

To make use of this function, write first load("distrib").


Next: , Previous: , Up: Top   [Contents][Index]

53 draw


Next: , Previous: , Up: draw-pkg   [Contents][Index]

53.1 Introduction to draw

draw is a Maxima-Gnuplot and a Maxima-vtk interface.

There are three main functions to be used at Maxima level:

Each scene can contain any number of objects and key=value pairs with options for the scene or the following objects.

A selection of useful objects a scene can be made up from are:

A short description of all draw commands and options including example plots (in the html and pdf version of this manual) can be found in the section See Functions and Variables for draw. A online version of the html manual can be found at maxima.sourceforge.net/docs/manual/maxima_singlepage.html#draw. More elaborated examples of this package can be found at the following locations:

http://tecnostats.net/Maxima/gnuplot
http://tecnostats.net/Maxima/vtk

Example:

(%i1) draw2d(
          title="Two simple plots",
          xlabel="x",ylabel="y",grid=true,

          color=red,key="A sinus",
          explicit(sin(x),x,1,10),
          color=blue,line_type=dots,key="A cosinus",
          explicit(cos(x),x,1,10)
)$
figures/draw_intro

You need Gnuplot 4.2 or newer to run draw; If you are using wxMaxima as a front end wxdraw, wxdraw2d and wxdraw3d are drop-in replacements for draw that do the same as draw, draw2d and draw3d but embed the resulting plot in the worksheet.

·

Category: Plotting · Category: Share packages · Category: Package draw


Next: , Previous: , Up: draw-pkg   [Contents][Index]

53.2 Functions and Variables for draw

53.2.1 Scenes

Scene constructor: gr2d (argument_1, ...)

Function gr2d builds an object describing a 2D scene. Arguments are graphic options, graphic objects, or lists containing both graphic options and objects. This scene is interpreted sequentially: graphic options affect those graphic objects placed on its right. Some graphic options affect the global appearance of the scene.

This is the list of graphic objects available for scenes in two dimensions: bars, ellipse, explicit, image, implicit, label, parametric, points, polar, polygon, quadrilateral, rectangle, triangle, vector and geomap (this one defined in package worldmap).

See also draw and draw2d.

(%i1) draw(
    gr2d(
        key="sin (x)",grid=[2,2],
        explicit(
            sin(x),
            x,0,2*%pi
        )
    ),
    gr2d(
        key="cos (x)",grid=[2,2],
        explicit(
            cos(x),
            x,0,2*%pi
        )
    )
 );
(%o1)           [gr2d(explicit), gr2d(explicit)]
figures/draw_scene

Scene constructor: gr3d (argument_1, ...)

Function gr3d builds an object describing a 3d scene. Arguments are graphic options, graphic objects, or lists containing both graphic options and objects. This scene is interpreted sequentially: graphic options affect those graphic objects placed on its right. Some graphic options affect the global appearance of the scene.

This is the list of graphic objects available for scenes in three dimensions:
cylindrical, elevation_grid, explicit, implicit, label, mesh, parametric,
parametric_surface, points, quadrilateral, spherical, triangle, tube,
vector, and geomap (this one defined in package worldmap).

See also draw and draw3d.

53.2.2 Functions

Function: draw (<arg_1>, ...)

Plots a series of scenes; its arguments are gr2d and/or gr3d objects, together with some options, or lists of scenes and options. By default, the scenes are put together in one column.

Besides scenes the function draw accepts the following global options: terminal, columns, dimensions, file_name and delay.

Functions draw2d and draw3d short cuts that can be used when only one scene is required, in two or three dimensions, respectively.

See also gr2d and gr3d.

Examples:

(%i1) scene1: gr2d(title="Ellipse",
                   nticks=300,
                   parametric(2*cos(t),5*sin(t),t,0,2*%pi))$
(%i2) scene2: gr2d(title="Triangle",
                   polygon([4,5,7],[6,4,2]))$
(%i3) draw(scene1, scene2, columns = 2)$
figures/draw_intro2
(%i1) scene1: gr2d(title="A sinus",
        grid=true,
        explicit(sin(t),t,0,2*%pi))$
(%i2) scene2: gr2d(title="A cosinus",
        grid=true,
        explicit(cos(t),t,0,2*%pi))$
(%i3) draw(scene1, scene2)$
figures/draw_intro3

The following two draw sentences are equivalent:

(%i1) draw(gr3d(explicit(x^2+y^2,x,-1,1,y,-1,1)));
(%o1)                          [gr3d(explicit)]
(%i2) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1));
(%o2)                          [gr3d(explicit)]

Creating an animated gif file:

(%i1) draw(
        delay     = 100,
        file_name = "zzz",
        terminal  = 'animated_gif,
        gr2d(explicit(x^2,x,-1,1)),
        gr2d(explicit(x^3,x,-1,1)),
        gr2d(explicit(x^4,x,-1,1)));
End of animation sequence
(%o1)          [gr2d(explicit), gr2d(explicit), gr2d(explicit)]
figures/draw_equiv

See also gr2d, gr3d, draw2d and draw3d.

Function: draw2d (argument_1, ...)

This function is a shortcut for draw(gr2d(options, ..., graphic_object, ...)).

It can be used to plot a unique scene in 2d, as can be seen in most examples below.

See also draw and gr2d.

Function: draw3d (argument_1, ...)

This function is a shortcut for draw(gr3d(options, ..., graphic_object, ...)).

It can be used to plot a unique scene in 3d, as can be seen in many examples below.

See also draw and gr3d.

Function: draw_file (graphic option, ..., graphic object, ...)

Saves the current plot into a file. Accepted graphics options are: terminal, dimensions and file_name.

Example:

(%i1) /* screen plot */
      draw(gr3d(explicit(x^2+y^2,x,-1,1,y,-1,1)))$
(%i2) /* same plot in eps format */
      draw_file(terminal  = eps,
                dimensions = [5,5]) $

Function: multiplot_mode (term)

This function enables Maxima to work in one-window multiplot mode with terminal term; accepted arguments for this function are screen, wxt, aquaterm, windows and none.

When multiplot mode is enabled, each call to draw sends a new plot to the same window, without erasing the previous ones. To disable the multiplot mode, write multiplot_mode(none).

When multiplot mode is enabled, global option terminal is blocked and you have to disable this working mode before changing to another terminal.

On Windows this feature requires Gnuplot 5.0 or newer. Note, that just plotting multiple expressions into the same plot doesn’t require multiplot: It can be done by just issuing multiple explicit or similar commands in a row.

Example:

(%i1) set_draw_defaults(
         xrange = [-1,1],
         yrange = [-1,1],
         grid   = true,
         title  = "Step by step plot" )$
(%i2) multiplot_mode(screen)$
(%i3) draw2d(color=blue,  explicit(x^2,x,-1,1))$
(%i4) draw2d(color=red,   explicit(x^3,x,-1,1))$
(%i5) draw2d(color=brown, explicit(x^4,x,-1,1))$
(%i6) multiplot_mode(none)$
figures/draw_multiplot

Function: set_draw_defaults (graphic option, ..., graphic object, ...)

Sets user graphics options. This function is useful for plotting a sequence of graphics with common graphics options. Calling this function without arguments removes user defaults.

Example:

(%i1) set_draw_defaults(
         xrange = [-10,10],
         yrange = [-2, 2],
         color  = blue,
         grid   = true)$
(%i2) /* plot with user defaults */
      draw2d(explicit(((1+x)**2/(1+x*x))-1,x,-10,10))$
(%i3) set_draw_defaults()$
(%i4) /* plot with standard defaults */
      draw2d(explicit(((1+x)**2/(1+x*x))-1,x,-10,10))$

53.2.3 Graphics options

Graphic option: adapt_depth

Default value: 10

adapt_depth is the maximum number of splittings used by the adaptive plotting routine.

This option is relevant only for 2d explicit functions.

See also nticks

Graphic option: allocation

Default value: false

With option allocation it is possible to place a scene in the output window at will; this is of interest in multiplots. When false, the scene is placed automatically, depending on the value assigned to option columns. In any other case, allocation must be set to a list of two pairs of numbers; the first corresponds to the position of the lower left corner of the scene, and the second pair gives the width and height of the plot. All quantities must be given in relative coordinates, between 0 and 1.

Examples:

In site graphics.

(%i1) draw(
        gr2d(
          explicit(x^2,x,-1,1)),
        gr2d(
          allocation = [[1/4, 1/4],[1/2, 1/2]],
          explicit(x^3,x,-1,1),
          grid = true) ) $
figures/draw_allocation

Multiplot with selected dimensions.

(%i1) draw(
        terminal = wxt,
        gr2d(
          grid=[5,5],
          allocation = [[0, 0],[1, 1/4]],
          explicit(x^2,x,-1,1)),
        gr3d(
          allocation = [[0, 1/4],[1, 3/4]],
          explicit(x^2+y^2,x,-1,1,y,-1,1) ))$
figures/draw_allocation2

See also option columns.

Graphic option: axis_3d

Default value: true

If axis_3d is true, the x, y and z axis are shown in 3d scenes.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw3d(axis_3d = false,
             explicit(sin(x^2+y^2),x,-2,2,y,-2,2) )$
figures/draw_axis3d

See also axis_bottom, axis_left, axis_top, and axis_right for axis in 2d.

Graphic option: axis_bottom

Default value: true

If axis_bottom is true, the bottom axis is shown in 2d scenes.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(axis_bottom = false,
             explicit(x^3,x,-1,1))$
figures/draw_axis_bottom

See also axis_left, axis_top, axis_right and axis_3d.

Graphic option: axis_left

Default value: true

If axis_left is true, the left axis is shown in 2d scenes.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(axis_left = false,
             explicit(x^3,x,-1,1))$

See also axis_bottom, axis_top, axis_right and axis_3d.

Graphic option: axis_right

Default value: true

If axis_right is true, the right axis is shown in 2d scenes.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(axis_right = false,
             explicit(x^3,x,-1,1))$

See also axis_bottom, axis_left, axis_top and axis_3d.

Graphic option: axis_top

Default value: true

If axis_top is true, the top axis is shown in 2d scenes.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(axis_top = false,
             explicit(x^3,x,-1,1))$

See also axis_bottom, axis_left, axis_right, and axis_3d.

Graphic option: background_color

Default value: white

Sets the background color for terminals. Default background color is white.

Since this is a global graphics option, its position in the scene description does not matter.

This option das not work with terminals epslatex and epslatex_standalone.

See also color

Graphic option: border

Default value: true

If border is true, borders of polygons are painted according to line_type and line_width.

This option affects the following graphic objects:

Example:

(%i1) draw2d(color       = brown,
             line_width  = 8,
             polygon([[3,2],[7,2],[5,5]]),
             border      = false,
             fill_color  = blue,
             polygon([[5,2],[9,2],[7,5]]) )$
figures/draw_border

Graphic option: capping

Default value: [false, false]

A list with two possible elements, true and false, indicating whether the extremes of a graphic object tube remain closed or open. By default, both extremes are left open.

Setting capping = false is equivalent to capping = [false, false], and capping = true is equivalent to capping = [true, true].

Example:

(%i1) draw3d(
        capping = [false, true],
        tube(0, 0, a, 1,
             a, 0, 8) )$
figures/draw_tube_extremes

Graphic option: cbrange

Default value: auto

If cbrange is auto, the range for the values which are colored when enhanced3d is not false is computed automatically. Values outside of the color range use color of the nearest extreme.

When enhanced3d or colorbox is false, option cbrange has no effect.

If the user wants a specific interval for the colored values, it must be given as a Maxima list, as in cbrange=[-2, 3].

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw3d (
        enhanced3d     = true,
        color          = green,
        cbrange = [-3,10],
        explicit(x^2+y^2, x,-2,2,y,-2,2)) $
figures/draw_cbrange

See also enhanced3d, colorbox and cbtics.

Graphic option: cbtics

Default value: auto

This graphic option controls the way tic marks are drawn on the colorbox when option enhanced3d is not false.

When enhanced3d or colorbox is false, option cbtics has no effect.

See xtics for a complete description.

Example :

(%i1) draw3d (
        enhanced3d = true,
        color      = green,
        cbtics  = {["High",10],["Medium",05],["Low",0]},
        cbrange = [0, 10],
        explicit(x^2+y^2, x,-2,2,y,-2,2)) $
figures/draw_cbtics

See also enhanced3d, colorbox and cbrange.

Graphic option: color

Default value: blue

color specifies the color for plotting lines, points, borders of polygons and labels.

Colors can be given as names or in hexadecimal rgb code. If a gnuplot version >= 5.0 is used and the terminal that is in use supports this rgba colors with transparency information are also supported.

Available color names are:

white            black            gray0            grey0 
gray10           grey10           gray20           grey20     
gray30           grey30           gray40           grey40     
gray50           grey50           gray60           grey60     
gray70           grey70           gray80           grey80
gray90           grey90           gray100          grey100 
gray             grey             light_gray       light_grey 
dark_gray        dark_grey        red              light_red 
dark_red         yellow           light_yellow     dark_yellow
green            light_green      dark_green       spring_green
forest_green     sea_green        blue             light_blue 
dark_blue        midnight_blue    navy             medium_blue 
royalblue        skyblue          cyan             light_cyan 
dark_cyan        magenta          light_magenta    dark_magenta
turquoise        light_turquoise  dark_turquoise   pink 
light_pink       dark_pink        coral            light_coral 
orange_red       salmon           light_salmon     dark_salmon 
aquamarine       khaki            dark_khaki       goldenrod 
light_goldenrod  dark_goldenrod   gold             beige 
brown            orange           dark_orange      violet 
dark_violet      plum             purple

Cromatic componentes in hexadecimal code are introduced in the form "#rrggbb".

Example:

(%i1) draw2d(explicit(x^2,x,-1,1), /* default is black */
             color = red,
             explicit(0.5 + x^2,x,-1,1),
             color = blue,
             explicit(1 + x^2,x,-1,1),
             color = light_blue,
             explicit(1.5 + x^2,x,-1,1),
             color = "#23ab0f",
             label(["This is a label",0,1.2])  )$
figures/draw_color
(%i1) draw2d(
             line_width=50,
             color="#FF0000",
             explicit(sin(x),x,0,10),
             color="#0000FF80",
             explicit(cos(x),x,0,10)
      );
figures/draw_color2
(%i1) H(p,p_0):=%i/(2*%pi*(p-p_0));
      draw2d(
          proportional_axes=xy,
          ip_grid=[150,150],
          grid=true,
          makelist(
              [
                  color=printf(false,"#~2,'0x~2,'0x~2,'0x",i*10,0,0),
                  key_pos=top_left,
                  key = if mod(i,5)=0 then sconcat("H=",i,"A/M") else "",
                  implicit(
                      cabs(H(x+%i*y,-1-%i)+H(x+%i*y,1+%i)-H(x+%i*y,1-%i)-H(x+%i*y,-1+%i))=i/10,
                      x,-3,3,
                      y,-3,3
                  )
              ],
              i,1,25
          )
      )$
figures/draw_color3

See also fill_color.

Graphic option: colorbox

Default value: true

If colorbox is true, a color scale without label is drawn together with image 2D objects, or coloured 3d objects. If colorbox is false, no color scale is shown. If colorbox is a string, a color scale with label is drawn.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

Color scale and images.

(%i1) im: apply('matrix,
                 makelist(makelist(random(200),i,1,30),i,1,30))$
(%i2) draw(
          gr2d(image(im,0,0,30,30)),
          gr2d(colorbox = false, image(im,0,0,30,30))
      )$
figures/draw_colorbox

Color scale and 3D coloured object.

(%i1) draw3d(
        colorbox   = "Magnitude",
        enhanced3d = true,
        explicit(x^2+y^2,x,-1,1,y,-1,1))$
figures/draw_colorbox2

See also palette_draw.

Graphic option: columns

Default value: 1

columns is the number of columns in multiple plots.

Since this is a global graphics option, its position in the scene description does not matter. It can be also used as an argument of function draw.

Example:

(%i1) scene1: gr2d(title="Ellipse",
                   nticks=30,
                   parametric(2*cos(t),5*sin(t),t,0,2*%pi))$
(%i2) scene2: gr2d(title="Triangle",
                   polygon([4,5,7],[6,4,2]))$
(%i3) draw(scene1, scene2, columns = 2)$
figures/draw_columns

Graphic option: contour

Default value: none

Option contour enables the user to select where to plot contour lines. Possible values are:

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw3d(explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
             contour_levels = 15,
             contour        = both,
             surface_hide   = true) $
figures/draw_contour
(%i1) draw3d(explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
             contour_levels = 15,
             contour        = map
      ) $
figures/draw_contour2

Graphic option: contour_levels

Default value: 5

This graphic option controls the way contours are drawn. contour_levels can be set to a positive integer number, a list of three numbers or an arbitrary set of numbers:

Since this is a global graphics option, its position in the scene description does not matter.

Examples:

Ten equally spaced contour lines. The actual number of levels can be adjusted to give simple labels.

(%i1) draw3d(color = green,
             explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
             contour_levels = 10,
             contour        = both,
             surface_hide   = true) $

From -8 to 8 in steps of 4.

(%i1) draw3d(color = green,
             explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
             contour_levels = [-8,4,8],
             contour        = both,
             surface_hide   = true) $

Isolines at levels -7, -6, 0.8 and 5.

(%i1) draw3d(color = green,
             explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3),
             contour_levels = {-7, -6, 0.8, 5},
             contour        = both,
             surface_hide   = true) $

See also contour.

Graphic option: data_file_name

Default value: "data.gnuplot"

This is the name of the file with the numeric data needed by Gnuplot to build the requested plot.

Since this is a global graphics option, its position in the scene description does not matter. It can be also used as an argument of function draw.

See example in gnuplot_file_name.

Graphic option: delay

Default value: 5

This is the delay in 1/100 seconds of frames in animated gif files.

Since this is a global graphics option, its position in the scene description does not matter. It can be also used as an argument of function draw.

Example:

(%i1) draw(
        delay     = 100,
        file_name = "zzz",
        terminal  = 'animated_gif,
        gr2d(explicit(x^2,x,-1,1)),
        gr2d(explicit(x^3,x,-1,1)),
        gr2d(explicit(x^4,x,-1,1)));
End of animation sequence
(%o2)          [gr2d(explicit), gr2d(explicit), gr2d(explicit)]

Option delay is only active in animated gif’s; it is ignored in any other case.

See also terminal, and dimensions.

Graphic option: dimensions

Default value: [600,500]

Dimensions of the output terminal. Its value is a list formed by the width and the height. The meaning of the two numbers depends on the terminal you are working with.

With terminals gif, animated_gif, png, jpg, svg, screen, wxt, and aquaterm, the integers represent the number of points in each direction. If they are not intergers, they are rounded.

With terminals eps, eps_color, pdf, and pdfcairo, both numbers represent hundredths of cm, which means that, by default, pictures in these formats are 6 cm in width and 5 cm in height.

Since this is a global graphics option, its position in the scene description does not matter. It can be also used as an argument of function draw.

Examples:

Option dimensions applied to file output and to wxt canvas.

(%i1) draw2d(
        dimensions = [300,300],
        terminal   = 'png,
        explicit(x^4,x,-1,1)) $
(%i2) draw2d(
        dimensions = [300,300],
        terminal   = 'wxt,
        explicit(x^4,x,-1,1)) $

Option dimensions applied to eps output. We want an eps file with A4 portrait dimensions.

(%i1) A4portrait: 100*[21, 29.7]$
(%i2) draw3d(
        dimensions = A4portrait,
        terminal   = 'eps,
        explicit(x^2-y^2,x,-2,2,y,-2,2)) $

Graphic option: draw_realpart

Default value: true

When true, functions to be drawn are considered as complex functions whose real part value should be plotted; when false, nothing will be plotted when the function does not give a real value.

This option affects objects explicit and parametric in 2D and 3D, and parametric_surface.

Example:

(%i1) draw2d(
        draw_realpart = false,
        explicit(sqrt(x^2  - 4*x) - x, x, -1, 5),
        color         = red,
        draw_realpart = true,
        parametric(x,sqrt(x^2  - 4*x) - x + 1, x, -1, 5) );

Graphic option: enhanced3d

Default value: none

If enhanced3d is none, surfaces are not colored in 3D plots. In order to get a colored surface, a list must be assigned to option enhanced3d, where the first element is an expression and the rest are the names of the variables or parameters used in that expression. A list such [f(x,y,z), x, y, z] means that point [x,y,z] of the surface is assigned number f(x,y,z), which will be colored according to the actual palette. For those 3D graphic objects defined in terms of parameters, it is possible to define the color number in terms of the parameters, as in [f(u), u], as in objects parametric and tube, or [f(u,v), u, v], as in object parametric_surface. While all 3D objects admit the model based on absolute coordinates, [f(x,y,z), x, y, z], only two of them, namely explicit and elevation_grid, accept also models defined on the [x,y] coordinates, [f(x,y), x, y]. 3D graphic object implicit accepts only the [f(x,y,z), x, y, z] model. Object points accepts also the [f(x,y,z), x, y, z] model, but when points have a chronological nature, model [f(k), k] is also valid, being k an ordering parameter.

When enhanced3d is assigned something different to none, options color and surface_hide are ignored.

The names of the variables defined in the lists may be different to those used in the definitions of the graphic objects.

In order to maintain back compatibility, enhanced3d = false is equivalent to enhanced3d = none, and enhanced3d = true is equivalent to enhanced3d = [z, x, y, z]. If an expression is given to enhanced3d, its variables must be the same used in the surface definition. This is not necessary when using lists.

See option palette to learn how palettes are specified.

Examples:

explicit object with coloring defined by the [f(x,y,z), x, y, z] model.

(%i1) draw3d(
         enhanced3d = [x-z/10,x,y,z],
         palette    = gray,
         explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3))$
figures/draw_enhanced3d

explicit object with coloring defined by the [f(x,y), x, y] model. The names of the variables defined in the lists may be different to those used in the definitions of the graphic objects; in this case, r corresponds to x, and s to y.

(%i1) draw3d(
         enhanced3d = [sin(r*s),r,s],
         explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3))$
figures/draw_enhanced3d2

parametric object with coloring defined by the [f(x,y,z), x, y, z] model.

(%i1) draw3d(
         nticks = 100,
         line_width = 2,
         enhanced3d = [if y>= 0 then 1 else 0, x, y, z],
         parametric(sin(u)^2,cos(u),u,u,0,4*%pi)) $
figures/draw_enhanced3d3

parametric object with coloring defined by the [f(u), u] model. In this case, (u-1)^2 is a shortcut for [(u-1)^2,u].

(%i1) draw3d(
         nticks = 60,
         line_width = 3,
         enhanced3d = (u-1)^2,
         parametric(cos(5*u)^2,sin(7*u),u-2,u,0,2))$
figures/draw_enhanced3d4

elevation_grid object with coloring defined by the [f(x,y), x, y] model.

(%i1) m: apply(
           matrix,
           makelist(makelist(cos(i^2/80-k/30),k,1,30),i,1,20)) $
(%i2) draw3d(
         enhanced3d = [cos(x*y*10),x,y],
         elevation_grid(m,-1,-1,2,2),
         xlabel = "x",
         ylabel = "y");
figures/draw_enhanced3d5

tube object with coloring defined by the [f(x,y,z), x, y, z] model.

(%i1) draw3d(
         enhanced3d = [cos(x-y),x,y,z],
         palette = gray,
         xu_grid = 50,
         tube(cos(a), a, 0, 1, a, 0, 4*%pi) )$
figures/draw_enhanced3d6

tube object with coloring defined by the [f(u), u] model. Here, enhanced3d = -a would be the shortcut for enhanced3d = [-foo,foo].

(%i1) draw3d(
         capping = [true, false],
         palette = [26,15,-2],
         enhanced3d = [-foo, foo],
         tube(a, a, a^2, 1, a, -2, 2) )$
figures/draw_enhanced3d7

implicit and points objects with coloring defined by the [f(x,y,z), x, y, z] model.

(%i1) draw3d(
         enhanced3d = [x-y,x,y,z],
         implicit((x^2+y^2+z^2-1)*(x^2+(y-1.5)^2+z^2-0.5)=0.015,
                  x,-1,1,y,-1.2,2.3,z,-1,1)) $
(%i2) m: makelist([random(1.0),random(1.0),random(1.0)],k,1,2000)$
figures/draw_enhanced3d9
(%i3) draw3d(
         point_type = filled_circle,
         point_size = 2,
         enhanced3d = [u+v-w,u,v,w],
         points(m) ) $
figures/draw_enhanced3d10

When points have a chronological nature, model [f(k), k] is also valid, being k an ordering parameter.

(%i1) m:makelist([random(1.0), random(1.0), random(1.0)],k,1,5)$
(%i2) draw3d(
         enhanced3d = [sin(j), j],
         point_size = 3,
         point_type = filled_circle,
         points_joined = true,
         points(m)) $
figures/draw_enhanced3d11

Graphic option: error_type

Default value: y

Depending on its value, which can be x, y, or xy, graphic object errors will draw points with horizontal, vertical, or both, error bars. When error_type=boxes, boxes will be drawn instead of crosses.

See also errors.

Graphic option: file_name

Default value: "maxima_out"

This is the name of the file where terminals png, jpg, gif, eps, eps_color, pdf, pdfcairo and svg will save the graphic.

Since this is a global graphics option, its position in the scene description does not matter. It can be also used as an argument of function draw.

Example:

(%i1) draw2d(file_name = "myfile",
             explicit(x^2,x,-1,1),
             terminal  = 'png)$

See also terminal, dimensions_draw.

Graphic option: fill_color

Default value: "red"

fill_color specifies the color for filling polygons and 2d explicit functions.

See color to learn how colors are specified.

Graphic option: fill_density

Default value: 0

fill_density is a number between 0 and 1 that specifies the intensity of the fill_color in bars objects.

See bars for examples.

Graphic option: filled_func

Default value: false

Option filled_func controls how regions limited by functions should be filled. When filled_func is true, the region bounded by the function defined with object explicit and the bottom of the graphic window is filled with fill_color. When filled_func contains a function expression, then the region bounded by this function and the function defined with object explicit will be filled. By default, explicit functions are not filled.

A useful special case is filled_func=0, which generates the region bond by the horizontal axis and the explicit function.

This option affects only the 2d graphic object explicit.

Example:

Region bounded by an explicit object and the bottom of the graphic window.

(%i1) draw2d(fill_color  = red,
             filled_func = true,
             explicit(sin(x),x,0,10) )$
figures/draw_filledfunc

Region bounded by an explicit object and the function defined by option filled_func. Note that the variable in filled_func must be the same as that used in explicit.

(%i1) draw2d(fill_color  = grey,
             filled_func = sin(x),
             explicit(-sin(x),x,0,%pi));
figures/draw_filledfunc2

See also fill_color and explicit.

Graphic option: font

Default value: "" (empty string)

This option can be used to set the font face to be used by the terminal. Only one font face and size can be used throughout the plot.

Since this is a global graphics option, its position in the scene description does not matter.

See also font_size.

Gnuplot doesn’t handle fonts by itself, it leaves this task to the support libraries of the different terminals, each one with its own philosophy about it. A brief summary follows:

The gnuplot documentation is an important source of information about terminals and fonts.

Graphic option: font_size

Default value: 10

This option can be used to set the font size to be used by the terminal. Only one font face and size can be used throughout the plot. font_size is active only when option font is not equal to the empty string.

Since this is a global graphics option, its position in the scene description does not matter.

See also font.

Graphic option: gnuplot_file_name

Default value: "maxout_xxx.gnuplot" with "xxx" being a number that is unique to each concurrently-running maxima process.

This is the name of the file with the necessary commands to be processed by Gnuplot.

Since this is a global graphics option, its position in the scene description does not matter. It can be also used as an argument of function draw.

Example:

(%i1) draw2d(
       file_name = "my_file",
       gnuplot_file_name = "my_commands_for_gnuplot",
       data_file_name    = "my_data_for_gnuplot",
       terminal          = png,
       explicit(x^2,x,-1,1)) $

See also data_file_name.

Graphic option: grid

Default value: false

If grid is not false, a grid will be drawn on the xy plane. If grid is assigned true, one grid line per tick of each axis is drawn. If grid is assigned a list nx,ny with [nx,ny] > [0,0] instead nx lines per tick of the x axis and ny lines per tick of the y axis are drawn.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(grid = true,
             explicit(exp(u),u,-2,2))$
figures/draw_grid
(%i1) draw2d(grid = [2,2],
             explicit(sin(x),x,0,2*%pi))$
figures/draw_grid2

Graphic option: head_angle

Default value: 45

head_angle indicates the angle, in degrees, between the arrow heads and the segment.

This option is relevant only for vector objects.

Example:

(%i1) draw2d(xrange      = [0,10],
             yrange      = [0,9],
             head_length = 0.7,
             head_angle  = 10,
             vector([1,1],[0,6]),
             head_angle  = 20,
             vector([2,1],[0,6]),
             head_angle  = 30,
             vector([3,1],[0,6]),
             head_angle  = 40,
             vector([4,1],[0,6]),
             head_angle  = 60,
             vector([5,1],[0,6]),
             head_angle  = 90,
             vector([6,1],[0,6]),
             head_angle  = 120,
             vector([7,1],[0,6]),
             head_angle  = 160,
             vector([8,1],[0,6]),
             head_angle  = 180,
             vector([9,1],[0,6]) )$
figures/draw_head_angle

See also head_both, head_length, and head_type.

Graphic option: head_both

Default value: false

If head_both is true, vectors are plotted with two arrow heads. If false, only one arrow is plotted.

This option is relevant only for vector objects.

Example:

(%i1) draw2d(xrange      = [0,8],
             yrange      = [0,8],
             head_length = 0.7,
             vector([1,1],[6,0]),
             head_both   = true,
             vector([1,7],[6,0]) )$
figures/draw_head_both

See also head_length, head_angle, and head_type.

Graphic option: head_length

Default value: 2

head_length indicates, in x-axis units, the length of arrow heads.

This option is relevant only for vector objects.

Example:

(%i1) draw2d(xrange      = [0,12],
             yrange      = [0,8],
             vector([0,1],[5,5]),
             head_length = 1,
             vector([2,1],[5,5]),
             head_length = 0.5,
             vector([4,1],[5,5]),
             head_length = 0.25,
             vector([6,1],[5,5]))$
figures/draw_head_length

See also head_both, head_angle, and head_type.

Graphic option: head_type

Default value: filled

head_type is used to specify how arrow heads are plotted. Possible values are: filled (closed and filled arrow heads), empty (closed but not filled arrow heads), and nofilled (open arrow heads).

This option is relevant only for vector objects.

Example:

(%i1) draw2d(xrange      = [0,12],
             yrange      = [0,10],
             head_length = 1,
             vector([0,1],[5,5]), /* default type */
             head_type = 'empty,
             vector([3,1],[5,5]),
             head_type = 'nofilled,
             vector([6,1],[5,5]))$
figures/draw_head_type

See also head_both, head_angle, and head_length.

Graphic option: interpolate_color

Default value: false

This option is relevant only when enhanced3d is not false.

When interpolate_color is false, surfaces are colored with homogeneous quadrangles. When true, color transitions are smoothed by interpolation.

interpolate_color also accepts a list of two numbers, [m,n]. For positive m and n, each quadrangle or triangle is interpolated m times and n times in the respective direction. For negative m and n, the interpolation frequency is chosen so that there will be at least |m| and |n| points drawn; you can consider this as a special gridding function. Zeros, i.e. interpolate_color=[0,0], will automatically choose an optimal number of interpolated surface points.

Also, interpolate_color=true is equivalent to interpolate_color=[0,0].

Examples:

Color interpolation with explicit functions.

(%i1) draw3d(
        enhanced3d   = sin(x*y),
        explicit(20*exp(-x^2-y^2)-10, x ,-3, 3, y, -3, 3)) $
figures/draw_interpolate_color
(%i2) draw3d(
        interpolate_color = true,
        enhanced3d   = sin(x*y),
        explicit(20*exp(-x^2-y^2)-10, x ,-3, 3, y, -3, 3)) $
figures/draw_interpolate_color2
(%i3) draw3d(
        interpolate_color = [-10,0],
        enhanced3d   = sin(x*y),
        explicit(20*exp(-x^2-y^2)-10, x ,-3, 3, y, -3, 3)) $
figures/draw_interpolate_color3

Color interpolation with the mesh graphic object.

Interpolating colors in parametric surfaces can give unexpected results.

(%i1) draw3d( 
        enhanced3d = true,
        mesh([[1,1,3],   [7,3,1],[12,-2,4],[15,0,5]],
             [[2,7,8],   [4,3,1],[10,5,8], [12,7,1]],
             [[-2,11,10],[6,9,5],[6,15,1], [20,15,2]])) $
figures/draw_interpolate_color4
(%i2) draw3d( 
        enhanced3d        = true,
        interpolate_color = true,
        mesh([[1,1,3],   [7,3,1],[12,-2,4],[15,0,5]],
             [[2,7,8],   [4,3,1],[10,5,8], [12,7,1]],
             [[-2,11,10],[6,9,5],[6,15,1], [20,15,2]])) $
figures/draw_interpolate_color5
(%i3) draw3d( 
        enhanced3d        = true,
        interpolate_color = true,
        view=map,
        mesh([[1,1,3],   [7,3,1],[12,-2,4],[15,0,5]],
             [[2,7,8],   [4,3,1],[10,5,8], [12,7,1]],
             [[-2,11,10],[6,9,5],[6,15,1], [20,15,2]])) $
figures/draw_interpolate_color6

See also enhanced3d.

Graphic option: ip_grid

Default value: [50, 50]

ip_grid sets the grid for the first sampling in implicit plots.

This option is relevant only for implicit objects.

Graphic option: ip_grid_in

Default value: [5, 5]

ip_grid_in sets the grid for the second sampling in implicit plots.

This option is relevant only for implicit objects.

Graphic option: key

Default value: "" (empty string)

key is the name of a function in the legend. If key is an empty string, no key is assigned to the function.

This option affects the following graphic objects:

Example:

(%i1) draw2d(key   = "Sinus",
             explicit(sin(x),x,0,10),
             key   = "Cosinus",
             color = red,
             explicit(cos(x),x,0,10) )$
figures/draw_key

Graphic option: key_pos

Default value: "" (empty string)

key_pos defines at which position the legend will be drawn. If key is an empty string, "top_right" is used. Available position specifiers are: top_left, top_center, top_right, center_left, center, center_right, bottom_left, bottom_center, and bottom_right.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(
        key_pos = top_left,
        key   = "x",
        explicit(x,  x,0,10),
        color= red,
        key   = "x squared",
        explicit(x^2,x,0,10))$
(%i3) draw3d(
        key_pos = center,
        key   = "x",
        explicit(x+y,x,0,10,y,0,10),
        color= red,
        key   = "x squared",
        explicit(x^2+y^2,x,0,10,y,0,10))$
figures/draw_key_pos

Graphic option: label_alignment

Default value: center

label_alignment is used to specify where to write labels with respect to the given coordinates. Possible values are: center, left, and right.

This option is relevant only for label objects.

Example:

(%i1) draw2d(xrange          = [0,10],
             yrange          = [0,10],
             points_joined   = true,
             points([[5,0],[5,10]]),
             color           = blue,
             label(["Centered alignment (default)",5,2]),
             label_alignment = 'left,
             label(["Left alignment",5,5]),
             label_alignment = 'right,
             label(["Right alignment",5,8]))$
figures/draw_label_alignment

See also label_orientation, and color

Graphic option: label_orientation

Default value: horizontal

label_orientation is used to specify orientation of labels. Possible values are: horizontal, and vertical.

This option is relevant only for label objects.

Example:

In this example, a dummy point is added to get an image. Package draw needs always data to draw an scene.

(%i1) draw2d(xrange     = [0,10],
             yrange     = [0,10],
             point_size = 0,
             points([[5,5]]),
             color      = navy,
             label(["Horizontal orientation (default)",5,2]),
             label_orientation = 'vertical,
             color             = "#654321",
             label(["Vertical orientation",1,5]))$
figures/draw_label_orientation

See also label_alignment and color

Graphic option: line_type

Default value: solid

line_type indicates how lines are displayed; possible values are solid and dots, both available in all terminals, and dashes, short_dashes, short_long_dashes, short_short_long_dashes, and dot_dash, which are not available in png, jpg, and gif terminals.

This option affects the following graphic objects:

Example:

(%i1) draw2d(line_type = dots,
             explicit(1 + x^2,x,-1,1),
             line_type = solid, /* default */
             explicit(2 + x^2,x,-1,1))$
figures/draw_line_type

See also line_width.

Graphic option: line_width

Default value: 1

line_width is the width of plotted lines. Its value must be a positive number.

This option affects the following graphic objects:

Example:

(%i1) draw2d(explicit(x^2,x,-1,1), /* default width */
             line_width = 5.5,
             explicit(1 + x^2,x,-1,1),
             line_width = 10,
             explicit(2 + x^2,x,-1,1))$
figures/draw_line_width

See also line_type.

Graphic option: logcb

Default value: false

If logcb is true, the tics in the colorbox will be drawn in the logarithmic scale.

When enhanced3d or colorbox is false, option logcb has no effect.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw3d (
        enhanced3d = true,
        color      = green,
        logcb = true,
        logz  = true,
        palette = [-15,24,-9],
        explicit(exp(x^2-y^2), x,-2,2,y,-2,2)) $
figures/draw_logcb

See also enhanced3d, colorbox and cbrange.

Graphic option: logx

Default value: false

If logx is true, the x axis will be drawn in the logarithmic scale.

Since this is a global graphics option, its position in the scene description does not matter, with the exception that it should be written before any 2D explicit object, so that draw can produce a better plot.

Example:

(%i1) draw2d(logx = true,
             explicit(log(x),x,0.01,5))$

See also logy, logx_secondary, logy_secondary, and logz.

Graphic option: logx_secondary

Default value: false

If logx_secondary is true, the secondary x axis will be drawn in the logarithmic scale.

This option is relevant only for 2d scenes.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(
        grid = true,
        key="x^2, linear scale",
        color=red,
        explicit(x^2,x,1,100),
        xaxis_secondary = true,
        xtics_secondary = true,
        logx_secondary  = true,
        key = "x^2, logarithmic x scale",
        color = blue,
        explicit(x^2,x,1,100) )$
figures/draw_logx_secondary

See also logx_draw, logy_draw, logy_secondary, and logz.

Graphic option: logy

Default value: false

If logy is true, the y axis will be drawn in the logarithmic scale.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(logy = true,
             explicit(exp(x),x,0,5))$

See also logx_draw, logx_secondary, logy_secondary, and logz.

Graphic option: logy_secondary

Default value: false

If logy_secondary is true, the secondary y axis will be drawn in the logarithmic scale.

This option is relevant only for 2d scenes.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(
        grid = true,
        key="x^2, linear scale",
        color=red,
        explicit(x^2,x,1,100),
        yaxis_secondary = true,
        ytics_secondary = true,
        logy_secondary  = true,
        key = "x^2, logarithmic y scale",
        color = blue,
        explicit(x^2,x,1,100) )$

See also logx_draw, logy_draw, logx_secondary, and logz.

Graphic option: logz

Default value: false

If logz is true, the z axis will be drawn in the logarithmic scale.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw3d(logz = true,
             explicit(exp(u^2+v^2),u,-2,2,v,-2,2))$

See also logx_draw and logy_draw.

Graphic option: nticks

Default value: 29

In 2d, nticks gives the initial number of points used by the adaptive plotting routine for explicit objects. It is also the number of points that will be shown in parametric and polar curves.

This option affects the following graphic objects:

See also adapt_depth

Example:

(%i1) draw2d(transparent = true,
             ellipse(0,0,4,2,0,180),
             nticks = 5,
             ellipse(0,0,4,2,180,180) )$
figures/draw_nticks

Graphic option: palette

Default value: color

palette indicates how to map gray levels onto color components. It works together with option enhanced3d in 3D graphics, who associates every point of a surfaces to a real number or gray level. It also works with gray images. With palette, levels are transformed into colors.

There are two ways for defining these transformations.

First, palette can be a vector of length three with components ranging from -36 to +36; each value is an index for a formula mapping the levels onto red, green and blue colors, respectively:

 0: 0               1: 0.5           2: 1
 3: x               4: x^2           5: x^3
 6: x^4             7: sqrt(x)       8: sqrt(sqrt(x))
 9: sin(90x)       10: cos(90x)     11: |x-0.5|
12: (2x-1)^2       13: sin(180x)    14: |cos(180x)|
15: sin(360x)      16: cos(360x)    17: |sin(360x)|
18: |cos(360x)|    19: |sin(720x)|  20: |cos(720x)|
21: 3x             22: 3x-1         23: 3x-2
24: |3x-1|         25: |3x-2|       26: (3x-1)/2 
27: (3x-2)/2       28: |(3x-1)/2|   29: |(3x-2)/2|
30: x/0.32-0.78125 31: 2*x-0.84     32: 4x;1;-2x+1.84;x/0.08-11.5
33: |2*x - 0.5|    34: 2*x          35: 2*x - 0.5
36: 2*x - 1

negative numbers mean negative colour component. palette = gray and palette = color are short cuts for palette = [3,3,3] and palette = [7,5,15], respectively.

Second, palette can be a user defined lookup table. In this case, the format for building a lookup table of length n is palette=[color_1, color_2, ..., color_n], where color_i is a well formed color (see option color) such that color_1 is assigned to the lowest gray level and color_n to the highest. The rest of colors are interpolated.

Since this is a global graphics option, its position in the scene description does not matter.

Examples:

It works together with option enhanced3d in 3D graphics.

(%i1) draw3d(
        enhanced3d = [z-x+2*y,x,y,z],
        palette = [32, -8, 17],
        explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3))$
figures/draw_palette

It also works with gray images.

(%i1) im: apply(
           'matrix,
            makelist(makelist(random(200),i,1,30),i,1,30))$
(%i2) /* palette = color, default */
      draw2d(image(im,0,0,30,30))$
(%i3) draw2d(palette = gray, image(im,0,0,30,30))$
(%i4) draw2d(palette = [15,20,-4],
             colorbox=false,
             image(im,0,0,30,30))$
figures/draw_palette2

palette can be a user defined lookup table. In this example, low values of x are colored in red, and higher values in yellow.

(%i1) draw3d(
         palette = [red, blue, yellow],
         enhanced3d = x,
         explicit(x^2+y^2,x,-1,1,y,-1,1)) $
figures/draw_palette3

See also colorbox and enhanced3d.

Graphic option: point_size

Default value: 1

point_size sets the size for plotted points. It must be a non negative number.

This option has no effect when graphic option point_type is set to dot.

This option affects the following graphic objects:

Example:

(%i1) draw2d(points(makelist([random(20),random(50)],k,1,10)),
        point_size = 5,
        points(makelist(k,k,1,20),makelist(random(30),k,1,20)))$
figures/draw_point_size

Graphic option: point_type

Default value: 1

point_type indicates how isolated points are displayed; the value of this option can be any integer index greater or equal than -1, or the name of a point style: $none (-1), dot (0), plus (1), multiply (2), asterisk (3), square (4), filled_square (5), circle (6), filled_circle (7), up_triangle (8), filled_up_triangle (9), down_triangle (10), filled_down_triangle (11), diamant (12) and filled_diamant (13).

This option affects the following graphic objects:

Example:

(%i1) draw2d(xrange = [0,10],
             yrange = [0,10],
             point_size = 3,
             point_type = diamant,
             points([[1,1],[5,1],[9,1]]),
             point_type = filled_down_triangle,
             points([[1,2],[5,2],[9,2]]),
             point_type = asterisk,
             points([[1,3],[5,3],[9,3]]),
             point_type = filled_diamant,
             points([[1,4],[5,4],[9,4]]),
             point_type = 5,
             points([[1,5],[5,5],[9,5]]),
             point_type = 6,
             points([[1,6],[5,6],[9,6]]),
             point_type = filled_circle,
             points([[1,7],[5,7],[9,7]]),
             point_type = 8,
             points([[1,8],[5,8],[9,8]]),
             point_type = filled_diamant,
             points([[1,9],[5,9],[9,9]]) )$
figures/draw_point_type

Graphic option: points_joined

Default value: false

When points_joined is true, points are joined by lines; when false, isolated points are drawn. A third possible value for this graphic option is impulses; in such case, vertical segments are drawn from points to the x-axis (2D) or to the xy-plane (3D).

This option affects the following graphic objects:

Example:

(%i1) draw2d(xrange        = [0,10],
             yrange        = [0,4],
             point_size    = 3,
             point_type    = up_triangle,
             color         = blue,
             points([[1,1],[5,1],[9,1]]),
             points_joined = true,
             point_type    = square,
             line_type     = dots,
             points([[1,2],[5,2],[9,2]]),
             point_type    = circle,
             color         = red,
             line_width    = 7,
             points([[1,3],[5,3],[9,3]]) )$
figures/draw_points_joined

Graphic option: proportional_axes

Default value: none

When proportional_axes is equal to xy or xyz, the aspect ratio of the axis units will be set to 1:1 resulting in a 2D or 3D scene that will be drawn with axes proportional to their relative lengths.

Since this is a global graphics option, its position in the scene description does not matter.

This option works with Gnuplot version 4.2.6 or greater.

Examples:

Single 2D plot.

(%i1) draw2d(
        ellipse(0,0,1,1,0,360),
        transparent=true,
        color = blue,
        line_width = 4,
        ellipse(0,0,2,1/2,0,360),
        proportional_axes = 'xy) $
figures/draw_proportional_axis

Multiplot.

(%i1) draw(
        terminal = wxt,
        gr2d(proportional_axes = 'xy,
             explicit(x^2,x,0,1)),
        gr2d(explicit(x^2,x,0,1),
             xrange = [0,1],
             yrange = [0,2],
             proportional_axes='xy),
        gr2d(explicit(x^2,x,0,1)))$
figures/draw_proportional_axis2

Graphic option: surface_hide

Default value: false

If surface_hide is true, hidden parts are not plotted in 3d surfaces.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw(columns=2,
           gr3d(explicit(exp(sin(x)+cos(x^2)),x,-3,3,y,-3,3)),
           gr3d(surface_hide = true,
                explicit(exp(sin(x)+cos(x^2)),x,-3,3,y,-3,3)) )$
figures/draw_surface_hide

Graphic option: terminal

Default value: screen

Selects the terminal to be used by Gnuplot; possible values are: screen (default), png, pngcairo, jpg, gif, eps, eps_color, epslatex, epslatex_standalone, svg, canvas, dumb, dumb_file, pdf, pdfcairo, wxt, animated_gif, multipage_pdfcairo, multipage_pdf, multipage_eps, multipage_eps_color, and aquaterm.

Terminals screen, wxt, windows and aquaterm can be also defined as a list with two elements: the name of the terminal itself and a non negative integer number. In this form, multiple windows can be opened at the same time, each with its corresponding number. This feature does not work in Windows platforms.

Since this is a global graphics option, its position in the scene description does not matter. It can be also used as an argument of function draw.

N.B. pdfcairo requires Gnuplot 4.3 or newer. pdf requires Gnuplot to be compiled with the option --enable-pdf and libpdf must be installed. The pdf library is available from: http://www.pdflib.com/en/download/pdflib-family/pdflib-lite/

Examples:

(%i1) /* screen terminal (default) */
      draw2d(explicit(x^2,x,-1,1))$
(%i2) /* png file */
      draw2d(terminal  = 'png,
             explicit(x^2,x,-1,1))$
(%i3) /* jpg file */
      draw2d(terminal   = 'jpg,
             dimensions = [300,300],
             explicit(x^2,x,-1,1))$
(%i4) /* eps file */
      draw2d(file_name = "myfile",
             explicit(x^2,x,-1,1),
             terminal  = 'eps)$
(%i5) /* pdf file */
      draw2d(file_name = "mypdf",
             dimensions = 100*[12.0,8.0],
             explicit(x^2,x,-1,1),
             terminal  = 'pdf)$
(%i6) /* wxwidgets window */
      draw2d(explicit(x^2,x,-1,1),
             terminal  = 'wxt)$

Multiple windows.

(%i1) draw2d(explicit(x^5,x,-2,2), terminal=[screen, 3])$
(%i2) draw2d(explicit(x^2,x,-2,2), terminal=[screen, 0])$

An animated gif file.

(%i1) draw(
        delay     = 100,
        file_name = "zzz",
        terminal  = 'animated_gif,
        gr2d(explicit(x^2,x,-1,1)),
        gr2d(explicit(x^3,x,-1,1)),
        gr2d(explicit(x^4,x,-1,1)));
End of animation sequence
(%o1)          [gr2d(explicit), gr2d(explicit), gr2d(explicit)]

Option delay is only active in animated gif’s; it is ignored in any other case.

Multipage output in eps format.

(%i1) draw(
        file_name = "parabol",
        terminal  = multipage_eps,
        dimensions = 100*[10,10],
        gr2d(explicit(x^2,x,-1,1)),
        gr3d(explicit(x^2+y^2,x,-1,1,y,-1,1))) $

See also file_name, dimensions_draw and delay.

Graphic option: title

Default value: "" (empty string)

Option title, a string, is the main title for the scene. By default, no title is written.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(explicit(exp(u),u,-2,2),
             title = "Exponential function")$
figures/draw_title

Graphic option: transform

Default value: none

If transform is none, the space is not transformed and graphic objects are drawn as defined. When a space transformation is desired, a list must be assigned to option transform. In case of a 2D scene, the list takes the form [f1(x,y), f2(x,y), x, y]. In case of a 3D scene, the list is of the form [f1(x,y,z), f2(x,y,z), f3(x,y,z), x, y, z].

The names of the variables defined in the lists may be different to those used in the definitions of the graphic objects.

Examples:

Rotation in 2D.

(%i1) th : %pi / 4$
(%i2) draw2d(
        color = "#e245f0",
        proportional_axes = 'xy,
        line_width = 8,
        triangle([3,2],[7,2],[5,5]),
        border     = false,
        fill_color = yellow,
        transform  = [cos(th)*x - sin(th)*y,
                      sin(th)*x + cos(th)*y, x, y],
        triangle([3,2],[7,2],[5,5]) )$
figures/draw_transform

Translation in 3D.

(%i1) draw3d(
        color     = "#a02c00",
        explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3),
        transform = [x+10,y+10,z+10,x,y,z],
        color     = blue,
        explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3) )$

Graphic option: transparent

Default value: false

If transparent is false, interior regions of polygons are filled according to fill_color.

This option affects the following graphic objects:

Example:

(%i1) draw2d(polygon([[3,2],[7,2],[5,5]]),
             transparent = true,
             color       = blue,
             polygon([[5,2],[9,2],[7,5]]) )$
figures/draw_transparent

Graphic option: unit_vectors

Default value: false

If unit_vectors is true, vectors are plotted with module 1. This is useful for plotting vector fields. If unit_vectors is false, vectors are plotted with its original length.

This option is relevant only for vector objects.

Example:

(%i1) draw2d(xrange      = [-1,6],
             yrange      = [-1,6],
             head_length = 0.1,
             vector([0,0],[5,2]),
             unit_vectors = true,
             color        = red,
             vector([0,3],[5,2]))$
figures/draw_unit_vectors

Graphic option: user_preamble

Default value: "" (empty string)

Expert Gnuplot users can make use of this option to fine tune Gnuplot’s behaviour by writing settings to be sent before the plot or splot command.

The value of this option must be a string or a list of strings (one per line).

Since this is a global graphics option, its position in the scene description does not matter.

Example:

Tell Gnuplot to draw axes and grid on top of graphics objects,

(%i1) draw2d(
        xaxis =true, xaxis_type=solid,
        yaxis =true, yaxis_type=solid,
        user_preamble="set grid front",
        region(x^2+y^2<1 ,x,-1.5,1.5,y,-1.5,1.5))$
figures/draw_user_preamble

Tell gnuplot to draw all contour lines in black

(%i1) draw3d(
          contour=both,
          surface_hide=true,enhanced3d=true,wired_surface=true,
          contour_levels=10,
          user_preamble="set for [i=1:8] linetype i dashtype i linecolor 0",
          explicit(sin(x)*cos(y),x,1,10,y,1,10)
      );
figures/draw_user_preamble2

Graphic option: view

Default value: [60,30]

A pair of angles, measured in degrees, indicating the view direction in a 3D scene. The first angle is the vertical rotation around the x axis, in the range [0, 360]. The second one is the horizontal rotation around the z axis, in the range [0, 360].

If option view is given the value map, the view direction is set to be perpendicular to the xy-plane.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw3d(view = [170, 50],
             enhanced3d = true,
             explicit(sin(x^2+y^2),x,-2,2,y,-2,2) )$
figures/draw_view
(%i2) draw3d(view = map,
             enhanced3d = true,
             explicit(sin(x^2+y^2),x,-2,2,y,-2,2) )$
figures/draw_view2

Graphic option: wired_surface

Default value: false

Indicates whether 3D surfaces in enhanced3d mode show the grid joinning the points or not.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw3d(
        enhanced3d    = [sin(x),x,y],
        wired_surface = true,
        explicit(x^2+y^2,x,-1,1,y,-1,1)) $
figures/draw_wired_surface

Graphic option: x_voxel

Default value: 10

x_voxel is the number of voxels in the x direction to be used by the marching cubes algorithm implemented by the 3d implicit object. It is also used by graphic object region.

Graphic option: xaxis

Default value: false

If xaxis is true, the x axis is drawn.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(explicit(x^3,x,-1,1),
             xaxis       = true,
             xaxis_color = blue)$
figures/draw_xaxis

See also xaxis_width, xaxis_type and xaxis_color.

Graphic option: xaxis_color

Default value: "black"

xaxis_color specifies the color for the x axis. See color to know how colors are defined.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(explicit(x^3,x,-1,1),
             xaxis       = true,
             xaxis_color = red)$

See also xaxis, xaxis_width and xaxis_type.

Graphic option: xaxis_secondary

Default value: false

If xaxis_secondary is true, function values can be plotted with respect to the second x axis, which will be drawn on top of the scene.

Note that this is a local graphics option which only affects to 2d plots.

Example:

(%i1) draw2d(
         key   = "Bottom x-axis",
         explicit(x+1,x,1,2),
         color = red,
         key   = "Above x-axis",
         xtics_secondary = true,
         xaxis_secondary = true,
         explicit(x^2,x,-1,1)) $
figures/draw_xaxis_secondary

See also xrange_secondary, xtics_secondary, xtics_rotate_secondary, xtics_axis_secondary and xaxis_secondary.

Graphic option: xaxis_type

Default value: dots

xaxis_type indicates how the x axis is displayed; possible values are solid and dots

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(explicit(x^3,x,-1,1),
             xaxis       = true,
             xaxis_type  = solid)$

See also xaxis, xaxis_width and xaxis_color.

Graphic option: xaxis_width

Default value: 1

xaxis_width is the width of the x axis. Its value must be a positive number.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(explicit(x^3,x,-1,1),
             xaxis       = true,
             xaxis_width = 3)$

See also xaxis, xaxis_type and xaxis_color.

Graphic option: xlabel

Default value: ""

Option xlabel, a string, is the label for the x axis. By default, the axis is labeled with string "x".

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(xlabel = "Time",
             explicit(exp(u),u,-2,2),
             ylabel = "Population")$

See also xlabel_secondary, ylabel, ylabel_secondary and zlabel_draw.

Graphic option: xlabel_secondary

Default value: "" (empty string)

Option xlabel_secondary, a string, is the label for the secondary x axis. By default, no label is written.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(
         xaxis_secondary=true,yaxis_secondary=true,
         xtics_secondary=true,ytics_secondary=true,
         xlabel_secondary="t[s]",
         ylabel_secondary="U[V]",
         explicit(sin(t),t,0,10) )$
figures/draw_ylabel_secondary

See also xlabel_draw, ylabel_draw, ylabel_secondary and zlabel_draw.

Graphic option: xrange

Default value: auto

If xrange is auto, the range for the x coordinate is computed automatically.

If the user wants a specific interval for x, it must be given as a Maxima list, as in xrange=[-2, 3].

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(xrange = [-3,5],
             explicit(x^2,x,-1,1))$

See also yrange and zrange.

Graphic option: xrange_secondary

Default value: auto

If xrange_secondary is auto, the range for the second x axis is computed automatically.

If the user wants a specific interval for the second x axis, it must be given as a Maxima list, as in xrange_secondary=[-2, 3].

Since this is a global graphics option, its position in the scene description does not matter.

See also xrange, yrange, zrange and yrange_secondary.

Graphic option: xtics

Default value: true

This graphic option controls the way tic marks are drawn on the x axis.

Since this is a global graphics option, its position in the scene description does not matter.

Examples:

Disable tics.

(%i1) draw2d(xtics = 'false,
             explicit(x^3,x,-1,1)  )$

Tics every 1/4 units.

(%i1) draw2d(xtics = 1/4,
             explicit(x^3,x,-1,1)  )$

Tics from -3/4 to 3/4 in steps of 1/8.

(%i1) draw2d(xtics = [-3/4,1/8,3/4],
             explicit(x^3,x,-1,1)  )$

Tics at points -1/2, -1/4 and 3/4.

(%i1) draw2d(xtics = {-1/2,-1/4,3/4},
             explicit(x^3,x,-1,1)  )$

Labeled tics.

(%i1) draw2d(xtics = {["High",0.75],["Medium",0],["Low",-0.75]},
             explicit(x^3,x,-1,1)  )$

See also ytics, and ztics.

Graphic option: xtics_axis

Default value: false

If xtics_axis is true, tic marks and their labels are plotted just along the x axis, if it is false tics are plotted on the border.

Since this is a global graphics option, its position in the scene description does not matter.

Graphic option: xtics_rotate

Default value: false

If xtics_rotate is true, tic marks on the x axis are rotated 90 degrees.

Since this is a global graphics option, its position in the scene description does not matter.

Graphic option: xtics_rotate_secondary

Default value: false

If xtics_rotate_secondary is true, tic marks on the secondary x axis are rotated 90 degrees.

Since this is a global graphics option, its position in the scene description does not matter.

Graphic option: xtics_secondary

Default value: auto

This graphic option controls the way tic marks are drawn on the second x axis.

See xtics for a complete description.

Graphic option: xtics_secondary_axis

Default value: false

If xtics_secondary_axis is true, tic marks and their labels are plotted just along the secondary x axis, if it is false tics are plotted on the border.

Since this is a global graphics option, its position in the scene description does not matter.

Graphic option: xu_grid

Default value: 30

xu_grid is the number of coordinates of the first variable (x in explicit and u in parametric 3d surfaces) to build the grid of sample points.

This option affects the following graphic objects:

Example:

(%i1) draw3d(xu_grid = 10,
             yv_grid = 50,
             explicit(x^2+y^2,x,-3,3,y,-3,3) )$

See also yv_grid.

Graphic option: xy_file

Default value: "" (empty string)

xy_file is the name of the file where the coordinates will be saved after clicking with the mouse button and hitting the ’x’ key. By default, no coordinates are saved.

Since this is a global graphics option, its position in the scene description does not matter.

Graphic option: xyplane

Default value: false

Allocates the xy-plane in 3D scenes. When xyplane is false, the xy-plane is placed automatically; when it is a real number, the xy-plane intersects the z-axis at this level. This option has no effect in 2D scenes.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw3d(xyplane = %e-2,
             explicit(x^2+y^2,x,-1,1,y,-1,1))$

Graphic option: y_voxel

Default value: 10

y_voxel is the number of voxels in the y direction to be used by the marching cubes algorithm implemented by the 3d implicit object. It is also used by graphic object region.

Graphic option: yaxis

Default value: false

If yaxis is true, the y axis is drawn.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(explicit(x^3,x,-1,1),
             yaxis       = true,
             yaxis_color = blue)$

See also yaxis_width, yaxis_type and yaxis_color.

Graphic option: yaxis_color

Default value: "black"

yaxis_color specifies the color for the y axis. See color to know how colors are defined.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(explicit(x^3,x,-1,1),
             yaxis       = true,
             yaxis_color = red)$

See also yaxis, yaxis_width and yaxis_type.

Graphic option: yaxis_secondary

Default value: false

If yaxis_secondary is true, function values can be plotted with respect to the second y axis, which will be drawn on the right side of the scene.

Note that this is a local graphics option which only affects to 2d plots.

Example:

(%i1) draw2d(
         explicit(sin(x),x,0,10),
         yaxis_secondary = true,
         ytics_secondary = true,
         color = blue,
         explicit(100*sin(x+0.1)+2,x,0,10));

See also yrange_secondary, ytics_secondary, ytics_rotate_secondary and ytics_axis_secondary

Graphic option: yaxis_type

Default value: dots

yaxis_type indicates how the y axis is displayed; possible values are solid and dots.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(explicit(x^3,x,-1,1),
             yaxis       = true,
             yaxis_type  = solid)$

See also yaxis, yaxis_width and yaxis_color.

Graphic option: yaxis_width

Default value: 1

yaxis_width is the width of the y axis. Its value must be a positive number.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(explicit(x^3,x,-1,1),
             yaxis       = true,
             yaxis_width = 3)$

See also yaxis, yaxis_type and yaxis_color.

Graphic option: ylabel

Default value: ""

Option ylabel, a string, is the label for the y axis. By default, the axis is labeled with string "y".

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(xlabel = "Time",
             ylabel = "Population",
             explicit(exp(u),u,-2,2) )$

See also xlabel_draw, xlabel_secondary, ylabel_secondary, and zlabel_draw.

Graphic option: ylabel_secondary

Default value: "" (empty string)

Option ylabel_secondary, a string, is the label for the secondary y axis. By default, no label is written.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(
        key_pos=bottom_right,
        key="current",
        xlabel="t[s]",
        ylabel="I[A]",ylabel_secondary="P[W]",
        explicit(sin(t),t,0,10),
        yaxis_secondary=true,
        ytics_secondary=true,
        color=red,key="Power",
        explicit((sin(t))^2,t,0,10)
    )$

See also xlabel_draw, xlabel_secondary, ylabel_draw and zlabel_draw.

Graphic option: yrange

Default value: auto

If yrange is auto, the range for the y coordinate is computed automatically.

If the user wants a specific interval for y, it must be given as a Maxima list, as in yrange=[-2, 3].

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(yrange = [-2,3],
             explicit(x^2,x,-1,1),
             xrange = [-3,3])$

See also xrange, yrange_secondary and zrange.

Graphic option: yrange_secondary

Default value: auto

If yrange_secondary is auto, the range for the second y axis is computed automatically.

If the user wants a specific interval for the second y axis, it must be given as a Maxima list, as in yrange_secondary=[-2, 3].

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw2d(
         explicit(sin(x),x,0,10),
         yaxis_secondary = true,
         ytics_secondary = true,
         yrange = [-3, 3],
         yrange_secondary = [-20, 20],
         color = blue,
         explicit(100*sin(x+0.1)+2,x,0,10)) $

See also xrange, yrange and zrange.

Graphic option: ytics

Default value: true

This graphic option controls the way tic marks are drawn on the y axis.

See xtics for a complete description.

Graphic option: ytics_axis

Default value: false

If ytics_axis is true, tic marks and their labels are plotted just along the y axis, if it is false tics are plotted on the border.

Since this is a global graphics option, its position in the scene description does not matter.

Graphic option: ytics_rotate

Default value: false

If ytics_rotate is true, tic marks on the y axis are rotated 90 degrees.

Since this is a global graphics option, its position in the scene description does not matter.

Graphic option: ytics_rotate_secondary

Default value: false

If ytics_rotate_secondary is true, tic marks on the secondary y axis are rotated 90 degrees.

Since this is a global graphics option, its position in the scene description does not matter.

Graphic option: ytics_secondary

Default value: auto

This graphic option controls the way tic marks are drawn on the second y axis.

See xtics for a complete description.

Graphic option: ytics_secondary_axis

Default value: false

If ytics_secondary_axis is true, tic marks and their labels are plotted just along the secondary y axis, if it is false tics are plotted on the border.

Since this is a global graphics option, its position in the scene description does not matter.

Graphic option: yv_grid

Default value: 30

yv_grid is the number of coordinates of the second variable (y in explicit and v in parametric 3d surfaces) to build the grid of sample points.

This option affects the following graphic objects:

Example:

(%i1) draw3d(xu_grid = 10,
             yv_grid = 50,
             explicit(x^2+y^2,x,-3,3,y,-3,3) )$
figures/draw_xugrid

See also xu_grid.

Graphic option: z_voxel

Default value: 10

z_voxel is the number of voxels in the z direction to be used by the marching cubes algorithm implemented by the 3d implicit object.

Graphic option: zaxis

Default value: false

If zaxis is true, the z axis is drawn in 3D plots. This option has no effect in 2D scenes.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1),
             zaxis       = true,
             zaxis_type  = solid,
             zaxis_color = blue)$

See also zaxis_width, zaxis_type and zaxis_color.

Graphic option: zaxis_color

Default value: "black"

zaxis_color specifies the color for the z axis. See color to know how colors are defined. This option has no effect in 2D scenes.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1),
             zaxis       = true,
             zaxis_type  = solid,
             zaxis_color = red)$

See also zaxis, zaxis_width and zaxis_type.

Graphic option: zaxis_type

Default value: dots

zaxis_type indicates how the z axis is displayed; possible values are solid and dots. This option has no effect in 2D scenes.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1),
             zaxis       = true,
             zaxis_type  = solid)$

See also zaxis, zaxis_width and zaxis_color.

Graphic option: zaxis_width

Default value: 1

zaxis_width is the width of the z axis. Its value must be a positive number. This option has no effect in 2D scenes.

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw3d(explicit(x^2+y^2,x,-1,1,y,-1,1),
             zaxis       = true,
             zaxis_type  = solid,
             zaxis_width = 3)$

See also zaxis, zaxis_type and zaxis_color.

Graphic option: zlabel

Default value: ""

Option zlabel, a string, is the label for the z axis. By default, the axis is labeled with string "z".

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw3d(zlabel = "Z variable",
             ylabel = "Y variable",
             explicit(sin(x^2+y^2),x,-2,2,y,-2,2),
             xlabel = "X variable" )$

See also xlabel_draw, ylabel_draw, and zlabel_rotate.

Graphic option: zlabel_rotate

Default value: "auto"

This graphics option allows to choose if the z axis label of 3d plots is drawn horizontal (false), vertical (true) or if maxima automatically chooses an orientation based on the length of the label (auto).

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw3d(
          explicit(sin(x)*sin(y),x,0,10,y,0,10),
          zlabel_rotate=false
      )$

See also zlabel_draw.

Graphic option: zrange

Default value: auto

If zrange is auto, the range for the z coordinate is computed automatically.

If the user wants a specific interval for z, it must be given as a Maxima list, as in zrange=[-2, 3].

Since this is a global graphics option, its position in the scene description does not matter.

Example:

(%i1) draw3d(yrange = [-3,3],
             zrange = [-2,5],
             explicit(x^2+y^2,x,-1,1,y,-1,1),
             xrange = [-3,3])$

See also xrange and yrange.

Graphic option: ztics

Default value: true

This graphic option controls the way tic marks are drawn on the z axis.

See xtics for a complete description.

Graphic option: ztics_axis

Default value: false

If ztics_axis is true, tic marks and their labels are plotted just along the z axis, if it is false tics are plotted on the border.

Since this is a global graphics option, its position in the scene description does not matter.

Graphic option: ztics_rotate

Default value: false

If ztics_rotate is true, tic marks on the z axis are rotated 90 degrees.

Since this is a global graphics option, its position in the scene description does not matter.

53.2.4 Graphics objects

Graphic object: bars ([x1,h1,w1], [x2,h2,w2, ...])

Draws vertical bars in 2D.

2D

bars ([x1,h1,w1], [x2,h2,w2, ...]) draws bars centered at values x1, x2, ... with heights h1, h2, ... and widths w1, w2, ...

This object is affected by the following graphic options: key, fill_color, fill_density and line_width.

Example:

(%i1) draw2d(
       key          = "Group A",
       fill_color   = blue,
       fill_density = 0.2,
       bars([0.8,5,0.4],[1.8,7,0.4],[2.8,-4,0.4]),
       key          = "Group B",
       fill_color   = red,
       fill_density = 0.6,
       line_width   = 4,
       bars([1.2,4,0.4],[2.2,-2,0.4],[3.2,5,0.4]),
       xaxis = true);
figures/draw_bars

Graphic object: cylindrical (radius, z, minz, maxz, azi, minazi, maxazi)

Draws 3D functions defined in cylindrical coordinates.

3D

cylindrical(radius, z, minz, maxz, azi, minazi, maxazi) plots the function radius(z, azi) defined in cylindrical coordinates, with variable z taking values from minz to maxz and azimuth azi taking values from minazi to maxazi.

This object is affected by the following graphic options: xu_grid, yv_grid, line_type, key, wired_surface, enhanced3d and color

Example:

(%i1) draw3d(cylindrical(1,z,-2,2,az,0,2*%pi))$
figures/draw_cylindrical

Graphic object: elevation_grid (mat,x0,y0,width,height)

Draws matrix mat in 3D space. z values are taken from mat, the abscissas range from x0 to x0 + width and ordinates from y0 to y0 + height. Element a(1,1) is projected on point (x0,y0+height), a(1,n) on (x0+width,y0+height), a(m,1) on (x0,y0), and a(m,n) on (x0+width,y0).

This object is affected by the following graphic options: line_type,, line_width key, wired_surface, enhanced3d and color

In older versions of Maxima, elevation_grid was called mesh. See also mesh.

Example:

(%i1) m: apply(
            matrix,
            makelist(makelist(random(10.0),k,1,30),i,1,20)) $
(%i2) draw3d(
         color = blue,
         elevation_grid(m,0,0,3,2),
         xlabel = "x",
         ylabel = "y",
         surface_hide = true);
figures/draw_elevation_grid

Graphic object: ellipse (xc, yc, a, b, ang1, ang2)

Draws ellipses and circles in 2D.

2D

ellipse (xc, yc, a, b, ang1, ang2) plots an ellipse centered at [xc, yc] with horizontal and vertical semi axis a and b, respectively, starting at angle ang1 with an amplitude equal to angle ang2.

This object is affected by the following graphic options: nticks, transparent, fill_color, border, line_width, line_type, key and color

Example:

(%i1) draw2d(transparent = false,
             fill_color  = red,
             color       = gray30,
             transparent = false,
             line_width  = 5,
             ellipse(0,6,3,2,270,-270),
             /* center (x,y), a, b, start & end in degrees */
             transparent = true,
             color       = blue,
             line_width  = 3,
             ellipse(2.5,6,2,3,30,-90),
             xrange      = [-3,6],
             yrange      = [2,9] )$
figures/draw_ellipse

Graphic object: errors ([x1, x2, …], [y1, y2, …])

Draws points with error bars, horizontally, vertically or both, depending on the value of option error_type.

2D

If error_type = x, arguments to errors must be of the form [x, y, xdelta] or [x, y, xlow, xhigh]. If error_type = y, arguments must be of the form [x, y, ydelta] or [x, y, ylow, yhigh]. If error_type = xy or error_type = boxes, arguments to errors must be of the form [x, y, xdelta, ydelta] or [x, y, xlow, xhigh, ylow, yhigh].

See also error_type.

This object is affected by the following graphic options: error_type, points_joined, line_width, key, line_type, color fill_density, xaxis_secondary and yaxis_secondary.

Option fill_density is only relevant when error_type=boxes.

Examples:

Horizontal error bars.

(%i1) draw2d(
        error_type = 'y,
        errors([[1,2,1], [3,5,3], [10,3,1], [17,6,2]]))$
figures/draw_errors

Vertical and horizontal error bars.

(%i1) draw2d(
        error_type = 'xy,
        points_joined = true,
        color = blue,
        errors([[1,2,1,2], [3,5,2,1], [10,3,1,1], [17,6,1/2,2]])); 
figures/draw_errors2

Graphic object: explicit
    explicit (expr,var,minval,maxval)
    explicit (fcn,var,minval,maxval)
    explicit (expr,var1,minval1,maxval1,var2,minval2,maxval2)
    explicit (fcn,var1,minval1,maxval1,var2,minval2,maxval2)

Draws explicit functions in 2D and 3D.

2D

explicit(fcn,var,minval,maxval) plots explicit function fcn, with variable var taking values from minval to maxval.

This object is affected by the following graphic options: nticks, adapt_depth, draw_realpart, line_width, line_type, key, filled_func, fill_color and color

Example:

(%i1) draw2d(line_width = 3,
             color      = blue,
             explicit(x^2,x,-3,3) )$
figures/draw_explicit
(%i2) draw2d(fill_color  = brown,
             filled_func = true,
             explicit(x^2,x,-3,3) )$
figures/draw_explicit2

3D

explicit(fcn, var1, minval1, maxval1, var2, minval2, maxval2) plots the explicit function fcn, with variable var1 taking values from minval1 to maxval1 and variable var2 taking values from minval2 to maxval2.

This object is affected by the following graphic options: draw_realpart, xu_grid, yv_grid, line_type, line_width, key, wired_surface, enhanced3d and color.

Example:

(%i1) draw3d(key   = "Gauss",
             color = "#a02c00",
             explicit(20*exp(-x^2-y^2)-10,x,-3,3,y,-3,3),
             yv_grid     = 10,
             color = blue,
             key   = "Plane",
             explicit(x+y,x,-5,5,y,-5,5),
             surface_hide = true)$
figures/draw_explicit3

See also filled_func for filled functions.

Graphic object: image (im,x0,y0,width,height)

Renders images in 2D.

2D

image (im,x0,y0,width,height) plots image im in the rectangular region from vertex (x0,y0) to (x0+width,y0+height) on the real plane. Argument im must be a matrix of real numbers, a matrix of vectors of length three or a picture object.

If im is a matrix of real numbers or a levels picture object, pixel values are interpreted according to graphic option palette, which is a vector of length three with components ranging from -36 to +36; each value is an index for a formula mapping the levels onto red, green and blue colors, respectively:

 0: 0               1: 0.5           2: 1
 3: x               4: x^2           5: x^3
 6: x^4             7: sqrt(x)       8: sqrt(sqrt(x))
 9: sin(90x)       10: cos(90x)     11: |x-0.5|
12: (2x-1)^2       13: sin(180x)    14: |cos(180x)|
15: sin(360x)      16: cos(360x)    17: |sin(360x)|
18: |cos(360x)|    19: |sin(720x)|  20: |cos(720x)|
21: 3x             22: 3x-1         23: 3x-2
24: |3x-1|         25: |3x-2|       26: (3x-1)/2
27: (3x-2)/2       28: |(3x-1)/2|   29: |(3x-2)/2|
30: x/0.32-0.78125                  31: 2*x-0.84
32: 4x;1;-2x+1.84;x/0.08-11.5
33: |2*x - 0.5|    34: 2*x          35: 2*x - 0.5
36: 2*x - 1

negative numbers mean negative colour component.

palette = gray and palette = color are short cuts for palette = [3,3,3] and palette = [7,5,15], respectively.

If im is a matrix of vectors of length three or an rgb picture object, they are interpreted as red, green and blue color components.

Examples:

If im is a matrix of real numbers, pixel values are interpreted according to graphic option palette.

(%i1) im: apply(
           'matrix,
            makelist(makelist(random(200),i,1,30),i,1,30))$
(%i2) /* palette = color, default */
      draw2d(image(im,0,0,30,30))$
figures/draw_image
(%i3) draw2d(palette = gray, image(im,0,0,30,30))$
figures/draw_image2
(%i4) draw2d(palette = [15,20,-4],
             colorbox=false,
             image(im,0,0,30,30))$
figures/draw_image3

See also colorbox.

If im is a matrix of vectors of length three, they are interpreted as red, green and blue color components.

(%i1) im: apply(
            'matrix,
             makelist(
               makelist([random(300),
                         random(300),
                         random(300)],i,1,30),i,1,30))$
(%i2) draw2d(image(im,0,0,30,30))$
figures/draw_image4

Package draw automatically loads package picture. In this example, a level picture object is built by hand and then rendered.

(%i1) im: make_level_picture([45,87,2,134,204,16],3,2);
(%o1)       picture(level, 3, 2, {Array:  #(45 87 2 134 204 16)})
(%i2) /* default color palette */
      draw2d(image(im,0,0,30,30))$
figures/draw_image5
(%i3) /* gray palette */
      draw2d(palette = gray,
             image(im,0,0,30,30))$
figures/draw_image6

An xpm file is read and then rendered.

(%i1) im: read_xpm("myfile.xpm")$
(%i2) draw2d(image(im,0,0,10,7))$

See also make_level_picture, make_rgb_picture and read_xpm.

http://www.telefonica.net/web2/biomates/maxima/gpdraw/image
contains more elaborated examples.

Graphic object: implicit
    implicit (fcn,x,xmin,xmax,y,ymin,ymax)
    implicit (fcn,x,xmin,xmax,y,ymin,ymax,z,zmin,zmax)

Draws implicit functions in 2D and 3D.

2D

implicit(fcn,x,xmin,xmax,y,ymin,ymax) plots the implicit function defined by fcn, with variable x taking values from xmin to xmax, and variable y taking values from ymin to ymax.

This object is affected by the following graphic options: ip_grid, ip_grid_in, line_width, line_type, key and color.

Example:

(%i1) draw2d(grid      = true,
             line_type = solid,
             key       = "y^2=x^3-2*x+1",
             implicit(y^2=x^3-2*x+1, x, -4,4, y, -4,4),
             line_type = dots,
             key       = "x^3+y^3 = 3*x*y^2-x-1",
             implicit(x^3+y^3 = 3*x*y^2-x-1, x,-4,4, y,-4,4),
             title     = "Two implicit functions" )$
figures/draw_implicit

3D

implicit (fcn,x,xmin,xmax, y,ymin,ymax, z,zmin,zmax) plots the implicit surface defined by fcn, with variable x taking values from xmin to xmax, variable y taking values from ymin to ymax and variable z taking values from zmin to zmax. This object implements the marching cubes algorithm.

This object is affected by the following graphic options: x_voxel, y_voxel, z_voxel, line_width, line_type, key, wired_surface, enhanced3d and color.

Example:

(%i1) draw3d(
        color=blue,
        implicit((x^2+y^2+z^2-1)*(x^2+(y-1.5)^2+z^2-0.5)=0.015,
                 x,-1,1,y,-1.2,2.3,z,-1,1),
        surface_hide=true);
figures/draw_implicit2

Graphic object: label
    label ([string,x,y],...)
    label ([string,x,y,z],...)

Writes labels in 2D and 3D.

Colored labels work only with Gnuplot 4.3 and up.

This object is affected by the following graphic options: label_alignment, label_orientation and color.

2D

label([string,x,y]) writes the string at point [x,y].

Example:

(%i1) draw2d(yrange = [0.1,1.4],
             color = red,
             label(["Label in red",0,0.3]),
             color = "#0000ff",
             label(["Label in blue",0,0.6]),
             color = light_blue,
             label(["Label in light-blue",0,0.9],
                   ["Another light-blue",0,1.2])  )$
figures/draw_label

3D

label([string,x,y,z]) writes the string at point [x,y,z].

Example:

(%i1) draw3d(explicit(exp(sin(x)+cos(x^2)),x,-3,3,y,-3,3),
             color = red,
             label(["UP 1",-2,0,3], ["UP 2",1.5,0,4]),
             color = blue,
             label(["DOWN 1",2,0,-3]) )$
figures/draw_label2

Graphic object: mesh (row_1,row_2,...)

Draws a quadrangular mesh in 3D.

3D

Argument row_i is a list of n 3D points of the form [[x_i1,y_i1,z_i1], ...,[x_in,y_in,z_in]], and all rows are of equal length. All these points define an arbitrary surface in 3D and in some sense it’s a generalization of the elevation_grid object.

This object is affected by the following graphic options: line_type, line_width, color, key, wired_surface, enhanced3d and transform.

Examples:

A simple example.

(%i1) draw3d( 
         mesh([[1,1,3],   [7,3,1],[12,-2,4],[15,0,5]],
              [[2,7,8],   [4,3,1],[10,5,8], [12,7,1]],
              [[-2,11,10],[6,9,5],[6,15,1], [20,15,2]])) $
figures/draw_mesh

Plotting a triangle in 3D.

(%i1) draw3d(
        line_width = 2,
        mesh([[1,0,0],[0,1,0]],
             [[0,0,1],[0,0,1]])) $
figures/draw_mesh2

Two quadrilaterals.

(%i1) draw3d(
        surface_hide = true,
        line_width   = 3,
        color = red,
        mesh([[0,0,0], [0,1,0]],
             [[2,0,2], [2,2,2]]),
        color = blue,
        mesh([[0,0,2], [0,1,2]],
             [[2,0,4], [2,2,4]])) $
figures/draw_mesh3

Graphic object: parametric
    parametric (xfun,yfun,par,parmin,parmax)
    parametric (xfun,yfun,zfun,par,parmin,parmax)

Draws parametric functions in 2D and 3D.

This object is affected by the following graphic options: nticks, line_width, line_type, key, color and enhanced3d.

2D

The command parametric(xfun, yfun, par, parmin, parmax) plots the parametric function [xfun, yfun], with parameter par taking values from parmin to parmax.

Example:

(%i1) draw2d(explicit(exp(x),x,-1,3),
             color = red,
             key   = "This is the parametric one!!",
             parametric(2*cos(rrr),rrr^2,rrr,0,2*%pi))$
figures/draw_parametric

3D

parametric(xfun, yfun, zfun, par, parmin, parmax) plots the parametric curve [xfun, yfun, zfun], with parameter par taking values from parmin to parmax.

Example:

(%i1) draw3d(explicit(exp(sin(x)+cos(x^2)),x,-3,3,y,-3,3),
             color = royalblue,
             parametric(cos(5*u)^2,sin(7*u),u-2,u,0,2),
             color      = turquoise,
             line_width = 2,
             parametric(t^2,sin(t),2+t,t,0,2),
             surface_hide = true,
             title = "Surface & curves" )$
figures/draw_parametric2

Graphic object: parametric_surface (xfun, yfun, zfun, par1, par1min, par1max, par2, par2min, par2max)

Draws parametric surfaces in 3D.

3D

The command parametric_surface(xfun, yfun, zfun, par1, par1min, par1max, par2, par2min, par2max) plots the parametric surface [xfun, yfun, zfun], with parameter par1 taking values from par1min to par1max and parameter par2 taking values from par2min to par2max.

This object is affected by the following graphic options: draw_realpart, xu_grid, yv_grid, line_type, line_width, key, wired_surface, enhanced3d and color.

Example:

(%i1) draw3d(title          = "Sea shell",
             xu_grid        = 100,
             yv_grid        = 25,
             view           = [100,20],
             surface_hide   = true,
             parametric_surface(0.5*u*cos(u)*(cos(v)+1),
                           0.5*u*sin(u)*(cos(v)+1),
                           u*sin(v) - ((u+3)/8*%pi)^2 - 20,
                           u, 0, 13*%pi, v, -%pi, %pi) )$
figures/draw_parametric3

Graphic object: points
    points ([[x1,y1], [x2,y2],...])
    points ([x1,x2,...], [y1,y2,...])
    points ([y1,y2,...])
    points ([[x1,y1,z1], [x2,y2,z2],...])
    points ([x1,x2,...], [y1,y2,...], [z1,z2,...])
    points (matrix)
    points (1d_y_array)
    points (1d_x_array, 1d_y_array)
    points (1d_x_array, 1d_y_array, 1d_z_array)
    points (2d_xy_array)
    points (2d_xyz_array)

Draws points in 2D and 3D.

This object is affected by the following graphic options: point_size, point_type, points_joined, line_width, key, line_type and color. In 3D mode, it is also affected by enhanced3d

2D

points ([[x1,y1], [x2,y2],...]) or points ([x1,x2,...], [y1,y2,...]) plots points [x1,y1], [x2,y2], etc. If abscissas are not given, they are set to consecutive positive integers, so that points ([y1,y2,...]) draws points [1,y1], [2,y2], etc. If matrix is a two-column or two-row matrix, points (matrix) draws the associated points. If matrix is a one-column or one-row matrix, abscissas are assigned automatically.

If 1d_y_array is a 1D lisp array of numbers, points (1d_y_array) plots them setting abscissas to consecutive positive integers. points (1d_x_array, 1d_y_array) plots points with their coordinates taken from the two arrays passed as arguments. If 2d_xy_array is a 2D array with two columns, or with two rows, points (2d_xy_array) plots the corresponding points on the plane.

Examples:

Two types of arguments for points, a list of pairs and two lists of separate coordinates.

(%i1) draw2d(
        key = "Small points",
        points(makelist([random(20),random(50)],k,1,10)),
        point_type    = circle,
        point_size    = 3,
        points_joined = true,
        key           = "Great points",
        points(makelist(k,k,1,20),makelist(random(30),k,1,20)),
        point_type    = filled_down_triangle,
        key           = "Automatic abscissas",
        color         = red,
        points([2,12,8]))$
figures/draw_points

Drawing impulses.

(%i1) draw2d(
        points_joined = impulses,
        line_width    = 2,
        color         = red,
        points(makelist([random(20),random(50)],k,1,10)))$
figures/draw_points2

Array with ordinates.

(%i1) a: make_array (flonum, 100) $
(%i2) for i:0 thru 99 do a[i]: random(1.0) $
(%i3) draw2d(points(a)) $
figures/draw_points3

Two arrays with separate coordinates.

(%i1) x: make_array (flonum, 100) $
(%i2) y: make_array (fixnum, 100) $
(%i3) for i:0 thru 99 do (
        x[i]: float(i/100),
        y[i]: random(10) ) $
(%i4) draw2d(points(x, y)) $
figures/draw_points4

A two-column 2D array.

(%i1) xy: make_array(flonum, 100, 2) $
(%i2) for i:0 thru 99 do (
        xy[i, 0]: float(i/100),
        xy[i, 1]: random(10) ) $
(%i3) draw2d(points(xy)) $
figures/draw_points5

Drawing an array filled with function read_array.

(%i1) a: make_array(flonum,100) $
(%i2) read_array (file_search ("pidigits.data"), a) $
(%i3) draw2d(points(a)) $

3D

points([[x1, y1, z1], [x2, y2, z2], ...]) or points([x1, x2, ...], [y1, y2, ...], [z1, z2,...]) plots points [x1, y1, z1], [x2, y2, z2], etc. If matrix is a three-column or three-row matrix, points (matrix) draws the associated points.

When arguments are lisp arrays, points (1d_x_array, 1d_y_array, 1d_z_array) takes coordinates from the three 1D arrays. If 2d_xyz_array is a 2D array with three columns, or with three rows, points (2d_xyz_array) plots the corresponding points.

Examples:

One tridimensional sample,

(%i1) load ("numericalio")$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) draw3d(title = "Daily average wind speeds",
             point_size = 2,
             points(args(submatrix (s2, 4, 5))) )$

Two tridimensional samples,

(%i1) load ("numericalio")$
(%i2) s2 : read_matrix (file_search ("wind.data"))$
(%i3) draw3d(
         title = "Daily average wind speeds. Two data sets",
         point_size = 2,
         key        = "Sample from stations 1, 2 and 3",
         points(args(submatrix (s2, 4, 5))),
         point_type = 4,
         key        = "Sample from stations 1, 4 and 5",
         points(args(submatrix (s2, 2, 3))) )$

Unidimensional arrays,

(%i1) x: make_array (fixnum, 10) $
(%i2) y: make_array (fixnum, 10) $
(%i3) z: make_array (fixnum, 10) $
(%i4) for i:0 thru 9 do (
        x[i]: random(10),
        y[i]: random(10),
        z[i]: random(10) ) $
(%i5) draw3d(points(x,y,z)) $
figures/draw_points6

Bidimensional colored array,

(%i1) xyz: make_array(fixnum, 10, 3) $
(%i2) for i:0 thru 9 do (
        xyz[i, 0]: random(10),
        xyz[i, 1]: random(10),
        xyz[i, 2]: random(10) ) $
(%i3) draw3d(
         enhanced3d = true,
         points_joined = true,
         points(xyz)) $
figures/draw_points7

Color numbers explicitly specified by the user.

(%i1) pts: makelist([t,t^2,cos(t)], t, 0, 15)$
(%i2) col_num: makelist(k, k, 1, length(pts))$
(%i3) draw3d(
        enhanced3d = ['part(col_num,k),k],
        point_size = 3,
        point_type = filled_circle,
        points(pts))$
figures/draw_points8

Graphic object: polar (radius,ang,minang,maxang)

Draws 2D functions defined in polar coordinates.

2D

polar (radius,ang,minang,maxang) plots function radius(ang) defined in polar coordinates, with variable ang taking values from minang to maxang.

This object is affected by the following graphic options: nticks, line_width, line_type, key and color.

Example:

(%i1) draw2d(user_preamble = "set grid polar",
             nticks        = 200,
             xrange        = [-5,5],
             yrange        = [-5,5],
             color         = blue,
             line_width    = 3,
             title         = "Hyperbolic Spiral",
             polar(10/theta,theta,1,10*%pi) )$
figures/draw_polar

Graphic object: polygon
    polygon ([[x1, y1], [x2, y2], …])
    polygon ([x1, x2, …], [y1, y2, …])

Draws polygons in 2D.

2D

The commands polygon([[x1, y1], [x2, y2], ...]) or polygon([x1, x2, ...], [y1, y2, ...]) plot on the plane a polygon with vertices [x1, y1], [x2, y2], etc.

This object is affected by the following graphic options: transparent, fill_color, border, line_width, key, line_type and color.

Example:

(%i1) draw2d(color      = "#e245f0",
             line_width = 8,
             polygon([[3,2],[7,2],[5,5]]),
             border      = false,
             fill_color  = yellow,
             polygon([[5,2],[9,2],[7,5]]) )$
figures/draw_polygon

Graphic object: quadrilateral (point_1, point_2, point_3, point_4)

Draws a quadrilateral.

2D

quadrilateral([x1, y1], [x2, y2], [x3, y3], [x4, y4]) draws a quadrilateral with vertices [x1, y1], [x2, y2], [x3, y3], and [x4, y4].

This object is affected by the following graphic options:
transparent, fill_color, border, line_width, key, xaxis_secondary, yaxis_secondary, line_type, transform and color.

Example:

(%i1) draw2d(
        quadrilateral([1,1],[2,2],[3,-1],[2,-2]))$
figures/draw_quadrilateral

3D

quadrilateral([x1, y1, z1], [x2, y2, z2], [x3, y3, z3], [x4, y4, z4]) draws a quadrilateral with vertices [x1, y1, z1], [x2, y2, z2], [x3, y3, z3], and [x4, y4, z4].

This object is affected by the following graphic options: line_type, line_width, color, key, enhanced3d and transform.

Graphic object: rectangle ([x1,y1], [x2,y2])

Draws rectangles in 2D.

2D

rectangle ([x1,y1], [x2,y2]) draws a rectangle with opposite vertices [x1,y1] and [x2,y2].

This object is affected by the following graphic options: transparent, fill_color, border, line_width, key, line_type and color.

Example:

(%i1) draw2d(fill_color  = red,
             line_width  = 6,
             line_type   = dots,
             transparent = false,
             fill_color  = blue,
             rectangle([-2,-2],[8,-1]), /* opposite vertices */
             transparent = true,
             line_type   = solid,
             line_width  = 1,
             rectangle([9,4],[2,-1.5]),
             xrange      = [-3,10],
             yrange      = [-3,4.5] )$
figures/draw_rectangle

Graphic object: region (expr,var1,minval1,maxval1,var2,minval2,maxval2)

Plots a region on the plane defined by inequalities.

2D expr is an expression formed by inequalities and boolean operators and, or, and not. The region is bounded by the rectangle defined by [minval1, maxval1] and [minval2, maxval2].

This object is affected by the following graphic options: fill_color, key, x_voxel and y_voxel.

Example:

(%i1) draw2d(
        x_voxel = 30,
        y_voxel = 30,
        region(x^2+y^2<1 and x^2+y^2 > 1/2,
               x, -1.5, 1.5, y, -1.5, 1.5));
figures/draw_region
Graphic object: spherical (radius, azi, minazi, maxazi, zen, minzen, maxzen)

Draws 3D functions defined in spherical coordinates.

3D

spherical(radius, azi, minazi, maxazi, zen, minzen, maxzen) plots the function radius(azi, zen) defined in spherical coordinates, with azimuth azi taking values from minazi to maxazi and zenith zen taking values from minzen to maxzen.

This object is affected by the following graphic options: xu_grid, yv_grid, line_type, key, wired_surface, enhanced3d and color.

Example:

(%i1) draw3d(spherical(1,a,0,2*%pi,z,0,%pi))$
figures/draw_spherical

Graphic object: triangle (point_1, point_2, point_3)

Draws a triangle.

2D

triangle ([x1,y1], [x2,y2], [x3,y3]) draws a triangle with vertices [x1,y1], [x2,y2], and [x3,y3].

This object is affected by the following graphic options:
transparent, fill_color, border, line_width, key, xaxis_secondary, yaxis_secondary, line_type, transform and color.

Example:

(%i1) draw2d(
        triangle([1,1],[2,2],[3,-1]))$
figures/draw_triangle

3D

triangle ([x1,y1,z1], [x2,y2,z2], [x3,y3,z3]) draws a triangle with vertices [x1,y1,z1], [x2,y2,z2], and [x3,y3,z3].

This object is affected by the following graphic options: line_type, line_width, color, key, enhanced3d and transform.

Graphic object: tube (xfun,yfun,zfun,rfun,p,pmin,pmax)

Draws a tube in 3D with varying diameter.

3D

[xfun,yfun,zfun] is the parametric curve with parameter p taking values from pmin to pmax. Circles of radius rfun are placed with their centers on the parametric curve and perpendicular to it.

This object is affected by the following graphic options: xu_grid, yv_grid, line_type, line_width, key, wired_surface, enhanced3d, color and capping.

Example:

(%i1) draw3d(
        enhanced3d = true,
        xu_grid = 50,
        tube(cos(a), a, 0, cos(a/10)^2,
             a, 0, 4*%pi) )$
figures/draw_tube

Graphic object: vector
    vector ([x,y], [dx,dy])
    vector ([x,y,z], [dx,dy,dz])

Draws vectors in 2D and 3D.

This object is affected by the following graphic options: head_both, head_length, head_angle, head_type, line_width, line_type, key and color.

2D

vector([x,y], [dx,dy]) plots vector [dx,dy] with origin in [x,y].

Example:

(%i1) draw2d(xrange      = [0,12],
             yrange      = [0,10],
             head_length = 1,
             vector([0,1],[5,5]), /* default type */
             head_type = 'empty,
             vector([3,1],[5,5]),
             head_both = true,
             head_type = 'nofilled,
             line_type = dots,
             vector([6,1],[5,5]))$
figures/draw_vector

3D

vector([x,y,z], [dx,dy,dz]) plots vector [dx,dy,dz] with origin in [x,y,z].

Example:

(%i1) draw3d(color = cyan,
             vector([0,0,0],[1,1,1]/sqrt(3)),
             vector([0,0,0],[1,-1,0]/sqrt(2)),
             vector([0,0,0],[1,1,-2]/sqrt(6)) )$
figures/draw_vector2


Next: , Previous: , Up: draw-pkg   [Contents][Index]

53.3 Functions and Variables for pictures

Function: get_pixel (pic,x,y)

Returns pixel from picture. Coordinates x and y range from 0 to width-1 and height-1, respectively.

Function: make_level_picture
    make_level_picture (data)
    make_level_picture (data,width,height)

Returns a levels picture object. make_level_picture (data) builds the picture object from matrix data. make_level_picture (data,width,height) builds the object from a list of numbers; in this case, both the width and the height must be given.

The returned picture object contains the following four parts:

  1. symbol level
  2. image width
  3. image height
  4. an integer array with pixel data ranging from 0 to 255. Argument data must contain only numbers ranged from 0 to 255; negative numbers are substituted by 0, and those which are greater than 255 are set to 255.

Example:

Level picture from matrix.

(%i1) make_level_picture(matrix([3,2,5],[7,-9,3000]));
(%o1)         picture(level, 3, 2, {Array:  #(3 2 5 7 0 255)})

Level picture from numeric list.

(%i1) make_level_picture([-2,0,54,%pi],2,2);
(%o1)            picture(level, 2, 2, {Array:  #(0 0 54 3)})

Function: make_rgb_picture (redlevel,greenlevel,bluelevel)

Returns an rgb-coloured picture object. All three arguments must be levels picture; with red, green and blue levels.

The returned picture object contains the following four parts:

  1. symbol rgb
  2. image width
  3. image height
  4. an integer array of length 3*width*height with pixel data ranging from 0 to 255. Each pixel is represented by three consecutive numbers (red, green, blue).

Example:

(%i1) red: make_level_picture(matrix([3,2],[7,260]));
(%o1)           picture(level, 2, 2, {Array:  #(3 2 7 255)})
(%i2) green: make_level_picture(matrix([54,23],[73,-9]));
(%o2)           picture(level, 2, 2, {Array:  #(54 23 73 0)})
(%i3) blue: make_level_picture(matrix([123,82],[45,32.5698]));
(%o3)          picture(level, 2, 2, {Array:  #(123 82 45 33)})
(%i4) make_rgb_picture(red,green,blue);
(%o4) picture(rgb, 2, 2, 
              {Array:  #(3 54 123 2 23 82 7 73 45 255 0 33)})

Function: negative_picture (pic)

Returns the negative of a (level or rgb) picture.

Function: picture_equalp (x,y)

Returns true in case of equal pictures, and false otherwise.

Function: picturep (x)

Returns true if the argument is a well formed image, and false otherwise.

Function: read_xpm (xpm_file)

Reads a file in xpm and returns a picture object.

Function: rgb2level (pic)

Transforms an rgb picture into a level one by averaging the red, green and blue channels.

Function: take_channel (im,color)

If argument color is red, green or blue, function take_channel returns the corresponding color channel of picture im. Example:

(%i1) red: make_level_picture(matrix([3,2],[7,260]));
(%o1)           picture(level, 2, 2, {Array:  #(3 2 7 255)})
(%i2) green: make_level_picture(matrix([54,23],[73,-9]));
(%o2)           picture(level, 2, 2, {Array:  #(54 23 73 0)})
(%i3) blue: make_level_picture(matrix([123,82],[45,32.5698]));
(%o3)          picture(level, 2, 2, {Array:  #(123 82 45 33)})
(%i4) make_rgb_picture(red,green,blue);
(%o4) picture(rgb, 2, 2, 
              {Array:  #(3 54 123 2 23 82 7 73 45 255 0 33)})
(%i5) take_channel(%,'green);  /* simple quote!!! */
(%o5)           picture(level, 2, 2, {Array:  #(54 23 73 0)})


Previous: , Up: draw-pkg   [Contents][Index]

53.4 Functions and Variables for worldmap

53.4.1 Variables and Functions

Global variable: boundaries_array

Default value: false

boundaries_array is where the graphic object geomap looks for boundaries coordinates.

Each component of boundaries_array is an array of floating point quantities, the coordinates of a polygonal segment or map boundary.

See also geomap.

Function: numbered_boundaries (nlist)

Draws a list of polygonal segments (boundaries), labeled by its numbers (boundaries_array coordinates). This is of great help when building new geographical entities.

Example:

Map of Europe labeling borders with their component number in boundaries_array.

(%i1) load(worldmap)$
(%i2) european_borders: 
           region_boundaries(-31.81,74.92,49.84,32.06)$
(%i3) numbered_boundaries(european_borders)$

Function: make_poly_continent
    make_poly_continent (continent_name)
    make_poly_continent (country_list)

Makes the necessary polygons to draw a colored continent or a list of countries.

Example:

(%i1) load(worldmap)$
(%i2) /* A continent */
      make_poly_continent(Africa)$
(%i3) apply(draw2d, %)$
figures/worldmap_make_poly_continent
(%i4) /* A list of countries */
      make_poly_continent([Germany,Denmark,Poland])$
(%i5) apply(draw2d, %)$
figures/worldmap_make_poly_continent2

Function: make_poly_country (country_name)

Makes the necessary polygons to draw a colored country. If islands exist, one country can be defined with more than just one polygon.

Example:

(%i1) load(worldmap)$
(%i2) make_poly_country(India)$
(%i3) apply(draw2d, %)$
figures/worldmap_make_poly_country

Function: make_polygon (nlist)

Returns a polygon object from boundary indices. Argument nlist is a list of components of boundaries_array.

Example:

Bhutan is defined by boundary numbers 171, 173 and 1143, so that make_polygon([171,173,1143]) appends arrays of coordinates boundaries_array[171], boundaries_array[173] and boundaries_array[1143] and returns a polygon object suited to be plotted by draw. To avoid an error message, arrays must be compatible in the sense that any two consecutive arrays have two coordinates in the extremes in common. In this example, the two first components of boundaries_array[171] are equal to the last two coordinates of boundaries_array[173], and the two first of boundaries_array[173] are equal to the two first of boundaries_array[1143]; in conclussion, boundary numbers 171, 173 and 1143 (in this order) are compatible and the colored polygon can be drawn.

(%i1) load(worldmap)$
(%i2) Bhutan;
(%o2)                        [[171, 173, 1143]]
(%i3) boundaries_array[171];
(%o3) {Array:  
       #(88.750549 27.14727 88.806351 27.25305 88.901367 27.282221
         88.917877 27.321039)}
(%i4) boundaries_array[173];
(%o4) {Array:
       #(91.659554 27.76511 91.6008 27.66666 91.598022 27.62499
         91.631348 27.536381 91.765533 27.45694 91.775253 27.4161 
         92.007751 27.471939 92.11441 27.28583 92.015259 27.168051
         92.015533 27.08083 92.083313 27.02277 92.112183 26.920271
         92.069977 26.86194 91.997192 26.85194 91.915253 26.893881
         91.916924 26.85416 91.8358 26.863331 91.712479 26.799999 
         91.542191 26.80444 91.492188 26.87472 91.418854 26.873329
         91.371353 26.800831 91.307457 26.778049 90.682457 26.77417
         90.392197 26.903601 90.344131 26.894159 90.143044 26.75333
         89.98996 26.73583 89.841919 26.70138 89.618301 26.72694 
         89.636093 26.771111 89.360786 26.859989 89.22081 26.81472
         89.110237 26.829161 88.921631 26.98777 88.873016 26.95499
         88.867737 27.080549 88.843307 27.108601 88.750549 
         27.14727)}
(%i5) boundaries_array[1143];
(%o5) {Array:  
       #(91.659554 27.76511 91.666924 27.88888 91.65831 27.94805 
         91.338028 28.05249 91.314972 28.096661 91.108856 27.971109
         91.015808 27.97777 90.896927 28.05055 90.382462 28.07972
         90.396088 28.23555 90.366074 28.257771 89.996353 28.32333
         89.83165 28.24888 89.58609 28.139999 89.35997 27.87166 
         89.225517 27.795 89.125793 27.56749 88.971077 27.47361
         88.917877 27.321039)}
(%i6) Bhutan_polygon: make_polygon([171,173,1143])$
(%i7) draw2d(Bhutan_polygon)$
figures/worldmap_make_polygon

Function: region_boundaries (x1,y1,x2,y2)

Detects polygonal segments of global variable boundaries_array fully contained in the rectangle with vertices (x1,y1) -upper left- and (x2,y2) -bottom right-.

Example:

Returns segment numbers for plotting southern Italy.

(%i1) load(worldmap)$
(%i2) region_boundaries(10.4,41.5,20.7,35.4);
(%o2)                [1846, 1863, 1864, 1881, 1888, 1894]
(%i3) draw2d(geomap(%))$
figures/worldmap_region_boundaries

Function: region_boundaries_plus (x1,y1,x2,y2)

Detects polygonal segments of global variable boundaries_array containing at least one vertex in the rectangle defined by vertices (x1,y1) -upper left- and (x2,y2) -bottom right-.

Example:

(%i1) load(worldmap)$
(%i2) region_boundaries_plus(10.4,41.5,20.7,35.4);
(%o2) [1060, 1062, 1076, 1835, 1839, 1844, 1846, 1858,
       1861, 1863, 1864, 1871, 1881, 1888, 1894, 1897]
(%i3) draw2d(geomap(%))$
figures/worldmap_region_boundaries_plus

53.4.2 Graphic objects

Graphic object: geomap
    geomap (numlist)
    geomap (numlist,3Dprojection)

Draws cartographic maps in 2D and 3D.

2D

This function works together with global variable boundaries_array.

Argument numlist is a list containing numbers or lists of numbers. All these numbers must be integers greater or equal than zero, representing the components of global array boundaries_array.

Each component of boundaries_array is an array of floating point quantities, the coordinates of a polygonal segment or map boundary.

geomap (numlist) flattens its arguments and draws the associated boundaries in boundaries_array.

This object is affected by the following graphic options: line_width, line_type and color.

Examples:

A simple map defined by hand:

(%i1) load(worldmap)$
(%i2) /* Vertices of boundary #0: {(1,1),(2,5),(4,3)} */
   ( bnd0: make_array(flonum,6),
     bnd0[0]:1.0, bnd0[1]:1.0, bnd0[2]:2.0,
     bnd0[3]:5.0, bnd0[4]:4.0, bnd0[5]:3.0 )$
(%i3) /* Vertices of boundary #1: {(4,3),(5,4),(6,4),(5,1)} */
   ( bnd1: make_array(flonum,8),
     bnd1[0]:4.0, bnd1[1]:3.0, bnd1[2]:5.0, bnd1[3]:4.0,
     bnd1[4]:6.0, bnd1[5]:4.0, bnd1[6]:5.0, bnd1[7]:1.0)$
(%i4) /* Vertices of boundary #2: {(5,1), (3,0), (1,1)} */
   ( bnd2: make_array(flonum,6),
     bnd2[0]:5.0, bnd2[1]:1.0, bnd2[2]:3.0,
     bnd2[3]:0.0, bnd2[4]:1.0, bnd2[5]:1.0 )$
(%i5) /* Vertices of boundary #3: {(1,1), (4,3)} */
   ( bnd3: make_array(flonum,4),
     bnd3[0]:1.0, bnd3[1]:1.0, bnd3[2]:4.0, bnd3[3]:3.0)$
(%i6) /* Vertices of boundary #4: {(4,3), (5,1)} */
   ( bnd4: make_array(flonum,4),
     bnd4[0]:4.0, bnd4[1]:3.0, bnd4[2]:5.0, bnd4[3]:1.0)$
(%i7) /* Pack all together in boundaries_array */
   ( boundaries_array: make_array(any,5),
     boundaries_array[0]: bnd0, boundaries_array[1]: bnd1,
     boundaries_array[2]: bnd2, boundaries_array[3]: bnd3,
     boundaries_array[4]: bnd4 )$
(%i8) draw2d(geomap([0,1,2,3,4]))$
figures/worldmap_geomap

The auxiliary package worldmap sets the global variable boundaries_array to real world boundaries in (longitude, latitude) coordinates. These data are in the public domain and come from https://web.archive.org/web/20100310124019/http://www-cger.nies.go.jp/grid-e/gridtxt/grid19.html. Package worldmap defines also boundaries for countries, continents and coastlines as lists with the necessary components of boundaries_array (see file share/draw/worldmap.mac for more information). Package worldmap automatically loads package worldmap.

(%i1) load(worldmap)$
(%i2) c1: gr2d(geomap([Canada,United_States,
                       Mexico,Cuba]))$
(%i3) c2: gr2d(geomap(Africa))$
(%i4) c3: gr2d(geomap([Oceania,China,Japan]))$
(%i5) c4: gr2d(geomap([France,Portugal,Spain,
                       Morocco,Western_Sahara]))$
(%i6) draw(columns  = 2,
           c1,c2,c3,c4)$
figures/worldmap_geomap2

Package worldmap is also useful for plotting countries as polygons. In this case, graphic object geomap is no longer necessary and the polygon object is used instead. Since lists are now used and not arrays, maps rendering will be slower. See also make_poly_country and make_poly_continent to understand the following code.

(%i1) load(worldmap)$
(%i2) mymap: append(
   [color      = white],  /* borders are white */
   [fill_color = red],             make_poly_country(Bolivia),
   [fill_color = cyan],            make_poly_country(Paraguay),
   [fill_color = green],           make_poly_country(Colombia),
   [fill_color = blue],            make_poly_country(Chile),
   [fill_color = "#23ab0f"],       make_poly_country(Brazil),
   [fill_color = goldenrod],       make_poly_country(Argentina),
   [fill_color = "midnight-blue"], make_poly_country(Uruguay))$
(%i3) apply(draw2d, mymap)$
figures/worldmap_geomap3

3D

geomap (numlist) projects map boundaries on the sphere of radius 1 centered at (0,0,0). It is possible to change the sphere or the projection type by using geomap (numlist,3Dprojection).

Available 3D projections:

figures/worldmap_geomap6

See also http://riotorto.users.sf.net/gnuplot/geomap for more elaborated examples.


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54 drawdf


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54.1 Introduction to drawdf

The function drawdf draws the direction field of a first-order Ordinary Differential Equation (ODE) or a system of two autonomous first-order ODE’s.

Since this is an additional package, in order to use it you must first load it with load("drawdf"). Drawdf is built upon the draw package, which requires Gnuplot 4.2.

To plot the direction field of a single ODE, the ODE must be written in the form:

       dy
       -- = F(x,y)
       dx

and the function F should be given as the argument for drawdf. If the independent and dependent variables are not x, and y, as in the equation above, then those two variables should be named explicitly in a list given as an argument to the drawdf command (see the examples).

To plot the direction field of a set of two autonomous ODE’s, they must be written in the form

       dx             dy
       -- = G(x,y)    -- = F(x,y) 
       dt             dt

and the argument for drawdf should be a list with the two functions G and F, in that order; namely, the first expression in the list will be taken to be the time derivative of the variable represented on the horizontal axis, and the second expression will be the time derivative of the variable represented on the vertical axis. Those two variables do not have to be x and y, but if they are not, then the second argument given to drawdf must be another list naming the two variables, first the one on the horizontal axis and then the one on the vertical axis.

If only one ODE is given, drawdf will implicitly admit x=t, and G(x,y)=1, transforming the non-autonomous equation into a system of two autonomous equations.

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54.2 Functions and Variables for drawdf

54.2.1 Functions

Function: drawdf
    drawdf (dydx, ...options and objects...)
    drawdf (dvdu, [u,v], ...options and objects...)
    drawdf (dvdu, [u,umin,umax], [v,vmin,vmax], ...options and objects...)
    drawdf ([dxdt,dydt], ...options and objects...)
    drawdf ([dudt,dvdt], [u,v], ...options and objects...)
    drawdf ([dudt,dvdt], [u,umin,umax], [v,vmin,vmax], ...options and objects...)

Function drawdf draws a 2D direction field with optional solution curves and other graphics using the draw package.

The first argument specifies the derivative(s), and must be either an expression or a list of two expressions. dydx, dxdt and dydt are expressions that depend on x and y. dvdu, dudt and dvdt are expressions that depend on u and v.

If the independent and dependent variables are not x and y, then their names must be specified immediately following the derivative(s), either as a list of two names [u,v], or as two lists of the form [u,umin,umax] and [v,vmin,vmax].

The remaining arguments are graphic options, graphic objects, or lists containing graphic options and objects, nested to arbitrary depth. The set of graphic options and objects supported by drawdf is a superset of those supported by draw2d and gr2d from the draw package.

The arguments are interpreted sequentially: graphic options affect all following graphic objects. Furthermore, graphic objects are drawn on the canvas in order specified, and may obscure graphics drawn earlier. Some graphic options affect the global appearance of the scene.

The additional graphic objects supported by drawdf include: solns_at, points_at, saddles_at, soln_at, point_at, and saddle_at.

The additional graphic options supported by drawdf include: field_degree, soln_arrows, field_arrows, field_grid, field_color, show_field, tstep, nsteps, duration, direction, field_tstep, field_nsteps, and field_duration.

Commonly used graphic objects inherited from the draw package include: explicit, implicit, parametric, polygon, points, vector, label, and all others supported by draw2d and gr2d.

Commonly used graphic options inherited from the draw package include:
points_joined, color, point_type, point_size, line_width, line_type, key, title, xlabel, ylabel, user_preamble, terminal, dimensions, file_name, and all others supported by draw2d and gr2d.

See also draw2d.

Users of wxMaxima or Imaxima may optionally use wxdrawdf, which is identical to drawdf except that the graphics are drawn within the notebook using wxdraw.

To make use of this function, write first load("drawdf").

Examples:

(%i1) load("drawdf")$
(%i2) drawdf(exp(-x)+y)$        /* default vars: x,y */
(%i3) drawdf(exp(-t)+y, [t,y])$ /* default range: [-10,10] */
(%i4) drawdf([y,-9*sin(x)-y/5], [x,1,5], [y,-2,2])$

For backward compatibility, drawdf accepts most of the parameters supported by plotdf.

(%i5) drawdf(2*cos(t)-1+y, [t,y], [t,-5,10], [y,-4,9],
             [trajectory_at,0,0])$

soln_at and solns_at draw solution curves passing through the specified points, using a slightly enhanced 4th-order Runge Kutta numerical integrator.

(%i6) drawdf(2*cos(t)-1+y, [t,-5,10], [y,-4,9],
             solns_at([0,0.1],[0,-0.1]),
             color=blue, soln_at(0,0))$

field_degree=2 causes the field to be composed of quadratic splines, based on the first and second derivatives at each grid point. field_grid=[COLS,ROWS] specifies the number of columns and rows in the grid.

(%i7) drawdf(2*cos(t)-1+y, [t,-5,10], [y,-4,9],
             field_degree=2, field_grid=[20,15],
             solns_at([0,0.1],[0,-0.1]),
             color=blue, soln_at(0,0))$

soln_arrows=true adds arrows to the solution curves, and (by default) removes them from the direction field. It also changes the default colors to emphasize the solution curves.

(%i8) drawdf(2*cos(t)-1+y, [t,-5,10], [y,-4,9],
             soln_arrows=true,
             solns_at([0,0.1],[0,-0.1],[0,0]))$

duration=40 specifies the time duration of numerical integration (default 10). Integration will also stop automatically if the solution moves too far away from the plotted region, or if the derivative becomes complex or infinite. Here we also specify field_degree=2 to plot quadratic splines. The equations below model a predator-prey system.

(%i9) drawdf([x*(1-x-y), y*(3/4-y-x/2)], [x,0,1.1], [y,0,1],
             field_degree=2, duration=40,
             soln_arrows=true, point_at(1/2,1/2),
             solns_at([0.1,0.2], [0.2,0.1], [1,0.8], [0.8,1],
                      [0.1,0.1], [0.6,0.05], [0.05,0.4],
                      [1,0.01], [0.01,0.75]))$

field_degree='solns causes the field to be composed of many small solution curves computed by 4th-order Runge Kutta, with better results in this case.

(%i10) drawdf([x*(1-x-y), y*(3/4-y-x/2)], [x,0,1.1], [y,0,1],
              field_degree='solns, duration=40,
              soln_arrows=true, point_at(1/2,1/2),
              solns_at([0.1,0.2], [0.2,0.1], [1,0.8],
                       [0.8,1], [0.1,0.1], [0.6,0.05],
                       [0.05,0.4], [1,0.01], [0.01,0.75]))$

saddles_at attempts to automatically linearize the equation at each saddle, and to plot a numerical solution corresponding to each eigenvector, including the separatrices. tstep=0.05 specifies the maximum time step for the numerical integrator (the default is 0.1). Note that smaller time steps will sometimes be used in order to keep the x and y steps small. The equations below model a damped pendulum.

(%i11) drawdf([y,-9*sin(x)-y/5], tstep=0.05,
              soln_arrows=true, point_size=0.5,
              points_at([0,0], [2*%pi,0], [-2*%pi,0]),
              field_degree='solns,
              saddles_at([%pi,0], [-%pi,0]))$

show_field=false suppresses the field entirely.

(%i12) drawdf([y,-9*sin(x)-y/5], tstep=0.05,
              show_field=false, soln_arrows=true,
              point_size=0.5,
              points_at([0,0], [2*%pi,0], [-2*%pi,0]),
              saddles_at([3*%pi,0], [-3*%pi,0],
                         [%pi,0], [-%pi,0]))$

drawdf passes all unrecognized parameters to draw2d or gr2d, allowing you to combine the full power of the draw package with drawdf.

(%i13) drawdf(x^2+y^2, [x,-2,2], [y,-2,2], field_color=gray,
              key="soln 1", color=black, soln_at(0,0),
              key="soln 2", color=red, soln_at(0,1),
              key="isocline", color=green, line_width=2,
              nticks=100, parametric(cos(t),sin(t),t,0,2*%pi))$

drawdf accepts nested lists of graphic options and objects, allowing convenient use of makelist and other function calls to generate graphics.

(%i14) colors : ['red,'blue,'purple,'orange,'green]$
(%i15) drawdf([x-x*y/2, (x*y - 3*y)/4],
              [x,2.5,3.5], [y,1.5,2.5],
              field_color = gray,
              makelist([ key   = concat("soln",k),
                         color = colors[k],
                         soln_at(3, 2 + k/20) ],
                       k,1,5))$


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55 dynamics


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55.1 The dynamics package

Package dynamics includes functions for 3D visualization, animations, graphical analysis of differential and difference equations and numerical solution of differential equations. The functions for differential equations are described in the section on Numerical Methods and the functions to plot the Mandelbrot and Julia sets are described in the section on Plotting.

All the functions in this package will be loaded automatically the first time they are used.

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55.2 Graphical analysis of discrete dynamical systems

Function: chaosgame ([[x1, y1]…[xm, ym]], [x0, y0], b, n, options, …);

Implements the so-called chaos game: the initial point (x0, y0) is plotted and then one of the m points [x1, y1]…xm, ym] will be selected at random. The next point plotted will be on the segment from the previous point plotted to the point chosen randomly, at a distance from the random point which will be b times that segment’s length. The procedure is repeated n times. The options are the same as for plot2d.

Example. A plot of Sierpinsky’s triangle:

(%i1) chaosgame([[0, 0], [1, 0], [0.5, sqrt(3)/2]], [0.1, 0.1], 1/2,
                 30000, [style, dots]);
figures/dynamics7

Function: evolution (F, y0, n, …, options, …);

Draws n+1 points in a two-dimensional graph, where the horizontal coordinates of the points are the integers 0, 1, 2, ..., n, and the vertical coordinates are the corresponding values y(n) of the sequence defined by the recurrence relation

        y(n+1) = F(y(n))

With initial value y(0) equal to y0. F must be an expression that depends only on one variable (in the example, it depend on y, but any other variable can be used), y0 must be a real number and n must be a positive integer. This function accepts the same options as plot2d.

Example.

(%i1) evolution(cos(y), 2, 11);
figures/dynamics1

Function: evolution2d ([F, G], [u, v], [u0, y0], n, options, …);

Shows, in a two-dimensional plot, the first n+1 points in the sequence of points defined by the two-dimensional discrete dynamical system with recurrence relations

        u(n+1) = F(u(n), v(n))    v(n+1) = G(u(n), v(n))

With initial values u0 and v0. F and G must be two expressions that depend only on two variables, u and v, which must be named explicitly in a list. The options are the same as for plot2d.

Example. Evolution of a two-dimensional discrete dynamical system:

(%i1) f: 0.6*x*(1+2*x)+0.8*y*(x-1)-y^2-0.9$
(%i2) g: 0.1*x*(1-6*x+4*y)+0.1*y*(1+9*y)-0.4$
(%i3) evolution2d([f,g], [x,y], [-0.5,0], 50000, [style,dots]);
figures/dynamics5

And an enlargement of a small region in that fractal:

(%i9) evolution2d([f,g], [x,y], [-0.5,0], 300000, [x,-0.8,-0.6],
                  [y,-0.4,-0.2], [style, dots]);
figures/dynamics6

Function: ifs ([r1, …, rm], [A1,…, Am], [[x1, y1], …, [xm, ym]], [x0, y0], n, options, …);

Implements the Iterated Function System method. This method is similar to the method described in the function chaosgame. but instead of shrinking the segment from the current point to the randomly chosen point, the 2 components of that segment will be multiplied by the 2 by 2 matrix Ai that corresponds to the point chosen randomly.

The random choice of one of the m attractive points can be made with a non-uniform probability distribution defined by the weights r1,...,rm. Those weights are given in cumulative form; for instance if there are 3 points with probabilities 0.2, 0.5 and 0.3, the weights r1, r2 and r3 could be 2, 7 and 10. The options are the same as for plot2d.

Example. Barnsley’s fern, obtained with 4 matrices and 4 points:

(%i1) a1: matrix([0.85,0.04],[-0.04,0.85])$
(%i2) a2: matrix([0.2,-0.26],[0.23,0.22])$
(%i3) a3: matrix([-0.15,0.28],[0.26,0.24])$
(%i4) a4: matrix([0,0],[0,0.16])$
(%i5) p1: [0,1.6]$
(%i6) p2: [0,1.6]$
(%i7) p3: [0,0.44]$
(%i8) p4: [0,0]$
(%i9) w: [85,92,99,100]$
(%i10) ifs(w, [a1,a2,a3,a4], [p1,p2,p3,p4], [5,0], 50000, [style,dots]);
figures/dynamics8

Function: orbits (F, y0, n1, n2, [x, x0, xf, xstep], options, …);

Draws the orbits diagram for a family of one-dimensional discrete dynamical systems, with one parameter x; that kind of diagram is used to study the bifurcations of a one-dimensional discrete system.

The function F(y) defines a sequence with a starting value of y0, as in the case of the function evolution, but in this case that function will also depend on a parameter x that will take values in the interval from x0 to xf with increments of xstep. Each value used for the parameter x is shown on the horizontal axis. The vertical axis will show the n2 values of the sequence y(n1+1),..., y(n1+n2+1) obtained after letting the sequence evolve n1 iterations. In addition to the options accepted by plot2d, it accepts an option pixels that sets up the maximum number of different points that will be represented in the vertical direction.

Example. Orbits diagram of the quadratic map, with a parameter a:

(%i1) orbits(x^2+a, 0, 50, 200, [a, -2, 0.25], [style, dots]);
figures/dynamics3

To enlarge the region around the lower bifurcation near x = -1.25 use:

(%i2) orbits(x^2+a, 0, 100, 400, [a,-1,-1.53], [x,-1.6,-0.8],
             [nticks, 400], [style,dots]);
figures/dynamics4

Function: staircase (F, y0, n,options,…);

Draws a staircase diagram for the sequence defined by the recurrence relation

        y(n+1) = F(y(n))

The interpretation and allowed values of the input parameters is the same as for the function evolution. A staircase diagram consists of a plot of the function F(y), together with the line G(y) = y. A vertical segment is drawn from the point (y0, y0) on that line until the point where it intersects the function F. From that point a horizontal segment is drawn until it reaches the point (y1, y1) on the line, and the procedure is repeated n times until the point (yn, yn) is reached. The options are the same as for plot2d.

Example.

(%i1) staircase(cos(y), 1, 11, [y, 0, 1.2]);
figures/dynamics2


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55.3 Visualization with VTK

Function scene creates 3D images and animations using the Visualization ToolKit (VTK) software. In order to use that function, Xmaxima and VTK should be installed in your system (including the TCL bindings of VTK, which in some system might come in a separate package).

Function: scene (objects, …, options, …);

Accepts an empty list or a list of several objects and options. The program launches Xmaxima, which opens an external window representing the given objects in a 3-dimensional space and applying the options given. Each object must belong to one of the following 4 classes: sphere, cube, cylinder or cone (see Scene objects). Objects are identified by giving their name or by a list in which the first element is the class name and the following elements are options for that object.

Example. A hexagonal pyramid with a blue background:

(%i1) scene(cone, [background,"#9980e5"])$
figures/scene1

By holding down the left button of the mouse while it is moved on the graphics window, the camera can be rotated showing different views of the pyramid. The two plot options elevation and azimuth can also be used to change the initial orientation of the viewing camera. The camera can be moved by holding the middle mouse button while moving it and holding the right-side mouse button while moving it up or down will zoom in or out.

Each object option should be a list starting with the option name, followed by its value. The list of allowed options can be found in the Scene object's options section.

Example. This will show a sphere falling to the ground and bouncing off without losing any energy. To start or pause the animation, press the play/pause button.

(%i1) p: makelist ([0,0,2.1- 9.8*t^2/2], t, 0, 0.64, 0.01)$

(%i2) p: append (p, reverse(p))$

(%i3) ball: [sphere, [radius,0.1], [thetaresolution,20],
  [phiresolution,20], [position,0,0,2.1], [color,red],
  [animate,position,p]]$

(%i4) ground: [cube, [xlength,2], [ylength,2], [zlength,0.2],
  [position,0,0,-0.1],[color,violet]]$

(%i5) scene (ball, ground, restart)$
figures/scene2

The restart option was used to make the animation restart automatically every time the last point in the position list is reached. The accepted values for the colors are the same as for the color option of plot2d.

55.3.1 Scene options

Scene option: azimuth [azimuth, angle]

Default value: 135

The rotation of the camera on the horizontal (x, y) plane. angle must be a real number; an angle of 0 means that the camera points in the direction of the y axis and the x axis will appear on the right.

Scene option: background [background, color]

Default value: black

The color of the graphics window’s background. It accepts color names or hexadecimal red-green-blue strings (see the color option of plot2d).

Scene option: elevation [elevation, angle]

Default value: 30

The vertical rotation of the camera. The angle must be a real number; an angle of 0 means that the camera points on the horizontal, and the default angle of 30 means that the camera is pointing 30 degrees down from the horizontal.

Scene option: height [height, pixels]

Default value: 500

The height, in pixels, of the graphics window. pixels must be a positive integer number.

Scene option: restart [restart, value]

Default value: false

A true value means that animations will restart automatically when the end of the list is reached. Writing just “restart” is equivalent to [restart, true].

Scene option: tstep [tstep, time]

Default value: 10

The amount of time, in mili-seconds, between iterations among consecutive animation frames. time must be a real number.

Scene option: width [width, pixels]

Default value: 500

The width, in pixels, of the graphics window. pixels must be a positive integer number.

Scene option: windowname [windowtitle, name]

Default value: .scene

name must be a string that can be used as the name of the Tk window created by Xmaxima for the scene graphics. The default value .scene implies that a new top level window will be created.

Scene option: windowtitle [windowtitle, name]

Default value: Xmaxima: scene

name must be a string that will be written in the title of the window created by scene.

55.3.2 Scene objects

Scene object: cone [cone, options]

Creates a regular pyramid with height equal to 1 and a hexagonal base with vertices 0.5 units away from the axis. Options height and radius can be used to change those defaults and option resolution can be used to change the number of edges of the base; higher values will make it look like a cone. By default, the axis will be along the x axis, the middle point of the axis will be at the origin and the vertex on the positive side of the x axis; use options orientation and center to change those defaults.

Example. This shows a pyramid that starts rotating around the z axis when the play button is pressed.

(%i1) scene([cone, [orientation,0,30,0], [tstep,100],
   [animate,orientation,makelist([0,30,i],i,5,360,5)]], restart)$

Scene object: cube [cube, options]

A cube with edges of 1 unit and faces parallel to the xy, xz and yz planes. The lengths of the three edges can be changed with options xlength, ylength and zlength, turning it into a rectangular box and the faces can be rotated with option orientation.

Scene object: cylinder [cylinder, options]

Creates a regular prism with height equal to 1 and a hexagonal base with vertices 0.5 units away from the axis. Options height and radius can be used to change those defaults and option resolution can be used to change the number of edges of the base; higher values will make it look like a cylinder. The default height can be changed with the option height. By default, the axis will be along the x axis and the middle point of the axis will be at the origin; use options orientation and center to change those defaults.

Scene object: sphere [sphere, options]

A sphere with default radius of 0.5 units and center at the origin.

55.3.3 Scene object’s options

Object option: animation [animation, property, positions]

property should be one of the following 4 object’s properties: origin, scale, position or orientation and positions should be a list of points. When the play button is pressed, the object property will be changed sequentially through all the values in the list, at intervals of time given by the option tstep. The rewind button can be used to point at the start of the sequence making the animation restart after the play button is pressed again.

See also track.

Object option: capping [capping, number]

Default value: 1

In a cone or a cylinder, it defines whether the base (or bases) will be shown. A value of 1 for number makes the base visible and a value of 0 makes it invisible.

Object option: center [center, point]

Default value: [0, 0, 0]

The coordinates of the object’s geometric center, with respect to its position. point can be a list with 3 real numbers, or 3 real numbers separated by commas. In a cylinder, cone or cube it will be at half its height and in a sphere at its center.

Object option: color [color, colorname]

Default value: white

The color of the object. It accepts color names or hexadecimal red-green-blue strings (see the color option of plot2d).

Object option: endphi [endphi, angle]

Default value: 180

In a sphere phi is the angle on the vertical plane that passes through the z axis, measured from the positive part of the z axis. angle must be a number between 0 and 180 that sets the final value of phi at which the surface will end. A value smaller than 180 will eliminate a part of the sphere’s surface.

See also startphi and phiresolution.

Object option: endtheta [endtheta, angle]

Default value: 360

In a sphere theta is the angle on the horizontal plane (longitude), measured from the positive part of the x axis. angle must be a number between 0 and 360 that sets the final value of theta at which the surface will end. A value smaller than 360 will eliminate a part of the sphere’s surface.

See also starttheta and thetaresolution.

Object option: height [height, value]

Default value: 1

value must be a positive number which sets the height of a cone or a cylinder.

Object option: linewidth [linewidth, value]

Default value: 1

The width of the lines, when option wireframe is used. value must be a positive number.

Object option: opacity [opacity, value]

Default value: 1

value must be a number between 0 and 1. The lower the number, the more transparent the object will become. The default value of 1 means a completely opaque object.

Object option: orientation [orientation, angles]

Default value: [0, 0, 0]

Three angles by which the object will be rotated with respect to the three axis. angles can be a list with 3 real numbers, or 3 real numbers separated by commas. Example: [0, 0, 90] rotates the x axis of the object to the y axis of the reference frame.

Object option: origin [origin, point]

Default value: [0, 0, 0]

The coordinates of the object’s origin, with respect to which its other dimensions are defined. point can be a list with 3 real numbers, or 3 real numbers separated by commas.

Object option: phiresolution [phiresolution, num]

Default value:

The number of sub-intervals into which the phi angle interval from startphi to endphi will be divided. num must be a positive integer.

See also startphi and endphi.

Object option: points [points]

Only the vertices of the triangulation used to render the surface will be shown. Example: [sphere, [points]]

See also surface and wireframe.

Object option: pointsize [pointsize, value]

Default value: 1

The size of the points, when option points is used. value must be a positive number.

Object option: position [position, point]

Default value: [0, 0, 0]

The coordinates of the object’s position. point can be a list with 3 real numbers, or 3 real numbers separated by commas.

Object option: radius [radius, value]

Default value: 0.5

The radius or a sphere or the distance from the axis to the base’s vertices in a cylinder or a cone. value must be a positive number.

Object option: resolution [resolution, number]

Default value: 6

number must be a integer greater than 2 that sets the number of edges in the base of a cone or a cylinder.

Object option: scale [scale, factors]

Default value: [1, 1, 1]

Three numbers by which the object will be scaled with respect to the three axis. factors can be a list with 3 real numbers, or 3 real numbers separated by commas. Example: [2, 0.5, 1] enlarges the object to twice its size in the x direction, reduces the dimensions in the y direction to half and leaves the z dimensions unchanged.

Object option: startphi [startphi, angle]

Default value: 0

In a sphere phi is the angle on the vertical plane that passes through the z axis, measured from the positive part of the z axis. angle must be a number between 0 and 180 that sets the initial value of phi at which the surface will start. A value bigger than 0 will eliminate a part of the sphere’s surface.

See also endphi and phiresolution.

Object option: starttheta [starttheta, angle]

Default value: 0

In a sphere theta is the angle on the horizontal plane (longitude), measured from the positive part of the x axis. angle must be a number between 0 and 360 that sets the initial value of theta at which the surface will start. A value bigger than 0 will eliminate a part of the sphere’s surface.

See also endtheta and thetaresolution.

Object option: surface [surface]

The surfaces of the object will be rendered and the lines and points of the triangulation used to build the surface will not be shown. This is the default behavior, which can be changed using either the option points or wireframe.

Object option: thetaresolution [thetaresolution, num]

Default value:

The number of sub-intervals into which the theta angle interval from starttheta to endtheta will be divided. num must be a positive integer.

See also starttheta and endtheta.

Object option: track [track, positions]

positions should be a list of points. When the play button is pressed, the object position will be changed sequentially through all the points in the list, at intervals of time given by the option tstep, leaving behind a track of the object’s trajectory. The rewind button can be used to point at the start of the sequence making the animation restart after the play button is pressed again.

Example. This will show the trajectory of a ball thrown with speed of 5 m/s, at an angle of 45 degrees, when the air resistance can be neglected:

(%i1) p: makelist ([0,4*t,4*t- 9.8*t^2/2], t, 0, 0.82, 0.01)$

(%i2) ball: [sphere, [radius,0.1], [color,red], [track,p]]$

(%i3) ground: [cube, [xlength,2], [ylength,4], [zlength,0.2],
      [position,0,1.5,-0.2],[color,green]]$

(%i4) scene (ball, ground)$

See also animation.

Object option: xlength [xlength, length]

Default value: 1

The height of a cube in the x direction. length must be a positive number. See also ylength and zlength.

Object option: ylength [ylength, length]

Default value: 1

The height of a cube in the y direction. length must be a positive number. See also xlength and zlength.

Object option: zlength [zlength, length]

Default value: 1

The height of a cube in z the direction. length must be a positive number. See also xlength and ylength.

Object option: wireframe [wireframe]

Only the edges of the triangulation used to render the surface will be shown. Example: [cube, [wireframe]]

See also surface and points.


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56 engineering-format

Engineering-format changes the way maxima outputs floating-point numbers to the notation engineers are used to: a*10^b with b dividable by three.


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56.1 Functions and Variables for engineering-format

Option variable: engineering_format_floats

Default value: true

This variable allows to temporarily switch off engineering-format.

(%i1) load("engineering-format");
(%o1) /home/gunter/src/maxima-code/share/contrib/engineering-for\
mat.lisp
(%i2) float(sin(10)/10000);
(%o2)                - 54.40211108893698e-6
(%i3) engineering_format_floats:false$
(%i4) float(sin(10)/10000);
(%o4)                - 5.440211108893698e-5

See also fpprintprec and float.

Option variable: engineering_format_min

Default value: 0.0

The minimum absolute value that isn’t automatically converted to the engineering format. See also engineering_format_max and engineering_format_floats.

(%i1) lst: float([.05,.5,5,500,5000,500000]);
(%o1)       [0.05, 0.5, 5.0, 500.0, 5000.0, 500000.0]
(%i2) load("engineering-format");
(%o2) /home/gunter/src/maxima-code/share/contrib/engineering-for\
mat.lisp
(%i3) lst;
(%o3) [50.0e-3, 500.0e-3, 5.0e+0, 500.0e+0, 5.0e+3, 500.0e+3]
(%i4) engineering_format_min:.1$
(%i5) engineering_format_max:1000$
(%i6) lst;
(%o6)     [50.0e-3, 0.5, 5.0, 500.0, 5.0e+3, 500.0e+3]

Option variable: engineering_format_max

Default value: 0.0

The maximum absolute value that isn’t automatically converted to the engineering format. See also engineering_format_min and engineering_format_floats.


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56.2 Known Bugs

The output routine of SBCL 1.3.0 has a bug that sometimes causes the exponent not to be dividable by three. The value of the displayed number is still valid in this case.


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57 ezunits


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57.1 Introduction to ezunits

ezunits is a package for working with dimensional quantities, including some functions for dimensional analysis. ezunits can carry out arithmetic operations on dimensional quantities and unit conversions. The built-in units include Systeme Internationale (SI) and US customary units, and other units can be declared. See also physical_constants, a collection of physical constants.

load("ezunits") loads this package. demo("ezunits") displays several examples. The convenience function known_units returns a list of the built-in and user-declared units, while display_known_unit_conversions displays the set of known conversions in an easy-to-read format.

An expression a ` b represents a dimensional quantity, with a indicating a nondimensional quantity and b indicating the dimensional units. A symbol can be used as a unit without declaring it as such; unit symbols need not have any special properties. The quantity and unit of an expression a ` b can be extracted by the qty and units functions, respectively.

A symbol may be declared to be a dimensional quantity, with specified quantity or specified units or both.

An expression a ` b `` c converts from unit b to unit c. ezunits has built-in conversions for SI base units, SI derived units, and some non-SI units. Unit conversions not already known to ezunits can be declared. The unit conversions known to ezunits are specified by the global variable known_unit_conversions, which comprises built-in and user-defined conversions. Conversions for products, quotients, and powers of units are derived from the set of known unit conversions.

As Maxima generally prefers exact numbers (integers or rationals) to inexact (float or bigfloat), so ezunits preserves exact numbers when they appear in dimensional quantities. All built-in unit conversions are expressed in terms of exact numbers; inexact numbers in declared conversions are coerced to exact.

There is no preferred system for display of units; input units are not converted to other units unless conversion is explicitly indicated. ezunits recognizes the prefixes m-, k-, M, and G- (for milli-, kilo-, mega-, and giga-) as applied to SI base units and SI derived units, but such prefixes are applied only when indicated by an explicit conversion.

Arithmetic operations on dimensional quantities are carried out by conventional rules for such operations.

ezunits does not require that units in a sum have the same dimensions; such terms are not added together, and no error is reported.

ezunits includes functions for elementary dimensional analysis, namely the fundamental dimensions and fundamental units of a dimensional quantity, and computation of dimensionless quantities and natural units. The functions for dimensional analysis were adapted from similar functions in another package, written by Barton Willis.

For the purpose of dimensional analysis, a list of fundamental dimensions and an associated list of fundamental units are maintained; by default the fundamental dimensions are length, mass, time, charge, temperature, and quantity, and the fundamental units are the associated SI units, but other fundamental dimensions and units can be declared.

·

Category: Physical units · Category: Share packages · Category: Package ezunits


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57.2 Introduction to physical_constants

physical_constants is a collection of physical constants, copied from CODATA 2006 recommended values (http://physics.nist.gov/constants). load ("physical_constants") loads this package, and loads ezunits also, if it is not already loaded.

A physical constant is represented as a symbol which has a property which is the constant value. The constant value is a dimensional quantity, as represented by ezunits. The function constvalue fetches the constant value; the constant value is not the ordinary value of the symbol, so symbols of physical constants persist in evaluated expressions until their values are fetched by constvalue.

physical_constants includes some auxiliary information, namely, a description string for each constant, an estimate of the error of its numerical value, and a property for TeX display. To identify physical constants, each symbol has the physical_constant property; propvars(physical_constant) therefore shows the list of all such symbols.

physical_constants comprises the following constants.

%c

speed of light in vacuum

%mu_0

magnetic constant

%e_0

electric constant

%Z_0

characteristic impedance of vacuum

%G

Newtonian constant of gravitation

%h

Planck constant

%h_bar

Planck constant

%m_P

Planck mass

%T_P

Planck temperature

%l_P

Planck length

%t_P

Planck time

%%e

elementary charge

%Phi_0

magnetic flux quantum

%G_0

conductance quantum

%K_J

Josephson constant

%R_K

von Klitzing constant

%mu_B

Bohr magneton

%mu_N

nuclear magneton

%alpha

fine-structure constant

%R_inf

Rydberg constant

%a_0

Bohr radius

%E_h

Hartree energy

%ratio_h_me

quantum of circulation

%m_e

electron mass

%N_A

Avogadro constant

%m_u

atomic mass constant

%F

Faraday constant

%R

molar gas constant

%%k

Boltzmann constant

%V_m

molar volume of ideal gas

%n_0

Loschmidt constant

%ratio_S0_R

Sackur-Tetrode constant (absolute entropy constant)

%sigma

Stefan-Boltzmann constant

%c_1

first radiation constant

%c_1L

first radiation constant for spectral radiance

%c_2

second radiation constant

%b

Wien displacement law constant

%b_prime

Wien displacement law constant

Reference: http://physics.nist.gov/constants

Examples:

The list of all symbols which have the physical_constant property.

(%i1) load ("physical_constants")$
(%i2) propvars (physical_constant);
(%o2) [%c, %mu_0, %e_0, %Z_0, %G, %h, %h_bar, %m_P, %T_P, %l_P, 
%t_P, %%e, %Phi_0, %G_0, %K_J, %R_K, %mu_B, %mu_N, %alpha, 
%R_inf, %a_0, %E_h, %ratio_h_me, %m_e, %N_A, %m_u, %F, %R, %%k, 
%V_m, %n_0, %ratio_S0_R, %sigma, %c_1, %c_1L, %c_2, %b, %b_prime]

Properties of the physical constant %c.

(%i1) load ("physical_constants")$
(%i2) constantp (%c);
(%o2)                         true
(%i3) get (%c, description);
(%o3)               speed of light in vacuum
(%i4) constvalue (%c);
                                      m
(%o4)                     299792458 ` -
                                      s
(%i5) get (%c, RSU);
(%o5)                           0
(%i6) tex (%c);
$$c$$
(%o6)                         false

The energy equivalent of 1 pound-mass. The symbol %c persists until its value is fetched by constvalue.

(%i1) load ("physical_constants")$
(%i2) m * %c^2;
                                2
(%o2)                         %c  m
(%i3) %, m = 1 ` lbm;
                              2
(%o3)                       %c  ` lbm
(%i4) constvalue (%);
                                            2
                                       lbm m
(%o4)              89875517873681764 ` ------
                                          2
                                         s
(%i5) E : % `` J;
Computing conversions to base units; may take a moment. 
                     366838848464007200
(%o5)                ------------------ ` J
                             9
(%i6) E `` GJ;
                      458548560580009
(%o6)                 --------------- ` GJ
                         11250000
(%i7) float (%);
(%o7)              4.0759872051556356e+7 ` GJ
·

Category: Physical units · Category: Share packages · Category: Package physical_constants


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57.3 Functions and Variables for ezunits

Operator: `

The dimensional quantity operator. An expression a ` b represents a dimensional quantity, with a indicating a nondimensional quantity and b indicating the dimensional units. A symbol can be used as a unit without declaring it as such; unit symbols need not have any special properties. The quantity and unit of an expression a ` b can be extracted by the qty and units functions, respectively.

Arithmetic operations on dimensional quantities are carried out by conventional rules for such operations.

ezunits does not require that units in a sum have the same dimensions; such terms are not added together, and no error is reported.

load ("ezunits") enables this operator.

Examples:

SI (Systeme Internationale) units.

(%i1) load ("ezunits")$
(%i2) foo : 10 ` m;
(%o2)                        10 ` m
(%i3) qty (foo);
(%o3)                          10
(%i4) units (foo);
(%o4)                           m
(%i5) dimensions (foo);
(%o5)                        length

"Customary" units.

(%i1) load ("ezunits")$
(%i2) bar : x ` acre;
(%o2)                       x ` acre
(%i3) dimensions (bar);
                                   2
(%o3)                        length
(%i4) fundamental_units (bar);
                                2
(%o4)                          m

Units ad hoc.

(%i1) load ("ezunits")$
(%i2) baz : 3 ` sheep + 8 ` goat + 1 ` horse;
(%o2)           8 ` goat + 3 ` sheep + 1 ` horse
(%i3) subst ([sheep = 3*goat, horse = 10*goat], baz);
(%o3)                       27 ` goat
(%i4) baz2 : 1000`gallon/fortnight;
                                gallon
(%o4)                   1000 ` ---------
                               fortnight
(%i5) subst (fortnight = 14*day, baz2);
                          500   gallon
(%o5)                     --- ` ------
                           7     day

Arithmetic operations on dimensional quantities.

(%i1) load ("ezunits")$
(%i2) 100 ` kg + 200 ` kg;
(%o2)                       300 ` kg
(%i3) 100 ` m^3 - 100 ` m^3;
                                  3
(%o3)                        0 ` m
(%i4) (10 ` kg) * (17 ` m/s^2);
                                 kg m
(%o4)                      170 ` ----
                                   2
                                  s
(%i5) (x ` m) / (y ` s);
                              x   m
(%o5)                         - ` -
                              y   s
(%i6) (a ` m)^2;
                              2    2
(%o6)                        a  ` m

Operator: ``

The unit conversion operator. An expression a ` b `` c converts from unit b to unit c. ezunits has built-in conversions for SI base units, SI derived units, and some non-SI units. Unit conversions not already known to ezunits can be declared. The unit conversions known to ezunits are specified by the global variable known_unit_conversions, which comprises built-in and user-defined conversions. Conversions for products, quotients, and powers of units are derived from the set of known unit conversions.

There is no preferred system for display of units; input units are not converted to other units unless conversion is explicitly indicated. ezunits does not attempt to simplify units by prefixes (milli-, centi-, deci-, etc) unless such conversion is explicitly indicated.

load ("ezunits") enables this operator.

Examples:

The set of known unit conversions.

(%i1) load ("ezunits")$
(%i2) display2d : false$
(%i3) known_unit_conversions;
(%o3) {acre = 4840*yard^2,Btu = 1055*J,cfm = feet^3/minute,
       cm = m/100,day = 86400*s,feet = 381*m/1250,ft = feet,
       g = kg/1000,gallon = 757*l/200,GHz = 1000000000*Hz,
       GOhm = 1000000000*Ohm,GPa = 1000000000*Pa,
       GWb = 1000000000*Wb,Gg = 1000000*kg,Gm = 1000000000*m,
       Gmol = 1000000*mol,Gs = 1000000000*s,ha = hectare,
       hectare = 100*m^2,hour = 3600*s,Hz = 1/s,inch = feet/12,
       km = 1000*m,kmol = 1000*mol,ks = 1000*s,l = liter,
       lbf = pound_force,lbm = pound_mass,liter = m^3/1000,
       metric_ton = Mg,mg = kg/1000000,MHz = 1000000*Hz,
       microgram = kg/1000000000,micrometer = m/1000000,
       micron = micrometer,microsecond = s/1000000,
       mile = 5280*feet,minute = 60*s,mm = m/1000,
       mmol = mol/1000,month = 2629800*s,MOhm = 1000000*Ohm,
       MPa = 1000000*Pa,ms = s/1000,MWb = 1000000*Wb,
       Mg = 1000*kg,Mm = 1000000*m,Mmol = 1000000000*mol,
       Ms = 1000000*s,ns = s/1000000000,ounce = pound_mass/16,
       oz = ounce,Ohm = s*J/C^2,
       pound_force = 32*ft*pound_mass/s^2,
       pound_mass = 200*kg/441,psi = pound_force/inch^2,
       Pa = N/m^2,week = 604800*s,Wb = J/A,yard = 3*feet,
       year = 31557600*s,C = s*A,F = C^2/J,GA = 1000000000*A,
       GC = 1000000000*C,GF = 1000000000*F,GH = 1000000000*H,
       GJ = 1000000000*J,GK = 1000000000*K,GN = 1000000000*N,
       GS = 1000000000*S,GT = 1000000000*T,GV = 1000000000*V,
       GW = 1000000000*W,H = J/A^2,J = m*N,kA = 1000*A,
       kC = 1000*C,kF = 1000*F,kH = 1000*H,kHz = 1000*Hz,
       kJ = 1000*J,kK = 1000*K,kN = 1000*N,kOhm = 1000*Ohm,
       kPa = 1000*Pa,kS = 1000*S,kT = 1000*T,kV = 1000*V,
       kW = 1000*W,kWb = 1000*Wb,mA = A/1000,mC = C/1000,
       mF = F/1000,mH = H/1000,mHz = Hz/1000,mJ = J/1000,
       mK = K/1000,mN = N/1000,mOhm = Ohm/1000,mPa = Pa/1000,
       mS = S/1000,mT = T/1000,mV = V/1000,mW = W/1000,
       mWb = Wb/1000,MA = 1000000*A,MC = 1000000*C,
       MF = 1000000*F,MH = 1000000*H,MJ = 1000000*J,
       MK = 1000000*K,MN = 1000000*N,MS = 1000000*S,
       MT = 1000000*T,MV = 1000000*V,MW = 1000000*W,
       N = kg*m/s^2,R = 5*K/9,S = 1/Ohm,T = J/(m^2*A),V = J/C,
       W = J/s}

Elementary unit conversions.

(%i1) load ("ezunits")$
(%i2) 1 ` ft `` m;
Computing conversions to base units; may take a moment. 
                            381
(%o2)                       ---- ` m
                            1250
(%i3) %, numer;
(%o3)                      0.3048 ` m
(%i4) 1 ` kg `` lbm;
                            441
(%o4)                       --- ` lbm
                            200
(%i5) %, numer;
(%o5)                      2.205 ` lbm
(%i6) 1 ` W `` Btu/hour;
                           720   Btu
(%o6)                      --- ` ----
                           211   hour
(%i7) %, numer;
                                        Btu
(%o7)               3.412322274881517 ` ----
                                        hour
(%i8) 100 ` degC `` degF;
(%o8)                      212 ` degF
(%i9) -40 ` degF `` degC;
(%o9)                     (- 40) ` degC
(%i10) 1 ` acre*ft `` m^3;
                        60228605349    3
(%o10)                  ----------- ` m
                         48828125
(%i11) %, numer;
                                          3
(%o11)                1233.48183754752 ` m

Coercing quantities in feet and meters to one or the other.

(%i1) load ("ezunits")$
(%i2) 100 ` m + 100 ` ft;
(%o2)                  100 ` m + 100 ` ft
(%i3) (100 ` m + 100 ` ft) `` ft;
Computing conversions to base units; may take a moment. 
                           163100
(%o3)                      ------ ` ft
                            381
(%i4) %, numer;
(%o4)                428.0839895013123 ` ft
(%i5) (100 ` m + 100 ` ft) `` m;
                            3262
(%o5)                       ---- ` m
                             25
(%i6) %, numer;
(%o6)                      130.48 ` m

Dimensional analysis to find fundamental dimensions and fundamental units.

(%i1) load ("ezunits")$
(%i2) foo : 1 ` acre * ft;
(%o2)                      1 ` acre ft
(%i3) dimensions (foo);
                                   3
(%o3)                        length
(%i4) fundamental_units (foo);
                                3
(%o4)                          m
(%i5) foo `` m^3;
Computing conversions to base units; may take a moment. 
                        60228605349    3
(%o5)                   ----------- ` m
                         48828125
(%i6) %, numer;
                                          3
(%o6)                 1233.48183754752 ` m

Declared unit conversions.

(%i1) load ("ezunits")$
(%i2) declare_unit_conversion (MMBtu = 10^6*Btu, kW = 1000*W);
(%o2)                         done
(%i3) declare_unit_conversion (kWh = kW*hour, MWh = 1000*kWh,
                               bell = 1800*s);
(%o3)                         done
(%i4) 1 ` kW*s `` MWh;
Computing conversions to base units; may take a moment. 
                             1
(%o4)                     ------- ` MWh
                          3600000
(%i5) 1 ` kW/m^2 `` MMBtu/bell/ft^2;
                       1306449      MMBtu
(%o5)                 ---------- ` --------
                      8242187500          2
                                   bell ft

Function: constvalue (x)

Shows the value and the units of one of the constants declared by package physical_constants, which includes a list of physical constants, or of a new constant declared in package ezunits (see declare_constvalue).

Note that constant values as recognized by constvalue are separate from values declared by numerval and recognized by constantp.

Example:

(%i1) load ("physical_constants")$
(%i2) constvalue (%G);
                                     3
                                    m
(%o2)                    6.67428 ` -----
                                       2
                                   kg s
(%i3) get ('%G, 'description);
(%o3)           Newtonian constant of gravitation

Function: declare_constvalue (a, x)

Declares the value of a constant to be used in package ezunits. This function should be loaded with load ("ezunits").

Example:

(%i1) load ("ezunits")$
(%i2) declare_constvalue (FOO, 100 ` lbm / acre);
                                 lbm
(%o2)                      100 ` ----
                                 acre
(%i3) FOO * (50 ` acre);
(%o3)                     50 FOO ` acre
(%i4) constvalue (%);
(%o4)                      5000 ` lbm

Function: remove_constvalue (a)

Reverts the effect of declare_constvalue. This function should be loaded with load ("ezunits").

Function: units (x)

Returns the units of a dimensional quantity x, or returns 1 if x is nondimensional.

x may be a literal dimensional expression a ` b, a symbol with declared units via declare_units, or an expression containing either or both of those.

This function should be loaded with load ("ezunits").

Example:

(%i1) load ("ezunits")$
(%i2) foo : 100 ` kg;
(%o2)                       100 ` kg
(%i3) bar : x ` m/s;
                                  m
(%o3)                         x ` -
                                  s
(%i4) units (foo);
(%o4)                          kg
(%i5) units (bar);
                                m
(%o5)                           -
                                s
(%i6) units (foo * bar);
                              kg m
(%o6)                         ----
                               s
(%i7) units (foo / bar);
                              kg s
(%o7)                         ----
                               m
(%i8) units (foo^2);
                                 2
(%o8)                          kg

Function: declare_units (a, u)

Declares that units should return units u for a, where u is an expression. This function should be loaded with load ("ezunits").

Example:

(%i1) load ("ezunits")$
(%i2) units (aa);
(%o2)                           1
(%i3) declare_units (aa, J);
(%o3)                           J
(%i4) units (aa);
(%o4)                           J
(%i5) units (aa^2);
                                2
(%o5)                          J
(%i6) foo : 100 ` kg;
(%o6)                       100 ` kg
(%i7) units (aa * foo);
(%o7)                         kg J

Function: qty (x)

Returns the nondimensional part of a dimensional quantity x, or returns x if x is nondimensional. x may be a literal dimensional expression a ` b, a symbol with declared quantity, or an expression containing either or both of those.

This function should be loaded with load ("ezunits").

Example:

(%i1) load ("ezunits")$
(%i2) foo : 100 ` kg;
(%o2)                       100 ` kg
(%i3) qty (foo);
(%o3)                          100
(%i4) bar : v ` m/s;
                                  m
(%o4)                         v ` -
                                  s
(%i5) foo * bar;
                                  kg m
(%o5)                     100 v ` ----
                                   s
(%i6) qty (foo * bar);
(%o6)                         100 v

Function: declare_qty (a, x)

Declares that qty should return x for symbol a, where x is a nondimensional quantity. This function should be loaded with load ("ezunits").

Example:

(%i1) load ("ezunits")$
(%i2) declare_qty (aa, xx);
(%o2)                          xx
(%i3) qty (aa);
(%o3)                          xx
(%i4) qty (aa^2);
                                 2
(%o4)                          xx
(%i5) foo : 100 ` kg;
(%o5)                       100 ` kg
(%i6) qty (aa * foo);
(%o6)                        100 xx

Function: unitp (x)

Returns true if x is a literal dimensional expression, a symbol declared dimensional, or an expression in which the main operator is declared dimensional. unitp returns false otherwise.

load ("ezunits") loads this function.

Examples:

unitp applied to a literal dimensional expression.

(%i1) load ("ezunits")$
(%i2) unitp (100 ` kg);
(%o2)                         true

unitp applied to a symbol declared dimensional.

(%i1) load ("ezunits")$
(%i2) unitp (foo);
(%o2)                         false
(%i3) declare (foo, dimensional);
(%o3)                         done
(%i4) unitp (foo);
(%o4)                         true

unitp applied to an expression in which the main operator is declared dimensional.

(%i1) load ("ezunits")$
(%i2) unitp (bar (x, y, z));
(%o2)                         false
(%i3) declare (bar, dimensional);
(%o3)                         done
(%i4) unitp (bar (x, y, z));
(%o4)                         true

Function: declare_unit_conversion (u = v, ...)

Appends equations u = v, ... to the list of unit conversions known to the unit conversion operator ``. u and v are both multiplicative terms, in which any variables are units, or both literal dimensional expressions.

At present, it is necessary to express conversions such that the left-hand side of each equation is a simple unit (not a multiplicative expression) or a literal dimensional expression with the quantity equal to 1 and the unit being a simple unit. This limitation might be relaxed in future versions.

known_unit_conversions is the list of known unit conversions.

This function should be loaded with load ("ezunits").

Examples:

Unit conversions expressed by equations of multiplicative terms.

(%i1) load ("ezunits")$
(%i2) declare_unit_conversion (nautical_mile = 1852 * m,
                               fortnight = 14 * day);
(%o2)                         done
(%i3) 100 ` nautical_mile / fortnight `` m/s;
Computing conversions to base units; may take a moment. 
                            463    m
(%o3)                       ---- ` -
                            3024   s

Unit conversions expressed by equations of literal dimensional expressions.

(%i1) load ("ezunits")$
(%i2) declare_unit_conversion (1 ` fluid_ounce = 2 ` tablespoon);
(%o2)                         done
(%i3) declare_unit_conversion (1 ` tablespoon = 3 ` teaspoon);
(%o3)                         done
(%i4) 15 ` fluid_ounce `` teaspoon;
Computing conversions to base units; may take a moment. 
(%o4)                     90 ` teaspoon

Function: declare_dimensions (a_1, d_1, ..., a_n, d_n)

Declares a_1, ..., a_n to have dimensions d_1, ..., d_n, respectively.

Each a_k is a symbol or a list of symbols. If it is a list, then every symbol in a_k is declared to have dimension d_k.

load ("ezunits") loads these functions.

Examples:

(%i1) load ("ezunits") $
(%i2) declare_dimensions ([x, y, z], length, [t, u], time);
(%o2)                         done
(%i3) dimensions (y^2/u);
                                   2
                             length
(%o3)                        -------
                              time
(%i4) fundamental_units (y^2/u);
0 errors, 0 warnings
                                2
                               m
(%o4)                          --
                               s

Function: remove_dimensions (a_1, ..., a_n)

Reverts the effect of declare_dimensions. This function should be loaded with load ("ezunits").

Function: declare_fundamental_dimensions (d_1, d_2, d_3, ...)
Function: remove_fundamental_dimensions (d_1, d_2, d_3, ...)
Global variable: fundamental_dimensions

declare_fundamental_dimensions declares fundamental dimensions. Symbols d_1, d_2, d_3, ... are appended to the list of fundamental dimensions, if they are not already on the list.

remove_fundamental_dimensions reverts the effect of declare_fundamental_dimensions.

fundamental_dimensions is the list of fundamental dimensions. By default, the list comprises several physical dimensions.

load ("ezunits") loads these functions.

Examples:

(%i1) load ("ezunits") $
(%i2) fundamental_dimensions;
(%o2) [length, mass, time, current, temperature, quantity]
(%i3) declare_fundamental_dimensions (money, cattle, happiness);
(%o3)                         done
(%i4) fundamental_dimensions;
(%o4) [length, mass, time, current, temperature, quantity, 
                                        money, cattle, happiness]
(%i5) remove_fundamental_dimensions (cattle, happiness);
(%o5)                         done
(%i6) fundamental_dimensions;
(%o6) [length, mass, time, current, temperature, quantity, money]

Function: declare_fundamental_units (u_1, d_1, ..., u_n, d_n)
Function: remove_fundamental_units (u_1, ..., u_n)

declare_fundamental_units declares u_1, ..., u_n to have dimensions d_1, ..., d_n, respectively. All arguments must be symbols.

After calling declare_fundamental_units, dimensions(u_k) returns d_k for each argument u_1, ..., u_n, and fundamental_units(d_k) returns u_k for each argument d_1, ..., d_n.

remove_fundamental_units reverts the effect of declare_fundamental_units.

load ("ezunits") loads these functions.

Examples:

(%i1) load ("ezunits") $
(%i2) declare_fundamental_dimensions (money, cattle, happiness);
(%o2)                         done
(%i3) declare_fundamental_units (dollar, money, goat, cattle,
                                 smile, happiness);
(%o3)                 [dollar, goat, smile]
(%i4) dimensions (100 ` dollar/goat/km^2);
                             money
(%o4)                    --------------
                                      2
                         cattle length
(%i5) dimensions (x ` smile/kg);
                            happiness
(%o5)                       ---------
                              mass
(%i6) fundamental_units (money*cattle/happiness);
0 errors, 0 warnings
                           dollar goat
(%o6)                      -----------
                              smile

Function: dimensions (x)
Function: dimensions_as_list (x)

dimensions returns the dimensions of the dimensional quantity x as an expression comprising products and powers of base dimensions.

dimensions_as_list returns the dimensions of the dimensional quantity x as a list, in which each element is an integer which indicates the power of the corresponding base dimension in the dimensions of x.

load ("ezunits") loads these functions.

Examples:

(%i1) load ("ezunits")$
(%i2) dimensions (1000 ` kg*m^2/s^3);
                                2
                          length  mass
(%o2)                     ------------
                                 3
                             time
(%i3) declare_units (foo, acre*ft/hour);
                             acre ft
(%o3)                        -------
                              hour
(%i4) dimensions (foo);
                                   3
                             length
(%o4)                        -------
                              time
(%i1) load ("ezunits")$
(%i2) fundamental_dimensions;
(%o2)  [length, mass, time, charge, temperature, quantity]
(%i3) dimensions_as_list (1000 ` kg*m^2/s^3);
(%o3)                 [2, 1, - 3, 0, 0, 0]
(%i4) declare_units (foo, acre*ft/hour);
                             acre ft
(%o4)                        -------
                              hour
(%i5) dimensions_as_list (foo);
(%o5)                 [3, 0, - 1, 0, 0, 0]

Function: fundamental_units
    fundamental_units (x)
    fundamental_units ()

fundamental_units(x) returns the units associated with the fundamental dimensions of x. as determined by dimensions(x).

x may be a literal dimensional expression a ` b, a symbol with declared units via declare_units, or an expression containing either or both of those.

fundamental_units() returns the list of all known fundamental units, as declared by declare_fundamental_units.

load ("ezunits") loads this function.

Examples:

(%i1) load ("ezunits")$
(%i2) fundamental_units ();
(%o2)                 [m, kg, s, A, K, mol]
(%i3) fundamental_units (100 ` mile/hour);
                                m
(%o3)                           -
                                s
(%i4) declare_units (aa, g/foot^2);
                                g
(%o4)                         -----
                                  2
                              foot
(%i5) fundamental_units (aa);
                               kg
(%o5)                          --
                                2
                               m

Function: dimensionless (L)

Returns a basis for the dimensionless quantities which can be formed from a list L of dimensional quantities.

load ("ezunits") loads this function.

Examples:

(%i1) load ("ezunits") $
(%i2) dimensionless ([x ` m, y ` m/s, z ` s]);
0 errors, 0 warnings
0 errors, 0 warnings
                               y z
(%o2)                         [---]
                                x

Dimensionless quantities derived from fundamental physical quantities. Note that the first element on the list is proportional to the fine-structure constant.

(%i1) load ("ezunits") $
(%i2) load ("physical_constants") $
(%i3) dimensionless([%h_bar, %m_e, %m_P, %%e, %c, %e_0]);
0 errors, 0 warnings
0 errors, 0 warnings
                              2
                           %%e        %m_e
(%o3)                [--------------, ----]
                      %c %e_0 %h_bar  %m_P

Function: natural_unit (expr, [v_1, ..., v_n])

Finds exponents e_1, ..., e_n such that dimension(expr) = dimension(v_1^e_1 ... v_n^e_n).

load ("ezunits") loads this function.

Examples:


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58 f90


Previous: , Up: f90-pkg   [Contents][Index]

58.1 Package f90

·

Category: Share packages · Category: Package f90

Option variable: f90_output_line_length_max

Default value: 65

f90_output_line_length_max is the maximum number of characters of Fortran code which are output by f90 per line. Longer lines of code are divided, and printed with an ampersand & at the end of an output line and an ampersand at the beginning of the following line.

f90_output_line_length_max must be a positive integer.

Example:

(%i1) load ("f90")$
(%i2) foo : expand ((xxx + yyy + 7)^4);
         4            3         3        2    2             2
(%o2) yyy  + 4 xxx yyy  + 28 yyy  + 6 xxx  yyy  + 84 xxx yyy
          2        3             2
 + 294 yyy  + 4 xxx  yyy + 84 xxx  yyy + 588 xxx yyy + 1372 yyy
      4         3          2
 + xxx  + 28 xxx  + 294 xxx  + 1372 xxx + 2401
(%i3) f90_output_line_length_max;
(%o3)                          65
(%i4) f90 ('foo = foo);
foo = yyy**4+4*xxx*yyy**3+28*yyy**3+6*xxx**2*yyy**2+84*xxx*yyy**2&
&+294*yyy**2+4*xxx**3*yyy+84*xxx**2*yyy+588*xxx*yyy+1372*yyy+xxx**&
&4+28*xxx**3+294*xxx**2+1372*xxx+2401
(%o4)                         false
(%i5) f90_output_line_length_max : 40 $
(%i6) f90 ('foo = foo);
foo = yyy**4+4*xxx*yyy**3+28*yyy**3+6*xx&
&x**2*yyy**2+84*xxx*yyy**2+294*yyy**2+4*x&
&xx**3*yyy+84*xxx**2*yyy+588*xxx*yyy+1372&
&*yyy+xxx**4+28*xxx**3+294*xxx**2+1372*xx&
&x+2401
(%o6)                         false

Function: f90 (expr_1, …, expr_n)

Prints one or more expressions expr_1, …, expr_n as a Fortran 90 program. Output is printed to the standard output.

f90 prints output in the so-called "free form" input format for Fortran 90: there is no special attention to column positions. Long lines are split at a fixed width with the ampersand & continuation character; the number of output characters per line, not including ampersands, is specified by f90_output_line_length_max. f90 outputs an ampersand at the end of a split line and another at the beginning of the next line.

load("f90") loads this function. See also the function fortran.

Examples:

(%i1) load ("f90")$
(%i2) foo : expand ((xxx + yyy + 7)^4);
         4            3         3        2    2             2
(%o2) yyy  + 4 xxx yyy  + 28 yyy  + 6 xxx  yyy  + 84 xxx yyy
          2        3             2
 + 294 yyy  + 4 xxx  yyy + 84 xxx  yyy + 588 xxx yyy + 1372 yyy
      4         3          2
 + xxx  + 28 xxx  + 294 xxx  + 1372 xxx + 2401
(%i3) f90 ('foo = foo);
foo = yyy**4+4*xxx*yyy**3+28*yyy**3+6*xxx**2*yyy**2+84*xxx*yyy**2&
&+294*yyy**2+4*xxx**3*yyy+84*xxx**2*yyy+588*xxx*yyy+1372*yyy+xxx**&
&4+28*xxx**3+294*xxx**2+1372*xxx+2401
(%o3)                         false

Multiple expressions. Capture standard output into a file via the with_stdout function.

(%i1) load ("f90")$
(%i2) foo : sin (3*x + 1) - cos (7*x - 2);
(%o2)              sin(3 x + 1) - cos(7 x - 2)
(%i3) with_stdout ("foo.f90",
                   f90 (x=0.25, y=0.625, 'foo=foo, 'stop, 'end));
(%o3)                         false
(%i4) printfile ("foo.f90");
x = 0.25
y = 0.625
foo = sin(3*x+1)-cos(7*x-2)
stop
end
(%o4)                        foo.f90


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59 finance


Next: , Previous: , Up: finance-pkg   [Contents][Index]

59.1 Introduction to finance

This is the Finance Package (Ver 0.1).

In all the functions, rate is the compound interest rate, num is the number of periods and must be positive and flow refers to cash flow so if you have an Output the flow is negative and positive for Inputs.

Note that before using the functions defined in this package, you have to load it writing load(finance)$.

Author: Nicolas Guarin Zapata.


Previous: , Up: finance-pkg   [Contents][Index]

59.2 Functions and Variables for finance

Function: days360 (year1,month1,day1,year2,month2,day2)

Calculates the distance between 2 dates, assuming 360 days years, 30 days months.

Example:

(%i1) load("finance")$
(%i2) days360(2008,12,16,2007,3,25);
(%o2)                      - 621

Function: fv (rate,PV,num)

We can calculate the future value of a Present one given a certain interest rate. rate is the interest rate, PV is the present value and num is the number of periods.

Example:

(%i1) load("finance")$
(%i2) fv(0.12,1000,3);
(%o2)                     1404.928

Function: pv (rate,FV,num)

We can calculate the present value of a Future one given a certain interest rate. rate is the interest rate, FV is the future value and num is the number of periods.

Example:

(%i1) load("finance")$
(%i2) pv(0.12,1000,3);
(%o2)                711.7802478134108

Function: graph_flow (val)

Plots the money flow in a time line, the positive values are in blue and upside; the negative ones are in red and downside. The direction of the flow is given by the sign of the value. val is a list of flow values.

Example:

(%i1) load("finance")$
(%i2) graph_flow([-5000,-3000,800,1300,1500,2000])$

Function: annuity_pv (rate,PV,num)

We can calculate the annuity knowing the present value (like an amount), it is a constant and periodic payment. rate is the interest rate, PV is the present value and num is the number of periods.

Example:

(%i1) load("finance")$
(%i2) annuity_pv(0.12,5000,10);
(%o2)                884.9208207992202

Function: annuity_fv (rate,FV,num)

We can calculate the annuity knowing the desired value (future value), it is a constant and periodic payment. rate is the interest rate, FV is the future value and num is the number of periods.

Example:

(%i1) load("finance")$
(%i2) annuity_fv(0.12,65000,10);
(%o2)                3703.970670389863

Function: geo_annuity_pv (rate,growing_rate,PV,num)

We can calculate the annuity knowing the present value (like an amount), in a growing periodic payment. rate is the interest rate, growing_rate is the growing rate, PV is the present value and num is the number of periods.

Example:

(%i1) load("finance")$
(%i2) geo_annuity_pv(0.14,0.05,5000,10);
(%o2)                802.6888176505123

Function: geo_annuity_fv (rate,growing_rate,FV,num)

We can calculate the annuity knowing the desired value (future value), in a growing periodic payment. rate is the interest rate, growing_rate is the growing rate, FV is the future value and num is the number of periods.

Example:

(%i1) load("finance")$
(%i2) geo_annuity_fv(0.14,0.05,5000,10);
(%o2)                216.5203395312695

Function: amortization (rate,amount,num)

Amortization table determined by a specific rate. rate is the interest rate, amount is the amount value, and num is the number of periods.

Example:

(%i1) load("finance")$
(%i2) amortization(0.05,56000,12)$
      "n"    "Balance"     "Interest"   "Amortization"  "Payment"      
     0.000     56000.000         0.000         0.000         0.000  
     1.000     52481.777      2800.000      3518.223      6318.223  
     2.000     48787.643      2624.089      3694.134      6318.223  
     3.000     44908.802      2439.382      3878.841      6318.223  
     4.000     40836.019      2245.440      4072.783      6318.223  
     5.000     36559.597      2041.801      4276.422      6318.223  
     6.000     32069.354      1827.980      4490.243      6318.223  
     7.000     27354.599      1603.468      4714.755      6318.223  
     8.000     22404.106      1367.730      4950.493      6318.223  
     9.000     17206.088      1120.205      5198.018      6318.223  
    10.000     11748.170       860.304      5457.919      6318.223  
    11.000      6017.355       587.408      5730.814      6318.223  
    12.000         0.000       300.868      6017.355      6318.223

Function: arit_amortization (rate,increment,amount,num)

The amortization table determined by a specific rate and with growing payment can be claculated by arit_amortization. Notice that the payment is not constant, it presents an arithmetic growing, increment is then the difference between two consecutive rows in the "Payment" column. rate is the interest rate, increment is the increment, amount is the amount value, and num is the number of periods.

Example:

(%i1) load("finance")$
(%i2) arit_amortization(0.05,1000,56000,12)$
      "n"    "Balance"     "Interest"   "Amortization"  "Payment"      
     0.000     56000.000         0.000         0.000         0.000  
     1.000     57403.679      2800.000     -1403.679      1396.321  
     2.000     57877.541      2870.184      -473.863      2396.321  
     3.000     57375.097      2893.877       502.444      3396.321  
     4.000     55847.530      2868.755      1527.567      4396.321  
     5.000     53243.586      2792.377      2603.945      5396.321  
     6.000     49509.443      2662.179      3734.142      6396.321  
     7.000     44588.594      2475.472      4920.849      7396.321  
     8.000     38421.703      2229.430      6166.892      8396.321  
     9.000     30946.466      1921.085      7475.236      9396.321  
    10.000     22097.468      1547.323      8848.998     10396.321  
    11.000     11806.020      1104.873     10291.448     11396.321  
    12.000        -0.000       590.301     11806.020     12396.321

Function: geo_amortization (rate,growing_rate,amount,num)

The amortization table determined by rate, amount, and number of periods can be found by geo_amortization. Notice that the payment is not constant, it presents a geometric growing, growing_rate is then the quotient between two consecutive rows in the "Payment" column. rate is the interest rate, amount is the amount value, and num is the number of periods.

Example:

(%i1) load("finance")$
(%i2) geo_amortization(0.05,0.03,56000,12)$
      "n"    "Balance"     "Interest"   "Amortization"  "Payment"      
     0.000     56000.000         0.000         0.000         0.000  
     1.000     53365.296      2800.000      2634.704      5434.704  
     2.000     50435.816      2668.265      2929.480      5597.745  
     3.000     47191.930      2521.791      3243.886      5765.677  
     4.000     43612.879      2359.596      3579.051      5938.648  
     5.000     39676.716      2180.644      3936.163      6116.807  
     6.000     35360.240      1983.836      4316.475      6300.311  
     7.000     30638.932      1768.012      4721.309      6489.321  
     8.000     25486.878      1531.947      5152.054      6684.000  
     9.000     19876.702      1274.344      5610.176      6884.520  
    10.000     13779.481       993.835      6097.221      7091.056  
    11.000      7164.668       688.974      6614.813      7303.787  
    12.000         0.000       358.233      7164.668      7522.901

Function: saving (rate,amount,num)

The table that represents the values in a constant and periodic saving can be found by saving. amount represents the desired quantity and num the number of periods to save.

Example:

(%i1) load("finance")$
(%i2) saving(0.15,12000,15)$
      "n"    "Balance"     "Interest"   "Payment"      
     0.000         0.000         0.000         0.000  
     1.000       252.205         0.000       252.205  
     2.000       542.240        37.831       252.205  
     3.000       875.781        81.336       252.205  
     4.000      1259.352       131.367       252.205  
     5.000      1700.460       188.903       252.205  
     6.000      2207.733       255.069       252.205  
     7.000      2791.098       331.160       252.205  
     8.000      3461.967       418.665       252.205  
     9.000      4233.467       519.295       252.205  
    10.000      5120.692       635.020       252.205  
    11.000      6141.000       768.104       252.205  
    12.000      7314.355       921.150       252.205  
    13.000      8663.713      1097.153       252.205  
    14.000     10215.474      1299.557       252.205  
    15.000     12000.000      1532.321       252.205

Function: npv (rate,val)

Calculates the present value of a value series to evaluate the viability in a project. val is a list of varying cash flows.

Example:

(%i1) load("finance")$
(%i2) npv(0.25,[100,500,323,124,300]);
(%o2)                714.4703999999999

Function: irr (val,IO)

IRR (Internal Rate of Return) is the value of rate which makes Net Present Value zero. flowValues is a list of varying cash flows, I0 is the initial investment.

Example:

(%i1) load("finance")$
(%i2) res:irr([-5000,0,800,1300,1500,2000],0)$
(%i3) rhs(res[1][1]);
(%o3)                .03009250374237132

Function: benefit_cost (rate,input,output)

Calculates the ratio Benefit/Cost. Benefit is the Net Present Value (NPV) of the inputs, and Cost is the Net Present Value (NPV) of the outputs. Notice that if there is not an input or output value in a specific period, the input/output would be a zero for that period. rate is the interest rate, input is a list of input values, and output is a list of output values.

Example:

(%i1) load("finance")$
(%i2) benefit_cost(0.24,[0,300,500,150],[100,320,0,180]);
(%o2)               1.427249324905784


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60 fractals


Next: , Previous: , Up: fractals-pkg   [Contents][Index]

60.1 Introduction to fractals

This package defines some well known fractals:

- with random IFS (Iterated Function System): the Sierpinsky triangle, a Tree and a Fern

- Complex Fractals: the Mandelbrot and Julia Sets

- the Koch snowflake sets

- Peano maps: the Sierpinski and Hilbert maps

Author: José Ramírez Labrador.

For questions, suggestions and bugs, please feel free to contact me at

pepe DOT ramirez AAATTT uca DOT es


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60.2 Definitions for IFS fractals

Some fractals can be generated by iterative applications of contractive affine transformations in a random way; see

Hoggar S. G., "Mathematics for computer graphics", Cambridge University Press 1994.

We define a list with several contractive affine transformations, and we randomly select the transformation in a recursive way. The probability of the choice of a transformation must be related with the contraction ratio.

You can change the transformations and find another fractal

Function: sierpinskiale (n)

Sierpinski Triangle: 3 contractive maps; .5 contraction constant and translations; all maps have the same contraction ratio. Argument n must be great enougth, 10000 or greater.

Example:

(%i1) load("fractals")$
(%i2) n: 10000$
(%i3) plot2d([discrete,sierpinskiale(n)], [style,dots])$

Function: treefale (n)

3 contractive maps all with the same contraction ratio. Argument n must be great enougth, 10000 or greater.

Example:

(%i1) load("fractals")$
(%i2) n: 10000$
(%i3) plot2d([discrete,treefale(n)], [style,dots])$

Function: fernfale (n)

4 contractive maps, the probability to choice a transformation must be related with the contraction ratio. Argument n must be great enougth, 10000 or greater.

Example:

(%i1) load("fractals")$
(%i2) n: 10000$
(%i3) plot2d([discrete,fernfale(n)], [style,dots])$


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60.3 Definitions for complex fractals

Function: mandelbrot_set (x, y)

Mandelbrot set.

Example:

This program is time consuming because it must make a lot of operations; the computing time is also related with the number of grid points.

(%i1) load("fractals")$
(%i2) plot3d (mandelbrot_set, [x, -2.5, 1], [y, -1.5, 1.5],
                [gnuplot_preamble, "set view map"],
                [gnuplot_pm3d, true],
                [grid, 150, 150])$

Function: julia_set (x, y)

Julia sets.

This program is time consuming because it must make a lot of operations; the computing time is also related with the number of grid points.

Example:

(%i1) load("fractals")$
(%i2) plot3d (julia_set, [x, -2, 1], [y, -1.5, 1.5],
                [gnuplot_preamble, "set view map"],
                [gnuplot_pm3d, true],
                [grid, 150, 150])$

See also julia_parameter.

Optional variable: julia_parameter

Default value: %i

Complex parameter for Julia fractals. Its default value is %i; we suggest the values -.745+%i*.113002, -.39054-%i*.58679, -.15652+%i*1.03225, -.194+%i*.6557 and .011031-%i*.67037.

Function: julia_sin (x, y)

While function julia_set implements the transformation julia_parameter+z^2, function julia_sin implements julia_parameter*sin(z). See source code for more details.

This program runs slowly because it calculates a lot of sines.

Example:

This program is time consuming because it must make a lot of operations; the computing time is also related with the number of grid points.

(%i1) load("fractals")$
(%i2) julia_parameter:1+.1*%i$
(%i3) plot3d (julia_sin, [x, -2, 2], [y, -3, 3], 
                [gnuplot_preamble, "set view map"],
                [gnuplot_pm3d, true],
                [grid, 150, 150])$

See also julia_parameter.


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60.4 Definitions for Koch snowflakes

Function: snowmap (ent, nn)

Koch snowflake sets. Function snowmap plots the snow Koch map over the vertex of an initial closed polygonal, in the complex plane. Here the orientation of the polygon is important. Argument nn is the number of recursive applications of Koch transformation; nn must be small (5 or 6).

Examples:

(%i1) load("fractals")$
(%i2) plot2d([discrete,
              snowmap([1,exp(%i*%pi*2/3),exp(-%i*%pi*2/3),1],4)])$
(%i3) plot2d([discrete,
              snowmap([1,exp(-%i*%pi*2/3),exp(%i*%pi*2/3),1],4)])$
(%i4) plot2d([discrete, snowmap([0,1,1+%i,%i,0],4)])$
(%i5) plot2d([discrete, snowmap([0,%i,1+%i,1,0],4)])$


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60.5 Definitions for Peano maps

Continuous curves that cover an area. Warning: the number of points exponentially grows with n.

Function: hilbertmap (nn)

Hilbert map. Argument nn must be small (5, for example). Maxima can crash if nn is 7 or greater.

Example:

(%i1) load("fractals")$
(%i2) plot2d([discrete,hilbertmap(6)])$

Function: sierpinskimap (nn)

Sierpinski map. Argument nn must be small (5, for example). Maxima can crash if nn is 7 or greater.

Example:

(%i1) load("fractals")$
(%i2) plot2d([discrete,sierpinskimap(6)])$


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61 ggf


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61.1 Functions and Variables for ggf

Option variable: GGFINFINITY

Default value: 3

This is an option variable for function ggf.

When computing the continued fraction of the generating function, a partial quotient having a degree (strictly) greater than GGFINFINITY will be discarded and the current convergent will be considered as the exact value of the generating function; most often the degree of all partial quotients will be 0 or 1; if you use a greater value, then you should give enough terms in order to make the computation accurate enough.

See also ggf.

Option variable: GGFCFMAX

Default value: 3

This is an option variable for function ggf.

When computing the continued fraction of the generating function, if no good result has been found (see the GGFINFINITY flag) after having computed GGFCFMAX partial quotients, the generating function will be considered as not being a fraction of two polynomials and the function will exit. Put freely a greater value for more complicated generating functions.

See also ggf.

Function: ggf (l)

Compute the generating function (if it is a fraction of two polynomials) of a sequence, its first terms being given. l is a list of numbers.

The solution is returned as a fraction of two polynomials. If no solution has been found, it returns with done.

This function is controlled by global variables GGFINFINITY and GGFCFMAX. See also GGFINFINITY and GGFCFMAX.

To use this function write first load("ggf").


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62 graphs


Next: , Previous: , Up: graphs-pkg   [Contents][Index]

62.1 Introduction to graphs

The graphs package provides graph and digraph data structure for Maxima. Graphs and digraphs are simple (have no multiple edges nor loops), although digraphs can have a directed edge from u to v and a directed edge from v to u.

Internally graphs are represented by adjacency lists and implemented as a lisp structures. Vertices are identified by their ids (an id is an integer). Edges/arcs are represented by lists of length 2. Labels can be assigned to vertices of graphs/digraphs and weights can be assigned to edges/arcs of graphs/digraphs.

There is a draw_graph function for drawing graphs. Graphs are drawn using a force based vertex positioning algorithm. draw_graph can also use graphviz programs available from http://www.graphviz.org. draw_graph is based on the maxima draw package.

To use the graphs package, first load it with load("graphs").

·

Category: Share packages · Category: Package graphs


Previous: , Up: graphs-pkg   [Contents][Index]

62.2 Functions and Variables for graphs

62.2.1 Building graphs

Function: create_graph
    create_graph (v_list, e_list)
    create_graph (n, e_list)
    create_graph (v_list, e_list, directed)

Creates a new graph on the set of vertices v_list and with edges e_list.

v_list is a list of vertices ([v1, v2,..., vn]) or a list of vertices together with vertex labels ([[v1,l1], [v2,l2],..., [vn,ln]]).

n is the number of vertices. Vertices will be identified by integers from 0 to n-1.

e_list is a list of edges ([e1, e2,..., em]) or a list of edges together with edge-weights ([[e1, w1], ..., [em, wm]]).

If directed is not false, a directed graph will be returned.

Example 1: create a cycle on 3 vertices:

(%i1) load ("graphs")$
(%i2) g : create_graph([1,2,3], [[1,2], [2,3], [1,3]])$
(%i3) print_graph(g)$
Graph on 3 vertices with 3 edges.
Adjacencies:
  3 :  1  2
  2 :  3  1
  1 :  3  2

Example 2: create a cycle on 3 vertices with edge weights:

(%i1) load ("graphs")$
(%i2) g : create_graph([1,2,3], [[[1,2], 1.0], [[2,3], 2.0],
                          [[1,3], 3.0]])$
(%i3) print_graph(g)$
Graph on 3 vertices with 3 edges.
Adjacencies:
  3 :  1  2
  2 :  3  1
  1 :  3  2

Example 3: create a directed graph:

(%i1) load ("graphs")$
(%i2) d : create_graph(
        [1,2,3,4],
        [
         [1,3], [1,4],
         [2,3], [2,4]
        ],
        'directed = true)$
(%i3) print_graph(d)$
Digraph on 4 vertices with 4 arcs.
Adjacencies:
  4 :
  3 :
  2 :  4  3
  1 :  4  3

Function: copy_graph (g)

Returns a copy of the graph g.

Function: circulant_graph (n, d)

Returns the circulant graph with parameters n and d.

Example:

(%i1) load ("graphs")$
(%i2) g : circulant_graph(10, [1,3])$
(%i3) print_graph(g)$
Graph on 10 vertices with 20 edges.
Adjacencies:
  9 :  2  6  0  8
  8 :  1  5  9  7
  7 :  0  4  8  6
  6 :  9  3  7  5
  5 :  8  2  6  4
  4 :  7  1  5  3
  3 :  6  0  4  2
  2 :  9  5  3  1
  1 :  8  4  2  0
  0 :  7  3  9  1

Function: clebsch_graph ()

Returns the Clebsch graph.

Function: complement_graph (g)

Returns the complement of the graph g.

Function: complete_bipartite_graph (n, m)

Returns the complete bipartite graph on n+m vertices.

Function: complete_graph (n)

Returns the complete graph on n vertices.

Function: cycle_digraph (n)

Returns the directed cycle on n vertices.

Function: cycle_graph (n)

Returns the cycle on n vertices.

Function: cuboctahedron_graph (n)

Returns the cuboctahedron graph.

Function: cube_graph (n)

Returns the n-dimensional cube.

Function: dodecahedron_graph ()

Returns the dodecahedron graph.

Function: empty_graph (n)

Returns the empty graph on n vertices.

Function: flower_snark (n)

Returns the flower graph on 4n vertices.

Example:

(%i1) load ("graphs")$
(%i2) f5 : flower_snark(5)$
(%i3) chromatic_index(f5);
(%o3)                           4

Function: from_adjacency_matrix (A)

Returns the graph represented by its adjacency matrix A.

Function: frucht_graph ()

Returns the Frucht graph.

Function: graph_product (g1, g1)

Returns the direct product of graphs g1 and g2.

Example:

(%i1) load ("graphs")$
(%i2) grid : graph_product(path_graph(3), path_graph(4))$
(%i3) draw_graph(grid)$

figures/graphs01
Function: graph_union (g1, g1)

Returns the union (sum) of graphs g1 and g2.

Function: grid_graph (n, m)

Returns the n x m grid.

Function: great_rhombicosidodecahedron_graph ()

Returns the great rhombicosidodecahedron graph.

Function: great_rhombicuboctahedron_graph ()

Returns the great rhombicuboctahedron graph.

Function: grotzch_graph ()

Returns the Grotzch graph.

Function: heawood_graph ()

Returns the Heawood graph.

Function: icosahedron_graph ()

Returns the icosahedron graph.

Function: icosidodecahedron_graph ()

Returns the icosidodecahedron graph.

Function: induced_subgraph (V, g)

Returns the graph induced on the subset V of vertices of the graph g.

Example:

(%i1) load ("graphs")$
(%i2) p : petersen_graph()$
(%i3) V : [0,1,2,3,4]$
(%i4) g : induced_subgraph(V, p)$
(%i5) print_graph(g)$
Graph on 5 vertices with 5 edges.
Adjacencies:
  4 :  3  0
  3 :  2  4
  2 :  1  3
  1 :  0  2
  0 :  1  4

Function: line_graph (g)

Returns the line graph of the graph g.

Function: make_graph
    make_graph (vrt, f)
    make_graph (vrt, f, oriented)

Creates a graph using a predicate function f.

vrt is a list/set of vertices or an integer. If vrt is an integer, then vertices of the graph will be integers from 1 to vrt.

f is a predicate function. Two vertices a and b will be connected if f(a,b)=true.

If directed is not false, then the graph will be directed.

Example 1:

(%i1) load("graphs")$
(%i2) g : make_graph(powerset({1,2,3,4,5}, 2), disjointp)$
(%i3) is_isomorphic(g, petersen_graph());
(%o3)                         true
(%i4) get_vertex_label(1, g);
(%o4)                        {1, 2}

Example 2:

(%i1) load("graphs")$
(%i2) f(i, j) := is (mod(j, i)=0)$
(%i3) g : make_graph(20, f, directed=true)$
(%i4) out_neighbors(4, g);
(%o4)                    [8, 12, 16, 20]
(%i5) in_neighbors(18, g);
(%o5)                    [1, 2, 3, 6, 9]

Function: mycielski_graph (g)

Returns the mycielskian graph of the graph g.

Function: new_graph ()

Returns the graph with no vertices and no edges.

Function: path_digraph (n)

Returns the directed path on n vertices.

Function: path_graph (n)

Returns the path on n vertices.

Function: petersen_graph
    petersen_graph ()
    petersen_graph (n, d)

Returns the petersen graph P_{n,d}. The default values for n and d are n=5 and d=2.

Function: random_bipartite_graph (a, b, p)

Returns a random bipartite graph on a+b vertices. Each edge is present with probability p.

Function: random_digraph (n, p)

Returns a random directed graph on n vertices. Each arc is present with probability p.

Function: random_regular_graph
    random_regular_graph (n)
    random_regular_graph (n, d)

Returns a random d-regular graph on n vertices. The default value for d is d=3.

Function: random_graph (n, p)

Returns a random graph on n vertices. Each edge is present with probability p.

Function: random_graph1 (n, m)

Returns a random graph on n vertices and random m edges.

Function: random_network (n, p, w)

Returns a random network on n vertices. Each arc is present with probability p and has a weight in the range [0,w]. The function returns a list [network, source, sink].

Example:

(%i1) load ("graphs")$
(%i2) [net, s, t] : random_network(50, 0.2, 10.0);
(%o2)                   [DIGRAPH, 50, 51]
(%i3) max_flow(net, s, t)$
(%i4) first(%);
(%o4)                   27.65981397932507

Function: random_tournament (n)

Returns a random tournament on n vertices.

Function: random_tree (n)

Returns a random tree on n vertices.

Function: small_rhombicosidodecahedron_graph ()

Returns the small rhombicosidodecahedron graph.

Function: small_rhombicuboctahedron_graph ()

Returns the small rhombicuboctahedron graph.

Function: snub_cube_graph ()

Returns the snub cube graph.

Function: snub_dodecahedron_graph ()

Returns the snub dodecahedron graph.

Function: truncated_cube_graph ()

Returns the truncated cube graph.

Function: truncated_dodecahedron_graph ()

Returns the truncated dodecahedron graph.

Function: truncated_icosahedron_graph ()

Returns the truncated icosahedron graph.

Function: truncated_tetrahedron_graph ()

Returns the truncated tetrahedron graph.

Function: tutte_graph ()

Returns the Tutte graph.

Function: underlying_graph (g)

Returns the underlying graph of the directed graph g.

Function: wheel_graph (n)

Returns the wheel graph on n+1 vertices.

62.2.2 Graph properties

Function: adjacency_matrix (gr)

Returns the adjacency matrix of the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) c5 : cycle_graph(4)$
(%i3) adjacency_matrix(c5);
                         [ 0  1  0  1 ]
                         [            ]
                         [ 1  0  1  0 ]
(%o3)                    [            ]
                         [ 0  1  0  1 ]
                         [            ]
                         [ 1  0  1  0 ]

Function: average_degree (gr)

Returns the average degree of vertices in the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) average_degree(grotzch_graph());
                               40
(%o2)                          --
                               11

Function: biconnected_components (gr)

Returns the (vertex sets of) 2-connected components of the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) g : create_graph(
            [1,2,3,4,5,6,7],
            [
             [1,2],[2,3],[2,4],[3,4],
             [4,5],[5,6],[4,6],[6,7]
            ])$
(%i3) biconnected_components(g);
(%o3)        [[6, 7], [4, 5, 6], [1, 2], [2, 3, 4]]
figures/graphs13

Function: bipartition (gr)

Returns a bipartition of the vertices of the graph gr or an empty list if gr is not bipartite.

Example:

(%i1) load ("graphs")$
(%i2) h : heawood_graph()$
(%i3) [A,B]:bipartition(h);
(%o3)  [[8, 12, 6, 10, 0, 2, 4], [13, 5, 11, 7, 9, 1, 3]]
(%i4) draw_graph(h, show_vertices=A, program=circular)$

figures/graphs02
Function: chromatic_index (gr)

Returns the chromatic index of the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) p : petersen_graph()$
(%i3) chromatic_index(p);
(%o3)                           4

Function: chromatic_number (gr)

Returns the chromatic number of the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) chromatic_number(cycle_graph(5));
(%o2)                           3
(%i3) chromatic_number(cycle_graph(6));
(%o3)                           2

Function: clear_edge_weight (e, gr)

Removes the weight of the edge e in the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) g : create_graph(3, [[[0,1], 1.5], [[1,2], 1.3]])$
(%i3) get_edge_weight([0,1], g);
(%o3)                          1.5
(%i4) clear_edge_weight([0,1], g)$
(%i5) get_edge_weight([0,1], g);
(%o5)                           1

Function: clear_vertex_label (v, gr)

Removes the label of the vertex v in the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) g : create_graph([[0,"Zero"], [1, "One"]], [[0,1]])$
(%i3) get_vertex_label(0, g);
(%o3)                         Zero
(%i4) clear_vertex_label(0, g);
(%o4)                         done
(%i5) get_vertex_label(0, g);
(%o5)                         false

Function: connected_components (gr)

Returns the (vertex sets of) connected components of the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) g: graph_union(cycle_graph(5), path_graph(4))$
(%i3) connected_components(g);
(%o3)            [[1, 2, 3, 4, 0], [8, 7, 6, 5]]

Function: diameter (gr)

Returns the diameter of the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) diameter(dodecahedron_graph());
(%o2)                           5

Function: edge_coloring (gr)

Returns an optimal coloring of the edges of the graph gr.

The function returns the chromatic index and a list representing the coloring of the edges of gr.

Example:

(%i1) load ("graphs")$
(%i2) p : petersen_graph()$
(%i3) [ch_index, col] : edge_coloring(p);
(%o3) [4, [[[0, 5], 3], [[5, 7], 1], [[0, 1], 1], [[1, 6], 2], 
[[6, 8], 1], [[1, 2], 3], [[2, 7], 4], [[7, 9], 2], [[2, 3], 2], 
[[3, 8], 3], [[5, 8], 2], [[3, 4], 1], [[4, 9], 4], [[6, 9], 3], 
[[0, 4], 2]]]
(%i4) assoc([0,1], col);
(%o4)                           1
(%i5) assoc([0,5], col);
(%o5)                           3

Function: degree_sequence (gr)

Returns the list of vertex degrees of the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) degree_sequence(random_graph(10, 0.4));
(%o2)            [2, 2, 2, 2, 2, 2, 3, 3, 3, 3]

Function: edge_connectivity (gr)

Returns the edge-connectivity of the graph gr.

See also min_edge_cut.

Function: edges (gr)

Returns the list of edges (arcs) in a (directed) graph gr.

Example:

(%i1) load ("graphs")$
(%i2) edges(complete_graph(4));
(%o2)   [[2, 3], [1, 3], [1, 2], [0, 3], [0, 2], [0, 1]]

Function: get_edge_weight
    get_edge_weight (e, gr)
    get_edge_weight (e, gr, ifnot)

Returns the weight of the edge e in the graph gr.

If there is no weight assigned to the edge, the function returns 1. If the edge is not present in the graph, the function signals an error or returns the optional argument ifnot.

Example:

(%i1) load ("graphs")$
(%i2) c5 : cycle_graph(5)$
(%i3) get_edge_weight([1,2], c5);
(%o3)                           1
(%i4) set_edge_weight([1,2], 2.0, c5);
(%o4)                         done
(%i5) get_edge_weight([1,2], c5);
(%o5)                          2.0

Function: get_vertex_label (v, gr)

Returns the label of the vertex v in the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) g : create_graph([[0,"Zero"], [1, "One"]], [[0,1]])$
(%i3) get_vertex_label(0, g);
(%o3)                         Zero

Function: graph_charpoly (gr, x)

Returns the characteristic polynomial (in variable x) of the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) p : petersen_graph()$
(%i3) graph_charpoly(p, x), factor;
                                   5        4
(%o3)               (x - 3) (x - 1)  (x + 2)

Function: graph_center (gr)

Returns the center of the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) g : grid_graph(5,5)$
(%i3) graph_center(g);
(%o3)                         [12]

Function: graph_eigenvalues (gr)

Returns the eigenvalues of the graph gr. The function returns eigenvalues in the same format as maxima eigenvalues function.

Example:

(%i1) load ("graphs")$
(%i2) p : petersen_graph()$
(%i3) graph_eigenvalues(p);
(%o3)               [[3, - 2, 1], [1, 4, 5]]

Function: graph_periphery (gr)

Returns the periphery of the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) g : grid_graph(5,5)$
(%i3) graph_periphery(g);
(%o3)                    [24, 20, 4, 0]

Function: graph_size (gr)

Returns the number of edges in the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) p : petersen_graph()$
(%i3) graph_size(p);
(%o3)                          15

Function: graph_order (gr)

Returns the number of vertices in the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) p : petersen_graph()$
(%i3) graph_order(p);
(%o3)                          10

Function: girth (gr)

Returns the length of the shortest cycle in gr.

Example:

(%i1) load ("graphs")$
(%i2) g : heawood_graph()$
(%i3) girth(g);
(%o3)                           6

Function: hamilton_cycle (gr)

Returns the Hamilton cycle of the graph gr or an empty list if gr is not hamiltonian.

Example:

(%i1) load ("graphs")$
(%i2) c : cube_graph(3)$
(%i3) hc : hamilton_cycle(c);
(%o3)              [7, 3, 2, 6, 4, 0, 1, 5, 7]
(%i4) draw_graph(c, show_edges=vertices_to_cycle(hc))$

figures/graphs03
Function: hamilton_path (gr)

Returns the Hamilton path of the graph gr or an empty list if gr does not have a Hamilton path.

Example:

(%i1) load ("graphs")$
(%i2) p : petersen_graph()$
(%i3) hp : hamilton_path(p);
(%o3)            [0, 5, 7, 2, 1, 6, 8, 3, 4, 9]
(%i4) draw_graph(p, show_edges=vertices_to_path(hp))$

figures/graphs04
Function: isomorphism (gr1, gr2)

Returns a an isomorphism between graphs/digraphs gr1 and gr2. If gr1 and gr2 are not isomorphic, it returns an empty list.

Example:

(%i1) load ("graphs")$
(%i2) clk5:complement_graph(line_graph(complete_graph(5)))$
(%i3) isomorphism(clk5, petersen_graph());
(%o3) [9 -> 0, 2 -> 1, 6 -> 2, 5 -> 3, 0 -> 4, 1 -> 5, 3 -> 6, 
                                          4 -> 7, 7 -> 8, 8 -> 9]

Function: in_neighbors (v, gr)

Returns the list of in-neighbors of the vertex v in the directed graph gr.

Example:

(%i1) load ("graphs")$
(%i2) p : path_digraph(3)$
(%i3) in_neighbors(2, p);
(%o3)                          [1]
(%i4) out_neighbors(2, p);
(%o4)                          []

Function: is_biconnected (gr)

Returns true if gr is 2-connected and false otherwise.

Example:

(%i1) load ("graphs")$
(%i2) is_biconnected(cycle_graph(5));
(%o2)                         true
(%i3) is_biconnected(path_graph(5));
(%o3)                         false

Function: is_bipartite (gr)

Returns true if gr is bipartite (2-colorable) and false otherwise.

Example:

(%i1) load ("graphs")$
(%i2) is_bipartite(petersen_graph());
(%o2)                         false
(%i3) is_bipartite(heawood_graph());
(%o3)                         true

Function: is_connected (gr)

Returns true if the graph gr is connected and false otherwise.

Example:

(%i1) load ("graphs")$
(%i2) is_connected(graph_union(cycle_graph(4), path_graph(3)));
(%o2)                         false

Function: is_digraph (gr)

Returns true if gr is a directed graph and false otherwise.

Example:

(%i1) load ("graphs")$
(%i2) is_digraph(path_graph(5));
(%o2)                         false
(%i3) is_digraph(path_digraph(5));
(%o3)                         true

Function: is_edge_in_graph (e, gr)

Returns true if e is an edge (arc) in the (directed) graph g and false otherwise.

Example:

(%i1) load ("graphs")$
(%i2) c4 : cycle_graph(4)$
(%i3) is_edge_in_graph([2,3], c4);
(%o3)                         true
(%i4) is_edge_in_graph([3,2], c4);
(%o4)                         true
(%i5) is_edge_in_graph([2,4], c4);
(%o5)                         false
(%i6) is_edge_in_graph([3,2], cycle_digraph(4));
(%o6)                         false

Function: is_graph (gr)

Returns true if gr is a graph and false otherwise.

Example:

(%i1) load ("graphs")$
(%i2) is_graph(path_graph(5));
(%o2)                         true
(%i3) is_graph(path_digraph(5));
(%o3)                         false

Function: is_graph_or_digraph (gr)

Returns true if gr is a graph or a directed graph and false otherwise.

Example:

(%i1) load ("graphs")$
(%i2) is_graph_or_digraph(path_graph(5));
(%o2)                         true
(%i3) is_graph_or_digraph(path_digraph(5));
(%o3)                         true

Function: is_isomorphic (gr1, gr2)

Returns true if graphs/digraphs gr1 and gr2 are isomorphic and false otherwise.

See also isomorphism.

Example:

(%i1) load ("graphs")$
(%i2) clk5:complement_graph(line_graph(complete_graph(5)))$
(%i3) is_isomorphic(clk5, petersen_graph());
(%o3)                         true

Function: is_planar (gr)

Returns true if gr is a planar graph and false otherwise.

The algorithm used is the Demoucron’s algorithm, which is a quadratic time algorithm.

Example:

(%i1) load ("graphs")$
(%i2) is_planar(dodecahedron_graph());
(%o2)                         true
(%i3) is_planar(petersen_graph());
(%o3)                         false
(%i4) is_planar(petersen_graph(10,2));
(%o4)                         true

Function: is_sconnected (gr)

Returns true if the directed graph gr is strongly connected and false otherwise.

Example:

(%i1) load ("graphs")$
(%i2) is_sconnected(cycle_digraph(5));
(%o2)                         true
(%i3) is_sconnected(path_digraph(5));
(%o3)                         false

Function: is_vertex_in_graph (v, gr)

Returns true if v is a vertex in the graph g and false otherwise.

Example:

(%i1) load ("graphs")$
(%i2) c4 : cycle_graph(4)$
(%i3) is_vertex_in_graph(0, c4);
(%o3)                         true
(%i4) is_vertex_in_graph(6, c4);
(%o4)                         false

Function: is_tree (gr)

Returns true if gr is a tree and false otherwise.

Example:

(%i1) load ("graphs")$
(%i2) is_tree(random_tree(4));
(%o2)                         true
(%i3) is_tree(graph_union(random_tree(4), random_tree(5)));
(%o3)                         false

Function: laplacian_matrix (gr)

Returns the laplacian matrix of the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) laplacian_matrix(cycle_graph(5));
                   [  2   - 1   0    0   - 1 ]
                   [                         ]
                   [ - 1   2   - 1   0    0  ]
                   [                         ]
(%o2)              [  0   - 1   2   - 1   0  ]
                   [                         ]
                   [  0    0   - 1   2   - 1 ]
                   [                         ]
                   [ - 1   0    0   - 1   2  ]

Function: max_clique (gr)

Returns a maximum clique of the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) g : random_graph(100, 0.5)$
(%i3) max_clique(g);
(%o3)          [6, 12, 31, 36, 52, 59, 62, 63, 80]

Function: max_degree (gr)

Returns the maximal degree of vertices of the graph gr and a vertex of maximal degree.

Example:

(%i1) load ("graphs")$
(%i2) g : random_graph(100, 0.02)$
(%i3) max_degree(g);
(%o3)                        [6, 79]
(%i4) vertex_degree(95, g);
(%o4)                           2

Function: max_flow (net, s, t)

Returns a maximum flow through the network net with the source s and the sink t.

The function returns the value of the maximal flow and a list representing the weights of the arcs in the optimal flow.

Example:

(%i1) load ("graphs")$
(%i2) net : create_graph(
  [1,2,3,4,5,6],
  [[[1,2], 1.0],
   [[1,3], 0.3],
   [[2,4], 0.2],
   [[2,5], 0.3],
   [[3,4], 0.1],
   [[3,5], 0.1],
   [[4,6], 1.0],
   [[5,6], 1.0]],
  directed=true)$
(%i3) [flow_value, flow] : max_flow(net, 1, 6);
(%o3) [0.7, [[[1, 2], 0.5], [[1, 3], 0.2], [[2, 4], 0.2], 
[[2, 5], 0.3], [[3, 4], 0.1], [[3, 5], 0.1], [[4, 6], 0.3], 
[[5, 6], 0.4]]]
(%i4) fl : 0$
(%i5) for u in out_neighbors(1, net)
     do fl : fl + assoc([1, u], flow)$
(%i6) fl;
(%o6)                          0.7

Function: max_independent_set (gr)

Returns a maximum independent set of the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) d : dodecahedron_graph()$
(%i3) mi : max_independent_set(d);
(%o3)             [0, 3, 5, 9, 10, 11, 18, 19]
(%i4) draw_graph(d, show_vertices=mi)$

figures/graphs05
Function: max_matching (gr)

Returns a maximum matching of the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) d : dodecahedron_graph()$
(%i3) m : max_matching(d);
(%o3) [[5, 7], [8, 9], [6, 10], [14, 19], [13, 18], [12, 17], 
                               [11, 16], [0, 15], [3, 4], [1, 2]]
(%i4) draw_graph(d, show_edges=m)$

figures/graphs06
Function: min_degree (gr)

Returns the minimum degree of vertices of the graph gr and a vertex of minimum degree.

Example:

(%i1) load ("graphs")$
(%i2) g : random_graph(100, 0.1)$
(%i3) min_degree(g);
(%o3)                        [3, 49]
(%i4) vertex_degree(21, g);
(%o4)                           9

Function: min_edge_cut (gr)

Returns the minimum edge cut in the graph gr.

See also edge_connectivity.

Function: min_vertex_cover (gr)

Returns the minimum vertex cover of the graph gr.

Function: min_vertex_cut (gr)

Returns the minimum vertex cut in the graph gr.

See also vertex_connectivity.

Function: minimum_spanning_tree (gr)

Returns the minimum spanning tree of the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) g : graph_product(path_graph(10), path_graph(10))$
(%i3) t : minimum_spanning_tree(g)$
(%i4) draw_graph(g, show_edges=edges(t))$

figures/graphs07
Function: neighbors (v, gr)

Returns the list of neighbors of the vertex v in the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) p : petersen_graph()$
(%i3) neighbors(3, p);
(%o3)                       [4, 8, 2]

Function: odd_girth (gr)

Returns the length of the shortest odd cycle in the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) g : graph_product(cycle_graph(4), cycle_graph(7))$
(%i3) girth(g);
(%o3)                           4
(%i4) odd_girth(g);
(%o4)                           7

Function: out_neighbors (v, gr)

Returns the list of out-neighbors of the vertex v in the directed graph gr.

Example:

(%i1) load ("graphs")$
(%i2) p : path_digraph(3)$
(%i3) in_neighbors(2, p);
(%o3)                          [1]
(%i4) out_neighbors(2, p);
(%o4)                          []

Function: planar_embedding (gr)

Returns the list of facial walks in a planar embedding of gr and false if gr is not a planar graph.

The graph gr must be biconnected.

The algorithm used is the Demoucron’s algorithm, which is a quadratic time algorithm.

Example:

(%i1) load ("graphs")$
(%i2) planar_embedding(grid_graph(3,3));
(%o2) [[3, 6, 7, 8, 5, 2, 1, 0], [4, 3, 0, 1], [3, 4, 7, 6], 
                                      [8, 7, 4, 5], [1, 2, 5, 4]]

Function: print_graph (gr)

Prints some information about the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) c5 : cycle_graph(5)$
(%i3) print_graph(c5)$
Graph on 5 vertices with 5 edges.
Adjacencies:
  4 :  0  3
  3 :  4  2
  2 :  3  1
  1 :  2  0
  0 :  4  1
(%i4) dc5 : cycle_digraph(5)$
(%i5) print_graph(dc5)$
Digraph on 5 vertices with 5 arcs.
Adjacencies:
  4 :  0
  3 :  4
  2 :  3
  1 :  2
  0 :  1
(%i6) out_neighbors(0, dc5);
(%o6)                          [1]

Function: radius (gr)

Returns the radius of the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) radius(dodecahedron_graph());
(%o2)                           5

Function: set_edge_weight (e, w, gr)

Assigns the weight w to the edge e in the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) g : create_graph([1, 2], [[[1,2], 1.2]])$
(%i3) get_edge_weight([1,2], g);
(%o3)                          1.2
(%i4) set_edge_weight([1,2], 2.1, g);
(%o4)                         done
(%i5) get_edge_weight([1,2], g);
(%o5)                          2.1

Function: set_vertex_label (v, l, gr)

Assigns the label l to the vertex v in the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) g : create_graph([[1, "One"], [2, "Two"]], [[1,2]])$
(%i3) get_vertex_label(1, g);
(%o3)                          One
(%i4) set_vertex_label(1, "oNE", g);
(%o4)                         done
(%i5) get_vertex_label(1, g);
(%o5)                          oNE

Function: shortest_path (u, v, gr)

Returns the shortest path from u to v in the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) d : dodecahedron_graph()$
(%i3) path : shortest_path(0, 7, d);
(%o3)                   [0, 1, 19, 13, 7]
(%i4) draw_graph(d, show_edges=vertices_to_path(path))$

figures/graphs08
Function: shortest_weighted_path (u, v, gr)

Returns the length of the shortest weighted path and the shortest weighted path from u to v in the graph gr.

The length of a weighted path is the sum of edge weights of edges in the path. If an edge has no weight, then it has a default weight 1.

Example:

(%i1) load ("graphs")$
(%i2) g: petersen_graph(20, 2)$
(%i3) for e in edges(g) do set_edge_weight(e, random(1.0), g)$
(%i4) shortest_weighted_path(0, 10, g);
(%o4) [2.575143920268482, [0, 20, 38, 36, 34, 32, 30, 10]]

Function: strong_components (gr)

Returns the strong components of a directed graph gr.

Example:

(%i1) load ("graphs")$
(%i2) t : random_tournament(4)$
(%i3) strong_components(t);
(%o3)                 [[1], [0], [2], [3]]
(%i4) vertex_out_degree(3, t);
(%o4)                           3

Function: topological_sort (dag)

Returns a topological sorting of the vertices of a directed graph dag or an empty list if dag is not a directed acyclic graph.

Example:

(%i1) load ("graphs")$
(%i2) g:create_graph(
         [1,2,3,4,5],
         [
          [1,2], [2,5], [5,3],
          [5,4], [3,4], [1,3]
         ],
         directed=true)$
(%i3) topological_sort(g);
(%o3)                    [1, 2, 5, 3, 4]

Function: vertex_connectivity (g)

Returns the vertex connectivity of the graph g.

See also min_vertex_cut.

Function: vertex_degree (v, gr)

Returns the degree of the vertex v in the graph gr.

Function: vertex_distance (u, v, gr)

Returns the length of the shortest path between u and v in the (directed) graph gr.

Example:

(%i1) load ("graphs")$
(%i2) d : dodecahedron_graph()$
(%i3) vertex_distance(0, 7, d);
(%o3)                           4
(%i4) shortest_path(0, 7, d);
(%o4)                   [0, 1, 19, 13, 7]

Function: vertex_eccentricity (v, gr)

Returns the eccentricity of the vertex v in the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) g:cycle_graph(7)$
(%i3) vertex_eccentricity(0, g);
(%o3)                           3

Function: vertex_in_degree (v, gr)

Returns the in-degree of the vertex v in the directed graph gr.

Example:

(%i1) load ("graphs")$
(%i2) p5 : path_digraph(5)$
(%i3) print_graph(p5)$
Digraph on 5 vertices with 4 arcs.
Adjacencies:
  4 :
  3 :  4
  2 :  3
  1 :  2
  0 :  1
(%i4) vertex_in_degree(4, p5);
(%o4)                           1
(%i5) in_neighbors(4, p5);
(%o5)                          [3]

Function: vertex_out_degree (v, gr)

Returns the out-degree of the vertex v in the directed graph gr.

Example:

(%i1) load ("graphs")$
(%i2) t : random_tournament(10)$
(%i3) vertex_out_degree(0, t);
(%o3)                           2
(%i4) out_neighbors(0, t);
(%o4)                        [7, 1]

Function: vertices (gr)

Returns the list of vertices in the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) vertices(complete_graph(4));
(%o2)                     [3, 2, 1, 0]

Function: vertex_coloring (gr)

Returns an optimal coloring of the vertices of the graph gr.

The function returns the chromatic number and a list representing the coloring of the vertices of gr.

Example:

(%i1) load ("graphs")$
(%i2) p:petersen_graph()$
(%i3) vertex_coloring(p);
(%o3) [3, [[0, 2], [1, 3], [2, 2], [3, 3], [4, 1], [5, 3], 
                                 [6, 1], [7, 1], [8, 2], [9, 2]]]

Function: wiener_index (gr)

Returns the Wiener index of the graph gr.

Example:

(%i2) wiener_index(dodecahedron_graph());
(%o2)                          500

62.2.3 Modifying graphs

Function: add_edge (e, gr)

Adds the edge e to the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) p : path_graph(4)$
(%i3) neighbors(0, p);
(%o3)                          [1]
(%i4) add_edge([0,3], p);
(%o4)                         done
(%i5) neighbors(0, p);
(%o5)                        [3, 1]

Function: add_edges (e_list, gr)

Adds all edges in the list e_list to the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) g : empty_graph(3)$
(%i3) add_edges([[0,1],[1,2]], g)$
(%i4) print_graph(g)$
Graph on 3 vertices with 2 edges.
Adjacencies:
  2 :  1
  1 :  2  0
  0 :  1

Function: add_vertex (v, gr)

Adds the vertex v to the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) g : path_graph(2)$
(%i3) add_vertex(2, g)$
(%i4) print_graph(g)$
Graph on 3 vertices with 1 edges.
Adjacencies:
  2 :
  1 :  0
  0 :  1

Function: add_vertices (v_list, gr)

Adds all vertices in the list v_list to the graph gr.

Function: connect_vertices (v_list, u_list, gr)

Connects all vertices from the list v_list with the vertices in the list u_list in the graph gr.

v_list and u_list can be single vertices or lists of vertices.

Example:

(%i1) load ("graphs")$
(%i2) g : empty_graph(4)$
(%i3) connect_vertices(0, [1,2,3], g)$
(%i4) print_graph(g)$
Graph on 4 vertices with 3 edges.
Adjacencies:
  3 :  0
  2 :  0
  1 :  0
  0 :  3  2  1

Function: contract_edge (e, gr)

Contracts the edge e in the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) g: create_graph(
      8, [[0,3],[1,3],[2,3],[3,4],[4,5],[4,6],[4,7]])$
(%i3) print_graph(g)$
Graph on 8 vertices with 7 edges.
Adjacencies:
  7 :  4
  6 :  4
  5 :  4
  4 :  7  6  5  3
  3 :  4  2  1  0
  2 :  3
  1 :  3
  0 :  3
(%i4) contract_edge([3,4], g)$
(%i5) print_graph(g)$
Graph on 7 vertices with 6 edges.
Adjacencies:
  7 :  3
  6 :  3
  5 :  3
  3 :  5  6  7  2  1  0
  2 :  3
  1 :  3
  0 :  3

Function: remove_edge (e, gr)

Removes the edge e from the graph gr.

Example:

(%i1) load ("graphs")$
(%i2) c3 : cycle_graph(3)$
(%i3) remove_edge([0,1], c3)$
(%i4) print_graph(c3)$
Graph on 3 vertices with 2 edges.
Adjacencies:
  2 :  0  1
  1 :  2
  0 :  2

Function: remove_vertex (v, gr)

Removes the vertex v from the graph gr.

62.2.4 Reading and writing to files

Function: dimacs_export
    dimacs_export (gr, fl)
    dimacs_export (gr, fl, comment1, ..., commentn)

Exports the graph into the file fl in the DIMACS format. Optional comments will be added to the top of the file.

Function: dimacs_import (fl)

Returns the graph from file fl in the DIMACS format.

Function: graph6_decode (str)

Returns the graph encoded in the graph6 format in the string str.

Function: graph6_encode (gr)

Returns a string which encodes the graph gr in the graph6 format.

Function: graph6_export (gr_list, fl)

Exports graphs in the list gr_list to the file fl in the graph6 format.

Function: graph6_import (fl)

Returns a list of graphs from the file fl in the graph6 format.

Function: sparse6_decode (str)

Returns the graph encoded in the sparse6 format in the string str.

Function: sparse6_encode (gr)

Returns a string which encodes the graph gr in the sparse6 format.

Function: sparse6_export (gr_list, fl)

Exports graphs in the list gr_list to the file fl in the sparse6 format.

Function: sparse6_import (fl)

Returns a list of graphs from the file fl in the sparse6 format.

62.2.5 Visualization

Function: draw_graph
    draw_graph (graph)
    draw_graph (graph, option1, ..., optionk)

Draws the graph using the draw-pkg package.

The algorithm used to position vertices is specified by the optional argument program. The default value is program=spring_embedding. draw_graph can also use the graphviz programs for positioning vertices, but graphviz must be installed separately.

Example 1:

(%i1) load ("graphs")$
(%i2) g:grid_graph(10,10)$
(%i3) m:max_matching(g)$
(%i4) draw_graph(g,
   spring_embedding_depth=100,
   show_edges=m, edge_type=dots,
   vertex_size=0)$
figures/graphs09

Example 2:

(%i1) load ("graphs")$
(%i2) g:create_graph(16,
    [
     [0,1],[1,3],[2,3],[0,2],[3,4],[2,4],
     [5,6],[6,4],[4,7],[6,7],[7,8],[7,10],[7,11],
     [8,10],[11,10],[8,9],[11,12],[9,15],[12,13],
     [10,14],[15,14],[13,14]
    ])$
(%i3) t:minimum_spanning_tree(g)$
(%i4) draw_graph(
    g,
    show_edges=edges(t),
    show_edge_width=4,
    show_edge_color=green,
    vertex_type=filled_square,
    vertex_size=2
    )$
figures/graphs10

Example 3:

(%i1) load ("graphs")$
(%i2) g:create_graph(16,
    [
     [0,1],[1,3],[2,3],[0,2],[3,4],[2,4],
     [5,6],[6,4],[4,7],[6,7],[7,8],[7,10],[7,11],
     [8,10],[11,10],[8,9],[11,12],[9,15],[12,13],
     [10,14],[15,14],[13,14]
    ])$
(%i3) mi : max_independent_set(g)$
(%i4) draw_graph(
    g,
    show_vertices=mi,
    show_vertex_type=filled_up_triangle,
    show_vertex_size=2,
    edge_color=cyan,
    edge_width=3,
    show_id=true,
    text_color=brown
    )$
figures/graphs11

Example 4:

(%i1) load ("graphs")$
(%i2) net : create_graph(
    [0,1,2,3,4,5],
    [
     [[0,1], 3], [[0,2], 2],
     [[1,3], 1], [[1,4], 3],
     [[2,3], 2], [[2,4], 2],
     [[4,5], 2], [[3,5], 2]
    ],
    directed=true
    )$
(%i3) draw_graph(
    net,
    show_weight=true,
    vertex_size=0,
    show_vertices=[0,5],
    show_vertex_type=filled_square,
    head_length=0.2,
    head_angle=10,
    edge_color="dark-green",
    text_color=blue
    )$
figures/graphs12

Example 5:

(%i1) load("graphs")$
(%i2) g: petersen_graph(20, 2);
(%o2)                         GRAPH
(%i3) draw_graph(g, redraw=true, program=planar_embedding);
(%o3)                         done
figures/graphs14

Example 6:

(%i1) load("graphs")$
(%i2) t: tutte_graph();
(%o2)                         GRAPH
(%i3) draw_graph(t, redraw=true, 
                    fixed_vertices=[1,2,3,4,5,6,7,8,9]);
(%o3)                         done
figures/graphs15

Option variable: draw_graph_program

Default value: spring_embedding

The default value for the program used to position vertices in draw_graph program.

draw_graph option: show_id

Default value: false

If true then ids of the vertices are displayed.

draw_graph option: show_label

Default value: false

If true then labels of the vertices are displayed.

draw_graph option: label_alignment

Default value: center

Determines how to align the labels/ids of the vertices. Can be left, center or right.

draw_graph option: show_weight

Default value: false

If true then weights of the edges are displayed.

draw_graph option: vertex_type

Default value: circle

Defines how vertices are displayed. See the point_type option for the draw package for possible values.

draw_graph option: vertex_size

The size of vertices.

draw_graph option: vertex_color

The color used for displaying vertices.

draw_graph option: show_vertices

Default value: []

Display selected vertices in the using a different color.

draw_graph option: show_vertex_type

Defines how vertices specified in show_vertices are displayed. See the point_type option for the draw package for possible values.

draw_graph option: show_vertex_size

The size of vertices in show_vertices.

draw_graph option: show_vertex_color

The color used for displaying vertices in the show_vertices list.

draw_graph option: vertex_partition

Default value: []

A partition [[v1,v2,...],...,[vk,...,vn]] of the vertices of the graph. The vertices of each list in the partition will be drawn in a different color.

draw_graph option: vertex_coloring

Specifies coloring of the vertices. The coloring col must be specified in the format as returned by vertex_coloring.

draw_graph option: edge_color

The color used for displaying edges.

draw_graph option: edge_width

The width of edges.

draw_graph option: edge_type

Defines how edges are displayed. See the line_type option for the draw package.

draw_graph option: show_edges

Display edges specified in the list e_list using a different color.

draw_graph option: show_edge_color

The color used for displaying edges in the show_edges list.

draw_graph option: show_edge_width

The width of edges in show_edges.

draw_graph option: show_edge_type

Defines how edges in show_edges are displayed. See the line_type option for the draw package.

draw_graph option: edge_partition

A partition [[e1,e2,...],...,[ek,...,em]] of edges of the graph. The edges of each list in the partition will be drawn using a different color.

draw_graph option: edge_coloring

The coloring of edges. The coloring must be specified in the format as returned by the function edge_coloring.

draw_graph option: redraw

Default value: false

If true, vertex positions are recomputed even if the positions have been saved from a previous drawing of the graph.

draw_graph option: head_angle

Default value: 15

The angle for the arrows displayed on arcs (in directed graphs).

draw_graph option: head_length

Default value: 0.1

The length for the arrows displayed on arcs (in directed graphs).

draw_graph option: spring_embedding_depth

Default value: 50

The number of iterations in the spring embedding graph drawing algorithm.

draw_graph option: terminal

The terminal used for drawing (see the terminal option in the draw package).

draw_graph option: file_name

The filename of the drawing if terminal is not screen.

draw_graph option: program

Defines the program used for positioning vertices of the graph. Can be one of the graphviz programs (dot, neato, twopi, circ, fdp), circular, spring_embedding or planar_embedding. planar_embedding is only available for 2-connected planar graphs. When program=spring_embedding, a set of vertices with fixed position can be specified with the fixed_vertices option.

draw_graph option: fixed_vertices

Specifies a list of vertices which will have positions fixed along a regular polygon. Can be used when program=spring_embedding.

Function: vertices_to_path (v_list)

Converts a list v_list of vertices to a list of edges of the path defined by v_list.

Function: vertices_to_cycle (v_list)

Converts a list v_list of vertices to a list of edges of the cycle defined by v_list.


Next: , Previous: , Up: Top   [Contents][Index]

63 grobner


Next: , Previous: , Up: Top   [Contents][Index]

63.1 Introduction to grobner

grobner is a package for working with Groebner bases in Maxima.

To use the following functions you must load the grobner.lisp package.

load(grobner);

A demo can be started by

demo("grobner.demo");

or

batch("grobner.demo")

Some of the calculation in the demo will take a lot of time therefore the output grobner-demo.output of the demo can be found in the same directory as the demo file.

63.1.1 Notes on the grobner package

The package was written by

Marek Rychlik

http://alamos.math.arizona.edu

and is released 2002-05-24 under the terms of the General Public License(GPL) (see file grobner.lisp). This documentation was extracted from the files

README, grobner.lisp, grobner.demo, grobner-demo.output

by Günter Nowak. Suggestions for improvement of the documentation can be discussed at the maxima-mailing-list maxima@math.utexas.edu. The code is a little bit out of date now. Modern implementation use the fast F4 algorithm described in

A new efficient algorithm for computing Gröbner bases (F4) 
Jean-Charles Faugère
LIP6/CNRS Université Paris VI 
January 20, 1999
·

Category: Groebner bases · Category: Share packages · Category: Package grobner

63.1.2 Implementations of admissible monomial orders in grobner


Previous: , Up: Top   [Contents][Index]

63.2 Functions and Variables for grobner

63.2.1 Global switches for grobner

Option variable: poly_monomial_order

Default value: lex

This global switch controls which monomial order is used in polynomial and Groebner Bases calculations. If not set, lex will be used.

Option variable: poly_coefficient_ring

Default value: expression_ring

This switch indicates the coefficient ring of the polynomials that will be used in grobner calculations. If not set, maxima’s general expression ring will be used. This variable may be set to ring_of_integers if desired.

Option variable: poly_primary_elimination_order

Default value: false

Name of the default order for eliminated variables in elimination-based functions. If not set, lex will be used.

Option variable: poly_secondary_elimination_order

Default value: false

Name of the default order for kept variables in elimination-based functions. If not set, lex will be used.

Option variable: poly_elimination_order

Default value: false

Name of the default elimination order used in elimination calculations. If set, it overrides the settings in variables poly_primary_elimination_order and poly_secondary_elimination_order. The user must ensure that this is a true elimination order valid for the number of eliminated variables.

Option variable: poly_return_term_list

Default value: false

If set to true, all functions in this package will return each polynomial as a list of terms in the current monomial order rather than a maxima general expression.

Option variable: poly_grobner_debug

Default value: false

If set to true, produce debugging and tracing output.

Option variable: poly_grobner_algorithm

Default value: buchberger

Possible values:

The name of the algorithm used to find the Groebner Bases.

Option variable: poly_top_reduction_only

Default value: false

If not false, use top reduction only whenever possible. Top reduction means that division algorithm stops after the first reduction.

63.2.2 Simple operators in grobner

poly_add, poly_subtract, poly_multiply and poly_expt are the arithmetical operations on polynomials. These are performed using the internal representation, but the results are converted back to the maxima general form.

Function: poly_add (poly1, poly2, varlist)

Adds two polynomials poly1 and poly2.


(%i1) poly_add(z+x^2*y,x-z,[x,y,z]);
                                    2
(%o1)                              x  y + x

Function: poly_subtract (poly1, poly2, varlist)

Subtracts a polynomial poly2 from poly1.


(%i1) poly_subtract(z+x^2*y,x-z,[x,y,z]);
                                      2
(%o1)                          2 z + x  y - x

Function: poly_multiply (poly1, poly2, varlist)

Returns the product of polynomials poly1 and poly2.


(%i2) poly_multiply(z+x^2*y,x-z,[x,y,z])-(z+x^2*y)*(x-z),expand;
(%o1)                                  0

Function: poly_s_polynomial (poly1, poly2, varlist)

Returns the syzygy polynomial (S-polynomial) of two polynomials poly1 and poly2.

Function: poly_primitive_part (poly1, varlist)

Returns the polynomial poly divided by the GCD of its coefficients.

(%i1) poly_primitive_part(35*y+21*x,[x,y]);
(%o1)                              5 y + 3 x

Function: poly_normalize (poly, varlist)

Returns the polynomial poly divided by the leading coefficient. It assumes that the division is possible, which may not always be the case in rings which are not fields.

63.2.3 Other functions in grobner

Function: poly_expand (poly, varlist)

This function parses polynomials to internal form and back. It is equivalent to expand(poly) if poly parses correctly to a polynomial. If the representation is not compatible with a polynomial in variables varlist, the result is an error. It can be used to test whether an expression correctly parses to the internal representation. The following examples illustrate that indexed and transcendental function variables are allowed.


(%i1) poly_expand((x-y)*(y+x),[x,y]);
                                     2    2
(%o1)                               x  - y
(%i2) poly_expand((y+x)^2,[x,y]);
                                2            2
(%o2)                          y  + 2 x y + x
(%i3) poly_expand((y+x)^5,[x,y]);
                  5      4         2  3       3  2      4      5
(%o3)            y  + 5 x y  + 10 x  y  + 10 x  y  + 5 x  y + x
(%i4) poly_expand(-1-x*exp(y)+x^2/sqrt(y),[x]);
                                          2
                                  y      x
(%o4)                       - x %e  + ------- - 1
                                       sqrt(y)

(%i5) poly_expand(-1-sin(x)^2+sin(x),[sin(x)]);
                                2
(%o5)                      - sin (x) + sin(x) - 1

Function: poly_expt (poly, number, varlist)

exponentitates poly by a positive integer number. If number is not a positive integer number an error will be raised.


(%i1) poly_expt(x-y,3,[x,y])-(x-y)^3,expand;
(%o1)                                  0

Function: poly_content (poly. varlist)

poly_content extracts the GCD of its coefficients


(%i1) poly_content(35*y+21*x,[x,y]);
(%o1)                                  7

Function: poly_pseudo_divide (poly, polylist, varlist)

Pseudo-divide a polynomial poly by the list of n polynomials polylist. Return multiple values. The first value is a list of quotients a. The second value is the remainder r. The third argument is a scalar coefficient c, such that c*poly can be divided by polylist within the ring of coefficients, which is not necessarily a field. Finally, the fourth value is an integer count of the number of reductions performed. The resulting objects satisfy the equation:

c*poly=sum(a[i]*polylist[i],i=1...n)+r.

Function: poly_exact_divide (poly1, poly2, varlist)

Divide a polynomial poly1 by another polynomial poly2. Assumes that exact division with no remainder is possible. Returns the quotient.

Function: poly_normal_form (poly, polylist, varlist)

poly_normal_form finds the normal form of a polynomial poly with respect to a set of polynomials polylist.

Function: poly_buchberger_criterion (polylist, varlist)

Returns true if polylist is a Groebner basis with respect to the current term order, by using the Buchberger criterion: for every two polynomials h1 and h2 in polylist the S-polynomial S(h1,h2) reduces to 0 modulo polylist.

Function: poly_buchberger (polylist_fl varlist)

poly_buchberger performs the Buchberger algorithm on a list of polynomials and returns the resulting Groebner basis.

63.2.4 Standard postprocessing of Groebner Bases

The k-th elimination Ideal I_k of an Ideal I over K[ x[1],...,x[n] ] is the ideal intersect(I, K[ x[k+1],...,x[n] ]).
The colon ideal I:J is the ideal {h|for all w in J: w*h in I}.
The ideal I:p^inf is the ideal {h| there is a n in N: p^n*h in I}.
The ideal I:J^inf is the ideal {h| there is a n in N and a p in J: p^n*h in I}.
The radical ideal sqrt(I) is the ideal {h| there is a n in N : h^n in I }.

Function: poly_reduction (polylist, varlist)

poly_reduction reduces a list of polynomials polylist, so that each polynomial is fully reduced with respect to the other polynomials.

Function: poly_minimization (polylist, varlist)

Returns a sublist of the polynomial list polylist spanning the same monomial ideal as polylist but minimal, i.e. no leading monomial of a polynomial in the sublist divides the leading monomial of another polynomial.

Function: poly_normalize_list (polylist, varlist)

poly_normalize_list applies poly_normalize to each polynomial in the list. That means it divides every polynomial in a list polylist by its leading coefficient.

Function: poly_grobner (polylist, varlist)

Returns a Groebner basis of the ideal span by the polynomials polylist. Affected by the global flags.

Function: poly_reduced_grobner (polylist, varlist)

Returns a reduced Groebner basis of the ideal span by the polynomials polylist. Affected by the global flags.

Function: poly_depends_p (poly, var, varlist)

poly_depends tests whether a polynomial depends on a variable var.

Function: poly_elimination_ideal (polylist, number, varlist)

poly_elimination_ideal returns the grobner basis of the number-th elimination ideal of an ideal specified as a list of generating polynomials (not necessarily Groebner basis).

Function: poly_colon_ideal (polylist1, polylist2, varlist)

Returns the reduced Groebner basis of the colon ideal

I(polylist1):I(polylist2)

where polylist1 and polylist2 are two lists of polynomials.

Function: poly_ideal_intersection (polylist1, polylist2, varlist)

poly_ideal_intersection returns the intersection of two ideals.

Function: poly_lcm (poly1, poly2, varlist)

Returns the lowest common multiple of poly1 and poly2.

Function: poly_gcd (poly1, poly2, varlist)

Returns the greatest common divisor of poly1 and poly2.

See also ezgcd, gcd, gcdex, and gcdivide.

Example:

(%i1) p1:6*x^3+19*x^2+19*x+6; 
                        3       2
(%o1)                6 x  + 19 x  + 19 x + 6
(%i2) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x;
                  5       4       3       2
(%o2)          6 x  + 13 x  + 12 x  + 13 x  + 6 x
(%i3) poly_gcd(p1, p2, [x]);
                            2
(%o3)                    6 x  + 13 x + 6

Function: poly_grobner_equal (polylist1, polylist2, varlist)

poly_grobner_equal tests whether two Groebner Bases generate the same ideal. Returns true if two lists of polynomials polylist1 and polylist2, assumed to be Groebner Bases, generate the same ideal, and false otherwise. This is equivalent to checking that every polynomial of the first basis reduces to 0 modulo the second basis and vice versa. Note that in the example below the first list is not a Groebner basis, and thus the result is false.

(%i1) poly_grobner_equal([y+x,x-y],[x,y],[x,y]);
(%o1)                         false

Function: poly_grobner_subsetp (polylist1, polylist2, varlist)

poly_grobner_subsetp tests whether an ideal generated by polylist1 is contained in the ideal generated by polylist2. For this test to always succeed, polylist2 must be a Groebner basis.

Function: poly_grobner_member (poly, polylist, varlist)

Returns true if a polynomial poly belongs to the ideal generated by the polynomial list polylist, which is assumed to be a Groebner basis. Returns false otherwise.

poly_grobner_member tests whether a polynomial belongs to an ideal generated by a list of polynomials, which is assumed to be a Groebner basis. Equivalent to normal_form being 0.

Function: poly_ideal_saturation1 (polylist, poly, varlist)

Returns the reduced Groebner basis of the saturation of the ideal

I(polylist):poly^inf

Geometrically, over an algebraically closed field, this is the set of polynomials in the ideal generated by polylist which do not identically vanish on the variety of poly.

Function: poly_ideal_saturation (polylist1, polylist2, varlist)

Returns the reduced Groebner basis of the saturation of the ideal

I(polylist1):I(polylist2)^inf

Geometrically, over an algebraically closed field, this is the set of polynomials in the ideal generated by polylist1 which do not identically vanish on the variety of polylist2.

Function: poly_ideal_polysaturation1 (polylist1, polylist2, varlist)

polylist2 ist a list of n polynomials [poly1,...,polyn]. Returns the reduced Groebner basis of the ideal

I(polylist):poly1^inf:...:polyn^inf

obtained by a sequence of successive saturations in the polynomials of the polynomial list polylist2 of the ideal generated by the polynomial list polylist1.

Function: poly_ideal_polysaturation (polylist, polylistlist, varlist)

polylistlist is a list of n list of polynomials [polylist1,...,polylistn]. Returns the reduced Groebner basis of the saturation of the ideal

I(polylist):I(polylist_1)^inf:...:I(polylist_n)^inf

Function: poly_saturation_extension (poly, polylist, varlist1, varlist2)

poly_saturation_extension implements the famous Rabinowitz trick.

Function: poly_polysaturation_extension (poly, polylist, varlist1, varlist2)


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64 impdiff


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64.1 Functions and Variables for impdiff

Function: implicit_derivative (f,indvarlist,orderlist,depvar)

This subroutine computes implicit derivatives of multivariable functions. f is an array function, the indexes are the derivative degree in the indvarlist order; indvarlist is the independent variable list; orderlist is the order desired; and depvar is the dependent variable.

To use this function write first load("impdiff").


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65 interpol


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65.1 Introduction to interpol

Package interpol defines the Lagrangian, the linear and the cubic splines methods for polynomial interpolation.

For comments, bugs or suggestions, please contact me at ’mario AT edu DOT xunta DOT es’.

·

Category: Numerical methods · Category: Share packages · Category: Package interpol


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65.2 Functions and Variables for interpol

Function: lagrange
    lagrange (points)
    lagrange (points, option)

Computes the polynomial interpolation by the Lagrangian method. Argument points must be either:

In the first two cases the pairs are ordered with respect to the first coordinate before making computations.

With the option argument it is possible to select the name for the independent variable, which is 'x by default; to define another one, write something like varname='z.

Note that when working with high degree polynomials, floating point evaluations are unstable.

See also linearinterpol, cspline, and ratinterpol.

Examples:

(%i1) load("interpol")$
(%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$
(%i3) lagrange(p);
       (x - 7) (x - 6) (x - 3) (x - 1)
(%o3)  -------------------------------
                     35
   (x - 8) (x - 6) (x - 3) (x - 1)
 - -------------------------------
                 12
   7 (x - 8) (x - 7) (x - 3) (x - 1)
 + ---------------------------------
                  30
   (x - 8) (x - 7) (x - 6) (x - 1)
 - -------------------------------
                 60
   (x - 8) (x - 7) (x - 6) (x - 3)
 + -------------------------------
                 84
(%i4) f(x):=''%;
               (x - 7) (x - 6) (x - 3) (x - 1)
(%o4)  f(x) := -------------------------------
                             35
   (x - 8) (x - 6) (x - 3) (x - 1)
 - -------------------------------
                 12
   7 (x - 8) (x - 7) (x - 3) (x - 1)
 + ---------------------------------
                  30
   (x - 8) (x - 7) (x - 6) (x - 1)
 - -------------------------------
                 60
   (x - 8) (x - 7) (x - 6) (x - 3)
 + -------------------------------
                 84
(%i5) /* Evaluate the polynomial at some points */
      expand(map(f,[2.3,5/7,%pi]));
                                  4          3           2
                    919062  73 %pi    701 %pi    8957 %pi
(%o5)  [- 1.567535, ------, ------- - -------- + ---------
                    84035     420       210         420
                                             5288 %pi   186
                                           - -------- + ---]
                                               105       5
(%i6) %,numer;
(%o6) [- 1.567535, 10.9366573451538, 2.89319655125692]
(%i7) load("draw")$  /* load draw package */
(%i8) /* Plot the polynomial together with points */
      draw2d(
        color      = red,
        key        = "Lagrange polynomial",
        explicit(f(x),x,0,10),
        point_size = 3,
        color      = blue,
        key        = "Sample points",
        points(p))$
(%i9) /* Change variable name */
      lagrange(p, varname=w);
       (w - 7) (w - 6) (w - 3) (w - 1)
(%o9)  -------------------------------
                     35
   (w - 8) (w - 6) (w - 3) (w - 1)
 - -------------------------------
                 12
   7 (w - 8) (w - 7) (w - 3) (w - 1)
 + ---------------------------------
                  30
   (w - 8) (w - 7) (w - 6) (w - 1)
 - -------------------------------
                 60
   (w - 8) (w - 7) (w - 6) (w - 3)
 + -------------------------------
                 84

Function: charfun2 (x, a, b)

The characteristic or indicator function on the half-open interval [a, b), that is, including a and excluding b.

When x >= a and x < b evaluates to true or false, charfun2 returns 1 or 0, respectively.

Otherwise, charfun2 returns a partially-evaluated result in terms of charfun.

Package interpol loads this function.

See also charfun.

Examples:

When x >= a and x < b evaluates to true or false, charfun2 returns 1 or 0, respectively.

(%i1) load ("interpol") $
(%i2) charfun2 (5, 0, 100);
(%o2)                           1
(%i3) charfun2 (-5, 0, 100);
(%o3)                           0

Otherwise, charfun2 returns a partially-evaluated result in terms of charfun.

(%i1) load ("interpol") $
(%i2) charfun2 (t, 0, 100);
(%o2)            charfun((0 <= t) and (t < 100))
(%i3) charfun2 (5, u, v);
(%o3)             charfun((u <= 5) and (5 < v))
(%i4) assume (v > u, u > 5);
(%o4)                    [v > u, u > 5]
(%i5) charfun2 (5, u, v);
(%o5)                           0

Function: linearinterpol
    linearinterpol (points)
    linearinterpol (points, option)

Computes the polynomial interpolation by the linear method. Argument points must be either:

In the first two cases the pairs are ordered with respect to the first coordinate before making computations.

With the option argument it is possible to select the name for the independent variable, which is 'x by default; to define another one, write something like varname='z.

See also lagrange, cspline, and ratinterpol.

Examples:

(%i1) load("interpol")$
(%i2) p: matrix([7,2],[8,3],[1,5],[3,2],[6,7])$
(%i3) linearinterpol(p);
        13   3 x
(%o3)  (-- - ---) charfun2(x, minf, 3)
        2     2
 + (x - 5) charfun2(x, 7, inf) + (37 - 5 x) charfun2(x, 6, 7)
    5 x
 + (--- - 3) charfun2(x, 3, 6)
     3

(%i4) f(x):=''%;
                13   3 x
(%o4)  f(x) := (-- - ---) charfun2(x, minf, 3)
                2     2
 + (x - 5) charfun2(x, 7, inf) + (37 - 5 x) charfun2(x, 6, 7)
    5 x
 + (--- - 3) charfun2(x, 3, 6)
     3
(%i5)  /* Evaluate the polynomial at some points */
       map(f,[7.3,25/7,%pi]);
                            62  5 %pi
(%o5)                 [2.3, --, ----- - 3]
                            21    3
(%i6) %,numer;
(%o6)  [2.3, 2.952380952380953, 2.235987755982989]
(%i7) load("draw")$  /* load draw package */
(%i8)  /* Plot the polynomial together with points */
       draw2d(
         color      = red,
         key        = "Linear interpolator",
         explicit(f(x),x,-5,20),
         point_size = 3,
         color      = blue,
         key        = "Sample points",
         points(args(p)))$
(%i9)  /* Change variable name */
       linearinterpol(p, varname='s);
       13   3 s
(%o9) (-- - ---) charfun2(s, minf, 3)
       2     2
 + (s - 5) charfun2(s, 7, inf) + (37 - 5 s) charfun2(s, 6, 7)
    5 s
 + (--- - 3) charfun2(s, 3, 6)
     3

Function: cspline
    cspline (points)
    cspline (points, option1, option2, ...)

Computes the polynomial interpolation by the cubic splines method. Argument points must be either:

In the first two cases the pairs are ordered with respect to the first coordinate before making computations.

There are three options to fit specific needs:

See also lagrange, linearinterpol, and ratinterpol.

Examples:

(%i1) load("interpol")$
(%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$
(%i3) /* Unknown first derivatives at the extremes
         is equivalent to natural cubic splines */
      cspline(p);
              3         2
        1159 x    1159 x    6091 x   8283
(%o3)  (------- - ------- - ------ + ----) charfun2(x, minf, 3)
         3288      1096      3288    1096
            3         2
      2587 x    5174 x    494117 x   108928
 + (- ------- + ------- - -------- + ------) charfun2(x, 7, inf)
       1644       137       1644      137
          3          2
    4715 x    15209 x    579277 x   199575
 + (------- - -------- + -------- - ------) charfun2(x, 6, 7)
     1644       274        1644      274
            3         2
      3287 x    2223 x    48275 x   9609
 + (- ------- + ------- - ------- + ----) charfun2(x, 3, 6)
       4932       274      1644     274

(%i4) f(x):=''%$
(%i5) /* Some evaluations */
      map(f,[2.3,5/7,%pi]), numer;
(%o5) [1.991460766423356, 5.823200187269903, 2.227405312429507]
(%i6) load("draw")$  /* load draw package */
(%i7) /* Plotting interpolating function */
      draw2d(
        color      = red,
        key        = "Cubic splines",
        explicit(f(x),x,0,10),
        point_size = 3,
        color      = blue,
        key        = "Sample points",
        points(p))$
(%i8) /* New call, but giving values at the derivatives */
      cspline(p,d1=0,dn=0);
              3          2
        1949 x    11437 x    17027 x   1247
(%o8)  (------- - -------- + ------- + ----) charfun2(x, minf, 3)
         2256       2256      2256     752
            3          2
      1547 x    35581 x    68068 x   173546
 + (- ------- + -------- - ------- + ------) charfun2(x, 7, inf)
        564       564        141      141
         3          2
    607 x    35147 x    55706 x   38420
 + (------ - -------- + ------- - -----) charfun2(x, 6, 7)
     188       564        141      47
            3         2
      3895 x    1807 x    5146 x   2148
 + (- ------- + ------- - ------ + ----) charfun2(x, 3, 6)
       5076       188      141      47
(%i8) /* Defining new interpolating function */
      g(x):=''%$
(%i9) /* Plotting both functions together */
      draw2d(
        color      = black,
        key        = "Cubic splines (default)",
        explicit(f(x),x,0,10),
        color      = red,
        key        = "Cubic splines (d1=0,dn=0)",
        explicit(g(x),x,0,10),
        point_size = 3,
        color      = blue,
        key        = "Sample points",
        points(p))$

Function: ratinterpol
    ratinterpol (points, numdeg)
    ratinterpol (points, numdeg, option1)

Generates a rational interpolator for data given by points and the degree of the numerator being equal to numdeg; the degree of the denominator is calculated automatically. Argument points must be either:

In the first two cases the pairs are ordered with respect to the first coordinate before making computations.

There is one option to fit specific needs:

See also lagrange, linearinterpol, cspline, minpack_lsquares, and lbfgs-pkg

Examples:

(%i1) load("interpol")$
(%i2) load("draw")$
(%i3) p:[[7.2,2.5],[8.5,2.1],[1.6,5.1],[3.4,2.4],[6.7,7.9]]$
(%i4) for k:0 thru length(p)-1 do                                     
        draw2d(
          explicit(ratinterpol(p,k),x,0,9),                      
          point_size = 3,                                        
          points(p),                                             
          title = concat("Degree of numerator = ",k),            
          yrange=[0,10])$


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66 lapack


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66.1 Introduction to lapack

lapack is a Common Lisp translation (via the program f2cl) of the Fortran library LAPACK, as obtained from the SLATEC project.

·

Category: Numerical methods · Category: Share packages · Category: Package lapack


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66.2 Functions and Variables for lapack

Function: dgeev
    dgeev (A)
    dgeev (A, right_p, left_p)

Computes the eigenvalues and, optionally, the eigenvectors of a matrix A. All elements of A must be integer or floating point numbers. A must be square (same number of rows and columns). A might or might not be symmetric.

dgeev(A) computes only the eigenvalues of A. dgeev(A, right_p, left_p) computes the eigenvalues of A and the right eigenvectors when right_p = true and the left eigenvectors when left_p = true.

A list of three items is returned. The first item is a list of the eigenvalues. The second item is false or the matrix of right eigenvectors. The third item is false or the matrix of left eigenvectors.

The right eigenvector v(j) (the j-th column of the right eigenvector matrix) satisfies

A . v(j) = lambda(j) . v(j)

where lambda(j) is the corresponding eigenvalue. The left eigenvector u(j) (the j-th column of the left eigenvector matrix) satisfies

u(j)**H . A = lambda(j) . u(j)**H

where u(j)**H denotes the conjugate transpose of u(j). The Maxima function ctranspose computes the conjugate transpose.

The computed eigenvectors are normalized to have Euclidean norm equal to 1, and largest component has imaginary part equal to zero.

Example:

(%i1) load ("lapack")$
(%i2) fpprintprec : 6;
(%o2)                           6
(%i3) M : matrix ([9.5, 1.75], [3.25, 10.45]);
                         [ 9.5   1.75  ]
(%o3)                    [             ]
                         [ 3.25  10.45 ]
(%i4) dgeev (M);
(%o4)          [[7.54331, 12.4067], false, false]
(%i5) [L, v, u] : dgeev (M, true, true);
                           [ - .666642  - .515792 ]
(%o5) [[7.54331, 12.4067], [                      ], 
                           [  .745378   - .856714 ]
                                        [ - .856714  - .745378 ]
                                        [                      ]]
                                        [  .515792   - .666642 ]
(%i6) D : apply (diag_matrix, L);
                      [ 7.54331     0    ]
(%o6)                 [                  ]
                      [    0     12.4067 ]
(%i7) M . v - v . D;
                [      0.0       - 8.88178E-16 ]
(%o7)           [                              ]
                [ - 8.88178E-16       0.0      ]
(%i8) transpose (u) . M - D . transpose (u);
                     [ 0.0  - 4.44089E-16 ]
(%o8)                [                    ]
                     [ 0.0       0.0      ]

Function: dgeqrf (A)

Computes the QR decomposition of the matrix A. All elements of A must be integer or floating point numbers. A may or may not have the same number of rows and columns.

A list of two items is returned. The first item is the matrix Q, which is a square, orthonormal matrix which has the same number of rows as A. The second item is the matrix R, which is the same size as A, and which has all elements equal to zero below the diagonal. The product Q . R, where "." is the noncommutative multiplication operator, is equal to A (ignoring floating point round-off errors).

(%i1) load ("lapack") $
(%i2) fpprintprec : 6 $
(%i3) M : matrix ([1, -3.2, 8], [-11, 2.7, 5.9]) $
(%i4) [q, r] : dgeqrf (M);
       [ - .0905357  .995893  ]
(%o4) [[                      ], 
       [  .995893    .0905357 ]
                               [ - 11.0454   2.97863   5.15148 ]
                               [                               ]]
                               [     0      - 2.94241  8.50131 ]
(%i5) q . r - M;
         [ - 7.77156E-16   1.77636E-15   - 8.88178E-16 ]
(%o5)    [                                             ]
         [      0.0       - 1.33227E-15   8.88178E-16  ]
(%i6) mat_norm (%, 1);
(%o6)                      3.10862E-15

Function: dgesv (A, b)

Computes the solution x of the linear equation A x = b, where A is a square matrix, and b is a matrix of the same number of rows as A and any number of columns. The return value x is the same size as b.

The elements of A and b must evaluate to real floating point numbers via float; thus elements may be any numeric type, symbolic numerical constants, or expressions which evaluate to floats. The elements of x are always floating point numbers. All arithmetic is carried out as floating point operations.

dgesv computes the solution via the LU decomposition of A.

Examples:

dgesv computes the solution of the linear equation A x = b.

(%i1) A : matrix ([1, -2.5], [0.375, 5]);
                               [   1    - 2.5 ]
(%o1)                          [              ]
                               [ 0.375    5   ]
(%i2) b : matrix ([1.75], [-0.625]);
                                  [  1.75   ]
(%o2)                             [         ]
                                  [ - 0.625 ]
(%i3) x : dgesv (A, b);
                            [  1.210526315789474  ]
(%o3)                       [                     ]
                            [ - 0.215789473684211 ]
(%i4) dlange (inf_norm, b - A.x);
(%o4)                                 0.0

b is a matrix with the same number of rows as A and any number of columns. x is the same size as b.

(%i1) A : matrix ([1, -0.15], [1.82, 2]);
                               [  1    - 0.15 ]
(%o1)                          [              ]
                               [ 1.82    2    ]
(%i2) b : matrix ([3.7, 1, 8], [-2.3, 5, -3.9]);
                              [  3.7   1    8   ]
(%o2)                         [                 ]
                              [ - 2.3  5  - 3.9 ]
(%i3) x : dgesv (A, b);
      [  3.103827540695117  1.20985481742191    6.781786185657722 ]
(%o3) [                                                           ]
      [ -3.974483062032557  1.399032116146062  -8.121425428948527 ]
(%i4) dlange (inf_norm, b - A . x);
(%o4)                       1.1102230246251565E-15

The elements of A and b must evaluate to real floating point numbers.

(%i1) A : matrix ([5, -%pi], [1b0, 11/17]);
                               [   5    - %pi ]
                               [              ]
(%o1)                          [         11   ]
                               [ 1.0b0   --   ]
                               [         17   ]
(%i2) b : matrix ([%e], [sin(1)]);
                                  [   %e   ]
(%o2)                             [        ]
                                  [ sin(1) ]
(%i3) x : dgesv (A, b);
                             [ 0.690375643155986 ]
(%o3)                        [                   ]
                             [ 0.233510982552952 ]
(%i4) dlange (inf_norm, b - A . x);
(%o4)                        2.220446049250313E-16

Function: dgesvd
    dgesvd (A)
    dgesvd (A, left_p, right_p)

Computes the singular value decomposition (SVD) of a matrix A, comprising the singular values and, optionally, the left and right singular vectors. All elements of A must be integer or floating point numbers. A might or might not be square (same number of rows and columns).

Let m be the number of rows, and n the number of columns of A. The singular value decomposition of A comprises three matrices, U, Sigma, and V^T, such that

A = U . Sigma . V^T

where U is an m-by-m unitary matrix, Sigma is an m-by-n diagonal matrix, and V^T is an n-by-n unitary matrix.

Let sigma[i] be a diagonal element of Sigma, that is, Sigma[i, i] = sigma[i]. The elements sigma[i] are the so-called singular values of A; these are real and nonnegative, and returned in descending order. The first min(m, n) columns of U and V are the left and right singular vectors of A. Note that dgesvd returns the transpose of V, not V itself.

dgesvd(A) computes only the singular values of A. dgesvd(A, left_p, right_p) computes the singular values of A and the left singular vectors when left_p = true and the right singular vectors when right_p = true.

A list of three items is returned. The first item is a list of the singular values. The second item is false or the matrix of left singular vectors. The third item is false or the matrix of right singular vectors.

Example:

(%i1) load ("lapack")$
(%i2) fpprintprec : 6;
(%o2)                           6
(%i3) M: matrix([1, 2, 3], [3.5, 0.5, 8], [-1, 2, -3], [4, 9, 7]);
                        [  1    2    3  ]
                        [               ]
                        [ 3.5  0.5   8  ]
(%o3)                   [               ]
                        [ - 1   2   - 3 ]
                        [               ]
                        [  4    9    7  ]
(%i4) dgesvd (M);
(%o4)      [[14.4744, 6.38637, .452547], false, false]
(%i5) [sigma, U, VT] : dgesvd (M, true, true);
(%o5) [[14.4744, 6.38637, .452547], 
[ - .256731  .00816168   .959029    - .119523 ]
[                                             ]
[ - .526456   .672116   - .206236   - .478091 ]
[                                             ], 
[  .107997   - .532278  - .0708315  - 0.83666 ]
[                                             ]
[ - .803287  - .514659  - .180867    .239046  ]
[ - .374486  - .538209  - .755044 ]
[                                 ]
[  .130623   - .836799   0.5317   ]]
[                                 ]
[ - .917986   .100488    .383672  ]
(%i6) m : length (U);
(%o6)                           4
(%i7) n : length (VT);
(%o7)                           3
(%i8) Sigma:
        genmatrix(lambda ([i, j], if i=j then sigma[i] else 0),
                  m, n);
                  [ 14.4744     0        0    ]
                  [                           ]
                  [    0     6.38637     0    ]
(%o8)             [                           ]
                  [    0        0     .452547 ]
                  [                           ]
                  [    0        0        0    ]
(%i9) U . Sigma . VT - M;
          [  1.11022E-15        0.0       1.77636E-15 ]
          [                                           ]
          [  1.33227E-15    1.66533E-15       0.0     ]
(%o9)     [                                           ]
          [ - 4.44089E-16  - 8.88178E-16  4.44089E-16 ]
          [                                           ]
          [  8.88178E-16    1.77636E-15   8.88178E-16 ]
(%i10) transpose (U) . U;
       [     1.0      5.55112E-17    2.498E-16     2.77556E-17  ]
       [                                                        ]
       [ 5.55112E-17      1.0       5.55112E-17    4.16334E-17  ]
(%o10) [                                                        ]
       [  2.498E-16   5.55112E-17       1.0       - 2.08167E-16 ]
       [                                                        ]
       [ 2.77556E-17  4.16334E-17  - 2.08167E-16       1.0      ]
(%i11) VT . transpose (VT);
          [      1.0           0.0      - 5.55112E-17 ]
          [                                           ]
(%o11)    [      0.0           1.0       5.55112E-17  ]
          [                                           ]
          [ - 5.55112E-17  5.55112E-17       1.0      ]

Function: dlange (norm, A)
Function: zlange (norm, A)

Computes a norm or norm-like function of the matrix A.

max

Compute max(abs(A(i, j))) where i and j range over the rows and columns, respectively, of A. Note that this function is not a proper matrix norm.

one_norm

Compute the L[1] norm of A, that is, the maximum of the sum of the absolute value of elements in each column.

inf_norm

Compute the L[inf] norm of A, that is, the maximum of the sum of the absolute value of elements in each row.

frobenius

Compute the Frobenius norm of A, that is, the square root of the sum of squares of the matrix elements.

Function: dgemm
    dgemm (A, B)
    dgemm (A, B, options)

Compute the product of two matrices and optionally add the product to a third matrix.

In the simplest form, dgemm(A, B) computes the product of the two real matrices, A and B.

In the second form, dgemm computes the alpha * A * B + beta * C where A, B, C are real matrices of the appropriate sizes and alpha and beta are real numbers. Optionally, A and/or B can be transposed before computing the product. The extra parameters are specifed by optional keyword arguments: The keyword arguments are optional and may be specified in any order. They all take the form key=val. The keyword arguments are:

C

The matrix C that should be added. The default is false, which means no matrix is added.

alpha

The product of A and B is multiplied by this value. The default is 1.

beta

If a matrix C is given, this value multiplies C before it is added. The default value is 0, which implies that C is not added, even if C is given. Hence, be sure to specify a non-zero value for beta.

transpose_a

If true, the transpose of A is used instead of A for the product. The default is false.

transpose_b

If true, the transpose of B is used instead of B for the product. The default is false.

(%i1) load ("lapack")$
(%i2) A : matrix([1,2,3],[4,5,6],[7,8,9]);
                                  [ 1  2  3 ]
                                  [         ]
(%o2)                             [ 4  5  6 ]
                                  [         ]
                                  [ 7  8  9 ]
(%i3) B : matrix([-1,-2,-3],[-4,-5,-6],[-7,-8,-9]);
                               [ - 1  - 2  - 3 ]
                               [               ]
(%o3)                          [ - 4  - 5  - 6 ]
                               [               ]
                               [ - 7  - 8  - 9 ]
(%i4) C : matrix([3,2,1],[6,5,4],[9,8,7]);
                                  [ 3  2  1 ]
                                  [         ]
(%o4)                             [ 6  5  4 ]
                                  [         ]
                                  [ 9  8  7 ]
(%i5) dgemm(A,B);
                         [ - 30.0   - 36.0   - 42.0  ]
                         [                           ]
(%o5)                    [ - 66.0   - 81.0   - 96.0  ]
                         [                           ]
                         [ - 102.0  - 126.0  - 150.0 ]
(%i6) A . B;
                            [ - 30   - 36   - 42  ]
                            [                     ]
(%o6)                       [ - 66   - 81   - 96  ]
                            [                     ]
                            [ - 102  - 126  - 150 ]
(%i7) dgemm(A,B,transpose_a=true);
                         [ - 66.0  - 78.0   - 90.0  ]
                         [                          ]
(%o7)                    [ - 78.0  - 93.0   - 108.0 ]
                         [                          ]
                         [ - 90.0  - 108.0  - 126.0 ]
(%i8) transpose(A) . B;
                           [ - 66  - 78   - 90  ]
                           [                    ]
(%o8)                      [ - 78  - 93   - 108 ]
                           [                    ]
                           [ - 90  - 108  - 126 ]
(%i9) dgemm(A,B,c=C,beta=1);
                         [ - 27.0  - 34.0   - 41.0  ]
                         [                          ]
(%o9)                    [ - 60.0  - 76.0   - 92.0  ]
                         [                          ]
                         [ - 93.0  - 118.0  - 143.0 ]
(%i10) A . B + C;
                            [ - 27  - 34   - 41  ]
                            [                    ]
(%o10)                      [ - 60  - 76   - 92  ]
                            [                    ]
                            [ - 93  - 118  - 143 ]
(%i11) dgemm(A,B,c=C,beta=1, alpha=-1);
                            [ 33.0   38.0   43.0  ]
                            [                     ]
(%o11)                      [ 72.0   86.0   100.0 ]
                            [                     ]
                            [ 111.0  134.0  157.0 ]
(%i12) -A . B + C;
                               [ 33   38   43  ]
                               [               ]
(%o12)                         [ 72   86   100 ]
                               [               ]
                               [ 111  134  157 ]

Function: zgeev
    zgeev (A)
    zgeev (A, right_p, left_p)

Like dgeev, but the matrix A is complex.

Function: zheev
    zheev (A)
    zheev (A, eigvec_p)

Like zheev, but the matrix A is assumed to be a square complex Hermitian matrix. If eigvec_p is true, then the eigenvectors of the matrix are also computed.

No check is made that the matrix A is, in fact, Hermitian.

A list of two items is returned, as in dgeev: a list of eigenvalues, and false or the matrix of the eigenvectors.


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67 lbfgs


Next: , Previous: , Up: Top   [Contents][Index]

67.1 Introduction to lbfgs

lbfgs is an implementation of the L-BFGS algorithm [1] to solve unconstrained minimization problems via a limited-memory quasi-Newton (BFGS) algorithm. It is called a limited-memory method because a low-rank approximation of the Hessian matrix inverse is stored instead of the entire Hessian inverse. The program was originally written in Fortran [2] by Jorge Nocedal, incorporating some functions originally written by Jorge J. Moré and David J. Thuente, and translated into Lisp automatically via the program f2cl. The Maxima package lbfgs comprises the translated code plus an interface function which manages some details.

References:

[1] D. Liu and J. Nocedal. "On the limited memory BFGS method for large scale optimization". Mathematical Programming B 45:503–528 (1989)

[2] http://netlib.org/opt/lbfgs_um.shar

·

Category: Numerical methods · Category: Optimization · Category: Share packages · Category: Package lbfgs


Previous: , Up: Top   [Contents][Index]

67.2 Functions and Variables for lbfgs

Function: lbfgs
    lbfgs (FOM, X, X0, epsilon, iprint)
    lbfgs ([FOM, grad] X, X0, epsilon, iprint)

Finds an approximate solution of the unconstrained minimization of the figure of merit FOM over the list of variables X, starting from initial estimates X0, such that norm(grad(FOM)) < epsilon*max(1, norm(X)).

grad, if present, is the gradient of FOM with respect to the variables X. grad may be a list or a function that returns a list, with one element for each element of X. If not present, the gradient is computed automatically by symbolic differentiation. If FOM is a function, the gradient grad must be supplied by the user.

The algorithm applied is a limited-memory quasi-Newton (BFGS) algorithm [1]. It is called a limited-memory method because a low-rank approximation of the Hessian matrix inverse is stored instead of the entire Hessian inverse. Each iteration of the algorithm is a line search, that is, a search along a ray in the variables X, with the search direction computed from the approximate Hessian inverse. The FOM is always decreased by a successful line search. Usually (but not always) the norm of the gradient of FOM also decreases.

iprint controls progress messages printed by lbfgs.

iprint[1]

iprint[1] controls the frequency of progress messages.

iprint[1] < 0

No progress messages.

iprint[1] = 0

Messages at the first and last iterations.

iprint[1] > 0

Print a message every iprint[1] iterations.

iprint[2]

iprint[2] controls the verbosity of progress messages.

iprint[2] = 0

Print out iteration count, number of evaluations of FOM, value of FOM, norm of the gradient of FOM, and step length.

iprint[2] = 1

Same as iprint[2] = 0, plus X0 and the gradient of FOM evaluated at X0.

iprint[2] = 2

Same as iprint[2] = 1, plus values of X at each iteration.

iprint[2] = 3

Same as iprint[2] = 2, plus the gradient of FOM at each iteration.

The columns printed by lbfgs are the following.

I

Number of iterations. It is incremented for each line search.

NFN

Number of evaluations of the figure of merit.

FUNC

Value of the figure of merit at the end of the most recent line search.

GNORM

Norm of the gradient of the figure of merit at the end of the most recent line search.

STEPLENGTH

An internal parameter of the search algorithm.

Additional information concerning details of the algorithm are found in the comments of the original Fortran code [2].

See also lbfgs_nfeval_max and lbfgs_ncorrections.

References:

[1] D. Liu and J. Nocedal. "On the limited memory BFGS method for large scale optimization". Mathematical Programming B 45:503–528 (1989)

[2] http://netlib.org/opt/lbfgs_um.shar

Examples:

The same FOM as computed by FGCOMPUTE in the program sdrive.f in the LBFGS package from Netlib. Note that the variables in question are subscripted variables. The FOM has an exact minimum equal to zero at u[k] = 1 for k = 1, ..., 8.

(%i1) load ("lbfgs")$
(%i2) t1[j] := 1 - u[j];
(%o2)                     t1  := 1 - u
                            j         j
(%i3) t2[j] := 10*(u[j + 1] - u[j]^2);
                                          2
(%o3)                t2  := 10 (u      - u )
                       j         j + 1    j
(%i4) n : 8;
(%o4)                           8
(%i5) FOM : sum (t1[2*j - 1]^2 + t2[2*j - 1]^2, j, 1, n/2);
                 2 2           2              2 2           2
(%o5) 100 (u  - u )  + (1 - u )  + 100 (u  - u )  + (1 - u )
            8    7           7           6    5           5
                     2 2           2              2 2           2
        + 100 (u  - u )  + (1 - u )  + 100 (u  - u )  + (1 - u )
                4    3           3           2    1           1
(%i6) lbfgs (FOM, '[u[1],u[2],u[3],u[4],u[5],u[6],u[7],u[8]],
       [-1.2, 1, -1.2, 1, -1.2, 1, -1.2, 1], 1e-3, [1, 0]);
*************************************************
  N=    8   NUMBER OF CORRECTIONS=25
       INITIAL VALUES
 F=  9.680000000000000D+01   GNORM=  4.657353755084533D+02
*************************************************
 I NFN   FUNC                    GNORM                   STEPLENGTH

 1   3   1.651479526340304D+01   4.324359291335977D+00   7.926153934390631D-04
 2   4   1.650209316638371D+01   3.575788161060007D+00   1.000000000000000D+00
 3   5   1.645461701312851D+01   6.230869903601577D+00   1.000000000000000D+00
 4   6   1.636867301275588D+01   1.177589920974980D+01   1.000000000000000D+00
 5   7   1.612153014409201D+01   2.292797147151288D+01   1.000000000000000D+00
 6   8   1.569118407390628D+01   3.687447158775571D+01   1.000000000000000D+00
 7   9   1.510361958398942D+01   4.501931728123679D+01   1.000000000000000D+00
 8  10   1.391077875774293D+01   4.526061463810630D+01   1.000000000000000D+00
 9  11   1.165625686278198D+01   2.748348965356907D+01   1.000000000000000D+00
10  12   9.859422687859144D+00   2.111494974231706D+01   1.000000000000000D+00
11  13   7.815442521732282D+00   6.110762325764183D+00   1.000000000000000D+00
12  15   7.346380905773044D+00   2.165281166715009D+01   1.285316401779678D-01
13  16   6.330460634066464D+00   1.401220851761508D+01   1.000000000000000D+00
14  17   5.238763939854303D+00   1.702473787619218D+01   1.000000000000000D+00
15  18   3.754016790406625D+00   7.981845727632704D+00   1.000000000000000D+00
16  20   3.001238402313225D+00   3.925482944745832D+00   2.333129631316462D-01
17  22   2.794390709722064D+00   8.243329982586480D+00   2.503577283802312D-01
18  23   2.563783562920545D+00   1.035413426522664D+01   1.000000000000000D+00
19  24   2.019429976373283D+00   1.065187312340952D+01   1.000000000000000D+00
20  25   1.428003167668592D+00   2.475962450735100D+00   1.000000000000000D+00
21  27   1.197874264859232D+00   8.441707983339661D+00   4.303451060697367D-01
22  28   9.023848942003913D-01   1.113189216665625D+01   1.000000000000000D+00
23  29   5.508226405855795D-01   2.380830599637816D+00   1.000000000000000D+00
24  31   3.902893258879521D-01   5.625595817143044D+00   4.834988416747262D-01
25  32   3.207542206881058D-01   1.149444645298493D+01   1.000000000000000D+00
26  33   1.874468266118200D-01   3.632482152347445D+00   1.000000000000000D+00
27  34   9.575763380282112D-02   4.816497449000391D+00   1.000000000000000D+00
28  35   4.085145106760390D-02   2.087009347116811D+00   1.000000000000000D+00
29  36   1.931106005512628D-02   3.886818624052740D+00   1.000000000000000D+00
30  37   6.894000636920714D-03   3.198505769992936D+00   1.000000000000000D+00
31  38   1.443296008850287D-03   1.590265460381961D+00   1.000000000000000D+00
32  39   1.571766574930155D-04   3.098257002223532D-01   1.000000000000000D+00
33  40   1.288011779655132D-05   1.207784334505595D-02   1.000000000000000D+00
34  41   1.806140190993455D-06   4.587890258846915D-02   1.000000000000000D+00
35  42   1.769004612050548D-07   1.790537363138099D-02   1.000000000000000D+00
36  43   3.312164244118216D-10   6.782068546986653D-04   1.000000000000000D+00

 THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.
 IFLAG = 0
(%o6) [u  = 1.000005339816132, u  = 1.000009942840108, 
        1                       2
u  = 1.000005339816132, u  = 1.000009942840108, 
 3                       4
u  = 1.000005339816132, u  = 1.000009942840108, 
 5                       6
u  = 1.000005339816132, u  = 1.000009942840108]
 7                       8

A regression problem. The FOM is the mean square difference between the predicted value F(X[i]) and the observed value Y[i]. The function F is a bounded monotone function (a so-called "sigmoidal" function). In this example, lbfgs computes approximate values for the parameters of F and plot2d displays a comparison of F with the observed data.

(%i1) load ("lbfgs")$
(%i2) FOM : '((1/length(X))*sum((F(X[i]) - Y[i])^2, i, 1,
                                                length(X)));
                               2
               sum((F(X ) - Y ) , i, 1, length(X))
                       i     i
(%o2)          -----------------------------------
                            length(X)
(%i3) X : [1, 2, 3, 4, 5];
(%o3)                    [1, 2, 3, 4, 5]
(%i4) Y : [0, 0.5, 1, 1.25, 1.5];
(%o4)                [0, 0.5, 1, 1.25, 1.5]
(%i5) F(x) := A/(1 + exp(-B*(x - C)));
                                   A
(%o5)            F(x) := ----------------------
                         1 + exp((- B) (x - C))
(%i6) ''FOM;
                A               2            A                2
(%o6) ((----------------- - 1.5)  + (----------------- - 1.25)
          - B (5 - C)                  - B (4 - C)
        %e            + 1            %e            + 1
            A             2            A               2
 + (----------------- - 1)  + (----------------- - 0.5)
      - B (3 - C)                - B (2 - C)
    %e            + 1          %e            + 1
             2
            A
 + --------------------)/5
      - B (1 - C)     2
   (%e            + 1)
(%i7) estimates : lbfgs (FOM, '[A, B, C], [1, 1, 1], 1e-4, [1, 0]);
*************************************************
  N=    3   NUMBER OF CORRECTIONS=25
       INITIAL VALUES
 F=  1.348738534246918D-01   GNORM=  2.000215531936760D-01
*************************************************

I  NFN  FUNC                    GNORM                   STEPLENGTH
1    3  1.177820636622582D-01   9.893138394953992D-02   8.554435968992371D-01  
2    6  2.302653892214013D-02   1.180098521565904D-01   2.100000000000000D+01  
3    8  1.496348495303004D-02   9.611201567691624D-02   5.257340567840710D-01  
4    9  7.900460841091138D-03   1.325041647391314D-02   1.000000000000000D+00  
5   10  7.314495451266914D-03   1.510670810312226D-02   1.000000000000000D+00  
6   11  6.750147275936668D-03   1.914964958023037D-02   1.000000000000000D+00  
7   12  5.850716021108202D-03   1.028089194579382D-02   1.000000000000000D+00  
8   13  5.778664230657800D-03   3.676866074532179D-04   1.000000000000000D+00  
9   14  5.777818823650780D-03   3.010740179797108D-04   1.000000000000000D+00  

 THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.
 IFLAG = 0
(%o7) [A = 1.461933911464101, B = 1.601593973254801, 
                                           C = 2.528933072164855]
(%i8) plot2d ([F(x), [discrete, X, Y]], [x, -1, 6]), ''estimates;
(%o8) 

Gradient of FOM is specified (instead of computing it automatically). Both the FOM and its gradient are passed as functions to lbfgs.

(%i1) load ("lbfgs")$
(%i2) F(a, b, c) := (a - 5)^2 + (b - 3)^4 + (c - 2)^6$
(%i3) define(F_grad(a, b, c),
             map (lambda ([x], diff (F(a, b, c), x)), [a, b, c]))$
(%i4) estimates : lbfgs ([F, F_grad],
                   [a, b, c], [0, 0, 0], 1e-4, [1, 0]);
*************************************************
  N=    3   NUMBER OF CORRECTIONS=25
       INITIAL VALUES
 F=  1.700000000000000D+02   GNORM=  2.205175729958953D+02
*************************************************

   I  NFN     FUNC                    GNORM                   STEPLENGTH

   1    2     6.632967565917637D+01   6.498411132518770D+01   4.534785987412505D-03  
   2    3     4.368890936228036D+01   3.784147651974131D+01   1.000000000000000D+00  
   3    4     2.685298972775191D+01   1.640262125898520D+01   1.000000000000000D+00  
   4    5     1.909064767659852D+01   9.733664001790506D+00   1.000000000000000D+00  
   5    6     1.006493272061515D+01   6.344808151880209D+00   1.000000000000000D+00  
   6    7     1.215263596054292D+00   2.204727876126877D+00   1.000000000000000D+00  
   7    8     1.080252896385329D-02   1.431637116951845D-01   1.000000000000000D+00  
   8    9     8.407195124830860D-03   1.126344579730008D-01   1.000000000000000D+00  
   9   10     5.022091686198525D-03   7.750731829225275D-02   1.000000000000000D+00  
  10   11     2.277152808939775D-03   5.032810859286796D-02   1.000000000000000D+00  
  11   12     6.489384688303218D-04   1.932007150271009D-02   1.000000000000000D+00  
  12   13     2.075791943844547D-04   6.964319310814365D-03   1.000000000000000D+00  
  13   14     7.349472666162258D-05   4.017449067849554D-03   1.000000000000000D+00  
  14   15     2.293617477985238D-05   1.334590390856715D-03   1.000000000000000D+00  
  15   16     7.683645404048675D-06   6.011057038099202D-04   1.000000000000000D+00  

 THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.
 IFLAG = 0
(%o4) [a = 5.000086823042934, b = 3.052395429705181, 
                                           c = 1.927980629919583]

Variable: lbfgs_nfeval_max

Default value: 100

lbfgs_nfeval_max is the maximum number of evaluations of the figure of merit (FOM) in lbfgs. When lbfgs_nfeval_max is reached, lbfgs returns the result of the last successful line search.

Variable: lbfgs_ncorrections

Default value: 25

lbfgs_ncorrections is the number of corrections applied to the approximate inverse Hessian matrix which is maintained by lbfgs.


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68 lindstedt


Previous: , Up: lindstedt-pkg   [Contents][Index]

68.1 Functions and Variables for lindstedt

Function: Lindstedt (eq,pvar,torder,ic)

This is a first pass at a Lindstedt code. It can solve problems with initial conditions entered, which can be arbitrary constants, (just not %k1 and %k2) where the initial conditions on the perturbation equations are z[i]=0, z'[i]=0 for i>0. ic is the list of initial conditions.

Problems occur when initial conditions are not given, as the constants in the perturbation equations are the same as the zero order equation solution. Also, problems occur when the initial conditions for the perturbation equations are not z[i]=0, z'[i]=0 for i>0, such as the Van der Pol equation.

Example:

(%i1) load("makeOrders")$

(%i2) load("lindstedt")$

(%i3) Lindstedt('diff(x,t,2)+x-(e*x^3)/6,e,2,[1,0]);
          2
         e  (cos(5 T) - 24 cos(3 T) + 23 cos(T))
(%o3) [[[---------------------------------------
                          36864
   e (cos(3 T) - cos(T))
 - --------------------- + cos(T)],
            192
          2
       7 e    e
T = (- ---- - -- + 1) t]]
       3072   16

To use this function write first load("makeOrders") and load("lindstedt").


Next: , Previous: , Up: Top   [Contents][Index]

69 linearalgebra


Next: , Previous: , Up: linearalgebra-pkg   [Contents][Index]

69.1 Introduction to linearalgebra

linearalgebra is a collection of functions for linear algebra.

Example:

(%i1) M : matrix ([1, 2], [1, 2]);
                            [ 1  2 ]
(%o1)                       [      ]
                            [ 1  2 ]
(%i2) nullspace (M);
                               [  1  ]
                               [     ]
(%o2)                     span([   1 ])
                               [ - - ]
                               [   2 ]
(%i3) columnspace (M);
                                [ 1 ]
(%o3)                      span([   ])
                                [ 1 ]
(%i4) ptriangularize (M - z*ident(2), z);
                         [ 1   2 - z   ]
(%o4)                    [             ]
                         [           2 ]
                         [ 0  3 z - z  ]
(%i5) M : matrix ([1, 2, 3], [4, 5, 6], [7, 8, 9]) - z*ident(3);
                     [ 1 - z    2      3   ]
                     [                     ]
(%o5)                [   4    5 - z    6   ]
                     [                     ]
                     [   7      8    9 - z ]
(%i6) MM : ptriangularize (M, z);
              [ 4  5 - z            6            ]
              [                                  ]
              [                2                 ]
              [     66        z    102 z   132   ]
              [ 0   --      - -- + ----- + ---   ]
(%o6)         [     49        7     49     49    ]
              [                                  ]
              [               3        2         ]
              [           49 z    245 z    147 z ]
              [ 0    0    ----- - ------ - ----- ]
              [            264      88      44   ]
(%i7) algebraic : true;
(%o7)                         true
(%i8) tellrat (MM [3, 3]);
                         3       2
(%o8)                  [z  - 15 z  - 18 z]
(%i9) MM : ratsimp (MM);
               [ 4  5 - z           6           ]
               [                                ]
               [                2               ]
(%o9)          [     66      7 z  - 102 z - 132 ]
               [ 0   --    - ------------------ ]
               [     49              49         ]
               [                                ]
               [ 0    0             0           ]
(%i10) nullspace (MM);
                        [        1         ]
                        [                  ]
                        [   2              ]
                        [  z  - 14 z - 16  ]
                        [  --------------  ]
(%o10)             span([        8         ])
                        [                  ]
                        [    2             ]
                        [   z  - 18 z - 12 ]
                        [ - -------------- ]
                        [         12       ]
(%i11) M : matrix ([1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12],
                   [13, 14, 15, 16]);
                       [ 1   2   3   4  ]
                       [                ]
                       [ 5   6   7   8  ]
(%o11)                 [                ]
                       [ 9   10  11  12 ]
                       [                ]
                       [ 13  14  15  16 ]
(%i12) columnspace (M);
                           [ 1  ]  [ 2  ]
                           [    ]  [    ]
                           [ 5  ]  [ 6  ]
(%o12)                span([    ], [    ])
                           [ 9  ]  [ 10 ]
                           [    ]  [    ]
                           [ 13 ]  [ 14 ]
(%i13) apply ('orthogonal_complement, args (nullspace (transpose (M))));
                           [ 0 ]  [  1  ]
                           [   ]  [     ]
                           [ 1 ]  [  0  ]
(%o13)                span([   ], [     ])
                           [ 2 ]  [ - 1 ]
                           [   ]  [     ]
                           [ 3 ]  [ - 2 ]
·

Category: Linear algebra · Category: Share packages · Category: Package linearalgebra


Previous: , Up: linearalgebra-pkg   [Contents][Index]

69.2 Functions and Variables for linearalgebra

Function: addmatrices (f, M_1, …, M_n)

Using the function f as the addition function, return the sum of the matrices M_1, …, M_n. The function f must accept any number of arguments (a Maxima nary function).

Examples:

(%i1) m1 : matrix([1,2],[3,4])$
(%i2) m2 : matrix([7,8],[9,10])$
(%i3) addmatrices('max,m1,m2);
(%o3) matrix([7,8],[9,10])
(%i4) addmatrices('max,m1,m2,5*m1);
(%o4) matrix([7,10],[15,20])

Function: blockmatrixp (M)

Return true if and only if M is a matrix and every entry of M is a matrix.

Function: columnop (M, i, j, theta)

If M is a matrix, return the matrix that results from doing the column operation C_i <- C_i - theta * C_j. If M doesn’t have a row i or j, signal an error.

Function: columnswap (M, i, j)

If M is a matrix, swap columns i and j. If M doesn’t have a column i or j, signal an error.

Function: columnspace (M)

If M is a matrix, return span (v_1, ..., v_n), where the set {v_1, ..., v_n} is a basis for the column space of M. The span of the empty set is {0}. Thus, when the column space has only one member, return span ().

Function: cholesky
    cholesky (M)
    cholesky (M, field)

Return the Cholesky factorization of the matrix selfadjoint (or hermitian) matrix M. The second argument defaults to ’generalring.’ For a description of the possible values for field, see lu_factor.

Function: ctranspose (M)

Return the complex conjugate transpose of the matrix M. The function ctranspose uses matrix_element_transpose to transpose each matrix element.

Function: diag_matrix (d_1, d_2, …, d_n)

Return a diagonal matrix with diagonal entries d_1, d_2, …, d_n. When the diagonal entries are matrices, the zero entries of the returned matrix are zero matrices of the appropriate size; for example:

(%i1) diag_matrix(diag_matrix(1,2),diag_matrix(3,4));

                            [ [ 1  0 ]  [ 0  0 ] ]
                            [ [      ]  [      ] ]
                            [ [ 0  2 ]  [ 0  0 ] ]
(%o1)                       [                    ]
                            [ [ 0  0 ]  [ 3  0 ] ]
                            [ [      ]  [      ] ]
                            [ [ 0  0 ]  [ 0  4 ] ]
(%i2) diag_matrix(p,q);

                                   [ p  0 ]
(%o2)                              [      ]
                                   [ 0  q ]

Function: dotproduct (u, v)

Return the dotproduct of vectors u and v. This is the same as conjugate (transpose (u)) . v. The arguments u and v must be column vectors.

Function: eigens_by_jacobi
    eigens_by_jacobi (A)
    eigens_by_jacobi (A, field_type)

Computes the eigenvalues and eigenvectors of A by the method of Jacobi rotations. A must be a symmetric matrix (but it need not be positive definite nor positive semidefinite). field_type indicates the computational field, either floatfield or bigfloatfield. If field_type is not specified, it defaults to floatfield.

The elements of A must be numbers or expressions which evaluate to numbers via float or bfloat (depending on field_type).

Examples:

(%i1) S: matrix([1/sqrt(2), 1/sqrt(2)],[-1/sqrt(2), 1/sqrt(2)]);
                     [     1         1    ]
                     [  -------   ------- ]
                     [  sqrt(2)   sqrt(2) ]
(%o1)                [                    ]
                     [      1        1    ]
                     [ - -------  ------- ]
                     [   sqrt(2)  sqrt(2) ]
(%i2) L : matrix ([sqrt(3), 0], [0, sqrt(5)]);
                      [ sqrt(3)     0    ]
(%o2)                 [                  ]
                      [    0     sqrt(5) ]
(%i3) M : S . L . transpose (S);
            [ sqrt(5)   sqrt(3)  sqrt(5)   sqrt(3) ]
            [ ------- + -------  ------- - ------- ]
            [    2         2        2         2    ]
(%o3)       [                                      ]
            [ sqrt(5)   sqrt(3)  sqrt(5)   sqrt(3) ]
            [ ------- - -------  ------- + ------- ]
            [    2         2        2         2    ]
(%i4) eigens_by_jacobi (M);
The largest percent change was 0.1454972243679
The largest percent change was 0.0
number of sweeps: 2
number of rotations: 1
(%o4) [[1.732050807568877, 2.23606797749979], 
                        [  0.70710678118655   0.70710678118655 ]
                        [                                      ]]
                        [ - 0.70710678118655  0.70710678118655 ]
(%i5) float ([[sqrt(3), sqrt(5)], S]);
(%o5) [[1.732050807568877, 2.23606797749979], 
                        [  0.70710678118655   0.70710678118655 ]
                        [                                      ]]
                        [ - 0.70710678118655  0.70710678118655 ]
(%i6) eigens_by_jacobi (M, bigfloatfield);
The largest percent change was 1.454972243679028b-1
The largest percent change was 0.0b0
number of sweeps: 2
number of rotations: 1
(%o6) [[1.732050807568877b0, 2.23606797749979b0], 
                [  7.071067811865475b-1   7.071067811865475b-1 ]
                [                                              ]]
                [ - 7.071067811865475b-1  7.071067811865475b-1 ]

Function: get_lu_factors (x)

When x = lu_factor (A), then get_lu_factors returns a list of the form [P, L, U], where P is a permutation matrix, L is lower triangular with ones on the diagonal, and U is upper triangular, and A = P L U.

Function: hankel
    hankel (col)
    hankel (col, row)

Return a Hankel matrix H. The first column of H is col; except for the first entry, the last row of H is row. The default for row is the zero vector with the same length as col.

Function: hessian (f, x)

Returns the Hessian matrix of f with respect to the list of variables x. The (i, j)-th element of the Hessian matrix is diff(f, x[i], 1, x[j], 1).

Examples:

(%i1) hessian (x * sin (y), [x, y]);
                     [   0       cos(y)   ]
(%o1)                [                    ]
                     [ cos(y)  - x sin(y) ]
(%i2) depends (F, [a, b]);
(%o2)                       [F(a, b)]
(%i3) hessian (F, [a, b]);
                        [   2      2   ]
                        [  d F    d F  ]
                        [  ---   ----- ]
                        [    2   da db ]
                        [  da          ]
(%o3)                   [              ]
                        [   2      2   ]
                        [  d F    d F  ]
                        [ -----   ---  ]
                        [ da db     2  ]
                        [         db   ]

Function: hilbert_matrix (n)

Return the n by n Hilbert matrix. When n isn’t a positive integer, signal an error.

Function: identfor
    identfor (M)
    identfor (M, fld)

Return an identity matrix that has the same shape as the matrix M. The diagonal entries of the identity matrix are the multiplicative identity of the field fld; the default for fld is generalring.

The first argument M should be a square matrix or a non-matrix. When M is a matrix, each entry of M can be a square matrix – thus M can be a blocked Maxima matrix. The matrix can be blocked to any (finite) depth.

See also zerofor

Function: invert_by_lu (M, (rng generalring))

Invert a matrix M by using the LU factorization. The LU factorization is done using the ring rng.

Function: jacobian (f, x)

Returns the Jacobian matrix of the list of functions f with respect to the list of variables x. The (i, j)-th element of the Jacobian matrix is diff(f[i], x[j]).

Examples:

(%i1) jacobian ([sin (u - v), sin (u * v)], [u, v]);
                  [ cos(v - u)  - cos(v - u) ]
(%o1)             [                          ]
                  [ v cos(u v)   u cos(u v)  ]
(%i2) depends ([F, G], [y, z]);
(%o2)                  [F(y, z), G(y, z)]
(%i3) jacobian ([F, G], [y, z]);
                           [ dF  dF ]
                           [ --  -- ]
                           [ dy  dz ]
(%o3)                      [        ]
                           [ dG  dG ]
                           [ --  -- ]
                           [ dy  dz ]

Function: kronecker_product (A, B)

Return the Kronecker product of the matrices A and B.

Function: listp
    listp (e, p)
    listp (e)

Given an optional argument p, return true if e is a Maxima list and p evaluates to true for every list element. When listp is not given the optional argument, return true if e is a Maxima list. In all other cases, return false.

Function: locate_matrix_entry (M, r_1, c_1, r_2, c_2, f, rel)

The first argument must be a matrix; the arguments r_1 through c_2 determine a sub-matrix of M that consists of rows r_1 through r_2 and columns c_1 through c_2.

Find a entry in the sub-matrix M that satisfies some property. Three cases:

(1) rel = 'bool and f a predicate:

Scan the sub-matrix from left to right then top to bottom, and return the index of the first entry that satisfies the predicate f. If no matrix entry satisfies f, return false.

(2) rel = 'max and f real-valued:

Scan the sub-matrix looking for an entry that maximizes f. Return the index of a maximizing entry.

(3) rel = 'min and f real-valued:

Scan the sub-matrix looking for an entry that minimizes f. Return the index of a minimizing entry.

Function: lu_backsub (M, b)

When M = lu_factor (A, field), then lu_backsub (M, b) solves the linear system A x = b.

The n by m matrix b, with n the number of rows of the matrix A, contains one right hand side per column. If there is only one right hand side then b must be a n by 1 matrix.

Each column of the matrix x=lu_backsub (M, b) is the solution corresponding to the respective column of b.

Examples:

(%i1) A : matrix ([1 - z, 3], [3, 8 - z]);
                               [ 1 - z    3   ]
(%o1)                          [              ]
                               [   3    8 - z ]
(%i2) M : lu_factor (A,generalring);
               [ 1 - z          3         ]
               [                          ]
(%o2)         [[   3              9       ], [1, 2], generalring]
               [ -----  (- z) - ----- + 8 ]
               [ 1 - z          1 - z     ]
(%i3) b : matrix([a],[c]);
                                     [ a ]
(%o3)                                [   ]
                                     [ c ]
(%i4) x : lu_backsub(M,b);
                           [               3 a     ]
                           [       3 (c - -----)   ]
                           [              1 - z    ]
                           [ a - ----------------- ]
                           [               9       ]
                           [     (- z) - ----- + 8 ]
                           [             1 - z     ]
                           [ --------------------- ]
(%o4)                      [         1 - z         ]
                           [                       ]
                           [            3 a        ]
                           [       c - -----       ]
                           [           1 - z       ]
                           [   -----------------   ]
                           [             9         ]
                           [   (- z) - ----- + 8   ]
                           [           1 - z       ]
(%i5) ratsimp(A . x - b);
                                     [ 0 ]
(%o5)                                [   ]
                                     [ 0 ]
(%i6) B : matrix([a,d],[c,f]);
                                   [ a  d ]
(%o6)                              [      ]
                                   [ c  f ]
(%i7) x : lu_backsub(M,B);
               [               3 a                    3 d     ]
               [       3 (c - -----)          3 (f - -----)   ]
               [              1 - z                  1 - z    ]
               [ a - -----------------  d - ----------------- ]
               [               9                      9       ]
               [     (- z) - ----- + 8      (- z) - ----- + 8 ]
               [             1 - z                  1 - z     ]
               [ ---------------------  --------------------- ]
(%o7)          [         1 - z                  1 - z         ]
               [                                              ]
               [            3 a                    3 d        ]
               [       c - -----              f - -----       ]
               [           1 - z                  1 - z       ]
               [   -----------------      -----------------   ]
               [             9                      9         ]
               [   (- z) - ----- + 8      (- z) - ----- + 8   ]
               [           1 - z                  1 - z       ]
(%i8) ratsimp(A . x - B);
                                   [ 0  0 ]
(%o8)                              [      ]
                                   [ 0  0 ]

Function: lu_factor (M, field)

Return a list of the form [LU, perm, fld], or [LU, perm, fld, lower-cnd upper-cnd], where

(1) The matrix LU contains the factorization of M in a packed form. Packed form means three things: First, the rows of LU are permuted according to the list perm. If, for example, perm is the list [3,2,1], the actual first row of the LU factorization is the third row of the matrix LU. Second, the lower triangular factor of m is the lower triangular part of LU with the diagonal entries replaced by all ones. Third, the upper triangular factor of M is the upper triangular part of LU.

(2) When the field is either floatfield or complexfield, the numbers lower-cnd and upper-cnd are lower and upper bounds for the infinity norm condition number of M. For all fields, the condition number might not be estimated; for such fields, lu_factor returns a two item list. Both the lower and upper bounds can differ from their true values by arbitrarily large factors. (See also mat_cond.)

The argument M must be a square matrix.

The optional argument fld must be a symbol that determines a ring or field. The pre-defined fields and rings are:

(a) generalring – the ring of Maxima expressions,

(b) floatfield – the field of floating point numbers of the type double,

(c) complexfield – the field of complex floating point numbers of the type double,

(d) crering – the ring of Maxima CRE expressions,

(e) rationalfield – the field of rational numbers,

(f) runningerror – track the all floating point rounding errors,

(g) noncommutingring – the ring of Maxima expressions where multiplication is the non-commutative dot operator.

When the field is floatfield, complexfield, or runningerror, the algorithm uses partial pivoting; for all other fields, rows are switched only when needed to avoid a zero pivot.

Floating point addition arithmetic isn’t associative, so the meaning of ’field’ differs from the mathematical definition.

A member of the field runningerror is a two member Maxima list of the form [x,n],where x is a floating point number and n is an integer. The relative difference between the ’true’ value of x and x is approximately bounded by the machine epsilon times n. The running error bound drops some terms that of the order the square of the machine epsilon.

There is no user-interface for defining a new field. A user that is familiar with Common Lisp should be able to define a new field. To do this, a user must define functions for the arithmetic operations and functions for converting from the field representation to Maxima and back. Additionally, for ordered fields (where partial pivoting will be used), a user must define functions for the magnitude and for comparing field members. After that all that remains is to define a Common Lisp structure mring. The file mring has many examples.

To compute the factorization, the first task is to convert each matrix entry to a member of the indicated field. When conversion isn’t possible, the factorization halts with an error message. Members of the field needn’t be Maxima expressions. Members of the complexfield, for example, are Common Lisp complex numbers. Thus after computing the factorization, the matrix entries must be converted to Maxima expressions.

See also get_lu_factors.

Examples:

(%i1) w[i,j] := random (1.0) + %i * random (1.0);
(%o1)          w     := random(1.) + %i random(1.)
                i, j
(%i2) showtime : true$
Evaluation took 0.00 seconds (0.00 elapsed)
(%i3) M : genmatrix (w, 100, 100)$
Evaluation took 7.40 seconds (8.23 elapsed)
(%i4) lu_factor (M, complexfield)$
Evaluation took 28.71 seconds (35.00 elapsed)
(%i5) lu_factor (M, generalring)$
Evaluation took 109.24 seconds (152.10 elapsed)
(%i6) showtime : false$

(%i7) M : matrix ([1 - z, 3], [3, 8 - z]); 
                        [ 1 - z    3   ]
(%o7)                   [              ]
                        [   3    8 - z ]
(%i8) lu_factor (M, generalring);
          [ 1 - z         3        ]
          [                        ]
(%o8)    [[   3            9       ], [1, 2], generalring]
          [ -----  - z - ----- + 8 ]
          [ 1 - z        1 - z     ]
(%i9) get_lu_factors (%);
                  [   1    0 ]  [ 1 - z         3        ]
        [ 1  0 ]  [          ]  [                        ]
(%o9)  [[      ], [   3      ], [                9       ]]
        [ 0  1 ]  [ -----  1 ]  [   0    - z - ----- + 8 ]
                  [ 1 - z    ]  [              1 - z     ]
(%i10) %[1] . %[2] . %[3];
                        [ 1 - z    3   ]
(%o10)                  [              ]
                        [   3    8 - z ]

Function: mat_cond
    mat_cond (M, 1)
    mat_cond (M, inf)

Return the p-norm matrix condition number of the matrix m. The allowed values for p are 1 and inf. This function uses the LU factorization to invert the matrix m. Thus the running time for mat_cond is proportional to the cube of the matrix size; lu_factor determines lower and upper bounds for the infinity norm condition number in time proportional to the square of the matrix size.

Function: mat_norm
    mat_norm (M, 1)
    mat_norm (M, inf)
    mat_norm (M, frobenius)

Return the matrix p-norm of the matrix M. The allowed values for p are 1, inf, and frobenius (the Frobenius matrix norm). The matrix M should be an unblocked matrix.

Function: matrixp
    matrixp (e, p)
    matrixp (e)

Given an optional argument p, return true if e is a matrix and p evaluates to true for every matrix element. When matrixp is not given an optional argument, return true if e is a matrix. In all other cases, return false.

See also blockmatrixp

Function: matrix_size (M)

Return a two member list that gives the number of rows and columns, respectively of the matrix M.

Function: mat_fullunblocker (M)

If M is a block matrix, unblock the matrix to all levels. If M is a matrix, return M; otherwise, signal an error.

Function: mat_trace (M)

Return the trace of the matrix M. If M isn’t a matrix, return a noun form. When M is a block matrix, mat_trace(M) returns the same value as does mat_trace(mat_unblocker(m)).

Function: mat_unblocker (M)

If M is a block matrix, unblock M one level. If M is a matrix, mat_unblocker (M) returns M; otherwise, signal an error.

Thus if each entry of M is matrix, mat_unblocker (M) returns an unblocked matrix, but if each entry of M is a block matrix, mat_unblocker (M) returns a block matrix with one less level of blocking.

If you use block matrices, most likely you’ll want to set matrix_element_mult to "." and matrix_element_transpose to 'transpose. See also mat_fullunblocker.

Example:

(%i1) A : matrix ([1, 2], [3, 4]);
                            [ 1  2 ]
(%o1)                       [      ]
                            [ 3  4 ]
(%i2) B : matrix ([7, 8], [9, 10]);
                            [ 7  8  ]
(%o2)                       [       ]
                            [ 9  10 ]
(%i3) matrix ([A, B]);
                     [ [ 1  2 ]  [ 7  8  ] ]
(%o3)                [ [      ]  [       ] ]
                     [ [ 3  4 ]  [ 9  10 ] ]
(%i4) mat_unblocker (%);
                         [ 1  2  7  8  ]
(%o4)                    [             ]
                         [ 3  4  9  10 ]

Function: nullspace (M)

If M is a matrix, return span (v_1, ..., v_n), where the set {v_1, ..., v_n} is a basis for the nullspace of M. The span of the empty set is {0}. Thus, when the nullspace has only one member, return span ().

Function: nullity (M)

If M is a matrix, return the dimension of the nullspace of M.

Function: orthogonal_complement (v_1, …, v_n)

Return span (u_1, ..., u_m), where the set {u_1, ..., u_m} is a basis for the orthogonal complement of the set (v_1, ..., v_n).

Each vector v_1 through v_n must be a column vector.

Function: polynomialp
    polynomialp (p, L, coeffp, exponp)
    polynomialp (p, L, coeffp)
    polynomialp (p, L)

Return true if p is a polynomial in the variables in the list L. The predicate coeffp must evaluate to true for each coefficient, and the predicate exponp must evaluate to true for all exponents of the variables in L. If you want to use a non-default value for exponp, you must supply coeffp with a value even if you want to use the default for coeffp.

The command polynomialp (p, L, coeffp) is equivalent to polynomialp (p, L, coeffp, 'nonnegintegerp) and the command polynomialp (p, L) is equivalent to polynomialp (p, L, 'constantp, 'nonnegintegerp).

The polynomial needn’t be expanded:

(%i1) polynomialp ((x + 1)*(x + 2), [x]);
(%o1)                         true
(%i2) polynomialp ((x + 1)*(x + 2)^a, [x]);
(%o2)                         false

An example using non-default values for coeffp and exponp:

(%i1) polynomialp ((x + 1)*(x + 2)^(3/2), [x], numberp, numberp);
(%o1)                         true
(%i2) polynomialp ((x^(1/2) + 1)*(x + 2)^(3/2), [x], numberp,
                                                        numberp);
(%o2)                         true

Polynomials with two variables:

(%i1) polynomialp (x^2 + 5*x*y + y^2, [x]);
(%o1)                         false
(%i2) polynomialp (x^2 + 5*x*y + y^2, [x, y]);
(%o2)                         true

Function: polytocompanion (p, x)

If p is a polynomial in x, return the companion matrix of p. For a monic polynomial p of degree n, we have p = (-1)^n charpoly (polytocompanion (p, x)).

When p isn’t a polynomial in x, signal an error.

Function: ptriangularize (M, v)

If M is a matrix with each entry a polynomial in v, return a matrix M2 such that

(1) M2 is upper triangular,

(2) M2 = E_n ... E_1 M, where E_1 through E_n are elementary matrices whose entries are polynomials in v,

(3) |det (M)| = |det (M2)|,

Note: This function doesn’t check that every entry is a polynomial in v.

Function: rowop (M, i, j, theta)

If M is a matrix, return the matrix that results from doing the row operation R_i <- R_i - theta * R_j. If M doesn’t have a row i or j, signal an error.

Function: linalg_rank (M)

Return the rank of the matrix M. This function is equivalent to function rank, but it uses a different algorithm: it finds the columnspace of the matrix and counts its elements, since the rank of a matrix is the dimension of its column space.

(%i1) linalg_rank(matrix([1,2],[2,4]));
(%o1)                           1
(%i2) linalg_rank(matrix([1,b],[c,d]));
(%o2)                           2

Function: rowswap (M, i, j)

If M is a matrix, swap rows i and j. If M doesn’t have a row i or j, signal an error.

Function: toeplitz
    toeplitz (col)
    toeplitz (col, row)

Return a Toeplitz matrix T. The first first column of T is col; except for the first entry, the first row of T is row. The default for row is complex conjugate of col. Example:

(%i1)  toeplitz([1,2,3],[x,y,z]);
                                  [ 1  y  z ]
                                  [         ]
(%o1)                             [ 2  1  y ]
                                  [         ]
                                  [ 3  2  1 ]
(%i2)  toeplitz([1,1+%i]);

                              [   1     1 - %I ]
(%o2)                         [                ]
                              [ %I + 1    1    ]

Function: vandermonde_matrix ([x_1, ..., x_n])

Return a n by n matrix whose i-th row is [1, x_i, x_i^2, ... x_i^(n-1)].

Function: zerofor
    zerofor (M)
    zerofor (M, fld)

Return a zero matrix that has the same shape as the matrix M. Every entry of the zero matrix is the additive identity of the field fld; the default for fld is generalring.

The first argument M should be a square matrix or a non-matrix. When M is a matrix, each entry of M can be a square matrix – thus M can be a blocked Maxima matrix. The matrix can be blocked to any (finite) depth.

See also identfor

Function: zeromatrixp (M)

If M is not a block matrix, return true if is (equal (e, 0)) is true for each element e of the matrix M. If M is a block matrix, return true if zeromatrixp evaluates to true for each element of e.


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70 lsquares


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70.1 Introduction to lsquares

lsquares is a collection of functions to implement the method of least squares to estimate parameters for a model from numerical data.

·

Category: Statistical estimation · Category: Share packages · Category: Package lsquares


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70.2 Functions and Variables for lsquares

Function: lsquares_estimates
    lsquares_estimates (D, x, e, a)
    lsquares_estimates (D, x, e, a, initial = L, tol = t)

Estimate parameters a to best fit the equation e in the variables x and a to the data D, as determined by the method of least squares. lsquares_estimates first seeks an exact solution, and if that fails, then seeks an approximate solution.

The return value is a list of lists of equations of the form [a = ..., b = ..., c = ...]. Each element of the list is a distinct, equivalent minimum of the mean square error.

The data D must be a matrix. Each row is one datum (which may be called a ‘record’ or ‘case’ in some contexts), and each column contains the values of one variable across all data. The list of variables x gives a name for each column of D, even the columns which do not enter the analysis. The list of parameters a gives the names of the parameters for which estimates are sought. The equation e is an expression or equation in the variables x and a; if e is not an equation, it is treated the same as e = 0.

Additional arguments to lsquares_estimates are specified as equations and passed on verbatim to the function lbfgs which is called to find estimates by a numerical method when an exact result is not found.

If some exact solution can be found (via solve), the data D may contain non-numeric values. However, if no exact solution is found, each element of D must have a numeric value. This includes numeric constants such as %pi and %e as well as literal numbers (integers, rationals, ordinary floats, and bigfloats). Numerical calculations are carried out with ordinary floating-point arithmetic, so all other kinds of numbers are converted to ordinary floats for calculations.

If lsquares_estimates needs excessive amounts of time or runs out of memory lsquares_estimates_approximate, which skips the attempt to find an exact solution, might still succeed.

load(lsquares) loads this function.

See also lsquares_estimates_exact, lsquares_estimates_approximate,
lsquares_mse, lsquares_residuals, and lsquares_residual_mse.

Examples:

A problem for which an exact solution is found.

(%i1) load ("lsquares")$
(%i2) M : matrix (
        [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                           [ 1  1  1 ]
                           [         ]
                           [ 3       ]
                           [ -  1  2 ]
                           [ 2       ]
                           [         ]
(%o2)                      [ 9       ]
                           [ -  2  1 ]
                           [ 4       ]
                           [         ]
                           [ 3  2  2 ]
                           [         ]
                           [ 2  2  1 ]
(%i3) lsquares_estimates (
         M, [z,x,y], (z+D)^2 = A*x+B*y+C, [A,B,C,D]);
                  59        27      10921        107
(%o3)     [[A = - --, B = - --, C = -----, D = - ---]]
                  16        16      1024         32

A problem for which no exact solution is found, so lsquares_estimates resorts to numerical approximation.

(%i1) load ("lsquares")$
(%i2) M : matrix ([1, 1], [2, 7/4], [3, 11/4], [4, 13/4]);
                            [ 1  1  ]
                            [       ]
                            [    7  ]
                            [ 2  -  ]
                            [    4  ]
                            [       ]
(%o2)                       [    11 ]
                            [ 3  -- ]
                            [    4  ]
                            [       ]
                            [    13 ]
                            [ 4  -- ]
                            [    4  ]
(%i3) lsquares_estimates (
  M, [x,y], y=a*x^b+c, [a,b,c], initial=[3,3,3], iprint=[-1,0]);
(%o3) [[a = 1.375751433061394, b = 0.7148891534417651, 
                                       c = - 0.4020908910062951]]

Exponential functions aren’t well-conditioned for least min square fitting. In case that fitting to them fails it might be possible to get rid of the exponential function using an logarithm.

(%i1) load ("lsquares")$
(%i2) yvalues:[1,3,5,60,200,203,80]$
(%i3) time:[1,2,4,5,6,8,10]$
(%i4) f:y=a*exp(b*t);
                                   b t
(%o4)                      y = a %e
(%i5) yvalues_log:log(yvalues)$
(%i6) f_log:log(subst(y=exp(y),f));
                                    b t
(%o6)                   y = log(a %e   )
(%i7) lsquares_estimates(
    transpose(matrix(yvalues_log,time)),
    [y,t],
    f_log,
    [a,b]
 );
*************************************************
  N=    2   NUMBER OF CORRECTIONS=25
       INITIAL VALUES
 F=  6.802906290754687D+00   GNORM=  2.851243373781393D+01
*************************************************

   I  NFN     FUNC                    GNORM                   STEPLENGTH

   1    3     1.141838765593467D+00   1.067358003667488D-01   1.390943719972406D-02  
   2    5     1.141118195694385D+00   1.237977833033414D-01   5.000000000000000D+00  
   3    6     1.136945723147959D+00   3.806696991691383D-01   1.000000000000000D+00  
   4    7     1.133958243220262D+00   3.865103550379243D-01   1.000000000000000D+00  
   5    8     1.131725773805499D+00   2.292258231154026D-02   1.000000000000000D+00  
   6    9     1.131625585698168D+00   2.664440547017370D-03   1.000000000000000D+00  
   7   10     1.131620564856599D+00   2.519366958715444D-04   1.000000000000000D+00  

 THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.
 IFLAG = 0
(%o7)   [[a = 1.155904145765554, b = 0.5772666876959847]]

Function: lsquares_estimates_exact (MSE, a)

Estimate parameters a to minimize the mean square error MSE, by constructing a system of equations and attempting to solve them symbolically via solve. The mean square error is an expression in the parameters a, such as that returned by lsquares_mse.

The return value is a list of lists of equations of the form [a = ..., b = ..., c = ...]. The return value may contain zero, one, or two or more elements. If two or more elements are returned, each represents a distinct, equivalent minimum of the mean square error.

See also lsquares_estimates, lsquares_estimates_approximate, lsquares_mse, lsquares_residuals, and lsquares_residual_mse.

Example:

(%i1) load ("lsquares")$
(%i2) M : matrix (
         [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                           [ 1  1  1 ]
                           [         ]
                           [ 3       ]
                           [ -  1  2 ]
                           [ 2       ]
                           [         ]
(%o2)                      [ 9       ]
                           [ -  2  1 ]
                           [ 4       ]
                           [         ]
                           [ 3  2  2 ]
                           [         ]
                           [ 2  2  1 ]
(%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
         5
        ====
        \                                         2     2
         >    ((- B M    ) - A M     + (M     + D)  - C)
        /            i, 3       i, 2     i, 1
        ====
        i = 1
(%o3)   -------------------------------------------------
                                5
(%i4) lsquares_estimates_exact (mse, [A, B, C, D]);
                  59        27      10921        107
(%o4)     [[A = - --, B = - --, C = -----, D = - ---]]
                  16        16      1024         32

Function: lsquares_estimates_approximate (MSE, a, initial = L, tol = t)

Estimate parameters a to minimize the mean square error MSE, via the numerical minimization function lbfgs. The mean square error is an expression in the parameters a, such as that returned by lsquares_mse.

The solution returned by lsquares_estimates_approximate is a local (perhaps global) minimum of the mean square error. For consistency with lsquares_estimates_exact, the return value is a nested list which contains one element, namely a list of equations of the form [a = ..., b = ..., c = ...].

Additional arguments to lsquares_estimates_approximate are specified as equations and passed on verbatim to the function lbfgs.

MSE must evaluate to a number when the parameters are assigned numeric values. This requires that the data from which MSE was constructed comprise only numeric constants such as %pi and %e and literal numbers (integers, rationals, ordinary floats, and bigfloats). Numerical calculations are carried out with ordinary floating-point arithmetic, so all other kinds of numbers are converted to ordinary floats for calculations.

load(lsquares) loads this function.

See also lsquares_estimates, lsquares_estimates_exact, lsquares_mse,
lsquares_residuals, and lsquares_residual_mse.

Example:

(%i1) load ("lsquares")$
(%i2) M : matrix (
         [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                           [ 1  1  1 ]
                           [         ]
                           [ 3       ]
                           [ -  1  2 ]
                           [ 2       ]
                           [         ]
(%o2)                      [ 9       ]
                           [ -  2  1 ]
                           [ 4       ]
                           [         ]
                           [ 3  2  2 ]
                           [         ]
                           [ 2  2  1 ]
(%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
         5
        ====
        \                                         2     2
         >    ((- B M    ) - A M     + (M     + D)  - C)
        /            i, 3       i, 2     i, 1
        ====
        i = 1
(%o3)   -------------------------------------------------
                                5
(%i4) lsquares_estimates_approximate (
        mse, [A, B, C, D], iprint = [-1, 0]);
(%o4) [[A = - 3.678504947401971, B = - 1.683070351177937, 
                 C = 10.63469950148714, D = - 3.340357993175297]]

Function: lsquares_mse (D, x, e)

Returns the mean square error (MSE), a summation expression, for the equation e in the variables x, with data D.

The MSE is defined as:

                    n
                   ====
               1   \                        2
               -    >    (lhs(e ) - rhs(e ))
               n   /           i         i
                   ====
                   i = 1

where n is the number of data and e[i] is the equation e evaluated with the variables in x assigned values from the i-th datum, D[i].

load(lsquares) loads this function.

Example:

(%i1) load ("lsquares")$
(%i2) M : matrix (
         [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                           [ 1  1  1 ]
                           [         ]
                           [ 3       ]
                           [ -  1  2 ]
                           [ 2       ]
                           [         ]
(%o2)                      [ 9       ]
                           [ -  2  1 ]
                           [ 4       ]
                           [         ]
                           [ 3  2  2 ]
                           [         ]
                           [ 2  2  1 ]
(%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
         5
        ====
        \                                         2     2
         >    ((- B M    ) - A M     + (M     + D)  - C)
        /            i, 3       i, 2     i, 1
        ====
        i = 1
(%o3)   -------------------------------------------------
                                5
(%i4) diff (mse, D);
(%o4) 
      5
     ====
     \                                                     2
   4  >    (M     + D) ((- B M    ) - A M     + (M     + D)  - C)
     /       i, 1             i, 3       i, 2     i, 1
     ====
     i = 1
   --------------------------------------------------------------
                                 5
(%i5) ''mse, nouns;
               2                 2         9 2               2
(%o5) (((D + 3)  - C - 2 B - 2 A)  + ((D + -)  - C - B - 2 A)
                                           4
           2               2         3 2               2
 + ((D + 2)  - C - B - 2 A)  + ((D + -)  - C - 2 B - A)
                                     2
           2             2
 + ((D + 1)  - C - B - A) )/5
(%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
           5
          ====
          \                 2                         2
           >    ((D + M    )  - C - M     B - M     A)
          /            i, 1          i, 3      i, 2
          ====
          i = 1
(%o3)     ---------------------------------------------
                                5
(%i4) diff (mse, D);
         5
        ====
        \                             2
      4  >    (D + M    ) ((D + M    )  - C - M     B - M     A)
        /           i, 1         i, 1          i, 3      i, 2
        ====
        i = 1
(%o4) ----------------------------------------------------------
                                  5
(%i5) ''mse, nouns;
               2                 2         9 2               2
(%o5) (((D + 3)  - C - 2 B - 2 A)  + ((D + -)  - C - B - 2 A)
                                           4
           2               2         3 2               2
 + ((D + 2)  - C - B - 2 A)  + ((D + -)  - C - 2 B - A)
                                     2
           2             2
 + ((D + 1)  - C - B - A) )/5

Function: lsquares_residuals (D, x, e, a)

Returns the residuals for the equation e with specified parameters a and data D.

D is a matrix, x is a list of variables, e is an equation or general expression; if not an equation, e is treated as if it were e = 0. a is a list of equations which specify values for any free parameters in e aside from x.

The residuals are defined as:

                        lhs(e ) - rhs(e )
                             i         i

where e[i] is the equation e evaluated with the variables in x assigned values from the i-th datum, D[i], and assigning any remaining free variables from a.

load(lsquares) loads this function.

Example:

(%i1) load ("lsquares")$
(%i2) M : matrix (
         [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                           [ 1  1  1 ]
                           [         ]
                           [ 3       ]
                           [ -  1  2 ]
                           [ 2       ]
                           [         ]
(%o2)                      [ 9       ]
                           [ -  2  1 ]
                           [ 4       ]
                           [         ]
                           [ 3  2  2 ]
                           [         ]
                           [ 2  2  1 ]
(%i3) a : lsquares_estimates (
          M, [z,x,y], (z+D)^2 = A*x+B*y+C, [A,B,C,D]);
                  59        27      10921        107
(%o3)     [[A = - --, B = - --, C = -----, D = - ---]]
                  16        16      1024         32
(%i4) lsquares_residuals (
          M, [z,x,y], (z+D)^2 = A*x+B*y+C, first(a));
                     13    13    13  13  13
(%o4)               [--, - --, - --, --, --]
                     64    64    32  64  64

Function: lsquares_residual_mse (D, x, e, a)

Returns the residual mean square error (MSE) for the equation e with specified parameters a and data D.

The residual MSE is defined as:

                    n
                   ====
               1   \                        2
               -    >    (lhs(e ) - rhs(e ))
               n   /           i         i
                   ====
                   i = 1

where e[i] is the equation e evaluated with the variables in x assigned values from the i-th datum, D[i], and assigning any remaining free variables from a.

load(lsquares) loads this function.

Example:

(%i1) load ("lsquares")$
(%i2) M : matrix (
         [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                           [ 1  1  1 ]
                           [         ]
                           [ 3       ]
                           [ -  1  2 ]
                           [ 2       ]
                           [         ]
(%o2)                      [ 9       ]
                           [ -  2  1 ]
                           [ 4       ]
                           [         ]
                           [ 3  2  2 ]
                           [         ]
                           [ 2  2  1 ]
(%i3) a : lsquares_estimates (
       M, [z,x,y], (z+D)^2 = A*x+B*y+C, [A,B,C,D]);
                  59        27      10921        107
(%o3)     [[A = - --, B = - --, C = -----, D = - ---]]
                  16        16      1024         32
(%i4) lsquares_residual_mse (
       M, [z,x,y], (z + D)^2 = A*x + B*y + C, first (a));
                              169
(%o4)                         ----
                              2560

Function: plsquares
    plsquares (Mat,VarList,depvars)
    plsquares (Mat,VarList,depvars,maxexpon)
    plsquares (Mat,VarList,depvars,maxexpon,maxdegree)

Multivariable polynomial adjustment of a data table by the "least squares" method. Mat is a matrix containing the data, VarList is a list of variable names (one for each Mat column, but use "-" instead of varnames to ignore Mat columns), depvars is the name of a dependent variable or a list with one or more names of dependent variables (which names should be in VarList), maxexpon is the optional maximum exponent for each independent variable (1 by default), and maxdegree is the optional maximum polynomial degree (maxexpon by default); note that the sum of exponents of each term must be equal or smaller than maxdegree, and if maxdgree = 0 then no limit is applied.

If depvars is the name of a dependent variable (not in a list), plsquares returns the adjusted polynomial. If depvars is a list of one or more dependent variables, plsquares returns a list with the adjusted polynomial(s). The Coefficients of Determination are displayed in order to inform about the goodness of fit, which ranges from 0 (no correlation) to 1 (exact correlation). These values are also stored in the global variable DETCOEF (a list if depvars is a list).

A simple example of multivariable linear adjustment:

(%i1) load("plsquares")$

(%i2) plsquares(matrix([1,2,0],[3,5,4],[4,7,9],[5,8,10]),
                [x,y,z],z);
     Determination Coefficient for z = .9897039897039897
                       11 y - 9 x - 14
(%o2)              z = ---------------
                              3

The same example without degree restrictions:

(%i3) plsquares(matrix([1,2,0],[3,5,4],[4,7,9],[5,8,10]),
                [x,y,z],z,1,0);
     Determination Coefficient for z = 1.0
                    x y + 23 y - 29 x - 19
(%o3)           z = ----------------------
                              6

How many diagonals does a N-sides polygon have? What polynomial degree should be used?

(%i4) plsquares(matrix([3,0],[4,2],[5,5],[6,9],[7,14],[8,20]),
                [N,diagonals],diagonals,5);
     Determination Coefficient for diagonals = 1.0
                                2
                               N  - 3 N
(%o4)              diagonals = --------
                                  2
(%i5) ev(%, N=9);   /* Testing for a 9 sides polygon */
(%o5)                 diagonals = 27

How many ways do we have to put two queens without they are threatened into a n x n chessboard?

(%i6) plsquares(matrix([0,0],[1,0],[2,0],[3,8],[4,44]),
                [n,positions],[positions],4);
     Determination Coefficient for [positions] = [1.0]
                         4       3      2
                      3 n  - 10 n  + 9 n  - 2 n
(%o6)    [positions = -------------------------]
                                  6
(%i7) ev(%[1], n=8); /* Testing for a (8 x 8) chessboard */
(%o7)                positions = 1288

An example with six dependent variables:

(%i8) mtrx:matrix([0,0,0,0,0,1,1,1],[0,1,0,1,1,1,0,0],
                  [1,0,0,1,1,1,0,0],[1,1,1,1,0,0,0,1])$
(%i8) plsquares(mtrx,[a,b,_And,_Or,_Xor,_Nand,_Nor,_Nxor],
                     [_And,_Or,_Xor,_Nand,_Nor,_Nxor],1,0);
      Determination Coefficient for
[_And, _Or, _Xor, _Nand, _Nor, _Nxor] =
[1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
(%o2) [_And = a b, _Or = - a b + b + a,
_Xor = - 2 a b + b + a, _Nand = 1 - a b,
_Nor = a b - b - a + 1, _Nxor = 2 a b - b - a + 1]

To use this function write first load("lsquares").


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71 minpack


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71.1 Introduction to minpack

Minpack is a Common Lisp translation (via f2cl) of the Fortran library MINPACK, as obtained from Netlib.

·

Category: Numerical methods · Category: Optimization · Category: Share packages · Category: Package minpack


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71.2 Functions and Variables for minpack

Function: minpack_lsquares (flist, varlist, guess [, tolerance, jacobian])

Compute the point that minimizes the sum of the squares of the functions in the list flist. The variables are in the list varlist. An initial guess of the optimum point must be provided in guess.

The optional keyword arguments, tolerance and jacobian provide some control over the algorithm. tolerance is the estimated relative error desired in the sum of squares. jacobian can be used to specify the Jacobian. If jacobian is not given or is true (the default), the Jacobian is computed from flist. If jacobian is false, a numerical approximation is used.

minpack_lsquares returns a list. The first item is the estimated solution; the second is the sum of squares, and the third indicates the success of the algorithm. The possible values are

0

improper input parameters.

1

algorithm estimates that the relative error in the sum of squares is at most tolerance.

2

algorithm estimates that the relative error between x and the solution is at most tolerance.

3

conditions for info = 1 and info = 2 both hold.

4

fvec is orthogonal to the columns of the jacobian to machine precision.

5

number of calls to fcn with iflag = 1 has reached 100*(n+1).

6

tol is too small. no further reduction in the sum of squares is possible.

7

tol is too small. no further improvement in the approximate solution x is possible.

/* Problem 6: Powell singular function */
(%i1) powell(x1,x2,x3,x4) := 
         [x1+10*x2, sqrt(5)*(x3-x4), (x2-2*x3)^2, 
              sqrt(10)*(x1-x4)^2]$
(%i2) minpack_lsquares(powell(x1,x2,x3,x4), [x1,x2,x3,x4], 
                       [3,-1,0,1]);
(%o2) [[1.652117596168394e-17, - 1.652117596168393e-18, 
        2.643388153869468e-18, 2.643388153869468e-18], 
       6.109327859207777e-34, 4] 
/* Same problem but use numerical approximation to Jacobian */
(%i3) minpack_lsquares(powell(x1,x2,x3,x4), [x1,x2,x3,x4], 
                       [3,-1,0,1], jacobian = false);
(%o3) [[5.060282149485331e-11, - 5.060282149491206e-12, 
        2.179447843547218e-11, 2.179447843547218e-11], 
       3.534491794847031e-21, 5]
Function: minpack_solve (flist, varlist, guess [, tolerance, jacobian])

Solve a system of n equations in n unknowns. The n equations are given in the list flist, and the unknowns are in varlist. An initial guess of the solution must be provided in guess.

The optional keyword arguments, tolerance and jacobian provide some control over the algorithm. tolerance is the estimated relative error desired in the sum of squares. jacobian can be used to specify the Jacobian. If jacobian is not given or is true (the default), the Jacobian is computed from flist. If jacobian is false, a numerical approximation is used.

minpack_solve returns a list. The first item is the estimated solution; the second is the sum of squares, and the third indicates the success of the algorithm. The possible values are

0

improper input parameters.

1

algorithm estimates that the relative error in the solution is at most tolerance.

2

number of calls to fcn with iflag = 1 has reached 100*(n+1).

3

tol is too small. no further reduction in the sum of squares is possible.

4

Iteration is not making good progress.


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72 makeOrders


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72.1 Functions and Variables for makeOrders

Function: makeOrders (indvarlist,orderlist)

Returns a list of all powers for a polynomial up to and including the arguments.

(%i1) load("makeOrders")$

(%i2) makeOrders([a,b],[2,3]);
(%o2) [[0, 0], [0, 1], [0, 2], [0, 3], [1, 0], [1, 1],
            [1, 2], [1, 3], [2, 0], [2, 1], [2, 2], [2, 3]]
(%i3) expand((1+a+a^2)*(1+b+b^2+b^3));
       2  3      3    3    2  2      2    2    2
(%o3) a  b  + a b  + b  + a  b  + a b  + b  + a  b + a b
                                                  2
                                           + b + a  + a + 1

where [0, 1] is associated with the term b and [2, 3] with a^2 b^3.

To use this function write first load("makeOrders").


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73 mnewton


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73.1 Introduction to mnewton

mnewton is an implementation of Newton’s method for solving nonlinear equations in one or more variables.

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73.2 Functions and Variables for mnewton

Option variable: newtonepsilon

Default value: 10.0^(-fpprec/2)

Precision to determine when the mnewton function has converged towards the solution. If newtonepsilon is a bigfloat, then mnewton computations are done with bigfloats. See also mnewton.

Option variable: newtonmaxiter

Default value: 50

Maximum number of iterations to stop the mnewton function if it does not converge or if it converges too slowly.

See also mnewton.

Function: mnewton (FuncList,VarList,GuessList)

Multiple nonlinear functions solution using the Newton method. FuncList is the list of functions to solve, VarList is the list of variable names, and GuessList is the list of initial approximations.

The solution is returned in the same format that solve() returns. If the solution is not found, [] is returned.

This function is controlled by global variables newtonepsilon and newtonmaxiter.

(%i1) load("mnewton")$

(%i2) mnewton([x1+3*log(x1)-x2^2, 2*x1^2-x1*x2-5*x1+1],
              [x1, x2], [5, 5]);
(%o2) [[x1 = 3.756834008012769, x2 = 2.779849592817897]]
(%i3) mnewton([2*a^a-5],[a],[1]);
(%o3)             [[a = 1.70927556786144]]
(%i4) mnewton([2*3^u-v/u-5, u+2^v-4], [u, v], [2, 2]);
(%o4) [[u = 1.066618389595407, v = 1.552564766841786]]

The variable newtonepsilon controls the precision of the approximations. It also controls if computations are performed with floats or bigfloats.

(%i1) load(mnewton)$

(%i2) (fpprec : 25, newtonepsilon : bfloat(10^(-fpprec+5)))$

(%i3) mnewton([2*3^u-v/u-5, u+2^v-4], [u, v], [2, 2]);
(%o3) [[u = 1.066618389595406772591173b0, 
                               v = 1.552564766841786450100418b0]]

To use this function write first load("mnewton"). See also newtonepsilon and newtonmaxiter.


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74 numericalio


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74.1 Introduction to numericalio

numericalio is a collection of functions to read and write files and streams. Functions for plain-text input and output can read and write numbers (integer, float, or bigfloat), symbols, and strings. Functions for binary input and output can read and write only floating-point numbers.

If there already exists a list, matrix, or array object to store input data, numericalio input functions can write data into that object. Otherwise, numericalio can guess, to some degree, the structure of an object to store the data, and return that object.

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Category: File input · Category: File output · Category: Share packages · Category: Package numericalio

74.1.1 Plain-text input and output

In plain-text input and output, it is assumed that each item to read or write is an atom: an integer, float, bigfloat, string, or symbol, and not a rational or complex number or any other kind of nonatomic expression. The numericalio functions may attempt to do something sensible faced with nonatomic expressions, but the results are not specified here and subject to change.

Atoms in both input and output files have the same format as in Maxima batch files or the interactive console. In particular, strings are enclosed in double quotes, backslash \ prevents any special interpretation of the next character, and the question mark ? is recognized at the beginning of a symbol to mean a Lisp symbol (as opposed to a Maxima symbol). No continuation character (to join broken lines) is recognized.

74.1.2 Separator flag values for input

The functions for plain-text input and output take an optional argument, separator_flag, that tells what character separates data.

For plain-text input, these values of separator_flag are recognized: comma for comma separated values, pipe for values separated by the vertical bar character |, semicolon for values separated by semicolon ;, and space for values separated by space or tab characters. If the file name ends in .csv and separator_flag is not specified, comma is assumed. If the file name ends in something other than .csv and separator_flag is not specified, space is assumed.

In plain-text input, multiple successive space and tab characters count as a single separator. However, multiple comma, pipe, or semicolon characters are significant. Successive comma, pipe, or semicolon characters (with or without intervening spaces or tabs) are considered to have false between the separators. For example, 1234,,Foo is treated the same as 1234,false,Foo.

74.1.3 Separator flag values for output

For plain-text output, tab, for values separated by the tab character, is recognized as a value of separator_flag, as well as comma, pipe, semicolon, and space.

In plain-text output, false atoms are written as such; a list [1234, false, Foo] is written 1234,false,Foo, and there is no attempt to collapse the output to 1234,,Foo.

74.1.4 Binary floating-point input and output

numericalio functions can read and write 8-byte IEEE 754 floating-point numbers. These numbers can be stored either least significant byte first or most significant byte first, according to the global flag set by assume_external_byte_order. If not specified, numericalio assumes the external byte order is most-significant byte first.

Other kinds of numbers are coerced to 8-byte floats; numericalio cannot read or write binary non-numeric data.

Some Lisp implementations do not recognize IEEE 754 special values (positive and negative infinity, not-a-number values, denormalized values). The effect of reading such values with numericalio is undefined.

numericalio includes functions to open a stream for reading or writing a stream of bytes.


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74.2 Functions and Variables for plain-text input and output

Function: read_matrix
    read_matrix (S)
    read_matrix (S, M)
    read_matrix (S, separator_flag)
    read_matrix (S, M, separator_flag)

read_matrix(S) reads the source S and returns its entire content as a matrix. The size of the matrix is inferred from the input data; each line of the file becomes one row of the matrix. If some lines have different lengths, read_matrix complains.

read_matrix(S, M) read the source S into the matrix M, until M is full or the source is exhausted. Input data are read into the matrix in row-major order; the input need not have the same number of rows and columns as M.

The source S may be a file name or a stream which for example allows skipping the very first line of a file (that may be useful, if you read CSV data, where the first line often contains the description of the columns):

s : openr("data.txt");
readline(s);  /* skip the first line */
M : read_matrix(s, 'comma);  /* read the following (comma-separated) lines into matrix M */
close(s);

The recognized values of separator_flag are comma, pipe, semicolon, and space. If separator_flag is not specified, the file is assumed space-delimited.

Function: read_array
    read_array (S, A)
    read_array (S, A, separator_flag)

Reads the source S into the array A, until A is full or the source is exhausted. Input data are read into the array in row-major order; the input need not conform to the dimensions of A.

The source S may be a file name or a stream.

The recognized values of separator_flag are comma, pipe, semicolon, and space. If separator_flag is not specified, the file is assumed space-delimited.

Function: read_hash_table
    read_hash_table (S, A)
    read_hash_table (S, A, separator_flag)

Reads the source S and returns its entire content as a hashed array. The source S may be a file name or a stream.

read_hash_table treats the first item on each line as a hash key, and associates the remainder of the line (as a list) with the key. For example, the line 567 12 17 32 55 is equivalent to A[567]: [12, 17, 32, 55]$. Lines need not have the same numbers of elements.

The recognized values of separator_flag are comma, pipe, semicolon, and space. If separator_flag is not specified, the file is assumed space-delimited.

Function: read_nested_list
    read_nested_list (S)
    read_nested_list (S, separator_flag)

Reads the source S and returns its entire content as a nested list. The source S may be a file name or a stream.

read_nested_list returns a list which has a sublist for each line of input. Lines need not have the same numbers of elements. Empty lines are not ignored: an empty line yields an empty sublist.

The recognized values of separator_flag are comma, pipe, semicolon, and space. If separator_flag is not specified, the file is assumed space-delimited.

Function: read_list
    read_list (S)
    read_list (S, L)
    read_list (S, separator_flag)
    read_list (S, L, separator_flag)

read_list(S) reads the source S and returns its entire content as a flat list.

read_list(S, L) reads the source S into the list L, until L is full or the source is exhausted.

The source S may be a file name or a stream.

The recognized values of separator_flag are comma, pipe, semicolon, and space. If separator_flag is not specified, the file is assumed space-delimited.

Function: write_data
    write_data (X, D)
    write_data (X, D, separator_flag)

Writes the object X to the destination D.

write_data writes a matrix in row-major order, with one line per row.

write_data writes an array created by array or make_array in row-major order, with a new line at the end of every slab. Higher-dimensional slabs are separated by additional new lines.

write_data writes a hashed array with each key followed by its associated list on one line.

write_data writes a nested list with each sublist on one line.

write_data writes a flat list all on one line.

The destination D may be a file name or a stream. When the destination is a file name, the global variable file_output_append governs whether the output file is appended or truncated. When the destination is a stream, no special action is taken by write_data after all the data are written; in particular, the stream remains open.

The recognized values of separator_flag are comma, pipe, semicolon, space, and tab. If separator_flag is not specified, the file is assumed space-delimited.


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74.3 Functions and Variables for binary input and output

Function: assume_external_byte_order (byte_order_flag)

Tells numericalio the byte order for reading and writing binary data. Two values of byte_order_flag are recognized: lsb which indicates least-significant byte first, also called little-endian byte order; and msb which indicates most-significant byte first, also called big-endian byte order.

If not specified, numericalio assumes the external byte order is most-significant byte first.

Function: openr_binary (file_name)

Returns an input stream of 8-bit unsigned bytes to read the file named by file_name.

Function: openw_binary (file_name)

Returns an output stream of 8-bit unsigned bytes to write the file named by file_name.

Function: opena_binary (file_name)

Returns an output stream of 8-bit unsigned bytes to append the file named by file_name.

Function: read_binary_matrix (S, M)

Reads binary 8-byte floating point numbers from the source S into the matrix M until M is full, or the source is exhausted. Elements of M are read in row-major order.

The source S may be a file name or a stream.

The byte order in elements of the source is specified by assume_external_byte_order.

Function: read_binary_array (S, A)

Reads binary 8-byte floating point numbers from the source S into the array A until A is full, or the source is exhausted. A must be an array created by array or make_array. Elements of A are read in row-major order.

The source S may be a file name or a stream.

The byte order in elements of the source is specified by assume_external_byte_order.

Function: read_binary_list
    read_binary_list (S)
    read_binary_list (S, L)

read_binary_list(S) reads the entire content of the source S as a sequence of binary 8-byte floating point numbers, and returns it as a list. The source S may be a file name or a stream.

read_binary_list(S, L) reads 8-byte binary floating point numbers from the source S until the list L is full, or the source is exhausted.

The byte order in elements of the source is specified by assume_external_byte_order.

Function: write_binary_data (X, D)

Writes the object X, comprising binary 8-byte IEEE 754 floating-point numbers, to the destination D. Other kinds of numbers are coerced to 8-byte floats. write_binary_data cannot write non-numeric data.

The object X may be a list, a nested list, a matrix, or an array created by array or make_array; X cannot be a hashed array or any other type of object. write_binary_data writes nested lists, matrices, and arrays in row-major order.

The destination D may be a file name or a stream. When the destination is a file name, the global variable file_output_append governs whether the output file is appended or truncated. When the destination is a stream, no special action is taken by write_binary_data after all the data are written; in particular, the stream remains open.

The byte order in elements of the destination is specified by assume_external_byte_order.


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75 odepack


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75.1 Introduction to ODEPACK

ODEPACK is a collection of Fortran solvers for the initial value problem for ordinary differential equation systems. It consists of nine solvers, namely a basic solver called LSODE and eight variants of it – LSODES, LSODA, LSODAR, LSODPK, LSODKR, LSODI, LSOIBT, and LSODIS. The collection is suitable for both stiff and nonstiff systems. It includes solvers for systems given in explicit form, dy/dt = f(t,y), and also solvers for systems given in linearly implicit form, A(t,y) dy/dt = g(t,y). Two of the solvers use general sparse matrix solvers for the linear systems that arise. Two others use iterative (preconditioned Krylov) methods instead of direct methods for these linear systems. The most recent addition is LSODIS, which solves implicit problems with general sparse treatment of all matrices involved.

9

References: [1] Fortran Code is from http://netlib.org/odepack/

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75.1.1 Getting Started

Of the eight variants of the solver, maxima currently only has an inteface the dlsode.

Let’s say we have this sytem of equations to solve:

  f1 = -.04d0*y1 + 1d4*y2*y3
  f3 = 3d7*y2*y2
  dy1/dt = f1
  dy2/dt = -f1 - f3
  dy3/dt = f3

The independent variable is t; the dependent variables are y1, y2, and y3,

To start the solution, set up the differential equations to solved:

load("dlsode");
f1: -.04d0*y1 + 1d4*y2*y3$
f3: 3d7*y2*y2$
f2: -f1 - f3$
fex: [f1, f2, f3];

Initialize the solver, where we have selected method 21

(%i6) state : dlsode_init(fex, ['t,y1,y2,y3], 21);
(%o6) [[f, #<Function "LAMBDA ($T $Y1 $Y2 $Y3)" {49DAC061}>], 
[vars, [t, y1, y2, y3]], [mf, 21], [neq, 3], [lrw, 58], [liw, 23], [rwork, {Li\
sp Array: #(0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
               0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
               0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
               0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0)}], 
[iwork, {Lisp Array: #(0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)}], 
[fjac, #<Function "LAMBDA ($T $Y1 $Y2 $Y3)" {49D52AC9}>]]

The arrays rwork and iwork carry state between calls to dlsode_step, so they should not be modified by the user. In fact, this state should not be modified by the user at all.

Now that the algorithm has been initialized we can compute solutions to the differential equation, using the state returned above.

For this example, we want to compute the solution at times 0.4*10^k for k from 0 to 11, with the initial values of 1, 0, 0 for the dependent variables and with a relative tolerance of 1d-4 and absolute tolerances of 1e-6, 1e-10, and 1d-6 for the dependent variables.

Then

y: [1d0, 0d0, 0d0];
t: 0d0;
rtol : 1d-4;
atol: [1d-6, 1d-10, 1d-6];
istate: 1;
t:0d0;
tout:.4d0;

for k : 1 thru 12 do
  block([],
    result: dlsode_step(y, t, tout, rtol, atol, istate, state),
    printf(true, "At t = ~12,4,2e   y = ~{~14,6,2e~}~%", result[1], result[2]),
    istate : result[3],
    tout : tout * 10);

This produces the output:

At t =   4.0000e-01   y =   9.851726e-01  3.386406e-05  1.479357e-02
At t =   4.0000e+00   y =   9.055142e-01  2.240418e-05  9.446344e-02
At t =   4.0000e+01   y =   7.158050e-01  9.184616e-06  2.841858e-01
At t =   4.0000e+02   y =   4.504846e-01  3.222434e-06  5.495122e-01
At t =   4.0000e+03   y =   1.831701e-01  8.940379e-07  8.168290e-01
At t =   4.0000e+04   y =   3.897016e-02  1.621193e-07  9.610297e-01
At t =   4.0000e+05   y =   4.935213e-03  1.983756e-08  9.950648e-01
At t =   4.0000e+06   y =   5.159269e-04  2.064759e-09  9.994841e-01
At t =   4.0000e+07   y =   5.306413e-05  2.122677e-10  9.999469e-01
At t =   4.0000e+08   y =   5.494530e-06  2.197824e-11  9.999945e-01
At t =   4.0000e+09   y =   5.129458e-07  2.051784e-12  9.999995e-01
At t =   4.0000e+10   y =  -7.170563e-08 -2.868225e-13  1.000000e+00

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75.2 Functions and Variables for odepack

Function: dlsode_init (fex, vars, method)

This must be called before running the solver. This function returns a state object for use in the solver. The user must not modify the state.

The ODE to be solved is given in fex, which is a list of the equations. vars is a list of independent variable and the dependent variables. The list of dependent variables must be in the same order as the equations if fex. Finally, method indicates the method to be used by the solver:

10

Nonstiff (Adams) method, no Jacobian used.

21

Stiff (BDF) method, user-supplied full Jacobian.

22

Stiff method, internally generated full Jacobian.

The returned state object is a list of lists. The sublist is a list of two elements:

f

The compiled function for the ODE.

vars

The list independent and dependent variables (vars).

mf

The method to be used (method).

neq

The number of equations.

lrw

Length of the work vector for real values.

liw

Length of the work vector for integer values.

rwork

Lisp array holding the real-valued work vector.

iwork

Lisp array holding the integer-valued work vector.

fjac

Compiled analytical Jacobian of the eqations

Function: dlsode_step (inity, t, tout, rtol, atol, istate, state)

Performs one step of the solver, returning the values of the independent and dependent variables, a succes or error code.

inity

For the first call (when istate = 1), the initial values

t

Current value of the independent value

tout

Next point where output is desired which must not be equal to t.

rtol

relative tolerance parameter

atol

Absolute tolerance parameter, scalar of vector. If scalar, it applies to all dependent variables. Otherwise it must be the tolerance for each dependent variable.

Use rtol = 0 for pure absolute error and use atol = 0 for pure relative error.

istate

1 for the first call to dlsode, 2 for subsequent calls.

state

state returned by dlsode_init.

The output is a list of the following items:

t

independent variable value

y

list of values of the dependent variables at time t.

istate

Integration status:

1

no work because tout = tt

2

successful result

-1

Excess work done on this call

-2

Excess accuracy requested

-3

Illegal input detected

-4

Repeated error test failures

-5

Repeated convergence failures (perhaps bad Jacobian or wrong choice of mf or tolerances)

-6

Error weight becase zero during problem (solution component i vanishded and atol(i) = 0.

info

association list of various bits of information:

n_steps

total steps taken thus far

n_f_eval

total number of function evals

n_j_eval

total number of Jacobian evals

method_order

method order

len_rwork

Actual length used for real work array

len_iwork

Actual length used for integer work array


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76 operatingsystem


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76.1 Introduction to operatingsystem

Package operatingsystem contains functions for operatingsystem-tasks, like file system operations.


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76.2 Directory operations

Function: chdir (dir)

Change to directory dir

Function: mkdir (dir)

Create directory dir

Function: rmdir (dir)

remove directory dir

Function: getcurrentdirectory ()

returns the current working directory.

See also directory.

Examples:

(%i1) load("operatingsystem")$
(%i2) mkdir("testdirectory")$
(%i3) chdir("testdirectory")$
(%i4) chdir("..")$
(%i5) rmdir("testdirectory")$

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76.3 File operations

Function: copy_file (file1, file2)

copies file file1 to file2

Function: rename_file (file1, file2)

renames file file1 to file2

Function: delete_file (file1)

deletes file file1


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76.4 Environment operations

Function: getenv (env)

Get the value of the environmentvariable env

Example:

(%i1) load("operatingsystem")$
(%i2) getenv("PATH");
(%o2) /usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin

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77 opsubst


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77.1 Functions and Variables for opsubst

Function: opsubst
    opsubst (f,g,e)
    opsubst (g=f,e)
    opsubst ([g1=f1,g2=f2,..., gn=fn],e)

The function opsubst is similar to the function subst, except that opsubst only makes substitutions for the operators in an expression. In general, When f is an operator in the expression e, substitute g for f in the expression e.

To determine the operator, opsubst sets inflag to true. This means opsubst substitutes for the internal, not the displayed, operator in the expression.

Examples:

(%i1) load ("opsubst")$

(%i2) opsubst(f,g,g(g(x)));
(%o2)                     f(f(x))
(%i3) opsubst(f,g,g(g));
(%o3)                       f(g)
(%i4) opsubst(f,g[x],g[x](z));
(%o4)                       f(z)
(%i5) opsubst(g[x],f, f(z));
(%o5)                      g (z)
                            x
(%i6) opsubst(tan, sin, sin(sin));
(%o6)                     tan(sin)
(%i7) opsubst([f=g,g=h],f(x));
(%o7)                       h(x)

Internally, Maxima does not use the unary negation, division, or the subtraction operators; thus:

(%i8) opsubst("+","-",a-b);
(%o8)                     a - b
(%i9) opsubst("f","-",-a);
(%o9)                      - a
(%i10) opsubst("^^","/",a/b);
                             a
(%o10)                       -
                             b

The internal representation of -a*b is *(-1,a,b); thus

(%i11) opsubst("[","*", -a*b);
(%o11)                  [- 1, a, b]

When either operator isn’t a Maxima symbol, generally some other function will signal an error:

(%i12) opsubst(a+b,f, f(x));

Improper name or value in functional position:
b + a
 -- an error.  Quitting.  To debug this try debugmode(true);

However, subscripted operators are allowed:

(%i13) opsubst(g[5],f, f(x));
(%o13)                     g (x)
                            5

To use this function write first load("opsubst").


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78 orthopoly


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78.1 Introduction to orthogonal polynomials

orthopoly is a package for symbolic and numerical evaluation of several kinds of orthogonal polynomials, including Chebyshev, Laguerre, Hermite, Jacobi, Legendre, and ultraspherical (Gegenbauer) polynomials. Additionally, orthopoly includes support for the spherical Bessel, spherical Hankel, and spherical harmonic functions.

For the most part, orthopoly follows the conventions of Abramowitz and Stegun Handbook of Mathematical Functions, Chapter 22 (10th printing, December 1972); additionally, we use Gradshteyn and Ryzhik, Table of Integrals, Series, and Products (1980 corrected and enlarged edition), and Eugen Merzbacher Quantum Mechanics (2nd edition, 1970).

Barton Willis of the University of Nebraska at Kearney (UNK) wrote the orthopoly package and its documentation. The package is released under the GNU General Public License (GPL).

·

Category: Orthogonal polynomials · Category: Share packages · Category: Package orthopoly

78.1.1 Getting Started with orthopoly

load ("orthopoly") loads the orthopoly package.

To find the third-order Legendre polynomial,

(%i1) legendre_p (3, x);
                      3             2
             5 (1 - x)    15 (1 - x)
(%o1)      - ---------- + ----------- - 6 (1 - x) + 1
                 2             2

To express this as a sum of powers of x, apply ratsimp or rat to the result.

(%i2) [ratsimp (%), rat (%)];
                        3           3
                     5 x  - 3 x  5 x  - 3 x
(%o2)/R/            [----------, ----------]
                         2           2

Alternatively, make the second argument to legendre_p (its “main” variable) a canonical rational expression (CRE).

(%i1) legendre_p (3, rat (x));
                              3
                           5 x  - 3 x
(%o1)/R/                   ----------
                               2

For floating point evaluation, orthopoly uses a running error analysis to estimate an upper bound for the error. For example,

(%i1) jacobi_p (150, 2, 3, 0.2);
(%o1) interval(- 0.062017037936715, 1.533267919277521E-11)

Intervals have the form interval (c, r), where c is the center and r is the radius of the interval. Since Maxima does not support arithmetic on intervals, in some situations, such as graphics, you want to suppress the error and output only the center of the interval. To do this, set the option variable orthopoly_returns_intervals to false.

(%i1) orthopoly_returns_intervals : false;
(%o1)                         false
(%i2) jacobi_p (150, 2, 3, 0.2);
(%o2)                  - 0.062017037936715

Refer to the section see Floating point Evaluation for more information.

Most functions in orthopoly have a gradef property; thus

(%i1) diff (hermite (n, x), x);
(%o1)                     2 n H     (x)
                               n - 1
(%i2) diff (gen_laguerre (n, a, x), x);
              (a)               (a)
           n L   (x) - (n + a) L     (x) unit_step(n)
              n                 n - 1
(%o2)      ------------------------------------------
                               x

The unit step function in the second example prevents an error that would otherwise arise by evaluating with n equal to 0.

(%i3) ev (%, n = 0);
(%o3)                           0

The gradef property only applies to the “main” variable; derivatives with respect other arguments usually result in an error message; for example

(%i1) diff (hermite (n, x), x);
(%o1)                     2 n H     (x)
                               n - 1
(%i2) diff (hermite (n, x), n);

Maxima doesn't know the derivative of hermite with respect the first
argument
 -- an error.  Quitting.  To debug this try debugmode(true);

Generally, functions in orthopoly map over lists and matrices. For the mapping to fully evaluate, the option variables doallmxops and listarith must both be true (the defaults). To illustrate the mapping over matrices, consider

(%i1) hermite (2, x);
                                     2
(%o1)                    - 2 (1 - 2 x )
(%i2) m : matrix ([0, x], [y, 0]);
                            [ 0  x ]
(%o2)                       [      ]
                            [ y  0 ]
(%i3) hermite (2, m);
               [                             2  ]
               [      - 2        - 2 (1 - 2 x ) ]
(%o3)          [                                ]
               [             2                  ]
               [ - 2 (1 - 2 y )       - 2       ]

In the second example, the i, j element of the value is hermite (2, m[i,j]); this is not the same as computing -2 + 4 m . m, as seen in the next example.

(%i4) -2 * matrix ([1, 0], [0, 1]) + 4 * m . m;
                    [ 4 x y - 2      0     ]
(%o4)               [                      ]
                    [     0      4 x y - 2 ]

If you evaluate a function at a point outside its domain, generally orthopoly returns the function unevaluated. For example,

(%i1) legendre_p (2/3, x);
(%o1)                        P   (x)
                              2/3

orthopoly supports translation into TeX; it also does two-dimensional output on a terminal.

(%i1) spherical_harmonic (l, m, theta, phi);
                          m
(%o1)                    Y (theta, phi)
                          l
(%i2) tex (%);
$$Y_{l}^{m}\left(\vartheta,\varphi\right)$$
(%o2)                         false
(%i3) jacobi_p (n, a, a - b, x/2);
                          (a, a - b) x
(%o3)                    P          (-)
                          n          2
(%i4) tex (%);
$$P_{n}^{\left(a,a-b\right)}\left({{x}\over{2}}\right)$$
(%o4)                         false

78.1.2 Limitations

When an expression involves several orthogonal polynomials with symbolic orders, it’s possible that the expression actually vanishes, yet Maxima is unable to simplify it to zero. If you divide by such a quantity, you’ll be in trouble. For example, the following expression vanishes for integers n greater than 1, yet Maxima is unable to simplify it to zero.

(%i1) (2*n - 1) * legendre_p (n - 1, x) * x - n * legendre_p (n, x)
      + (1 - n) * legendre_p (n - 2, x);
(%o1)  (2 n - 1) P     (x) x - n P (x) + (1 - n) P     (x)
                  n - 1           n               n - 2

For a specific n, we can reduce the expression to zero.

(%i2) ev (% ,n = 10, ratsimp);
(%o2)                           0

Generally, the polynomial form of an orthogonal polynomial is ill-suited for floating point evaluation. Here’s an example.

(%i1) p : jacobi_p (100, 2, 3, x)$

(%i2) subst (0.2, x, p);
(%o2)                3.4442767023833592E+35
(%i3) jacobi_p (100, 2, 3, 0.2);
(%o3)  interval(0.18413609135169, 6.8990300925815987E-12)
(%i4) float(jacobi_p (100, 2, 3, 2/10));
(%o4)                   0.18413609135169

The true value is about 0.184; this calculation suffers from extreme subtractive cancellation error. Expanding the polynomial and then evaluating, gives a better result.

(%i5) p : expand(p)$
(%i6) subst (0.2, x, p);
(%o6) 0.18413609766122982

This isn’t a general rule; expanding the polynomial does not always result in an expression that is better suited for numerical evaluation. By far, the best way to do numerical evaluation is to make one or more of the function arguments floating point numbers. By doing that, specialized floating point algorithms are used for evaluation.

Maxima’s float function is somewhat indiscriminate; if you apply float to an expression involving an orthogonal polynomial with a symbolic degree or order parameter, these parameters may be converted into floats; after that, the expression will not evaluate fully. Consider

(%i1) assoc_legendre_p (n, 1, x);
                               1
(%o1)                         P (x)
                               n
(%i2) float (%);
                              1.0
(%o2)                        P   (x)
                              n
(%i3) ev (%, n=2, x=0.9);
                             1.0
(%o3)                       P   (0.9)
                             2

The expression in (%o3) will not evaluate to a float; orthopoly doesn’t recognize floating point values where it requires an integer. Similarly, numerical evaluation of the pochhammer function for orders that exceed pochhammer_max_index can be troublesome; consider

(%i1) x :  pochhammer (1, 10), pochhammer_max_index : 5;
(%o1)                         (1)
                                 10

Applying float doesn’t evaluate x to a float

(%i2) float (x);
(%o2)                       (1.0)
                                 10.0

To evaluate x to a float, you’ll need to bind pochhammer_max_index to 11 or greater and apply float to x.

(%i3) float (x), pochhammer_max_index : 11;
(%o3)                       3628800.0

The default value of pochhammer_max_index is 100; change its value after loading orthopoly.

Finally, be aware that reference books vary on the definitions of the orthogonal polynomials; we’ve generally used the conventions of Abramowitz and Stegun.

Before you suspect a bug in orthopoly, check some special cases to determine if your definitions match those used by orthopoly. Definitions often differ by a normalization; occasionally, authors use “shifted” versions of the functions that makes the family orthogonal on an interval other than (-1, 1). To define, for example, a Legendre polynomial that is orthogonal on (0, 1), define

(%i1) shifted_legendre_p (n, x) := legendre_p (n, 2*x - 1)$

(%i2) shifted_legendre_p (2, rat (x));
                            2
(%o2)/R/                 6 x  - 6 x + 1
(%i3) legendre_p (2, rat (x));
                               2
                            3 x  - 1
(%o3)/R/                    --------
                               2

78.1.3 Floating point Evaluation

Most functions in orthopoly use a running error analysis to estimate the error in floating point evaluation; the exceptions are the spherical Bessel functions and the associated Legendre polynomials of the second kind. For numerical evaluation, the spherical Bessel functions call SLATEC functions. No specialized method is used for numerical evaluation of the associated Legendre polynomials of the second kind.

The running error analysis ignores errors that are second or higher order in the machine epsilon (also known as unit roundoff). It also ignores a few other errors. It’s possible (although unlikely) that the actual error exceeds the estimate.

Intervals have the form interval (c, r), where c is the center of the interval and r is its radius. The center of an interval can be a complex number, and the radius is always a positive real number.

Here is an example.

(%i1) fpprec : 50$

(%i2) y0 : jacobi_p (100, 2, 3, 0.2);
(%o2) interval(0.1841360913516871, 6.8990300925815987E-12)
(%i3) y1 : bfloat (jacobi_p (100, 2, 3, 1/5));
(%o3) 1.8413609135168563091370224958913493690868904463668b-1

Let’s test that the actual error is smaller than the error estimate

(%i4) is (abs (part (y0, 1) - y1) < part (y0, 2));
(%o4)                         true

Indeed, for this example the error estimate is an upper bound for the true error.

Maxima does not support arithmetic on intervals.

(%i1) legendre_p (7, 0.1) + legendre_p (8, 0.1);
(%o1) interval(0.18032072148437508, 3.1477135311021797E-15)
        + interval(- 0.19949294375000004, 3.3769353084291579E-15)

A user could define arithmetic operators that do interval math. To define interval addition, we can define

(%i1) infix ("@+")$

(%i2) "@+"(x,y) := interval (part (x, 1) + part (y, 1), part (x, 2)
      + part (y, 2))$

(%i3) legendre_p (7, 0.1) @+ legendre_p (8, 0.1);
(%o3) interval(- 0.019172222265624955, 6.5246488395313372E-15)

The special floating point routines get called when the arguments are complex. For example,

(%i1) legendre_p (10, 2 + 3.0*%i);
(%o1) interval(- 3.876378825E+7 %i - 6.0787748E+7, 
                                           1.2089173052721777E-6)

Let’s compare this to the true value.

(%i1) float (expand (legendre_p (10, 2 + 3*%i)));
(%o1)          - 3.876378825E+7 %i - 6.0787748E+7

Additionally, when the arguments are big floats, the special floating point routines get called; however, the big floats are converted into double floats and the final result is a double.

(%i1) ultraspherical (150, 0.5b0, 0.9b0);
(%o1) interval(- 0.043009481257265, 3.3750051301228864E-14)

78.1.4 Graphics and orthopoly

To plot expressions that involve the orthogonal polynomials, you must do two things:

  1. Set the option variable orthopoly_returns_intervals to false,
  2. Quote any calls to orthopoly functions.

If function calls aren’t quoted, Maxima evaluates them to polynomials before plotting; consequently, the specialized floating point code doesn’t get called. Here is an example of how to plot an expression that involves a Legendre polynomial.

(%i1) plot2d ('(legendre_p (5, x)), [x, 0, 1]),
                        orthopoly_returns_intervals : false;
(%o1)
figures/orthopoly1

The entire expression legendre_p (5, x) is quoted; this is different than just quoting the function name using 'legendre_p (5, x).

·

Category: Plotting

78.1.5 Miscellaneous Functions

The orthopoly package defines the Pochhammer symbol and a unit step function. orthopoly uses the Kronecker delta function and the unit step function in gradef statements.

To convert Pochhammer symbols into quotients of gamma functions, use makegamma.

(%i1) makegamma (pochhammer (x, n));
                          gamma(x + n)
(%o1)                     ------------
                            gamma(x)
(%i2) makegamma (pochhammer (1/2, 1/2));
                                1
(%o2)                       ---------
                            sqrt(%pi)

Derivatives of the Pochhammer symbol are given in terms of the psi function.

(%i1) diff (pochhammer (x, n), x);
(%o1)             (x)  (psi (x + n) - psi (x))
                     n     0             0
(%i2) diff (pochhammer (x, n), n);
(%o2)                   (x)  psi (x + n)
                           n    0

You need to be careful with the expression in (%o1); the difference of the psi functions has polynomials when x = -1, -2, .., -n. These polynomials cancel with factors in pochhammer (x, n) making the derivative a degree n - 1 polynomial when n is a positive integer.

The Pochhammer symbol is defined for negative orders through its representation as a quotient of gamma functions. Consider

(%i1) q : makegamma (pochhammer (x, n));
                          gamma(x + n)
(%o1)                     ------------
                            gamma(x)
(%i2) sublis ([x=11/3, n= -6], q);
                               729
(%o2)                        - ----
                               2240

Alternatively, we can get this result directly.

(%i1) pochhammer (11/3, -6);
                               729
(%o1)                        - ----
                               2240

The unit step function is left-continuous; thus

(%i1) [unit_step (-1/10), unit_step (0), unit_step (1/10)];
(%o1)                       [0, 0, 1]

If you need a unit step function that is neither left or right continuous at zero, define your own using signum; for example,

(%i1) xunit_step (x) := (1 + signum (x))/2$

(%i2) [xunit_step (-1/10), xunit_step (0), xunit_step (1/10)];
                                1
(%o2)                       [0, -, 1]
                                2

Do not redefine unit_step itself; some code in orthopoly requires that the unit step function be left-continuous.

78.1.6 Algorithms

Generally, orthopoly does symbolic evaluation by using a hypergeometic representation of the orthogonal polynomials. The hypergeometic functions are evaluated using the (undocumented) functions hypergeo11 and hypergeo21. The exceptions are the half-integer Bessel functions and the associated Legendre function of the second kind. The half-integer Bessel functions are evaluated using an explicit representation, and the associated Legendre function of the second kind is evaluated using recursion.

For floating point evaluation, we again convert most functions into a hypergeometic form; we evaluate the hypergeometic functions using forward recursion. Again, the exceptions are the half-integer Bessel functions and the associated Legendre function of the second kind. Numerically, the half-integer Bessel functions are evaluated using the SLATEC code.


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78.2 Functions and Variables for orthogonal polynomials

Function: assoc_legendre_p (n, m, x)

The associated Legendre function of the first kind of degree n and order m.

Reference: Abramowitz and Stegun, equations 22.5.37, page 779, 8.6.6 (second equation), page 334, and 8.2.5, page 333.

Function: assoc_legendre_q (n, m, x)

The associated Legendre function of the second kind of degree n and order m.

Reference: Abramowitz and Stegun, equation 8.5.3 and 8.1.8.

Function: chebyshev_t (n, x)

The Chebyshev polynomial of the first kind of degree n.

Reference: Abramowitz and Stegun, equation 22.5.47, page 779.

Function: chebyshev_u (n, x)

The Chebyshev polynomial of the second kind of degree n.

Reference: Abramowitz and Stegun, equation 22.5.48, page 779.

Function: gen_laguerre (n, a, x)

The generalized Laguerre polynomial of degree n.

Reference: Abramowitz and Stegun, equation 22.5.54, page 780.

Function: hermite (n, x)

The Hermite polynomial of degree n.

Reference: Abramowitz and Stegun, equation 22.5.55, page 780.

Function: intervalp (e)

Return true if the input is an interval and return false if it isn’t.

Function: jacobi_p (n, a, b, x)

The Jacobi polynomial.

The Jacobi polynomials are actually defined for all a and b; however, the Jacobi polynomial weight (1 - x)^a (1 + x)^b isn’t integrable for a <= -1 or b <= -1.

Reference: Abramowitz and Stegun, equation 22.5.42, page 779.

Function: laguerre (n, x)

The Laguerre polynomial of degree n.

Reference: Abramowitz and Stegun, equations 22.5.16 and 22.5.54, page 780.

Function: legendre_p (n, x)

The Legendre polynomial of the first kind of degree n.

Reference: Abramowitz and Stegun, equations 22.5.50 and 22.5.51, page 779.

Function: legendre_q (n, x)

The Legendre function of the second kind of degree n.

Reference: Abramowitz and Stegun, equations 8.5.3 and 8.1.8.

Function: orthopoly_recur (f, args)

Returns a recursion relation for the orthogonal function family f with arguments args. The recursion is with respect to the polynomial degree.

(%i1) orthopoly_recur (legendre_p, [n, x]);
                    (2 n + 1) P (x) x - n P     (x)
                               n           n - 1
(%o1)   P     (x) = -------------------------------
         n + 1                   n + 1

The second argument to orthopoly_recur must be a list with the correct number of arguments for the function f; if it isn’t, Maxima signals an error.

(%i1) orthopoly_recur (jacobi_p, [n, x]);

Function jacobi_p needs 4 arguments, instead it received 2
 -- an error.  Quitting.  To debug this try debugmode(true);

Additionally, when f isn’t the name of one of the families of orthogonal polynomials, an error is signalled.

(%i1) orthopoly_recur (foo, [n, x]);

A recursion relation for foo isn't known to Maxima
 -- an error.  Quitting.  To debug this try debugmode(true);

Variable: orthopoly_returns_intervals

Default value: true

When orthopoly_returns_intervals is true, floating point results are returned in the form interval (c, r), where c is the center of an interval and r is its radius. The center can be a complex number; in that case, the interval is a disk in the complex plane.

Function: orthopoly_weight (f, args)

Returns a three element list; the first element is the formula of the weight for the orthogonal polynomial family f with arguments given by the list args; the second and third elements give the lower and upper endpoints of the interval of orthogonality. For example,

(%i1) w : orthopoly_weight (hermite, [n, x]);
                            2
                         - x
(%o1)                 [%e    , - inf, inf]
(%i2) integrate(w[1]*hermite(3, x)*hermite(2, x), x, w[2], w[3]);
(%o2)                           0

The main variable of f must be a symbol; if it isn’t, Maxima signals an error.

Function: pochhammer (x, n)

The Pochhammer symbol. For nonnegative integers n with n <= pochhammer_max_index, the expression pochhammer (x, n) evaluates to the product x (x + 1) (x + 2) ... (x + n - 1) when n > 0 and to 1 when n = 0. For negative n, pochhammer (x, n) is defined as (-1)^n / pochhammer (1 - x, -n). Thus

(%i1) pochhammer (x, 3);
(%o1)                   x (x + 1) (x + 2)
(%i2) pochhammer (x, -3);
                                 1
(%o2)               - -----------------------
                      (1 - x) (2 - x) (3 - x)

To convert a Pochhammer symbol into a quotient of gamma functions, (see Abramowitz and Stegun, equation 6.1.22) use makegamma; for example

(%i1) makegamma (pochhammer (x, n));
                          gamma(x + n)
(%o1)                     ------------
                            gamma(x)

When n exceeds pochhammer_max_index or when n is symbolic, pochhammer returns a noun form.

(%i1) pochhammer (x, n);
(%o1)                         (x)
                                 n

Variable: pochhammer_max_index

Default value: 100

pochhammer (n, x) expands to a product if and only if n <= pochhammer_max_index.

Examples:

(%i1) pochhammer (x, 3), pochhammer_max_index : 3;
(%o1)                   x (x + 1) (x + 2)
(%i2) pochhammer (x, 4), pochhammer_max_index : 3;
(%o2)                         (x)
                                 4

Reference: Abramowitz and Stegun, equation 6.1.16, page 256.

Function: spherical_bessel_j (n, x)

The spherical Bessel function of the first kind.

Reference: Abramowitz and Stegun, equations 10.1.8, page 437 and 10.1.15, page 439.

Function: spherical_bessel_y (n, x)

The spherical Bessel function of the second kind.

Reference: Abramowitz and Stegun, equations 10.1.9, page 437 and 10.1.15, page 439.

Function: spherical_hankel1 (n, x)

The spherical Hankel function of the first kind.

Reference: Abramowitz and Stegun, equation 10.1.36, page 439.

Function: spherical_hankel2 (n, x)

The spherical Hankel function of the second kind.

Reference: Abramowitz and Stegun, equation 10.1.17, page 439.

Function: spherical_harmonic (n, m, x, y)

The spherical harmonic function.

Reference: Merzbacher 9.64.

Function: unit_step (x)

The left-continuous unit step function; thus unit_step (x) vanishes for x <= 0 and equals 1 for x > 0.

If you want a unit step function that takes on the value 1/2 at zero, use (1 + signum (x))/2.

Function: ultraspherical (n, a, x)

The ultraspherical polynomial (also known as the Gegenbauer polynomial).

Reference: Abramowitz and Stegun, equation 22.5.46, page 779.


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79 ratpow

The package ratpow provides functions that find the exponents of the denominator in a CRE polynomial. If the exponents in the denominator are needed instead ratdenom can be used to extract this denominator first. Returned coefficients are in CRE form except for numbers.

In order to get a list of vars in a CRE polynomial showratvars can be used.

For information about CREs see also rat, ratdisrep and showratvars.


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79.1 Functions and Variables for ratpow

Function: ratp_hipow (expr, x)

Finds the highest power of the main variable in ratnumer(expr)

(%i1) load("ratpow")$
(%i2) ratp_hipow( x^(5/2) + x^2 , x);
(%o2)                           2
(%i3) ratp_hipow( x^(5/2) + x^2 , sqrt(x));
(%o3)                           5

Function: ratp_lopow (expr, x)

Finds the lowest power of the main variable in ratnumer(expr)

(%i1) load("ratpow")$
(%i2) ratp_lopow( x^5 + x^2 , x);
(%o2)                           2

The following example will return 0 since 1 equals x^0:

(%i1) load("ratpow")$
(%i2) ratp_lopow( x^5 + x^2 + 1, x);
(%o2)                           0

The CRE form of the following equation contains sqrt(x) and x. Since they are interpreted as independent variables ratp_lopow returns 0 in this case:

(%i1) load("ratpow")$
(%i2) g:sqrt(x)^5 + sqrt(x)^2;
                             5/2
(%o2)                       x    + x
(%i3) showratvars(g);
                              1/2
(%o3)                       [x   , x]
(%i4) ratp_lopow( g, x);
(%o4)                           0
(%i5) ratp_lopow( g, sqrt(x));
(%o5)                           0

Function: ratp_coeffs (expr, x)

Generates a list of powers and coefficients of the main variable ratnumer(expr).

(%i1) load("ratpow")$
(%i2) ratp_coeffs( 4*x^3 + x + sqrt(x), x);
(%o2)/R/         [[3, 4], [1, 1], [0, sqrt(x)]]

Function: ratp_dense_coeffs (expr, x)

Generates a list of coefficients in ratnumer(expr); returned coefficients are in CRE form except for numbers.

(%i1) load("ratpow")$
(%i2) ratp_dense_coeffs( 4*x^3 + x + sqrt(x), x);
(%o2)/R/               [4, 0, 1, sqrt(x)]


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80 romberg


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80.1 Functions and Variables for romberg

Function: romberg
    romberg (expr, x, a, b)
    romberg (F, a, b)

Computes a numerical integration by Romberg’s method.

romberg(expr, x, a, b) returns an estimate of the integral integrate(expr, x, a, b). expr must be an expression which evaluates to a floating point value when x is bound to a floating point value.

romberg(F, a, b) returns an estimate of the integral integrate(F(x), x, a, b) where x represents the unnamed, sole argument of F; the actual argument is not named x. F must be a Maxima or Lisp function which returns a floating point value when the argument is a floating point value. F may name a translated or compiled Maxima function.

The accuracy of romberg is governed by the global variables rombergabs and rombergtol. romberg terminates successfully when the absolute difference between successive approximations is less than rombergabs, or the relative difference in successive approximations is less than rombergtol. Thus when rombergabs is 0.0 (the default) only the relative error test has any effect on romberg.

romberg halves the stepsize at most rombergit times before it gives up; the maximum number of function evaluations is therefore 2^rombergit. If the error criterion established by rombergabs and rombergtol is not satisfied, romberg prints an error message. romberg always makes at least rombergmin iterations; this is a heuristic intended to prevent spurious termination when the integrand is oscillatory.

romberg repeatedly evaluates the integrand after binding the variable of integration to a specific value (and not before). This evaluation policy makes it possible to nest calls to romberg, to compute multidimensional integrals. However, the error calculations do not take the errors of nested integrations into account, so errors may be underestimated. Also, methods devised especially for multidimensional problems may yield the same accuracy with fewer function evaluations.

See also Introduction to QUADPACK, a collection of numerical integration functions.

Examples:

A 1-dimensional integration.

(%i1) f(x) := 1/((x - 1)^2 + 1/100) + 1/((x - 2)^2 + 1/1000)
              + 1/((x - 3)^2 + 1/200);
                    1                 1                1
(%o1) f(x) := -------------- + --------------- + --------------
                     2    1           2    1            2    1
              (x - 1)  + ---   (x - 2)  + ----   (x - 3)  + ---
                         100              1000              200
(%i2) rombergtol : 1e-6;
(%o2)                 9.999999999999999e-7
(%i3) rombergit : 15;
(%o3)                          15
(%i4) estimate : romberg (f(x), x, -5, 5);
(%o4)                   173.6730736617464
(%i5) exact : integrate (f(x), x, -5, 5);
        3/2          3/2      3/2          3/2
(%o5) 10    atan(7 10   ) + 10    atan(3 10   )
      3/2         9/2       3/2         5/2
 + 5 2    atan(5 2   ) + 5 2    atan(5 2   ) + 10 atan(60)
 + 10 atan(40)
(%i6) abs (estimate - exact) / exact, numer;
(%o6)                 7.552722451569877e-11

A 2-dimensional integration, implemented by nested calls to romberg.

(%i1) g(x, y) := x*y / (x + y);
                                    x y
(%o1)                   g(x, y) := -----
                                   x + y
(%i2) rombergtol : 1e-6;
(%o2)                 9.999999999999999e-7
(%i3) estimate : romberg (romberg (g(x, y), y, 0, x/2), x, 1, 3);
(%o3)                  0.8193023962835647
(%i4) assume (x > 0);
(%o4)                        [x > 0]
(%i5) integrate (integrate (g(x, y), y, 0, x/2), x, 1, 3);
                                           3
                                     2 log(-) - 1
                    9                      2        9
(%o5)      (- 9 log(-)) + 9 log(3) + ------------ + -
                    2                     6         2
(%i6) exact : radcan (%);
                    26 log(3) - 26 log(2) - 13
(%o6)             - --------------------------
                                3
(%i7) abs (estimate - exact) / exact, numer;
(%o7)                 1.371197987185102e-10

Option variable: rombergabs

Default value: 0.0

The accuracy of romberg is governed by the global variables rombergabs and rombergtol. romberg terminates successfully when the absolute difference between successive approximations is less than rombergabs, or the relative difference in successive approximations is less than rombergtol. Thus when rombergabs is 0.0 (the default) only the relative error test has any effect on romberg.

See also rombergit and rombergmin.

Option variable: rombergit

Default value: 11

romberg halves the stepsize at most rombergit times before it gives up; the maximum number of function evaluations is therefore 2^rombergit. romberg always makes at least rombergmin iterations; this is a heuristic intended to prevent spurious termination when the integrand is oscillatory.

See also rombergabs and rombergtol.

Option variable: rombergmin

Default value: 0

romberg always makes at least rombergmin iterations; this is a heuristic intended to prevent spurious termination when the integrand is oscillatory.

See also rombergit, rombergabs, and rombergtol.

Option variable: rombergtol

Default value: 1e-4

The accuracy of romberg is governed by the global variables rombergabs and rombergtol. romberg terminates successfully when the absolute difference between successive approximations is less than rombergabs, or the relative difference in successive approximations is less than rombergtol. Thus when rombergabs is 0.0 (the default) only the relative error test has any effect on romberg.

See also rombergit and rombergmin.


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81 simplex


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81.1 Introduction to simplex

simplex is a package for linear optimization using the simplex algorithm.

Example:

(%i1) load("simplex")$
(%i2) minimize_lp(x+y, [3*x+2*y>2, x+4*y>3]);
                  9        7       1
(%o2)            [--, [y = --, x = -]]
                  10       10      5
·

Category: Numerical methods · Category: Optimization · Category: Share packages · Category: Package simplex

81.1.1 Tests for simplex

There are some tests in the directory share/simplex/Tests.

81.1.1.1 klee_minty

The function klee_minty produces input for linear_program, for which exponential time for solving is required without scaling.

Example:

load(klee_minty)$
apply(linear_program, klee_minty(6));

A better approach:

epsilon_sx : 0$
scale_sx : true$
apply(linear_program, klee_minty(10));

81.1.1.2 NETLIB

Some smaller problems from netlib (http://www.netlib.org/lp/data/) test suite are converted to a format, readable by Maxima. Problems are adlittle, afiro, kb2, sc50a and share2b. Each problem has three input files in CSV format for matrix A and vectors b and c.

Example:

A : read_matrix("adlittle_A.csv", 'csv)$
b : read_list("adlittle_b.csv", 'csv)$
c : read_list("adlittle_c.csv", 'csv)$
linear_program(A, b, c)$
%[2];
=> 225494.9631623802

Results:

PROBLEM        MINIMUM                  SCALING
adlittle       +2.2549496316E+05        false
afiro          -4.6475314286E+02        false
kb2            -1.7499001299E+03        true
sc50a          -6.4575077059E+01        false
share2b        -4.1573518187E+02        false

The Netlib website http://www.netlib.org/lp/data/readme lists the values as

PROBLEM        MINIMUM
adlittle       +2.2549496316E+05
afiro          -4.6475314286E+02
kb2            -1.7499001299E+03
sc50a          -6.4575077059E+01
share2b        -4.1573224074E+02

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81.2 Functions and Variables for simplex

Option variable: epsilon_lp

Default value: 10^-8

Epsilon used for numerical computations in linear_program; it is set to 0 in linear_program when all inputs are rational.

Example:

(%i1) load(simplex)$

(%i2) minimize_lp(-x, [1e-9*x + y <= 1], [x,y]);
Warning: linear_program(A,b,c): non-rat inputs found, epsilon_lp= 1.0e-8
Warning: Solution may be incorrect.
(%o2)                        Problem not bounded!
(%i3) minimize_lp(-x, [10^-9*x + y <= 1], [x,y]);
(%o3)               [- 1000000000, [y = 0, x = 1000000000]]
(%i4) minimize_lp(-x, [1e-9*x + y <= 1], [x,y]), epsilon_lp=0;
(%o4)     [- 9.999999999999999e+8, [y = 0, x = 9.999999999999999e+8]]

See also: linear_program, ratnump.

Function: linear_program (A, b, c)

linear_program is an implementation of the simplex algorithm. linear_program(A, b, c) computes a vector x for which c.x is minimum possible among vectors for which A.x = b and x >= 0. Argument A is a matrix and arguments b and c are lists.

linear_program returns a list which contains the minimizing vector x and the minimum value c.x. If the problem is not bounded, it returns "Problem not bounded!" and if the problem is not feasible, it returns "Problem not feasible!".

To use this function first load the simplex package with load(simplex);.

Example:

(%i2) A: matrix([1,1,-1,0], [2,-3,0,-1], [4,-5,0,0])$
(%i3) b: [1,1,6]$
(%i4) c: [1,-2,0,0]$
(%i5) linear_program(A, b, c);
                   13     19        3
(%o5)            [[--, 4, --, 0], - -]
                   2      2         2

See also: minimize_lp, scale_lp, and epsilon_lp.

Function: maximize_lp (obj, cond, [pos])

Maximizes linear objective function obj subject to some linear constraints cond. See minimize_lp for detailed description of arguments and return value.

See also: minimize_lp.

Function: minimize_lp (obj, cond, [pos])

Minimizes a linear objective function obj subject to some linear constraints cond. cond a list of linear equations or inequalities. In strict inequalities > is replaced by >= and < by <=. The optional argument pos is a list of decision variables which are assumed to be positive.

If the minimum exists, minimize_lp returns a list which contains the minimum value of the objective function and a list of decision variable values for which the minimum is attained. If the problem is not bounded, minimize_lp returns "Problem not bounded!" and if the problem is not feasible, it returns "Problem not feasible!".

The decision variables are not assumed to be non-negative by default. If all decision variables are non-negative, set nonnegative_lp to true or include all in the optional argument pos. If only some of decision variables are positive, list them in the optional argument pos (note that this is more efficient than adding constraints).

minimize_lp uses the simplex algorithm which is implemented in maxima linear_program function.

To use this function first load the simplex package with load(simplex);.

Examples:

(%i1) minimize_lp(x+y, [3*x+y=0, x+2*y>2]);
                      4       6        2
(%o1)                [-, [y = -, x = - -]]
                      5       5        5
(%i2) minimize_lp(x+y, [3*x+y>0, x+2*y>2]), nonnegative_lp=true;
(%o2)                [1, [y = 1, x = 0]]
(%i3) minimize_lp(x+y, [3*x+y>0, x+2*y>2], all);
(%o3)                         [1, [y = 1, x = 0]]
(%i4) minimize_lp(x+y, [3*x+y=0, x+2*y>2]), nonnegative_lp=true;
(%o4)                Problem not feasible!
(%i5) minimize_lp(x+y, [3*x+y>0]);
(%o5)                Problem not bounded!

There is also a limited ability to solve linear programs with symbolic constants.

(%i1) declare(c,constant)$
(%i2) maximize_lp(x+y, [y<=-x/c+3, y<=-x+4], [x, y]), epsilon_lp=0;
Is (c-1)*c positive, negative or zero?
p;
Is c*(2*c-1) positive, negative or zero?
p;
Is c positive or negative?
p;
Is c-1 positive, negative or zero?
p;
Is 2*c-1 positive, negative or zero?
p;
Is 3*c-4 positive, negative or zero?
p;
                                 1                1
(%o2)                 [4, [x = -----, y = 3 - ---------]]
                                   1               1
                               1 - -          (1 - -) c
                                   c               c
(%i1) (assume(c>4/3), declare(c,constant))$
(%i2) maximize_lp(x+y, [y<=-x/c+3, y<=-x+4], [x, y]), epsilon_lp=0;
                                 1                1
(%o2)                 [4, [x = -----, y = 3 - ---------]]
                                   1               1
                               1 - -          (1 - -) c
                                   c               c

See also: maximize_lp, nonnegative_lp, epsilon_lp.

Option variable: nonnegative_lp
Option variable: nonegative_lp

Default value: false

If nonnegative_lp is true all decision variables to minimize_lp and maximize_lp are assumed to be non-negative. nonegative_lp is a deprecated alias.

See also: minimize_lp.

Option variable: scale_lp

Default value: false

When scale_lp is true, linear_program scales its input so that the maximum absolute value in each row or column is 1.

Variable: pivot_count_sx

After linear_program returns, pivot_count_sx is the number of pivots in last computation.

Variable: pivot_max_sx

pivot_max_sx is the maximum number of pivots allowed by linear_program.


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82 simplification


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82.1 Introduction to simplification

The directory maxima/share/simplification contains several scripts which implement simplification rules and functions, and also some functions not related to simplification.


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82.2 Package absimp

The absimp package contains pattern-matching rules that extend the built-in simplification rules for the abs and signum functions. absimp respects relations established with the built-in assume function and by declarations such as mode_declare (m, even, n, odd) for even or odd integers.

absimp defines unitramp and unitstep functions in terms of abs and signum.

load ("absimp") loads this package. demo ("absimp") shows a demonstration of this package.

Examples:

(%i1) load ("absimp")$
(%i2) (abs (x))^2;
                                2
(%o2)                          x
(%i3) diff (abs (x), x);
                               x
(%o3)                        ------
                             abs(x)
(%i4) cosh (abs (x));
(%o4)                        cosh(x)
·

Category: Simplification functions · Category: Rules and patterns · Category: Share packages · Category: Package absimp


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82.3 Package facexp

The facexp package contains several related functions that provide the user with the ability to structure expressions by controlled expansion. This capability is especially useful when the expression contains variables that have physical meaning, because it is often true that the most economical form of such an expression can be obtained by fully expanding the expression with respect to those variables, and then factoring their coefficients. While it is true that this procedure is not difficult to carry out using standard Maxima functions, additional fine-tuning may also be desirable, and these finishing touches can be more difficult to apply.

The function facsum and its related forms provide a convenient means for controlling the structure of expressions in this way. Another function, collectterms, can be used to add two or more expressions that have already been simplified to this form, without resimplifying the whole expression again. This function may be useful when the expressions are very large.

load ("facexp") loads this package. demo ("facexp") shows a demonstration of this package.

·

Category: Expressions · Category: Share packages · Category: Package facexp

Function: facsum (expr, arg_1, ..., arg_n)

Returns a form of expr which depends on the arguments arg_1, ..., arg_n. The arguments can be any form suitable for ratvars, or they can be lists of such forms. If the arguments are not lists, then the form returned is fully expanded with respect to the arguments, and the coefficients of the arguments are factored. These coefficients are free of the arguments, except perhaps in a non-rational sense.

If any of the arguments are lists, then all such lists are combined into a single list, and instead of calling factor on the coefficients of the arguments, facsum calls itself on these coefficients, using this newly constructed single list as the new argument list for this recursive call. This process can be repeated to arbitrary depth by nesting the desired elements in lists.

It is possible that one may wish to facsum with respect to more complicated subexpressions, such as log (x + y). Such arguments are also permissible.

Occasionally the user may wish to obtain any of the above forms for expressions which are specified only by their leading operators. For example, one may wish to facsum with respect to all log’s. In this situation, one may include among the arguments either the specific log’s which are to be treated in this way, or alternatively, either the expression operator (log) or 'operator (log). If one wished to facsum the expression expr with respect to the operators op_1, ..., op_n, one would evaluate facsum (expr, operator (op_1, ..., op_n)). The operator form may also appear inside list arguments.

In addition, the setting of the switches facsum_combine and nextlayerfactor may affect the result of facsum.

Global variable: nextlayerfactor

Default value: false

When nextlayerfactor is true, recursive calls of facsum are applied to the factors of the factored form of the coefficients of the arguments.

When false, facsum is applied to each coefficient as a whole whenever recusive calls to facsum occur.

Inclusion of the atom nextlayerfactor in the argument list of facsum has the effect of nextlayerfactor: true, but for the next level of the expression only. Since nextlayerfactor is always bound to either true or false, it must be presented single-quoted whenever it appears in the argument list of facsum.

Global variable: facsum_combine

Default value: true

facsum_combine controls the form of the final result returned by facsum when its argument is a quotient of polynomials. If facsum_combine is false then the form will be returned as a fully expanded sum as described above, but if true, then the expression returned is a ratio of polynomials, with each polynomial in the form described above.

The true setting of this switch is useful when one wants to facsum both the numerator and denominator of a rational expression, but does not want the denominator to be multiplied through the terms of the numerator.

Function: factorfacsum (expr, arg_1, ... arg_n)

Returns a form of expr which is obtained by calling facsum on the factors of expr with arg_1, ... arg_n as arguments. If any of the factors of expr is raised to a power, both the factor and the exponent will be processed in this way.

Function: collectterms (expr, arg_1, …, arg_n)

Collects all terms that contain arg_1 ... arg_n. If several expressions have been simplified with the following functions facsum, factorfacsum, factenexpand, facexpten or factorfacexpten, and they are to be added together, it may be desirable to combine them using the function collecterms. collecterms can take as arguments all of the arguments that can be given to these other associated functions with the exception of nextlayerfactor, which has no effect on collectterms. The advantage of collectterms is that it returns a form similar to facsum, but since it is adding forms that have already been processed by facsum, it does not need to repeat that effort. This capability is especially useful when the expressions to be summed are very large.

See also factor.

Example:

(%i1) (exp(x)+2)*x+exp(x);
                             x          x
(%o1)                   x (%e  + 2) + %e
(%i2) collectterms(expand(%),exp(x));
                                  x
(%o2)                   (x + 1) %e  + 2 x


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82.4 Package functs

·

Category: Share packages · Category: Package functs

Function: rempart (expr, n)

Removes part n from the expression expr.

If n is a list of the form [l, m] then parts l thru m are removed.

To use this function write first load(functs).

Function: wronskian ([f_1, ..., f_n], x)

Returns the Wronskian matrix of the list of expressions [f_1, ..., f_n] in the variable x. The determinant of the Wronskian matrix is the Wronskian determinant of the list of expressions.

To use wronskian, first load(functs). Example:

(%i1) load ("functs")$
(%i2) wronskian([f(x), g(x)],x);
                    [   f(x)       g(x)    ]
                    [                      ]
(%o2)               [ d          d         ]
                    [ -- (f(x))  -- (g(x)) ]
                    [ dx         dx        ]

Function: tracematrix (M)

Returns the trace (sum of the diagonal elements) of matrix M.

To use this function write first load(functs).

Function: rational (z)

Multiplies numerator and denominator of z by the complex conjugate of denominator, thus rationalizing the denominator. Returns canonical rational expression (CRE) form if given one, else returns general form.

To use this function write first load(functs).

Function: nonzeroandfreeof (x, expr)

Returns true if expr is nonzero and freeof (x, expr) returns true. Returns false otherwise.

To use this function write first load(functs).

Function: linear (expr, x)

When expr is an expression of the form a*x + b where a is nonzero, and a and b are free of x, linear returns a list of three equations, one for each of the three formal variables b, a, and x. Otherwise, linear returns false.

load(antid) loads this function.

Example:

(%i1) load ("antid");
(%o1)  /maxima/share/integration/antid.mac
(%i2) linear ((1 - w)*(1 - x)*z, z);
(%o2) [bargumentb = 0, aargumenta = (w - 1) x - w + 1, 
                                                  xargumentx = z]
(%i3) linear (cos(u - v) + cos(u + v), u);
(%o3)                         false

Function: gcdivide (p, q)

When the option variable takegcd is true which is the default, gcdivide divides the polynomials p and q by their greatest common divisor and returns the ratio of the results. gcdivde calls the function ezgcd to divide the polynomials by the greatest common divisor.

When takegcd is false, gcdivide returns the ratio p/q.

To use this function write first load(functs).

See also ezgcd, gcd, gcdex, and poly_gcd.

Example:

(%i1) load(functs)$

(%i2) p1:6*x^3+19*x^2+19*x+6; 
                        3       2
(%o2)                6 x  + 19 x  + 19 x + 6
(%i3) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x;
                  5       4       3       2
(%o3)          6 x  + 13 x  + 12 x  + 13 x  + 6 x
(%i4) gcdivide(p1, p2);
                             x + 1
(%o4)                        ------
                              3
                             x  + x
(%i5) takegcd:false;
(%o5)                         false
(%i6) gcdivide(p1, p2);
                       3       2
                    6 x  + 19 x  + 19 x + 6
(%o6)          ----------------------------------
                  5       4       3       2
               6 x  + 13 x  + 12 x  + 13 x  + 6 x
(%i7) ratsimp(%);
                             x + 1
(%o7)                        ------
                              3
                             x  + x

Function: arithmetic (a, d, n)

Returns the n-th term of the arithmetic series a, a + d, a + 2*d, ..., a + (n - 1)*d.

To use this function write first load(functs).

Function: geometric (a, r, n)

Returns the n-th term of the geometric series a, a*r, a*r^2, ..., a*r^(n - 1).

To use this function write first load(functs).

Function: harmonic (a, b, c, n)

Returns the n-th term of the harmonic series a/b, a/(b + c), a/(b + 2*c), ..., a/(b + (n - 1)*c).

To use this function write first load(functs).

Function: arithsum (a, d, n)

Returns the sum of the arithmetic series from 1 to n.

To use this function write first load(functs).

Function: geosum (a, r, n)

Returns the sum of the geometric series from 1 to n. If n is infinity (inf) then a sum is finite only if the absolute value of r is less than 1.

To use this function write first load(functs).

Function: gaussprob (x)

Returns the Gaussian probability function %e^(-x^2/2) / sqrt(2*%pi).

To use this function write first load(functs).

Function: gd (x)

Returns the Gudermannian function 2*atan(%e^x)-%pi/2.

To use this function write first load(functs).

Function: agd (x)

Returns the inverse Gudermannian function log (tan (%pi/4 + x/2)).

To use this function write first load(functs).

Function: vers (x)

Returns the versed sine 1 - cos (x).

To use this function write first load(functs).

Function: covers (x)

Returns the coversed sine 1 - sin (x).

To use this function write first load(functs).

Function: exsec (x)

Returns the exsecant sec (x) - 1.

To use this function write first load(functs).

Function: hav (x)

Returns the haversine (1 - cos(x))/2.

To use this function write first load(functs).

Function: combination (n, r)

Returns the number of combinations of n objects taken r at a time.

To use this function write first load(functs).

Function: permutation (n, r)

Returns the number of permutations of r objects selected from a set of n objects.

To use this function write first load(functs).


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82.5 Package ineq

The ineq package contains simplification rules for inequalities.

Example session:

(%i1) load(ineq)$
tellsimp: warning: rule will treat '+
                          ' as noncommutative and nonassociative.
tellsimp: warning: rule will treat '+
                          ' as noncommutative and nonassociative.
tellsimp: warning: rule will treat '+
                          ' as noncommutative and nonassociative.
tellsimp: warning: rule will treat '+
                          ' as noncommutative and nonassociative.
tellsimp: warning: rule will treat '+
                          ' as noncommutative and nonassociative.
tellsimp: warning: rule will treat '+
                          ' as noncommutative and nonassociative.
tellsimp: warning: rule will treat '+
                          ' as noncommutative and nonassociative.
tellsimp: warning: rule will treat '+
                          ' as noncommutative and nonassociative.
(%i2) a>=4;  /* a sample inequality */
(%o2)                        a >= 4
(%o3)                     b + a > c + 4
(%o4)                       7 x < 7 y
(%o5)                    - 2 x <= - 6 z
                                 2
(%o6)                      1 <= a  + 1
(%o8)                       2 x < 3 x
(%o9)                        a >= b
(%o10)                   a + 3 >= b + 3
(%o11)                       a >= b
(%o12)                     a >= c - b
(%o13)                     b + a >= c
(%o14)                 (- c) + b + a >= 0
(%o15)                   c - b - a <= 0
                               2
(%o16)                  (z - 1)  > - 2 z
                            2
(%o17)                     z  + 1 > 0
(%o18)                        true
(%i19) (b>c)+%; /* add a second, strict inequality */

Be careful about using parentheses around the inequalities: when the user types in (A > B) + (C = 5) the result is A + C > B + 5, but A > B + C = 5 is a syntax error, and (A > B + C) = 5 is something else entirely.

Do disprule (all) to see a complete listing of the rule definitions.

The user will be queried if Maxima is unable to decide the sign of a quantity multiplying an inequality.

The most common mis-feature is illustrated by:

(%i1) eq: a > b;
(%o1)                         a > b
(%i2) 2*eq;
(%o2)                       2 (a > b)
(%i3) % - eq;
(%o3)                         a > b

Another problem is 0 times an inequality; the default to have this turn into 0 has been left alone. However, if you type X*some_inequality and Maxima asks about the sign of X and you respond zero (or z), the program returns X*some_inequality and not use the information that X is 0. You should do ev (%, x: 0) in such a case, as the database will only be used for comparison purposes in decisions, and not for the purpose of evaluating X.

The user may note a slower response when this package is loaded, as the simplifier is forced to examine more rules than without the package, so you might wish to remove the rules after making use of them. Do kill (rules) to eliminate all of the rules (including any that you might have defined); or you may be more selective by killing only some of them; or use remrule on a specific rule.

Note that if you load this package after defining your own rules you will clobber your rules that have the same name. The rules in this package are: *rule1, ..., *rule8, +rule1, ..., +rule18, and you must enclose the rulename in quotes to refer to it, as in remrule ("+", "+rule1") to specifically remove the first rule on "+" or disprule ("*rule2") to display the definition of the second multiplicative rule.

·

Category: Simplification functions · Category: Rules and patterns · Category: Share packages · Category: Package ineq


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82.6 Package rducon

·

Category: Expressions · Category: Share packages · Category: Package rducon

Function: reduce_consts (expr)

Replaces constant subexpressions of expr with constructed constant atoms, saving the definition of all these constructed constants in the list of equations const_eqns, and returning the modified expr. Those parts of expr are constant which return true when operated on by the function constantp. Hence, before invoking reduce_consts, one should do

declare ([objects to be given the constant property], constant)$

to set up a database of the constant quantities occurring in your expressions.

If you are planning to generate Fortran output after these symbolic calculations, one of the first code sections should be the calculation of all constants. To generate this code segment, do

map ('fortran, const_eqns)$

Variables besides const_eqns which affect reduce_consts are:

const_prefix (default value: xx) is the string of characters used to prefix all symbols generated by reduce_consts to represent constant subexpressions.

const_counter (default value: 1) is the integer index used to generate unique symbols to represent each constant subexpression found by reduce_consts.

load ("rducon") loads this function. demo ("rducon") shows a demonstration of this function.


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82.7 Package scifac

·

Category: Expressions · Category: Share packages · Category: Package scifac

Function: gcfac (expr)

gcfac is a factoring function that attempts to apply the same heuristics which scientists apply in trying to make expressions simpler. gcfac is limited to monomial-type factoring. For a sum, gcfac does the following:

  1. Factors over the integers.
  2. Factors out the largest powers of terms occurring as coefficients, regardless of the complexity of the terms.
  3. Uses (1) and (2) in factoring adjacent pairs of terms.
  4. Repeatedly and recursively applies these techniques until the expression no longer changes.

Item (3) does not necessarily do an optimal job of pairwise factoring because of the combinatorially-difficult nature of finding which of all possible rearrangements of the pairs yields the most compact pair-factored result.

load ("scifac") loads this function. demo ("scifac") shows a demonstration of this function.


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82.8 Package sqdnst

Function: sqrtdenest (expr)

Denests sqrt of simple, numerical, binomial surds, where possible. E.g.

(%i1) load ("sqdnst")$
(%i2) sqrt(sqrt(3)/2+1)/sqrt(11*sqrt(2)-12);
                             sqrt(3)
                        sqrt(------- + 1)
                                2
(%o2)                 ---------------------
                      sqrt(11 sqrt(2) - 12)
(%i3) sqrtdenest(%);
                           sqrt(3)   1
                           ------- + -
                              2      2
(%o3)                     -------------
                             1/4    3/4
                          3 2    - 2

Sometimes it helps to apply sqrtdenest more than once, on such as (19601-13860 sqrt(2))^(7/4).

load ("sqdnst") loads this function.


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83 solve_rec


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83.1 Introduction to solve_rec

solve_rec is a package for solving linear recurrences with polynomial coefficients.

A demo is available with demo("solve_rec");.

Example:

(%i1) load("solve_rec")$
(%i2) solve_rec((n+4)*s[n+2] + s[n+1] - (n+1)*s[n], s[n]);
                                    n
                 %k  (2 n + 3) (- 1)          %k
                   1                            2
(%o2)       s  = -------------------- + ---------------
             n     (n + 1) (n + 2)      (n + 1) (n + 2)
·

Category: Linear recurrences · Category: Share packages · Category: Package solve_rec


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83.2 Functions and Variables for solve_rec

Function: reduce_order (rec, sol, var)

Reduces the order of linear recurrence rec when a particular solution sol is known. The reduced reccurence can be used to get other solutions.

Example:

(%i3) rec: x[n+2] = x[n+1] + x[n]/n;
                                      x
                                       n
(%o3)               x      = x      + --
                     n + 2    n + 1   n
(%i4) solve_rec(rec, x[n]);
WARNING: found some hypergeometrical solutions! 
(%o4)                    x  = %k  n
                          n     1
(%i5) reduce_order(rec, n, x[n]);
(%t5)                    x  = n %z
                          n       n

                           n - 1
                           ====
                           \
(%t6)                %z  =  >     %u
                       n   /        %j
                           ====
                           %j = 0

(%o6)             (- n - 2) %u     - %u
                              n + 1     n
(%i6) solve_rec((n+2)*%u[n+1] + %u[n], %u[n]);
                                     n
                            %k  (- 1)
                              1
(%o6)                 %u  = ----------
                        n    (n + 1)!

So the general solution is

             n - 1
             ====        j
             \      (- 1)
       %k  n  >    -------- + %k  n
         2   /     (j + 1)!     1
             ====
             j = 0

Option variable: simplify_products

Default value: true

If simplify_products is true, solve_rec will try to simplify products in result.

See also: solve_rec.

Function: simplify_sum (expr)

Tries to simplify all sums appearing in expr to a closed form.

To use this function first load the simplify_sum package with load(simplify_sum).

Example:

(%i1) load("simplify_sum")$
(%i2) sum(binomial(n+k,k)/2^k,k,1,n)+sum(binomial(2*n,2*k),k,1,n);
        n                          n
       ====                       ====
       \     binomial(n + k, k)   \
(%o2)   >    ------------------ +  >    binomial(2 n, 2 k)
       /              k           /
       ====          2            ====
       k = 1                      k = 1
(%i3) simplify_sum(%);

                         2 n - 1    n
(%o3)                   2        + 2  - 2

Function: solve_rec (eqn, var, [init])

Solves for hypergeometrical solutions to linear recurrence eqn with polynomials coefficient in variable var. Optional arguments init are initial conditions.

solve_rec can solve linear recurrences with constant coefficients, finds hypergeometrical solutions to homogeneous linear recurrences with polynomial coefficients, rational solutions to linear recurrences with polynomial coefficients and can solve Ricatti type recurrences.

Note that the running time of the algorithm used to find hypergeometrical solutions is exponential in the degree of the leading and trailing coefficient.

To use this function first load the solve_rec package with load(solve_rec);.

Example of linear recurrence with constant coefficients:

(%i2) solve_rec(a[n]=a[n-1]+a[n-2]+n/2^n, a[n]);
                        n          n
           (sqrt(5) - 1)  %k  (- 1)
                            1           n
(%o2) a  = ------------------------- - ----
       n               n                  n
                      2                5 2
                                                n
                                   (sqrt(5) + 1)  %k
                                                    2    2
                                 + ------------------ - ----
                                            n              n
                                           2            5 2

Example of linear recurrence with polynomial coefficients:

(%i7) 2*x*(x+1)*y[x] - (x^2+3*x-2)*y[x+1] + (x-1)*y[x+2];
                         2
(%o7) (x - 1) y      - (x  + 3 x - 2) y      + 2 x (x + 1) y
               x + 2                   x + 1                x
(%i8) solve_rec(%, y[x], y[1]=1, y[3]=3);
                              x
                           3 2    x!
(%o9)                 y  = ---- - --
                       x    4     2

Example of Ricatti type recurrence:

(%i2) x*y[x+1]*y[x] - y[x+1]/(x+2) + y[x]/(x-1) = 0;
                            y         y
                             x + 1     x
(%o2)         x y  y      - ------ + ----- = 0
                 x  x + 1   x + 2    x - 1
(%i3) solve_rec(%, y[x], y[3]=5)$
(%i4) ratsimp(minfactorial(factcomb(%)));
                                   3
                               30 x  - 30 x
(%o4) y  = - -------------------------------------------------
       x        6      5       4       3       2
             5 x  - 3 x  - 25 x  + 15 x  + 20 x  - 12 x - 1584

See also: solve_rec_rat, simplify_products and product_use_gamma.

Function: solve_rec_rat (eqn, var, [init])

Solves for rational solutions to linear recurrences. See solve_rec for description of arguments.

To use this function first load the solve_rec package with load(solve_rec);.

Example:

(%i1) (x+4)*a[x+3] + (x+3)*a[x+2] - x*a[x+1] + (x^2-1)*a[x];
(%o1)  (x + 4) a      + (x + 3) a      - x a
                x + 3            x + 2      x + 1
                                                   2
                                               + (x  - 1) a
                                                            x
(%i2) solve_rec_rat(% = (x+2)/(x+1), a[x]);
                       1
(%o2)      a  = ---------------
            x   (x - 1) (x + 1)

See also: solve_rec.

Option variable: product_use_gamma

Default value: true

When simplifying products, solve_rec introduces gamma function into the expression if product_use_gamma is true.

See also: simplify_products, solve_rec.

Function: summand_to_rec
    summand_to_rec (summand, k, n)
    summand_to_rec (summand, [k, lo, hi], n)

Returns the recurrence sattisfied by the sum

     hi
    ====
    \
     >     summand
    /
    ====
  k = lo

where summand is hypergeometrical in k and n. If lo and hi are omitted, they are assumed to be lo = -inf and hi = inf.

To use this function first load the simplify_sum package with load(simplify_sum).

Example:

(%i1) load("simplify_sum")$
(%i2) summand: binom(n,k);
(%o2)                           binomial(n, k)
(%i3) summand_to_rec(summand,k,n);
(%o3)                      2 sm  - sm      = 0
                               n     n + 1
(%i7) summand: binom(n, k)/(k+1);
                                binomial(n, k)
(%o7)                           --------------
                                    k + 1
(%i8) summand_to_rec(summand, [k, 0, n], n);
(%o8)               2 (n + 1) sm  - (n + 2) sm      = - 1
                                n             n + 1


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84 stats


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84.1 Introduction to stats

Package stats contains a set of classical statistical inference and hypothesis testing procedures.

All these functions return an inference_result Maxima object which contains the necessary results for population inferences and decision making.

Global variable stats_numer controls whether results are given in floating point or symbolic and rational format; its default value is true and results are returned in floating point format.

Package descriptive contains some utilities to manipulate data structures (lists and matrices); for example, to extract subsamples. It also contains some examples on how to use package numericalio to read data from plain text files. See descriptive and numericalio for more details.

Package stats loads packages descriptive, distrib and inference_result.

For comments, bugs or suggestions, please contact the author at

’mario AT edu DOT xunta DOT es’.

·

Category: Statistical inference · Category: Share packages · Category: Package stats


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84.2 Functions and Variables for inference_result

Function: inference_result (title, values, numbers)

Constructs an inference_result object of the type returned by the stats functions. Argument title is a string with the name of the procedure; values is a list with elements of the form symbol = value and numbers is a list with positive integer numbers ranging from one to length(values), indicating which values will be shown by default.

Example:

This is a simple example showing results concerning a rectangle. The title of this object is the string "Rectangle", it stores five results, named 'base, 'height, 'diagonal, 'area, and 'perimeter, but only the first, second, fifth, and fourth will be displayed. The 'diagonal is stored in this object, but it is not displayed; to access its value, make use of function take_inference.

(%i1) load(inference_result)$
(%i2) b: 3$ h: 2$
(%i3) inference_result("Rectangle",
                        ['base=b,
                         'height=h,
                         'diagonal=sqrt(b^2+h^2),
                         'area=b*h,
                         'perimeter=2*(b+h)],
                        [1,2,5,4] );
                        |   Rectangle
                        |
                        |    base = 3
                        |
(%o3)                   |   height = 2
                        |
                        | perimeter = 10
                        |
                        |    area = 6
(%i4) take_inference('diagonal,%);
(%o4)                        sqrt(13)

See also take_inference.

Function: inferencep (obj)

Returns true or false, depending on whether obj is an inference_result object or not.

Function: items_inference (obj)

Returns a list with the names of the items stored in obj, which must be an inference_result object.

Example:

The inference_result object stores two values, named 'pi and 'e, but only the second is displayed. The items_inference function returns the names of all items, no matter they are displayed or not.

(%i1) load(inference_result)$
(%i2) inference_result("Hi", ['pi=%pi,'e=%e],[2]);
                            |   Hi
(%o2)                       |
                            | e = %e
(%i3) items_inference(%);
(%o3)                        [pi, e]

Function: take_inference
    take_inference (n, obj)
    take_inference (name, obj)
    take_inference (list, obj)

Returns the n-th value stored in obj if n is a positive integer, or the item named name if this is the name of an item. If the first argument is a list of numbers and/or symbols, function take_inference returns a list with the corresponding results.

Example:

Given an inference_result object, function take_inference is called in order to extract some information stored in it.

(%i1) load(inference_result)$
(%i2) b: 3$ h: 2$
(%i3) sol: inference_result("Rectangle",
                            ['base=b,
                             'height=h,
                             'diagonal=sqrt(b^2+h^2),
                             'area=b*h,
                             'perimeter=2*(b+h)],
                            [1,2,5,4] );
                        |   Rectangle
                        |
                        |    base = 3
                        |
(%o3)                   |   height = 2
                        |
                        | perimeter = 10
                        |
                        |    area = 6
(%i4) take_inference('base,sol);
(%o4)                           3
(%i5) take_inference(5,sol);
(%o5)                          10
(%i6) take_inference([1,'diagonal],sol);
(%o6)                     [3, sqrt(13)]
(%i7) take_inference(items_inference(sol),sol);
(%o7)                [3, 2, sqrt(13), 6, 10]

See also inference_result, and take_inference.


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84.3 Functions and Variables for stats

Option variable: stats_numer

Default value: true

If stats_numer is true, inference statistical functions return their results in floating point numbers. If it is false, results are given in symbolic and rational format.

Function: test_mean
    test_mean (x)
    test_mean (x, options ...)

This is the mean t-test. Argument x is a list or a column matrix containing a one dimensional sample. It also performs an asymptotic test based on the Central Limit Theorem if option 'asymptotic is true.

Options:

The output of function test_mean is an inference_result Maxima object showing the following results:

  1. 'mean_estimate: the sample mean.
  2. 'conf_level: confidence level selected by the user.
  3. 'conf_interval: confidence interval for the population mean.
  4. 'method: inference procedure.
  5. 'hypotheses: null and alternative hypotheses to be tested.
  6. 'statistic: value of the sample statistic used for testing the null hypothesis.
  7. 'distribution: distribution of the sample statistic, together with its parameter(s).
  8. 'p_value: p-value of the test.

Examples:

Performs an exact t-test with unknown variance. The null hypothesis is H_0: mean=50 against the one sided alternative H_1: mean<50; according to the results, the p-value is too great, there are no evidence for rejecting H_0.

(%i1) load("stats")$
(%i2) data: [78,64,35,45,45,75,43,74,42,42]$
(%i3) test_mean(data,'conflevel=0.9,'alternative='less,'mean=50);
          |                 MEAN TEST
          |
          |            mean_estimate = 54.3
          |
          |              conf_level = 0.9
          |
          | conf_interval = [minf, 61.51314273502712]
          |
(%o3)     |  method = Exact t-test. Unknown variance.
          |
          | hypotheses = H0: mean = 50 , H1: mean < 50
          |
          |       statistic = .8244705235071678
          |
          |       distribution = [student_t, 9]
          |
          |        p_value = .7845100411786889

This time Maxima performs an asymptotic test, based on the Central Limit Theorem. The null hypothesis is H_0: equal(mean, 50) against the two sided alternative H_1: not equal(mean, 50); according to the results, the p-value is very small, H_0 should be rejected in favor of the alternative H_1. Note that, as indicated by the Method component, this procedure should be applied to large samples.

(%i1) load("stats")$
(%i2) test_mean([36,118,52,87,35,256,56,178,57,57,89,34,25,98,35,
              98,41,45,198,54,79,63,35,45,44,75,42,75,45,45,
              45,51,123,54,151],
              'asymptotic=true,'mean=50);
          |                       MEAN TEST
          |
          |           mean_estimate = 74.88571428571429
          |
          |                   conf_level = 0.95
          |
          | conf_interval = [57.72848600856194, 92.04294256286663]
          |
(%o2)     |    method = Large sample z-test. Unknown variance.
          |
          |       hypotheses = H0: mean = 50 , H1: mean # 50
          |
          |             statistic = 2.842831192874313
          |
          |             distribution = [normal, 0, 1]
          |
          |             p_value = .004471474652002261

Function: test_means_difference
    test_means_difference (x1, x2)
    test_means_difference (x1, x2, options ...)

This is the difference of means t-test for two samples. Arguments x1 and x2 are lists or column matrices containing two independent samples. In case of different unknown variances (see options 'dev1, 'dev2 and 'varequal bellow), the degrees of freedom are computed by means of the Welch approximation. It also performs an asymptotic test based on the Central Limit Theorem if option 'asymptotic is set to true.

Options:

The output of function test_means_difference is an inference_result Maxima object showing the following results:

  1. 'diff_estimate: the difference of means estimate.
  2. 'conf_level: confidence level selected by the user.
  3. 'conf_interval: confidence interval for the difference of means.
  4. 'method: inference procedure.
  5. 'hypotheses: null and alternative hypotheses to be tested.
  6. 'statistic: value of the sample statistic used for testing the null hypothesis.
  7. 'distribution: distribution of the sample statistic, together with its parameter(s).
  8. 'p_value: p-value of the test.

Examples:

The equality of means is tested with two small samples x and y, against the alternative H_1: m_1>m_2, being m_1 and m_2 the populations means; variances are unknown and supposed to be different.

(%i1) load("stats")$
(%i2) x: [20.4,62.5,61.3,44.2,11.1,23.7]$
(%i3) y: [1.2,6.9,38.7,20.4,17.2]$
(%i4) test_means_difference(x,y,'alternative='greater);
            |              DIFFERENCE OF MEANS TEST
            |
            |         diff_estimate = 20.31999999999999
            |
            |                 conf_level = 0.95
            |
            |    conf_interval = [- .04597417812882298, inf]
            |
(%o4)       |        method = Exact t-test. Welch approx.
            |
            | hypotheses = H0: mean1 = mean2 , H1: mean1 > mean2
            |
            |           statistic = 1.838004300728477
            |
            |    distribution = [student_t, 8.62758740184604]
            |
            |            p_value = .05032746527991905

The same test as before, but now variances are supposed to be equal.

(%i1) load("stats")$
(%i2) x: [20.4,62.5,61.3,44.2,11.1,23.7]$
(%i3) y: matrix([1.2],[6.9],[38.7],[20.4],[17.2])$
(%i4) test_means_difference(x,y,'alternative='greater,
                                                 'varequal=true);
            |              DIFFERENCE OF MEANS TEST
            |
            |         diff_estimate = 20.31999999999999
            |
            |                 conf_level = 0.95
            |
            |     conf_interval = [- .7722627696897568, inf]
            |
(%o4)       |   method = Exact t-test. Unknown equal variances
            |
            | hypotheses = H0: mean1 = mean2 , H1: mean1 > mean2
            |
            |           statistic = 1.765996124515009
            |
            |           distribution = [student_t, 9]
            |
            |            p_value = .05560320992529344

Function: test_variance
    test_variance (x)
    test_variance (x, options, ...)

This is the variance chi^2-test. Argument x is a list or a column matrix containing a one dimensional sample taken from a normal population.

Options:

The output of function test_variance is an inference_result Maxima object showing the following results:

  1. 'var_estimate: the sample variance.
  2. 'conf_level: confidence level selected by the user.
  3. 'conf_interval: confidence interval for the population variance.
  4. 'method: inference procedure.
  5. 'hypotheses: null and alternative hypotheses to be tested.
  6. 'statistic: value of the sample statistic used for testing the null hypothesis.
  7. 'distribution: distribution of the sample statistic, together with its parameter.
  8. 'p_value: p-value of the test.

Examples:

It is tested whether the variance of a population with unknown mean is equal to or greater than 200.

(%i1) load("stats")$
(%i2) x: [203,229,215,220,223,233,208,228,209]$
(%i3) test_variance(x,'alternative='greater,'variance=200);
             |                  VARIANCE TEST
             |
             |              var_estimate = 110.75
             |
             |                conf_level = 0.95
             |
             |     conf_interval = [57.13433376937479, inf]
             |
(%o3)        | method = Variance Chi-square test. Unknown mean.
             |
             |    hypotheses = H0: var = 200 , H1: var > 200
             |
             |                 statistic = 4.43
             |
             |             distribution = [chi2, 8]
             |
             |           p_value = .8163948512777689

Function: test_variance_ratio
    test_variance_ratio (x1, x2)
    test_variance_ratio (x1, x2, options ...)

This is the variance ratio F-test for two normal populations. Arguments x1 and x2 are lists or column matrices containing two independent samples.

Options:

The output of function test_variance_ratio is an inference_result Maxima object showing the following results:

  1. 'ratio_estimate: the sample variance ratio.
  2. 'conf_level: confidence level selected by the user.
  3. 'conf_interval: confidence interval for the variance ratio.
  4. 'method: inference procedure.
  5. 'hypotheses: null and alternative hypotheses to be tested.
  6. 'statistic: value of the sample statistic used for testing the null hypothesis.
  7. 'distribution: distribution of the sample statistic, together with its parameters.
  8. 'p_value: p-value of the test.

Examples:

The equality of the variances of two normal populations is checked against the alternative that the first is greater than the second.

(%i1) load("stats")$
(%i2) x: [20.4,62.5,61.3,44.2,11.1,23.7]$
(%i3) y: [1.2,6.9,38.7,20.4,17.2]$
(%i4) test_variance_ratio(x,y,'alternative='greater);
              |              VARIANCE RATIO TEST
              |
              |       ratio_estimate = 2.316933391522034
              |
              |               conf_level = 0.95
              |
              |    conf_interval = [.3703504689507268, inf]
              |
(%o4)         | method = Variance ratio F-test. Unknown means.
              |
              | hypotheses = H0: var1 = var2 , H1: var1 > var2
              |
              |         statistic = 2.316933391522034
              |
              |            distribution = [f, 5, 4]
              |
              |          p_value = .2179269692254457

Function: test_proportion
    test_proportion (x, n)
    test_proportion (x, n, options ...)

Inferences on a proportion. Argument x is the number of successes in n trials in a Bernoulli experiment with unknown probability.

Options:

The output of function test_proportion is an inference_result Maxima object showing the following results:

  1. 'sample_proportion: the sample proportion.
  2. 'conf_level: confidence level selected by the user.
  3. 'conf_interval: Wilson confidence interval for the proportion.
  4. 'method: inference procedure.
  5. 'hypotheses: null and alternative hypotheses to be tested.
  6. 'statistic: value of the sample statistic used for testing the null hypothesis.
  7. 'distribution: distribution of the sample statistic, together with its parameters.
  8. 'p_value: p-value of the test.

Examples:

Performs an exact test. The null hypothesis is H_0: p=1/2 against the one sided alternative H_1: p<1/2.

(%i1) load("stats")$
(%i2) test_proportion(45, 103, alternative = less);
         |            PROPORTION TEST              
         |                                         
         | sample_proportion = .4368932038834951   
         |                                         
         |           conf_level = 0.95             
         |                                         
         | conf_interval = [0, 0.522714149150231]  
         |                                         
(%o2)    |     method = Exact binomial test.       
         |                                         
         | hypotheses = H0: p = 0.5 , H1: p < 0.5  
         |                                         
         |             statistic = 45              
         |                                         
         |  distribution = [binomial, 103, 0.5]    
         |                                         
         |      p_value = .1184509388901454 

A two sided asymptotic test. Confidence level is 99/100.

(%i1) load("stats")$
(%i2) fpprintprec:7$
(%i3) test_proportion(45, 103, 
                  conflevel = 99/100, asymptotic=true);
      |                 PROPORTION TEST                  
      |                                                  
      |           sample_proportion = .43689             
      |                                                  
      |                conf_level = 0.99                 
      |                                                  
      |        conf_interval = [.31422, .56749]          
      |                                                  
(%o3) | method = Asympthotic test with Yates correction.
      |                                                  
      |     hypotheses = H0: p = 0.5 , H1: p # 0.5       
      |                                                  
      |               statistic = .43689                 
      |                                                  
      |      distribution = [normal, 0.5, .048872]       
      |                                                  
      |                p_value = .19662

Function: test_proportions_difference
    test_proportions_difference (x1, n1, x2, n2)
    test_proportions_difference (x1, n1, x2, n2, options …)

Inferences on the difference of two proportions. Argument x1 is the number of successes in n1 trials in a Bernoulli experiment in the first population, and x2 and n2 are the corresponding values in the second population. Samples are independent and the test is asymptotic.

Options:

The output of function test_proportions_difference is an inference_result Maxima object showing the following results:

  1. 'proportions: list with the two sample proportions.
  2. 'conf_level: confidence level selected by the user.
  3. 'conf_interval: Confidence interval for the difference of proportions p1 - p2.
  4. 'method: inference procedure and warning message in case of any of the samples sizes is less than 10.
  5. 'hypotheses: null and alternative hypotheses to be tested.
  6. 'statistic: value of the sample statistic used for testing the null hypothesis.
  7. 'distribution: distribution of the sample statistic, together with its parameters.
  8. 'p_value: p-value of the test.

Examples:

A machine produced 10 defective articles in a batch of 250. After some maintenance work, it produces 4 defective in a batch of 150. In order to know if the machine has improved, we test the null hypothesis H0:p1=p2, against the alternative H0:p1>p2, where p1 and p2 are the probabilities for one produced article to be defective before and after maintenance. According to the p value, there is not enough evidence to accept the alternative.

(%i1) load("stats")$
(%i2) fpprintprec:7$
(%i3) test_proportions_difference(10, 250, 4, 150,
                                alternative = greater);
      |       DIFFERENCE OF PROPORTIONS TEST         
      |                                              
      |       proportions = [0.04, .02666667]        
      |                                              
      |              conf_level = 0.95               
      |                                              
      |      conf_interval = [- .02172761, 1]        
      |                                              
(%o3) | method = Asymptotic test. Yates correction.  
      |                                              
      |   hypotheses = H0: p1 = p2 , H1: p1 > p2     
      |                                              
      |            statistic = .01333333             
      |                                              
      |    distribution = [normal, 0, .01898069]     
      |                                              
      |             p_value = .2411936 

Exact standard deviation of the asymptotic normal distribution when the data are unknown.

(%i1) load("stats")$
(%i2) stats_numer: false$
(%i3) sol: test_proportions_difference(x1,n1,x2,n2)$
(%i4) last(take_inference('distribution,sol));
               1    1                  x2 + x1
              (-- + --) (x2 + x1) (1 - -------)
               n2   n1                 n2 + n1
(%o4)    sqrt(---------------------------------)
                           n2 + n1

Function: test_sign
    test_sign (x)
    test_sign (x, options …)

This is the non parametric sign test for the median of a continuous population. Argument x is a list or a column matrix containing a one dimensional sample.

Options:

The output of function test_sign is an inference_result Maxima object showing the following results:

  1. 'med_estimate: the sample median.
  2. 'method: inference procedure.
  3. 'hypotheses: null and alternative hypotheses to be tested.
  4. 'statistic: value of the sample statistic used for testing the null hypothesis.
  5. 'distribution: distribution of the sample statistic, together with its parameter(s).
  6. 'p_value: p-value of the test.

Examples:

Checks whether the population from which the sample was taken has median 6, against the alternative H_1: median > 6.

(%i1) load("stats")$
(%i2) x: [2,0.1,7,1.8,4,2.3,5.6,7.4,5.1,6.1,6]$
(%i3) test_sign(x,'median=6,'alternative='greater);
               |                  SIGN TEST
               |
               |              med_estimate = 5.1
               |
               |      method = Non parametric sign test.
               |
(%o3)          | hypotheses = H0: median = 6 , H1: median > 6
               |
               |                statistic = 7
               |
               |      distribution = [binomial, 10, 0.5]
               |
               |         p_value = .05468749999999989

Function: test_signed_rank
    test_signed_rank (x)
    test_signed_rank (x, options …)

This is the Wilcoxon signed rank test to make inferences about the median of a continuous population. Argument x is a list or a column matrix containing a one dimensional sample. Performs normal approximation if the sample size is greater than 20, or if there are zeroes or ties.

See also pdf_rank_test and cdf_rank_test

Options:

The output of function test_signed_rank is an inference_result Maxima object with the following results:

  1. 'med_estimate: the sample median.
  2. 'method: inference procedure.
  3. 'hypotheses: null and alternative hypotheses to be tested.
  4. 'statistic: value of the sample statistic used for testing the null hypothesis.
  5. 'distribution: distribution of the sample statistic, together with its parameter(s).
  6. 'p_value: p-value of the test.

Examples:

Checks the null hypothesis H_0: median = 15 against the alternative H_1: median > 15. This is an exact test, since there are no ties.

(%i1) load("stats")$
(%i2) x: [17.1,15.9,13.7,13.4,15.5,17.6]$
(%i3) test_signed_rank(x,median=15,alternative=greater);
                 |             SIGNED RANK TEST
                 |
                 |           med_estimate = 15.7
                 |
                 |           method = Exact test
                 |
(%o3)            | hypotheses = H0: med = 15 , H1: med > 15
                 |
                 |              statistic = 14
                 |
                 |     distribution = [signed_rank, 6]
                 |
                 |            p_value = 0.28125

Checks the null hypothesis H_0: equal(median, 2.5) against the alternative H_1: not equal(median, 2.5). This is an approximated test, since there are ties.

(%i1) load("stats")$
(%i2) y:[1.9,2.3,2.6,1.9,1.6,3.3,4.2,4,2.4,2.9,1.5,3,2.9,4.2,3.1]$
(%i3) test_signed_rank(y,median=2.5);
             |                 SIGNED RANK TEST
             |
             |                med_estimate = 2.9
             |
             |          method = Asymptotic test. Ties
             |
(%o3)        |    hypotheses = H0: med = 2.5 , H1: med # 2.5
             |
             |                 statistic = 76.5
             |
             | distribution = [normal, 60.5, 17.58195097251724]
             |
             |           p_value = .3628097734643669

Function: test_rank_sum
    test_rank_sum (x1, x2)
    test_rank_sum (x1, x2, option)

This is the Wilcoxon-Mann-Whitney test for comparing the medians of two continuous populations. The first two arguments x1 and x2 are lists or column matrices with the data of two independent samples. Performs normal approximation if any of the sample sizes is greater than 10, or if there are ties.

Option:

The output of function test_rank_sum is an inference_result Maxima object with the following results:

  1. 'method: inference procedure.
  2. 'hypotheses: null and alternative hypotheses to be tested.
  3. 'statistic: value of the sample statistic used for testing the null hypothesis.
  4. 'distribution: distribution of the sample statistic, together with its parameters.
  5. 'p_value: p-value of the test.

Examples:

Checks whether populations have similar medians. Samples sizes are small and an exact test is made.

(%i1) load("stats")$
(%i2) x:[12,15,17,38,42,10,23,35,28]$
(%i3) y:[21,18,25,14,52,65,40,43]$
(%i4) test_rank_sum(x,y);
              |                 RANK SUM TEST
              |
              |              method = Exact test
              |
              | hypotheses = H0: med1 = med2 , H1: med1 # med2
(%o4)         |
              |                 statistic = 22
              |
              |        distribution = [rank_sum, 9, 8]
              |
              |          p_value = .1995886466474702

Now, with greater samples and ties, the procedure makes normal approximation. The alternative hypothesis is H_1: median1 < median2.

(%i1) load("stats")$
(%i2) x: [39,42,35,13,10,23,15,20,17,27]$
(%i3) y: [20,52,66,19,41,32,44,25,14,39,43,35,19,56,27,15]$
(%i4) test_rank_sum(x,y,'alternative='less);
             |                  RANK SUM TEST
             |
             |          method = Asymptotic test. Ties
             |
             |  hypotheses = H0: med1 = med2 , H1: med1 < med2
(%o4)        |
             |                 statistic = 48.5
             |
             | distribution = [normal, 79.5, 18.95419580097078]
             |
             |           p_value = .05096985666598441

Function: test_normality (x)

Shapiro-Wilk test for normality. Argument x is a list of numbers, and sample size must be greater than 2 and less or equal than 5000, otherwise, function test_normality signals an error message.

Reference:

[1] Algorithm AS R94, Applied Statistics (1995), vol.44, no.4, 547-551

The output of function test_normality is an inference_result Maxima object with the following results:

  1. 'statistic: value of the W statistic.
  2. 'p_value: p-value under normal assumption.

Examples:

Checks for the normality of a population, based on a sample of size 9.

(%i1) load("stats")$
(%i2) x:[12,15,17,38,42,10,23,35,28]$
(%i3) test_normality(x);
                       |      SHAPIRO - WILK TEST
                       |
(%o3)                  | statistic = .9251055695162436
                       |
                       |  p_value = .4361763918860381

Function: linear_regression
    linear_regression (x)
    linear_regression (x option)

Multivariate linear regression, y_i = b0 + b1*x_1i + b2*x_2i + ... + bk*x_ki + u_i, where u_i are N(0,sigma) independent random variables. Argument x must be a matrix with more than one column. The last column is considered as the responses (y_i).

Option:

The output of function linear_regression is an inference_result Maxima object with the following results:

  1. 'b_estimation: regression coefficients estimates.
  2. 'b_covariances: covariance matrix of the regression coefficients estimates.
  3. b_conf_int: confidence intervals of the regression coefficients.
  4. b_statistics: statistics for testing coefficient.
  5. b_p_values: p-values for coefficient tests.
  6. b_distribution: probability distribution for coefficient tests.
  7. v_estimation: unbiased variance estimator.
  8. v_conf_int: variance confidence interval.
  9. v_distribution: probability distribution for variance test.
  10. residuals: residuals.
  11. adc: adjusted determination coefficient.
  12. aic: Akaike’s information criterion.
  13. bic: Bayes’s information criterion.

Only items 1, 4, 5, 6, 7, 8, 9 and 11 above, in this order, are shown by default. The rest remain hidden until the user makes use of functions items_inference and take_inference.

Example:

Fitting a linear model to a trivariate sample. The last column is considered as the responses (y_i).

(%i2) load("stats")$
(%i3) X:matrix(
    [58,111,64],[84,131,78],[78,158,83],
    [81,147,88],[82,121,89],[102,165,99],
    [85,174,101],[102,169,102])$
(%i4) fpprintprec: 4$
(%i5) res: linear_regression(X);
             |       LINEAR REGRESSION MODEL         
             |                                       
             | b_estimation = [9.054, .5203, .2397]  
             |                                       
             | b_statistics = [.6051, 2.246, 1.74]   
             |                                       
             | b_p_values = [.5715, .07466, .1423]   
             |                                       
(%o5)        |   b_distribution = [student_t, 5]     
             |                                       
             |         v_estimation = 35.27          
             |                                       
             |     v_conf_int = [13.74, 212.2]       
             |                                       
             |      v_distribution = [chi2, 5]       
             |                                       
             |             adc = .7922               
(%i6) items_inference(res);
(%o6) [b_estimation, b_covariances, b_conf_int, b_statistics, 
b_p_values, b_distribution, v_estimation, v_conf_int, 
v_distribution, residuals, adc, aic, bic]
(%i7) take_inference('b_covariances, res);
                  [  223.9    - 1.12   - .8532  ]
                  [                             ]
(%o7)             [ - 1.12    .05367   - .02305 ]
                  [                             ]
                  [ - .8532  - .02305   .01898  ]
(%i8) take_inference('bic, res);
(%o8)                          30.98
(%i9) load("draw")$
(%i10) draw2d(
    points_joined = true,
    grid = true,
    points(take_inference('residuals, res)) )$


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84.4 Functions and Variables for special distributions

Function: pdf_signed_rank (x, n)

Probability density function of the exact distribution of the signed rank statistic. Argument x is a real number and n a positive integer.

See also test_signed_rank.

Function: cdf_signed_rank (x, n)

Cumulative density function of the exact distribution of the signed rank statistic. Argument x is a real number and n a positive integer.

See also test_signed_rank.

Function: pdf_rank_sum (x, n, m)

Probability density function of the exact distribution of the rank sum statistic. Argument x is a real number and n and m are both positive integers.

See also test_rank_sum.

Function: cdf_rank_sum (x, n, m)

Cumulative density function of the exact distribution of the rank sum statistic. Argument x is a real number and n and m are both positive integers.

See also test_rank_sum.


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85 stirling


Previous: , Up: stirling-pkg   [Contents][Index]

85.1 Functions and Variables for stirling

Function: stirling
    stirling (z,n)
    stirling (z,n,pred)

Replace gamma(x) with the O(1/x^(2n-1)) Stirling formula. when n isn’t a nonnegative integer, signal an error. With the optional third argument pred, the Stirling formula is applied only when pred is true.

Reference: Abramowitz & Stegun, " Handbook of mathematical functions", 6.1.40.

Examples:

(%i1) load ("stirling")$

(%i2) stirling(gamma(%alpha+x)/gamma(x),1);
       1/2 - x             x + %alpha - 1/2
(%o2) x        (x + %alpha)
                                   1           1
                            --------------- - ---- - %alpha
                            12 (x + %alpha)   12 x
                          %e
(%i3) taylor(%,x,inf,1);
                    %alpha       2    %alpha
          %alpha   x       %alpha  - x       %alpha
(%o3)/T/ x       + -------------------------------- + . . .
                                 2 x
(%i4) map('factor,%);
                                       %alpha - 1
         %alpha   (%alpha - 1) %alpha x
(%o4)   x       + -------------------------------
                                  2

The function stirling knows the difference between the variable ’gamma’ and the function gamma:

(%i5) stirling(gamma + gamma(x),0);
                                    x - 1/2   - x
(%o5)    gamma + sqrt(2) sqrt(%pi) x        %e
(%i6) stirling(gamma(y) + gamma(x),0);
                         y - 1/2   - y
(%o6) sqrt(2) sqrt(%pi) y        %e
                                              x - 1/2   - x
                         + sqrt(2) sqrt(%pi) x        %e

To apply the Stirling formula only to terms that involve the variable k, use an optional third argument; for example

(%i7) makegamma(pochhammer(a,k)/pochhammer(b,k));
(%o7) (gamma(b)*gamma(k+a))/(gamma(a)*gamma(k+b))
(%i8) stirling(%,1, lambda([s], not(freeof(k,s))));
(%o8) (%e^(b-a)*gamma(b)*(k+a)^(k+a-1/2)*(k+b)^(-k-b+1/2))/gamma(a)

The terms gamma(a) and gamma(b) are free of k, so the Stirling formula was not applied to these two terms.

To use this function write first load("stirling").


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86 stringproc


Next: , Previous: , Up: stringproc-pkg   [Contents][Index]

86.1 Introduction to String Processing

The package stringproc contains functions for processing strings and characters including formatting, encoding and data streams. This package is completed by some tools for cryptography, e.g. base64 and hash functions.

It can be directly loaded via load(stringproc) or automatically by using one of its functions.

For questions and bug reports please contact the author. The following command prints his e-mail-address.

printf(true, "~{~a~}@gmail.com", split(sdowncase("Volker van Nek")))$

A string is constructed by typing e.g. "Text". When the option variable stringdisp is set to false, which is the default, the double quotes won’t be printed. stringp is a test, if an object is a string.

(%i1) str: "Text";
(%o1)                         Text
(%i2) stringp(str);
(%o2)                         true

Characters are represented by a string of length 1. charp is the corresponding test.

(%i1) char: "e";
(%o1)                           e
(%i2) charp(char);
(%o2)                         true

In Maxima position indices in strings are like in list 1-indexed which results to the following consistency.

(%i1) is(charat("Lisp",1) = charlist("Lisp")[1]);
(%o1)                         true

A string may contain Maxima expressions. These can be parsed with parse_string.

(%i1) map(parse_string, ["42" ,"sqrt(2)", "%pi"]);
(%o1)                   [42, sqrt(2), %pi]
(%i2) map('float, %);
(%o2)        [42.0, 1.414213562373095, 3.141592653589793]

Strings can be processed as characters or in binary form as octets. Functions for conversions are string_to_octets and octets_to_string. Usable encodings depend on the platform, the application and the underlying Lisp. (The following shows Maxima in GNU/Linux, compiled with SBCL.)

(%i1) obase: 16.$
(%i2) string_to_octets("$£€", "cp1252");
(%o2)                     [24, 0A3, 80]
(%i3) string_to_octets("$£€", "utf-8");
(%o3)               [24, 0C2, 0A3, 0E2, 82, 0AC]

Strings may be written to character streams or as octets to binary streams. The following example demonstrates file in and output of characters.

openw returns an output stream to a file, printf writes formatted to that file and by e.g. close all characters contained in the stream are written to the file.

(%i1) s: openw("file.txt");
(%o1)                #<output stream file.txt>
(%i2) printf(s, "~%~d ~f ~a ~a ~f ~e ~a~%", 
42, 1.234, sqrt(2), %pi, 1.0e-2, 1.0e-2, 1.0b-2)$
(%i3) close(s)$

openr then returns an input stream from the previously used file and readline returns the line read as a string. The string may be tokenized by e.g. split or tokens and finally parsed by parse_string.

(%i4) s: openr("file.txt");
(%o4)                 #<input stream file.txt>
(%i5) readline(s);
(%o5)          42 1.234 sqrt(2) %pi 0.01 1.0E-2 1.0b-2
(%i6) map(parse_string, split(%));
(%o6)       [42, 1.234, sqrt(2), %pi, 0.01, 0.01, 1.0b-2]
(%i7) close(s)$
·

Category: Strings · Category: Share packages · Category: Package stringproc


Next: , Previous: , Up: stringproc-pkg   [Contents][Index]

86.2 Input and Output

Example: Formatted printing to a file.

(%i1) s: openw("file.txt");
(%o1)                      #<output stream file.txt>
(%i2) control: 
"~2tAn atom: ~20t~a~%~2tand a list: ~20t~{~r ~}~%~2t\
and an integer: ~20t~d~%"$
(%i3) printf( s,control, 'true,[1,2,3],42 )$
(%o3)                                false
(%i4) close(s);
(%o4)                                true
(%i5) s: openr("file.txt");
(%o5)                      #<input stream file.txt>
(%i6) while stringp( tmp:readline(s) ) do print(tmp)$
  An atom:          true 
  and a list:       one two three  
  and an integer:   42 
(%i7) close(s)$
Function: close (stream)

Closes stream and returns true if stream had been open.

Function: flength (stream)

stream has to be an open stream from or to a file. flength then returns the number of bytes which are currently present in this file.

Example: See writebyte .

Function: flush_output (stream)

Flushes stream where stream has to be an output stream to a file.

Example: See writebyte .

Function: fposition
    fposition (stream)
    fposition (stream, pos)

Returns the current position in stream, if pos is not used. If pos is used, fposition sets the position in stream. stream has to be a stream from or to a file and pos has to be a positive number.

Positions in data streams are like in strings or lists 1-indexed, i.e. the first element in stream is in position 1.

Function: freshline
    freshline ()
    freshline (stream)

Writes a new line to the standard output stream if the position is not at the beginning of a line und returns true. Using the optional argument stream the new line is written to that stream. There are some cases, where freshline() does not work as expected.

See also newline.

Function: get_output_stream_string (stream)

Returns a string containing all the characters currently present in stream which must be an open string-output stream. The returned characters are removed from stream.

Example: See make_string_output_stream .

Function: make_string_input_stream
    make_string_input_stream (string)
    make_string_input_stream (string, start)
    make_string_input_stream (string, start, end)

Returns an input stream which contains parts of string and an end of file. Without optional arguments the stream contains the entire string and is positioned in front of the first character. start and end define the substring contained in the stream. The first character is available at position 1.

(%i1) istream : make_string_input_stream("text", 1, 4);
(%o1)              #<string-input stream from "text">
(%i2) (while (c : readchar(istream)) # false do sprint(c), newline())$
t e x 
(%i3) close(istream)$

Function: make_string_output_stream ()

Returns an output stream that accepts characters. Characters currently present in this stream can be retrieved by get_output_stream_string.

(%i1) ostream : make_string_output_stream();
(%o1)               #<string-output stream 09622ea0>
(%i2) printf(ostream, "foo")$

(%i3) printf(ostream, "bar")$

(%i4) string : get_output_stream_string(ostream);
(%o4)                            foobar
(%i5) printf(ostream, "baz")$

(%i6) string : get_output_stream_string(ostream);
(%o6)                              baz
(%i7) close(ostream)$

Function: newline
    newline ()
    newline (stream)

Writes a new line to the standard output stream. Using the optional argument stream the new line is written to that stream. There are some cases, where newline() does not work as expected.

See sprint for an example of using newline().

Function: opena (file)

Returns a character output stream to file. If an existing file is opened, opena appends elements at the end of file.

For binary output see opena_binary .

Function: openr
    openr (file)
    openr (file, encoding)

Returns a character input stream to file. openr assumes that file already exists. If reading the file results in a lisp error about its encoding passing the correct string as the argument encoding might help. The available encodings and their names depend on the lisp being used. For sbcl a list of suitable strings can be found at http://www.sbcl.org/manual/#External-Formats.

For binary input see openr_binary . See also close and openw.

(%i1) istream : openr("data.txt","EUC-JP");
(%o1)     #<FD-STREAM for "file /home/gunter/data.txt" {10099A3AE3}>
(%i2) close(istream);
(%o2)                                true

Function: openw (file)

Returns a character output stream to file. If file does not exist, it will be created. If an existing file is opened, openw destructively modifies file.

For binary output see openw_binary .

See also close and openr.

Function: printf
    printf (dest, string)
    printf (dest, string, expr_1, ..., expr_n)

Produces formatted output by outputting the characters of control-string string and observing that a tilde introduces a directive. The character after the tilde, possibly preceded by prefix parameters and modifiers, specifies what kind of formatting is desired. Most directives use one or more elements of the arguments expr_1, ..., expr_n to create their output.

If dest is a stream or true, then printf returns false. Otherwise, printf returns a string containing the output. By default the streams stdin, stdout and stderr are defined. If maxima is running as a server (which is the normal case if maxima communicating with a graphical user interface) setup-client will define old_stdout and old_stderr, too.

printf provides the Common Lisp function format in Maxima. The following example illustrates the general relation between these two functions.

(%i1) printf(true, "R~dD~d~%", 2, 2);
R2D2
(%o1)                                false
(%i2) :lisp (format t "R~dD~d~%" 2 2)
R2D2
NIL

The following description is limited to a rough sketch of the possibilities of printf. The Lisp function format is described in detail in many reference books. Of good help is e.g. the free available online-manual "Common Lisp the Language" by Guy L. Steele. See chapter 22.3.3 there.

   ~%       new line
   ~&       fresh line
   ~t       tab
   ~$       monetary
   ~d       decimal integer
   ~b       binary integer
   ~o       octal integer
   ~x       hexadecimal integer
   ~br      base-b integer
   ~r       spell an integer
   ~p       plural
   ~f       floating point
   ~e       scientific notation
   ~g       ~f or ~e, depending upon magnitude
   ~h       bigfloat
   ~a       uses Maxima function string
   ~s       like ~a, but output enclosed in "double quotes"
   ~~       ~
   ~<       justification, ~> terminates
   ~(       case conversion, ~) terminates 
   ~[       selection, ~] terminates 
   ~{       iteration, ~} terminates

The directive ~h for bigfloat is no Lisp-standard and is therefore illustrated below.

Note that the directive ~* is not supported.

If dest is a stream or true, then printf returns false. Otherwise, printf returns a string containing the output.

(%i1) printf( false, "~a ~a ~4f ~a ~@r", 
              "String",sym,bound,sqrt(12),144), bound = 1.234;
(%o1)                 String sym 1.23 2*sqrt(3) CXLIV
(%i2) printf( false,"~{~a ~}",["one",2,"THREE"] );
(%o2)                          one 2 THREE 
(%i3) printf(true,"~{~{~9,1f ~}~%~}",mat ),
          mat = args(matrix([1.1,2,3.33],[4,5,6],[7,8.88,9]))$
      1.1       2.0       3.3 
      4.0       5.0       6.0 
      7.0       8.9       9.0 
(%i4) control: "~:(~r~) bird~p ~[is~;are~] singing."$
(%i5) printf( false,control, n,n,if n=1 then 1 else 2 ), n=2;
(%o5)                    Two birds are singing.

The directive ~h has been introduced to handle bigfloats.

~w,d,e,x,o,p@H
 w : width
 d : decimal digits behind floating point
 e : minimal exponent digits
 x : preferred exponent
 o : overflow character
 p : padding character
 @ : display sign for positive numbers
(%i1) fpprec : 1000$
(%i2) printf(true, "|~h|~%", 2.b0^-64)$
|0.0000000000000000000542101086242752217003726400434970855712890625|
(%i3) fpprec : 26$
(%i4) printf(true, "|~h|~%", sqrt(2))$
|1.4142135623730950488016887|
(%i5) fpprec : 24$
(%i6) printf(true, "|~h|~%", sqrt(2))$
|1.41421356237309504880169|
(%i7) printf(true, "|~28h|~%", sqrt(2))$
|   1.41421356237309504880169|
(%i8) printf(true, "|~28,,,,,'*h|~%", sqrt(2))$
|***1.41421356237309504880169|
(%i9) printf(true, "|~,18h|~%", sqrt(2))$
|1.414213562373095049|
(%i10) printf(true, "|~,,,-3h|~%", sqrt(2))$
|1414.21356237309504880169b-3|
(%i11) printf(true, "|~,,2,-3h|~%", sqrt(2))$
|1414.21356237309504880169b-03|
(%i12) printf(true, "|~20h|~%", sqrt(2))$
|1.41421356237309504880169|
(%i13) printf(true, "|~20,,,,'+h|~%", sqrt(2))$
|++++++++++++++++++++|

Function: readbyte (stream)

Removes and returns the first byte in stream which must be a binary input stream. If the end of file is encountered readbyte returns false.

Example: Read the first 16 bytes from a file encrypted with AES in OpenSSL.

(%i1) ibase: obase: 16.$

(%i2) in: openr_binary("msg.bin");
(%o2)                       #<input stream msg.bin>
(%i3) (L:[],  thru 16. do push(readbyte(in), L),  L:reverse(L));
(%o3) [53, 61, 6C, 74, 65, 64, 5F, 5F, 88, 56, 0DE, 8A, 74, 0FD, 0AD, 0F0]
(%i4) close(in);
(%o4)                                true
(%i5) map(ascii, rest(L,-8));
(%o5)                      [S, a, l, t, e, d, _, _]
(%i6) salt: octets_to_number(rest(L,8));
(%o6)                          8856de8a74fdadf0

Function: readchar (stream)

Removes and returns the first character in stream. If the end of file is encountered readchar returns false.

Example: See make_string_input_stream.

Function: readline (stream)

Returns a string containing all characters starting at the current position in stream up to the end of the line or false if the end of the file is encountered.

Function: sprint (expr_1, …, expr_n)

Evaluates and displays its arguments one after the other ‘on a line’ starting at the leftmost position. The expressions are printed with a space character right next to the number, and it disregards line length. newline() might be used for line breaking.

Example: Sequential printing with sprint. Creating a new line with newline().

(%i1) for n:0 thru 19 do sprint(fib(n))$
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181
(%i2) for n:0 thru 22 do ( 
         sprint(fib(n)), 
         if mod(n,10) = 9 then newline() )$
0 1 1 2 3 5 8 13 21 34 
55 89 144 233 377 610 987 1597 2584 4181 
6765 10946 17711 

Function: writebyte (byte, stream)

Writes byte to stream which must be a binary output stream. writebyte returns byte.

Example: Write some bytes to a binary file output stream. In this example all bytes correspond to printable characters and are printed by printfile. The bytes remain in the stream until flush_output or close have been called.

(%i1) ibase: obase: 16.$

(%i2) bytes: map(cint, charlist("GNU/Linux"));
(%o2)                [47, 4E, 55, 2F, 4C, 69, 6E, 75, 78]
(%i3) out: openw_binary("test.bin");
(%o3)                      #<output stream test.bin>
(%i4) for i thru 3 do writebyte(bytes[i], out);
(%o4)                                done
(%i5) printfile("test.bin")$

(%i6) flength(out);
(%o6)                                  0
(%i7) flush_output(out);
(%o7)                                true
(%i8) flength(out);
(%o8)                                  3
(%i9) printfile("test.bin")$
GNU
(%i0A) for b in rest(bytes,3) do writebyte(b, out);
(%o0A)                               done
(%i0B) close(out);
(%o0B)                               true
(%i0C) printfile("test.bin")$
GNU/Linux


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86.3 Characters

Characters are strings of length 1.

Function: adjust_external_format ()

Prints information about the current external format of the Lisp reader and in case the external format encoding differs from the encoding of the application which runs Maxima adjust_external_format tries to adjust the encoding or prints some help or instruction. adjust_external_format returns true when the external format has been changed and false otherwise.

Functions like cint, unicode, octets_to_string and string_to_octets need UTF-8 as the external format of the Lisp reader to work properly over the full range of Unicode characters.

Examples (Maxima on Windows, March 2016): Using adjust_external_format when the default external format is not equal to the encoding provided by the application.

1. Command line Maxima

In case a terminal session is preferred it is recommended to use Maxima compiled with SBCL. Here Unicode support is provided by default and calls to adjust_external_format are unnecessary.

If Maxima is compiled with CLISP or GCL it is recommended to change the terminal encoding from CP850 to CP1252. adjust_external_format prints some help.

CCL reads UTF-8 while the terminal input is CP850 by default. CP1252 is not supported by CCL. adjust_external_format prints instructions for changing the terminal encoding and external format both to iso-8859-1.

2. wxMaxima

In wxMaxima SBCL reads CP1252 by default but the input from the application is UTF-8 encoded. Adjustment is needed.

Calling adjust_external_format and restarting Maxima permanently changes the default external format to UTF-8.

(%i1)adjust_external_format();
The line
(setf sb-impl::*default-external-format* :utf-8)
has been appended to the init file
C:/Users/Username/.sbclrc
Please restart Maxima to set the external format to UTF-8.
(%i1) false

Restarting Maxima.

(%i1) adjust_external_format();
The external format is currently UTF-8
and has not been changed.
(%i1) false

Function: alphacharp (char)

Returns true if char is an alphabetic character.

To identify a non-US-ASCII character as an alphabetic character the underlying Lisp must provide full Unicode support. E.g. a German umlaut is detected as an alphabetic character with SBCL in GNU/Linux but not with GCL. (In Windows Maxima, when compiled with SBCL, must be set to UTF-8. See adjust_external_format for more.)

Example: Examination of non-US-ASCII characters.

The underlying Lisp (SBCL, GNU/Linux) is able to convert the typed character into a Lisp character and to examine.

(%i1) alphacharp("ü");
(%o1)                          true

In GCL this is not possible. An error break occurs.

(%i1) alphacharp("u");
(%o1)                          true
(%i2) alphacharp("ü");

package stringproc: ü cannot be converted into a Lisp character.
 -- an error.

Function: alphanumericp (char)

Returns true if char is an alphabetic character or a digit (only corresponding US-ASCII characters are regarded as digits).

Note: See remarks on alphacharp.

Function: ascii (int)

Returns the US-ASCII character corresponding to the integer int which has to be less than 128.

See unicode for converting code points larger than 127.

Examples:

(%i1) for n from 0 thru 127 do ( 
        ch: ascii(n), 
        if alphacharp(ch) then sprint(ch),
        if n = 96 then newline() )$
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 
a b c d e f g h i j k l m n o p q r s t u v w x y z

Function: cequal (char_1, char_2)

Returns true if char_1 and char_2 are the same character.

Function: cequalignore (char_1, char_2)

Like cequal but ignores case which is only possible for non-US-ASCII characters when the underlying Lisp is able to recognize a character as an alphabetic character. See remarks on alphacharp.

Function: cgreaterp (char_1, char_2)

Returns true if the code point of char_1 is greater than the code point of char_2.

Function: cgreaterpignore (char_1, char_2)

Like cgreaterp but ignores case which is only possible for non-US-ASCII characters when the underlying Lisp is able to recognize a character as an alphabetic character. See remarks on alphacharp.

Function: charp (obj)

Returns true if obj is a Maxima-character. See introduction for example.

Function: cint (char)

Returns the Unicode code point of char which must be a Maxima character, i.e. a string of length 1.

Examples: The hexadecimal code point of some characters (Maxima with SBCL on GNU/Linux).

(%i1) obase: 16.$
(%i2) map(cint, ["$","£","€"]);
(%o2)                           [24, 0A3, 20AC]

Warning: It is not possible to enter characters corresponding to code points larger than 16 bit in wxMaxima with SBCL on Windows when the external format has not been set to UTF-8. See adjust_external_format.

CMUCL doesn’t process these characters as one character. cint then returns false. Converting a character to a code point via UTF-8-octets may serve as a workaround:

utf8_to_unicode(string_to_octets(character));

See utf8_to_unicode, string_to_octets.

Function: clessp (char_1, char_2)

Returns true if the code point of char_1 is less than the code point of char_2.

Function: clesspignore (char_1, char_2)

Like clessp but ignores case which is only possible for non-US-ASCII characters when the underlying Lisp is able to recognize a character as an alphabetic character. See remarks on alphacharp.

Function: constituent (char)

Returns true if char is a graphic character but not a space character. A graphic character is a character one can see, plus the space character. (constituent is defined by Paul Graham. See Paul Graham, ANSI Common Lisp, 1996, page 67.)

(%i1) for n from 0 thru 255 do ( 
tmp: ascii(n), if constituent(tmp) then sprint(tmp) )$
! " #  %  ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B
C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c
d e f g h i j k l m n o p q r s t u v w x y z { | } ~

Function: digitcharp (char)

Returns true if char is a digit where only the corresponding US-ASCII-character is regarded as a digit.

Function: lowercasep (char)

Returns true if char is a lowercase character.

Note: See remarks on alphacharp.

Variable: newline

The newline character (ASCII-character 10).

Variable: space

The space character.

Variable: tab

The tab character.

Function: unicode (arg)

Returns the character defined by arg which might be a Unicode code point or a name string if the underlying Lisp provides full Unicode support.

Example: Characters defined by hexadecimal code points (Maxima with SBCL on GNU/Linux).

(%i1) ibase: 16.$
(%i2) map(unicode, [24, 0A3, 20AC]);
(%o2)                            [$, £, €]

Warning: In wxMaxima with SBCL on Windows it is not possible to convert code points larger than 16 bit to characters when the external format has not been set to UTF-8. See adjust_external_format for more information.

CMUCL doesn’t process code points larger than 16 bit. In these cases unicode returns false. Converting a code point to a character via UTF-8 octets may serve as a workaround:

octets_to_string(unicode_to_utf8(code_point));

See octets_to_string, unicode_to_utf8.

In case the underlying Lisp provides full Unicode support the character might be specified by its name. The following is possible in ECL, CLISP and SBCL, where in SBCL on Windows the external format has to be set to UTF-8. unicode(name) is supported by CMUCL too but again limited to 16 bit characters.

The string argument to unicode is basically the same string returned by printf using the "~@c" specifier. But as shown below the prefix "#\" must be omitted. Underlines might be replaced by spaces and uppercase letters by lowercase ones.

Example (continued): Characters defined by names (Maxima with SBCL on GNU/Linux).

(%i3) printf(false, "~@c", unicode(0DF));
(%o3)                    #\LATIN_SMALL_LETTER_SHARP_S
(%i4) unicode("LATIN_SMALL_LETTER_SHARP_S");
(%o4)                                  ß
(%i5) unicode("Latin small letter sharp s");
(%o5)                                  ß

Function: unicode_to_utf8 (code_point)

Returns a list containing the UTF-8 code corresponding to the Unicode code_point.

Examples: Converting Unicode code points to UTF-8 and vice versa.

(%i1) ibase: obase: 16.$
(%i2) map(cint, ["$","£","€"]);
(%o2)                           [24, 0A3, 20AC]
(%i3) map(unicode_to_utf8, %);
(%o3)                 [[24], [0C2, 0A3], [0E2, 82, 0AC]]
(%i4) map(utf8_to_unicode, %);
(%o4)                           [24, 0A3, 20AC]

Function: uppercasep (char)

Returns true if char is an uppercase character.

Note: See remarks on alphacharp.

Variable: us_ascii_only

This option variable affects Maxima when the character encoding provided by the application which runs Maxima is UTF-8 but the external format of the Lisp reader is not equal to UTF-8.

On GNU/Linux this is true when Maxima is built with GCL and on Windows in wxMaxima with GCL- and SBCL-builds. With SBCL it is recommended to change the external format to UTF-8. Setting us_ascii_only is unnecessary then. See adjust_external_format for details.

us_ascii_only is false by default. Maxima itself then (i.e. in the above described situation) parses the UTF-8 encoding.

When us_ascii_only is set to true it is assumed that all strings used as arguments to string processing functions do not contain Non-US-ASCII characters. Given that promise, Maxima avoids parsing UTF-8 and strings can be processed more efficiently.

Function: utf8_to_unicode (list)

Returns a Unicode code point corresponding to the list which must contain the UTF-8 encoding of a single character.

Examples: See unicode_to_utf8.


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86.4 String Processing

Position indices in strings are 1-indexed like in Maxima lists. See example in charat.

Function: charat (string, n)

Returns the n-th character of string. The first character in string is returned with n = 1.

(%i1) charat("Lisp",1);
(%o1)                           L
(%i2) charlist("Lisp")[1];
(%o2)                           L

Function: charlist (string)

Returns the list of all characters in string.

(%i1) charlist("Lisp");
(%o1)                     [L, i, s, p]

Function: eval_string (str)

Parse the string str as a Maxima expression and evaluate it. The string str may or may not have a terminator (dollar sign $ or semicolon ;). Only the first expression is parsed and evaluated, if there is more than one.

Complain if str is not a string.

Examples:

(%i1) eval_string ("foo: 42; bar: foo^2 + baz");
(%o1)                       42
(%i2) eval_string ("(foo: 42, bar: foo^2 + baz)");
(%o2)                   baz + 1764

See also parse_string.

Function: parse_string (str)

Parse the string str as a Maxima expression (do not evaluate it). The string str may or may not have a terminator (dollar sign $ or semicolon ;). Only the first expression is parsed, if there is more than one.

Complain if str is not a string.

Examples:

(%i1) parse_string ("foo: 42; bar: foo^2 + baz");
(%o1)                    foo : 42
(%i2) parse_string ("(foo: 42, bar: foo^2 + baz)");
                                   2
(%o2)          (foo : 42, bar : foo  + baz)

See also eval_string.

Function: scopy (string)

Returns a copy of string as a new string.

Function: sdowncase
    sdowncase (string)
    sdowncase (string, start)
    sdowncase (string, start, end)

Like supcase but uppercase characters are converted to lowercase.

Function: sequal (string_1, string_2)

Returns true if string_1 and string_2 contain the same sequence of characters.

Function: sequalignore (string_1, string_2)

Like sequal but ignores case which is only possible for non-US-ASCII characters when the underlying Lisp is able to recognize a character as an alphabetic character. See remarks on alphacharp.

Function: sexplode (string)

sexplode is an alias for function charlist.

Function: simplode
    simplode (list)
    simplode (list, delim)

simplode takes a list of expressions and concatenates them into a string. If no delimiter delim is specified, simplode uses no delimiter. delim can be any string.

Examples:

(%i1) simplode(["xx[",3,"]:",expand((x+y)^3)]);
(%o1)             xx[3]:y^3+3*x*y^2+3*x^2*y+x^3
(%i2) simplode( sexplode("stars")," * " );
(%o2)                   s * t * a * r * s
(%i3) simplode( ["One","more","coffee."]," " );
(%o3)                   One more coffee.

Function: sinsert (seq, string, pos)

Returns a string that is a concatenation of substring(string, 1, pos-1), the string seq and substring (string, pos). Note that the first character in string is in position 1.

Examples:

(%i1) s: "A submarine."$
(%i2) concat( substring(s,1,3),"yellow ",substring(s,3) );
(%o2)                  A yellow submarine.
(%i3) sinsert("hollow ",s,3);
(%o3)                  A hollow submarine.

Function: sinvertcase
    sinvertcase (string)
    sinvertcase (string, start)
    sinvertcase (string, start, end)

Returns string except that each character from position start to end is inverted. If end is not given, all characters from start to the end of string are replaced.

Examples:

(%i1) sinvertcase("sInvertCase");
(%o1)                      SiNVERTcASE

Function: slength (string)

Returns the number of characters in string.

Function: smake (num, char)

Returns a new string with a number of num characters char.

Example:

(%i1) smake(3,"w");
(%o1)                          www

Function: smismatch
    smismatch (string_1, string_2)
    smismatch (string_1, string_2, test)

Returns the position of the first character of string_1 at which string_1 and string_2 differ or false. Default test function for matching is sequal. If smismatch should ignore case, use sequalignore as test.

Example:

(%i1) smismatch("seven","seventh");
(%o1)                           6

Function: split
    split (string)
    split (string, delim)
    split (string, delim, multiple)

Returns the list of all tokens in string. Each token is an unparsed string. split uses delim as delimiter. If delim is not given, the space character is the default delimiter. multiple is a boolean variable with true by default. Multiple delimiters are read as one. This is useful if tabs are saved as multiple space characters. If multiple is set to false, each delimiter is noted.

Examples:

(%i1) split("1.2   2.3   3.4   4.5");
(%o1)                 [1.2, 2.3, 3.4, 4.5]
(%i2) split("first;;third;fourth",";",false);
(%o2)               [first, , third, fourth]

Function: sposition (char, string)

Returns the position of the first character in string which matches char. The first character in string is in position 1. For matching characters ignoring case see ssearch.

Function: sremove
    sremove (seq, string)
    sremove (seq, string, test)
    sremove (seq, string, test, start)
    sremove (seq, string, test, start, end)

Returns a string like string but without all substrings matching seq. Default test function for matching is sequal. If sremove should ignore case while searching for seq, use sequalignore as test. Use start and end to limit searching. Note that the first character in string is in position 1.

Examples:

(%i1) sremove("n't","I don't like coffee.");
(%o1)                   I do like coffee.
(%i2) sremove ("DO ",%,'sequalignore);
(%o2)                    I like coffee.

Function: sremovefirst
    sremovefirst (seq, string)
    sremovefirst (seq, string, test)
    sremovefirst (seq, string, test, start)
    sremovefirst (seq, string, test, start, end)

Like sremove except that only the first substring that matches seq is removed.

Function: sreverse (string)

Returns a string with all the characters of string in reverse order.

Function: ssearch
    ssearch (seq, string)
    ssearch (seq, string, test)
    ssearch (seq, string, test, start)
    ssearch (seq, string, test, start, end)

Returns the position of the first substring of string that matches the string seq. Default test function for matching is sequal. If ssearch should ignore case, use sequalignore as test. Use start and end to limit searching. Note that the first character in string is in position 1.

Example:

(%i1) ssearch("~s","~{~S ~}~%",'sequalignore);
(%o1)                                  4

Function: ssort
    ssort (string)
    ssort (string, test)

Returns a string that contains all characters from string in an order such there are no two successive characters c and d such that test (c, d) is false and test (d, c) is true. Default test function for sorting is clessp. The set of test functions is {clessp, clesspignore, cgreaterp, cgreaterpignore, cequal, cequalignore}.

Examples:

(%i1) ssort("I don't like Mondays.");
(%o1)                    '.IMaddeiklnnoosty
(%i2) ssort("I don't like Mondays.",'cgreaterpignore);
(%o2)                 ytsoonnMlkIiedda.'   

Function: ssubst
    ssubst (new, old, string)
    ssubst (new, old, string, test)
    ssubst (new, old, string, test, start)
    ssubst (new, old, string, test, start, end)

Returns a string like string except that all substrings matching old are replaced by new. old and new need not to be of the same length. Default test function for matching is sequal. If ssubst should ignore case while searching for old, use sequalignore as test. Use start and end to limit searching. Note that the first character in string is in position 1.

Examples:

(%i1) ssubst("like","hate","I hate Thai food. I hate green tea.");
(%o1)          I like Thai food. I like green tea.
(%i2) ssubst("Indian","thai",%,'sequalignore,8,12);
(%o2)         I like Indian food. I like green tea.

Function: ssubstfirst
    ssubstfirst (new, old, string)
    ssubstfirst (new, old, string, test)
    ssubstfirst (new, old, string, test, start)
    ssubstfirst (new, old, string, test, start, end)

Like subst except that only the first substring that matches old is replaced.

Function: strim (seq,string)

Returns a string like string, but with all characters that appear in seq removed from both ends.

Examples:

(%i1) "/* comment */"$
(%i2) strim(" /*",%);
(%o2)                        comment
(%i3) slength(%);
(%o3)                           7

Function: striml (seq, string)

Like strim except that only the left end of string is trimmed.

Function: strimr (seq, string)

Like strim except that only the right end of string is trimmed.

Function: stringp (obj)

Returns true if obj is a string. See introduction for example.

Function: substring
    substring (string, start)
    substring (string, start, end)

Returns the substring of string beginning at position start and ending at position end. The character at position end is not included. If end is not given, the substring contains the rest of the string. Note that the first character in string is in position 1.

Examples:

(%i1) substring("substring",4);
(%o1)                        string
(%i2) substring(%,4,6);
(%o2)                          in

Function: supcase
    supcase (string)
    supcase (string, start)
    supcase (string, start, end)

Returns string except that lowercase characters from position start to end are replaced by the corresponding uppercase ones. If end is not given, all lowercase characters from start to the end of string are replaced.

Example:

(%i1) supcase("english",1,2);
(%o1)                        English

Function: tokens
    tokens (string)
    tokens (string, test)

Returns a list of tokens, which have been extracted from string. The tokens are substrings whose characters satisfy a certain test function. If test is not given, constituent is used as the default test. {constituent, alphacharp, digitcharp, lowercasep, uppercasep, charp, characterp, alphanumericp} is the set of test functions. (The Lisp-version of tokens is written by Paul Graham. ANSI Common Lisp, 1996, page 67.)

Examples:

(%i1) tokens("24 October 2005");
(%o1)                  [24, October, 2005]
(%i2) tokens("05-10-24",'digitcharp);
(%o2)                     [05, 10, 24]
(%i3) map(parse_string,%);
(%o3)                      [5, 10, 24]


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86.5 Octets and Utilities for Cryptography

Function: base64 (arg)

Returns the base64-representation of arg as a string. The argument arg may be a string, a non-negative integer or a list of octets.

Examples:

(%i1) base64: base64("foo bar baz");
(%o1)                          Zm9vIGJhciBiYXo=
(%i2) string: base64_decode(base64);
(%o2)                            foo bar baz
(%i3) obase: 16.$
(%i4) integer: base64_decode(base64, 'number);
(%o4)                       666f6f206261722062617a
(%i5) octets: base64_decode(base64, 'list);
(%o5)            [66, 6F, 6F, 20, 62, 61, 72, 20, 62, 61, 7A]
(%i6) ibase: 16.$
(%i7) base64(octets);
(%o7)                          Zm9vIGJhciBiYXo=

Note that if arg contains umlauts (resp. octets larger than 127) the resulting base64-string is platform dependend. However the decoded string will be equal to the original.

Function: base64_decode
    base64_decode (base64-string)
    base64_decode (base64-string, return-type)

By default base64_decode decodes the base64-string back to the original string.

The optional argument return-type allows base64_decode to alternatively return the corresponding number or list of octets. return-type may be string, number or list.

Example: See base64.

Function: crc24sum
    crc24sum (octets)
    crc24sum (octets, return-type)

By default crc24sum returns the CRC24 checksum of an octet-list as a string.

The optional argument return-type allows crc24sum to alternatively return the corresponding number or list of octets. return-type may be string, number or list.

Example:

-----BEGIN PGP SIGNATURE-----
Version: GnuPG v2.0.22 (GNU/Linux)

iQEcBAEBAgAGBQJVdCTzAAoJEG/1Mgf2DWAqCSYH/AhVFwhu1D89C3/QFcgVvZTM
wnOYzBUURJAL/cT+IngkLEpp3hEbREcugWp+Tm6aw3R4CdJ7G3FLxExBH/5KnDHi
rBQu+I7+3ySK2hpryQ6Wx5J9uZSa4YmfsNteR8up0zGkaulJeWkS4pjiRM+auWVe
vajlKZCIK52P080DG7Q2dpshh4fgTeNwqCuCiBhQ73t8g1IaLdhDN6EzJVjGIzam
/spqT/sTo6sw8yDOJjvU+Qvn6/mSMjC/YxjhRMaQt9EMrR1AZ4ukBF5uG1S7mXOH
WdiwkSPZ3gnIBhM9SuC076gLWZUNs6NqTeE3UzMjDAFhH3jYk1T7mysCvdtIkms=
=WmeC
-----END PGP SIGNATURE-----
(%i1) ibase : obase : 16.$
(%i2) sig64 : sconcat(
 "iQEcBAEBAgAGBQJVdCTzAAoJEG/1Mgf2DWAqCSYH/AhVFwhu1D89C3/QFcgVvZTM",
 "wnOYzBUURJAL/cT+IngkLEpp3hEbREcugWp+Tm6aw3R4CdJ7G3FLxExBH/5KnDHi",
 "rBQu+I7+3ySK2hpryQ6Wx5J9uZSa4YmfsNteR8up0zGkaulJeWkS4pjiRM+auWVe",
 "vajlKZCIK52P080DG7Q2dpshh4fgTeNwqCuCiBhQ73t8g1IaLdhDN6EzJVjGIzam",
 "/spqT/sTo6sw8yDOJjvU+Qvn6/mSMjC/YxjhRMaQt9EMrR1AZ4ukBF5uG1S7mXOH",
 "WdiwkSPZ3gnIBhM9SuC076gLWZUNs6NqTeE3UzMjDAFhH3jYk1T7mysCvdtIkms=" )$
(%i3) octets: base64_decode(sig64, 'list)$
(%i4) crc24: crc24sum(octets, 'list);
(%o4)                          [5A, 67, 82]
(%i5) base64(crc24);
(%o5)                              WmeC

Function: md5sum
    md5sum (arg)
    md5sum (arg, return-type)

Returns the MD5 checksum of a string, a non-negative integer or a list of octets. The default return value is a string containing 32 hex characters.

The optional argument return-type allows md5sum to alternatively return the corresponding number or list of octets. return-type may be string, number or list.

Examples:

(%i1) ibase: obase: 16.$
(%i2) msg: "foo bar baz"$
(%i3) string: md5sum(msg);
(%o3)                  ab07acbb1e496801937adfa772424bf7
(%i4) integer: md5sum(msg, 'number);
(%o4)                 0ab07acbb1e496801937adfa772424bf7
(%i5) octets: md5sum(msg, 'list);
(%o5)        [0AB,7,0AC,0BB,1E,49,68,1,93,7A,0DF,0A7,72,42,4B,0F7]
(%i6) sdowncase( printf(false, "~{~2,'0x~^:~}", octets) );
(%o6)           ab:07:ac:bb:1e:49:68:01:93:7a:df:a7:72:42:4b:f7

Note that in case arg contains German umlauts or other non-ASCII characters (resp. octets larger than 127) the MD5 checksum is platform dependend.

Function: mgf1_sha1
    mgf1_sha1 (seed, len)
    mgf1_sha1 (seed, len, return-type)

Returns a pseudo random number of variable length. By default the returned value is a number with a length of len octets.

The optional argument return-type allows mgf1_sha1 to alternatively return the corresponding list of len octets. return-type may be number or list.

The computation of the returned value is described in RFC 3447, appendix B.2.1 MGF1. SHA1 ist used as hash function, i.e. the randomness of the computed number relies on the randomness of SHA1 hashes.

Example:

(%i1) ibase: obase: 16.$
(%i2) number: mgf1_sha1(4711., 8);
(%o2)                        0e0252e5a2a42fea1
(%i3) octets: mgf1_sha1(4711., 8, 'list);
(%o3)                  [0E0,25,2E,5A,2A,42,0FE,0A1]

Function: number_to_octets (number)

Returns an octet-representation of number as a list of octets. The number must be a non-negative integer.

Example:

(%i1) ibase : obase : 16.$
(%i2) octets: [0ca,0fe,0ba,0be]$
(%i3) number: octets_to_number(octets);
(%o3)                            0cafebabe
(%i4) number_to_octets(number);
(%o4)                      [0CA, 0FE, 0BA, 0BE]

Function: octets_to_number (octets)

Returns a number by concatenating the octets in the list of octets.

Example: See number_to_octets.

Function: octets_to_oid (octets)

Computes an object identifier (OID) from the list of octets.

Example: RSA encryption OID

(%i1) ibase : obase : 16.$
(%i2) oid: octets_to_oid([2A,86,48,86,0F7,0D,1,1,1]);
(%o2)                      1.2.840.113549.1.1.1
(%i3) oid_to_octets(oid);
(%o3)               [2A, 86, 48, 86, 0F7, 0D, 1, 1, 1]

Function: octets_to_string
    octets_to_string (octets)
    octets_to_string (octets, encoding)

Decodes the list of octets into a string according to current system defaults. When decoding octets corresponding to Non-US-ASCII characters the result depends on the platform, application and underlying Lisp.

Example: Using system defaults (Maxima compiled with GCL, which uses no format definition and simply passes through the UTF-8-octets encoded by the GNU/Linux terminal).

(%i1) octets: string_to_octets("abc");
(%o1)                            [61, 62, 63]
(%i2) octets_to_string(octets);
(%o2)                                 abc
(%i3) ibase: obase: 16.$
(%i4) unicode(20AC);
(%o4)                                  €
(%i5) octets: string_to_octets(%);
(%o5)                           [0E2, 82, 0AC]
(%i6) octets_to_string(octets);
(%o6)                                  €
(%i7) utf8_to_unicode(octets);
(%o7)                                20AC

In case the external format of the Lisp reader is equal to UTF-8 the optional argument encoding allows to set the encoding for the octet to string conversion. If necessary see adjust_external_format for changing the external format.

Some names of supported encodings (see corresponding Lisp manual for more):
CCL, CLISP, SBCL: utf-8, ucs-2be, ucs-4be, iso-8859-1, cp1252, cp850
CMUCL: utf-8, utf-16-be, utf-32-be, iso8859-1, cp1252
ECL: utf-8, ucs-2be, ucs-4be, iso-8859-1, windows-cp1252, dos-cp850

Example (continued): Using the optional encoding argument (Maxima compiled with SBCL, GNU/Linux terminal).

(%i8) string_to_octets("€", "ucs-2be");
(%o8)                              [20, 0AC]

Function: oid_to_octets (oid-string)

Convertes an object identifier (OID) to a list of octets.

Example: See octets_to_oid.

Function: sha1sum
    sha1sum (arg)
    sha1sum (arg, return-type)

Returns the SHA1 fingerprint of a string, a non-negative integer or a list of octets. The default return value is a string containing 40 hex characters.

The optional argument return-type allows sha1sum to alternatively return the corresponding number or list of octets. return-type may be string, number or list.

Example:

(%i1) ibase: obase: 16.$
(%i2) msg: "foo bar baz"$
(%i3) string: sha1sum(msg);
(%o3)              c7567e8b39e2428e38bf9c9226ac68de4c67dc39
(%i4) integer: sha1sum(msg, 'number);
(%o4)             0c7567e8b39e2428e38bf9c9226ac68de4c67dc39
(%i5) octets: sha1sum(msg, 'list);
(%o5)  [0C7,56,7E,8B,39,0E2,42,8E,38,0BF,9C,92,26,0AC,68,0DE,4C,67,0DC,39]
(%i6) sdowncase( printf(false, "~{~2,'0x~^:~}", octets) );
(%o6)     c7:56:7e:8b:39:e2:42:8e:38:bf:9c:92:26:ac:68:de:4c:67:dc:39

Note that in case arg contains German umlauts or other non-ASCII characters (resp. octets larger than 127) the SHA1 fingerprint is platform dependend.

Function: sha256sum
    sha256sum (arg)
    sha256sum (arg, return-type)

Returns the SHA256 fingerprint of a string, a non-negative integer or a list of octets. The default return value is a string containing 64 hex characters.

The optional argument return-type allows sha256sum to alternatively return the corresponding number or list of octets (see sha1sum).

Example:

(%i1) string: sha256sum("foo bar baz");
(%o1)  dbd318c1c462aee872f41109a4dfd3048871a03dedd0fe0e757ced57dad6f2d7

Note that in case arg contains German umlauts or other non-ASCII characters (resp. octets larger than 127) the SHA256 fingerprint is platform dependend.

Function: string_to_octets
    string_to_octets (string)
    string_to_octets (string, encoding)

Encodes a string into a list of octets according to current system defaults. When encoding strings containing Non-US-ASCII characters the result depends on the platform, application and underlying Lisp.

In case the external format of the Lisp reader is equal to UTF-8 the optional argument encoding allows to set the encoding for the string to octet conversion. If necessary see adjust_external_format for changing the external format.

See octets_to_string for examples and some more information.


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87 to_poly_solve


Previous: , Up: to_poly_solve-pkg   [Contents][Index]

87.1 Functions and Variables for to_poly_solve

The packages to_poly and to_poly_solve are experimental; the specifications of the functions in these packages might change or the some of the functions in these packages might be merged into other Maxima functions.

Barton Willis (Professor of Mathematics, University of Nebraska at Kearney) wrote the to_poly and to_poly_solve packages and the English language user documentation for these packages.

Operator: %and

The operator %and is a simplifying nonshort-circuited logical conjunction. Maxima simplifies an %and expression to either true, false, or a logically equivalent, but simplified, expression. The operator %and is associative, commutative, and idempotent. Thus when %and returns a noun form, the arguments of %and form a non-redundant sorted list; for example

(%i1) a %and (a %and b);
(%o1)                       a %and b

If one argument to a conjunction is the explicit the negation of another argument, %and returns false:

(%i2) a %and (not a);
(%o2)                         false

If any member of the conjunction is false, the conjunction simplifies to false even if other members are manifestly non-boolean; for example

(%i3) 42 %and false;
(%o3)                         false

Any argument of an %and expression that is an inequation (that is, an inequality or equation), is simplified using the Fourier elimination package. The Fourier elimination simplifier has a pre-processor that converts some, but not all, nonlinear inequations into linear inequations; for example the Fourier elimination code simplifies abs(x) + 1 > 0 to true, so

(%i4) (x < 1) %and (abs(x) + 1 > 0);
(%o4)                         x < 1

Notes

Limitations The conjunction %and simplifies inequations locally, not globally. This means that conjunctions such as

(%i5) (x < 1) %and (x > 1);
(%o5)                 (x > 1) %and (x < 1)

do not simplify to false. Also, the Fourier elimination code ignores the fact database;

(%i6) assume(x > 5);
(%o6)                        [x > 5]
(%i7) (x > 1) %and (x > 2);
(%o7)                 (x > 1) %and (x > 2)

Finally, nonlinear inequations that aren’t easily converted into an equivalent linear inequation aren’t simplified.

There is no support for distributing %and over %or; neither is there support for distributing a logical negation over %and.

To use load(to_poly_solve)

Related functions %or, %if, and, or, not

Status The operator %and is experimental; the specifications of this function might change and its functionality might be merged into other Maxima functions.

Operator: %if (bool, a, b)

The operator %if is a simplifying conditional. The conditional bool should be boolean-valued. When the conditional is true, return the second argument; when the conditional is false, return the third; in all other cases, return a noun form.

Maxima inequations (either an inequality or an equality) are not boolean-valued; for example, Maxima does not simplify 5 < 6 to true, and it does not simplify 5 = 6 to false; however, in the context of a conditional to an %if statement, Maxima automatically attempts to determine the truth value of an inequation. Examples:

(%i1) f : %if(x # 1, 2, 8);
(%o1)                 %if(x - 1 # 0, 2, 8)
(%i2) [subst(x = -1,f), subst(x=1,f)];
(%o2)                        [2, 8]

If the conditional involves an inequation, Maxima simplifies it using the Fourier elimination package.

Notes

(%i3) %if(42,1,2);
(%o3)                     %if(42, 1, 2)

Limitations The Fourier elimination code only simplifies nonlinear inequations that are readily convertible to an equivalent linear inequation.

To use: load(to_poly_solve)

Status: The operator %if is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Operator: %or

The operator %or is a simplifying nonshort-circuited logical disjunction. Maxima simplifies an %or expression to either true, false, or a logically equivalent, but simplified, expression. The operator %or is associative, commutative, and idempotent. Thus when %or returns a noun form, the arguments of %or form a non-redundant sorted list; for example

(%i1) a %or (a %or b);
(%o1)                        a %or b

If one member of the disjunction is the explicit the negation of another member, %or returns true:

(%i2) a %or (not a);
(%o2)                         true

If any member of the disjunction is true, the disjunction simplifies to true even if other members of the disjunction are manifestly non-boolean; for example

(%i3) 42 %or true;
(%o3)                         true

Any argument of an %or expression that is an inequation (that is, an inequality or equation), is simplified using the Fourier elimination package. The Fourier elimination code simplifies abs(x) + 1 > 0 to true, so we have

(%i4) (x < 1) %or (abs(x) + 1 > 0);
(%o4)                         true

Notes

Limitations The conjunction %or simplifies inequations locally, not globally. This means that conjunctions such as

(%i1) (x < 1) %or (x >= 1);
(%o1) (x > 1) %or (x >= 1)

do not simplify to true. Further, the Fourier elimination code ignores the fact database;

(%i2) assume(x > 5);
(%o2)                        [x > 5]
(%i3) (x > 1) %and (x > 2);
(%o3)                 (x > 1) %and (x > 2)

Finally, nonlinear inequations that aren’t easily converted into an equivalent linear inequation aren’t simplified.

The algorithm that looks for terms that cannot both be false is weak; also there is no support for distributing %or over %and; neither is there support for distributing a logical negation over %or.

To use load(to_poly_solve)

Related functions %or, %if, and, or, not

Status The operator %or is experimental; the specifications of this function might change and its functionality might be merged into other Maxima functions.

Function: complex_number_p (x)

The predicate complex_number_p returns true if its argument is either a + %i * b, a, %i b, or %i, where a and b are either rational or floating point numbers (including big floating point); for all other inputs, complex_number_p returns false; for example

(%i1) map('complex_number_p,[2/3, 2 + 1.5 * %i, %i]);
(%o1)                  [true, true, true]
(%i2) complex_number_p((2+%i)/(5-%i));
(%o2)                         false
(%i3) complex_number_p(cos(5 - 2 * %i));
(%o3)                         false

Related functions isreal_p

To use load(to_poly_solve)

Status The operator complex_number_p is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: compose_functions (l)

The function call compose_functions(l) returns a lambda form that is the composition of the functions in the list l. The functions are applied from right to left; for example

(%i1) compose_functions([cos, exp]);
                                        %g151
(%o1)             lambda([%g151], cos(%e     ))
(%i2) %(x);
                                  x
(%o2)                       cos(%e )

When the function list is empty, return the identity function:

(%i3) compose_functions([]);
(%o3)                lambda([%g152], %g152)
(%i4)  %(x);
(%o4)                           x

Notes

(%i5) compose_functions([a < b]);

funmake: first argument must be a symbol, subscripted symbol,
string, or lambda expression; found: a < b
#0: compose_functions(l=[a < b])(to_poly_solve.mac line 40)
 -- an error. To debug this try: debugmode(true);

To use load(to_poly_solve)

Status The function compose_functions is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: dfloat (x)

The function dfloat is a similar to float, but the function dfloat applies rectform when float fails to evaluate to an IEEE double floating point number; thus

(%i1) float(4.5^(1 + %i));
                               %i + 1
(%o1)                       4.5
(%i2) dfloat(4.5^(1 + %i));
(%o2)        4.48998802962884 %i + .3000124893895671

Notes

Related functions float, bfloat

To use load(to_poly_solve)

Status The function dfloat is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: elim (l, x)

The function elim eliminates the variables in the set or list x from the equations in the set or list l. Each member of x must be a symbol; the members of l can either be equations, or expressions that are assumed to equal zero.

The function elim returns a list of two lists; the first is the list of expressions with the variables eliminated; the second is the list of pivots; thus, the second list is a list of expressions that elim used to eliminate the variables.

Here is a example of eliminating between linear equations:

(%i1) elim(set(x + y + z = 1, x - y  - z = 8, x - z = 1), 
           set(x,y));
(%o1)            [[2 z - 7], [y + 7, z - x + 1]]

Eliminating x and y yields the single equation 2 z - 7 = 0; the equations y + 7 = 0 and z - z + 1 = 1 were used as pivots. Eliminating all three variables from these equations, triangularizes the linear system:

(%i2) elim(set(x + y + z = 1, x - y  - z = 8, x - z = 1),
           set(x,y,z));
(%o2)           [[], [2 z - 7, y + 7, z - x + 1]]

Of course, the equations needn’t be linear:

(%i3) elim(set(x^2 - 2 * y^3 = 1,  x - y = 5), [x,y]);
                     3    2
(%o3)       [[], [2 y  - y  - 10 y - 24, y - x + 5]]

The user doesn’t control the order the variables are eliminated. Instead, the algorithm uses a heuristic to attempt to choose the best pivot and the best elimination order.

Notes

Related functions elim_allbut, eliminate_using, eliminate

Option variables resultant

To use load(to_poly)

Status The function elim is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: elim_allbut (l, x)

This function is similar to elim, except that it eliminates all the variables in the list of equations l except for those variables that in in the list x

(%i1) elim_allbut([x+y = 1, x - 5*y = 1],[]);
(%o1)                 [[], [y, y + x - 1]]
(%i2) elim_allbut([x+y = 1, x - 5*y = 1],[x]);
(%o2)                [[x - 1], [y + x - 1]]

To use load(to_poly)

Option variables resultant

Related functions elim, eliminate_using, eliminate

Status The function elim_allbut is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: eliminate_using (l, e, x)

Using e as the pivot, eliminate the symbol x from the list or set of equations in l. The function eliminate_using returns a set.

(%i1) eq : [x^2 - y^2 - z^3 , x*y - z^2 - 5, x - y + z];
               3    2    2     2
(%o1)      [- z  - y  + x , - z  + x y - 5, z - y + x]
(%i2) eliminate_using(eq,first(eq),z);
        3              2      2      3    2
(%o2) {y  + (1 - 3 x) y  + 3 x  y - x  - x , 
                        4    3  3       2  2             4
                       y  - x  y  + 13 x  y  - 75 x y + x  + 125}
(%i3) eliminate_using(eq,second(eq),z);
        2            2       4    3  3       2  2             4
(%o3) {y  - 3 x y + x  + 5, y  - x  y  + 13 x  y  - 75 x y + x
                                                           + 125}
(%i4) eliminate_using(eq, third(eq),z);
        2            2       3              2      2      3    2
(%o4) {y  - 3 x y + x  + 5, y  + (1 - 3 x) y  + 3 x  y - x  - x }

Option variables resultant

Related functions elim, eliminate, elim_allbut

To use load(to_poly)

Status The function eliminate_using is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: fourier_elim ([eq1, eq2, …], [var1, var, …])

Fourier elimination is the analog of Gauss elimination for linear inequations (equations or inequalities). The function call fourier_elim([eq1, eq2, ...], [var1, var2, ...]) does Fourier elimination on a list of linear inequations [eq1, eq2, ...] with respect to the variables [var1, var2, ...]; for example

(%i1) fourier_elim([y-x < 5, x - y < 7, 10 < y],[x,y]);
(%o1)            [y - 5 < x, x < y + 7, 10 < y]
(%i2) fourier_elim([y-x < 5, x - y < 7, 10 < y],[y,x]);
(%o2)        [max(10, x - 7) < y, y < x + 5, 5 < x]

Eliminating first with respect to x and second with respect to y yields lower and upper bounds for x that depend on y, and lower and upper bounds for y that are numbers. Eliminating in the other order gives x dependent lower and upper bounds for y, and numerical lower and upper bounds for x.

When necessary, fourier_elim returns a disjunction of lists of inequations:

(%i3) fourier_elim([x # 6],[x]);
(%o3)                  [x < 6] or [6 < x]

When the solution set is empty, fourier_elim returns emptyset, and when the solution set is all reals, fourier_elim returns universalset; for example

(%i4) fourier_elim([x < 1, x > 1],[x]);
(%o4)                       emptyset
(%i5) fourier_elim([minf < x, x < inf],[x]);
(%o5)                     universalset

For nonlinear inequations, fourier_elim returns a (somewhat) simplified list of inequations:

(%i6) fourier_elim([x^3 - 1 > 0],[x]);
               2                             2
(%o6) [1 < x, x  + x + 1 > 0] or [x < 1, - (x  + x + 1) > 0]
(%i7) fourier_elim([cos(x) < 1/2],[x]);
(%o7)                  [1 - 2 cos(x) > 0]

Instead of a list of inequations, the first argument to fourier_elim may be a logical disjunction or conjunction:

(%i8) fourier_elim((x + y < 5) and (x - y >8),[x,y]);
                                              3
(%o8)            [y + 8 < x, x < 5 - y, y < - -]
                                              2
(%i9) fourier_elim(((x + y < 5) and x < 1) or  (x - y >8),[x,y]);
(%o9)          [y + 8 < x] or [x < min(1, 5 - y)]

The function fourier_elim supports the inequation operators <, <=, >, >=, #, and =.

The Fourier elimination code has a preprocessor that converts some nonlinear inequations that involve the absolute value, minimum, and maximum functions into linear in equations. Additionally, the preprocessor handles some expressions that are the product or quotient of linear terms:

(%i10) fourier_elim([max(x,y) > 6, x # 8, abs(y-1) > 12],[x,y]);
(%o10) [6 < x, x < 8, y < - 11] or [8 < x, y < - 11]
 or [x < 8, 13 < y] or [x = y, 13 < y] or [8 < x, x < y, 13 < y]
 or [y < x, 13 < y]
(%i11) fourier_elim([(x+6)/(x-9) <= 6],[x]);
(%o11)           [x = 12] or [12 < x] or [x < 9]
(%i12) fourier_elim([x^2 - 1 # 0],[x]);
(%o12)      [- 1 < x, x < 1] or [1 < x] or [x < - 1]

To use load(fourier_elim)

Function: isreal_p (e)

The predicate isreal_p returns true when Maxima is able to determine that e is real-valued on the entire real line; it returns false when Maxima is able to determine that e isn’t real-valued on some nonempty subset of the real line; and it returns a noun form for all other cases.

(%i1) map('isreal_p, [-1, 0, %i, %pi]);
(%o1)               [true, true, false, true]

Maxima variables are assumed to be real; thus

(%i2) isreal_p(x);
(%o2)                         true

The function isreal_p examines the fact database:

(%i3) declare(z,complex)$

(%i4) isreal_p(z);
(%o4)                      isreal_p(z)

Limitations Too often, isreal_p returns a noun form when it should be able to return false; a simple example: the logarithm function isn’t real-valued on the entire real line, so isreal_p(log(x)) should return false; however

(%i5) isreal_p(log(x));
(%o5)                   isreal_p(log(x))

To use load(to_poly_solve)

Related functions complex_number_p

Status The function isreal_p is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: new_variable (type)

Return a unique symbol of the form %[z,n,r,c,g]k, where k is an integer. The allowed values for type are integer, natural_number, real, natural_number, and general. (By natural number, we mean the nonnegative integers; thus zero is a natural number. Some, but not all,definitions of natural number exclude zero.)

When type isn’t one of the allowed values, type defaults to general. For integers, natural numbers, and complex numbers, Maxima automatically appends this information to the fact database.

(%i1) map('new_variable,
          ['integer, 'natural_number, 'real, 'complex, 'general]);
(%o1)          [%z144, %n145, %r146, %c147, %g148]
(%i2) nicedummies(%);
(%o2)               [%z0, %n0, %r0, %c0, %g0]
(%i3) featurep(%z0, 'integer);
(%o3)                         true
(%i4) featurep(%n0, 'integer);
(%o4)                         true
(%i5) is(%n0 >= 0);
(%o5)                         true
(%i6) featurep(%c0, 'complex);
(%o6)                         true

Note Generally, the argument to new_variable should be quoted. The quote will protect against errors similar to

(%i7) integer : 12$

(%i8) new_variable(integer);
(%o8)                         %g149
(%i9) new_variable('integer);
(%o9)                         %z150

Related functions nicedummies

To use load(to_poly_solve)

Status The function new_variable is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: nicedummies

Starting with zero, the function nicedummies re-indexes the variables in an expression that were introduced by new_variable;

(%i1) new_variable('integer) + 52 * new_variable('integer);
(%o1)                   52 %z136 + %z135
(%i2) new_variable('integer) - new_variable('integer);
(%o2)                     %z137 - %z138
(%i3) nicedummies(%);
(%o3)                       %z0 - %z1

Related functions new_variable

To use load(to_poly_solve)

Status The function nicedummies is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: parg (x)

The function parg is a simplifying version of the complex argument function carg; thus

(%i1) map('parg,[1,1+%i,%i, -1 + %i, -1]);
                        %pi  %pi  3 %pi
(%o1)               [0, ---, ---, -----, %pi]
                         4    2     4

Generally, for a non-constant input, parg returns a noun form; thus

(%i2) parg(x + %i * sqrt(x));
(%o2)                 parg(x + %i sqrt(x))

When sign can determine that the input is a positive or negative real number, parg will return a non-noun form for a non-constant input. Here are two examples:

(%i3) parg(abs(x));
(%o3) 0
(%i4) parg(-x^2-1);
(%o4)                          %pi

Note The sign function mostly ignores the variables that are declared to be complex (declare(x,complex)); for variables that are declared to be complex, the parg can return incorrect values; for example

(%i1) declare(x,complex)$

(%i2) parg(x^2 + 1);
(%o2) 0

Related function carg, isreal_p

To use load(to_poly_solve)

Status The function parg is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: real_imagpart_to_conjugate (e)

The function real_imagpart_to_conjugate replaces all occurrences of realpart and imagpart to algebraically equivalent expressions involving the conjugate.

(%i1) declare(x, complex)$

(%i2) real_imagpart_to_conjugate(realpart(x) +  imagpart(x) = 3);
          conjugate(x) + x   %i (x - conjugate(x))
(%o2)     ---------------- - --------------------- = 3
                 2                     2

To use load(to_poly_solve)

Status The function real_imagpart_to_conjugate is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: rectform_log_if_constant (e)

The function rectform_log_if_constant converts all terms of the form log(c) to rectform(log(c)), where c is either a declared constant expression or explicitly declared constant

(%i1) rectform_log_if_constant(log(1-%i) - log(x - %i));
                                 log(2)   %i %pi
(%o1)            - log(x - %i) + ------ - ------
                                   2        4
(%i2) declare(a,constant, b,constant)$

(%i3) rectform_log_if_constant(log(a + %i*b));
                       2    2
                  log(b  + a )
(%o3)             ------------ + %i atan2(b, a)
                       2

To use load(to_poly_solve)

Status The function rectform_log_if_constant is experimental; the specifications of this function might change might change and its functionality might be merged into other Maxima functions.

Function: simp_inequality (e)

The function simp_inequality applies some simplifications to conjunctions and disjunctions of inequations.

Limitations The function simp_inequality is limited in at least two ways; first, the simplifications are local; thus

(%i1) simp_inequality((x > minf) %and (x < 0));
(%o1) (x>1) %and (x<1)

And second, simp_inequality doesn’t consult the fact database:

(%i2) assume(x > 0)$

(%i3) simp_inequality(x > 0);
(%o3)                         x > 0

To use load(fourier_elim)

Status The function simp_inequality is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: standardize_inverse_trig (e)

This function applies the identities cot(x) = atan(1/x), acsc(x) = asin(1/x), and similarly for asec, acoth, acsch and asech to an expression. See Abramowitz and Stegun, Eqs. 4.4.6 through 4.4.8 and 4.6.4 through 4.6.6.

To use load(to_poly_solve)

Status The function standardize_inverse_trig is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: subst_parallel (l, e)

When l is a single equation or a list of equations, substitute the right hand side of each equation for the left hand side. The substitutions are made in parallel; for example

(%i1) load(to_poly_solve)$

(%i2) subst_parallel([x=y,y=x], [x,y]);
(%o2)                        [y, x]

Compare this to substitutions made serially:

(%i3) subst([x=y,y=x],[x,y]);
(%o3)                        [x, x]

The function subst_parallel is similar to sublis except that subst_parallel allows for substitution of nonatoms; for example

(%i4) subst_parallel([x^2 = a, y = b], x^2 * y);
(%o4)                          a b
(%i5) sublis([x^2 = a, y = b], x^2 * y);

                                                             2
sublis: left-hand side of equation must be a symbol; found: x
 -- an error. To debug this try: debugmode(true);

The substitutions made by subst_parallel are literal, not semantic; thus subst_parallel does not recognize that x * y is a subexpression of x^2 * y

(%i6) subst_parallel([x * y = a], x^2 * y);
                               2
(%o6)                         x  y

The function subst_parallel completes all substitutions before simplifications. This allows for substitutions into conditional expressions where errors might occur if the simplifications were made earlier:

(%i7) subst_parallel([x = 0], %if(x < 1, 5, log(x)));
(%o7)                           5
(%i8) subst([x = 0], %if(x < 1, 5, log(x)));

log: encountered log(0).
 -- an error. To debug this try: debugmode(true);

Related functions subst, sublis, ratsubst

To use load(to_poly_solve_extra.lisp)

Status The function subst_parallel is experimental; the specifications of this function might change might change and its functionality might be merged into other Maxima functions.

Function: to_poly (e, l)

The function to_poly attempts to convert the equation e into a polynomial system along with inequality constraints; the solutions to the polynomial system that satisfy the constraints are solutions to the equation e. Informally, to_poly attempts to polynomialize the equation e; an example might clarify:

(%i1) load(to_poly_solve)$

(%i2) to_poly(sqrt(x) = 3, [x]);
                            2
(%o2) [[%g130 - 3, x = %g130 ], 
                      %pi                               %pi
                   [- --- < parg(%g130), parg(%g130) <= ---], []]
                       2                                 2

The conditions -%pi/2<parg(%g130),parg(%g130)<=%pi/2 tell us that %g130 is in the range of the square root function. When this is true, the solution set to sqrt(x) = 3 is the same as the solution set to %g130-3,x=%g130^2.

To polynomialize trigonometric expressions, it is necessary to introduce a non algebraic substitution; these non algebraic substitutions are returned in the third list returned by to_poly; for example

(%i3) to_poly(cos(x),[x]);
                2                                 %i x
(%o3)    [[%g131  + 1], [2 %g131 # 0], [%g131 = %e    ]]

Constant terms aren’t polynomializied unless the number one is a member of the variable list; for example

(%i4) to_poly(x = sqrt(5),[x]);
(%o4)                [[x - sqrt(5)], [], []]
(%i5) to_poly(x = sqrt(5),[1,x]);
                            2
(%o5) [[x - %g132, 5 = %g132 ], 
                      %pi                               %pi
                   [- --- < parg(%g132), parg(%g132) <= ---], []]
                       2                                 2

To generate a polynomial with sqrt(5) + sqrt(7) as one of its roots, use the commands

(%i6) first(elim_allbut(first(to_poly(x = sqrt(5) + sqrt(7),
                                      [1,x])), [x]));
                          4       2
(%o6)                   [x  - 24 x  + 4]

Related functions to_poly_solve

To use load(to_poly)

Status: The function to_poly is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Function: to_poly_solve (e, l, [options])

The function to_poly_solve tries to solve the equations e for the variables l. The equation(s) e can either be a single expression or a set or list of expressions; similarly, l can either be a single symbol or a list of set of symbols. When a member of e isn’t explicitly an equation, for example x^2 -1, the solver asummes that the expression vanishes.

The basic strategy of to_poly_solve is to convert the input into a polynomial form and to call algsys on the polynomial system. Internally to_poly_solve defaults algexact to true. To change the default for algexact, append ’algexact=false to the to_poly_solve argument list.

When to_poly_solve is able to determine the solution set, each member of the solution set is a list in a %union object:

(%i1) load(to_poly_solve)$

(%i2) to_poly_solve(x*(x-1) = 0, x);
(%o2)               %union([x = 0], [x = 1])

When to_poly_solve is unable to determine the solution set, a %solve nounform is returned (in this case, a warning is printed)

(%i3) to_poly_solve(x^k + 2* x + 1 = 0, x);

Nonalgebraic argument given to 'to_poly'
unable to solve
                          k
(%o3)            %solve([x  + 2 x + 1 = 0], [x])

Subsitution into a %solve nounform can sometimes result in the solution

(%i4) subst(k = 2, %);
(%o4)                   %union([x = - 1])

Especially for trigonometric equations, the solver sometimes needs to introduce an arbitrary integer. These arbitrary integers have the form %zXXX, where XXX is an integer; for example

(%i5) to_poly_solve(sin(x) = 0, x);
(%o5)   %union([x = 2 %pi %z33 + %pi], [x = 2 %pi %z35])

To re-index these variables to zero, use nicedummies:

(%i6) nicedummies(%);
(%o6)    %union([x = 2 %pi %z0 + %pi], [x = 2 %pi %z1])

Occasionally, the solver introduces an arbitrary complex number of the form %cXXX or an arbitrary real number of the form %rXXX. The function nicedummies will re-index these identifiers to zero.

The solution set sometimes involves simplifing versions of various of logical operators including %and, %or, or %if for conjunction, disjuntion, and implication, respectively; for example

(%i7) sol : to_poly_solve(abs(x) = a, x);
(%o7) %union(%if(isnonnegative_p(a), [x = - a], %union()), 
                      %if(isnonnegative_p(a), [x = a], %union()))
(%i8) subst(a = 42, sol);
(%o8)             %union([x = - 42], [x = 42])
(%i9) subst(a = -42, sol);
(%o9)                       %union()

The empty set is represented by %union().

The function to_poly_solve is able to solve some, but not all, equations involving rational powers, some nonrational powers, absolute values, trigonometric functions, and minimum and maximum. Also, some it can solve some equations that are solvable in in terms of the Lambert W function; some examples:

(%i1) load(to_poly_solve)$

(%i2) to_poly_solve(set(max(x,y) = 5, x+y = 2), set(x,y));
(%o2)      %union([x = - 3, y = 5], [x = 5, y = - 3])
(%i3) to_poly_solve(abs(1-abs(1-x)) = 10,x);
(%o3)             %union([x = - 10], [x = 12])
(%i4) to_poly_solve(set(sqrt(x) + sqrt(y) = 5, x + y = 10),
                    set(x,y));
                     3/2               3/2
                    5    %i - 10      5    %i + 10
(%o4) %union([x = - ------------, y = ------------], 
                         2                 2
                                3/2                 3/2
                               5    %i + 10        5    %i - 10
                          [x = ------------, y = - ------------])
                                    2                   2
(%i5) to_poly_solve(cos(x) * sin(x) = 1/2,x,
                    'simpfuncs = ['expand, 'nicedummies]);
                                         %pi
(%o5)              %union([x = %pi %z0 + ---])
                                          4
(%i6) to_poly_solve(x^(2*a) + x^a + 1,x);
                                        2 %i %pi %z81
                                        -------------
                                  1/a         a
                  (sqrt(3) %i - 1)    %e
(%o6) %union([x = -----------------------------------], 
                                  1/a
                                 2
                                                  2 %i %pi %z83
                                                  -------------
                                            1/a         a
                          (- sqrt(3) %i - 1)    %e
                     [x = -------------------------------------])
                                           1/a
                                          2
(%i7) to_poly_solve(x * exp(x) = a, x);
(%o7)              %union([x = lambert_w(a)])

For linear inequalities, to_poly_solve automatically does Fourier elimination:

(%i8) to_poly_solve([x + y < 1, x - y >= 8], [x,y]);
                               7
(%o8) %union([x = y + 8, y < - -], 
                               2
                                                              7
                                 [y + 8 < x, x < 1 - y, y < - -])
                                                              2

Each optional argument to to_poly_solve must be an equation; generally, the order of these options does not matter.

The to_poly_solve uses data stored in the hashed array one_to_one_reduce to solve equations of the form f(a) = f(b). The assignment one_to_one_reduce['f,'f] : lambda([a,b], a=b) tells to_poly_solve that the solution set of f(a) = f(b) equals the solution set of a=b; for example

(%i13) one_to_one_reduce['f,'f] : lambda([a,b], a=b)$

(%i14) to_poly_solve(f(x^2-1) = f(0),x);
(%o14)             %union([x = - 1], [x = 1])

More generally, the assignment one_to_one_reduce['f,'g] : lambda([a,b], w(a, b) = 0 tells to_poly_solve that the solution set of f(a) = f(b) equals the solution set of w(a,b) = 0; for example

(%i15) one_to_one_reduce['f,'g] : lambda([a,b], a = 1 + b/2)$

(%i16) to_poly_solve(f(x) - g(x),x);
(%o16)                   %union([x = 2])

Additionally, the function to_poly_solve uses data stored in the hashed array function_inverse to solve equations of the form f(a) = b. The assignment function_inverse['f] : lambda([s], g(s)) informs to_poly_solve that the solution set to f(x) = b equals the solution set to x = g(b); two examples:

(%i17) function_inverse['Q] : lambda([s], P(s))$

(%i18) to_poly_solve(Q(x-1) = 2009,x);
(%o18)              %union([x = P(2009) + 1])
(%i19) function_inverse['G] : lambda([s], s+new_variable(integer));
(%o19)       lambda([s], s + new_variable(integer))
(%i20) to_poly_solve(G(x - a) = b,x);
(%o20)             %union([x = b + a + %z125])

Notes

(%i1) to_poly_solve([x^2 + y^2 + x * y = 5, x * y = 8],
                    [x^2 + y^2, x * y]);
                                  2    2
(%o1)           %union([x y = 8, y  + x  = - 3])
(%i1) declare(x,complex)$

(%i2) to_poly_solve(x + (5 + %i) * conjugate(x) = 1, x);
                                   %i + 21
(%o2)              %union([x = - -----------])
                                 25 %i - 125
(%i3) declare(y,complex)$

(%i4) to_poly_solve(set(conjugate(x) - y = 42 + %i,
                        x + conjugate(y) = 0), set(x,y));
                           %i - 42        %i + 42
(%o4)        %union([x = - -------, y = - -------])
                              2              2

Relevant option variables algexact, resultant, algebraic

Related functions to_poly

To use load(to_poly_solve)

Status: The function to_poly_solve is experimental; its specifications might change and its functionality might be merged into other Maxima functions.

Operator: %union (soln_1, soln_2, soln_3, ...)
Operator: %union ()

%union(soln_1, soln_2, soln_3, ...) represents the union of its arguments, each of which represents a solution set, as determined by to_poly_solve. %union() represents the empty set.

In many cases, a solution is a list of equations [x = ..., y = ..., z = ...] where x, y, and z are one or more unknowns. In such cases, to_poly_solve returns a %union expression containing one or more such lists.

The solution set sometimes involves simplifing versions of various of logical operators including %and, %or, or %if for conjunction, disjuntion, and implication, respectively.

Examples:

%union(...) represents the union of its arguments, each of which represents a solution set, as determined by to_poly_solve. In many cases, a solution is a list of equations.

(%i1) load ("to_poly_solve") $
(%i2) to_poly_solve ([sqrt(x^2 - y^2), x + y], [x, y]);
(%o2)    %union([x = 0, y = 0], [x = %c13, y = - %c13])

%union() represents the empty set.

(%i1) load ("to_poly_solve") $
(%i2) to_poly_solve (abs(x) = -1, x);
(%o2)                       %union()

The solution set sometimes involves simplifing versions of various of logical operators.

(%i1) load ("to_poly_solve") $
(%i2) sol : to_poly_solve (abs(x) = a, x);
(%o2) %union(%if(isnonnegative_p(a), [x = - a], %union()), 
                      %if(isnonnegative_p(a), [x = a], %union()))
(%i3) subst (a = 42, sol);
(%o3)             %union([x = - 42], [x = 42])
(%i4) subst (a = -42, sol);
(%o4)                       %union()

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88 unit


Next: , Previous: , Up: unit-pkg   [Contents][Index]

88.1 Introduction to Units

The unit package enables the user to convert between arbitrary units and work with dimensions in equations. The functioning of this package is radically different from the original Maxima units package - whereas the original was a basic list of definitions, this package uses rulesets to allow the user to chose, on a per dimension basis, what unit final answers should be rendered in. It will separate units instead of intermixing them in the display, allowing the user to readily identify the units associated with a particular answer. It will allow a user to simplify an expression to its fundamental Base Units, as well as providing fine control over simplifying to derived units. Dimensional analysis is possible, and a variety of tools are available to manage conversion and simplification options. In addition to customizable automatic conversion, units also provides a traditional manual conversion option.

Note - when unit conversions are inexact Maxima will make approximations resulting in fractions. This is a consequence of the techniques used to simplify units. The messages warning of this type of substitution are disabled by default in the case of units (normally they are on) since this situation occurs frequently and the warnings clutter the output. (The existing state of ratprint is restored after unit conversions, so user changes to that setting will be preserved otherwise.) If the user needs this information for units, they can set unitverbose:on to reactivate the printing of warnings from the unit conversion process.

unit is included in Maxima in the share/contrib/unit directory. It obeys normal Maxima package loading conventions:

(%i1) load("unit")$
*******************************************************************
*                       Units version 0.50                        *
*          Definitions based on the NIST Reference on             *
*              Constants, Units, and Uncertainty                  *
*       Conversion factors from various sources including         *
*                   NIST and the GNU units package                *
*******************************************************************

Redefining necessary functions...
WARNING: DEFUN/DEFMACRO: redefining function TOPLEVEL-MACSYMA-EVAL ...
WARNING: DEFUN/DEFMACRO: redefining function MSETCHK ...
WARNING: DEFUN/DEFMACRO: redefining function KILL1 ...
WARNING: DEFUN/DEFMACRO: redefining function NFORMAT ...
Initializing unit arrays...
Done.

The WARNING messages are expected and not a cause for concern - they indicate the unit package is redefining functions already defined in Maxima proper. This is necessary in order to properly handle units. The user should be aware that if other changes have been made to these functions by other packages those changes will be overwritten by this loading process.

The unit.mac file also loads a lisp file unit-functions.lisp which contains the lisp functions needed for the package.

Clifford Yapp is the primary author. He has received valuable assistance from Barton Willis of the University of Nebraska at Kearney (UNK), Robert Dodier, and other intrepid folk of the Maxima mailing list.

There are probably lots of bugs. Let me know. float and numer don’t do what is expected.

TODO : dimension functionality, handling of temperature, showabbr and friends. Show examples with addition of quantities containing units.

·

Category: Physical units · Category: Share packages · Category: Package unit


Previous: , Up: unit-pkg   [Contents][Index]

88.2 Functions and Variables for Units

Function: setunits (list)

By default, the unit package does not use any derived dimensions, but will convert all units to the seven fundamental dimensions using MKS units.

(%i2) N;
                                     kg m
(%o2)                                ----
                                       2
                                      s
(%i3) dyn;
                                   1      kg m
(%o3)                           (------) (----)
                                 100000     2
                                           s
(%i4) g;
                                    1
(%o4)                             (----) (kg)
                                   1000
(%i5) centigram*inch/minutes^2;
                                  127        kg m
(%o5)                       (-------------) (----)
                             1800000000000     2
                                              s

In some cases this is the desired behavior. If the user wishes to use other units, this is achieved with the setunits command:

(%i6) setunits([centigram,inch,minute]);
(%o6)                                done
(%i7) N;
                            1800000000000   %in cg
(%o7)                      (-------------) (------)
                                 127            2
                                            %min
(%i8) dyn;
                               18000000   %in cg
(%o8)                         (--------) (------)
                                 127          2
                                          %min
(%i9) g;
(%o9)                             (100) (cg)
(%i10) centigram*inch/minutes^2;
                                    %in cg
(%o10)                              ------
                                        2
                                    %min

The setting of units is quite flexible. For example, if we want to get back to kilograms, meters, and seconds as defaults for those dimensions we can do:

(%i11) setunits([kg,m,s]);
(%o11)                               done
(%i12) centigram*inch/minutes^2;
                                  127        kg m
(%o12)                      (-------------) (----)
                             1800000000000     2
                                              s

Derived units are also handled by this command:

(%i17) setunits(N);
(%o17)                               done
(%i18) N;
(%o18)                                 N
(%i19) dyn;
                                    1
(%o19)                           (------) (N)
                                  100000
(%i20) kg*m/s^2;
(%o20)                                 N
(%i21) centigram*inch/minutes^2;
                                    127
(%o21)                        (-------------) (N)
                               1800000000000

Notice that the unit package recognized the non MKS combination of mass, length, and inverse time squared as a force, and converted it to Newtons. This is how Maxima works in general. If, for example, we prefer dyne to Newtons, we simply do the following:

(%i22) setunits(dyn);
(%o22)                               done
(%i23) kg*m/s^2;
(%o23)                          (100000) (dyn)
(%i24) centigram*inch/minutes^2;
                                  127
(%o24)                         (--------) (dyn)
                                18000000

To discontinue simplifying to any force, we use the uforget command:

(%i26) uforget(dyn);
(%o26)                               false
(%i27) kg*m/s^2;
                                     kg m
(%o27)                               ----
                                       2
                                      s
(%i28) centigram*inch/minutes^2;
                                  127        kg m
(%o28)                      (-------------) (----)
                             1800000000000     2
                                              s

This would have worked equally well with uforget(N) or uforget(%force).

See also uforget. To use this function write first load("unit").

Function: uforget (list)

By default, the unit package converts all units to the seven fundamental dimensions using MKS units. This behavior can be changed with the setunits command. After that, the user can restore the default behavior for a particular dimension by means of the uforget command:

(%i13) setunits([centigram,inch,minute]);
(%o13)                               done
(%i14) centigram*inch/minutes^2;
                                    %in cg
(%o14)                              ------
                                        2
                                    %min
(%i15) uforget([cg,%in,%min]);
(%o15)                      [false, false, false]
(%i16) centigram*inch/minutes^2;
                                  127        kg m
(%o16)                      (-------------) (----)
                             1800000000000     2
                                              s

uforget operates on dimensions, not units, so any unit of a particular dimension will work. The dimension itself is also a legal argument.

See also setunits. To use this function write first load("unit").

Function: convert (expr, list)

When resetting the global environment is overkill, there is the convert command, which allows one time conversions. It can accept either a single argument or a list of units to use in conversion. When a convert operation is done, the normal global evaluation system is bypassed, in order to avoid the desired result being converted again. As a consequence, for inexact calculations "rat" warnings will be visible if the global environment controlling this behavior (ratprint) is true. This is also useful for spot-checking the accuracy of a global conversion. Another feature is convert will allow a user to do Base Dimension conversions even if the global environment is set to simplify to a Derived Dimension.

(%i2) kg*m/s^2;
                                     kg m
(%o2)                                ----
                                       2
                                      s
(%i3) convert(kg*m/s^2,[g,km,s]);
                                     g km
(%o3)                                ----
                                       2
                                      s
(%i4) convert(kg*m/s^2,[g,inch,minute]);

`rat' replaced 39.37007874015748 by 5000/127 = 39.37007874015748
                              18000000000   %in g
(%o4)                        (-----------) (-----)
                                  127           2
                                            %min
(%i5) convert(kg*m/s^2,[N]);
(%o5)                                  N
(%i6) convert(kg*m^2/s^2,[N]);
(%o6)                                 m N
(%i7) setunits([N,J]);
(%o7)                                done
(%i8) convert(kg*m^2/s^2,[N]);
(%o8)                                 m N
(%i9) convert(kg*m^2/s^2,[N,inch]);

`rat' replaced 39.37007874015748 by 5000/127 = 39.37007874015748
                                 5000
(%o9)                           (----) (%in N)
                                 127
(%i10) convert(kg*m^2/s^2,[J]);
(%o10)                                 J
(%i11) kg*m^2/s^2;
(%o11)                                 J
(%i12) setunits([g,inch,s]);
(%o12)                               done
(%i13) kg*m/s^2;
(%o13)                                 N
(%i14) uforget(N);
(%o14)                               false
(%i15) kg*m/s^2;
                                5000000   %in g
(%o15)                         (-------) (-----)
                                  127       2
                                           s
(%i16) convert(kg*m/s^2,[g,inch,s]);

`rat' replaced 39.37007874015748 by 5000/127 = 39.37007874015748
                                5000000   %in g
(%o16)                         (-------) (-----)
                                  127       2
                                           s

See also setunits and uforget. To use this function write first load("unit").

Optional variable: usersetunits

Default value: none

If a user wishes to have a default unit behavior other than that described, they can make use of maxima-init.mac and the usersetunits variable. The unit package will check on startup to see if this variable has been assigned a list. If it has, it will use setunits on that list and take the units from that list to be defaults. uforget will revert to the behavior defined by usersetunits over its own defaults. For example, if we have a maxima-init.mac file containing:

usersetunits : [N,J];

we would see the following behavior:

(%i1) load("unit")$
*******************************************************************
*                       Units version 0.50                        *
*          Definitions based on the NIST Reference on             *
*              Constants, Units, and Uncertainty                  *
*       Conversion factors from various sources including         *
*                   NIST and the GNU units package                *
*******************************************************************

Redefining necessary functions...
WARNING: DEFUN/DEFMACRO: redefining function
 TOPLEVEL-MACSYMA-EVAL ...
WARNING: DEFUN/DEFMACRO: redefining function MSETCHK ...
WARNING: DEFUN/DEFMACRO: redefining function KILL1 ...
WARNING: DEFUN/DEFMACRO: redefining function NFORMAT ...
Initializing unit arrays...
Done.
User defaults found...
User defaults initialized.
(%i2) kg*m/s^2;
(%o2)                                  N
(%i3) kg*m^2/s^2;
(%o3)                                  J
(%i4) kg*m^3/s^2;
(%o4)                                 J m
(%i5) kg*m*km/s^2;
(%o5)                             (1000) (J)
(%i6) setunits([dyn,eV]);
(%o6)                                done
(%i7) kg*m/s^2;
(%o7)                           (100000) (dyn)
(%i8) kg*m^2/s^2;
(%o8)                     (6241509596477042688) (eV)
(%i9) kg*m^3/s^2;
(%o9)                    (6241509596477042688) (eV m)
(%i10) kg*m*km/s^2;
(%o10)                   (6241509596477042688000) (eV)
(%i11) uforget([dyn,eV]);
(%o11)                           [false, false]
(%i12) kg*m/s^2;
(%o12)                                 N
(%i13) kg*m^2/s^2;
(%o13)                                 J
(%i14) kg*m^3/s^2;
(%o14)                                J m
(%i15) kg*m*km/s^2;
(%o15)                            (1000) (J)

Without usersetunits, the initial inputs would have been converted to MKS, and uforget would have resulted in a return to MKS rules. Instead, the user preferences are respected in both cases. Notice these can still be overridden if desired. To completely eliminate this simplification - i.e. to have the user defaults reset to factory defaults - the dontusedimension command can be used. uforget can restore user settings again, but only if usedimension frees it for use. Alternately, kill(usersetunits) will completely remove all knowledge of the user defaults from the session. Here are some examples of how these various options work.

(%i2) kg*m/s^2;
(%o2)                                  N
(%i3) kg*m^2/s^2;
(%o3)                                  J
(%i4) setunits([dyn,eV]);
(%o4)                                done
(%i5) kg*m/s^2;
(%o5)                           (100000) (dyn)
(%i6) kg*m^2/s^2;
(%o6)                     (6241509596477042688) (eV)
(%i7) uforget([dyn,eV]);
(%o7)                          [false, false]
(%i8) kg*m/s^2;
(%o8)                                  N
(%i9) kg*m^2/s^2;
(%o9)                                  J
(%i10) dontusedimension(N);
(%o10)                             [%force]
(%i11) dontusedimension(J);
(%o11)                         [%energy, %force]
(%i12) kg*m/s^2;
                                     kg m
(%o12)                               ----
                                       2
                                      s
(%i13) kg*m^2/s^2;
                                         2
                                     kg m
(%o13)                               -----
                                       2
                                      s
(%i14) setunits([dyn,eV]);
(%o14)                               done
(%i15) kg*m/s^2;
                                     kg m
(%o15)                               ----
                                       2
                                      s
(%i16) kg*m^2/s^2;
                                         2
                                     kg m
(%o16)                               -----
                                       2
                                      s
(%i17) uforget([dyn,eV]);
(%o17)                         [false, false]
(%i18) kg*m/s^2;
                                     kg m
(%o18)                               ----
                                       2
                                      s
(%i19) kg*m^2/s^2;
                                         2
                                     kg m
(%o19)                               -----
                                       2
                                      s
(%i20) usedimension(N);
Done.  To have Maxima simplify to this dimension, use
setunits([unit]) to select a unit.
(%o20)                               true
(%i21) usedimension(J);
Done.  To have Maxima simplify to this dimension, use
setunits([unit]) to select a unit.
(%o21)                               true
(%i22) kg*m/s^2;
                                     kg m
(%o22)                               ----
                                       2
                                      s
(%i23) kg*m^2/s^2;
                                         2
                                     kg m
(%o23)                               -----
                                       2
                                      s
(%i24) setunits([dyn,eV]);
(%o24)                               done
(%i25) kg*m/s^2;
(%o25)                          (100000) (dyn)
(%i26) kg*m^2/s^2;
(%o26)                    (6241509596477042688) (eV)
(%i27) uforget([dyn,eV]);
(%o27)                           [false, false]
(%i28) kg*m/s^2;
(%o28)                                 N
(%i29) kg*m^2/s^2;
(%o29)                                 J
(%i30) kill(usersetunits);
(%o30)                               done
(%i31) uforget([dyn,eV]);
(%o31)                          [false, false]
(%i32) kg*m/s^2;
                                     kg m
(%o32)                               ----
                                       2
                                      s
(%i33) kg*m^2/s^2;
                                         2
                                     kg m
(%o33)                               -----
                                       2
                                      s

Unfortunately this wide variety of options is a little confusing at first, but once the user grows used to them they should find they have very full control over their working environment.

Function: metricexpandall (x)

Rebuilds global unit lists automatically creating all desired metric units. x is a numerical argument which is used to specify how many metric prefixes the user wishes defined. The arguments are as follows, with each higher number defining all lower numbers’ units:

           0 - none. Only base units
           1 - kilo, centi, milli
(default)  2 - giga, mega, kilo, hecto, deka, deci, centi, milli,
               micro, nano
           3 - peta, tera, giga, mega, kilo, hecto, deka, deci,
               centi, milli, micro, nano, pico, femto
           4 - all

Normally, Maxima will not define the full expansion since this results in a very large number of units, but metricexpandall can be used to rebuild the list in a more or less complete fashion. The relevant variable in the unit.mac file is %unitexpand.

Variable: %unitexpand

Default value: 2

This is the value supplied to metricexpandall during the initial loading of unit.


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89 wrstcse


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89.1 Introduction to wrstcse

wrstcse is a naive go at interval arithmetics is powerful enough to perform worst case calculations that appear in engineering by applying all combinations of tolerances to all parameters.

This approach isn’t guaranteed to find the exact combination of parameters that results in the worst-case. But it avoids the problems that make a true interval arithmetics affected by the halting problem as an equation can have an infinite number of local minima and maxima and it might be impossible to algorithmically determine which one is the global one.

Tolerances are applied to parameters by providing the parameter with a tol[n] that wrstcase will vary between -1 and 1. Using the same n for two parameters will make both parameters tolerate in the same way.

load ("wrstcse") loads this package.

·

Category: Share packages · Category: Package wrstcse


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89.2 Functions and Variables for wrstcse

Function: wc_typicalvalues (expression, [num])

Returns what happens if all tolerances (that are represented by tol [n] that can vary from 0 to 1) happen to be 0.

Example:

(%i1) load("wrstcse")$
(%i2) vals: [
   R_1= 1000.0*(1+tol[1]*.01),
   R_2= 2000.0*(1+tol[2]*.01)
 ];
(%o2) [R_1 = 1000.0 (0.01 tol  + 1), 
                             1
                                    R_2 = 2000.0 (0.01 tol  + 1)]
                                                          2
(%i3) divider:U_Out=U_In*R_1/(R_1+R_2);
                                R_1 U_In
(%o3)                   U_Out = ---------
                                R_2 + R_1
(%i4) wc_typicalvalues(vals);
(%o4)             [R_1 = 1000.0, R_2 = 2000.0]
(%i5) wc_typicalvalues(subst(vals,divider));
(%o5)            U_Out = 0.3333333333333333 U_In
Function: wc_inputvalueranges (expression, [num])

Convenience function: Displays a list which parameter can vary between which values.

Example:

(%i1) load("wrstcse")$
(%i2) vals: [
   R_1= 1000.0*(1+tol[1]*.01),
   R_2= 2000.0*(1+tol[2]*.01)
 ];
(%o2) [R_1 = 1000.0 (0.01 tol  + 1), 
                             1
                                    R_2 = 2000.0 (0.01 tol  + 1)]
                                                          2
(%i3) wc_inputvalueranges(vals);
        [ R_1  min = 990.0   typ = 1000.0  max = 1010.0 ]
(%o3)   [                                               ]
        [ R_2  min = 1980.0  typ = 2000.0  max = 2020.0 ]
Function: wc_systematic (expression, [num])

Systematically introduces num values per parameter into expression and returns a list of the result. If no num is given, num defaults to 3.

See also wc_montecarlo.

Example:

(%i1) load("wrstcse")$
(%i2) vals: [
   R_1= 1000.0*(1+tol[1]*.01),
   R_2= 2000.0*(1+tol[2]*.01)
 ];
(%o2) [R_1 = 1000.0 (0.01 tol  + 1), 
                             1
                                    R_2 = 2000.0 (0.01 tol  + 1)]
                                                          2
(%i3) divider: U_Out=U_In*(R_1)/(R_1+R_2);
                                R_1 U_In
(%o3)                   U_Out = ---------
                                R_2 + R_1
(%i4) wc_systematic(subst(vals,rhs(divider)));
(%o4) [0.3333333333333334 U_In, 0.3311036789297659 U_In, 
0.3289036544850498 U_In, 0.3355704697986577 U_In, 
0.3333333333333333 U_In, 0.3311258278145696 U_In, 
0.3377926421404682 U_In, 0.3355481727574751 U_In, 
0.3333333333333333 U_In]
Function: wc_montecarlo (expression, num)

Introduces num random values per parameter into expression and returns a list of the result.

See also wc_systematic.

Example:

(%i1) load("wrstcse")$
(%i2) vals: [
   R_1= 1000.0*(1+tol[1]*.01),
   R_2= 2000.0*(1+tol[2]*.01)
 ];
(%o2) [R_1 = 1000.0 (0.01 tol  + 1), 
                             1
                                    R_2 = 2000.0 (0.01 tol  + 1)]
                                                          2
(%i3) divider: U_Out=U_In*(R_1)/(R_1+R_2);
                                R_1 U_In
(%o3)                   U_Out = ---------
                                R_2 + R_1
(%i4) wc_montecarlo(subst(vals,rhs(divider)),10);
(%o4) [0.3365488313167528 U_In, 0.3339089445851889 U_In, 
0.314651402884122 U_In, 0.3447359711624277 U_In, 
0.3294005710066001 U_In, 0.3330897542463686 U_In, 
0.3397591863729343 U_In, 0.3227030530673181 U_In, 
0.3385512773502185 U_In, 0.314764470912582 U_In]
Function: wc_mintypmax (expr, [n])

Prints the minimum, maximum and typical value of expr. If n is positive, n values for each parameter will be tried systematically. If n is negative, -n random values are used instead. If no n is given, 3 is assumed.

Example:

(%i1) load("wrstcse")$
(%i2) ratprint:false$
(%i3) vals: [
   R_1= 1000.0*(1+tol[1]*.01),
   R_2= 1000.0*(1+tol[2]*.01)
 ];
(%o3) [R_1 = 1000.0 (0.01 tol  + 1), 
                             1
                                    R_2 = 1000.0 (0.01 tol  + 1)]
                                                          2
(%i4) assume(U_In>0);
(%o4)                      [U_In > 0]
(%i5) divider:U_Out=U_In*R_1/(R_1+R_2);
                                R_1 U_In
(%o5)                   U_Out = ---------
                                R_2 + R_1
(%i6) lhs(divider)=wc_mintypmax(subst(vals,rhs(divider)));
(%o6) U_Out = [min = 0.495 U_In, typ = 0.5 U_In, 
                                                max = 0.505 U_In]
Function: wc_tolappend (list)

Appends two list of parameters with tolerances renumbering the tolerances of both lists so they don’t coincide.

Example:

(%i1) load("wrstcse")$
(%i2) val_a: [
   R_1= 1000.0*(1+tol[1]*.01),
   R_2= 1000.0*(1+tol[2]*.01)
 ];
(%o2) [R_1 = 1000.0 (0.01 tol  + 1), 
                             1
                                    R_2 = 1000.0 (0.01 tol  + 1)]
                                                          2
(%i3) val_b: [
   R_3= 1000.0*(1+tol[1]*.01),
   R_4= 1000.0*(1+tol[2]*.01)
 ];
(%o3) [R_3 = 1000.0 (0.01 tol  + 1), 
                             1
                                    R_4 = 1000.0 (0.01 tol  + 1)]
                                                          2
(%i4) wc_tolappend(val_a,val_b);
(%o4) [R_1 = 1000.0 (0.01 tol  + 1), 
                             2
R_2 = 1000.0 (0.01 tol  + 1), R_3 = 1000.0 (0.01 tol  + 1), 
                      1                             4
R_4 = 1000.0 (0.01 tol  + 1)]
                      3
Function: wc_mintypmax2tol (tolname, minval, typval, maxval)

Generates a parameter that uses the tolerance tolname that tolerates between the given values.

Example:

(%i1) load("wrstcse")$
(%i2) V_F: U_Diode=wc_mintypmax2tol(tol[1],.5,.75,.82);
                                          2
(%o2) U_Diode = (- 0.09000000000000002 tol ) + 0.16 tol  + 0.75
                                          1            1
(%i3) lhs(V_F)=wc_mintypmax(rhs(V_F));
(%o3) U_Diode = [min = 0.5, typ = 0.75, max = 0.8199999999999998]

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90 zeilberger


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90.1 Introduction to zeilberger

zeilberger is a implementation of Zeilberger’s algorithm for definite hypergeometric summation, and also Gosper’s algorithm for indefinite hypergeometric summation.

zeilberger makes use of the "filtering" optimization method developed by Axel Riese.

zeilberger was developed by Fabrizio Caruso.

load ("zeilberger") loads this package.

·

Category: Sums and products · Category: Share packages · Category: Package zeilberger

90.1.1 The indefinite summation problem

zeilberger implements Gosper’s algorithm for indefinite hypergeometric summation. Given a hypergeometric term F_k in k we want to find its hypergeometric anti-difference, that is, a hypergeometric term f_k such that

F_k = f_(k+1) - f_k.

90.1.2 The definite summation problem

zeilberger implements Zeilberger’s algorithm for definite hypergeometric summation. Given a proper hypergeometric term (in n and k) F_(n,k) and a positive integer d we want to find a d-th order linear recurrence with polynomial coefficients (in n) for F_(n,k) and a rational function R in n and k such that

a_0 F_(n,k) + ... + a_d F_(n+d),k = Delta_k(R(n,k) F_(n,k)),

where Delta_k is the k-forward difference operator, i.e., Delta_k(t_k) := t_(k+1) - t_k.

90.1.3 Verbosity levels

There are also verbose versions of the commands which are called by adding one of the following prefixes:

Summary

Just a summary at the end is shown

Verbose

Some information in the intermidiate steps

VeryVerbose

More information

Extra

Even more information including information on the linear system in Zeilberger’s algorithm

For example:
GosperVerbose, parGosperVeryVerbose, ZeilbergerExtra, AntiDifferenceSummary.


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90.2 Functions and Variables for zeilberger

Function: AntiDifference (F_k, k)

Returns the hypergeometric anti-difference of F_k, if it exists.
Otherwise AntiDifference returns no_hyp_antidifference.

Function: Gosper (F_k, k)

Returns the rational certificate R(k) for F_k, that is, a rational function such that F_k = R(k+1) F_(k+1) - R(k) F_k, if it exists. Otherwise, Gosper returns no_hyp_sol.

Function: GosperSum (F_k, k, a, b)

Returns the summmation of F_k from k = a to k = b if F_k has a hypergeometric anti-difference. Otherwise, GosperSum returns nongosper_summable.

Examples:

(%i1) load ("zeilberger")$
(%i2) GosperSum ((-1)^k*k / (4*k^2 - 1), k, 1, n);
Dependent equations eliminated:  (1)
                           3       n + 1
                      (n + -) (- 1)
                           2               1
(%o2)               - ------------------ - -
                                  2        4
                      2 (4 (n + 1)  - 1)
(%i3) GosperSum (1 / (4*k^2 - 1), k, 1, n);
                                3
                          - n - -
                                2       1
(%o3)                  -------------- + -
                                2       2
                       4 (n + 1)  - 1
(%i4) GosperSum (x^k, k, 1, n);
                          n + 1
                         x          x
(%o4)                    ------ - -----
                         x - 1    x - 1
(%i5) GosperSum ((-1)^k*a! / (k!*(a - k)!), k, 1, n);
                                n + 1
                a! (n + 1) (- 1)              a!
(%o5)       - ------------------------- - ----------
              a (- n + a - 1)! (n + 1)!   a (a - 1)!
(%i6) GosperSum (k*k!, k, 1, n);
Dependent equations eliminated:  (1)
(%o6)                     (n + 1)! - 1
(%i7) GosperSum ((k + 1)*k! / (k + 1)!, k, 1, n);
                  (n + 1) (n + 2) (n + 1)!
(%o7)             ------------------------ - 1
                          (n + 2)!
(%i8) GosperSum (1 / ((a - k)!*k!), k, 1, n);
(%o8)                  NON_GOSPER_SUMMABLE

Function: parGosper (F_(n,k), k, n, d)

Attempts to find a d-th order recurrence for F_(n,k).

The algorithm yields a sequence [s_1, s_2, ..., s_m] of solutions. Each solution has the form

[R(n, k), [a_0, a_1, ..., a_d]].

parGosper returns [] if it fails to find a recurrence.

Function: Zeilberger (F_(n,k), k, n)

Attempts to compute the indefinite hypergeometric summation of F_(n,k).

Zeilberger first invokes Gosper, and if that fails to find a solution, then invokes parGosper with order 1, 2, 3, ..., up to MAX_ORD. If Zeilberger finds a solution before reaching MAX_ORD, it stops and returns the solution.

The algorithms yields a sequence [s_1, s_2, ..., s_m] of solutions. Each solution has the form

[R(n,k), [a_0, a_1, ..., a_d]].

Zeilberger returns [] if it fails to find a solution.

Zeilberger invokes Gosper only if Gosper_in_Zeilberger is true.

90.3 General global variables

Global variable: MAX_ORD

Default value: 5

MAX_ORD is the maximum recurrence order attempted by Zeilberger.

Global variable: simplified_output

Default value: false

When simplified_output is true, functions in the zeilberger package attempt further simplification of the solution.

Global variable: linear_solver

Default value: linsolve

linear_solver names the solver which is used to solve the system of equations in Zeilberger’s algorithm.

Global variable: warnings

Default value: true

When warnings is true, functions in the zeilberger package print warning messages during execution.

Global variable: Gosper_in_Zeilberger

Default value: true

When Gosper_in_Zeilberger is true, the Zeilberger function calls Gosper before calling parGosper. Otherwise, Zeilberger goes immediately to parGosper.

Global variable: trivial_solutions

Default value: true

When trivial_solutions is true, Zeilberger returns solutions which have certificate equal to zero, or all coefficients equal to zero.

90.4 Variables related to the modular test

Global variable: mod_test

Default value: false

When mod_test is true, parGosper executes a modular test for discarding systems with no solutions.

Global variable: modular_linear_solver

Default value: linsolve

modular_linear_solver names the linear solver used by the modular test in parGosper.

Global variable: ev_point

Default value: big_primes[10]

ev_point is the value at which the variable n is evaluated when executing the modular test in parGosper.

Global variable: mod_big_prime

Default value: big_primes[1]

mod_big_prime is the modulus used by the modular test in parGosper.

Global variable: mod_threshold

Default value: 4

mod_threshold is the greatest order for which the modular test in parGosper is attempted.


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91 Error and warning messages

This chapter provides detailed information about the meaning of some error messages or on how to recover from errors.


Next: , Up: Error and warning messages   [Contents][Index]

91.1 Error messages


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91.1.1 apply: no such "list" element

One common cause for this error message is that square brackets operator ([ ]) was used trying to access a list element that whose element number was < 1 or > length(list).

· Category: Error messages


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91.1.2 argument must be a non-atomic expression

This normally means that a list, a set or something else that consists of more than one element was expected. One possible cause for this error message is a construct of the following type:

(%i1) l:[1,2,3];
(%o1)                       [1, 2, 3]
(%i2) append(l,4);
append: argument must be a non-atomic expression; found 4
 -- an error. To debug this try: debugmode(true);

The correct way to append variables or numbers to a list is to wrap them in a single-element list first:

(%i1) l:[1,2,3];
(%o1)                       [1, 2, 3]
(%i2) append(l,[4]);
(%o2)                     [1, 2, 3, 4]
·

Category: Error messages


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91.1.3 assignment: cannot assign to <function name>

Maxima supports several assignment operators. When trying to define a function := has to be used.

· Category: Error messages


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91.1.4 expt: undefined: 0 to a negative exponent.

This message notifies about a classical division by zero error.

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91.1.5 incorrect syntax: , is not a prefix operator

This might be caused by a command starting with a comma (,) or by one comma being directly followed by another one..

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91.1.6 incorrect syntax: Illegal use of delimiter )

Common reasons for this error appearing are a closing parenthesis without an opening one or a closing parenthesis directly preceded by a comma.

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91.1.7 loadfile: failed to load <filename>

This error message normally indicates that the file exists, but can not be read. If the file is present and readable there is another possible for this error message: Maxima can compile packages to native binary files in order to make them run faster. If after compiling the file something in the system has changed in a way that makes it incompatible with the binary the binary the file cannot be loaded any more. Maxima normally puts binary files it creates from its own packages in a folder named binary within the folder whose name it is printed after typing:

(%i1) maxima_userdir;
(%o1)                 /home/gunter/.maxima

If this directory is missing maxima will recreate it again as soon as it has to compile a package.

· Category: Error messages


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91.1.8 makelist: second argument must evaluate to a number

makelist expects the second argument to be the name of the variable whose value is to be stepped. This time instead of the name of a still-undefined variable maxima has found something else, possibly a list or the name of a list.

·

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91.1.9 Only symbols can be bound

The most probable cause for this error is that there was an attempt to either use a number or a variable whose numerical value is known as a loop counter.

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91.1.10 operators of arguments must all be the same

One possible reason for this error message to appear is a try to use append in order to add an equation to a list:

(%i1) l:[a=1,b=2,c=3];
(%o1)                 [a = 1, b = 2, c = 3]
(%i2) append(l,d=5);
append: operators of arguments must all be the same.
 -- an error. To debug this try: debugmode(true);

In order to add an equation to a list it has to be wrapped in a single-element list first:

(%i1) l:[a=1,b=2,c=3];
(%o1)                 [a = 1, b = 2, c = 3]
(%i2) append(l,[d=5]);
(%o2)             [a = 1, b = 2, c = 3, d = 5]
·

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91.1.11 Out of memory

Lisp typically handles several types of memory containing at least one stack and a heap that contains user objects. To avoid running out of memory several approaches might be useful:

·

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91.1.12 part: fell off the end

part() was used to access the nth item in something that has less than n items.

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91.1.13 undefined variable (draw or plot)

A function could not be plotted since it still contained a variable maxima doesn’t know the value of.

In order to find out which variable this could be it is sometimes helpful to temporarily replace the name of the drawing command (draw2d, plot2d or similar) by a random name (for example ddraw2d) that doesn’t coincide with the name of an existing function to make maxima print out what parameters the drawing command sees.

(%i1) load("draw")$
(%i2) f(x):=sin(omega*t);
(%o2) f(x) := sin(omega t)
(%i3) draw2d(
        explicit(
          f(x),
          x,1,10
        )
      );
draw2d (explicit): non defined variable
 -- an error. To debug this try: debugmode(true);
(%i4) ddraw2d(
        explicit(
          f(x),
          x,1,10
        )
      );
(%o4) ddraw2d(explicit(sin(omega t), x, 1, 10))
·

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91.1.14 VTK is not installed, which is required for Scene

This might either mean that VTK is actually not installed - or cannot be found by maxima - or that maxima has no write access to the directory whose name is output if the following maxima command is entered:

(%i1) maxima_tempdir;
(%o1)                         /tmp
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91.2 Warning messages


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91.2.1 Encountered undefined variable <x> in translation

A function was compiled but the type of the variable x was not known. This means that the compiled command contains additional code that makes it retain all the flexibility maxima provides in respect to this variable. If x isn’t meant as a variable name but just a named option to a command prepending the named option by a single quote (') should resolve this issue.

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91.2.2 Rat: replaced <x> by <y> = <z>

Floating-point numbers provide a maximum number of digits that is typically high, but still limited. Good examples that this limitation might be too low even for harmless-looking examples include Wilkinson’s Polynomial, The Rump polynomial and the fact that an exact 1/10 cannot be expressed as a binary floating-point number. In places where the floating-point error might add up or hinder terms from cancelling each other out maxima therefore by default replaces them with exact fractions. See also ratprint, ratepsilon, bftorat, fpprintprec and rationalize.

· Category: Warning messages


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Appendix A Function and Variable Index

Jump to:   !   #   $   %   '   *   +   -   .   /   :   ;   <   =   >   ?   @   [   ]   ^   _   `   |   ~  
A   B   C   D   E   F   G   H   I   J   K   L   M   N   O   P   Q   R   S   T   U   V   W   X   Y   Z  
Index Entry  Section

!
!: Combinatorial Functions
!!: Combinatorial Functions

#
#: Operators for Equations

$
$: Functions and Variables for Command Line

%
%: Functions and Variables for Command Line
%%: Functions and Variables for Command Line
%and: Functions and Variables for to_poly_solve
%c: Functions and Variables for contrib_ode
%e: Functions and Variables for Constants
%edispflag: Functions and Variables for Display
%emode: Root Exponential and Logarithmic Functions
%enumer: Root Exponential and Logarithmic Functions
%e_to_numlog: Root Exponential and Logarithmic Functions
%f: Hypergeometric Functions
%gamma: Functions and Variables for Constants
%i: Functions and Variables for Constants
%iargs: Functions and Variables for Trigonometric
%if: Functions and Variables for to_poly_solve
%k1: Functions and Variables for contrib_ode
%k2: Functions and Variables for contrib_ode
%m: Hypergeometric Functions
%or: Functions and Variables for to_poly_solve
%phi: Functions and Variables for Constants
%pi: Functions and Variables for Constants
%piargs: Functions and Variables for Trigonometric
%rnum: Functions and Variables for Equations
%rnum_list: Functions and Variables for Equations
%s: Bessel Functions
%th: Functions and Variables for Command Line
%union: Functions and Variables for to_poly_solve
%union: Functions and Variables for to_poly_solve
%unitexpand: Functions and Variables for Units
%w: Hypergeometric Functions

'
': Functions and Variables for Evaluation
'': Functions and Variables for Evaluation

*
*: Arithmetic operators
**: Arithmetic operators

+
+: Arithmetic operators

-
-: Arithmetic operators

.
.: Arithmetic operators

/
/: Arithmetic operators

:
:: Assignment operators
::: Assignment operators
::=: Assignment operators
:=: Assignment operators

;
;: Functions and Variables for Command Line

<
<: Relational operators
<=: Relational operators

=
=: Operators for Equations

>
>: Relational operators
>=: Relational operators

?
?: Functions and Variables for Command Line
??: Functions and Variables for Command Line

@
@: Functions and Variables for Structures

[
[: Functions and Variables for Lists

]
]: Functions and Variables for Lists

^
^: Arithmetic operators
^^: Arithmetic operators

_
_: Functions and Variables for Command Line
__: Functions and Variables for Command Line

`
`: Functions and Variables for ezunits
``: Functions and Variables for ezunits

|
|: Functions and Variables for itensor

~
~: Functions and Variables for itensor

A
abasep: Functions and Variables for atensor
abs: Functions for Numbers
absboxchar: Functions and Variables for Display
absint: Functions and Variables for Fourier series
absolute_real_time: Functions and Variables for Runtime Environment
acos: Functions and Variables for Trigonometric
acosh: Functions and Variables for Trigonometric
acot: Functions and Variables for Trigonometric
acoth: Functions and Variables for Trigonometric
acsc: Functions and Variables for Trigonometric
acsch: Functions and Variables for Trigonometric
activate: Functions and Variables for Facts
activecontexts: Functions and Variables for Facts
adapt_depth: Plotting Options
adapt_depth: Functions and Variables for draw
addcol: Functions and Variables for Matrices and Linear Algebra
Addition: Arithmetic operators
additive: Functions and Variables for Simplification
addmatrices: Functions and Variables for linearalgebra
addrow: Functions and Variables for Matrices and Linear Algebra
add_edge: Functions and Variables for graphs
add_edges: Functions and Variables for graphs
add_vertex: Functions and Variables for graphs
add_vertices: Functions and Variables for graphs
adim: Functions and Variables for atensor
adjacency_matrix: Functions and Variables for graphs
adjoin: Functions and Variables for Sets
adjoint: Functions and Variables for Matrices and Linear Algebra
adjust_external_format: Characters
af: Functions and Variables for atensor
aform: Functions and Variables for atensor
agd: Package functs
airy_ai: Airy Functions
airy_bi: Airy Functions
airy_dai: Airy Functions
airy_dbi: Airy Functions
algebraic: Functions and Variables for Polynomials
algepsilon: Functions and Variables for Equations
algexact: Functions and Variables for Equations
algsys: Functions and Variables for Equations
alg_type: Functions and Variables for atensor
alias: Functions and Variables for Expressions
aliases: Functions and Variables for Expressions
allbut: Functions and Variables for Expressions
allocation: Functions and Variables for draw
allroots: Functions and Variables for Equations
allsym: Functions and Variables for itensor
all_dotsimp_denoms: Functions and Variables for Affine
alphabetic: Functions and Variables for Properties
alphacharp: Characters
alphanumericp: Characters
amortization: Functions and Variables for finance
and: Logical operators
animation: Visualization with VTK
annuity_fv: Functions and Variables for finance
annuity_pv: Functions and Variables for finance
antid: Functions and Variables for Differentiation
antidiff: Functions and Variables for Differentiation
AntiDifference: Functions and Variables for zeilberger
antisymmetric: Functions and Variables for Simplification
append: Functions and Variables for Lists
appendfile: Functions and Variables for File Input and Output
apply: Functions and Variables for Function Definition
apply1: Functions and Variables for Rules and Patterns
apply2: Functions and Variables for Rules and Patterns
applyb1: Functions and Variables for Rules and Patterns
apply_cycles: Functions and Variables for Combinatorics
apropos: Functions and Variables for Help
args: Functions and Variables for Expressions
arithmetic: Package functs
arithsum: Package functs
arit_amortization: Functions and Variables for finance
array: Functions and Variables for Arrays
arrayapply: Functions and Variables for Arrays
arrayinfo: Functions and Variables for Arrays
arraymake: Functions and Variables for Arrays
arrays: Functions and Variables for Arrays
arraysetapply: Functions and Variables for Arrays
ascii: Characters
asec: Functions and Variables for Trigonometric
asech: Functions and Variables for Trigonometric
asin: Functions and Variables for Trigonometric
asinh: Functions and Variables for Trigonometric
askexp: Functions and Variables for Miscellaneous Options
askinteger: Functions and Variables for Facts
asksign: Functions and Variables for Facts
assoc: Functions and Variables for Lists
assoc_legendre_p: Functions and Variables for orthogonal polynomials
assoc_legendre_q: Functions and Variables for orthogonal polynomials
assume: Functions and Variables for Facts
assumescalar: Functions and Variables for Facts
assume_external_byte_order: Functions and Variables for binary input and output
assume_pos: Functions and Variables for Facts
assume_pos_pred: Functions and Variables for Facts
asymbol: Functions and Variables for atensor
asympa: Introduction to asympa
at: Functions and Variables for Differentiation
atan: Functions and Variables for Trigonometric
atan2: Functions and Variables for Trigonometric
atanh: Functions and Variables for Trigonometric
atensimp: Functions and Variables for atensor
atom: Functions and Variables for Expressions
atomgrad: Functions and Variables for Differentiation
atrig1: Functions and Variables for Trigonometric
atvalue: Functions and Variables for Differentiation
augcoefmatrix: Functions and Variables for Matrices and Linear Algebra
augmented_lagrangian_method: Functions and Variables for augmented_lagrangian
av: Functions and Variables for atensor
average_degree: Functions and Variables for graphs
axes: Plotting Options
axis_3d: Functions and Variables for draw
axis_bottom: Functions and Variables for draw
axis_left: Functions and Variables for draw
axis_right: Functions and Variables for draw
axis_top: Functions and Variables for draw
azimuth: Plotting Options
azimuth: Visualization with VTK

B
background: Visualization with VTK
background_color: Functions and Variables for draw
backslash: Introduction to Strings
backsubst: Functions and Variables for Equations
backtrace: Functions and Variables for Program Flow
bars: Functions and Variables for draw
barsplot: Functions and Variables for statistical graphs
barsplot_description: Functions and Variables for statistical graphs
base64: Octets and Utilities for Cryptography
base64_decode: Octets and Utilities for Cryptography
bashindices: Functions and Variables for Sums and Products
batch: Functions and Variables for File Input and Output
batchload: Functions and Variables for File Input and Output
bc2: Functions and Variables for Differential Equations
bdvac: Functions and Variables for ctensor
belln: Functions and Variables for Sets
benefit_cost: Functions and Variables for finance
berlefact: Functions and Variables for Polynomials
bern: Functions and Variables for Number Theory
bernpoly: Functions and Variables for Number Theory
bernstein_approx: Functions and Variables for Bernstein
bernstein_expand: Functions and Variables for Bernstein
bernstein_explicit: Functions and Variables for Bernstein
bernstein_poly: Functions and Variables for Bernstein
besselexpand: Bessel Functions
bessel_i: Bessel Functions
bessel_j: Bessel Functions
bessel_k: Bessel Functions
bessel_simplify: Functions and Variables for contrib_ode
bessel_y: Bessel Functions
beta: Gamma and factorial Functions
beta_args_sum_to_integer: Gamma and factorial Functions
beta_expand: Gamma and factorial Functions
beta_incomplete: Gamma and factorial Functions
beta_incomplete_generalized: Gamma and factorial Functions
beta_incomplete_regularized: Gamma and factorial Functions
bezout: Functions and Variables for Polynomials
bfallroots: Functions and Variables for Equations
bffac: Gamma and factorial Functions
bfhzeta: Functions and Variables for Number Theory
bfloat: Functions and Variables for Numbers
bfloatp: Functions and Variables for Numbers
bfpsi: Gamma and factorial Functions
bfpsi0: Gamma and factorial Functions
bftorat: Functions and Variables for Numbers
bftrunc: Functions and Variables for Numbers
bfzeta: Functions and Variables for Number Theory
bf_fft: Functions and Variables for fast Fourier transform
bf_find_root: Functions for numerical solution of equations
bf_find_root: Functions for numerical solution of equations
bf_fmin_cobyla: Functions and Variables for cobyla
bf_inverse_fft: Functions and Variables for fast Fourier transform
bf_inverse_real_fft: Functions and Variables for fast Fourier transform
bf_real_fft: Functions and Variables for fast Fourier transform
biconnected_components: Functions and Variables for graphs
bimetric: Functions and Variables for ctensor
bindtest: Functions and Variables for Properties
binomial: Combinatorial Functions
bipartition: Functions and Variables for graphs
bit_and: Functions and Variables for bitwise
bit_length: Functions and Variables for bitwise
bit_lsh: Functions and Variables for bitwise
bit_not: Functions and Variables for bitwise
bit_onep: Functions and Variables for bitwise
bit_or: Functions and Variables for bitwise
bit_rsh: Functions and Variables for bitwise
bit_xor: Functions and Variables for bitwise
block: Functions and Variables for Function Definition
blockmatrixp: Functions and Variables for linearalgebra
bode_gain: Functions and Variables for bode
bode_phase: Functions and Variables for bode
border: Functions and Variables for draw
bothcoef: Functions and Variables for Polynomials
boundaries_array: Functions and Variables for worldmap
box: Functions and Variables for Expressions
box: Plotting Options
boxchar: Functions and Variables for Expressions
boxplot: Functions and Variables for statistical graphs
boxplot_description: Functions and Variables for statistical graphs
break: Functions and Variables for Function Definition
breakup: Functions and Variables for Equations
bug_report: Functions and Variables for Bug Detection and Reporting
buildq: Macros
build_info: Functions and Variables for Bug Detection and Reporting
build_sample: Functions and Variables for data manipulation
burn: Functions and Variables for Number Theory

C
cabs: Functions for Complex Numbers
canform: Functions and Variables for itensor
canten: Functions and Variables for itensor
capping: Functions and Variables for draw
capping: Visualization with VTK
cardinality: Functions and Variables for Sets
carg: Functions for Complex Numbers
cartan: Functions and Variables for Differentiation
cartesian_product: Functions and Variables for Sets
catch: Functions and Variables for Function Definition
cauchysum: Functions and Variables for Series
cauchy_matrix: Functions and Variables for Matrices and Linear Algebra
cbffac: Gamma and factorial Functions
cbrange: Functions and Variables for draw
cbtics: Functions and Variables for draw
cdf_bernoulli: Functions and Variables for discrete distributions
cdf_beta: Functions and Variables for continuous distributions
cdf_binomial: Functions and Variables for discrete distributions
cdf_cauchy: Functions and Variables for continuous distributions
cdf_chi2: Functions and Variables for continuous distributions
cdf_continuous_uniform: Functions and Variables for continuous distributions
cdf_discrete_uniform: Functions and Variables for discrete distributions
cdf_empirical: Functions and Variables for descriptive statistics
cdf_exp: Functions and Variables for continuous distributions
cdf_f: Functions and Variables for continuous distributions
cdf_gamma: Functions and Variables for continuous distributions
cdf_general_finite_discrete: Functions and Variables for discrete distributions
cdf_geometric: Functions and Variables for discrete distributions
cdf_gumbel: Functions and Variables for continuous distributions
cdf_hypergeometric: Functions and Variables for discrete distributions
cdf_laplace: Functions and Variables for continuous distributions
cdf_logistic: Functions and Variables for continuous distributions
cdf_lognormal: Functions and Variables for continuous distributions
cdf_negative_binomial: Functions and Variables for discrete distributions
cdf_noncentral_chi2: Functions and Variables for continuous distributions
cdf_noncentral_student_t: Functions and Variables for continuous distributions
cdf_normal: Functions and Variables for continuous distributions
cdf_pareto: Functions and Variables for continuous distributions
cdf_poisson: Functions and Variables for discrete distributions
cdf_rank_sum: Functions and Variables for special distributions
cdf_rayleigh: Functions and Variables for continuous distributions
cdf_signed_rank: Functions and Variables for special distributions
cdf_student_t: Functions and Variables for continuous distributions
cdf_weibull: Functions and Variables for continuous distributions
cdisplay: Functions and Variables for ctensor
ceiling: Functions for Numbers
center: Visualization with VTK
central_moment: Functions and Variables for descriptive statistics
cequal: Characters
cequalignore: Characters
cf: Functions and Variables for Number Theory
cfdisrep: Functions and Variables for Number Theory
cfexpand: Functions and Variables for Number Theory
cflength: Functions and Variables for Number Theory
cframe_flag: Functions and Variables for ctensor
cgeodesic: Functions and Variables for ctensor
cgreaterp: Characters
cgreaterpignore: Characters
changename: Functions and Variables for itensor
changevar: Functions and Variables for Integration
chaosgame: Graphical analysis of discrete dynamical systems
charat: String Processing
charfun: Functions and Variables for Predicates
charfun2: Functions and Variables for interpol
charlist: String Processing
charp: Characters
charpoly: Functions and Variables for Matrices and Linear Algebra
chdir: Directory operations
chebyshev_t: Functions and Variables for orthogonal polynomials
chebyshev_u: Functions and Variables for orthogonal polynomials
checkdiv: Functions and Variables for ctensor
check_overlaps: Functions and Variables for Affine
chinese: Functions and Variables for Number Theory
cholesky: Functions and Variables for linearalgebra
christof: Functions and Variables for ctensor
chromatic_index: Functions and Variables for graphs
chromatic_number: Functions and Variables for graphs
cint: Characters
circulant_graph: Functions and Variables for graphs
clear_edge_weight: Functions and Variables for graphs
clear_rules: Functions and Variables for Rules and Patterns
clear_vertex_label: Functions and Variables for graphs
clebsch_gordan: Functions and Variables for clebsch_gordan
clebsch_graph: Functions and Variables for graphs
clessp: Characters
clesspignore: Characters
close: Input and Output
closefile: Functions and Variables for File Input and Output
cmetric: Functions and Variables for ctensor
cnonmet_flag: Functions and Variables for ctensor
coeff: Functions and Variables for Polynomials
coefmatrix: Functions and Variables for Matrices and Linear Algebra
cograd: Functions and Variables for ctensor
col: Functions and Variables for Matrices and Linear Algebra
collapse: Functions and Variables for Expressions
collectterms: Package facexp
color: Plotting Options
color: Functions and Variables for draw
color: Visualization with VTK
colorbox: Functions and Variables for draw
color_bar: Plotting Options
color_bar_tics: Plotting Options
columnop: Functions and Variables for linearalgebra
columns: Functions and Variables for draw
columnspace: Functions and Variables for linearalgebra
columnswap: Functions and Variables for linearalgebra
columnvector: Functions and Variables for Matrices and Linear Algebra
combination: Package functs
combine: Functions and Variables for Simplification
commutative: Functions and Variables for Simplification
comp2pui: Functions and Variables for Symmetries
compare: Functions and Variables for Predicates
compfile: Functions and Variables for Function Definition
compile: Functions and Variables for Function Definition
compile_file: Functions and Variables for Function Definition
complement_graph: Functions and Variables for graphs
complete_bipartite_graph: Functions and Variables for graphs
complete_graph: Functions and Variables for graphs
complex: Functions and Variables for Properties
complex_number_p: Functions and Variables for to_poly_solve
components: Functions and Variables for itensor
compose_functions: Functions and Variables for to_poly_solve
concan: Functions and Variables for itensor
concat: Functions and Variables for Strings
cone: Visualization with VTK
conjugate: Functions for Complex Numbers
conmetderiv: Functions and Variables for itensor
connected_components: Functions and Variables for graphs
connect_vertices: Functions and Variables for graphs
cons: Functions and Variables for Lists
constant: Functions and Variables for Properties
constantp: Functions and Variables for Properties
constituent: Characters
constvalue: Functions and Variables for ezunits
cont2part: Functions and Variables for Symmetries
content: Functions and Variables for Polynomials
context: Functions and Variables for Facts
contexts: Functions and Variables for Facts
continuous_freq: Functions and Variables for data manipulation
contortion: Functions and Variables for ctensor
contour: Functions and Variables for draw
contour_levels: Functions and Variables for draw
contour_plot: Functions and Variables for Plotting
contract: Functions and Variables for itensor
contract: Functions and Variables for Symmetries
contract_edge: Functions and Variables for graphs
contragrad: Functions and Variables for ctensor
contrib_ode: Functions and Variables for contrib_ode
convert: Functions and Variables for Units
coord: Functions and Variables for itensor
copy: Functions and Variables for Expressions
copylist: Functions and Variables for Lists
copymatrix: Functions and Variables for Matrices and Linear Algebra
copy_file: File operations
copy_graph: Functions and Variables for graphs
cor: Functions and Variables for descriptive statistics
cos: Functions and Variables for Trigonometric
cosh: Functions and Variables for Trigonometric
cosnpiflag: Functions and Variables for Fourier series
cot: Functions and Variables for Trigonometric
coth: Functions and Variables for Trigonometric
cov: Functions and Variables for descriptive statistics
cov1: Functions and Variables for descriptive statistics
covdiff: Functions and Variables for itensor
covect: Functions and Variables for Matrices and Linear Algebra
covers: Package functs
crc24sum: Octets and Utilities for Cryptography
create_graph: Functions and Variables for graphs
create_list: Functions and Variables for Lists
csc: Functions and Variables for Trigonometric
csch: Functions and Variables for Trigonometric
csetup: Functions and Variables for ctensor
cspline: Functions and Variables for interpol
ctaylor: Functions and Variables for ctensor
ctaypov: Functions and Variables for ctensor
ctaypt: Functions and Variables for ctensor
ctayswitch: Functions and Variables for ctensor
ctayvar: Functions and Variables for ctensor
ctorsion_flag: Functions and Variables for ctensor
ctransform: Functions and Variables for ctensor
ctranspose: Functions and Variables for linearalgebra
ctrgsimp: Functions and Variables for ctensor
ct_coords: Functions and Variables for ctensor
ct_coordsys: Functions and Variables for ctensor
cube: Visualization with VTK
cube_graph: Functions and Variables for graphs
cuboctahedron_graph: Functions and Variables for graphs
current_let_rule_package: Functions and Variables for Rules and Patterns
cv: Functions and Variables for descriptive statistics
cyclep: Functions and Variables for Combinatorics
cycle_digraph: Functions and Variables for graphs
cycle_graph: Functions and Variables for graphs
cylinder: Visualization with VTK
cylindrical: Functions and Variables for draw

D
data_file_name: Functions and Variables for draw
days360: Functions and Variables for finance
dblint: Functions and Variables for Integration
deactivate: Functions and Variables for Facts
debugmode: Functions and Variables for Debugging
declare: Functions and Variables for Properties
declare_constvalue: Functions and Variables for ezunits
declare_dimensions: Functions and Variables for ezunits
declare_fundamental_dimensions: Functions and Variables for ezunits
declare_fundamental_units: Functions and Variables for ezunits
declare_qty: Functions and Variables for ezunits
declare_translated: Functions and Variables for Function Definition
declare_units: Functions and Variables for ezunits
declare_unit_conversion: Functions and Variables for ezunits
declare_weights: Functions and Variables for Affine
decode_time: Functions and Variables for Runtime Environment
decreasing: Functions and Variables for Properties
decsym: Functions and Variables for itensor
default_let_rule_package: Functions and Variables for Rules and Patterns
defcon: Functions and Variables for itensor
define: Functions and Variables for Function Definition
define_alt_display: Functions and Variables for alt-display
define_opproperty: Functions and Variables for Simplification
define_variable: Functions and Variables for Function Definition
defint: Functions and Variables for Integration
defmatch: Functions and Variables for Rules and Patterns
defrule: Functions and Variables for Rules and Patterns
defstruct: Functions and Variables for Structures
deftaylor: Functions and Variables for Series
degree_sequence: Functions and Variables for graphs
del: Functions and Variables for Differentiation
delay: Functions and Variables for draw
delete: Functions and Variables for Lists
deleten: Functions and Variables for ctensor
delete_file: File operations
delta: Functions and Variables for Differentiation
demo: Functions and Variables for Help
demoivre: Functions and Variables for Simplification
demoivre: Functions and Variables for Simplification
denom: Functions and Variables for Polynomials
dependencies: Functions and Variables for Differentiation
dependencies: Functions and Variables for Differentiation
depends: Functions and Variables for Differentiation
derivabbrev: Functions and Variables for Differentiation
derivdegree: Functions and Variables for Differentiation
derivlist: Functions and Variables for Differentiation
derivsubst: Functions and Variables for Differentiation
describe: Functions and Variables for Help
desolve: Functions and Variables for Differential Equations
determinant: Functions and Variables for Matrices and Linear Algebra
detout: Functions and Variables for Matrices and Linear Algebra
dfloat: Functions and Variables for to_poly_solve
dgauss_a: Functions and Variables for contrib_ode
dgauss_b: Functions and Variables for contrib_ode
dgeev: Functions and Variables for lapack
dgemm: Functions and Variables for lapack
dgeqrf: Functions and Variables for lapack
dgesv: Functions and Variables for lapack
dgesvd: Functions and Variables for lapack
diag: Functions and Variables for diag
diagmatrix: Functions and Variables for Matrices and Linear Algebra
diagmatrixp: Functions and Variables for ctensor
diagmetric: Functions and Variables for ctensor
diag_matrix: Functions and Variables for linearalgebra
diameter: Functions and Variables for graphs
diff: Functions and Variables for Differentiation
diff: Functions and Variables for Differentiation
diff: Functions and Variables for itensor
digitcharp: Characters
dim: Functions and Variables for ctensor
dimacs_export: Functions and Variables for graphs
dimacs_import: Functions and Variables for graphs
dimension: Functions and Variables for Equations
dimensionless: Functions and Variables for ezunits
dimensions: Functions and Variables for draw
dimensions: Functions and Variables for ezunits
dimensions_as_list: Functions and Variables for ezunits
direct: Functions and Variables for Symmetries
directory: Functions and Variables for File Input and Output
discrete_freq: Functions and Variables for data manipulation
disjoin: Functions and Variables for Sets
disjointp: Functions and Variables for Sets
disolate: Functions and Variables for Expressions
disp: Functions and Variables for Display
dispcon: Functions and Variables for itensor
dispflag: Functions and Variables for Equations
dispform: Functions and Variables for Expressions
dispfun: Functions and Variables for Function Definition
dispJordan: Functions and Variables for diag
display: Functions and Variables for Display
display2d: Functions and Variables for Display
display_format_internal: Functions and Variables for Display
disprule: Functions and Variables for Rules and Patterns
dispterms: Functions and Variables for Display
distrib: Functions and Variables for Simplification
distribute_over: Functions and Variables for Simplification
divide: Functions and Variables for Polynomials
Division: Arithmetic operators
divisors: Functions and Variables for Sets
divsum: Functions and Variables for Number Theory
dkummer_m: Functions and Variables for contrib_ode
dkummer_u: Functions and Variables for contrib_ode
dlange: Functions and Variables for lapack
dlsode_init: Functions and Variables for odepack
dlsode_step: Functions and Variables for odepack
do: Functions and Variables for Program Flow
doallmxops: Functions and Variables for Matrices and Linear Algebra
dodecahedron_graph: Functions and Variables for graphs
domain: Functions and Variables for Simplification
domxexpt: Functions and Variables for Matrices and Linear Algebra
domxmxops: Functions and Variables for Matrices and Linear Algebra
domxnctimes: Functions and Variables for Matrices and Linear Algebra
dontfactor: Functions and Variables for Matrices and Linear Algebra
doscmxops: Functions and Variables for Matrices and Linear Algebra
doscmxplus: Functions and Variables for Matrices and Linear Algebra
dot0nscsimp: Functions and Variables for Matrices and Linear Algebra
dot0simp: Functions and Variables for Matrices and Linear Algebra
dot1simp: Functions and Variables for Matrices and Linear Algebra
dotassoc: Functions and Variables for Matrices and Linear Algebra
dotconstrules: Functions and Variables for Matrices and Linear Algebra
dotdistrib: Functions and Variables for Matrices and Linear Algebra
dotexptsimp: Functions and Variables for Matrices and Linear Algebra
dotident: Functions and Variables for Matrices and Linear Algebra
dotproduct: Functions and Variables for linearalgebra
dotscrules: Functions and Variables for Matrices and Linear Algebra
dotsimp: Functions and Variables for Affine
dpart: Functions and Variables for Expressions
draw: Functions and Variables for draw
draw2d: Functions and Variables for draw
draw3d: Functions and Variables for draw
drawdf: Functions and Variables for drawdf
draw_file: Functions and Variables for draw
draw_graph: Functions and Variables for graphs
draw_graph_program: Functions and Variables for graphs
draw_realpart: Functions and Variables for draw
dscalar: Functions and Variables for ctensor

E
echelon: Functions and Variables for Matrices and Linear Algebra
edges: Functions and Variables for graphs
edge_color: Functions and Variables for graphs
edge_coloring: Functions and Variables for graphs
edge_coloring: Functions and Variables for graphs
edge_connectivity: Functions and Variables for graphs
edge_partition: Functions and Variables for graphs
edge_type: Functions and Variables for graphs
edge_width: Functions and Variables for graphs
eigens_by_jacobi: Functions and Variables for linearalgebra
eigenvalues: Functions and Variables for Matrices and Linear Algebra
eigenvectors: Functions and Variables for Matrices and Linear Algebra
eighth: Functions and Variables for Lists
einstein: Functions and Variables for ctensor
eivals: Functions and Variables for Matrices and Linear Algebra
eivects: Functions and Variables for Matrices and Linear Algebra
elapsed_real_time: Functions and Variables for Runtime Environment
elapsed_run_time: Functions and Variables for Runtime Environment
ele2comp: Functions and Variables for Symmetries
ele2polynome: Functions and Variables for Symmetries
ele2pui: Functions and Variables for Symmetries
elem: Functions and Variables for Symmetries
elementp: Functions and Variables for Sets
elevation: Plotting Options
elevation: Visualization with VTK
elevation_grid: Functions and Variables for draw
elim: Functions and Variables for to_poly_solve
eliminate: Functions and Variables for Polynomials
eliminate_using: Functions and Variables for to_poly_solve
elim_allbut: Functions and Variables for to_poly_solve
ellipse: Functions and Variables for draw
elliptic_e: Functions and Variables for Elliptic Integrals
elliptic_ec: Functions and Variables for Elliptic Integrals
elliptic_eu: Functions and Variables for Elliptic Integrals
elliptic_f: Functions and Variables for Elliptic Integrals
elliptic_kc: Functions and Variables for Elliptic Integrals
elliptic_pi: Functions and Variables for Elliptic Integrals
ematrix: Functions and Variables for Matrices and Linear Algebra
emptyp: Functions and Variables for Sets
empty_graph: Functions and Variables for graphs
encode_time: Functions and Variables for Runtime Environment
endcons: Functions and Variables for Lists
endphi: Visualization with VTK
endtheta: Visualization with VTK
engineering_format_floats: Functions and Variables for engineering-format
engineering_format_max: Functions and Variables for engineering-format
engineering_format_min: Functions and Variables for engineering-format
enhanced3d: Functions and Variables for draw
entermatrix: Functions and Variables for Matrices and Linear Algebra
entertensor: Functions and Variables for itensor
entier: Functions for Numbers
epsilon_lp: Functions and Variables for simplex
equal: Functions and Variables for Predicates
equalp: Functions and Variables for Fourier series
equiv_classes: Functions and Variables for Sets
erf: Error Function
erfc: Error Function
erfflag: Functions and Variables for Integration
erfi: Error Function
erf_generalized: Error Function
erf_representation: Error Function
errcatch: Functions and Variables for Program Flow
error: Functions and Variables for Program Flow
error: Functions and Variables for Program Flow
errormsg: Functions and Variables for Program Flow
errormsg: Functions and Variables for Program Flow
errors: Functions and Variables for draw
error_size: Functions and Variables for Program Flow
error_syms: Functions and Variables for Program Flow
error_type: Functions and Variables for draw
euler: Functions and Variables for Number Theory
ev: Functions and Variables for Evaluation
eval: Functions and Variables for Evaluation
eval_string: String Processing
even: Functions and Variables for Properties
evenfun: Functions and Variables for Simplification
evenp: Functions and Variables for Numbers
every: Functions and Variables for Sets
evflag: Functions and Variables for Evaluation
evfun: Functions and Variables for Evaluation
evolution: Graphical analysis of discrete dynamical systems
evolution2d: Graphical analysis of discrete dynamical systems
evundiff: Functions and Variables for itensor
ev_point: Functions and Variables for zeilberger
example: Functions and Variables for Help
exp: Root Exponential and Logarithmic Functions
expand: Functions and Variables for Simplification
expandwrt: Functions and Variables for Simplification
expandwrt_denom: Functions and Variables for Simplification
expandwrt_factored: Functions and Variables for Simplification
expintegral_chi: Exponential Integrals
expintegral_ci: Exponential Integrals
expintegral_e: Exponential Integrals
expintegral_e1: Exponential Integrals
expintegral_ei: Exponential Integrals
expintegral_e_simplify: Functions and Variables for contrib_ode
expintegral_li: Exponential Integrals
expintegral_shi: Exponential Integrals
expintegral_si: Exponential Integrals
expintexpand: Exponential Integrals
expintrep: Exponential Integrals
explicit: Functions and Variables for draw
explose: Functions and Variables for Symmetries
expon: Functions and Variables for Simplification
exponentialize: Functions and Variables for Simplification
exponentialize: Functions and Variables for Simplification
Exponentiation: Arithmetic operators
expop: Functions and Variables for Simplification
express: Functions and Variables for Differentiation
expt: Functions and Variables for Display
exptdispflag: Functions and Variables for Display
exptisolate: Functions and Variables for Expressions
exptsubst: Functions and Variables for Expressions
exsec: Package functs
extdiff: Functions and Variables for itensor
extract_linear_equations: Functions and Variables for Affine
extremal_subset: Functions and Variables for Sets
ezgcd: Functions and Variables for Polynomials

F
f90: Package f90
f90_output_line_length_max: Package f90
facexpand: Functions and Variables for Polynomials
facsum: Package facexp
facsum_combine: Package facexp
factcomb: Combinatorial Functions
factlim: Combinatorial Functions
factor: Functions and Variables for Polynomials
factorfacsum: Package facexp
factorflag: Functions and Variables for Polynomials
factorial: Combinatorial Functions
factorial_expand: Combinatorial Functions
factorout: Functions and Variables for Polynomials
factorsum: Functions and Variables for Polynomials
factors_only: Functions and Variables for Number Theory
factor_max_degree: Functions and Variables for Polynomials
factor_max_degree_print_warning: Functions and Variables for Polynomials
facts: Functions and Variables for Facts
false: Functions and Variables for Constants
fasttimes: Functions and Variables for Polynomials
fast_central_elements: Functions and Variables for Affine
fast_linsolve: Functions and Variables for Affine
fb: Functions and Variables for ctensor
feature: Functions and Variables for Properties
featurep: Functions and Variables for Properties
features: Functions and Variables for Properties
fernfale: Definitions for IFS fractals
fft: Functions and Variables for fast Fourier transform
fib: Functions and Variables for Number Theory
fibtophi: Functions and Variables for Number Theory
fifth: Functions and Variables for Lists
filename_merge: Functions and Variables for File Input and Output
file_name: Functions and Variables for draw
file_name: Functions and Variables for graphs
file_output_append: Functions and Variables for File Input and Output
file_search: Functions and Variables for File Input and Output
file_search_demo: Functions and Variables for File Input and Output
file_search_lisp: Functions and Variables for File Input and Output
file_search_maxima: Functions and Variables for File Input and Output
file_search_tests: Functions and Variables for File Input and Output
file_search_usage: Functions and Variables for File Input and Output
file_type: Functions and Variables for File Input and Output
file_type_lisp: Functions and Variables for File Input and Output
file_type_maxima: Functions and Variables for File Input and Output
fillarray: Functions and Variables for Arrays
filled_func: Functions and Variables for draw
fill_color: Functions and Variables for draw
fill_density: Functions and Variables for draw
findde: Functions and Variables for ctensor
find_root: Functions for numerical solution of equations
find_root: Functions for numerical solution of equations
find_root_abs: Functions for numerical solution of equations
find_root_error: Functions for numerical solution of equations
find_root_rel: Functions for numerical solution of equations
first: Functions and Variables for Lists
firstn: Functions and Variables for Lists
fix: Functions for Numbers
fixed_vertices: Functions and Variables for graphs
flatten: Functions and Variables for Sets
flength: Input and Output
flipflag: Functions and Variables for itensor
float: Functions and Variables for Numbers
float2bf: Functions and Variables for Numbers
floatnump: Functions and Variables for Numbers
floor: Functions for Numbers
flower_snark: Functions and Variables for graphs
flush: Functions and Variables for itensor
flush1deriv: Functions and Variables for itensor
flushd: Functions and Variables for itensor
flushnd: Functions and Variables for itensor
flush_output: Input and Output
fmin_cobyla: Functions and Variables for cobyla
font: Functions and Variables for draw
font_size: Functions and Variables for draw
for: Functions and Variables for Program Flow
forget: Functions and Variables for Facts
fortindent: Functions and Variables for Fortran Output
fortran: Functions and Variables for Fortran Output
fortspaces: Functions and Variables for Fortran Output
fourcos: Functions and Variables for Fourier series
fourexpand: Functions and Variables for Fourier series
fourier: Functions and Variables for Fourier series
fourier_elim: Functions and Variables for to_poly_solve
fourint: Functions and Variables for Fourier series
fourintcos: Functions and Variables for Fourier series
fourintsin: Functions and Variables for Fourier series
foursimp: Functions and Variables for Fourier series
foursin: Functions and Variables for Fourier series
fourth: Functions and Variables for Lists
fposition: Input and Output
fpprec: Functions and Variables for Numbers
fpprintprec: Functions and Variables for Numbers
frame_bracket: Functions and Variables for ctensor
freeof: Functions and Variables for Expressions
freshline: Input and Output
fresnel_c: Error Function
fresnel_s: Error Function
from: Functions and Variables for Program Flow
from_adjacency_matrix: Functions and Variables for graphs
frucht_graph: Functions and Variables for graphs
fullmap: Functions and Variables for Function Definition
fullmapl: Functions and Variables for Function Definition
fullratsimp: Functions and Variables for Polynomials
fullratsubst: Functions and Variables for Polynomials
fullsetify: Functions and Variables for Sets
full_listify: Functions and Variables for Sets
funcsolve: Functions and Variables for Equations
functions: Functions and Variables for Function Definition
fundamental_dimensions: Functions and Variables for ezunits
fundamental_units: Functions and Variables for ezunits
fundef: Functions and Variables for Function Definition
funmake: Functions and Variables for Function Definition
funp: Functions and Variables for Fourier series
fv: Functions and Variables for finance

G
gamma: Gamma and factorial Functions
gammalim: Gamma and factorial Functions
gamma_expand: Gamma and factorial Functions
gamma_incomplete: Gamma and factorial Functions
gamma_incomplete_generalized: Gamma and factorial Functions
gamma_incomplete_lower: Gamma and factorial Functions
gamma_incomplete_regularized: Gamma and factorial Functions
garbage_collect: Garbage Collection
gaussprob: Package functs
gauss_a: Functions and Variables for contrib_ode
gauss_b: Functions and Variables for contrib_ode
gcd: Functions and Variables for Polynomials
gcdex: Functions and Variables for Polynomials
gcdivide: Package functs
gcfac: Package scifac
gcfactor: Functions and Variables for Polynomials
gd: Package functs
gdet: Functions and Variables for ctensor
generalized_lambert_w: Functions and Variables for Special Functions
genfact: Combinatorial Functions
genindex: Functions and Variables for Miscellaneous Options
genmatrix: Functions and Variables for Matrices and Linear Algebra
gensumnum: Functions and Variables for Miscellaneous Options
gensym: Functions and Variables for Miscellaneous Options
gen_laguerre: Functions and Variables for orthogonal polynomials
geomap: Functions and Variables for worldmap
geometric: Package functs
geometric_mean: Functions and Variables for descriptive statistics
geomview_command: Functions and Variables for Plotting
geosum: Package functs
geo_amortization: Functions and Variables for finance
geo_annuity_fv: Functions and Variables for finance
geo_annuity_pv: Functions and Variables for finance
get: Functions and Variables for Properties
getcurrentdirectory: Directory operations
getenv: Environment operations
get_edge_weight: Functions and Variables for graphs
get_lu_factors: Functions and Variables for linearalgebra
get_output_stream_string: Input and Output
get_pixel: Functions and Variables for pictures
get_plot_option: Functions and Variables for Plotting
get_tex_environment: Functions and Variables for TeX Output
get_tex_environment_default: Functions and Variables for TeX Output
get_vertex_label: Functions and Variables for graphs
gfactor: Functions and Variables for Polynomials
gfactorsum: Functions and Variables for Polynomials
ggf: Functions and Variables for ggf
GGFCFMAX: Functions and Variables for ggf
GGFINFINITY: Functions and Variables for ggf
girth: Functions and Variables for graphs
globalsolve: Functions and Variables for Equations
global_variances: Functions and Variables for descriptive statistics
gnuplot_close: Gnuplot_pipes Format Functions
gnuplot_command: Functions and Variables for Plotting
gnuplot_curve_styles: Gnuplot Options
gnuplot_curve_titles: Gnuplot Options
gnuplot_default_term_command: Gnuplot Options
gnuplot_dumb_term_command: Gnuplot Options
gnuplot_file_args: Functions and Variables for Plotting
gnuplot_file_name: Functions and Variables for draw
gnuplot_out_file: Gnuplot Options
gnuplot_pdf_term_command: Gnuplot Options
gnuplot_pm3d: Gnuplot Options
gnuplot_png_term_command: Gnuplot Options
gnuplot_postamble: Gnuplot Options
gnuplot_preamble: Gnuplot Options
gnuplot_ps_term_command: Gnuplot Options
gnuplot_replot: Gnuplot_pipes Format Functions
gnuplot_reset: Gnuplot_pipes Format Functions
gnuplot_restart: Gnuplot_pipes Format Functions
gnuplot_start: Gnuplot_pipes Format Functions
gnuplot_svg_term_command: Gnuplot Options
gnuplot_term: Gnuplot Options
gnuplot_view_args: Functions and Variables for Plotting
go: Functions and Variables for Program Flow
Gosper: Functions and Variables for zeilberger
GosperSum: Functions and Variables for zeilberger
Gosper_in_Zeilberger: Functions and Variables for zeilberger
gr2d: Functions and Variables for draw
gr3d: Functions and Variables for draw
gradef: Functions and Variables for Differentiation
gradefs: Functions and Variables for Differentiation
gramschmidt: Functions and Variables for Matrices and Linear Algebra
graph6_decode: Functions and Variables for graphs
graph6_encode: Functions and Variables for graphs
graph6_export: Functions and Variables for graphs
graph6_import: Functions and Variables for graphs
graph_center: Functions and Variables for graphs
graph_charpoly: Functions and Variables for graphs
graph_eigenvalues: Functions and Variables for graphs
graph_flow: Functions and Variables for finance
graph_order: Functions and Variables for graphs
graph_periphery: Functions and Variables for graphs
graph_product: Functions and Variables for graphs
graph_size: Functions and Variables for graphs
graph_union: Functions and Variables for graphs
Greater than: Relational operators
Greater than or equal: Relational operators
great_rhombicosidodecahedron_graph: Functions and Variables for graphs
great_rhombicuboctahedron_graph: Functions and Variables for graphs
grid: Plotting Options
grid: Functions and Variables for draw
grid2d: Plotting Options
grid_graph: Functions and Variables for graphs
grind: Functions and Variables for Display
grind: Functions and Variables for Display
grobner_basis: Functions and Variables for Affine
grotzch_graph: Functions and Variables for graphs

H
halfangles: Functions and Variables for Trigonometric
hamilton_cycle: Functions and Variables for graphs
hamilton_path: Functions and Variables for graphs
hankel: Functions and Variables for linearalgebra
hankel_1: Bessel Functions
hankel_2: Bessel Functions
harmonic: Package functs
harmonic_mean: Functions and Variables for descriptive statistics
hav: Package functs
head_angle: Functions and Variables for draw
head_angle: Functions and Variables for graphs
head_both: Functions and Variables for draw
head_length: Functions and Variables for draw
head_length: Functions and Variables for graphs
head_type: Functions and Variables for draw
heawood_graph: Functions and Variables for graphs
height: Visualization with VTK
height: Visualization with VTK
Help: Functions and Variables for Help
hermite: Functions and Variables for orthogonal polynomials
hessian: Functions and Variables for linearalgebra
hgfred: Functions and Variables for Special Functions
hilbertmap: Definitions for Peano maps
hilbert_matrix: Functions and Variables for linearalgebra
hipow: Functions and Variables for Polynomials
histogram: Functions and Variables for statistical graphs
histogram_description: Functions and Variables for statistical graphs
hodge: Functions and Variables for itensor
horner: Functions for numerical solution of equations
hypergeometric: Hypergeometric Functions
hypergeometric_representation: Error Function
hypergeometric_simp: Functions and Variables for Special Functions

I
ibase: Functions and Variables for Display
ic1: Functions and Variables for Differential Equations
ic2: Functions and Variables for Differential Equations
icc1: Functions and Variables for itensor
icc2: Functions and Variables for itensor
ichr1: Functions and Variables for itensor
ichr2: Functions and Variables for itensor
icosahedron_graph: Functions and Variables for graphs
icosidodecahedron_graph: Functions and Variables for graphs
icounter: Functions and Variables for itensor
icurvature: Functions and Variables for itensor
ic_convert: Functions and Variables for itensor
ident: Functions and Variables for Matrices and Linear Algebra
identfor: Functions and Variables for linearalgebra
identity: Functions and Variables for Sets
idiff: Functions and Variables for itensor
idim: Functions and Variables for itensor
idummy: Functions and Variables for itensor
idummyx: Functions and Variables for itensor
ieqn: Functions and Variables for Equations
ieqnprint: Functions and Variables for Equations
if: Functions and Variables for Program Flow
ifactors: Functions and Variables for Number Theory
ifb: Functions and Variables for itensor
ifc1: Functions and Variables for itensor
ifc2: Functions and Variables for itensor
ifg: Functions and Variables for itensor
ifgi: Functions and Variables for itensor
ifr: Functions and Variables for itensor
iframes: Functions and Variables for itensor
iframe_bracket_form: Functions and Variables for itensor
ifri: Functions and Variables for itensor
ifs: Graphical analysis of discrete dynamical systems
igcdex: Functions and Variables for Number Theory
igeodesic_coords: Functions and Variables for itensor
igeowedge_flag: Functions and Variables for itensor
ikt1: Functions and Variables for itensor
ikt2: Functions and Variables for itensor
ilt: Functions and Variables for Integration
image: Functions and Variables for draw
imaginary: Functions and Variables for Properties
imagpart: Functions for Complex Numbers
imetric: Functions and Variables for itensor
imetric: Functions and Variables for itensor
implicit: Functions and Variables for draw
implicit_derivative: Functions and Variables for impdiff
implicit_plot: Functions and Variables for Plotting
in: Functions and Variables for Program Flow
inchar: Functions and Variables for Command Line
increasing: Functions and Variables for Properties
ind: Functions and Variables for Constants
indexed_tensor: Functions and Variables for itensor
indices: Functions and Variables for itensor
induced_subgraph: Functions and Variables for graphs
inf: Functions and Variables for Constants
inferencep: Functions and Variables for inference_result
inference_result: Functions and Variables for inference_result
infeval: Functions and Variables for Evaluation
infinity: Functions and Variables for Constants
infix: User defined operators
inflag: Functions and Variables for Expressions
infolists: Functions and Variables for Command Line
info_display: Functions and Variables for alt-display
init_atensor: Functions and Variables for atensor
init_ctensor: Functions and Variables for ctensor
inm: Functions and Variables for itensor
inmc1: Functions and Variables for itensor
inmc2: Functions and Variables for itensor
innerproduct: Functions and Variables for Matrices and Linear Algebra
inpart: Functions and Variables for Expressions
inprod: Functions and Variables for Matrices and Linear Algebra
inrt: Functions and Variables for Number Theory
intanalysis: Functions and Variables for Integration
integer: Functions and Variables for Properties
integerp: Functions and Variables for Numbers
integervalued: Functions and Variables for Properties
integer_partitions: Functions and Variables for Sets
integrate: Functions and Variables for Integration
integrate_use_rootsof: Functions and Variables for Integration
integration_constant: Functions and Variables for Integration
integration_constant_counter: Functions and Variables for Integration
interpolate_color: Functions and Variables for draw
intersect: Functions and Variables for Sets
intersection: Functions and Variables for Sets
intervalp: Functions and Variables for orthogonal polynomials
intfaclim: Functions and Variables for Polynomials
intopois: Functions and Variables for Poisson series
intosum: Functions and Variables for Sums and Products
invariant1: Functions and Variables for ctensor
invariant2: Functions and Variables for ctensor
inverse_fft: Functions and Variables for fast Fourier transform
inverse_jacobi_cd: Functions and Variables for Elliptic Functions
inverse_jacobi_cn: Functions and Variables for Elliptic Functions
inverse_jacobi_cs: Functions and Variables for Elliptic Functions
inverse_jacobi_dc: Functions and Variables for Elliptic Functions
inverse_jacobi_dn: Functions and Variables for Elliptic Functions
inverse_jacobi_ds: Functions and Variables for Elliptic Functions
inverse_jacobi_nc: Functions and Variables for Elliptic Functions
inverse_jacobi_nd: Functions and Variables for Elliptic Functions
inverse_jacobi_ns: Functions and Variables for Elliptic Functions
inverse_jacobi_sc: Functions and Variables for Elliptic Functions
inverse_jacobi_sd: Functions and Variables for Elliptic Functions
inverse_jacobi_sn: Functions and Variables for Elliptic Functions
inverse_real_fft: Functions and Variables for fast Fourier transform
invert: Functions and Variables for Matrices and Linear Algebra
invert_by_adjoint: Functions and Variables for Matrices and Linear Algebra
invert_by_lu: Functions and Variables for linearalgebra
inv_mod: Functions and Variables for Number Theory
in_neighbors: Functions and Variables for graphs
ip_grid: Functions and Variables for draw
ip_grid_in: Functions and Variables for draw
irr: Functions and Variables for finance
irrational: Functions and Variables for Properties
is: Functions and Variables for Facts
ishow: Functions and Variables for itensor
isolate: Functions and Variables for Expressions
isolate_wrt_times: Functions and Variables for Expressions
isomorphism: Functions and Variables for graphs
isqrt: Functions and Variables for Number Theory
isreal_p: Functions and Variables for to_poly_solve
is_biconnected: Functions and Variables for graphs
is_bipartite: Functions and Variables for graphs
is_connected: Functions and Variables for graphs
is_digraph: Functions and Variables for graphs
is_edge_in_graph: Functions and Variables for graphs
is_graph: Functions and Variables for graphs
is_graph_or_digraph: Functions and Variables for graphs
is_isomorphic: Functions and Variables for graphs
is_planar: Functions and Variables for graphs
is_sconnected: Functions and Variables for graphs
is_tree: Functions and Variables for graphs
is_vertex_in_graph: Functions and Variables for graphs
items_inference: Functions and Variables for inference_result
iterations: Plotting Options
itr: Functions and Variables for itensor

J
jacobi: Functions and Variables for Number Theory
jacobian: Functions and Variables for linearalgebra
jacobi_cd: Functions and Variables for Elliptic Functions
jacobi_cn: Functions and Variables for Elliptic Functions
jacobi_cs: Functions and Variables for Elliptic Functions
jacobi_dc: Functions and Variables for Elliptic Functions
jacobi_dn: Functions and Variables for Elliptic Functions
jacobi_ds: Functions and Variables for Elliptic Functions
jacobi_nc: Functions and Variables for Elliptic Functions
jacobi_nd: Functions and Variables for Elliptic Functions
jacobi_ns: Functions and Variables for Elliptic Functions
jacobi_p: Functions and Variables for orthogonal polynomials
jacobi_sc: Functions and Variables for Elliptic Functions
jacobi_sd: Functions and Variables for Elliptic Functions
jacobi_sn: Functions and Variables for Elliptic Functions
JF: Functions and Variables for diag
join: Functions and Variables for Lists
jordan: Functions and Variables for diag
julia: Functions and Variables for Plotting
julia_parameter: Definitions for complex fractals
julia_set: Definitions for complex fractals
julia_sin: Definitions for complex fractals

K
kdels: Functions and Variables for itensor
kdelta: Functions and Variables for itensor
keepfloat: Functions and Variables for Polynomials
key: Functions and Variables for draw
key_pos: Functions and Variables for draw
kill: Functions and Variables for Command Line
killcontext: Functions and Variables for Facts
kinvariant: Functions and Variables for ctensor
km: Functions and Variables for descriptive statistics
kostka: Functions and Variables for Symmetries
kronecker_product: Functions and Variables for linearalgebra
kron_delta: Functions and Variables for Sets
kt: Functions and Variables for ctensor
kummer_m: Functions and Variables for contrib_ode
kummer_u: Functions and Variables for contrib_ode
kurtosis: Functions and Variables for descriptive statistics
kurtosis_bernoulli: Functions and Variables for discrete distributions
kurtosis_beta: Functions and Variables for continuous distributions
kurtosis_binomial: Functions and Variables for discrete distributions
kurtosis_chi2: Functions and Variables for continuous distributions
kurtosis_continuous_uniform: Functions and Variables for continuous distributions
kurtosis_discrete_uniform: Functions and Variables for discrete distributions
kurtosis_exp: Functions and Variables for continuous distributions
kurtosis_f: Functions and Variables for continuous distributions
kurtosis_gamma: Functions and Variables for continuous distributions
kurtosis_general_finite_discrete: Functions and Variables for discrete distributions
kurtosis_geometric: Functions and Variables for discrete distributions
kurtosis_gumbel: Functions and Variables for continuous distributions
kurtosis_hypergeometric: Functions and Variables for discrete distributions
kurtosis_laplace: Functions and Variables for continuous distributions
kurtosis_logistic: Functions and Variables for continuous distributions
kurtosis_lognormal: Functions and Variables for continuous distributions
kurtosis_negative_binomial: Functions and Variables for discrete distributions
kurtosis_noncentral_chi2: Functions and Variables for continuous distributions
kurtosis_noncentral_student_t: Functions and Variables for continuous distributions
kurtosis_normal: Functions and Variables for continuous distributions
kurtosis_pareto: Functions and Variables for continuous distributions
kurtosis_poisson: Functions and Variables for discrete distributions
kurtosis_rayleigh: Functions and Variables for continuous distributions
kurtosis_student_t: Functions and Variables for continuous distributions
kurtosis_weibull: Functions and Variables for continuous distributions

L
label: Plotting Options
label: Functions and Variables for draw
labels: Functions and Variables for Command Line
labels: Functions and Variables for Command Line
label_alignment: Functions and Variables for draw
label_alignment: Functions and Variables for graphs
label_orientation: Functions and Variables for draw
lagrange: Functions and Variables for interpol
laguerre: Functions and Variables for orthogonal polynomials
lambda: Functions and Variables for Function Definition
lambert_w: Functions and Variables for Special Functions
laplace: Functions and Variables for Differentiation
laplacian_matrix: Functions and Variables for graphs
lassociative: Functions and Variables for Simplification
last: Functions and Variables for Lists
lastn: Functions and Variables for Lists
lbfgs: Functions and Variables for lbfgs
lbfgs_ncorrections: Functions and Variables for lbfgs
lbfgs_nfeval_max: Functions and Variables for lbfgs
lc2kdt: Functions and Variables for itensor
lcm: Functions and Variables for Number Theory
lc_l: Functions and Variables for itensor
lc_u: Functions and Variables for itensor
ldefint: Functions and Variables for Integration
ldisp: Functions and Variables for Display
ldisplay: Functions and Variables for Display
leftjust: Functions and Variables for Display
legend: Plotting Options
legendre_p: Functions and Variables for orthogonal polynomials
legendre_q: Functions and Variables for orthogonal polynomials
leinstein: Functions and Variables for ctensor
length: Functions and Variables for Lists
Less than: Relational operators
Less than or equal: Relational operators
let: Functions and Variables for Rules and Patterns
letrat: Functions and Variables for Rules and Patterns
letrules: Functions and Variables for Rules and Patterns
letsimp: Functions and Variables for Rules and Patterns
let_rule_packages: Functions and Variables for Rules and Patterns
levi_civita: Functions and Variables for itensor
lfg: Functions and Variables for ctensor
lfreeof: Functions and Variables for Expressions
lg: Functions and Variables for ctensor
lgtreillis: Functions and Variables for Symmetries
lhospitallim: Functions and Variables for Limits
lhs: Functions and Variables for Equations
li: Root Exponential and Logarithmic Functions
liediff: Functions and Variables for itensor
limit: Functions and Variables for Limits
limsubst: Functions and Variables for Limits
linalg_rank: Functions and Variables for linearalgebra
Lindstedt: Functions and Variables for lindstedt
linear: Functions and Variables for Simplification
linear: Package functs
linearinterpol: Functions and Variables for interpol
linear_program: Functions and Variables for simplex
linear_regression: Functions and Variables for stats
linear_solver: Functions and Variables for zeilberger
linechar: Functions and Variables for Command Line
linel: Functions and Variables for Display
linenum: Functions and Variables for Command Line
linewidth: Visualization with VTK
line_graph: Functions and Variables for graphs
line_type: Functions and Variables for draw
line_width: Functions and Variables for draw
linsolve: Functions and Variables for Equations
linsolvewarn: Functions and Variables for Equations
linsolve_params: Functions and Variables for Equations
lispdisp: Functions and Variables for Display
List delimiters: Functions and Variables for Lists
listarith: Functions and Variables for Lists
listarray: Functions and Variables for Arrays
listconstvars: Functions and Variables for Expressions
listdummyvars: Functions and Variables for Expressions
listify: Functions and Variables for Sets
listoftens: Functions and Variables for itensor
listofvars: Functions and Variables for Expressions
listp: Functions and Variables for Lists
listp: Functions and Variables for linearalgebra
list_correlations: Functions and Variables for descriptive statistics
list_matrix_entries: Functions and Variables for Matrices and Linear Algebra
list_nc_monomials: Functions and Variables for Affine
lmax: Functions for Numbers
lmin: Functions for Numbers
lmxchar: Functions and Variables for Matrices and Linear Algebra
load: Functions and Variables for File Input and Output
loadfile: Functions and Variables for File Input and Output
loadprint: Functions and Variables for File Input and Output
load_pathname: Functions and Variables for File Input and Output
local: Functions and Variables for Function Definition
locate_matrix_entry: Functions and Variables for linearalgebra
log: Root Exponential and Logarithmic Functions
logabs: Root Exponential and Logarithmic Functions
logarc: Root Exponential and Logarithmic Functions
logarc: Root Exponential and Logarithmic Functions
logcb: Functions and Variables for draw
logconcoeffp: Root Exponential and Logarithmic Functions
logcontract: Root Exponential and Logarithmic Functions
logexpand: Root Exponential and Logarithmic Functions
lognegint: Root Exponential and Logarithmic Functions
logsimp: Root Exponential and Logarithmic Functions
logx: Plotting Options
logx: Functions and Variables for draw
logx_secondary: Functions and Variables for draw
logy: Plotting Options
logy: Functions and Variables for draw
logy_secondary: Functions and Variables for draw
logz: Functions and Variables for draw
log_gamma: Gamma and factorial Functions
lopow: Functions and Variables for Polynomials
lorentz_gauge: Functions and Variables for itensor
lowercasep: Characters
lpart: Functions and Variables for Expressions
lratsubst: Functions and Variables for Polynomials
lreduce: Functions and Variables for Lists
lriem: Functions and Variables for ctensor
lriemann: Functions and Variables for ctensor
lsquares_estimates: Functions and Variables for lsquares
lsquares_estimates_approximate: Functions and Variables for lsquares
lsquares_estimates_exact: Functions and Variables for lsquares
lsquares_mse: Functions and Variables for lsquares
lsquares_residuals: Functions and Variables for lsquares
lsquares_residual_mse: Functions and Variables for lsquares
lsum: Functions and Variables for Sums and Products
ltreillis: Functions and Variables for Symmetries
lucas: Functions and Variables for Number Theory
lu_backsub: Functions and Variables for linearalgebra
lu_factor: Functions and Variables for linearalgebra

M
m1pbranch: Functions and Variables for Numbers
macroexpand: Macros
macroexpand1: Macros
macroexpansion: Functions and Variables for Function Definition
macros: Macros
mainvar: Functions and Variables for Expressions
makebox: Functions and Variables for itensor
makefact: Gamma and factorial Functions
makegamma: Gamma and factorial Functions
makelist: Functions and Variables for Lists
makeOrders: Functions and Variables for makeOrders
makeset: Functions and Variables for Sets
make_array: Functions and Variables for Arrays
make_graph: Functions and Variables for graphs
make_level_picture: Functions and Variables for pictures
make_polygon: Functions and Variables for worldmap
make_poly_continent: Functions and Variables for worldmap
make_poly_country: Functions and Variables for worldmap
make_random_state: Random Numbers
make_rgb_picture: Functions and Variables for pictures
make_string_input_stream: Input and Output
make_string_output_stream: Input and Output
make_transform: Functions and Variables for Plotting
mandelbrot: Functions and Variables for Plotting
mandelbrot_set: Definitions for complex fractals
manual_demo: Functions and Variables for Help
map: Functions and Variables for Program Flow
mapatom: Functions and Variables for Program Flow
maperror: Functions and Variables for Program Flow
maplist: Functions and Variables for Program Flow
mapprint: Functions and Variables for Program Flow
matchdeclare: Functions and Variables for Rules and Patterns
matchfix: User defined operators
mathml_display: Functions and Variables for alt-display
matrix: Functions and Variables for Matrices and Linear Algebra
matrixexp: Functions and Variables for Matrices and Linear Algebra
matrixmap: Functions and Variables for Matrices and Linear Algebra
matrixp: Functions and Variables for Matrices and Linear Algebra
matrixp: Functions and Variables for linearalgebra
matrix_element_add: Functions and Variables for Matrices and Linear Algebra
matrix_element_mult: Functions and Variables for Matrices and Linear Algebra
matrix_element_transpose: Functions and Variables for Matrices and Linear Algebra
matrix_size: Functions and Variables for linearalgebra
mattrace: Functions and Variables for Matrices and Linear Algebra
mat_cond: Functions and Variables for linearalgebra
mat_fullunblocker: Functions and Variables for linearalgebra
mat_function: Functions and Variables for diag
mat_norm: Functions and Variables for linearalgebra
mat_trace: Functions and Variables for linearalgebra
mat_unblocker: Functions and Variables for linearalgebra
max: Functions for Numbers
maxapplydepth: Functions and Variables for Rules and Patterns
maxapplyheight: Functions and Variables for Rules and Patterns
maxima_tempdir: Functions and Variables for Runtime Environment
maxima_userdir: Functions and Variables for Runtime Environment
maximize_lp: Functions and Variables for simplex
maxnegex: Functions and Variables for Simplification
maxposex: Functions and Variables for Simplification
maxpsifracdenom: Gamma and factorial Functions
maxpsifracnum: Gamma and factorial Functions
maxpsinegint: Gamma and factorial Functions
maxpsiposint: Gamma and factorial Functions
maxtayorder: Functions and Variables for Series
max_clique: Functions and Variables for graphs
max_degree: Functions and Variables for graphs
max_flow: Functions and Variables for graphs
max_independent_set: Functions and Variables for graphs
max_matching: Functions and Variables for graphs
MAX_ORD: Functions and Variables for zeilberger
maybe: Functions and Variables for Facts
md5sum: Octets and Utilities for Cryptography
mean: Functions and Variables for descriptive statistics
mean_bernoulli: Functions and Variables for discrete distributions
mean_beta: Functions and Variables for continuous distributions
mean_binomial: Functions and Variables for discrete distributions
mean_chi2: Functions and Variables for continuous distributions
mean_continuous_uniform: Functions and Variables for continuous distributions
mean_deviation: Functions and Variables for descriptive statistics
mean_discrete_uniform: Functions and Variables for discrete distributions
mean_exp: Functions and Variables for continuous distributions
mean_f: Functions and Variables for continuous distributions
mean_gamma: Functions and Variables for continuous distributions
mean_general_finite_discrete: Functions and Variables for discrete distributions
mean_geometric: Functions and Variables for discrete distributions
mean_gumbel: Functions and Variables for continuous distributions
mean_hypergeometric: Functions and Variables for discrete distributions
mean_laplace: Functions and Variables for continuous distributions
mean_logistic: Functions and Variables for continuous distributions
mean_lognormal: Functions and Variables for continuous distributions
mean_negative_binomial: Functions and Variables for discrete distributions
mean_noncentral_chi2: Functions and Variables for continuous distributions
mean_noncentral_student_t: Functions and Variables for continuous distributions
mean_normal: Functions and Variables for continuous distributions
mean_pareto: Functions and Variables for continuous distributions
mean_poisson: Functions and Variables for discrete distributions
mean_rayleigh: Functions and Variables for continuous distributions
mean_student_t: Functions and Variables for continuous distributions
mean_weibull: Functions and Variables for continuous distributions
median: Functions and Variables for descriptive statistics
median_deviation: Functions and Variables for descriptive statistics
member: Functions and Variables for Lists
mesh: Functions and Variables for draw
mesh_lines_color: Plotting Options
method: Functions and Variables for contrib_ode
metricexpandall: Functions and Variables for Units
mgf1_sha1: Octets and Utilities for Cryptography
min: Functions for Numbers
minf: Functions and Variables for Constants
minfactorial: Combinatorial Functions
minimalPoly: Functions and Variables for diag
minimize_lp: Functions and Variables for simplex
minimum_spanning_tree: Functions and Variables for graphs
minor: Functions and Variables for Matrices and Linear Algebra
minpack_lsquares: Functions and Variables for minpack
minpack_solve: Functions and Variables for minpack
min_degree: Functions and Variables for graphs
min_edge_cut: Functions and Variables for graphs
min_vertex_cover: Functions and Variables for graphs
min_vertex_cut: Functions and Variables for graphs
mkdir: Directory operations
mnewton: Functions and Variables for mnewton
mod: Functions and Variables for Number Theory
ModeMatrix: Functions and Variables for diag
mode_checkp: Functions and Variables for Function Definition
mode_check_errorp: Functions and Variables for Function Definition
mode_check_warnp: Functions and Variables for Function Definition
mode_declare: Functions and Variables for Function Definition
mode_identity: Functions and Variables for Function Definition
modular_linear_solver: Functions and Variables for zeilberger
modulus: Functions and Variables for Polynomials
mod_big_prime: Functions and Variables for zeilberger
mod_test: Functions and Variables for zeilberger
mod_threshold: Functions and Variables for zeilberger
moebius: Functions and Variables for Sets
mon2schur: Functions and Variables for Symmetries
mono: Functions and Variables for Affine
monomial_dimensions: Functions and Variables for Affine
multibernstein_poly: Functions and Variables for Bernstein
multinomial: Functions and Variables for Symmetries
multinomial_coeff: Functions and Variables for Sets
Multiplication: Arithmetic operators
multiplicative: Functions and Variables for Simplification
multiplicities: Functions and Variables for Equations
multiplot_mode: Functions and Variables for draw
multi_display_for_texinfo: Functions and Variables for alt-display
multi_elem: Functions and Variables for Symmetries
multi_orbit: Functions and Variables for Symmetries
multi_pui: Functions and Variables for Symmetries
multsym: Functions and Variables for Symmetries
multthru: Functions and Variables for Simplification
mycielski_graph: Functions and Variables for graphs
myoptions: Functions and Variables for Command Line

N
nary: Functions and Variables for Simplification
nary: User defined operators
natural_unit: Functions and Variables for ezunits
ncexpt: Functions and Variables for Display
ncharpoly: Functions and Variables for Matrices and Linear Algebra
nc_degree: Functions and Variables for Affine
negative_picture: Functions and Variables for pictures
negdistrib: Functions and Variables for Simplification
negsumdispflag: Functions and Variables for Display
neighbors: Functions and Variables for graphs
new: Functions and Variables for Structures
newcontext: Functions and Variables for Facts
newdet: Functions and Variables for Matrices and Linear Algebra
newline: Characters
newline: Input and Output
newton: Functions for numerical solution of equations
newtonepsilon: Functions and Variables for mnewton
newtonmaxiter: Functions and Variables for mnewton
new_graph: Functions and Variables for graphs
new_variable: Functions and Variables for to_poly_solve
next: Functions and Variables for Program Flow
nextlayerfactor: Package facexp
next_prime: Functions and Variables for Number Theory
nicedummies: Functions and Variables for to_poly_solve
niceindices: Functions and Variables for Series
niceindicespref: Functions and Variables for Series
ninth: Functions and Variables for Lists
nm: Functions and Variables for ctensor
nmc: Functions and Variables for ctensor
noeval: Functions and Variables for Evaluation
nofix: User defined operators
nolabels: Functions and Variables for Command Line
nonarray: Functions and Variables for Properties
noncentral_moment: Functions and Variables for descriptive statistics
nonegative_lp: Functions and Variables for simplex
noninteger: Functions and Variables for Properties
nonmetricity: Functions and Variables for ctensor
nonnegative_lp: Functions and Variables for simplex
nonnegintegerp: Functions and Variables for Numbers
nonscalar: Functions and Variables for Properties
nonscalarp: Functions and Variables for Properties
nonzeroandfreeof: Package functs
not: Logical operators
notequal: Functions and Variables for Predicates
noun: Functions and Variables for Expressions
noundisp: Functions and Variables for Expressions
nounify: Functions and Variables for Expressions
nouns: Functions and Variables for Evaluation
np: Functions and Variables for ctensor
npi: Functions and Variables for ctensor
nptetrad: Functions and Variables for ctensor
npv: Functions and Variables for finance
nroots: Functions and Variables for Equations
nterms: Functions and Variables for Expressions
ntermst: Functions and Variables for ctensor
nthroot: Functions and Variables for Equations
nticks: Plotting Options
nticks: Functions and Variables for draw
ntrig: Functions and Variables for Trigonometric
nullity: Functions and Variables for linearalgebra
nullspace: Functions and Variables for linearalgebra
num: Functions and Variables for Polynomials
numbered_boundaries: Functions and Variables for worldmap
numberp: Functions and Variables for Numbers
number_to_octets: Octets and Utilities for Cryptography
numer: Functions and Variables for Numbers
numerval: Functions and Variables for Numbers
numer_pbranch: Functions and Variables for Numbers
numfactor: Gamma and factorial Functions
num_distinct_partitions: Functions and Variables for Sets
num_partitions: Functions and Variables for Sets
nusum: Functions and Variables for Series
nzeta: Functions and Variables for Special Functions
nzetai: Functions and Variables for Special Functions
nzetar: Functions and Variables for Special Functions

O
obase: Functions and Variables for Display
octets_to_number: Octets and Utilities for Cryptography
octets_to_oid: Octets and Utilities for Cryptography
octets_to_string: Octets and Utilities for Cryptography
odd: Functions and Variables for Properties
oddfun: Functions and Variables for Simplification
oddp: Functions and Variables for Numbers
odd_girth: Functions and Variables for graphs
ode2: Functions and Variables for Differential Equations
odelin: Functions and Variables for contrib_ode
ode_check: Functions and Variables for contrib_ode
oid_to_octets: Octets and Utilities for Cryptography
op: Functions and Variables for Expressions
opacity: Visualization with VTK
opena: Input and Output
opena_binary: Functions and Variables for binary input and output
openr: Input and Output
openr_binary: Functions and Variables for binary input and output
openw: Input and Output
openw_binary: Functions and Variables for binary input and output
operatorp: Functions and Variables for Expressions
opproperties: Functions and Variables for Simplification
opsubst: Functions and Variables for Expressions
opsubst: Functions and Variables for opsubst
optimize: Functions and Variables for Expressions
optimprefix: Functions and Variables for Expressions
optionset: Functions and Variables for Command Line
or: Logical operators
orbit: Functions and Variables for Symmetries
orbits: Graphical analysis of discrete dynamical systems
ordergreat: Functions and Variables for Expressions
ordergreatp: Functions and Variables for Expressions
orderless: Functions and Variables for Expressions
orderlessp: Functions and Variables for Expressions
orientation: Visualization with VTK
origin: Visualization with VTK
orthogonal_complement: Functions and Variables for linearalgebra
orthopoly_recur: Functions and Variables for orthogonal polynomials
orthopoly_returns_intervals: Functions and Variables for orthogonal polynomials
orthopoly_weight: Functions and Variables for orthogonal polynomials
outative: Functions and Variables for Simplification
outchar: Functions and Variables for Command Line
outermap: Functions and Variables for Program Flow
outofpois: Functions and Variables for Poisson series
out_neighbors: Functions and Variables for graphs

P
packagefile: Functions and Variables for Miscellaneous Options
pade: Functions and Variables for Series
palette: Plotting Options
palette: Functions and Variables for draw
parabolic_cylinder_d: Parabolic Cylinder Functions
parametric: Functions and Variables for draw
parametric_surface: Functions and Variables for draw
parg: Functions and Variables for to_poly_solve
parGosper: Functions and Variables for zeilberger
parse_string: String Processing
parse_timedate: Functions and Variables for Runtime Environment
part: Functions and Variables for Expressions
part2cont: Functions and Variables for Symmetries
partfrac: Functions and Variables for Number Theory
partition: Functions and Variables for Expressions
partition_set: Functions and Variables for Sets
partpol: Functions and Variables for Symmetries
partswitch: Functions and Variables for Expressions
pathname_directory: Functions and Variables for File Input and Output
pathname_name: Functions and Variables for File Input and Output
pathname_type: Functions and Variables for File Input and Output
path_digraph: Functions and Variables for graphs
path_graph: Functions and Variables for graphs
pdf_bernoulli: Functions and Variables for discrete distributions
pdf_beta: Functions and Variables for continuous distributions
pdf_binomial: Functions and Variables for discrete distributions
pdf_cauchy: Functions and Variables for continuous distributions
pdf_chi2: Functions and Variables for continuous distributions
pdf_continuous_uniform: Functions and Variables for continuous distributions
pdf_discrete_uniform: Functions and Variables for discrete distributions
pdf_exp: Functions and Variables for continuous distributions
pdf_f: Functions and Variables for continuous distributions
pdf_file: Plotting Options
pdf_gamma: Functions and Variables for continuous distributions
pdf_general_finite_discrete: Functions and Variables for discrete distributions
pdf_geometric: Functions and Variables for discrete distributions
pdf_gumbel: Functions and Variables for continuous distributions
pdf_hypergeometric: Functions and Variables for discrete distributions
pdf_laplace: Functions and Variables for continuous distributions
pdf_logistic: Functions and Variables for continuous distributions
pdf_lognormal: Functions and Variables for continuous distributions
pdf_negative_binomial: Functions and Variables for discrete distributions
pdf_noncentral_chi2: Functions and Variables for continuous distributions
pdf_noncentral_student_t: Functions and Variables for continuous distributions
pdf_normal: Functions and Variables for continuous distributions
pdf_pareto: Functions and Variables for continuous distributions
pdf_poisson: Functions and Variables for discrete distributions
pdf_rank_sum: Functions and Variables for special distributions
pdf_rayleigh: Functions and Variables for continuous distributions
pdf_signed_rank: Functions and Variables for special distributions
pdf_student_t: Functions and Variables for continuous distributions
pdf_weibull: Functions and Variables for continuous distributions
pearson_skewness: Functions and Variables for descriptive statistics
permanent: Functions and Variables for Matrices and Linear Algebra
permp: Functions and Variables for Combinatorics
perms: Functions and Variables for Combinatorics
perms_lex: Functions and Variables for Combinatorics
permult: Functions and Variables for Combinatorics
permut: Functions and Variables for Symmetries
permutation: Package functs
permutations: Functions and Variables for Sets
permute: Functions and Variables for Combinatorics
perm_cycles: Functions and Variables for Combinatorics
perm_decomp: Functions and Variables for Combinatorics
perm_inverse: Functions and Variables for Combinatorics
perm_length: Functions and Variables for Combinatorics
perm_lex_next: Functions and Variables for Combinatorics
perm_lex_rank: Functions and Variables for Combinatorics
perm_lex_unrank: Functions and Variables for Combinatorics
perm_next: Functions and Variables for Combinatorics
perm_parity: Functions and Variables for Combinatorics
perm_rank: Functions and Variables for Combinatorics
perm_undecomp: Functions and Variables for Combinatorics
perm_unrank: Functions and Variables for Combinatorics
petersen_graph: Functions and Variables for graphs
petrov: Functions and Variables for ctensor
pfeformat: Functions and Variables for Display
phiresolution: Visualization with VTK
pickapart: Functions and Variables for Expressions
picturep: Functions and Variables for pictures
picture_equalp: Functions and Variables for pictures
piece: Functions and Variables for Expressions
piechart: Functions and Variables for statistical graphs
piechart_description: Functions and Variables for statistical graphs
pivot_count_sx: Functions and Variables for simplex
pivot_max_sx: Functions and Variables for simplex
planar_embedding: Functions and Variables for graphs
playback: Functions and Variables for Command Line
plog: Root Exponential and Logarithmic Functions
plot2d: Functions and Variables for Plotting
plot3d: Functions and Variables for Plotting
plotdf: Functions for numerical solution of differential equations
ploteq: Functions for numerical solution of differential equations
plot_format: Plotting Options
plot_options: Functions and Variables for Plotting
plot_realpart: Plotting Options
plsquares: Functions and Variables for lsquares
png_file: Plotting Options
pochhammer: Functions and Variables for orthogonal polynomials
pochhammer_max_index: Functions and Variables for orthogonal polynomials
points: Visualization with VTK
points: Functions and Variables for draw
pointsize: Visualization with VTK
points_joined: Functions and Variables for draw
point_size: Functions and Variables for draw
point_type: Plotting Options
point_type: Functions and Variables for draw
poisdiff: Functions and Variables for Poisson series
poisexpt: Functions and Variables for Poisson series
poisint: Functions and Variables for Poisson series
poislim: Functions and Variables for Poisson series
poismap: Functions and Variables for Poisson series
poisplus: Functions and Variables for Poisson series
poissimp: Functions and Variables for Poisson series
poisson: Functions and Variables for Poisson series
poissubst: Functions and Variables for Poisson series
poistimes: Functions and Variables for Poisson series
poistrim: Functions and Variables for Poisson series
polar: Functions and Variables for draw
polarform: Functions for Complex Numbers
polartorect: Functions and Variables for fast Fourier transform
polar_to_xy: Functions and Variables for Plotting
polydecomp: Functions and Variables for Polynomials
polyfactor: Functions and Variables for Equations
polygon: Functions and Variables for draw
polymod: Functions and Variables for Polynomials
polynome2ele: Functions and Variables for Symmetries
polynomialp: Functions and Variables for linearalgebra
polytocompanion: Functions and Variables for linearalgebra
poly_add: Functions and Variables for grobner
poly_buchberger: Functions and Variables for grobner
poly_buchberger_criterion: Functions and Variables for grobner
poly_coefficient_ring: Functions and Variables for grobner
poly_colon_ideal: Functions and Variables for grobner
poly_content: Functions and Variables for grobner
poly_depends_p: Functions and Variables for grobner
poly_elimination_ideal: Functions and Variables for grobner
poly_elimination_order: Functions and Variables for grobner
poly_exact_divide: Functions and Variables for grobner
poly_expand: Functions and Variables for grobner
poly_expt: Functions and Variables for grobner
poly_gcd: Functions and Variables for grobner
poly_grobner: Functions and Variables for grobner
poly_grobner_algorithm: Functions and Variables for grobner
poly_grobner_debug: Functions and Variables for grobner
poly_grobner_equal: Functions and Variables for grobner
poly_grobner_member: Functions and Variables for grobner
poly_grobner_subsetp: Functions and Variables for grobner
poly_ideal_intersection: Functions and Variables for grobner
poly_ideal_polysaturation: Functions and Variables for grobner
poly_ideal_polysaturation1: Functions and Variables for grobner
poly_ideal_saturation: Functions and Variables for grobner
poly_ideal_saturation1: Functions and Variables for grobner
poly_lcm: Functions and Variables for grobner
poly_minimization: Functions and Variables for grobner
poly_monomial_order: Functions and Variables for grobner
poly_multiply: Functions and Variables for grobner
poly_normalize: Functions and Variables for grobner
poly_normalize_list: Functions and Variables for grobner
poly_normal_form: Functions and Variables for grobner
poly_polysaturation_extension: Functions and Variables for grobner
poly_primary_elimination_order: Functions and Variables for grobner
poly_primitive_part: Functions and Variables for grobner
poly_pseudo_divide: Functions and Variables for grobner
poly_reduced_grobner: Functions and Variables for grobner
poly_reduction: Functions and Variables for grobner
poly_return_term_list: Functions and Variables for grobner
poly_saturation_extension: Functions and Variables for grobner
poly_secondary_elimination_order: Functions and Variables for grobner
poly_subtract: Functions and Variables for grobner
poly_s_polynomial: Functions and Variables for grobner
poly_top_reduction_only: Functions and Variables for grobner
pop: Functions and Variables for Lists
posfun: Functions and Variables for Properties
position: Visualization with VTK
postfix: User defined operators
potential: Functions and Variables for Integration
powerdisp: Functions and Variables for Display
powerseries: Functions and Variables for Series
powerset: Functions and Variables for Sets
power_mod: Functions and Variables for Number Theory
pred: Functions and Variables for Evaluation
prederror: Functions and Variables for Program Flow
prefix: User defined operators
prev_prime: Functions and Variables for Number Theory
primep: Functions and Variables for Number Theory
primep_number_of_tests: Functions and Variables for Number Theory
primes: Functions and Variables for Number Theory
principal_components: Functions and Variables for descriptive statistics
print: Functions and Variables for Display
printf: Input and Output
printfile: Functions and Variables for File Input and Output
printpois: Functions and Variables for Poisson series
printprops: Functions and Variables for Properties
print_graph: Functions and Variables for graphs
prodrac: Functions and Variables for Symmetries
product: Functions and Variables for Sums and Products
product_use_gamma: Functions and Variables for solve_rec
program: Functions and Variables for graphs
programmode: Functions and Variables for Equations
prompt: Functions and Variables for Command Line
properties: Functions and Variables for Properties
proportional_axes: Functions and Variables for draw
props: Functions and Variables for Properties
propvars: Functions and Variables for Properties
psexpand: Functions and Variables for Series
psi: Gamma and factorial Functions
psi: Functions and Variables for ctensor
psubst: Functions and Variables for Expressions
ps_file: Plotting Options
ptriangularize: Functions and Variables for linearalgebra
pui: Functions and Variables for Symmetries
pui2comp: Functions and Variables for Symmetries
pui2ele: Functions and Variables for Symmetries
pui2polynome: Functions and Variables for Symmetries
puireduc: Functions and Variables for Symmetries
pui_direct: Functions and Variables for Symmetries
push: Functions and Variables for Lists
put: Functions and Variables for Properties
pv: Functions and Variables for finance

Q
qput: Functions and Variables for Properties
qrange: Functions and Variables for descriptive statistics
qty: Functions and Variables for ezunits
quadrilateral: Functions and Variables for draw
quad_control: Functions and Variables for QUADPACK
quad_qag: Functions and Variables for QUADPACK
quad_qagi: Functions and Variables for QUADPACK
quad_qagp: Functions and Variables for QUADPACK
quad_qags: Functions and Variables for QUADPACK
quad_qawc: Functions and Variables for QUADPACK
quad_qawf: Functions and Variables for QUADPACK
quad_qawo: Functions and Variables for QUADPACK
quad_qaws: Functions and Variables for QUADPACK
quantile: Functions and Variables for descriptive statistics
quantile_bernoulli: Functions and Variables for discrete distributions
quantile_beta: Functions and Variables for continuous distributions
quantile_binomial: Functions and Variables for discrete distributions
quantile_cauchy: Functions and Variables for continuous distributions
quantile_chi2: Functions and Variables for continuous distributions
quantile_continuous_uniform: Functions and Variables for continuous distributions
quantile_discrete_uniform: Functions and Variables for discrete distributions
quantile_exp: Functions and Variables for continuous distributions
quantile_f: Functions and Variables for continuous distributions
quantile_gamma: Functions and Variables for continuous distributions
quantile_general_finite_discrete: Functions and Variables for discrete distributions
quantile_geometric: Functions and Variables for discrete distributions
quantile_gumbel: Functions and Variables for continuous distributions
quantile_hypergeometric: Functions and Variables for discrete distributions
quantile_laplace: Functions and Variables for continuous distributions
quantile_logistic: Functions and Variables for continuous distributions
quantile_lognormal: Functions and Variables for continuous distributions
quantile_negative_binomial: Functions and Variables for discrete distributions
quantile_noncentral_chi2: Functions and Variables for continuous distributions
quantile_noncentral_student_t: Functions and Variables for continuous distributions
quantile_normal: Functions and Variables for continuous distributions
quantile_pareto: Functions and Variables for continuous distributions
quantile_poisson: Functions and Variables for discrete distributions
quantile_rayleigh: Functions and Variables for continuous distributions
quantile_student_t: Functions and Variables for continuous distributions
quantile_weibull: Functions and Variables for continuous distributions
quartile_skewness: Functions and Variables for descriptive statistics
quit: Functions and Variables for Command Line
qunit: Functions and Variables for Number Theory
quotient: Functions and Variables for Polynomials

R
racah_v: Functions and Variables for clebsch_gordan
racah_w: Functions and Variables for clebsch_gordan
radcan: Functions and Variables for Simplification
radexpand: Functions and Variables for Simplification
radius: Visualization with VTK
radius: Functions and Variables for graphs
radsubstflag: Functions and Variables for Polynomials
random: Random Numbers
random_bernoulli: Functions and Variables for discrete distributions
random_beta: Functions and Variables for continuous distributions
random_binomial: Functions and Variables for discrete distributions
random_bipartite_graph: Functions and Variables for graphs
random_cauchy: Functions and Variables for continuous distributions
random_chi2: Functions and Variables for continuous distributions
random_continuous_uniform: Functions and Variables for continuous distributions
random_digraph: Functions and Variables for graphs
random_discrete_uniform: Functions and Variables for discrete distributions
random_exp: Functions and Variables for continuous distributions
random_f: Functions and Variables for continuous distributions
random_gamma: Functions and Variables for continuous distributions
random_general_finite_discrete: Functions and Variables for discrete distributions
random_geometric: Functions and Variables for discrete distributions
random_graph: Functions and Variables for graphs
random_graph1: Functions and Variables for graphs
random_gumbel: Functions and Variables for continuous distributions
random_hypergeometric: Functions and Variables for discrete distributions
random_laplace: Functions and Variables for continuous distributions
random_logistic: Functions and Variables for continuous distributions
random_lognormal: Functions and Variables for continuous distributions
random_negative_binomial: Functions and Variables for discrete distributions
random_network: Functions and Variables for graphs
random_noncentral_chi2: Functions and Variables for continuous distributions
random_noncentral_student_t: Functions and Variables for continuous distributions
random_normal: Functions and Variables for continuous distributions
random_pareto: Functions and Variables for continuous distributions
random_perm: Functions and Variables for Combinatorics
random_permutation: Functions and Variables for Sets
random_poisson: Functions and Variables for discrete distributions
random_rayleigh: Functions and Variables for continuous distributions
random_regular_graph: Functions and Variables for graphs
random_student_t: Functions and Variables for continuous distributions
random_tournament: Functions and Variables for graphs
random_tree: Functions and Variables for graphs
random_weibull: Functions and Variables for continuous distributions
range: Functions and Variables for descriptive statistics
rank: Functions and Variables for Matrices and Linear Algebra
rassociative: Functions and Variables for Simplification
rat: Functions and Variables for Polynomials
ratalgdenom: Functions and Variables for Polynomials
ratchristof: Functions and Variables for ctensor
ratcoef: Functions and Variables for Polynomials
ratdenom: Functions and Variables for Polynomials
ratdenomdivide: Functions and Variables for Polynomials
ratdiff: Functions and Variables for Polynomials
ratdisrep: Functions and Variables for Polynomials
rateinstein: Functions and Variables for ctensor
ratepsilon: Functions and Variables for Numbers
ratexpand: Functions and Variables for Polynomials
ratexpand: Functions and Variables for Polynomials
ratfac: Functions and Variables for Polynomials
ratinterpol: Functions and Variables for interpol
rational: Functions and Variables for Properties
rational: Package functs
rationalize: Functions and Variables for Numbers
ratmx: Functions and Variables for Matrices and Linear Algebra
ratnumer: Functions and Variables for Polynomials
ratnump: Functions and Variables for Numbers
ratp: Functions and Variables for Polynomials
ratprint: Functions and Variables for Polynomials
ratp_coeffs: Functions and Variables for ratpow
ratp_dense_coeffs: Functions and Variables for ratpow
ratp_hipow: Functions and Variables for ratpow
ratp_lopow: Functions and Variables for ratpow
ratriemann: Functions and Variables for ctensor
ratsimp: Functions and Variables for Polynomials
ratsimp: Functions and Variables for Polynomials
ratsimpexpons: Functions and Variables for Polynomials
ratsubst: Functions and Variables for Polynomials
ratvars: Functions and Variables for Polynomials
ratvars: Functions and Variables for Polynomials
ratvars: Functions and Variables for Polynomials
ratvarswitch: Functions and Variables for Polynomials
ratweight: Functions and Variables for Polynomials
ratweights: Functions and Variables for Polynomials
ratweyl: Functions and Variables for ctensor
ratwtlvl: Functions and Variables for Polynomials
read: Functions and Variables for Command Line
readbyte: Input and Output
readchar: Input and Output
readline: Input and Output
readonly: Functions and Variables for Command Line
read_array: Functions and Variables for plain-text input and output
read_binary_array: Functions and Variables for binary input and output
read_binary_list: Functions and Variables for binary input and output
read_binary_matrix: Functions and Variables for binary input and output
read_hash_table: Functions and Variables for plain-text input and output
read_list: Functions and Variables for plain-text input and output
read_matrix: Functions and Variables for plain-text input and output
read_nested_list: Functions and Variables for plain-text input and output
read_xpm: Functions and Variables for pictures
real: Functions and Variables for Properties
realonly: Functions and Variables for Equations
realpart: Functions for Complex Numbers
realroots: Functions and Variables for Equations
real_fft: Functions and Variables for fast Fourier transform
real_imagpart_to_conjugate: Functions and Variables for to_poly_solve
rearray: Functions and Variables for Arrays
rectangle: Functions and Variables for draw
rectform: Functions for Complex Numbers
rectform_log_if_constant: Functions and Variables for to_poly_solve
recttopolar: Functions and Variables for fast Fourier transform
rediff: Functions and Variables for itensor
redraw: Functions and Variables for graphs
reduce_consts: Package rducon
reduce_order: Functions and Variables for solve_rec
refcheck: Functions and Variables for Debugging
region: Functions and Variables for draw
region_boundaries: Functions and Variables for worldmap
region_boundaries_plus: Functions and Variables for worldmap
rem: Functions and Variables for Properties
remainder: Functions and Variables for Polynomials
remarray: Functions and Variables for Arrays
rembox: Functions and Variables for Expressions
remcomps: Functions and Variables for itensor
remcon: Functions and Variables for itensor
remcoord: Functions and Variables for itensor
remfun: Functions and Variables for Fourier series
remfunction: Functions and Variables for Function Definition
remlet: Functions and Variables for Rules and Patterns
remove: Functions and Variables for Properties
remove_constvalue: Functions and Variables for ezunits
remove_dimensions: Functions and Variables for ezunits
remove_edge: Functions and Variables for graphs
remove_fundamental_dimensions: Functions and Variables for ezunits
remove_fundamental_units: Functions and Variables for ezunits
remove_plot_option: Functions and Variables for Plotting
remove_vertex: Functions and Variables for graphs
rempart: Package functs
remrule: Functions and Variables for Rules and Patterns
remsym: Functions and Variables for itensor
remvalue: Functions and Variables for Miscellaneous Options
rename: Functions and Variables for itensor
rename_file: File operations
reset: Functions and Variables for Command Line
reset_displays: Functions and Variables for alt-display
residue: Functions and Variables for Integration
resolution: Visualization with VTK
resolvante: Functions and Variables for Symmetries
resolvante_alternee1: Functions and Variables for Symmetries
resolvante_bipartite: Functions and Variables for Symmetries
resolvante_diedrale: Functions and Variables for Symmetries
resolvante_klein: Functions and Variables for Symmetries
resolvante_klein3: Functions and Variables for Symmetries
resolvante_produit_sym: Functions and Variables for Symmetries
resolvante_unitaire: Functions and Variables for Symmetries
resolvante_vierer: Functions and Variables for Symmetries
rest: Functions and Variables for Lists
restart: Visualization with VTK
resultant: Functions and Variables for Polynomials
resultant: Functions and Variables for Polynomials
return: Functions and Variables for Program Flow
reveal: Functions and Variables for Expressions
reverse: Functions and Variables for Lists
revert: Functions and Variables for Series
revert2: Functions and Variables for Series
rgb2level: Functions and Variables for pictures
rhs: Functions and Variables for Equations
ric: Functions and Variables for ctensor
ricci: Functions and Variables for ctensor
riem: Functions and Variables for ctensor
riemann: Functions and Variables for ctensor
rinvariant: Functions and Variables for ctensor
risch: Functions and Variables for Integration
rk: Functions for numerical solution of differential equations
rmdir: Directory operations
rmxchar: Functions and Variables for Matrices and Linear Algebra
rncombine: Functions and Variables for Miscellaneous Options
romberg: Functions and Variables for romberg
rombergabs: Functions and Variables for romberg
rombergit: Functions and Variables for romberg
rombergmin: Functions and Variables for romberg
rombergtol: Functions and Variables for romberg
room: Functions and Variables for Runtime Environment
rootsconmode: Functions and Variables for Equations
rootscontract: Functions and Variables for Equations
rootsepsilon: Functions and Variables for Equations
round: Functions for Numbers
row: Functions and Variables for Matrices and Linear Algebra
rowop: Functions and Variables for linearalgebra
rowswap: Functions and Variables for linearalgebra
rreduce: Functions and Variables for Lists
run_testsuite: Functions and Variables for Bug Detection and Reporting
run_viewer: Plotting Options

S
same_xy: Plotting Options
same_xyz: Plotting Options
save: Functions and Variables for File Input and Output
savedef: Functions and Variables for Function Definition
savefactors: Functions and Variables for Polynomials
saving: Functions and Variables for finance
scalar: Functions and Variables for Properties
scalarmatrixp: Functions and Variables for Matrices and Linear Algebra
scalarp: Functions and Variables for Properties
scale: Visualization with VTK
scaled_bessel_i: Bessel Functions
scaled_bessel_i0: Bessel Functions
scaled_bessel_i1: Bessel Functions
scalefactors: Functions and Variables for Matrices and Linear Algebra
scale_lp: Functions and Variables for simplex
scanmap: Functions and Variables for Program Flow
scatterplot: Functions and Variables for statistical graphs
scatterplot_description: Functions and Variables for statistical graphs
scene: Visualization with VTK
schur2comp: Functions and Variables for Symmetries
sconcat: Functions and Variables for Strings
scopy: String Processing
scsimp: Functions and Variables for Simplification
scurvature: Functions and Variables for ctensor
sdowncase: String Processing
sec: Functions and Variables for Trigonometric
sech: Functions and Variables for Trigonometric
second: Functions and Variables for Lists
sequal: String Processing
sequalignore: String Processing
setcheck: Functions and Variables for Debugging
setcheckbreak: Functions and Variables for Debugging
setdifference: Functions and Variables for Sets
setelmx: Functions and Variables for Matrices and Linear Algebra
setequalp: Functions and Variables for Sets
setify: Functions and Variables for Sets
setp: Functions and Variables for Sets
setunits: Functions and Variables for Units
setup_autoload: Functions and Variables for Miscellaneous Options
setval: Functions and Variables for Debugging
set_alt_display: Functions and Variables for alt-display
set_draw_defaults: Functions and Variables for draw
set_edge_weight: Functions and Variables for graphs
set_partitions: Functions and Variables for Sets
set_plot_option: Functions and Variables for Plotting
set_prompt: Functions and Variables for alt-display
set_random_state: Random Numbers
set_tex_environment: Functions and Variables for TeX Output
set_tex_environment_default: Functions and Variables for TeX Output
set_up_dot_simplifications: Functions and Variables for Affine
set_vertex_label: Functions and Variables for graphs
seventh: Functions and Variables for Lists
sexplode: String Processing
sf: Functions and Variables for atensor
sha1sum: Octets and Utilities for Cryptography
sha256sum: Octets and Utilities for Cryptography
share_testsuite_files: Functions and Variables for Bug Detection and Reporting
shortest_path: Functions and Variables for graphs
shortest_weighted_path: Functions and Variables for graphs
show: Functions and Variables for itensor
showcomps: Functions and Variables for itensor
showratvars: Functions and Variables for Polynomials
showtime: Functions and Variables for Command Line
show_edges: Functions and Variables for graphs
show_edge_color: Functions and Variables for graphs
show_edge_type: Functions and Variables for graphs
show_edge_width: Functions and Variables for graphs
show_id: Functions and Variables for graphs
show_label: Functions and Variables for graphs
show_vertex_color: Functions and Variables for graphs
show_vertex_size: Functions and Variables for graphs
show_vertex_type: Functions and Variables for graphs
show_vertices: Functions and Variables for graphs
show_weight: Functions and Variables for graphs
sierpinskiale: Definitions for IFS fractals
sierpinskimap: Definitions for Peano maps
sign: Functions and Variables for Facts
signum: Functions for Numbers
similaritytransform: Functions and Variables for Matrices and Linear Algebra
simp: Functions and Variables for Simplification
simplified_output: Functions and Variables for zeilberger
simplify_products: Functions and Variables for solve_rec
simplify_sum: Functions and Variables for solve_rec
simplode: String Processing
simpmetderiv: Functions and Variables for itensor
simpproduct: Functions and Variables for Sums and Products
simpsum: Functions and Variables for Sums and Products
simp_inequality: Functions and Variables for to_poly_solve
simtran: Functions and Variables for Matrices and Linear Algebra
sin: Functions and Variables for Trigonometric
sinh: Functions and Variables for Trigonometric
sinnpiflag: Functions and Variables for Fourier series
sinsert: String Processing
sinvertcase: String Processing
sixth: Functions and Variables for Lists
skewness: Functions and Variables for descriptive statistics
skewness_bernoulli: Functions and Variables for discrete distributions
skewness_beta: Functions and Variables for continuous distributions
skewness_binomial: Functions and Variables for discrete distributions
skewness_chi2: Functions and Variables for continuous distributions
skewness_continuous_uniform: Functions and Variables for continuous distributions
skewness_discrete_uniform: Functions and Variables for discrete distributions
skewness_exp: Functions and Variables for continuous distributions
skewness_f: Functions and Variables for continuous distributions
skewness_gamma: Functions and Variables for continuous distributions
skewness_general_finite_discrete: Functions and Variables for discrete distributions
skewness_geometric: Functions and Variables for discrete distributions
skewness_gumbel: Functions and Variables for continuous distributions
skewness_hypergeometric: Functions and Variables for discrete distributions
skewness_laplace: Functions and Variables for continuous distributions
skewness_logistic: Functions and Variables for continuous distributions
skewness_lognormal: Functions and Variables for continuous distributions
skewness_negative_binomial: Functions and Variables for discrete distributions
skewness_noncentral_chi2: Functions and Variables for continuous distributions
skewness_noncentral_student_t: Functions and Variables for continuous distributions
skewness_normal: Functions and Variables for continuous distributions
skewness_pareto: Functions and Variables for continuous distributions
skewness_poisson: Functions and Variables for discrete distributions
skewness_rayleigh: Functions and Variables for continuous distributions
skewness_student_t: Functions and Variables for continuous distributions
skewness_weibull: Functions and Variables for continuous distributions
slength: String Processing
smake: String Processing
small_rhombicosidodecahedron_graph: Functions and Variables for graphs
small_rhombicuboctahedron_graph: Functions and Variables for graphs
smax: Functions and Variables for descriptive statistics
smin: Functions and Variables for descriptive statistics
smismatch: String Processing
snowmap: Definitions for Koch snowflakes
snub_cube_graph: Functions and Variables for graphs
snub_dodecahedron_graph: Functions and Variables for graphs
solve: Functions and Variables for Equations
solvedecomposes: Functions and Variables for Equations
solveexplicit: Functions and Variables for Equations
solvefactors: Functions and Variables for Equations
solvenullwarn: Functions and Variables for Equations
solveradcan: Functions and Variables for Equations
solvetrigwarn: Functions and Variables for Equations
solve_rec: Functions and Variables for solve_rec
solve_rec_rat: Functions and Variables for solve_rec
some: Functions and Variables for Sets
somrac: Functions and Variables for Symmetries
sort: Functions and Variables for Lists
space: Characters
sparse: Functions and Variables for Matrices and Linear Algebra
sparse6_decode: Functions and Variables for graphs
sparse6_encode: Functions and Variables for graphs
sparse6_export: Functions and Variables for graphs
sparse6_import: Functions and Variables for graphs
specint: Functions and Variables for Special Functions
sphere: Visualization with VTK
spherical: Functions and Variables for draw
spherical_bessel_j: Functions and Variables for orthogonal polynomials
spherical_bessel_y: Functions and Variables for orthogonal polynomials
spherical_hankel1: Functions and Variables for orthogonal polynomials
spherical_hankel2: Functions and Variables for orthogonal polynomials
spherical_harmonic: Functions and Variables for orthogonal polynomials
spherical_to_xyz: Functions and Variables for Plotting
splice: Macros
split: String Processing
sposition: String Processing
spring_embedding_depth: Functions and Variables for graphs
sprint: Input and Output
sqfr: Functions and Variables for Polynomials
sqrt: Root Exponential and Logarithmic Functions
sqrtdenest: Package sqdnst
sqrtdispflag: Functions and Variables for Display
sremove: String Processing
sremovefirst: String Processing
sreverse: String Processing
ssearch: String Processing
ssort: String Processing
sstatus: Functions and Variables for Runtime Environment
ssubst: String Processing
ssubstfirst: String Processing
staircase: Graphical analysis of discrete dynamical systems
standardize: Functions and Variables for data manipulation
standardize_inverse_trig: Functions and Variables for to_poly_solve
stardisp: Functions and Variables for Display
starplot: Functions and Variables for statistical graphs
starplot_description: Functions and Variables for statistical graphs
startphi: Visualization with VTK
starttheta: Visualization with VTK
stats_numer: Functions and Variables for stats
status: Functions and Variables for Runtime Environment
std: Functions and Variables for descriptive statistics
std1: Functions and Variables for descriptive statistics
std_bernoulli: Functions and Variables for discrete distributions
std_beta: Functions and Variables for continuous distributions
std_binomial: Functions and Variables for discrete distributions
std_chi2: Functions and Variables for continuous distributions
std_continuous_uniform: Functions and Variables for continuous distributions
std_discrete_uniform: Functions and Variables for discrete distributions
std_exp: Functions and Variables for continuous distributions
std_f: Functions and Variables for continuous distributions
std_gamma: Functions and Variables for continuous distributions
std_general_finite_discrete: Functions and Variables for discrete distributions
std_geometric: Functions and Variables for discrete distributions
std_gumbel: Functions and Variables for continuous distributions
std_hypergeometric: Functions and Variables for discrete distributions
std_laplace: Functions and Variables for continuous distributions
std_logistic: Functions and Variables for continuous distributions
std_lognormal: Functions and Variables for continuous distributions
std_negative_binomial: Functions and Variables for discrete distributions
std_noncentral_chi2: Functions and Variables for continuous distributions
std_noncentral_student_t: Functions and Variables for continuous distributions
std_normal: Functions and Variables for continuous distributions
std_pareto: Functions and Variables for continuous distributions
std_poisson: Functions and Variables for discrete distributions
std_rayleigh: Functions and Variables for continuous distributions
std_student_t: Functions and Variables for continuous distributions
std_weibull: Functions and Variables for continuous distributions
stemplot: Functions and Variables for statistical graphs
step: Functions and Variables for Program Flow
stirling: Functions and Variables for stirling
stirling1: Functions and Variables for Sets
stirling2: Functions and Variables for Sets
strim: String Processing
striml: String Processing
strimr: String Processing
string: Functions and Variables for Strings
stringdisp: Functions and Variables for Strings
stringout: Functions and Variables for File Input and Output
stringp: String Processing
string_to_octets: Octets and Utilities for Cryptography
strong_components: Functions and Variables for graphs
structures: Functions and Variables for Structures
struve_h: Struve Functions
struve_l: Struve Functions
style: Plotting Options
sublis: Functions and Variables for Expressions
sublist: Functions and Variables for Lists
sublist_indices: Functions and Variables for Lists
sublis_apply_lambda: Functions and Variables for Expressions
submatrix: Functions and Variables for Matrices and Linear Algebra
subnumsimp: Functions and Variables for Expressions
subsample: Functions and Variables for data manipulation
Subscript operator: Functions and Variables for Lists
subset: Functions and Variables for Sets
subsetp: Functions and Variables for Sets
subst: Functions and Variables for Expressions
substinpart: Functions and Variables for Expressions
substpart: Functions and Variables for Expressions
substring: String Processing
subst_parallel: Functions and Variables for to_poly_solve
Subtraction: Arithmetic operators
subvar: Functions and Variables for Arrays
subvarp: Functions and Variables for Arrays
sum: Functions and Variables for Sums and Products
sumcontract: Functions and Variables for Sums and Products
sumexpand: Functions and Variables for Sums and Products
summand_to_rec: Functions and Variables for solve_rec
sumsplitfact: Combinatorial Functions
supcase: String Processing
supcontext: Functions and Variables for Facts
surface: Visualization with VTK
surface_hide: Functions and Variables for draw
svg_file: Plotting Options
symbolp: Functions and Variables for Expressions
symmdifference: Functions and Variables for Sets
symmetric: Functions and Variables for Simplification
symmetricp: Functions and Variables for ctensor
system: Functions and Variables for Runtime Environment

T
t: Plotting Options
tab: Characters
take_channel: Functions and Variables for pictures
take_inference: Functions and Variables for inference_result
tan: Functions and Variables for Trigonometric
tanh: Functions and Variables for Trigonometric
taylor: Functions and Variables for Series
taylordepth: Functions and Variables for Series
taylorinfo: Functions and Variables for Series
taylorp: Functions and Variables for Series
taylor_logexpand: Functions and Variables for Series
taylor_order_coefficients: Functions and Variables for Series
taylor_simplifier: Functions and Variables for Series
taylor_truncate_polynomials: Functions and Variables for Series
taytorat: Functions and Variables for Series
tcl_output: Functions and Variables for Miscellaneous Options
tcontract: Functions and Variables for Symmetries
tellrat: Functions and Variables for Polynomials
tellsimp: Functions and Variables for Rules and Patterns
tellsimpafter: Functions and Variables for Rules and Patterns
tensorkill: Functions and Variables for ctensor
tentex: Functions and Variables for itensor
tenth: Functions and Variables for Lists
terminal: Functions and Variables for draw
terminal: Functions and Variables for graphs
testsuite_files: Functions and Variables for Bug Detection and Reporting
test_mean: Functions and Variables for stats
test_means_difference: Functions and Variables for stats
test_normality: Functions and Variables for stats
test_proportion: Functions and Variables for stats
test_proportions_difference: Functions and Variables for stats
test_rank_sum: Functions and Variables for stats
test_sign: Functions and Variables for stats
test_signed_rank: Functions and Variables for stats
test_variance: Functions and Variables for stats
test_variance_ratio: Functions and Variables for stats
tex: Functions and Variables for TeX Output
tex1: Functions and Variables for TeX Output
texput: Functions and Variables for TeX Output
tex_display: Functions and Variables for alt-display
thetaresolution: Visualization with VTK
third: Functions and Variables for Lists
throw: Functions and Variables for Program Flow
thru: Functions and Variables for Program Flow
time: Functions and Variables for Runtime Environment
timedate: Functions and Variables for Runtime Environment
timer: Functions and Variables for Debugging
timer_devalue: Functions and Variables for Debugging
timer_info: Functions and Variables for Debugging
title: Plotting Options
title: Functions and Variables for draw
tldefint: Functions and Variables for Integration
tlimit: Functions and Variables for Limits
tlimswitch: Functions and Variables for Limits
todd_coxeter: Functions and Variables for Groups
toeplitz: Functions and Variables for linearalgebra
tokens: String Processing
topological_sort: Functions and Variables for graphs
totaldisrep: Functions and Variables for Polynomials
totalfourier: Functions and Variables for Fourier series
totient: Functions and Variables for Number Theory
to_lisp: Functions and Variables for Command Line
to_poly: Functions and Variables for to_poly_solve
to_poly_solve: Functions and Variables for to_poly_solve
tpartpol: Functions and Variables for Symmetries
tr: Functions and Variables for ctensor
trace: Functions and Variables for Debugging
tracematrix: Package functs
trace_options: Functions and Variables for Debugging
track: Visualization with VTK
transcompile: Functions and Variables for Function Definition
transform: Functions and Variables for draw
transform_sample: Functions and Variables for data manipulation
transform_xy: Plotting Options
translate: Functions and Variables for Function Definition
translate_fast_arrays: Functions and Variables for Arrays
translate_file: Functions and Variables for Function Definition
transparent: Functions and Variables for draw
transpose: Functions and Variables for Matrices and Linear Algebra
transrun: Functions and Variables for Function Definition
treefale: Definitions for IFS fractals
tree_reduce: Functions and Variables for Lists
treillis: Functions and Variables for Symmetries
treinat: Functions and Variables for Symmetries
triangle: Functions and Variables for draw
triangularize: Functions and Variables for Matrices and Linear Algebra
trigexpand: Functions and Variables for Trigonometric
trigexpandplus: Functions and Variables for Trigonometric
trigexpandtimes: Functions and Variables for Trigonometric
triginverses: Functions and Variables for Trigonometric
trigrat: Functions and Variables for Trigonometric
trigreduce: Functions and Variables for Trigonometric
trigsign: Functions and Variables for Trigonometric
trigsimp: Functions and Variables for Trigonometric
trivial_solutions: Functions and Variables for zeilberger
true: Functions and Variables for Constants
trunc: Functions and Variables for Series
truncate: Functions for Numbers
truncated_cube_graph: Functions and Variables for graphs
truncated_dodecahedron_graph: Functions and Variables for graphs
truncated_icosahedron_graph: Functions and Variables for graphs
truncated_tetrahedron_graph: Functions and Variables for graphs
tr_array_as_ref: Functions and Variables for Function Definition
tr_bound_function_applyp: Functions and Variables for Function Definition
tr_file_tty_messagesp: Functions and Variables for Function Definition
tr_float_can_branch_complex: Functions and Variables for Function Definition
tr_function_call_default: Functions and Variables for Function Definition
tr_numer: Functions and Variables for Function Definition
tr_optimize_max_loop: Functions and Variables for Function Definition
tr_semicompile: Functions and Variables for Function Definition
tr_state_vars: Functions and Variables for Function Definition
tr_warnings_get: Functions and Variables for Function Definition
tr_warn_bad_function_calls: Functions and Variables for Function Definition
tr_warn_fexpr: Functions and Variables for Function Definition
tr_warn_meval: Functions and Variables for Function Definition
tr_warn_mode: Functions and Variables for Function Definition
tr_warn_undeclared: Functions and Variables for Function Definition
tr_warn_undefined_variable: Functions and Variables for Function Definition
tstep: Visualization with VTK
ttyoff: Functions and Variables for Display
tube: Functions and Variables for draw
tutte_graph: Functions and Variables for graphs

U
ueivects: Functions and Variables for Matrices and Linear Algebra
ufg: Functions and Variables for ctensor
uforget: Functions and Variables for Units
ug: Functions and Variables for ctensor
ultraspherical: Functions and Variables for orthogonal polynomials
und: Functions and Variables for Constants
underlying_graph: Functions and Variables for graphs
undiff: Functions and Variables for itensor
unicode: Characters
unicode_to_utf8: Characters
union: Functions and Variables for Sets
unique: Functions and Variables for Lists
uniteigenvectors: Functions and Variables for Matrices and Linear Algebra
unitp: Functions and Variables for ezunits
units: Functions and Variables for ezunits
unitvector: Functions and Variables for Matrices and Linear Algebra
unit_step: Functions and Variables for orthogonal polynomials
unit_vectors: Functions and Variables for draw
unknown: Functions and Variables for Predicates
unless: Functions and Variables for Program Flow
unorder: Functions and Variables for Expressions
unsum: Functions and Variables for Series
untellrat: Functions and Variables for Polynomials
untimer: Functions and Variables for Debugging
untrace: Functions and Variables for Debugging
uppercasep: Characters
uric: Functions and Variables for ctensor
uricci: Functions and Variables for ctensor
uriem: Functions and Variables for ctensor
uriemann: Functions and Variables for ctensor
usersetunits: Functions and Variables for Units
user_preamble: Functions and Variables for draw
use_fast_arrays: Functions and Variables for Arrays
us_ascii_only: Characters
utf8_to_unicode: Characters
uvect: Functions and Variables for Matrices and Linear Algebra

V
values: Functions and Variables for Command Line
vandermonde_matrix: Functions and Variables for linearalgebra
var: Functions and Variables for descriptive statistics
var1: Functions and Variables for descriptive statistics
var_bernoulli: Functions and Variables for discrete distributions
var_beta: Functions and Variables for continuous distributions
var_binomial: Functions and Variables for discrete distributions
var_chi2: Functions and Variables for continuous distributions
var_continuous_uniform: Functions and Variables for continuous distributions
var_discrete_uniform: Functions and Variables for discrete distributions
var_exp: Functions and Variables for continuous distributions
var_f: Functions and Variables for continuous distributions
var_gamma: Functions and Variables for continuous distributions
var_general_finite_discrete: Functions and Variables for discrete distributions
var_geometric: Functions and Variables for discrete distributions
var_gumbel: Functions and Variables for continuous distributions
var_hypergeometric: Functions and Variables for discrete distributions
var_laplace: Functions and Variables for continuous distributions
var_logistic: Functions and Variables for continuous distributions
var_lognormal: Functions and Variables for continuous distributions
var_negative_binomial: Functions and Variables for discrete distributions
var_noncentral_chi2: Functions and Variables for continuous distributions
var_noncentral_student_t: Functions and Variables for continuous distributions
var_normal: Functions and Variables for continuous distributions
var_pareto: Functions and Variables for continuous distributions
var_poisson: Functions and Variables for discrete distributions
var_rayleigh: Functions and Variables for continuous distributions
var_student_t: Functions and Variables for continuous distributions
var_weibull: Functions and Variables for continuous distributions
vector: Functions and Variables for draw
vectorpotential: Functions and Variables for Matrices and Linear Algebra
vectorsimp: Functions and Variables for Matrices and Linear Algebra
vect_cross: Functions and Variables for Matrices and Linear Algebra
verbify: Functions and Variables for Expressions
verbose: Functions and Variables for Series
vers: Package functs
vertex_color: Functions and Variables for graphs
vertex_coloring: Functions and Variables for graphs
vertex_coloring: Functions and Variables for graphs
vertex_connectivity: Functions and Variables for graphs
vertex_degree: Functions and Variables for graphs
vertex_distance: Functions and Variables for graphs
vertex_eccentricity: Functions and Variables for graphs
vertex_in_degree: Functions and Variables for graphs
vertex_out_degree: Functions and Variables for graphs
vertex_partition: Functions and Variables for graphs
vertex_size: Functions and Variables for graphs
vertex_type: Functions and Variables for graphs
vertices: Functions and Variables for graphs
vertices_to_cycle: Functions and Variables for graphs
vertices_to_path: Functions and Variables for graphs
view: Functions and Variables for draw

W
warning: Functions and Variables for Program Flow
warnings: Functions and Variables for zeilberger
wc_inputvalueranges: Functions and Variables for wrstcse
wc_mintypmax: Functions and Variables for wrstcse
wc_mintypmax2tol: Functions and Variables for wrstcse
wc_montecarlo: Functions and Variables for wrstcse
wc_systematic: Functions and Variables for wrstcse
wc_tolappend: Functions and Variables for wrstcse
wc_typicalvalues: Functions and Variables for wrstcse
weyl: Functions and Variables for ctensor
weyl: Functions and Variables for ctensor
wheel_graph: Functions and Variables for graphs
while: Functions and Variables for Program Flow
width: Visualization with VTK
wiener_index: Functions and Variables for graphs
wigner_3j: Functions and Variables for clebsch_gordan
wigner_6j: Functions and Variables for clebsch_gordan
wigner_9j: Functions and Variables for clebsch_gordan
windowname: Visualization with VTK
windowtitle: Visualization with VTK
wired_surface: Functions and Variables for draw
wireframe: Visualization with VTK
with_stdout: Functions and Variables for File Input and Output
writebyte: Input and Output
writefile: Functions and Variables for File Input and Output
write_binary_data: Functions and Variables for binary input and output
write_data: Functions and Variables for plain-text input and output
wronskian: Package functs

X
x: Plotting Options
xaxis: Functions and Variables for draw
xaxis_color: Functions and Variables for draw
xaxis_secondary: Functions and Variables for draw
xaxis_type: Functions and Variables for draw
xaxis_width: Functions and Variables for draw
xlabel: Plotting Options
xlabel: Functions and Variables for draw
xlabel_secondary: Functions and Variables for draw
xlength: Visualization with VTK
xrange: Functions and Variables for draw
xrange_secondary: Functions and Variables for draw
xreduce: Functions and Variables for Lists
xthru: Functions and Variables for Simplification
xtics: Plotting Options
xtics: Functions and Variables for draw
xtics_axis: Functions and Variables for draw
xtics_rotate: Functions and Variables for draw
xtics_rotate_secondary: Functions and Variables for draw
xtics_secondary: Functions and Variables for draw
xtics_secondary_axis: Functions and Variables for draw
xu_grid: Functions and Variables for draw
xyplane: Functions and Variables for draw
xy_file: Functions and Variables for draw
xy_scale: Plotting Options
x_voxel: Functions and Variables for draw

Y
y: Plotting Options
yaxis: Functions and Variables for draw
yaxis_color: Functions and Variables for draw
yaxis_secondary: Functions and Variables for draw
yaxis_type: Functions and Variables for draw
yaxis_width: Functions and Variables for draw
ylabel: Plotting Options
ylabel: Functions and Variables for draw
ylabel_secondary: Functions and Variables for draw
ylength: Visualization with VTK
yrange: Functions and Variables for draw
yrange_secondary: Functions and Variables for draw
ytics: Plotting Options
ytics: Functions and Variables for draw
ytics_axis: Functions and Variables for draw
ytics_rotate: Functions and Variables for draw
ytics_rotate_secondary: Functions and Variables for draw
ytics_secondary: Functions and Variables for draw
ytics_secondary_axis: Functions and Variables for draw
yv_grid: Functions and Variables for draw
yx_ratio: Plotting Options
y_voxel: Functions and Variables for draw

Z
z: Plotting Options
zaxis: Functions and Variables for draw
zaxis_color: Functions and Variables for draw
zaxis_type: Functions and Variables for draw
zaxis_width: Functions and Variables for draw
Zeilberger: Functions and Variables for zeilberger
zeroa: Functions and Variables for Constants
zerob: Functions and Variables for Constants
zerobern: Functions and Variables for Number Theory
zeroequiv: Functions and Variables for Predicates
zerofor: Functions and Variables for linearalgebra
zeromatrix: Functions and Variables for Matrices and Linear Algebra
zeromatrixp: Functions and Variables for linearalgebra
zeta: Functions and Variables for Number Theory
zeta%pi: Functions and Variables for Number Theory
zgeev: Functions and Variables for lapack
zheev: Functions and Variables for lapack
zlabel: Plotting Options
zlabel: Functions and Variables for draw
zlabel_rotate: Functions and Variables for draw
zlange: Functions and Variables for lapack
zlength: Visualization with VTK
zmin: Plotting Options
zn_add_table: Functions and Variables for Number Theory
zn_carmichael_lambda: Functions and Variables for Number Theory
zn_characteristic_factors: Functions and Variables for Number Theory
zn_determinant: Functions and Variables for Number Theory
zn_factor_generators: Functions and Variables for Number Theory
zn_invert_by_lu: Functions and Variables for Number Theory
zn_log: Functions and Variables for Number Theory
zn_mult_table: Functions and Variables for Number Theory
zn_nth_root: Functions and Variables for Number Theory
zn_order: Functions and Variables for Number Theory
zn_power_table: Functions and Variables for Number Theory
zn_primroot: Functions and Variables for Number Theory
zn_primroot_limit: Functions and Variables for Number Theory
zn_primroot_p: Functions and Variables for Number Theory
zn_primroot_pretest: Functions and Variables for Number Theory
zn_primroot_verbose: Functions and Variables for Number Theory
zrange: Functions and Variables for draw
ztics: Functions and Variables for draw
ztics_axis: Functions and Variables for draw
ztics_rotate: Functions and Variables for draw
z_voxel: Functions and Variables for draw

Jump to:   !   #   $   %   '   *   +   -   .   /   :   ;   <   =   >   ?   @   [   ]   ^   _   `   |   ~  
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Table of Contents


Previous:   [Contents][Index]

B Documentation Categories

Category: Airy functions

Introduction to Special Functions · airy_ai · airy_bi · airy_dai · airy_dbi

Category: Algebraic equations

%rnum · %rnum_list · algepsilon · algexact · algsys · backsubst · breakup · dispflag · eliminate · bf_find_root · find_root · find_root_abs · find_root_error · find_root_rel · funcsolve · multiplicities · newton · programmode · realonly · solve · solvedecomposes · solveexplicit · solvefactors · solvenullwarn · solveradcan · solvetrigwarn

Category: Arrays

array · arrayapply · arrayinfo · arraymake · arrays · arraysetapply · fillarray · listarray · make_array · rearray · remarray · subvar · translate_fast_arrays · use_fast_arrays

Category: Bessel functions

%f · %s · Introduction to Special Functions · bessel_i · bessel_j · bessel_k · bessel_simplify · bessel_y · besselexpand · hankel_1 · hankel_2 · scaled_bessel_i · scaled_bessel_i0 · scaled_bessel_i1 · spherical_bessel_j · spherical_bessel_y · spherical_hankel1 · spherical_hankel2

Category: Binary operations

bit_and · bit_length · bit_lsh · bit_not · bit_onep · bit_or · bit_rsh · bit_xor

Category: Complex variables

Introduction to Numbers · cabs · carg · cbffac · conjugate · demoivre · exponentialize · imagpart · plog · polarform · polartorect · realpart · rectform · recttopolar · residue

Category: Console interaction

% · %th · ? · ?? · Documentation · Interrupts · _ · appendfile · closefile · demo · describe · entermatrix · example · ibase · kill · labels · labels · linenum · myoptions · obase · optionset · playback · printfile · prompt · quit · ratprint · read · readonly · refcheck · setcheck · setcheckbreak · setval · to_lisp · writefile

Category: Constants

%e · %gamma · %i · %phi · %pi · constant · constantp · false · ind · inf · infinity · minf · true · und · zeroa · zerob

Category: Continued fractions

cf · cfdisrep · cfexpand · cflength

Category: Debugging

Keyword Commands · Source Level Debugging · backtrace · break · bug_report · build_info · debugmode · error_size · error_syms · room · run_testsuite · share_testsuite_files · showtime · testsuite_files · time · timer · timer_devalue · timer_info · trace · trace_options · untimer · untrace

Category: Declarations and inferences

activate · activecontexts · additive · alias · aliases · alphabetic · antisymmetric · askexp · askinteger · asksign · assume · assume_pos · assume_pos_pred · assumescalar · atvalue · commutative · compare · constant · context · contexts · deactivate · declare · decreasing · increasing · dependencies · depends · even · odd · facts · featurep · features · forget · get · gradef · gradefs · infix · infolists · integer · noninteger · integervalued · is · killcontext · lassociative · linear · mainvar · matchdeclare · maybe · multiplicative · newcontext · nonscalar · numerval · outative · posfun · printprops · properties · props · propvars · put · qput · rassociative · irrational · rational · complex · imaginary · real · rem · remove · scalar · sign · supcontext · symmetric

Category: Descriptive statistics

Introduction to descriptive

Category: Differential calculus

atomgrad · del · depends · derivabbrev · derivdegree · derivlist · derivsubst · diff · express · gradef · gradefs · hessian · implicit_derivative · jacobian · vect_cross · wronskian

Category: Differential equations

Introduction to contrib_ode · Introduction to drawdf · Introduction to numerical solution of differential equations · Lindstedt · at · atvalue · bc2 · desolve · ic1 · ic2 · laplace · ode2 · plotdf · ploteq · rk

Category: Differential geometry

cartan

Category: Display flags and variables

%edispflag · absboxchar · dispflag · display2d · display_format_internal · error_size · error_syms · exptdispflag · fpprintprec · grind · inchar · labels · leftjust · linechar · linel · linenum · lispdisp · lmxchar · negsumdispflag · noundisp · obase · outchar · pfeformat · powerdisp · psexpand · rmxchar · showtime · sqrtdispflag · stardisp · stringdisp · ttyoff

Category: Display functions

disp · dispcon · dispfun · display · disprule · dispterms · engineering_format_floats · engineering_format_max · engineering_format_min · grind · labels · ldisp · ldisplay · playback · print · printpois · printprops · reveal · show · showratvars

Category: Dynamical systems

The dynamics package

Category: Elliptic functions

Introduction to Elliptic Functions and Integrals · inverse_jacobi_cd · inverse_jacobi_cn · inverse_jacobi_cs · inverse_jacobi_dc · inverse_jacobi_dn · inverse_jacobi_ds · inverse_jacobi_nc · inverse_jacobi_nd · inverse_jacobi_ns · inverse_jacobi_sc · inverse_jacobi_sd · inverse_jacobi_sn · jacobi_cd · jacobi_cn · jacobi_cs · jacobi_dc · jacobi_dn · jacobi_ds · jacobi_nc · jacobi_nd · jacobi_ns · jacobi_sc · jacobi_sd · jacobi_sn

Category: Elliptic integrals

elliptic_e · elliptic_ec · elliptic_eu · elliptic_f · elliptic_kc · elliptic_pi

Category: Error messages

0 to a negative exponent · Comma is not a prefix operator · Illegal use of delimiter · No such list element · Only symbols can be bound · Operators of arguments must all be the same · VTK is not installed · argument must be a non-atomic expression · cannot assign to function · loadfile failed to load · makelist second argument must evaluate to a number · out of memory · part fell off the end · undefined variable during plotting · undefined variable during translation

Category: Evaluation flags

%enumer · detout · eval · evflag · evfun · float · infeval · noeval · nouns · numer · pred · simp

Category: Evaluation

· · : · :: · Nouns and Verbs · at · derivlist · ev · kill · refcheck · remvalue · values

Category: Exponential Integrals

expintegral_chi · expintegral_ci · expintegral_e · expintegral_e1 · expintegral_e_simplify · expintegral_ei · expintegral_li · expintegral_shi · expintegral_si · expintexpand · expintrep

Category: Exponential and logarithm functions

%e_to_numlog · %edispflag · %emode · %enumer · exp · exptsubst · li · log · logabs · logarc · logconcoeffp · logcontract · logexpand · lognegint · logsimp · plog · polarform · taylor_logexpand

Category: Expressions

Package facexp · Package rducon · Package scifac · append · args · arrayapply · arraymake · arraysetapply · assoc · atom · block · box · boxchar · collapse · collectterms · combine · concat · cons · copy · delete · denom · derivdegree · derivsubst · disolate · dispform · distrib · dontfactor · dpart · eighth · endcons · expand · expandwrt · expandwrt_denom · expandwrt_factored · expon · expop · exptdispflag · exptisolate · exptsubst · facsum · facsum_combine · factorfacsum · factorout · factorsum · fifth · first · firstn · fourth · freeof · fullmap · fullmapl · funmake · gcfac · gfactorsum · hipow · inflag · inpart · intosum · isolate · isolate_wrt_times · last · lastn · length · lfreeof · lhs · linear · listconstvars · listdummyvars · listofvars · lopow · lpart · m1pbranch · mainvar · maxnegex · maxposex · member · multiplicative · multthru · nextlayerfactor · ninth · nonarray · nonzeroandfreeof · nterms · num · numfactor · op · opsubst · opsubst · optimize · optimprefix · ordergreat · orderless · ordergreatp · orderlessp · part · partition · partswitch · pickapart · piece · pop · psubst · push · rational · reduce_consts · rembox · rempart · rest · reveal · reverse · rhs · rncombine · rootsconmode · sconcat · second · seventh · sixth · sqrtdenest · sublis · sublis_apply_lambda · subnumsimp · subst · substinpart · substpart · subvar · tenth · third · unorder · xthru

Category: File input

Introduction to numericalio · batch · batchload · close · demo · directory · file_search · file_search_demo · file_search_lisp · file_search_maxima · file_search_tests · file_search_usage · file_type · filename_merge · flength · fposition · load · load_pathname · loadfile · loadprint · openr · openr_binary · pathname_directory · pathname_name · pathname_type · printfile · read_array · read_binary_array · read_binary_list · read_binary_matrix · read_hash_table · read_list · read_matrix · read_nested_list · readbyte · readchar · readline · setup_autoload

Category: File output

Introduction to numericalio · appendfile · close · closefile · draw · draw2d · draw3d · draw_file · file_output_append · filename_merge · flength · flush_output · fposition · freshline · multiplot_mode · newline · opena · opena_binary · openw · openw_binary · printf · save · stringout · tex · with_stdout · write_binary_data · write_data · writebyte · writefile

Category: Fourier transform

Introduction to Fourier series · Introduction to fast Fourier transform

Category: Function application

apply · fullmap · fullmapl · funmake · macroexpand · macroexpand1 · macroexpansion · map · maperror · maplist · mapprint · maxapplydepth · maxapplyheight · outermap · scanmap

Category: Function definition

::= · := · Function · buildq · define · dispfun · functions · fundef · lambda · local · macros · remfunction · setup_autoload · splice

Category: Gamma and factorial functions

!! · beta · beta_args_sum_to_integer · beta_expand · beta_incomplete · beta_incomplete_generalized · beta_incomplete_regularized · bffac · bfpsi · bfpsi0 · cbffac · factcomb · factlim · ! · factorial · factorial_expand · gamma · gamma_expand · gamma_incomplete · gamma_incomplete_generalized · gamma_incomplete_lower · gamma_incomplete_regularized · gammalim · genfact · log_gamma · makefact · makegamma · maxpsifracdenom · maxpsifracnum · maxpsinegint · maxpsiposint · pochhammer · pochhammer_max_index · psi · stirling · sumsplitfact

Category: Generating functions

ggf

Category: Global flags

debugmode · derivabbrev · engineering_format_floats · engineering_format_max · engineering_format_min · file_output_append · listarith · loadprint · logabs · m1pbranch · macroexpansion · nolabels · optionset · ratvarswitch · refcheck · setcheck · setcheckbreak · timer_devalue · use_fast_arrays

Category: Global variables

% · %% · _ · __ · adim · aform · aliases · arrays · asymbol · dependencies · f90_output_line_length_max · file_search_demo · file_search_lisp · file_search_maxima · file_search_tests · file_search_usage · functions · infolists · macros · manual_demo · maxima_tempdir · maxima_userdir · myoptions · newline · opproperties · prompt · props · setval · share_testsuite_files · space · structures · tab · testsuite_files · us_ascii_only · values

Category: Groebner bases

Introduction to Affine · Introduction to grobner

Category: Group theory

Introduction to Symmetries · todd_coxeter

Category: Help

? · ?? · apropos · demo · describe · example · manual_demo

Category: Hyperbolic functions

%iargs · acosh · acoth · acsch · asech · asinh · atanh · cosh · coth · csch · demoivre · exponentialize · sech · sinh · tanh

Category: Hypergeometric functions

hypergeometric_simp

Category: Integers

divisors · gcfactor · integer_partitions · intfaclim · modulus · moebius · multinomial_coeff · num_distinct_partitions · num_partitions · stirling1 · stirling2

Category: Integral calculus

Introduction to QUADPACK · absint · antid · antidiff · changevar · dblint · defint · erfflag · intanalysis · integrate · integrate_use_rootsof · integration_constant · integration_constant_counter · ldefint · logabs · residue · risch · tldefint

Category: Integral equations

ieqn · ieqnprint

Category: Laplace transform

delta · desolve · ilt · laplace · specint

Category: Limits

lhospitallim · limit · limsubst · tlimit · tlimswitch

Category: Linear algebra

Introduction to linearalgebra

Category: Linear equations

augcoefmatrix · coefmatrix · dgesv · echelon · globalsolve · linsolve · linsolve_params · linsolvewarn · triangularize

Category: Linear recurrences

Introduction to solve_rec

Category: Lists

[ · ] · append · assoc · cons · copylist · create_list · delete · eighth · endcons · fifth · first · firstn · flatten · fourth · fullsetify · join · last · lastn · length · listarith · listp · lmax · lmin · lreduce · makelist · member · ninth · permut · permutations · pop · push · random_permutation · rest · reverse · rreduce · second · setify · seventh · sixth · some · sort · sublist · sublist_indices · tenth · third · tree_reduce · xreduce

Category: Mathematical functions

abs · agd · ceiling · charfun · combination · covers · delta · entier · exsec · fix · floor · gaussprob · gd · hav · isqrt · lmax · lmin · max · min · mod · permutation · round · signum · sqrt · sqrtdispflag · truncate · unit_step · vers

Category: Matrices

addcol · addrow · adjoint · augcoefmatrix · cauchy_matrix · charpoly · coefmatrix · col · columnvector · covect · copymatrix · determinant · detout · diag · diagmatrix · doallmxops · domxexpt · domxmxops · domxnctimes · doscmxops · doscmxplus · echelon · eigen · ematrix · entermatrix · genmatrix · ident · invert · list_matrix_entries · lmxchar · matrix · matrix_element_add · matrix_element_mult · matrix_element_transpose · matrixexp · matrixmap · matrixp · mattrace · minor · ncharpoly · newdet · nonscalar · nonscalarp · permanent · rank · ratmx · row · scalarmatrixp · scalarp · setelmx · sparse · submatrix · tracematrix · transpose · triangularize · zeromatrix

Category: Matrix decompositions

cholesky · eigens_by_jacobi · lu_factor

Category: Nouns and verbs

Nouns and Verbs · noun · noundisp · nounify · nouns · verbify

Category: Number theory

bern · bernpoly · bfhzeta · bfzeta · binomial · bit_and · bit_length · bit_lsh · bit_not · bit_onep · bit_or · bit_rsh · bit_xor · burn · chinese · divsum · euler · factors_only · fib · fibtophi · ifactors · igcdex · inrt · inv_mod · jacobi · lcm · lucas · minfactorial · next_prime · power_mod · prev_prime · primep · primep_number_of_tests · primes · qunit · totient · zerobern · zeta · zeta%pi · zn_add_table · zn_carmichael_lambda · zn_characteristic_factors · zn_determinant · zn_factor_generators · zn_invert_by_lu · zn_log · zn_mult_table · zn_nth_root · zn_order · zn_power_table · zn_primroot · zn_primroot_limit · zn_primroot_p · zn_primroot_pretest · zn_primroot_verbose

Category: Numerical evaluation

bffac · bfhzeta · bfloat · bfloatp · bfpsi · bfpsi0 · bftorat · bftrunc · bfzeta · cbffac · float · float2bf · floatnump · fpprec · fpprintprec · keepfloat · numer · numer_pbranch · numerval · ratepsilon · rationalize · ratprint · stats_numer

Category: Numerical methods

Introduction to ODEPACK · Introduction to QUADPACK · Introduction to cobyla · Introduction to fast Fourier transform · Introduction to interpol · Introduction to lapack · Introduction to lbfgs · Introduction to minpack · Introduction to mnewton · Introduction to numerical solution of differential equations · Introduction to simplex · allroots · augmented_lagrangian_method · bfallroots · bf_find_root · find_root · find_root_abs · find_root_error · find_root_rel · horner · linear_program · lsquares_estimates · lsquares_estimates_approximate · maximize_lp · minimize_lp · newton · nroots · plotdf · ploteq · plsquares · polyfactor · quad_control · quad_qag · quad_qagi · quad_qagp · quad_qags · quad_qawc · quad_qawf · quad_qawo · quad_qaws · random · realroots · rk · romberg · rootsepsilon

Category: Operators

!! · # · · · ** · * · + · - · / · ^ · . · : · :: · ::= · := · < · <= · > · >= · = · @ · Introduction to operators · [ · ] · ^^ · additive · and · antisymmetric · commutative · define_opproperty · equal · express · ! · factorial · infix · lassociative · linear · matchfix · nary · nofix · not · notequal · op · operatorp · opproperties · or · outative · posfun · postfix · prefix · rassociative · symmetric · | · ~

Category: Optimization

Introduction to cobyla · Introduction to lbfgs · Introduction to minpack · Introduction to simplex · augmented_lagrangian_method

Category: Orthogonal polynomials

Introduction to orthogonal polynomials

Category: Package absimp

Package absimp

Category: Package affine

Introduction to Affine · all_dotsimp_denoms · check_overlaps · declare_weights · dotsimp · extract_linear_equations · fast_central_elements · fast_linsolve · grobner_basis · list_nc_monomials · mono · monomial_dimensions · nc_degree · set_up_dot_simplifications

Category: Package alt-display

define_alt_display · info_display · mathml_display · multi_display_for_texinfo · reset_displays · set_alt_display · set_prompt · tex_display

Category: Package antid

linear

Category: Package asympa

asympa

Category: Package atensor

Introduction to atensor · abasep · adim · af · aform · alg_type · asymbol · atensimp · av · init_atensor · sf

Category: Package atrig1

atrig1

Category: Package augmented_lagrangian

augmented_lagrangian_method

Category: Package bitwise

bit_and · bit_length · bit_lsh · bit_not · bit_onep · bit_or · bit_rsh · bit_xor

Category: Package bode

bode_gain · bode_phase

Category: Package cobyla

Introduction to cobyla

Category: Package combinatorics

Package combinatorics · apply_cycles · cyclep · perm_cycles · perm_decomp · perm_inverse · perm_length · perm_lex_next · perm_lex_rank · perm_lex_unrank · perm_next · perm_parity · perm_rank · perm_undecomp · perm_unrank · permp · perms · perms_lex · permult · permute · random_perm

Category: Package contrib_ode

%c · %k1 · %k2 · Introduction to contrib_ode · bessel_simplify · contrib_ode · dgauss_a · dgauss_b · dkummer_m · dkummer_u · expintegral_e_simplify · gauss_a · gauss_b · kummer_m · kummer_u · method · ode_check · odelin

Category: Package ctensor

Introduction to ctensor · bdvac · bimetric · cdisplay · cframe_flag · cgeodesic · checkdiv · christof · cmetric · cnonmet_flag · cograd · contortion · contragrad · csetup · ct_coords · ct_coordsys · ctaylor · ctaypov · ctaypt · ctayswitch · ctayvar · ctorsion_flag · ctransform · ctrgsimp · deleten · diagmatrixp · diagmetric · dim · dscalar · einstein · fb · findde · frame_bracket · gdet · ic_convert · init_ctensor · invariant1 · invariant2 · kinvariant · kt · leinstein · lfg · lg · lriem · lriemann · nm · nmc · nonmetricity · np · npi · nptetrad · ntermst · petrov · psi · ratchristof · rateinstein · ratriemann · ratweyl · ric · ricci · riem · riemann · rinvariant · scurvature · symmetricp · tensorkill · tr · ufg · ug · uric · uricci · uriem · uriemann · weyl · weyl

Category: Package descriptive

Introduction to descriptive · barsplot · barsplot_description · boxplot · boxplot_description · build_sample · cdf_empirical · central_moment · continuous_freq · cor · cov · cov1 · cv · discrete_freq · geometric_mean · global_variances · harmonic_mean · histogram · histogram_description · km · kurtosis · list_correlations · mean · mean_deviation · median · median_deviation · noncentral_moment · pearson_skewness · piechart · piechart_description · principal_components · qrange · quantile · quartile_skewness · range · scatterplot · scatterplot_description · skewness · smax · smin · standardize · starplot · starplot_description · std · std1 · stemplot · subsample · transform_sample · var · var1

Category: Package diag

JF · ModeMatrix · diag · dispJordan · jordan · mat_function · minimalPoly

Category: Package distrib

Introduction to distrib · cdf_bernoulli · cdf_beta · cdf_binomial · cdf_cauchy · cdf_chi2 · cdf_continuous_uniform · cdf_discrete_uniform · cdf_exp · cdf_f · cdf_gamma · cdf_general_finite_discrete · cdf_geometric · cdf_gumbel · cdf_hypergeometric · cdf_laplace · cdf_logistic · cdf_lognormal · cdf_negative_binomial · cdf_noncentral_chi2 · cdf_noncentral_student_t · cdf_normal · cdf_pareto · cdf_poisson · cdf_rayleigh · cdf_student_t · cdf_weibull · kurtosis_bernoulli · kurtosis_beta · kurtosis_binomial · kurtosis_chi2 · kurtosis_continuous_uniform · kurtosis_discrete_uniform · kurtosis_exp · kurtosis_f · kurtosis_gamma · kurtosis_general_finite_discrete · kurtosis_geometric · kurtosis_gumbel · kurtosis_gumbel · kurtosis_hypergeometric · kurtosis_laplace · kurtosis_logistic · kurtosis_lognormal · kurtosis_negative_binomial · kurtosis_noncentral_chi2 · kurtosis_noncentral_student_t · kurtosis_normal · kurtosis_pareto · kurtosis_poisson · kurtosis_rayleigh · kurtosis_student_t · kurtosis_weibull · mean_bernoulli · mean_beta · mean_binomial · mean_chi2 · mean_continuous_uniform · mean_discrete_uniform · mean_exp · mean_f · mean_gamma · mean_general_finite_discrete · mean_geometric · mean_gumbel · mean_hypergeometric · mean_laplace · mean_logistic · mean_lognormal · mean_negative_binomial · mean_noncentral_chi2 · mean_noncentral_student_t · mean_normal · mean_pareto · mean_poisson · mean_rayleigh · mean_student_t · mean_weibull · pdf_bernoulli · pdf_beta · pdf_binomial · pdf_cauchy · pdf_chi2 · pdf_continuous_uniform · pdf_discrete_uniform · pdf_exp · pdf_f · pdf_gamma · pdf_general_finite_discrete · pdf_geometric · pdf_gumbel · pdf_hypergeometric · pdf_laplace · pdf_logistic · pdf_lognormal · pdf_negative_binomial · pdf_noncentral_chi2 · pdf_noncentral_student_t · pdf_normal · pdf_pareto · pdf_poisson · pdf_rayleigh · pdf_student_t · pdf_weibull · quantile_bernoulli · quantile_beta · quantile_binomial · quantile_cauchy · quantile_chi2 · quantile_continuous_uniform · quantile_discrete_uniform · quantile_exp · quantile_f · quantile_gamma · quantile_general_finite_discrete · quantile_geometric · quantile_gumbel · quantile_hypergeometric · quantile_laplace · quantile_logistic · quantile_lognormal · quantile_negative_binomial · quantile_noncentral_chi2 · quantile_noncentral_student_t · quantile_normal · quantile_pareto · quantile_poisson · quantile_rayleigh · quantile_student_t · quantile_weibull · random_bernoulli · random_beta · random_binomial · random_cauchy · random_chi2 · random_continuous_uniform · random_discrete_uniform · random_exp · random_f · random_gamma · random_general_finite_discrete · random_geometric · random_gumbel · random_hypergeometric · random_laplace · random_logistic · random_lognormal · random_negative_binomial · random_noncentral_chi2 · random_noncentral_student_t · random_normal · random_pareto · random_poisson · random_rayleigh · random_student_t · random_weibull · skewness_bernoulli · skewness_beta · skewness_binomial · skewness_chi2 · skewness_continuous_uniform · skewness_discrete_uniform · skewness_exp · skewness_f · skewness_gamma · skewness_general_finite_discrete · skewness_geometric · skewness_gumbel · skewness_hypergeometric · skewness_laplace · skewness_logistic · skewness_lognormal · skewness_negative_binomial · skewness_noncentral_chi2 · skewness_noncentral_student_t · skewness_normal · skewness_pareto · skewness_poisson · skewness_rayleigh · skewness_student_t · skewness_weibull · std_bernoulli · std_beta · std_binomial · std_chi2 · std_continuous_uniform · std_discrete_uniform · std_exp · std_f · std_gamma · std_general_finite_discrete · std_geometric · std_gumbel · std_hypergeometric · std_laplace · std_logistic · std_lognormal · std_negative_binomial · std_noncentral_chi2 · std_noncentral_student_t · std_normal · std_pareto · std_poisson · std_rayleigh · std_student_t · std_weibull · var_bernoulli · var_beta · var_binomial · var_chi2 · var_continuous_uniform · var_discrete_uniform · var_exp · var_f · var_gamma · var_general_finite_discrete · var_geometric · var_gumbel · var_hypergeometric · var_laplace · var_logistic · var_lognormal · var_negative_binomial · var_noncentral_chi2 · var_noncentral_student_t · var_normal · var_pareto · var_poisson · var_rayleigh · var_student_t · var_weibull

Category: Package draw

Introduction to draw · Introduction to drawdf · adapt_depth · allocation · axis_3d · axis_bottom · axis_left · axis_right · axis_top · background_color · bars · border · boundaries_array · capping · cbrange · cbtics · color · colorbox · columns · contour · contour_levels · cylindrical · data_file_name · delay · dimensions · draw · draw2d · draw3d · draw_file · draw_realpart · elevation_grid · ellipse · enhanced3d · errors · explicit · file_name · fill_color · filled_func · font · font_size · geomap · get_pixel · gnuplot_file_name · gr2d · gr3d · grid · head_angle · head_both · head_length · head_type · image · implicit · interpolate_color · ip_grid · ip_grid_in · key · key_pos · label · label_alignment · label_orientation · line_type · line_width · logcb · logx · logx_secondary · logy · logy_secondary · logz · make_level_picture · make_poly_continent · make_poly_country · make_polygon · make_rgb_picture · mesh · multiplot_mode · negative_picture · nticks · numbered_boundaries · palette · parametric · parametric_surface · picture_equalp · picturep · point_size · point_type · points · points_joined · polar · polygon · proportional_axes · quadrilateral · read_xpm · rectangle · region_boundaries · region_boundaries_plus · rgb2level · set_draw_defaults · spherical · surface_hide · take_channel · terminal · title · transform · transparent · triangle · tube · unit_vectors · user_preamble · vector · view · wired_surface · x_voxel · xaxis · xaxis_color · xaxis_secondary · xaxis_type · xaxis_width · xlabel · xlabel_secondary · xrange · xrange_secondary · xtics · xtics_axis · xtics_rotate · xtics_rotate_secondary · xtics_secondary · xtics_secondary_axis · xu_grid · xy_file · xyplane · y_voxel · yaxis · yaxis_color · yaxis_secondary · yaxis_type · yaxis_width · ylabel · ylabel_secondary · yrange · yrange_secondary · ytics · ytics_axis · ytics_rotate · ytics_rotate_secondary · ytics_secondary · ytics_secondary_axis · yv_grid · z_voxel · zaxis · zaxis_color · zaxis_type · zaxis_width · zlabel · zlabel_rotate · zrange · ztics · ztics_axis · ztics_rotate

Category: Package drawdf

Introduction to drawdf · drawdf

Category: Package dynamics

The dynamics package · animation · azimuth · background · capping · center · chaosgame · color · cone · cube · cylinder · elevation · endphi · endtheta · evolution · evolution2d · height · height · ifs · julia · linewidth · mandelbrot · opacity · orbits · orientation · origin · phiresolution · points · pointsize · position · radius · resolution · restart · scale · scene · sphere · staircase · startphi · starttheta · surface · thetaresolution · track · tstep · width · windowname · windowtitle · wireframe · xlength · ylength · zlength

Category: Package eigen

eigen · eigenvalues · eivals · eigenvectors · eivects · gramschmidt · innerproduct · inprod · similaritytransform · simtran · ueivects · uniteigenvectors · unitvector · uvect

Category: Package ezunits

Introduction to ezunits · · · constvalue · declare_constvalue · declare_dimensions · declare_fundamental_dimensions · fundamental_dimensions · remove_fundamental_dimensions · declare_fundamental_units · remove_fundamental_units · declare_qty · declare_unit_conversion · declare_units · dimensionless · dimensions · dimensions_as_list · fundamental_units · natural_unit · qty · remove_constvalue · remove_dimensions · unitp · units

Category: Package f90

Package f90 · f90 · f90_output_line_length_max

Category: Package facexp

Package facexp · collectterms · facsum · facsum_combine · factorfacsum · nextlayerfactor

Category: Package fft

Introduction to fast Fourier transform · bf_fft · bf_inverse_fft · bf_inverse_real_fft · bf_real_fft · fft · inverse_fft · inverse_real_fft · polartorect · real_fft · recttopolar

Category: Package finance

amortization · annuity_fv · annuity_pv · arit_amortization · benefit_cost · days360 · fv · geo_amortization · geo_annuity_fv · geo_annuity_pv · graph_flow · irr · npv · pv · saving

Category: Package fourie

Introduction to Fourier series · absint · cosnpiflag · equalp · fourcos · fourexpand · fourier · fourint · fourintcos · fourintsin · foursimp · foursin · funp · remfun · sinnpiflag · totalfourier

Category: Package fractals

fernfale · hilbertmap · julia_parameter · julia_set · julia_sin · mandelbrot_set · sierpinskiale · sierpinskimap · snowmap · treefale

Category: Package functs

Package functs · agd · arithmetic · arithsum · combination · covers · exsec · gaussprob · gcdivide · gd · geometric · geosum · harmonic · hav · nonzeroandfreeof · permutation · rational · rempart · tracematrix · vers · wronskian

Category: Package ggf

GGFCFMAX · GGFINFINITY · ggf

Category: Package graphs - constructions

circulant_graph · clebsch_graph · complement_graph · complete_bipartite_graph · complete_graph · copy_graph · cube_graph · cuboctahedron_graph · cycle_digraph · cycle_graph · dodecahedron_graph · empty_graph · flower_snark · from_adjacency_matrix · frucht_graph · graph_product · graph_union · great_rhombicosidodecahedron_graph · great_rhombicuboctahedron_graph · grid_graph · grotzch_graph · heawood_graph · icosahedron_graph · icosidodecahedron_graph · induced_subgraph · line_graph · make_graph · mycielski_graph · new_graph · path_digraph · path_graph · petersen_graph · random_bipartite_graph · random_digraph · random_graph · random_graph1 · random_network · random_regular_graph · random_tournament · random_tree · small_rhombicosidodecahedron_graph · small_rhombicuboctahedron_graph · snub_cube_graph · snub_dodecahedron_graph · truncated_cube_graph · truncated_dodecahedron_graph · truncated_icosahedron_graph · truncated_tetrahedron_graph · tutte_graph · underlying_graph · wheel_graph

Category: Package graphs - draw_graphs options

draw_graph_program · edge_color · edge_coloring · edge_partition · edge_type · edge_width · file_name · fixed_vertices · head_angle · head_length · label_alignment · program · redraw · show_edge_color · show_edge_type · show_edge_width · show_edges · show_id · show_label · show_vertex_color · show_vertex_size · show_vertex_type · show_vertices · show_weight · spring_embedding_depth · terminal · vertex_color · vertex_coloring · vertex_partition · vertex_size · vertex_type

Category: Package graphs - io

dimacs_export · dimacs_import · graph6_decode · graph6_encode · graph6_export · graph6_import · sparse6_decode · sparse6_encode · sparse6_export · sparse6_import

Category: Package graphs - modifications

add_edge · add_edges · add_vertex · add_vertices · connect_vertices · contract_edge · remove_edge

Category: Package graphs - properties

adjacency_matrix · average_degree · biconnected_components · bipartition · chromatic_index · chromatic_number · clear_edge_weight · clear_vertex_label · connected_components · degree_sequence · diameter · edge_coloring · edge_connectivity · edges · get_edge_weight · get_vertex_label · girth · graph_center · graph_charpoly · graph_eigenvalues · graph_order · graph_periphery · graph_size · hamilton_cycle · hamilton_path · in_neighbors · is_biconnected · is_bipartite · is_connected · is_digraph · is_edge_in_graph · is_graph · is_graph_or_digraph · is_isomorphic · is_planar · is_sconnected · is_tree · is_vertex_in_graph · isomorphism · laplacian_matrix · max_clique · max_degree · max_flow · max_independent_set · max_matching · min_degree · min_edge_cut · min_vertex_cover · min_vertex_cut · minimum_spanning_tree · neighbors · odd_girth · out_neighbors · planar_embedding · radius · set_edge_weight · set_vertex_label · shortest_path · shortest_weighted_path · strong_components · topological_sort · vertex_coloring · vertex_connectivity · vertex_degree · vertex_distance · vertex_eccentricity · vertex_in_degree · vertex_out_degree · vertices · wiener_index

Category: Package graphs

Introduction to graphs · add_edge · add_edges · add_vertex · add_vertices · adjacency_matrix · average_degree · biconnected_components · bipartition · chromatic_index · chromatic_number · circulant_graph · clear_edge_weight · clear_vertex_label · clebsch_graph · complement_graph · complete_bipartite_graph · complete_graph · connect_vertices · connected_components · contract_edge · copy_graph · create_graph · cube_graph · cuboctahedron_graph · cycle_digraph · cycle_graph · degree_sequence · diameter · dimacs_export · dimacs_import · dodecahedron_graph · draw_graph · draw_graph_program · edge_color · edge_coloring · edge_coloring · edge_connectivity · edge_partition · edge_type · edge_width · edges · empty_graph · file_name · fixed_vertices · flower_snark · from_adjacency_matrix · frucht_graph · get_edge_weight · get_vertex_label · girth · graph6_decode · graph6_encode · graph6_export · graph6_import · graph_center · graph_charpoly · graph_eigenvalues · graph_order · graph_periphery · graph_product · graph_size · graph_union · great_rhombicosidodecahedron_graph · great_rhombicuboctahedron_graph · grid_graph · grotzch_graph · hamilton_cycle · hamilton_path · head_angle · head_length · heawood_graph · icosahedron_graph · icosidodecahedron_graph · in_neighbors · induced_subgraph · is_biconnected · is_bipartite · is_connected · is_digraph · is_edge_in_graph · is_graph · is_graph_or_digraph · is_isomorphic · is_planar · is_sconnected · is_tree · is_vertex_in_graph · isomorphism · label_alignment · laplacian_matrix · line_graph · make_graph · max_clique · max_degree · max_flow · max_independent_set · max_matching · min_degree · min_edge_cut · min_vertex_cover · min_vertex_cut · minimum_spanning_tree · mycielski_graph · neighbors · new_graph · odd_girth · out_neighbors · path_digraph · path_graph · petersen_graph · planar_embedding · print_graph · program · radius · random_bipartite_graph · random_digraph · random_graph · random_graph1 · random_network · random_regular_graph · random_tournament · random_tree · redraw · remove_edge · remove_vertex · set_edge_weight · set_vertex_label · shortest_path · shortest_weighted_path · show_edge_color · show_edge_type · show_edge_width · show_edges · show_id · show_label · show_vertex_color · show_vertex_size · show_vertex_type · show_vertices · show_weight · small_rhombicosidodecahedron_graph · small_rhombicuboctahedron_graph · snub_cube_graph · snub_dodecahedron_graph · sparse6_decode · sparse6_encode · sparse6_export · sparse6_import · spring_embedding_depth · strong_components · terminal · topological_sort · truncated_cube_graph · truncated_dodecahedron_graph · truncated_icosahedron_graph · truncated_tetrahedron_graph · tutte_graph · underlying_graph · vertex_color · vertex_coloring · vertex_coloring · vertex_connectivity · vertex_degree · vertex_distance · vertex_eccentricity · vertex_in_degree · vertex_out_degree · vertex_partition · vertex_size · vertex_type · vertices · vertices_to_cycle · vertices_to_path · wheel_graph · wiener_index

Category: Package grobner

Introduction to grobner · poly_add · poly_buchberger · poly_buchberger_criterion · poly_coefficient_ring · poly_colon_ideal · poly_content · poly_depends_p · poly_elimination_ideal · poly_elimination_order · poly_exact_divide · poly_expand · poly_expt · poly_gcd · poly_grobner · poly_grobner_algorithm · poly_grobner_debug · poly_grobner_equal · poly_grobner_member · poly_grobner_subsetp · poly_ideal_intersection · poly_ideal_polysaturation · poly_ideal_polysaturation1 · poly_ideal_saturation · poly_ideal_saturation1 · poly_lcm · poly_minimization · poly_monomial_order · poly_multiply · poly_normal_form · poly_normalize · poly_normalize_list · poly_polysaturation_extension · poly_primary_elimination_order · poly_primitive_part · poly_pseudo_divide · poly_reduced_grobner · poly_reduction · poly_return_term_list · poly_s_polynomial · poly_saturation_extension · poly_secondary_elimination_order · poly_subtract · poly_top_reduction_only

Category: Package impdiff

implicit_derivative

Category: Package implicit_plot

implicit_plot

Category: Package ineq

Package ineq

Category: Package interpol

Introduction to interpol · charfun2 · cspline · lagrange · linearinterpol · ratinterpol

Category: Package itensor

Introduction to itensor · allsym · canform · canten · changename · components · concan · conmetderiv · contract · coord · covdiff · decsym · defcon · diff · entertensor · evundiff · extdiff · flipflag · flush · flush1deriv · flushd · flushnd · hodge · ic_convert · icc1 · icc2 · ichr1 · ichr2 · icounter · icurvature · idiff · idim · idummy · idummyx · ifb · ifc1 · ifc2 · ifg · ifgi · ifr · iframe_bracket_form · iframes · ifri · igeodesic_coords · igeowedge_flag · ikt1 · ikt2 · imetric · indexed_tensor · indices · inm · inmc1 · inmc2 · ishow · itr · kdels · kdelta · lc2kdt · lc_l · lc_u · levi_civita · liediff · listoftens · lorentz_gauge · makebox · rediff · remcomps · remcon · remcoord · remsym · rename · show · showcomps · simpmetderiv · tentex · undiff · | · ~

Category: Package lapack

Introduction to lapack · dgeev · dgemm · dgeqrf · dgesv · dgesvd · dlange · zlange · zgeev · zheev

Category: Package lbfgs

Introduction to lbfgs · lbfgs · lbfgs_ncorrections · lbfgs_nfeval_max

Category: Package lindstedt

Lindstedt

Category: Package linearalgebra

Introduction to linearalgebra · addmatrices · blockmatrixp · cholesky · columnop · columnspace · columnswap · ctranspose · diag_matrix · dotproduct · eigens_by_jacobi · get_lu_factors · hankel · hessian · hilbert_matrix · identfor · invert_by_lu · jacobian · kronecker_product · linalg_rank · listp · locate_matrix_entry · lu_backsub · lu_factor · mat_cond · mat_fullunblocker · mat_norm · mat_trace · mat_unblocker · matrix_size · matrixp · nonnegintegerp · nullity · nullspace · orthogonal_complement · polynomialp · polytocompanion · ptriangularize · rowop · rowswap · toeplitz · vandermonde_matrix · zerofor · zeromatrixp

Category: Package lsquares

Introduction to lsquares · lsquares_estimates · lsquares_estimates_approximate · lsquares_estimates_exact · lsquares_mse · lsquares_residual_mse · lsquares_residuals · plsquares

Category: Package makeOrders

makeOrders

Category: Package minpack

Introduction to minpack

Category: Package mnewton

Introduction to mnewton · mnewton · newtonepsilon · newtonmaxiter

Category: Package nchrpl

mattrace · ncharpoly

Category: Package ntrig

ntrig

Category: Package numericalio

Introduction to numericalio · assume_external_byte_order · opena_binary · openr_binary · openw_binary · read_array · read_binary_array · read_binary_list · read_binary_matrix · read_hash_table · read_list · read_matrix · read_nested_list · write_binary_data · write_data

Category: Package odepack

Introduction to ODEPACK

Category: Package opsubst

opsubst

Category: Package orthopoly

Introduction to orthogonal polynomials · assoc_legendre_p · assoc_legendre_q · chebyshev_t · chebyshev_u · gen_laguerre · hermite · intervalp · jacobi_p · laguerre · legendre_p · legendre_q · orthopoly_recur · orthopoly_returns_intervals · orthopoly_weight · pochhammer · pochhammer_max_index · spherical_bessel_j · spherical_bessel_y · spherical_hankel1 · spherical_hankel2 · spherical_harmonic · ultraspherical · unit_step

Category: Package physical_constants

Introduction to physical_constants

Category: Package quadpack

Introduction to QUADPACK · quad_control · quad_qag · quad_qagi · quad_qagp · quad_qags · quad_qawc · quad_qawf · quad_qawo · quad_qaws

Category: Package ratpow

ratp_coeffs · ratp_dense_coeffs · ratp_hipow · ratp_lopow

Category: Package rducon

Package rducon · reduce_consts

Category: Package romberg

romberg · rombergabs · rombergit · rombergmin · rombergtol

Category: Package scifac

Package scifac · gcfac

Category: Package simplex

Introduction to simplex · epsilon_lp · linear_program · maximize_lp · minimize_lp · nonegative_lp · nonnegative_lp · pivot_count_sx · pivot_max_sx · scale_lp

Category: Package solve_rec

Introduction to solve_rec · product_use_gamma · reduce_order · simplify_products · simplify_sum · solve_rec · solve_rec_rat · summand_to_rec

Category: Package sqdnst

sqrtdenest

Category: Package stats

Introduction to stats · cdf_rank_sum · cdf_signed_rank · inference_result · inferencep · items_inference · linear_regression · pdf_rank_sum · pdf_signed_rank · stats_numer · take_inference · test_mean · test_means_difference · test_normality · test_proportion · test_proportions_difference · test_rank_sum · test_sign · test_signed_rank · test_variance · test_variance_ratio

Category: Package stirling

stirling

Category: Package stringproc

Introduction to String Processing · adjust_external_format · alphacharp · alphanumericp · ascii · base64 · base64_decode · cequal · cequalignore · cgreaterp · cgreaterpignore · charat · charlist · charp · cint · clessp · clesspignore · close · constituent · crc24sum · digitcharp · eval_string · flength · flush_output · fposition · freshline · get_output_stream_string · lowercasep · make_string_input_stream · make_string_output_stream · md5sum · mgf1_sha1 · newline · newline · number_to_octets · octets_to_number · octets_to_oid · octets_to_string · oid_to_octets · opena · openr · openw · parse_string · printf · readbyte · readchar · readline · scopy · sdowncase · sequal · sequalignore · sexplode · sha1sum · sha256sum · simplode · sinsert · sinvertcase · slength · smake · smismatch · space · split · sposition · sprint · sremove · sremovefirst · sreverse · ssearch · ssort · ssubst · ssubstfirst · strim · striml · strimr · string_to_octets · stringp · substring · supcase · tab · tokens · unicode · unicode_to_utf8 · uppercasep · us_ascii_only · utf8_to_unicode · writebyte

Category: Package sym

Introduction to Symmetries · comp2pui · cont2part · contract · direct · ele2comp · ele2polynome · ele2pui · elem · explose · kostka · lgtreillis · ltreillis · mon2schur · multi_elem · multi_orbit · multi_pui · multinomial · multsym · orbit · part2cont · partpol · permut · polynome2ele · prodrac · pui · pui2comp · pui2ele · pui2polynome · pui_direct · puireduc · resolvante · resolvante_alternee1 · resolvante_bipartite · resolvante_diedrale · resolvante_klein · resolvante_klein3 · resolvante_produit_sym · resolvante_unitaire · resolvante_vierer · schur2comp · somrac · tcontract · tpartpol · treillis · treinat

Category: Package unit

%unitexpand · Introduction to Units · convert · metricexpandall · setunits · uforget · usersetunits

Category: Package vect

Vectors · scalefactors · vect_cross · vectorpotential · vectorsimp

Category: Package wrstcse

Introduction to wrstcse

Category: Package zeilberger

AntiDifference · Gosper · GosperSum · Gosper_in_Zeilberger · Introduction to zeilberger · MAX_ORD · Zeilberger · Zeilberger · ev_point · linear_solver · mod_big_prime · mod_test · mod_threshold · modular_linear_solver · parGosper · parGosper · simplified_output · trivial_solutions · warnings

Category: Physical units

Introduction to Units · Introduction to ezunits · Introduction to physical_constants

Category: Plotting

Introduction to Plotting · Introduction to draw · Introduction to drawdf · Introduction to numerical solution of differential equations · Introduction to orthogonal polynomials · Plotting Formats · adapt_depth · animation · axes · azimuth · azimuth · background · barsplot · barsplot_description · bode_gain · bode_phase · box · boxplot · boxplot_description · capping · center · chaosgame · color · color · color_bar · color_bar_tics · cone · contour_plot · cube · cylinder · elevation · elevation · endphi · endtheta · evolution · evolution2d · geomview_command · gnuplot_close · gnuplot_command · gnuplot_curve_styles · gnuplot_curve_titles · gnuplot_default_term_command · gnuplot_dumb_term_command · gnuplot_file_args · gnuplot_out_file · gnuplot_pdf_term_command · gnuplot_pm3d · gnuplot_png_term_command · gnuplot_postamble · gnuplot_preamble · gnuplot_ps_term_command · gnuplot_replot · gnuplot_reset · gnuplot_restart · gnuplot_start · gnuplot_svg_term_command · gnuplot_term · gnuplot_view_args · grid · grid2d · height · height · histogram · histogram_description · ifs · implicit_plot · iterations · julia · label · legend · linewidth · logx · logy · make_transform · mandelbrot · mesh_lines_color · nticks · opacity · orbits · orientation · origin · palette · pdf_file · phiresolution · piechart · piechart_description · plot2d · plot3d · plot_format · plot_options · plot_realpart · plotdf · ploteq · png_file · point_type · points · pointsize · polar_to_xy · position · ps_file · radius · remove_plot_option · resolution · restart · run_viewer · same_xy · same_xyz · scale · scatterplot · scatterplot_description · scene · set_plot_option · sphere · spherical_to_xyz · staircase · starplot · starplot_description · startphi · starttheta · stemplot · style · surface · svg_file · t · thetaresolution · title · track · transform_xy · tstep · width · windowname · windowtitle · wireframe · x · xlabel · xlength · xtics · xy_scale · y · ylabel · ylength · ytics · yx_ratio · z · zlabel · zlength · zmin

Category: Poisson series

intopois · outofpois · poisdiff · poisexpt · poisint · poislim · poismap · poisplus · poissimp · poisson · poissubst · poistimes · poistrim · printpois

Category: Polynomials

Introduction to Affine · Introduction to Polynomials · Introduction to Symmetries · allroots · berlefact · bezout · bfallroots · bothcoef · coeff · content · divide · eliminate · ezgcd · facexpand · factor · factor_max_degree · factor_max_degree_print_warning · factorflag · fasttimes · gcd · gcdex · gcdivide · gfactor · lratsubst · makeOrders · multiplicities · nroots · nthroot · polydecomp · polyfactor · polymod · programmode · quotient · ratcoef · realroots · remainder · resultant · resultant · rootsepsilon · savefactors · sqfr · tellrat · untellrat

Category: Power series

deftaylor · maxtayorder · pade · powerseries · revert · revert2 · taylor · taylor_logexpand · taylor_order_coefficients · taylor_simplifier · taylor_truncate_polynomials · taylordepth · taylorinfo · taylorp · taytorat · trunc · verbose

Category: Predicate functions

abasep · alphacharp · alphanumericp · atom · bfloatp · blockmatrixp · cequal · cequalignore · cgreaterp · cgreaterpignore · charp · clessp · clesspignore · constantp · constituent · diagmatrixp · digitcharp · disjointp · elementp · emptyp · evenp · featurep · floatnump · if · integerp · intervalp · is · listp · listp · lowercasep · mapatom · matrixp · matrixp · maybe · member · nonnegintegerp · nonscalarp · numberp · oddp · operatorp · ordergreatp · orderlessp · picture_equalp · picturep · poly_depends_p · poly_grobner_subsetp · polynomialp · prederror · primep · ratnump · ratp · scalarp · sequal · sequalignore · setequalp · setp · stringp · subsetp · subvarp · symbolp · symmetricp · taylorp · unknown · uppercasep · zeroequiv · zeromatrixp · zn_primroot_p

Category: Programming

Function · Lisp and Maxima · block · catch · do · for · from · in · next · step · thru · unless · while · errcatch · error · errormsg · errormsg · garbage_collect · go · if · local · prederror · return · sstatus · status · throw · warning

Category: Random numbers

make_random_state · random · random_bernoulli · random_beta · random_binomial · random_cauchy · random_chi2 · random_continuous_uniform · random_discrete_uniform · random_exp · random_f · random_gamma · random_general_finite_discrete · random_geometric · random_gumbel · random_hypergeometric · random_laplace · random_logistic · random_lognormal · random_negative_binomial · random_noncentral_chi2 · random_noncentral_student_t · random_normal · random_pareto · random_poisson · random_rayleigh · random_student_t · random_weibull · set_random_state

Category: Rational expressions

Introduction to Polynomials · fullratsimp · fullratsubst · gcd · gcdex · lratsubst · rat · ratcoef · ratdenom · ratdenomdivide · ratdiff · ratdisrep · ratepsilon · ratexpand · ratfac · ratmx · ratnumer · ratnump · ratp · ratp_coeffs · ratp_dense_coeffs · ratp_hipow · ratp_lopow · ratprint · ratsimp · ratsimpexpons · ratsubst · ratvars · ratvarswitch · ratweight · ratweights · ratwtlvl · showratvars · taytorat · tellrat · totaldisrep · untellrat

Category: Rules and patterns

Introduction to Rules and Patterns · Package absimp · Package ineq · apply1 · apply2 · applyb1 · clear_rules · current_let_rule_package · default_let_rule_package · defmatch · defrule · disprule · let · let_rule_packages · letrat · letrules · letsimp · matchdeclare · remlet · remrule · tellsimp · tellsimpafter

Category: Session management

Introduction for Runtime Environment · batch · batchload · kill · load · loadfile · myoptions · nolabels · optionset · reset · save · stringout

Category: Sets

Introduction to Sets · adjoin · belln · cardinality · cartesian_product · disjoin · disjointp · elementp · emptyp · equiv_classes · every · extremal_subset · flatten · full_listify · intersect · intersection · listify · lmax · lmin · makeset · partition_set · permutations · powerset · random_permutation · set_partitions · setdifference · setequalp · setp · some · subset · subsetp · symmdifference · tree_reduce · union · xreduce

Category: Share packages

Introduction to Affine · Introduction to Fourier series · Introduction to ODEPACK · Introduction to QUADPACK · Introduction to String Processing · Introduction to Symmetries · Introduction to Units · Introduction to atensor · Introduction to cobyla · Introduction to contrib_ode · Introduction to ctensor · Introduction to descriptive · Introduction to distrib · Introduction to draw · Introduction to drawdf · Introduction to ezunits · Introduction to fast Fourier transform · Introduction to graphs · Introduction to grobner · Introduction to interpol · Introduction to itensor · Introduction to lapack · Introduction to lbfgs · Introduction to linearalgebra · Introduction to lsquares · Introduction to minpack · Introduction to mnewton · Introduction to numericalio · Introduction to orthogonal polynomials · Introduction to physical_constants · Introduction to simplex · Introduction to solve_rec · Introduction to stats · Introduction to wrstcse · Introduction to zeilberger · Lindstedt · Package absimp · Package combinatorics · Package f90 · Package facexp · Package functs · Package ineq · Package rducon · Package scifac · The dynamics package · Vectors · augmented_lagrangian_method · diag · dimension · eigen · engineering_format_floats · engineering_format_max · engineering_format_min · ggf · implicit_derivative · implicit_plot · makeOrders · opsubst · sqrtdenest · stirling

Category: Simplification flags and variables

%e_to_numlog · %emode · %iargs · %piargs · algebraic · besselexpand · beta_args_sum_to_integer · beta_expand · ctrgsimp · distribute_over · domain · dot0nscsimp · dot0simp · dot1simp · dotassoc · dotconstrules · dotdistrib · dotexptsimp · dotident · dotscrules · evflag · gamma_expand · gammalim · halfangles · logarc · logconcoeffp · logexpand · lognegint · logsimp · negdistrib · radexpand · radsubstflag · ratalgdenom · ratdenomdivide · ratsimpexpons · rootsconmode · scalarmatrixp · simpproduct · simpsum · sumexpand · sumsplitfact · trigexpandplus · trigexpandtimes · triginverses · trigsign

Category: Simplification functions

Package absimp · Package ineq · atensimp · foursimp · fullratsimp · hypergeometric_simp · logarc · radcan · ratsimp · rootscontract · scsimp · simplify_sum · trigexpand · trigrat · trigreduce · trigsimp · unknown · vectorsimp

Category: Simplification

Introduction to Rules and Patterns · define_opproperty · lassociative · linear · multiplicative · opproperties

Category: Special functions

%f · %m · %s · %w · Introduction to Special Functions · airy_ai · airy_bi · airy_dai · airy_dbi · bessel_i · bessel_j · bessel_k · bessel_simplify · bessel_y · besselexpand · erf · erf_generalized · erfc · erfi · expintegral_chi · expintegral_ci · expintegral_e · expintegral_e1 · expintegral_e_simplify · expintegral_ei · expintegral_li · expintegral_shi · expintegral_si · fresnel_c · fresnel_s · gamma · gamma_incomplete · gamma_incomplete_generalized · gamma_incomplete_lower · gamma_incomplete_regularized · generalized_lambert_w · hankel_1 · hankel_2 · hypergeometric_simp · lambert_w · log_gamma · nzeta · nzetai · nzetar · parabolic_cylinder_d · scaled_bessel_i0 · scaled_bessel_i1 · struve_h · struve_l

Category: Statistical estimation

Introduction to lsquares · linear_regression

Category: Statistical functions

Introduction to distrib

Category: Statistical inference

Introduction to stats

Category: Strings

Introduction to String Processing · concat · sconcat · string

Category: Structures

@ · defstruct · new · structures

Category: Sums and products

Introduction to zeilberger · arithmetic · arithsum · bashindices · cauchysum · genindex · gensumnum · geometric · geosum · harmonic · lsum · niceindices · niceindicespref · nusum · product · simplify_sum · simpproduct · simpsum · sum · sumcontract · sumexpand · unsum

Category: Syntax

Comments · Identifiers · Introduction to Strings · Introduction to operators · infix · matchfix · nary · nofix · postfix · prefix

Category: TeX output

get_tex_environment · set_tex_environment · get_tex_environment_default · set_tex_environment_default · tentex · tex · texput

Category: Tensors

Introduction to atensor · Introduction to ctensor · Introduction to itensor

Category: Time and date functions

absolute_real_time · decode_time · elapsed_real_time · elapsed_run_time · encode_time · parse_timedate · timedate

Category: Translation and compilation

compfile · compile · compile_file · declare_translated · define_variable · f90 · f90_output_line_length_max · fortindent · fortran · fortspaces · mode_declare · mode_identity · tr_warnings_get · translate · translate_file

Category: Translation flags and variables

mode_check_errorp · mode_check_warnp · mode_checkp · packagefile · savedef · tr_array_as_ref · tr_bound_function_applyp · tr_file_tty_messagesp · tr_float_can_branch_complex · tr_function_call_default · tr_numer · tr_optimize_max_loop · tr_semicompile · tr_state_vars · tr_warn_bad_function_calls · tr_warn_fexpr · tr_warn_meval · tr_warn_mode · tr_warn_undeclared · tr_warn_undefined_variable · transcompile · translate_fast_arrays · transrun

Category: Trigonometric functions

%iargs · %piargs · Introduction to Trigonometric · acos · acot · acsc · asec · asin · atan · atan2 · atrig1 · cos · cot · csc · demoivre · exponentialize · foursimp · halfangles · ntrig · sec · sin · tan · trigexpand · trigexpandplus · trigexpandtimes · triginverses · trigrat · trigreduce · trigsign · trigsimp

Category: Vectors

Vectors · eigen · express · nonscalar · nonscalarp · scalarp

Category: Warning messages

replaced x by y

Category: clebsch_gordan

clebsch_gordan · racah_v · racah_w · wigner_3j · wigner_6j · wigner_9j


Footnotes

(1)

Applied Mathematics and Programming Division, K.U. Leuven

(2)

Applied Mathematics and Programming Division, K.U. Leuven

(3)

Institut für Mathematik, T.U. Wien

(4)

National Bureau of Standards, Washington, D.C., U.S.A

(5)

http://www.netlib.org/slatec

(6)

http://www.netlib.org/quadpack

(7)

R. Piessens, E. de Doncker-Kapenga, C.W. Uberhuber, and D.K. Kahaner. QUADPACK: A Subroutine Package for Automatic Integration. Berlin: Springer-Verlag, 1983, ISBN 0387125531.

(8)

Readers using the info reader in Emacs will see the actual prompt strings; other readers will see the colorized output

(9)

From http://netlib.org/odepack/opkd-sum