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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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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])$
(%i2) plot2d ([atan(x), erf(x), tanh(x)], [x, -5, 5], [y, -1.5, 2])$
(%i3) plot3d (sin(sqrt(x^2 + y^2))/sqrt(x^2 + y^2), [x, -12, 12], [y, -12, 12])$
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2.1 Functions and Variables for Bug Detection and Reporting |
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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
.
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.
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
..
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 ""
.
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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3.1 Documentation | ||
3.2 Functions and Variables for Help |
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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.
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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]
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) _
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.
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
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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4.1 Introduction to Command Line | ||
4.2 Functions and Variables for Command Line | ||
4.3 Functions and Variables for Display |
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__
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.
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
_
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.
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 $B) (%i8) _; (%o8) b + a (%i9) a + b; (%o9) 42 (%i10) ev (_); (%o10) 42
%
is the output expression (e.g., %o1
, %o2
, %o3
,
…) most recently computed by Maxima, whether or not it was displayed.
%
is recognized by batch
and load
. In a file processed
by batch
, %
has the same meaning as at the interactive prompt.
In a file processed by load
, %
is bound to the output expression
most recently computed at the interactive prompt or in a batch file; %
is not bound to output expressions in the file being processed.
In compound statements, namely block
, lambda
, or
(s_1, ..., s_n)
, %%
is the value of the previous
statement.
At the first statement in a compound statement, or outside of a compound
statement, %%
is undefined.
%%
is recognized by batch
and load
, and it has the
same meaning as at the interactive prompt.
See also %
.
Examples:
The following two examples yield the same result.
(%i1) block (integrate (x^5, x), ev (%%, x=2) - ev (%%, x=1)); 21 (%o1) -- 2 (%i2) block ([prev], prev: integrate (x^5, x), ev (prev, x=2) - ev (prev, x=1)); 21 (%o2) -- 2
A compound statement may comprise other compound statements. Whether a
statement be simple or compound, %%
is the value of the previous
statement.
(%i3) block (block (a^n, %%*42), %%/6); n (%o3) 7 a
Within a compound statement, the value of %%
may be inspected at a break
prompt, which is opened by executing the break
function. For example,
entering %%;
in the following example yields 42
.
(%i4) block (a: 42, break ())$ Entering a Maxima break point. Type 'exit;' to resume. _%%; 42 _
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.
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
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
.
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
.
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
The semicolon ;
terminates an input expression,
and the resulting output is displayed.
See also $
.
Example:
(%i1) 1 + 2 + 3; (%o1) 6
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
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
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.)
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.
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.
The variable labels
is the list of input, output, and intermediate
expression labels, including all previous labels if inchar
,
outchar
, or linechar
were redefined.
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
.
The line number of the current pair of input and output expressions.
Default value: []
myoptions
is the list of all options ever reset by the user,
whether or not they get reset to their default value.
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.
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
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
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)
.
Default value: _
prompt
is the prompt symbol of the demo
function,
playback (slow)
mode, and the Maxima break loop (as invoked by
break
).
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.
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)
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
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.
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
.
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 $X)) MAXIMA> (to-maxima) Returning to Maxima (%o2) true (%i3) result; (%o3) x (x + 1)
Initial value: []
values
is a list of all bound user variables (not Maxima options or
switches). The list comprises symbols bound by :
, or ::
.
If the value of a variable is removed with the commands kill
,
remove
, or remvalue
the variable is deleted from
values
.
See functions
for a list of user defined functions.
Examples:
First, values
shows the symbols a
, b
, and c
, but
not d
, it is not bound to a value, and not the user function f
.
The values are removed from the variables. values
is the empty list.
(%i1) [a:99, b:: a-90, c:a-b, d, f(x):=x^2]; 2 (%o1) [99, 9, 90, d, f(x) := x ] (%i2) values; (%o2) [a, b, c] (%i3) [kill(a), remove(b,value), remvalue(c)]; (%o3) [done, done, [c]] (%i4) values; (%o4) []
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Default value: false
When %edispflag
is true
, Maxima displays %e
to a negative
exponent as a quotient. For example, %e^-x
is displayed as
1/%e^x
. See also exptdispflag
.
Example:
(%i1) %e^-10; - 10 (%o1) %e (%i2) %edispflag:true$ (%i3) %e^-10; 1 (%o3) ---- 10 %e
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!
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.
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
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
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)
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
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
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.
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
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)
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.
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
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
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
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
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
.
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
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.
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
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)
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
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)
Default value: true
When sqrtdispflag
is false
, causes sqrt
to display with
exponent 1/2.
Default value: false
When stardisp
is true
, multiplication is
displayed with an asterisk *
between operands.
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
.
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
5.1 Numbers | ||
5.2 Strings | ||
5.3 Constants | ||
5.4 Lists | ||
5.5 Arrays | ||
5.6 Structures |
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5.1.1 Introduction to Numbers | ||
5.1.2 Functions and Variables for Numbers |
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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.
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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).
Returns true
if expr is a bigfloat number, otherwise false
.
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
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
.
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
.
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.
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).
Returns true
if expr is a floating point number, otherwise
false
.
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
.
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.
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
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
Return true
if and only if n >= 0
and n is an integer.
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]
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]
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
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.
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
.
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.
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
Returns true
if expr is a literal integer or ratio of literal
integers, otherwise false
.
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5.2.1 Introduction to Strings | ||
5.2.2 Functions and Variables for Strings |
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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;
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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
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
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.
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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5.3.1 Functions and Variables for Constants |
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%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.
%i
represents the imaginary unit, sqrt(- 1).
false
represents the Boolean constant of the same name.
Maxima implements false
by the value NIL
in Lisp.
The Euler-Mascheroni constant, 0.5772156649015329 ....
ind
represents a bounded, indefinite result.
See also limit
.
Example:
(%i1) limit (sin(1/x), x, 0); (%o1) ind
inf
represents real positive infinity.
infinity
represents complex infinity.
minf
represents real minus (i.e., negative) infinity.
%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
%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.
true
represents the Boolean constant of the same name.
Maxima implements true
by the value T
in Lisp.
und
represents an undefined result.
See also limit
.
Example:
(%i1) limit (x*sin(x), x, inf); (%o1) und
zeroa
represents an infinitesimal above zero. zeroa
can be used
in expressions. limit
simplifies expressions which contain
infinitesimals.
Example:
limit
simplifies expressions which contain infinitesimals:
(%i1) limit(zeroa); (%o1) 0 (%i2) limit(x+zeroa); (%o2) x
zerob
represents an infinitesimal below zero. zerob
can be used
in expressions. limit
simplifies expressions which contain
infinitesimals.
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5.4.1 Introduction to Lists | ||
5.4.2 Functions and Variables for Lists | ||
5.4.3 Performance considerations for Lists |
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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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[
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 n
th element of a list
may need an amount of time that is roughly proportional to n
,
See section 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 section 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
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.
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
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
Returns a copy of the list list.
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
.
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 ]
Returns the 8'th item of expression or list expr.
See first
for more details.
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
Returns the 5'th item of expression or list expr.
See first
for more details.
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 second
…
tenth
yield the second through the tenth part of their input argument.
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.
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
Returns the 4'th item of expression or list expr.
See first
for more details.
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]
Returns the last part (term, row, element, etc.) of the expr.
See also lastn
.
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.
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
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.
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.
Returns true
if expr is a list else false
.
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
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]
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
Returns the 9'th item of expression or list expr.
See first
for more details.
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
.
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]
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 ^.
(%i1) rest(a+b+c); (%o1) b+a (%i2) rest(a+b+c,2); (%o2) a (%i3) rest(a+b+c,-2); (%o3) c
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
.
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
Returns the 2'nd item of expression or list expr.
See first
for more details.
Returns the 7'th item of expression or list expr.
See first
for more details.
Returns the 6'th item of expression or list expr.
See first
for more details.
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]
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]
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]
Returns the 10'th item of expression or list expr.
See first
for more details.
Returns the 3'rd item of expression or list expr.
See first
for more details.
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)
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]
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]
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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
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Maxima supports 3 array-like constructs:
(%i1) a["feww"]:1; (%o1) 1 (%i2) a[qqwdqwd]:3; (%o2) 3 (%i3) a[5]:99; (%o3) 99 (%i4) a[qqwdqwd]; (%o4) 3 (%i5) a[5]; (%o5) 99 (%i6) a["feww"]; (%o6) 1
Since lisp handles hashed arrays and memoizing functions
similar to arrays
many of the functions that can be applied to arrays can be applied to them, as well.
makelist
allow for fast addition and removal
of elements, can be created without knowing their final size.
5.5.1 Functions and Variables for Arrays |
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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
.
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.
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 )]
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
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]
Assigns x to A[i_1, ..., i_n]
,
where A is an array and i_1, …, i_n are integers.
arraysetapply
evaluates its arguments.
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))}
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]]
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
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.
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.
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
Returns true
if expr is a subscripted variable, for example
a[i]
.
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
.
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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5.6.1 Introduction to Structures | ||
5.6.2 Functions and Variables for Structures |
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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.
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structures
is the list of user-defined structures defined by defstruct
.
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)]
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)
@
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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6.1 Introduction to Expressions | ||
6.2 Nouns and Verbs | ||
6.3 Identifiers | ||
6.4 Inequality | ||
6.5 Functions and Variables for Expressions |
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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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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)
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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
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Maxima has the inequality operators <
, <=
, >=
, >
,
#
, and notequal
. See if
for a description of conditional
expressions.
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provides an alternate name for a (user or system) function, variable, array, etc. Any even number of arguments may be used.
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
.)
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,
…
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]
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).
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)- -----------------
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.
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))
.
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.
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")$
.
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
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" """
Default value: false
exptisolate
, when true
, causes isolate (expr, var)
to
examine exponents of atoms (such as %e
) which contain var
.
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
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
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 for
… in
construct,
map
, fullmap
, maplist
, reveal
and
pickapart
.
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)
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.
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
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.
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]
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
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
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.
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
.)
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
Default value: false
When noundisp
is true
, nouns display with
a single quote. This switch is always true
when displaying function
definitions.
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
.
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).
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
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.
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
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.
Default value: %
optimprefix
is the prefix used for generated symbols by
the optimize
command.
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
.
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
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.
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)]]
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.
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
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.
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)
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
Replaces parts of expr at the specified integer depth with descriptive summaries.
Sum(n)
where n is the number of operands of the sum.
Product(n)
where n is the number of operands of the product.
Expt
.
Quotient
.
Negterm
.
List(n)
where n ist the number of
elements of the list.
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
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)
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.
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
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)
.
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
.
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
.
Returns true
if expr is a symbol, else false
.
See also Identifiers.
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]
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
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
7.1 Introduction to operators | ||
7.2 Arithmetic operators | ||
7.3 Relational operators | ||
7.4 Logical operators | ||
7.5 Operators for Equations | ||
7.6 Assignment operators | ||
7.7 User defined operators |
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
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.
negation - a
factorial a!
exponentiation a^b
addition a + b
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)
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
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 ]
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
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
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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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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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.
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.
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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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
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.
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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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
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
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
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.
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
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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
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
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.
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.
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.
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.
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8.1 Functions and Variables for Evaluation |
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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)
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
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.
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.
sum
, cause evaluation of occurrences
of those names in expr as though they were verbs.
F(x)
) may be defined
locally for the purpose of this evaluation of expr by giving
F(x) := expression
as an argument to ev
.
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
.
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.
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))
.
eval
in the arguments, steps (3) and (4) are
repeated.
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
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
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
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))
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.
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.
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
.
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
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9.1 Introduction to Simplification | ||
9.2 Functions and Variables for Simplification |
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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 section 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 section 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 section 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 section simplification.
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.
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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)
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)
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
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)
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.
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
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])
Default value: real
When domain
is set to complex
, sqrt (x^2)
will remain
sqrt (x^2)
instead of returning abs(x)
.
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
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
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'.
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.
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'.
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.
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.
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.
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)
.
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
Default value: 1000
maxnegex
is the largest negative exponent which will
be expanded by the expand
command, see also maxposex
.
Default value: 1000
maxposex
is the largest exponent which will be
expanded with the expand
command, see also maxnegex
.
declare(f, multiplicative)
tells the Maxima simplifier that f
is multiplicative.
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(...))
.
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
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
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)
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)
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]
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
declare(f, outative)
tells the Maxima simplifier that constant factors
in the argument of f
can be pulled out.
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.
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)
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
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)
.
radexpand
is all
or assume (x > 0)
has been executed,
sqrt(x^2)
simplifies to x
.
radexpand
is true
and domain
is real
(its default), sqrt(x^2)
simplifies to abs(x)
.
radexpand
is false
, or radexpand
is true
and
domain
is complex
, sqrt(x^2)
is not simplified.
Note that domain
only matters when radexpand
is true
.
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)))
.
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.
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)
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
.
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 ratsimp
ing 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
ratsimp
ed.
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)
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
10.1 Functions for Numbers | ||
10.2 Functions for Complex Numbers | ||
10.3 Combinatorial Functions | ||
10.4 Root, Exponential and Logarithmic Functions | ||
10.5 Trigonometric Functions | ||
10.6 Random Numbers |
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
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
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
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.
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
A synonym for entier (x)
.
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
.
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
.
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)
.
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)
.
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
.
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
.
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
.
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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:
(%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))
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
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
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)
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
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
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
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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)
.
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)
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)!
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
Default value: 100000
factlim
specifies the highest factorial which is
automatically expanded. If it is -1 then all integers are expanded.
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.
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!!
.
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)
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)
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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
.
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
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.
(%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
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))
Represents the polylogarithm function of order s and argument z, defined by the infinite series
$${\rm Li}_s \left(z\right) = \sum_{k=1}^\infty {z^k \over k^s}$$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]
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
.
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.
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
.
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))
.
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))
.
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
Default value: false
If true
implements the rule
log(-n)
-> log(n)+%i*%pi
for n
a positive integer.
Default value: true
If false
then no simplification of %e
to a
power containing log
's is done.
Represents the principal branch of the complex-valued natural
logarithm with -%pi
< carg(x)
<= +%pi
.
The square root of x. It is represented internally by
x^(1/2)
. See also rootscontract
and radexpand
.
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10.5.1 Introduction to Trigonometric | ||
10.5.2 Functions and Variables for Trigonometric |
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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.
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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) ]
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)
- Arc Cosine.
- Hyperbolic Arc Cosine.
- Arc Cotangent.
- Hyperbolic Arc Cotangent.
- Arc Cosecant.
- Hyperbolic Arc Cosecant.
- Arc Secant.
- Hyperbolic Arc Secant.
- Arc Sine.
- Hyperbolic Arc Sine.
- Arc Tangent.
- yields the value of atan(y/x)
in the interval -%pi
to %pi
.
- Hyperbolic Arc Tangent.
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.
- Cosine.
- Hyperbolic Cosine.
- Cotangent.
- Hyperbolic Cotangent.
- Cosecant.
- Hyperbolic Cosecant.
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)
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
.
- Secant.
- Hyperbolic Secant.
- Sine.
- Hyperbolic Sine.
- Tangent.
- Hyperbolic Tangent.
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)
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
.
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
.
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.
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
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
.
Employs the identities
\(\sin\left(x\right)^2 + \cos\left(x\right)^2 = 1\)
and
\(\cosh\left(x\right)^2 - \sinh\left(x\right)^2 = 1\)
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
.
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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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.
Copies s to the random number generator state.
set_random_state
always returns done
.
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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11.1 Introduction to Maximas Database | ||
11.2 Functions and Variables for Properties | ||
11.3 Functions and Variables for Facts | ||
11.4 Functions and Variables for Predicates |
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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~]
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
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.
Example:
(%i1) declare(c, constant); (%o1) done (%i2) constantp(c); (%o2) true (%i3) c : x; (%o3) x (%i4) constantp(c); (%o4) false
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)
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.
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
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
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.
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
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
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.
(%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]]
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
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
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
Makes atoms behave as does a list or matrix with respect to the dot operator.
See also declare
.
Returns true
if expr is a non-scalar, i.e., it contains
atoms declared as non-scalars, lists, or matrices.
declare (f, posfun)
declares f
to be a positive function.
is (f(x) > 0)
yields true
.
See also declare
.
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
.
Returns a list of the names of all the properties associated with the atom a.
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.
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.
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.
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
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
declare(a, rational)
or declare(a, irrational)
tells
Maxima to recognize a as a rational or irrational real variable.
See also declare
.
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
.
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.
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.
declare(a, scalar)
tells Maxima to consider a a scalar
variable.
See also declare
.
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
.
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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.
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.
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.
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
.
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
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.
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.
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
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.
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
.
Deactivates the specified contexts context_1, …, context_n.
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.
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.
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
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
.
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
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.
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).
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.
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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]
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.
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
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
Returns true
if and only if expr contains an operator or function
not recognized by the Maxima simplifier.
Tests whether the expression expr in the variable v is equivalent
to zero, returning true
, false
, or dontknow
.
zeroequiv
has these restrictions:
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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12.1 Introduction to Plotting | ||
12.2 Plotting Formats | ||
12.3 Functions and Variables for Plotting | ||
12.4 Plotting Options | ||
12.5 Gnuplot Options | ||
12.6 Gnuplot_pipes Format Functions |
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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.
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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:
Used to launch the external program gnuplot, which must be installed in
your system. All plotting commands and data are saved into the file
maxout_xxx.gnuplot
.
This format is not available in Windows platforms.
It is similar to the gnuplot
format except that the commands are sent
to gnuplot through a pipe, while the data are saved into the file
maxout_xxx.gnuplot_pipes
. A single gnuplot process is kept open
and subsequent plot commands will be sent to the same process, replacing
previous plots, unless the gnuplot pipe is closed with the function
gnuplot_close
. When this format is used, the function
gnuplot_replot
can be used to modify a plot that has already
displayed on the screen.
This format is only used to plot to the screen; whenever graphic files are
created, the format is silently switched to gnuplot
and
the commands needed to create the graphic file are saved with the data
in file maxout_xxx.gnuplot
.
Mgnuplot is a Tk-based wrapper around gnuplot. It is included in the Maxima distribution. Mgnuplot offers a rudimentary GUI for gnuplot, but has fewer overall features than the plain gnuplot interface. Mgnuplot requires an external gnuplot installation and, in Unix systems, the Tcl/Tk system.
Xmaxima is a Tcl/Tk graphical interface for Maxima that can also be used
to display plots created when Maxima is run from the console or from
other graphical interfaces. To use this format, the xmaxima program,
which is distributed together with Maxima, must be installed. If
Maxima is being run from the Xmaxima console, the data and commands are
passed to xmaxima through the same socket used for the communication
between Maxima and the Xmaxima console. When used from a terminal or
from graphical interfaces different from Xmaxima, the commands and data
are saved in the file maxout_xxx.xmaxima
and xmaxima is run with the
name of that file as argument.
In previous versions this format used to be called openmath
; that
old name still works as a synonym for xmaxima
.
Geomview, a Motif based interactive 3D viewing program for Unix, can also be used to display plots created by Maxima. To use this format, the geomview program must be installed.
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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])$
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"])$
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"$
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.
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"$
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".
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
.
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])$
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])$
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.
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])$
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.
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])$
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])$
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"])$
A plot with a logarithmic scale in the vertical axis:
(%i1) plot2d (exp(3*s), [s, -2, 2], logy)$
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])$
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]])$
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)$
A plot of 200 random numbers between 0 and 9:
(%i1) plot2d ([discrete, makelist ( random(10), 200)])$
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]])$
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])$
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)"])$
See also the section about Plotting 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])$
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])$
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])$
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])$
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])$
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])$
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])$
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])$
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])$
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])$
See also the section about Plotting 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
.
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.
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]]
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.
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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].
Default value: 5
The maximum number of splittings used by the adaptive plotting routine.
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.
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
.
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.
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.
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.
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.
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
.
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
.
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
.
Default value: 9
Number of iterations made by the programs mandelbrot and julia.
Writes one or several labels in the points with x, y coordinates indicated after each label.
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.
Makes the horizontal axes to be scaled logarithmically. It can be either true or false.
Makes the vertical axes to be scaled logarithmically. It can be either true or false.
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.
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.
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
.
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.
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
.
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
.
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.
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.
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.
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.
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
.
It can be either true or false. If true, the scales used in the 3 axes of a 3d plot will be the same.
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
.
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.
Default range for parametric plots.
Defines a title that will be written at the top of the plot.
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
.
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.
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
.
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.
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]
.
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.
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
.
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
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
.
Used in plot3d
to set the effective range of values of z that will be
shown in the plot.
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
.
In 3d plots, the value of z that will be at the bottom of the plot box.
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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.
Sets the output terminal type for gnuplot. The argument terminal_name can be a string or one of the following 3 special symbols
Gnuplot output is displayed in a separate graphical window and the
gnuplot terminal used will be specified by the value of the option
gnuplot_default_term_command
.
Gnuplot output is saved to a file maxout_xxx.gnuplot
using "ASCII
art" approximation to graphics. If the option gnuplot_out_file
is
set to filename, the plot will be saved there, instead of the
default maxout_xxx.gnuplot
. The settings for the "dumb" terminal of
Gnuplot are given by the value of option
gnuplot_dumb_term_command
. If option run_viewer
is set
to true and the plot_format is gnuplot
that ASCII representation
will also be shown in the Maxima or Xmaxima console.
Gnuplot generates commands in the PostScript page description language.
If the option gnuplot_out_file
is set to filename, gnuplot
writes the PostScript commands to filename. Otherwise, it is
saved as maxplot.ps
file. The settings for this terminal are given by the value of the option gnuplot_dumb_term_command
.
Gnuplot can generate output in many other graphical formats such as png,
jpeg, svg etc. To use those formats, option gnuplot_term
can be
set to any supported gnuplot term name (which must be a symbol) or even a
full gnuplot term specification with any valid options (which must be a string). For
example [gnuplot_term, png]
creates output in PNG (Portable
Network Graphics) format while [gnuplot_term, "png size
1000,1000"]
creates PNG of 1000 x 1000 pixels size. If the option
gnuplot_out_file
is set to filename, gnuplot writes the
output to filename. Otherwise, it is saved as
maxplot.term
file, where term is gnuplot terminal
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
.
With a value of false
, it can be used to disable the use of PM3D
mode, which is enabled by default.
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
.
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
.
[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"
.
[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.
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.
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.
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.
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.
This is an obsolete option that has been replaced legend
described
above.
This is an obsolete option that has been replaced by style
.
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Opens the pipe to gnuplot used for plotting with the gnuplot_pipes
format. Is not necessary to manually open the pipe before plotting.
Closes the pipe to gnuplot which is used with the gnuplot_pipes
format.
Closes the pipe to gnuplot which is used with the gnuplot_pipes
format and opens a new pipe.
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.
Resets the state of gnuplot used with the gnuplot_pipes
format. To
update the gnuplot window call gnuplot_replot
after gnuplot_reset
.
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13.1 Comments | ||
13.2 Files | ||
13.3 Functions and Variables for File Input and Output | ||
13.4 Functions and Variables for TeX Output | ||
13.5 Functions and Variables for Fortran Output |
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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
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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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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
.
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,
is like demo
)demo(filename)
.
In this case batch
searches for filename in the list
file_search_demo
. See demo
.
batch(filename,
is like test
)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.
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.
Closes the transcript file opened by writefile
or appendfile
.
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.
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.
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.
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
.
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]
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
.
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
.
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.
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
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
.
Default value: true
loadprint
tells whether to print a message when a file is loaded.
loadprint
is true
, always print a message.
loadprint
is 'loadfile
, print a message only if
a file is loaded by the function loadfile
.
loadprint
is 'autoload
,
print a message only if a file is automatically loaded.
See setup_autoload
.
loadprint
is false
, never print a message.
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.
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
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.
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.
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.
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
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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Note that the built-in TeX output functionality of wxMaxima makes no use of the functions described here but uses its own implementation instead.
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) $$x\,y\,z$$
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
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
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
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
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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.
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
Default value: false
When fortspaces
is true
, fortran
fills out
each printed line with spaces to 80 columns.
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14.1 Introduction to Polynomials | ||
14.2 Functions and Variables for Polynomials |
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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.
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Default value: false
algebraic
must be set to true
in order for the simplification of
algebraic integers to take effect.
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.
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
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
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
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]
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
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.
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]
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]
Default value: true
facexpand
controls whether the irreducible factors returned by
factor
are in expanded (the default) or recursive (normal CRE) form.
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)
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
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
Default value: false
When factorflag
is false
, suppresses the factoring of
integer factors of rational expressions.
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)
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) )
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
.
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
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")$
subst
can carry out multiple substitutions.
lratsubst
is analogous to subst
.
(%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
fullratsubst
is equivalent to ratsubst
except that it recurses until its result stops changing.
(%i5) ratsubst (b*a, a^2, a^3); 2 (%o5) a b (%i6) fullratsubst (b*a, a^2, a^3); 2 (%o6) a b
fullratsubst
also accepts a list of equations or a single
equation as first argument.
(%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
fullratsubst
may cause an indefinite recursion.
(%i9) errcatch (fullratsubst (b*a^2, a^2, a^3)); *** - Lisp stack overflow. RESET
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]
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
Factors the Gaussian integer n over the Gaussian integers, i.e., numbers
of the form a + b
where a and b are
rational integers (i.e., ordinary integers). Factors are normalized by making
a and b non-negative.
%i
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)
is similar to factorsum
but applies gfactor
instead
of factor
.
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
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.
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]
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)
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")$
subst
can carry out multiple substitutions.
lratsubst
is analogous to subst
.
(%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
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
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)
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)]
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
.
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
.
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
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.
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
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
.
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
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
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
.
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
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.
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
.
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.
Default value: true
When ratprint
is true
,
a message informing the user of the conversion of floating point numbers
to rational numbers is displayed.
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
Default value: false
When ratsimpexpons
is true
,
ratsimp
is applied to the exponents of expressions during simplification.
Default value: false
radsubstflag
, if true
, permits ratsubst
to make
substitutions such as u
for sqrt (x)
in x
.
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
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.
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 $Y)
Assigns a weight w_i to the variable x_i.
This causes a term to be replaced by 0 if its weight exceeds the
value of the variable ratwtlvl
(default yields no truncation).
The weight of a term is the sum of the products of the
weight of a variable in the term times its power.
For example, the weight of 3 x_1^2 x_2
is 2 w_1 + w_2
.
Truncation according to ratwtlvl
is carried out only when multiplying
or exponentiating canonical rational expressions (CRE).
ratweight ()
returns the cumulative list of weight assignments.
Note: The ratfac
and ratweight
schemes are incompatible and may
not both be used at the same time.
Examples:
(%i1) ratweight (a, 1, b, 1); (%o1) [a, 1, b, 1] (%i2) expr1: rat(a + b + 1)$ (%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
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.
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.
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
.
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
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.
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.
Returns a list of the canonical rational expression (CRE) variables in
expression expr
.
See also ratvars
.
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)
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]
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.
Removes tellrat
properties from x_1, …, x_n.
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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
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The Bessel function of the first kind of order v and argument z.
bessel_j
is defined as
although the infinite series is not used for computations.
The Bessel function of the second kind of order v and argument z.
$$Y_\nu(z) = {{\cos \left(\pi\,v\right)\,J_{v}(z)-J_{-v}(z)}\over{ \sin \left(\pi\,v\right)}}$$when v is not an integer. When v is an integer n, the limit as v approaches n is taken.
The modified Bessel function of the first kind of order v and argument z.
bessel_i
is defined as
$$I_\nu(z) = \sum_{k=0}^{\infty } {{1\over{k!\,\Gamma
\left(v+k+1\right)}} {\left(z\over 2\right)^{v+2\,k}}}$$
although the infinite series is not used for computations.
The modified Bessel function of the second kind of order v and argument z.
bessel_k
is defined as
$$K_\nu(z) = {{\pi\,\csc
\left(\pi\,v\right)\,\left(I_{-v}(z)-I_{v}(z)\right)}\over{2}}$$
when v is not an integer. If v is an integer n, then the limit as v approaches n is taken.
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
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
.
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)
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)
.
Identical to scaled_bessel_i(0,z)
.
Identical to scaled_bessel_i(1,z)
.
Lommel's little s[u,v](z) function. Probably Gradshteyn & Ryzhik 8.570.1.
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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.
The Airy function Ai(x). (A&S 10.4.2)
The derivative diff (airy_ai(x), x)
is airy_dai(x)
.
The derivative of the Airy function Ai airy_ai(x)
.
See airy_ai
.
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
.
The derivative of the Airy Bi function airy_bi(x)
.
See airy_ai
and airy_bi
.
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The gamma function and the related beta, psi and incomplete gamma functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 6.
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.
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.
Complex bigfloat factorial.
load ("bffac")
loads this function.
The basic definition of the gamma function (A&S 6.1.1) is
$$\Gamma\left(z\right)=\int_{0}^{\infty }{t^{z-1}\,e^ {- t }\;dt}$$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
.
The natural logarithm of the gamma function.
The lower incomplete gamma function (A&S 6.5.2):
$$\gamma\left(a , z\right)=\int_{0}^{z}{t^{a-1}\,e^ {- t }\;dt}$$See also gamma_incomplete
(upper incomplete gamma function).
The incomplete upper gamma function (A&S 6.5.3):
$$\Gamma\left(a , z\right)=\int_{z}^{\infty }{t^{a-1}\,e^ {- t }\;dt}$$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
.
The regularized incomplete upper gamma function (A&S 6.5.1):
$${\it Q}\left(a , z\right)={{\Gamma\left(a , z\right)}\over{\Gamma\left(a\right)}}$$See also gamma_expand
for controlling how
gamma_incomplete
is expressed in terms of elementary functions
and erfc
.
Also see gamma_incomplete
.
The generalized incomplete gamma function.
$$\Gamma\left(a , z_{1}, z_{2}\right)=\int_{z_{1}}^{z_{2}}{t^{a-1}\,e^ {- t }\;dt}$$Also see gamma_incomplete
and gamma_incomplete_regularized
.
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
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.
Transforms instances of binomial, factorial, and beta functions in expr into gamma functions.
See also makefact
.
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
The basic definition of the incomplete beta function (A&S 6.6.1) is
$$I_z(a,b) = \int_0^z (1-t)^{b-1} t^{a-1}\, dt$$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
$$z^{n-1}\sum_{k=0}^{m-1} {(1-m)_k z^k\over{k! (n-k)}}$$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)
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
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)
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.
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.
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.
Default value: 20
maxpsiposint
is the largest positive value for which
psi[n](x)
will try to compute an exact value.
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.
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.
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.
Transforms instances of binomial, gamma, and beta functions in expr into factorials.
See also makegamma
.
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
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
The Exponential Integral and related funtions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 5
The Exponential Integral E1(z) (A&S 5.1.1) defined as
$$E_1(z) = \int_z^\infty {e^{-t} \over t} dt$$with \(|\arg\ z| < \pi\).
The Exponential Integral Ei(z) (A&S 5.1.2)
The Exponential Integral Li(z) (A&S 5.1.3)
The Exponential Integral En(z) (A&S 5.1.4) defined as
$$E_n(z) = \int_1^\infty {e^{-zt} \over t^n} dt$$with ${\rm Re}\ z > 0$ and $n = 0, 1, 2, \ldots$.
The Exponential Integral Si(z) (A&S 5.2.1) defined as
$${\rm Si}(z) = \int_0^z {\sin t \over t}\, dt$$The Exponential Integral Ci(z) (A&S 5.2.2) defined as
$${\rm Ci}(z) = \gamma + \log z + \int_0^z {{\cos t - 1} \over t} dt$$with \(|\arg\ z| < \pi\).
The Exponential Integral Shi(z) (A&S 5.2.3) defined as
$${\rm Shi}(z) = \int_0^z {\sinh t \over t} dt$$The Exponential Integral Chi(z) (A&S 5.2.4) defined as
$${\rm Chi}(z) = \gamma + \log z + \int_0^z {{\cosh t - 1} \over t} dt$$with \(|\arg\ z| < \pi\).
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.
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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The Error function and related funtions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 7
The Error Function erf(z) (A&S 7.1.1)
See also flag erfflag
.
The Complementary Error Function erfc(z) (A&S 7.1.2)
erfc(z) = 1-erf(z)
The Imaginary Error Function.
erfi(z) = -%i*erf(%i*z)
Generalized Error function Erf(z1,z2)
The Fresnel Integral, A&S 7.3.1: $$C(z) = \int_0^z \cos{\pi t^2\over{2}}\, dt$$
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
.
The Fresnel Integral, A&S 7.3.2: $$S(z) = \int_0^z \sin{\pi t^2\over{2}}\, dt$$
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
.
Default value: false
When T erfc, erfi, erf_generalized, fresnel_s and fresnel_c are transformed to erf.
Default value: false
Enables transformation to a Hypergeometric representation for fresnel_s and fresnel_c
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The Struve functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 12.
The Struve Function H of order v and argument z. (A&S 12.1.1)
The Modified Struve Function L of order v and argument z. (A&S 12.2.1)
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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
.
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)
Whittaker W function. (A&S 13.1.33)
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
.
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
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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
.
The parabolic cylinder function parabolic_cylinder_d(v,z)
. (A&S 19.3.1)
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Compute the Laplace transform of expr with respect to the variable t. The integrand expr may contain special functions.
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
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)
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
The principal branch of Lambert's W function W(z), the solution of
z = W(z) * exp(W(z))
. (DLMF 4.13)
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)
.
The Plasma Dispersion Function
nzeta(z) = %i*sqrt(%pi)*exp(-z^2)*(1-erf(-%i*z))
Returns realpart(nzeta(z))
.
Returns imagpart(nzeta(z))
.
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16.1 Introduction to Elliptic Functions and Integrals | ||
16.2 Functions and Variables for Elliptic Functions | ||
16.3 Functions and Variables for Elliptic Integrals |
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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: $$m = k^2$$ and $$k = \sin \alpha$$
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.
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Elliptic functions can be defined in many different ways, including as
inverses of elliptic functions and ratios of theta functions, In the
following, we express the elliptic functions as Fourier series where
$$q = e^{-\pi K'(m)/K(m)}$$
and
$$v = {{\pi u}\over{2 K(m)}}$$
with K(m) being elliptic_kc(m)
and
K'(m) being elliptic_kc(1-m)
.
The Jacobian elliptic function sn(u,m): $${\rm sn}(u|m) = \frac{2\pi}{\sqrt{m} K} \sum_{n=0}^{\infty} \frac{q^{n+1/2}}{1-q^{2n+1}}\sin[(2n+1)v]$$
The Jacobian elliptic function cn(u,m).
The Jacobian elliptic function dn(u,m).
The Jacobian elliptic function ns(u,m) = 1/sn(u,m).
The Jacobian elliptic function sc(u,m) = sn(u,m)/cn(u,m).
The Jacobian elliptic function sd(u,m) = sn(u,m)/dn(u,m).
The Jacobian elliptic function nc(u,m) = 1/cn(u,m).
The Jacobian elliptic function cs(u,m) = cn(u,m)/sn(u,m).
The Jacobian elliptic function cd(u,m) = cn(u,m)/dn(u,m).
The Jacobian elliptic function nd(u,m) = 1/dn(u,m).
The Jacobian elliptic function ds(u,m) = dn(u,m)/sn(u,m).
The Jacobian elliptic function dc(u,m) = dn(u,m)/cn(u,m).
The inverse of the Jacobian elliptic function \({\rm sn}(u,m)\). This can also be represented by $${\rm sn}^{-1}(u, m) = \int_0^u {dt\over\sqrt{(1-t^2)(1-mt^2)}}$$
The inverse of the Jacobian elliptic function cn(u,m).
The inverse of the Jacobian elliptic function dn(u,m).
The inverse of the Jacobian elliptic function ns(u,m).
The inverse of the Jacobian elliptic function sc(u,m).
The inverse of the Jacobian elliptic function sd(u,m).
The inverse of the Jacobian elliptic function nc(u,m).
The inverse of the Jacobian elliptic function cs(u,m).
The inverse of the Jacobian elliptic function cd(u,m).
The inverse of the Jacobian elliptic function nd(u,m).
The inverse of the Jacobian elliptic function ds(u,m).
The inverse of the Jacobian elliptic function dc(u,m).
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The incomplete elliptic integral of the first kind, defined as
$$F(\phi|m) = \int_0^\phi {{d\theta}\over{\sqrt{1 - m\sin^2\theta}}}$$See also elliptic_e and elliptic_kc.
The incomplete elliptic integral of the second kind, defined as
$$E(\phi|m) = \int_0^\phi \sqrt{1 - m\sin^2\theta}\,d\theta$$This is Legendre's form for the elliptic integral of the second kind. See also elliptic_f and elliptic_ec.
The incomplete elliptic integral of the second kind, defined as
$$Eu(u,m) = \int_0^u {\rm dn}(v, m)\,dv = \int_0^\tau \sqrt{{1-m t^2}\over{1-t^2}}\,dt$$where \(\tau = {\rm sn}(u, m)\). This is Jacobi's epsilon function.
This is related to elliptic_e
by
See also elliptic_e.
The incomplete elliptic integral of the third kind, defined as
$$\Pi(n; \phi|m) = \int_0^\phi {{d\theta}\over{(1-n\sin^2 \theta)\sqrt{1 - m\sin^2\theta}}}$$The complete elliptic integral of the first kind, defined as
$$K(m) = \int_0^{{\pi}\over{2}} {{d\theta}\over{\sqrt{1 - m\sin^2\theta}}}$$For certain values of m, the value of the integral is known in
terms of Gamma functions. Use makegamma
to evaluate them.
The complete elliptic integral of the second kind, defined as
$$E(m) = \int_0^{{\pi}\over{2}} \sqrt{1 - m\sin^2\theta}\,d\theta$$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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17.1 Functions and Variables for Limits |
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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)
.
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.
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.
Take the limit of the Taylor series expansion of expr
in x
at val
from direction dir
.
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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18.1 Functions and Variables for Differentiation |
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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 /
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 /
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
atomgrad
is the atomic gradient property of an expression.
This property is assigned by gradef
.
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
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.
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)
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
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])
.
(%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)]
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
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
.
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
Causes only differentiations with respect to
the indicated variables, within the ev
command.
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)
.
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.
When diff
is present as an evflag
in call to ev
,
all differentiations indicated in expr
are carried out.
Applies the scalar d'Alembertian to the scalar function f.
load ("ctensor")
loads this function.
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]
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.
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
.
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)
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
19.1 Introduction to Integration | ||
19.2 Functions and Variables for Integration | ||
19.3 Introduction to QUADPACK | ||
19.4 Functions and Variables for QUADPACK |
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
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)
.
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
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
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
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.
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.
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.
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
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
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:
(%i1) integrate (sin(x)^3, x); 3 cos (x) (%o1) ------- - cos(x) 3 (%i2) integrate (x/ sqrt (b^2 - x^2), x); 2 2 (%o2) - sqrt(b - x ) (%i3) integrate (cos(x)^2 * exp(x), x, 0, %pi); %pi 3 %e 3 (%o3) ------- - - 5 5 (%i4) integrate (x^2 * exp(-x^2), x, minf, inf); sqrt(%pi) (%o4) --------- 2
assume
and interactive query.
(%i1) assume (a > 1)$ (%i2) integrate (x**a/(x+1)**(5/2), x, 0, inf); 2 a + 2 Is ------- an integer? 5 no; Is 2 a - 3 positive, negative, or zero? neg; 3 (%o2) beta(a + 1, - - a) 2
gradef
, and one using the
derivation diff(r(x))
of an unspecified function r(x)
.
(%i3) gradef (q(x), sin(x**2)); (%o3) q(x) (%i4) diff (log (q (r (x))), x); d 2 (-- (r(x))) sin(r (x)) dx (%o4) ---------------------- q(r(x)) (%i5) integrate (%, x); (%o5) log(q(r(x)))
'integrate
noun form. In this example, Maxima
can extract one factor of the denominator of a rational function, but cannot
factor the remainder or otherwise find its integral. grind
shows the
noun form 'integrate
in the result. See also
integrate_use_rootsof
for more on integrals of rational functions.
(%i1) expand ((x-4) * (x^3+2*x+1)); 4 3 2 (%o1) x - 4 x + 2 x - 7 x - 4 (%i2) integrate (1/%, x); / 2 [ x + 4 x + 18 I ------------- dx ] 3 log(x - 4) / x + 2 x + 1 (%o2) ---------- - ------------------ 73 73 (%i3) grind (%); log(x-4)/73-('integrate((x^2+4*x+18)/(x^3+2*x+1),x))/73$
f_1
in this
example contains the noun form of integrate
. The quote-quote operator
''
causes the integral to be evaluated, and the result becomes the
body of f_2
.
f_2 (a) := "(integrate (x^3, x, 1, a));
(%i1) f_1 (a) := integrate (x^3, x, 1, a); 3 (%o1) f_1(a) := integrate(x , x, 1, a) (%i2) ev (f_1 (7), nouns); (%o2) 600 (%i3) /* Note parentheses around integrate(...) here */ f_2 (a) := ''(integrate (x^3, x, 1, a)); 4 a 1 (%o3) f_2(a) := -- - - 4 4 (%i4) f_2 (7); (%o4) 600
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
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
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.
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.
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.
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
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)
Equivalent to ldefint
with tlimswitch
set to true
.
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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).
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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
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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
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
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
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.
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
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
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
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
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
The weight function w is selected by trig:
cos
sin
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
Integration of
\(\cos \left(\omega \, x\right) \, f\left(x\right)\)
or
\(\sin \left(\omega \, x\right) \, f\left(x\right)\)
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:
The weight function w is selected by trig:
cos
sin
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
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:
The weight function w is selected by wfun:
1
2
3
4
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
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
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
.
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.
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20.1 Functions and Variables for Equations |
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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
.
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
Default value: 10^8
algepsilon
is used by algsys
.
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.
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:
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.
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
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)
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.
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]
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]
dimen
is a package for dimensional analysis.
load ("dimen")
loads this package.
demo ("dimen")
displays a short demonstration.
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
.
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.
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
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.
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).
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
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]
Default value: true
When linsolvewarn
is true
, linsolve
prints a message
"Dependent equations eliminated".
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.
Default value: not_set_yet
multiplicities
is set to a list of the multiplicities of the individual
solutions returned by solve
or realroots
.
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
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
.
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.
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.
Default value: false
When realonly
is true
, algsys
returns only those solutions
which are free of %i
.
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]
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
Default value: true
rootsconmode
governs the behavior of the rootscontract
command.
See rootscontract
for details.
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
Default value: 1.0e-7
rootsepsilon
is the tolerance which establishes the confidence interval
for the roots found by the realroots
function.
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.
Default value: true
When solvedecomposes
is true
, solve
calls
polydecomp
if asked to solve polynomials.
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.
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.
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
[]
.
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.
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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21.1 Introduction to Differential Equations | ||
21.2 Functions and Variables for Differential Equations |
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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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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.
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
.
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.
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.
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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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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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
.
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
.
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
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]
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.
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.)
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.
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.
Computes the forward fast Fourier transform of a real-valued input
returning a bigfloat result. This is the bigfloat version of
real_fft
.
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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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
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
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
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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The Ordinary Differential Equations (ODE) solved by the functions in this section should have the form, $${{dy}\over{dx}} = F(x,y)$$ 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 $${{dx}\over{dt}} = G(x,y,t) \qquad {{dy}\over{dt}} = F(x,y,t)$$
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.
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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:
forward
, to make the independent variable increase
nsteps
times, with increments tstep
, backward
, to
make the independent variable decrease, or both
that will lead to
an integral curve that extends nsteps
forward, and nsteps
backward. The keywords right
and left
can be used as
synonyms for forward
and backward
.
The default value is both
.
versus_t
is given any value
different from 0, the second plot window will be displayed. The second
plot window includes another menu, similar to the menu of the main plot
window.
The default value is 0.
name=value
.
name=min:max
Examples:
(%i1) plotdf(exp(-x)+y,[trajectory_at,2,-0.1])$
(%i1) plotdf(x-y^2,[xfun,"sqrt(x);-sqrt(x)"], [trajectory_at,-1,3], [direction,forward], [y,-5,5], [x,-4,16])$
The graph also shows the function y = sqrt(x).
(%i1) plotdf([v,-k*z/m], [z,v], [parameters,"m=2,k=2"], [sliders,"m=1:5"], [trajectory_at,6,0])$
(%i1) plotdf([y,-(k*x + c*y + b*x^3)/m], [parameters,"k=-1,m=1.0,c=0,b=1"], [sliders,"k=-2:2,m=-1:1"],[tstep,0.1])$
(%i1) plotdf([w,-g*sin(a)/l - b*w/m/l], [a,w], [parameters,"g=9.8,l=0.5,m=0.3,b=0.05"], [trajectory_at,1.05,-9],[tstep,0.01], [a,-10,2], [w,-14,14], [direction,forward], [nsteps,300], [sliders,"m=0.1:1"], [versus_t,1])$
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).
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
$${{dx}\over{dt}} = t - x^2$$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:
$$\cases{{\displaystyle{dx}\over\displaystyle{dt}} = 4-x^2-4y^2 &\cr &\cr {\displaystyle{dy}\over\displaystyle{dt}} = y^2-x^2+1}$$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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23.1 Introduction to Matrices and Linear Algebra | ||
23.2 Functions and Variables for Matrices and Linear Algebra |
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23.1.1 Dot | ||
23.1.2 Matrices | ||
23.1.3 Vectors | ||
23.1.4 eigen |
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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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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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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.
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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
.
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Appends the column(s) given by the one or more lists (or matrices) onto the matrix M.
Appends the row(s) given by the one or more lists (or matrices) onto the matrix M.
Returns the adjoint of the matrix M. The adjoint matrix is the transpose of the matrix of cofactors of M.
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 ]
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 ]
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)
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 ]
Returns the i'th column of the matrix M. The return value is a matrix.
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 ]
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.
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
.
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
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.
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.
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) ]
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.
Default value: false
When domxnctimes
is true
, non-commutative products of
matrices are carried out.
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.
Default value: false
When doscmxops
is true
, scalar-matrix operations are
carried out.
Default value: false
When doscmxplus
is true
, scalar-matrix operations yield
a matrix result. This switch is not subsumed under doallmxops
.
Default value: true
When dot0nscsimp
is true
, a non-commutative product of zero
and a nonscalar term is simplified to a commutative product.
Default value: true
When dot0simp
is true
,
a non-commutative product of zero and
a scalar term is simplified to a commutative product.
Default value: true
When dot1simp
is true
,
a non-commutative product of one and
another term is simplified to a commutative product.
Default value: true
When dotassoc
is true
, an expression (A.B).C
simplifies to
A.(B.C)
.
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.
Default value: false
When dotdistrib
is true
, an expression A.(B + C)
simplifies
to A.B + A.C
.
Default value: true
When dotexptsimp
is true
, an expression A.A
simplifies to
A^^2
.
Default value: 1
dotident
is the value returned by X^^0
.
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)
.
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 ]
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.
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
Returns an m by n matrix, all elements of which
are zero except for the [i, j]
element which is x.
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) ]
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 ]
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]
Returns an n by n identity matrix.
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
.
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
.
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
.
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]
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 ]
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 ]
matrixexp
.
(%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 ]
b^^m
is the same as b^m
.
(%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 ]
Calculates the matrix exponential
$$e^{M V}$$
. 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: $$e^M=\sum_{k=0}^\infty{{M^k}\over{k!}}$$
Returns a matrix with element i,j
equal to f(M[i,j])
.
Returns true
if expr is a matrix, otherwise false
.
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 ]
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) ) ]
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 ]
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.
Returns the i, j minor of the matrix M. That is, M with row i and column j removed.
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.
Computes the determinant of the matrix M by the Johnson-Gentleman tree
minor algorithm. newdet
returns the result in CRE form.
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
.
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.
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.
Returns the i'th row of the matrix M. The return value is a matrix.
Default value: ]
rmxchar
is the character drawn on the right-hand side of a matrix.
See also lmxchar
.
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.
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
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.
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
.
Default value: false
When sparse
is true
, and if ratmx
is true
, then
determinant
will use special routines for computing sparse determinants.
Returns a new matrix composed of the matrix M with rows i_1, …, i_m deleted, and columns j_1, …, j_n deleted.
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)
.
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 ]
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
.
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
.
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.
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
.
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.)
Returns an m by n matrix, all elements of which are zero.
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
24.1 Introduction to Affine | ||
24.2 Functions and Variables for Affine |
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
affine
is a package to work with groups of polynomials.
Category: Polynomials · Category: Groebner bases · Category: Share packages · Category: Package affine
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
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.
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.
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.
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.
Returns the degree of a noncommutative polynomial p. See declare_weights
.
load("affine")
loads this function.
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.
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.
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.
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.
Compute the Hilbert series through degree n for the current algebra.
load("affine")
loads this function.
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.
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.
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
.
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25.1 Introduction to itensor | ||
25.2 Functions and Variables for itensor |
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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.
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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!
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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
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Displays the contraction properties of its arguments as were given to
defcon
. dispcon (all)
displays all the contraction properties
which were defined.
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
).
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.
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
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).
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.
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
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.
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
).
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
.
Removes all the contraction properties
from the (tensor_1, ..., tensor_n). remcon(all)
removes all contraction
properties from all indexed objects.
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.
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
.
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
Unbinds all values from tensor which were assigned with the
components
function.
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.
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
).
Default value: %
Is the prefix for dummy indices (see the example under indices
).
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: %
).
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.
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
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.
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
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
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
.
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
.
Similar to canten
but also performs index contraction.
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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
.
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]], []]]
Removes all symmetry properties from tensor which has m covariant indices and n contravariant indices.
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.
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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.
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.
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
Evaluates all occurrences of the idiff
command in the tensorial
expression ten.
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)
.
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
Set to zero, in expr, all occurrences of the tensor_i that have no derivative indices.
Set to zero, in expr, all occurrences of the tensor_i that have derivative indices.
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
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.
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.
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)
.
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
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.
Set to zero, in expr
, all occurrences of tensor
that have
exactly one derivative index.
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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.
Sets the dimensions of the metric. Also initializes the antisymmetry properties of the Levi-Civita symbols for the given dimension.
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
.
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
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
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)
Imposes the Lorentz condition by substituting 0 for all indexed objects in expr that have a derivative index identical to a contravariant index.
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
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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
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.
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
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.
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.
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
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
The frame field. Contracts with the inverse frame field (ifri
) to
form the frame metric (ifg
).
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.
The frame metric. Defaults to kdelta
, but can be changed using
components
.
The inverse frame metric. Contracts with the frame metric (ifg
)
to kdelta
.
Default value: true
Specifies how the frame bracket (ifb
) is computed.
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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.
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
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.)
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.)
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.)
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.)
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
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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.
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
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.
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
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
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.
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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.
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.
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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
.
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
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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
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26.1 Introduction to ctensor | ||
26.2 Functions and Variables for ctensor |
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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
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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.
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.
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
.
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.
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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
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.
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.
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.
Returns the scalar curvature (obtained by contracting the Ricci tensor) of the Riemannian manifold with the given metric.
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.
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.
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.
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
.
Computes the contravariant components of the Riemann
curvature tensor as array elements uriem[i,j,k,l]
. These are displayed
if dis is true
.
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.
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.
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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()
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.
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When the variable cframe_flag
is set to true, the ctensor
package
performs its calculations using a moving frame.
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
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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
.
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
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.
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.
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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
.
Computes the (2,1) contortion coefficients from the torsion tensor tr.
Computes the (2,1) nonmetricity coefficients from the nonmetricity vector nm.
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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
.)
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]]
Computes the covariant gradient of a scalar function allowing the
user to choose the corresponding vector name as the example under
contragrad
illustrates.
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
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
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
.
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.
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
.
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
.
*** 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
.
*** NOT YET IMPLEMENTED ***
generates the field equations of Rosen's bimetric theory. The field
equations are the components of an array named rosen
.
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Returns true
if the first n rows and n columns of M
form a diagonal matrix or (2D) array.
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.
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.
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
Returns a new list consisting of L with the n'th element deleted.
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ctensor
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
.
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.
Causes trigonometric simplifications to be used when tensors are computed. Presently,
ctrgsimp
affects only computations involving a moving frame.
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.
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
.
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
.
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.
Variable used for Taylor-series expansion if ctayswitch
is set to
true
.
Maximum power used in Taylor-series expansion when ctayswitch
is
set to true
.
Point around which Taylor-series expansion is carried out when
ctayswitch
is set to true
.
The determinant of the metric tensor lg
. Computed by cmetric
when
cframe_flag
is set to false
.
Causes rational simplification to be applied by christof
.
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.
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.
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.
The covariant frame metric. By default, it is initialized to the 4-dimensional Lorentz frame with signature (+,+,+,-). Used when cframe_flag
is true
.
The inverse frame metric. Computed from lfg
when cmetric
is called while cframe_flag
is set to true
.
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
.
The covariant Riemann tensor. Computed by lriemann
.
The contravariant Riemann tensor. Computed by uriemann
.
The mixed Ricci-tensor. Computed by ricci
.
The contravariant Ricci-tensor. Computed by uricci
.
The metric tensor. This tensor must be specified (as a dim
by dim
matrix)
before other computations can be performed.
The inverse of the metric tensor. Computed by cmetric
.
The Weyl tensor. Computed by weyl
.
Frame bracket coefficients, as computed by frame_bracket
.
The Kretchmann invariant. Computed by rinvariant
.
A Newman-Penrose null tetrad. Computed by nptetrad
.
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 ]
User-supplied rank-3 tensor representing torsion. Used by contortion
.
The contortion tensor, computed from tr
by contortion
.
User-supplied nonmetricity vector. Used by nonmetricity
.
The nonmetricity coefficients, computed from nm
by nonmetricity
.
Variable indicating if the tensor package has been initialized. Set and used by
csetup
, reset by init_ctensor
.
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
.
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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()
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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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27.1 Introduction to atensor | ||
27.2 Functions and Variables for atensor |
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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.
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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)
.
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.
The algebra type. Valid values are universal
, grassmann
,
clifford
, symmetric
, symplectic
and lie_envelop
.
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
.
Default value: ident(3)
Default values for the bilinear forms sf
, af
, and
av
. The default is the identity matrix ident(3)
.
Default value: v
The symbol for base vectors.
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
.
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
.
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
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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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.
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)
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
.
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.
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
.
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
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.
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
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
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
.
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
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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.
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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
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
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.
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
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
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
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)]
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
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.
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
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
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
.
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]]
Returns true
if expr is a Taylor series,
and false
otherwise.
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.
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.
Simplifies coefficients of the power series expr.
taylor
calls this function.
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.
Converts expr from taylor
form to canonical rational expression
(CRE) form. The effect is the same as rat (ratdisrep (expr))
, but
faster.
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
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)
Default value: false
When verbose
is true
,
powerseries
prints progress messages.
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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.
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Returns true
if equal (x, y)
otherwise false
(doesn't give an error message like equal (x, y)
would do in this case).
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.
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.
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.
Returns a list of the Fourier coefficients of f(x)
defined
on the interval [-p, p]
.
Simplifies sin (n %pi)
to 0 if sinnpiflag
is true
and
cos (n %pi)
to (-1)^n
if cosnpiflag
is true
.
Default value: true
See foursimp
.
Default value: true
See foursimp
.
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
.
Returns the Fourier cosine coefficients for f(x)
defined on
[0, p]
.
Returns the Fourier sine coefficients for f(x)
defined on
[0, p]
.
Returns fourexpand (foursimp (fourier (f, x, p)),
x, p, 'inf)
.
Constructs and returns a list of the Fourier integral coefficients of
f(x)
defined on [minf, inf]
.
Returns the Fourier cosine integral coefficients for f(x)
on [0, inf]
.
Returns the Fourier sine integral coefficients for f(x)
on
[0, inf]
.
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Converts a into a Poisson encoding.
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.
Differentiates a with respect to b. b must occur only in the trig arguments or only in the coefficients.
Functionally identical to intopois (a^b)
.
b must be a positive integer.
Integrates in a similarly restricted sense (to poisdiff
). Non-periodic
terms in b are dropped if b is in the trig arguments.
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)].
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.
Is functionally identical to intopois (a + b)
.
Converts a into a Poisson series for a in general representation.
The symbol /P/
follows the line label of Poisson series
expressions.
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)
.
Is functionally identical to intopois (a*b)
.
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.
Prints a Poisson series in a readable format. In common
with outofpois
, it will convert a into a Poisson encoding first, if
necessary.
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29.1 Functions and Variables for Number Theory |
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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
Returns the n'th Bernoulli polynomial in the variable x.
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.
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
$$\zeta \left(s,h\right) = \sum_{k=0}^\infty {1 \over \left(k+h\right)^{s}}$$load ("bffac")
loads this function.
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
.
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
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:
(%i1) cf ([5, 3, 1]*[11, 9, 7] + [3, 7]/[4, 3, 2]); (%o1) [59, 17, 2, 1, 1, 1, 27] (%i2) cf ((3/17)*[1, -2, 5]/sqrt(11) + (8/13)); (%o2) [0, 1, 1, 1, 3, 2, 1, 4, 1, 9, 1, 9, 2]
cflength
controls how many periods of the continued fraction
are computed for algebraic, irrational numbers.
(%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]
cfdisrep
.
(%i1) cflength: 3$ (%i2) cfdisrep (cf (sqrt (3)))$ (%i3) ev (%, numer); (%o3) 1.731707317073171
cf
.
(%i1) cf ([1,1,1,1,1,2] * 3); (%o1) [4, 1, 5, 2] (%i2) cf ([1,1,1,1,1,2]) * 3; (%o2) [3, 3, 3, 3, 3, 6]
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
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
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]
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
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]
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
.
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]
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
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]
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]
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]
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
Returns the "integer square root" of the absolute value of x, which is an integer.
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]
Returns the least common multiple of its arguments. The arguments may be general expressions as well as integers.
load ("functs")
loads this function.
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]
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
Returns the smallest prime bigger than n.
(%i1) next_prime(27); (%o1) 29
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)
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
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.
Default value: 25
Number of Miller-Rabin's tests used in primep
.
Returns the list of all primes from start to end.
(%i1) primes(3, 7); (%o1) [3, 5, 7]
Returns the greatest prime smaller than n.
(%i1) prev_prime(27); (%o1) 23
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
Returns the number of integers less than or equal to n which are relatively prime to n.
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
.
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.
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
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)
Shows an addition table of all elements in (Z/nZ).
See also zn_mult_table
, zn_power_table
.
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]]
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
.
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
Returns a list containing factor generators corresponding to the characteristic factors of the totient of n.
For remarks and examples see zn_characteristic_factors
.
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 ]
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
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 ]
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]
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]
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 ]
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
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.
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]]
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.
Default value: false
Controls whether zn_primroot
prints a message when reaching
zn_primroot_limit
.
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30.1 Introduction to Symmetries | ||
30.2 Functions and Variables for Symmetries |
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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:
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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)]
goes from the elementary symmetric functions to the complete functions.
Similar to comp2ele
and comp2pui
.
Other functions for changing bases: comp2ele
.
Goes from the elementary symmetric functions to the compete functions.
Similar to comp2ele
and comp2pui
.
Other functions for changing bases: comp2ele
.
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
.
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
\(x_1,x_2,\ldots\)
(%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
.
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
.
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
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
.
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
.
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
.
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
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
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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]]
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
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
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
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]]
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
.
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
.
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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
).
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.
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
.
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.
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 ]
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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
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
.
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
.
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
.
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
.
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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
.
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)
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
.
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
.
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
.
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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:
(x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x1 - (x1*x3 + x3*x5 + x5*x2 + x2*x4 + x4*x1))^2
general,
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
.
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
.
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
.
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
.
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
.
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
.
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
.
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
.
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
.
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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!)
.
returns the list of permutations of the list L.
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31.1 Functions and Variables for Groups |
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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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32.1 Introduction for Runtime Environment | ||
32.2 Interrupts | ||
32.3 Functions and Variables for Runtime Environment |
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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.
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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.
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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.
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.
Prints out a description of the state of storage and
stack management in Maxima. room
calls the Lisp function of
the same name.
room ()
prints out a moderate description.
room (true)
prints out a verbose description.
room (false)
prints out a terse description.
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
.
feature
) feature
, item)
Returns information about the presence or absence of certain system-dependent features.
status (feature)
returns a list of system features. These include Lisp
version, operating system type, etc. The list may vary from one Lisp type to
another.
status (feature, item)
returns true
if item is on the
list of items returned by status (feature)
and false
otherwise.
status
quotes the argument item. The quote-quote operator
''
defeats quotation. A feature whose name contains a special
character, such as a hyphen, must be given as a string argument. For example,
status (feature, "ansi-cl")
.
See also sstatus
.
The variable features
contains a list of features which apply to
mathematical expressions. See features
and featurep
for more
information.
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)&")
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.
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
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:
YYYY-MM-DD[ T]hh:mm:ss[,.]nnn
YYYY-MM-DD[ T]hh:mm:ss
YYYY-MM-DD
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:
[+-]hh:mm
[+-]hhmm
[+-]hh
Z
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
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
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
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
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
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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33.1 Introduction to Miscellaneous Options | ||
33.2 Share | ||
33.3 Functions and Variables for Miscellaneous Options |
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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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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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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.
Default value: i
genindex
is the alphabetic prefix used to generate the
next variable of summation when necessary.
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.
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
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.
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
.
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.
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
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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34.1 Introduction to Rules and Patterns | ||
34.2 Functions and Variables for Rules and Patterns |
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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.
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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
.
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
.
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.
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.
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
.
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)]
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
.
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) -> []]
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:
letsimp
will search for literally unless previous to calling
letsimp
the matchdeclare
function is used to associate a
predicate with the atom. In this case letsimp
will match the atom to
any term of a product satisfying the predicate.
sin(x)
, n!
, f(x,y)
, etc. As with atoms
above letsimp
will look for a literal match unless matchdeclare
is used to associate a predicate with the argument of the kernel.
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
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)!
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
.
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
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
.
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)]
Default value: 10000
maxapplydepth
is the maximum depth to which apply1
and apply2
will delve.
Default value: 10000
maxapplyheight
is the maximum height to which applyb1
will reach before giving up.
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
.
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)]
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)
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
Executes kill (rules)
and then resets the next rule number to 1
for addition +
, multiplication *
, and exponentiation ^
.
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35.1 Introduction to Sets | ||
35.2 Functions and Variables for Sets |
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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.
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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.
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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
.
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Stavros Macrakis of Cambridge, Massachusetts and Barton Willis of the University of Nebraska at Kearney (UNK) wrote the Maxima set functions and their documentation.
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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}
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)]
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
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]}
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}
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
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)}
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
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]
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}}
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
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)}
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);
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])]))
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])}
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
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]}
intersect
is the same as intersection
, which see.
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) {}
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
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})
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)}
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}
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
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)
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)
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}]
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]}
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) {{}}
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]
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) {}
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
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}
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
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
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}}
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
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:
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
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.
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)
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}
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
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}
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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36.1 Introduction to Function Definition | ||
36.2 Function | ||
36.3 Macros | ||
36.4 Functions and Variables for Function Definition |
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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
.
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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
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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)
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
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
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
.
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<>
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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.
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 ]
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.
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.
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.
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
.
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:
mode_declare
or a statement like the following one:
put(x_1, bigfloat, numerical_type)
'
tells the compiler that the text is meant as an option.
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.
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)
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:
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.
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
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
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)
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]]
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]
Returns the definition of the function f.
The argument may be
::=
),
:=
or define
),
memoizing function
(defined with :=
or define
, but enclosing arguments in square brackets [ ]
),
:=
or define
,
but enclosing some arguments in square brackets and others in parentheses
( )
),
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.
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.
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
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
lambda
quotes its arguments, lambda expression i
below does
not define a "multiply by a
" function. Such a function can be defined
via buildq
, as in lambda expression i2
below.
(%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
[L]
as the sole or final argument.
The arguments appear within the function body as a list.
(%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
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
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
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
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.
Default value: false
When mode_check_errorp
is true
, mode_declare
calls
error.
Default value: true
When mode_check_warnp
is true
, mode errors are
described.
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.
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.
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.
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.
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.
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
.
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.
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
.
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.
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.
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
.
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.
Default value: false
When tr_numer
is true
, numer
properties are used for
atoms which have them, e.g. %pi
.
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.
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.
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.
Prints a list of warnings which have been given by the translator during the current translation.
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.
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.
Default value: compfile
- Gives a warning if the function meval
gets called. If meval
is
called that indicates problems in the translation.
Default value: all
- Gives a warning when variables are assigned values inappropriate for their mode.
Default value: compile
- Determines when to send warnings about undeclared variables to the TTY.
Default value: all
- Gives a warning when undefined global variables are seen.
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.
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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37.1 Lisp and Maxima | ||
37.2 Garbage Collection | ||
37.3 Introduction to Program Flow | ||
37.4 Functions and Variables for Program Flow |
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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.
$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 $FOO
.
|$FOO|
and |$foo|
correspond to Maxima foo
and FOO
,
respectively.
|$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 '$foo 'a 'b) ((MTIMES SIMP) A B)
Some Lisp functions are shadowed in the Maxima package, namely the following.
complement continue // float functionp array exp listen signum atan asin acos asinh acosh atanh tanh cosh sinh tan break gcd
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One of the advantages of using lisp is that it uses "Garbage Collection". In other words it automatically takes care of freeing memory occupied for example of intermediate results that were used during symbolic computation.
Garbage Collection avoids many errors frequently found in C programs (memory being freed too early, multiple times or not at all).
Tries to manually trigger the lisp's garbage collection. This rarely is necessary as the lisp will employ an excellent algorithm for determining when to start garbage collection.
If maxima knows how to do manually trigger the garbage collection for the
current lisp garbage_collect
returns true
, else false
.
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Maxima provides a do
loop for iteration, as well as more primitive
constructs such as go
.
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Prints the call stack, that is, the list of functions which called the currently active function.
backtrace ()
prints the entire call stack.
backtrace (n)
prints the n most recent
functions, including the currently active function.
backtrace
can be called from a script, a function, or the interactive
prompt (not only in a debugging context).
Examples:
backtrace ()
prints the entire call stack.
(%i1) h(x) := g(x/7)$ (%i2) g(x) := f(x-11)$ (%i3) f(x) := e(x^2)$ (%i4) e(x) := (backtrace(), 2*x + 13)$ (%i5) h(10); #0: e(x=4489/49) #1: f(x=-67/7) #2: g(x=10/7) #3: h(x=10) 9615 (%o5) ---- 49
backtrace (n)
prints the n most recent
functions, including the currently active function.
(%i1) h(x) := (backtrace(1), g(x/7))$ (%i2) g(x) := (backtrace(1), f(x-11))$ (%i3) f(x) := (backtrace(1), e(x^2))$ (%i4) e(x) := (backtrace(1), 2*x + 13)$ (%i5) h(10); #0: h(x=10) #0: g(x=10/7) #0: f(x=-67/7) #0: e(x=4489/49) 9615 (%o5) ---- 49
The do
statement is used for performing iteration. The general
form of the do
statements maxima supports is:
for variable: initial_value step increment
thru limit do body
for variable: initial_value step increment
while condition do body
for variable: initial_value step increment
unless condition do body
for variable in list do body
If the loop is expected to generate a list as output the command
makelist
may be the appropriate command to use instead,
See section 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
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
.
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.
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.
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);
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
.
Reprints the most recent error message.
The variable error
holds the message,
and errormsg
formats and prints it.
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
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
.
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
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]
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.
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.
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.
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
).
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.
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
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.
Evaluates expr and throws the value back to the most recent
catch
. throw
is used with catch
as a nonlocal return
mechanism.
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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38.1 Source Level Debugging | ||
38.2 Keyword Commands | ||
38.3 Functions and Variables for Debugging |
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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
.
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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
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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.
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.
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
.
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
.
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
.
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
.
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.
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
.
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
.
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.
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:
noprint
Do not print a message at function entry and exit.
break
Put a breakpoint before the function is entered,
and after the function is exited. See break
.
lisp_print
Display arguments and return values as Lisp objects.
info
Print -> true
at function entry and exit.
errorcatch
Catch errors, giving the option to signal an error,
retry the function call, or specify a return value.
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);
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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39.1 Introduction to alt-display | ||
39.2 Functions and Variables for alt-display |
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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 $X) is not of type (OR SYMBOL STRING FUNCTION). *alt-display2d* reset to nil. -- an error. To debug this try: debugmode(true); (%i10) x; (%o10) x
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This function is similar to define
: it evaluates its arguments
and expands into a function definition. The function is a
function of a single input input. For convenience, a substitution
is applied to expr after evaluation, to provide easy access to
Lisp variable names.
Set a time-stamp on each prompt:
(%i1) load("alt-display.mac")$ (%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
.
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
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>
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}$${{x}\over{y^2+x^2}}$$
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
Resets the prompt prefix and suffix to the empty string, and sets both 1-d and 2-d display functions to the default.
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.
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 Emacs(8).
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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40.1 Introduction to asympa | ||
40.2 Functions and variables for asympa |
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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.
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41.1 Functions and Variables for augmented_lagrangian |
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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
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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42.1 Functions and Variables for Bernstein |
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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")
.
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.
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")
.
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")
.
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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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.
43.1 Functions and Variables for bitwise |
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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
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
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
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
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
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
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
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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44.1 Functions and Variables for bode |
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Function to draw Bode gain plots.
Examples (1 through 7 from
http://www.swarthmore.edu/NatSci/echeeve1/Ref/Bode/BodeHow.html,
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 to draw Bode phase plots.
Examples (1 through 7 from
http://www.swarthmore.edu/NatSci/echeeve1/Ref/Bode/BodeHow.html,
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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45.1 Introduction to celine |
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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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46.1 Functions and Variables for clebsch_gordan |
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Compute the Clebsch-Gordan coefficient <j1, j2, m1, m2 | j, m>.
Compute Racah's V coefficient (computed in terms of a related Clebsch-Gordan coefficient).
Compute Racah's W coefficient (computed in terms of a Wigner 6j symbol)
Compute Wigner's 3j symbol (computed in terms of a related Clebsch-Gordan coefficient).
Compute Wigner's 6j symbol.
Compute Wigner's 9j symbol.
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47.1 Introduction to cobyla | ||
47.2 Functions and Variables for cobyla | ||
47.3 Examples for cobyla |
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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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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)
maxfun
The maximum number of function evaluations. (Defaults to 1000).
On return, a vector is given:
var = value
for each of the
variables listed in X.
load("fmin_cobyla")
loads this function.
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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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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48.1 Package combinatorics | ||
48.2 Functions and Variables for Combinatorics |
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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")
.
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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
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
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]]
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]]
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]
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
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]
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
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]
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]
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
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
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]
Returns the n-degree permutation at position i (from 1 to n!) in the Trotter-Johnson ordering of permutations.
Example:
(%i1) load("combinatorics")$ (%i2) perm_unrank (8, 19729); (%o2) [4, 6, 3, 1, 7, 5, 2, 8]
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
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]]
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]]
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]
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
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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49.1 Introduction to contrib_ode | ||
49.2 Functions and Variables for contrib_ode | ||
49.3 Possible improvements to contrib_ode | ||
49.4 Test cases for contrib_ode | ||
49.5 References for contrib_ode |
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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:
%t
, or
%u
.
%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.
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Returns a list of solutions of the ODE eqn with independent variable x and dependent variable y.
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
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
The variable method
is set to the successful solution
method.
%c
is the integration constant for first order ODEs.
%k1
is the first integration constant for second order ODEs.
%k2
is the second integration constant for second order ODEs.
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 gauss_a
.
The derivative with respect to x of gauss_a(a, b, c, x)
.
The derivative with respect to x of gauss_b(a, b, c, 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.
Kummer's U function, as defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Section 13.1.3.
See kummer_m
.
The derivative with respect to x of kummer_m(a, b, x)
.
The derivative with respect to x of kummer_u(a, b, x)
.
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
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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These routines are work in progress. I still need to:
ode1_factor
to work for multiple roots.
ode1_factor
to attempt to solve higher
order factors. At present it only attemps to solve linear factors.
ode1_lagrange
to prefer real roots over
complex roots.
ode1_lie
. There are quite a
few problems with it: some parts are unimplemented; some test cases
seem to run forever; other test cases crash; yet others return very
complex "solutions". I wonder if it really ready for release yet.
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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.
ode1_clairault
finds all known solutions,
including singular solutions, of the Clairault equations in Murphy and
Kamke.
ode1_lie
are overly complex and
impossible to check.
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50.1 Introduction to descriptive | ||
50.2 Functions and Variables for data manipulation | ||
50.3 Functions and Variables for descriptive statistics | ||
50.4 Functions and Variables for statistical graphs |
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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 ]
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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) $
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]]
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]]
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.
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
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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This is the sample mean, defined as $${\bar{x}={1\over{n}}{\sum_{i=1}^{n}{x_{i}}}}$$
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]
This is the sample variance, defined as $${{1}\over{n}}{\sum_{i=1}^{n}{(x_{i}-\bar{x})^2}}$$
Example:
(%i1) load ("descriptive")$ (%i2) s1 : read_list (file_search ("pidigits.data"))$ (%i3) var (s1), numer; (%o3) 8.425899999999999
See also function var1
.
This is the sample variance, defined as $${{1\over{n-1}}{\sum_{i=1}^{n}{(x_{i}-\bar{x})^2}}}$$
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
.
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]
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]
The non central moment of order k, defined as $${{1\over{n}}{\sum_{i=1}^{n}{x_{i}^k}}}$$
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
.
The central moment of order k, defined as $${{1\over{n}}{\sum_{i=1}^{n}{(x_{i}-\bar{x})^k}}}$$
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
.
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]
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
.
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
.
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]
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
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]
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
.
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
.
The mean deviation, defined as $${{1\over{n}}{\sum_{i=1}^{n}{|x_{i}-\bar{x}|}}}$$
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
.
The median deviation, defined as
$${{1\over{n}}{\sum_{i=1}^{n}{|x_{i}-med|}}}$$
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
.
The harmonic mean, defined as $${{n}\over{\sum_{i=1}^{n}{{{1}\over{x_{i}}}}}}$$
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
.
The geometric mean, defined as $$\left(\prod_{i=1}^{n}{x_{i}}\right)^{{{1}\over{n}}}$$
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
.
The kurtosis coefficient, defined as $${{1\over{n s^4}}{\sum_{i=1}^{n}{(x_{i}-\bar{x})^4}}-3}$$
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]
The skewness coefficient, defined as $${{1\over{n s^3}}{\sum_{i=1}^{n}{(x_{i}-\bar{x})^3}}}$$
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]
Pearson's skewness coefficient, defined as $${{3\,\left(\bar{x}-med\right)}\over{s}}$$ 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]
The quartile skewness coefficient, defined as $${{c_{{{3}\over{4}}}-2\,c_{{{1}\over{2}}}+c_{{{1}\over{4}}}}\over{c _{{{3}\over{4}}}-c_{{{1}\over{4}}}}}$$ 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
.
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
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)) $
The covariance matrix of the multivariate sample, defined as $${S={1\over{n}}{\sum_{j=1}^{n}{\left(X_{j}-\bar{X}\right)\,\left(X_{j}-\bar{X}\right)'}}}$$ 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
.
The covariance matrix of the multivariate sample, defined as $${{1\over{n-1}}{\sum_{j=1}^{n}{\left(X_{j}-\bar{X}\right)\,\left(X_{j}-\bar{X}\right)'}}}$$ 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
returns a list of global variance measures:
trace(S_1)
,
trace(S_1)/p
,
determinant(S_1)
,
sqrt(determinant(S_1))
,
determinant(S_1)^(1/p)
, (defined in: Peña, D. (2002) Análisis de datos multivariantes; McGraw-Hill, Madrid.)
determinant(S_1)^(1/(2*p))
.
where p is the dimension of the multivariate random variable and S_1 the covariance matrix returned by cov1
.
Option:
'data
, default 'true
, indicates whether the input matrix contains the sample data,
in which case the covariance matrix cov1
must be calculated, or not, and then the covariance
matrix (symmetric) must be given, instead of the data.
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]
The correlation matrix of the multivariate sample.
Option:
'data
, default 'true
, indicates whether the input matrix contains the sample data,
in which case the covariance matrix cov1
must be calculated, or not, and then the covariance
matrix (symmetric) must be given, instead of the data.
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 ]
Function list_correlations
returns a list of correlation measures:
Option:
'data
, default 'true
, indicates whether the input matrix contains the sample data,
in which case the covariance matrix cov1
must be calculated, or not, and then the covariance
matrix (symmetric) must be given, instead of the data.
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 ]
Calculates the principal componentes of a multivariate sample. Principal components are used in multivariate statistical analysis to reduce the dimensionality of the sample.
Option:
'data
, default 'true
, indicates whether the input matrix contains the sample data,
in which case the covariance matrix cov1
must be calculated, or not, and then the covariance
matrix (symmetric) must be given, instead of the data.
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 ]
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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:
3/4
): relative width of rectangles. This
value must be in the range [0,1]
.
clustered
): indicates how multiple samples are
shown. Valid values are: clustered
and stacked
.
1
): a positive integer number representing
the gap between two consecutive groups of bars.
[]
): a list of colors for multiple samples.
When there are more samples than specified colors, the extra necessary colors
are chosen at random. See color
to learn more about them.
absolute
): indicates the scale of the
ordinates. Possible values are: absolute
, relative
,
and percent
.
orderlessp
): possible values are orderlessp
or ordergreatp
,
indicating how statistical outcomes should be ordered on the x-axis.
[]
): a list with the strings to be used in the legend.
When the list length is other than 0 or the number of samples, an error message is returned.
0
): indicates where the plot begins to be plotted on the
x axis.
draw
options, except xtics
, which is
internally assigned by barsplot
.
If you want to set your own values for this option or want to build
complex scenes, make use of barsplot_description
. See example below.
key
, color_draw
,
fill_color
, fill_density
and line_width
.
See also
barsplot
.
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
See also functions histogram
and piechart
.
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))$
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:
3/4
): relative width of boxes.
This value must be in the range [0,1]
.
vertical
): possible values: vertical
and horizontal
.
inf
): positive coefficient of the interquartilic range
to set outliers boundaries.
1
): circle size for isolated outliers.
draw
options, except points_joined
, point_size
, point_type
,
xtics
, ytics
, xrange
, and yrange
, which are
internally assigned by boxplot
.
If you want to set your own values for this options or want to build
complex scenes, make use of boxplot_description
.
draw
options: key
, color
,
and line_width
.
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
creates a graphic object
suitable for creating complex scenes, together with other
graphic objects.
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:
10
): number of classes of the histogram, or
a list indicating the limits of the classes and the number of them, or
only the limits. This option also accepts bounds for varying bin widths, or
a symbol with the name of one of the three optimal algorithms available for
the number of classes: 'fd
(Freedman, D. and Diaconis, P. (1981) On the
histogram as a density estimator: L_2 theory. Zeitschrift fuer
Wahrscheinlichkeitstheorie und verwandte Gebiete 57, 453-476.), 'scott
(Scott, D. W. (1979) On optimal and data-based histograms. Biometrika 66,
605-610.), and 'sturges
(Sturges, H. A. (1926) The choice of a class
interval. Journal of the American Statistical Association 21, 65-66).
absolute
): indicates the scale of the
ordinates. Possible values are: absolute
, relative
, percent
,
and density
. With density
, the histogram area has a total area of one.
auto
): format of the histogram tics. Possible
values are: auto
, endpoints
, intervals
, or a list
of labels.
draw
options, except xrange
, yrange
,
and xtics
, which are internally assigned by histogram
.
If you want to set your own values for these options, make use of
histogram_description
. See examples bellow.
key
, color
,
fill_color
, fill_density
and line_width
. See also
barsplot
.
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
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))$
Similar to barsplot
, but plots sectors instead of rectangles.
Available options are:
[]
): a list of colors for sectors.
When there are more sectors than specified colors, the extra necessary colors
are chosen at random. See color
to learn more about them.
[0,0]
): diagram's center.
1
): diagram's radius.
draw
options, except key
, which is
internally assigned by piechart
.
If you want to set your own values for this option or want to build
complex scenes, make use of piechart_description
.
draw
options: key
, color
,
fill_density
and line_width
. See also
ellipse
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
creates a graphic object
suitable for creating complex scenes, together with other
graphic objects.
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
creates a graphic object
suitable for creating complex scenes, together with other
graphic objects.
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:
[]
): a list of colors for multiple samples.
When there are more samples than specified colors, the extra necessary colors
are chosen at random. See color
to learn more about them.
absolute
): indicates the scale of the
radii. Possible values are: absolute
and relative
.
orderlessp
): possible values are orderlessp
or ordergreatp
,
indicating how statistical outcomes should be ordered.
[]
): a list with the strings to be used in the legend.
When the list length is other than 0 or the number of samples, an error message is returned.
[0,0]
): diagram's center.
1
): diagram's radius.
draw
options, except points_joined
, point_type
,
and key
, which are internally assigned by starplot
.
If you want to set your own values for this options or want to build
complex scenes, make use of starplot_description
.
draw
option: line_width
.
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
creates a graphic object
suitable for creating complex scenes, together with other
graphic objects.
Plots stem and leaf diagrams.
Unique available option is:
1
): indicates the unit of the leaves; must be a
power of 10.
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
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51.1 Functions and Variables for diag |
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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")
.
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")
.
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
.
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
.
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
.
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
.
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")
.
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52.1 Introduction to distrib | ||
52.2 Functions and Variables for continuous distributions | ||
52.3 Functions and Variables for discrete distributions |
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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'.
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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")
.
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
.
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
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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
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")
.
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")
.
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
Returns the standard deviation of a Student random variable t(n), with n>2. To make use of this function, write first load("distrib")
.
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")
.
Returns the kurtosis coefficient of a Student random variable t(n), with n>4. To make use of this function, write first load("distrib")
.
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")
.
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
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
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")
.
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
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")
.
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")
.
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")
.
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")
.
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")
.
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
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
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
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
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
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)
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)
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
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")
.
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")
.
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")
.
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.
Returns the mean of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
Returns the variance of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
Returns the standard deviation of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
Returns the skewness coefficient of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
Returns the kurtosis coefficient of a noncentral Chi-square random variable nc_Chi^2(n,ncp), with n>0 and noncentrality parameter ncp>=0.
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")
.
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")
.
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
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
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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
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
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
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
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
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
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
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
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")
.
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")
.
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
.
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
Returns the mean of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load("distrib")
.
Returns the variance of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load("distrib")
.
Returns the standard deviation of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load("distrib")
.
Returns the skewness coefficient of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load("distrib")
.
Returns the kurtosis coefficient of a Lognormal(m,s) random variable, with s>0. To make use of this function, write first load("distrib")
.
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")
.
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")
.
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
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")
.
Returns the mean of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib")
.
Returns the variance of a Gamma(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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
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")
.
Returns the mean of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib")
.
Returns the variance of a Beta(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
Returns the mean of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first load("distrib")
.
Returns the variance of a Continuous Uniform(a,b) random variable, with a<b. To make use of this function, write first load("distrib")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
Returns the mean of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib")
.
Returns the variance of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib")
.
Returns the standard deviation of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib")
.
Returns the skewness coefficient of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib")
.
Returns the kurtosis coefficient of a Logistic(a,b) random variable , with b>0. To make use of this function, write first load("distrib")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
Returns the mean of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib")
.
Returns the variance of a Weibull(a,b) random variable, with a,b>0. To make use of this function, write first load("distrib")
.
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")
.
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")
.
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")
.
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")
.
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
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
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
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
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
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
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
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
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")
.
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")
.
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")
.
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")
.
Returns the mean of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib")
.
Returns the variance of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib")
.
Returns the standard deviation of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib")
.
Returns the skewness coefficient of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib")
.
Returns the kurtosis coefficient of a Laplace(a,b) random variable, with b>0. To make use of this function, write first load("distrib")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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
.
Returns the variance of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first load("distrib")
.
Returns the standard deviation of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first load("distrib")
.
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.
Returns the kurtosis coefficient of a Gumbel(a,b) random variable, with b>0. To make use of this function, write first load("distrib")
.
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")
.
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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
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
Returns the q-quantile of a general finite discrete random variable, with vector probabilities v.
See pdf_general_finite_discrete
for more details.
Returns the mean of a general finite discrete random variable, with vector probabilities v.
See pdf_general_finite_discrete
for more details.
Returns the variance of a general finite discrete random variable, with vector probabilities v.
See pdf_general_finite_discrete
for more details.
Returns the standard deviation of a general finite discrete random variable, with vector probabilities v.
See pdf_general_finite_discrete
for more details.
Returns the skewness coefficient of a general finite discrete random variable, with vector probabilities v.
See pdf_general_finite_discrete
for more details.
Returns the kurtosis coefficient of a general finite discrete random variable, with vector probabilities v.
See pdf_general_finite_discrete
for more details.
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]
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")
.
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
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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
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")
.
Returns the mean of a Poisson(m) random variable, with m>0. To make use of this function, write first load("distrib")
.
Returns the variance of a Poisson(m) random variable, with m>0. To make use of this function, write first load("distrib")
.
Returns the standard deviation of a Poisson(m) random variable, with m>0. To make use of this function, write first load("distrib")
.
Returns the skewness coefficient of a Poisson(m) random variable, with m>0. To make use of this function, write first load("distrib")
.
Returns the kurtosis coefficient of a Poisson random variable Poi(m), with m>0. To make use of this function, write first load("distrib")
.
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")
.
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
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")
.
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")
.
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
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
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)
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)
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
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")
.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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")
.
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53.1 Introduction to draw | ||
53.2 Functions and Variables for draw | ||
53.3 Functions and Variables for pictures | ||
53.4 Functions and Variables for worldmap |
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draw
is a Maxima-Gnuplot and a Maxima-vtk interface.
There are three main functions to be used at Maxima level:
draw2d
, draws a single 2D scene.
draw3d
, draws a single 3D scene.
draw
, can be filled with multiple gr2d
and gr3d
commands that each creates a draw scene all sharing the same window.
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:
explicit
plots a function.
implicit
plots all points an equation is true at.
points
plots points that are connected by lines if the option
points_joined
was set to true
in a previous line of the
current scene.
parametric
allows to specify separate expressions that calculate
the x, y (and in 3d plots also for the z) variable.
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 section 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) )$
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.
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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
).
(%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)]
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
).
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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.
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)$
(%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)$
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)]See also
gr2d
, gr3d
, draw2d
and draw3d
.
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.
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.
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]) $
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)$
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))$
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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
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) ) $
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) ))$
See also option columns
.
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) )$
See also axis_bottom
, axis_left
, axis_top
, and axis_right
for axis in 2d.
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))$
See also axis_left
, axis_top
, axis_right
and axis_3d
.
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
.
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
.
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
.
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
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]]) )$
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) )$
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)) $
See also enhanced3d
, colorbox
and 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)) $
See also enhanced3d
, colorbox
and cbrange
.
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]) )$
(%i1) draw2d( line_width=50, color="#FF0000", explicit(sin(x),x,0,10), color="#0000FF80", explicit(cos(x),x,0,10) );
(%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 ) )$
See also fill_color
.
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)) )$Color scale and 3D coloured object.
(%i1) draw3d( colorbox = "Magnitude", enhanced3d = true, explicit(x^2+y^2,x,-1,1,y,-1,1))$
See also palette_draw
.
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)$
Default value: none
Option contour
enables the user to select where to plot contour lines.
Possible values are:
none
:
no contour lines are plotted.
base
:
contour lines are projected on the xy plane.
surface
:
contour lines are plotted on the surface.
both
:
two contour lines are plotted: on the xy plane and on the surface.
map
:
contour lines are projected on the xy plane, and the view point is
set just in the vertical.
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) $
(%i1) draw3d(explicit(20*exp(-x^2-y^2)-10,x,0,2,y,-3,3), contour_levels = 15, contour = map ) $
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:
contour_levels
is bounded to positive integer n,
n contour lines will be drawn at equal intervals. By default, five
equally spaced contours are plotted.
contour_levels
is bounded to a list of length three of the
form [lowest,s,highest]
, contour lines are plotted from lowest
to highest
in steps of s
.
contour_levels
is bounded to a set of numbers of the
form {n1, n2, ...}
, contour lines are plotted at values n1
,
n2
, ...
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
.
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
.
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
.
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)) $
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) );
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))$
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))$
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)) $
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))$
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");
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) )$
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) )$
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)$
(%i3) draw3d( point_type = filled_circle, point_size = 2, enhanced3d = [u+v-w,u,v,w], points(m) ) $
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)) $
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
.
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
.
Default value: "red"
fill_color
specifies the color for filling polygons and
2d explicit
functions.
See color
to learn how colors are specified.
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.
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) )$
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));See also
fill_color
and explicit
.
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:
Example:
(%i1) draw2d(font = "Arial", font_size = 20, label(["Arial font, size 20",1,1]))$
GDFONTPATH
; in this case, it is only necessary to
set option font
to the font's name. It is also possible to
give the complete path to the font file.
Examples:
Option font
can be given the complete path to the font file:
(%i1) path: "/usr/share/fonts/truetype/freefont/" $ (%i2) file: "FreeSerifBoldItalic.ttf" $ (%i3) draw2d( font = concat(path, file), font_size = 20, color = red, label(["FreeSerifBoldItalic font, size 20",1,1]), terminal = png)$
If environment variable GDFONTPATH
is set to the
path where font files are allocated, it is possible to
set graphic option font
to the name of the font.
(%i1) draw2d( font = "FreeSerifBoldItalic", font_size = 20, color = red, label(["FreeSerifBoldItalic font, size 20",1,1]), terminal = png)$
"Times-Roman"
, "Times-Italic"
, "Times-Bold"
,
"Times-BoldItalic"
,"Helvetica"
, "Helvetica-Oblique"
, "Helvetica-Bold"
,"Helvetic-BoldOblique"
, "Courier"
,
"Courier-Oblique"
, "Courier-Bold"
,"Courier-BoldOblique"
.
Example:
(%i1) draw2d( font = "Courier-Oblique", font_size = 15, label(["Courier-Oblique font, size 15",1,1]), terminal = eps)$
fontconfig
utility.
"Times-Roman"
.
The gnuplot documentation is an important source of information about terminals and fonts.
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
.
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
.
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))$
(%i1) draw2d(grid = [2,2], explicit(sin(x),x,0,2*%pi))$
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]) )$
See also head_both
, head_length
, and head_type
.
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]) )$
See also head_length
, head_angle
, and head_type
.
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]))$
See also head_both
, head_angle
, and 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]))$
See also head_both
, head_angle
, and head_length
.
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)) $
(%i2) draw3d( interpolate_color = true, enhanced3d = sin(x*y), explicit(20*exp(-x^2-y^2)-10, x ,-3, 3, y, -3, 3)) $
(%i3) draw3d( interpolate_color = [-10,0], enhanced3d = sin(x*y), explicit(20*exp(-x^2-y^2)-10, x ,-3, 3, y, -3, 3)) $
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]])) $
(%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]])) $
(%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]])) $
See also enhanced3d
.
Default value: [50, 50]
ip_grid
sets the grid for the first sampling in implicit plots.
This option is relevant only for implicit
objects.
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.
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:
gr2d
: points
, polygon
, rectangle
,
ellipse
, vector
, explicit
, implicit
,
parametric
and polar
.
gr3d
: points
, explicit
, parametric
and parametric_surface
.
Example:
(%i1) draw2d(key = "Sinus", explicit(sin(x),x,0,10), key = "Cosinus", color = red, explicit(cos(x),x,0,10) )$
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))$
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]))$
See also label_orientation
, and color
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]))$
See also label_alignment
and color
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:
gr2d
: points
, polygon
, rectangle
,
ellipse
, vector
, explicit
, implicit
,
parametric
and polar
.
gr3d
: points
, explicit
, parametric
and parametric_surface
.
Example:
(%i1) draw2d(line_type = dots, explicit(1 + x^2,x,-1,1), line_type = solid, /* default */ explicit(2 + x^2,x,-1,1))$
See also 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:
gr2d
: points
, polygon
, rectangle
,
ellipse
, vector
, explicit
, implicit
,
parametric
and polar
.
gr3d
: points
and parametric
.
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))$
See also line_type
.
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)) $
See also enhanced3d
, colorbox
and cbrange
.
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
.
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) )$
See also logx_draw
, logy_draw
, logy_secondary
, and logz
.
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
.
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
.
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))$
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:
gr2d
: ellipse
, explicit
, parametric
and polar
.
gr3d
: parametric
.
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) )$
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))$
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))$
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)) $
See also colorbox
and enhanced3d
.
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)))$
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: $none
(-1), dot
(0), plus
(1), multiply
(2),
asterisk
(3), square
(4), filled_square
(5), circle
(6),
filled_circle
(7), up_triangle
(8), filled_up_triangle
(9),
down_triangle
(10), filled_down_triangle
(11), diamant
(12) and
filled_diamant
(13).
This option affects the following graphic objects:
Example:
(%i1) draw2d(xrange = [0,10], yrange = [0,10], point_size = 3, point_type = diamant, points([[1,1],[5,1],[9,1]]), point_type = filled_down_triangle, points([[1,2],[5,2],[9,2]]), point_type = asterisk, points([[1,3],[5,3],[9,3]]), point_type = filled_diamant, points([[1,4],[5,4],[9,4]]), point_type = 5, points([[1,5],[5,5],[9,5]]), point_type = 6, points([[1,6],[5,6],[9,6]]), point_type = filled_circle, points([[1,7],[5,7],[9,7]]), point_type = 8, points([[1,8],[5,8],[9,8]]), point_type = filled_diamant, points([[1,9],[5,9],[9,9]]) )$
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]]) )$
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) $
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)))$
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)) )$
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
.
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")$
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]) )$
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) )$
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]]) )$
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]))$
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))$
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) );
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) )$
(%i2) draw3d(view = map, enhanced3d = true, explicit(sin(x^2+y^2),x,-2,2,y,-2,2) )$
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)) $
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
.
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)$
See also xaxis_width
, xaxis_type
and 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
.
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)) $
See also xrange_secondary
, xtics_secondary
, xtics_rotate_secondary
,
xtics_axis_secondary
and xaxis_secondary
.
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
.
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
.
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
.
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) )$
See also xlabel_draw
, ylabel_draw
, ylabel_secondary
and zlabel_draw
.
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))$
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
.
Default value: true
This graphic option controls the way tic marks are drawn on the x axis.
xtics
is bounded to symbol true, tic marks are
drawn automatically.
xtics
is bounded to symbol false, tic marks are
not drawn.
xtics
is bounded to a positive number, this is the distance
between two consecutive tic marks.
xtics
is bounded to a list of length three of the
form [start,incr,end]
, tic marks are plotted from start
to end
at intervals of length incr
.
xtics
is bounded to a set of numbers of the
form {n1, n2, ...}
, tic marks are plotted at values n1
,
n2
, ...
xtics
is bounded to a set of pairs of the
form {["label1", n1], ["label2", n2], ...}
, tic marks corresponding to values n1
,
n2
, ... are labeled with "label1"
, "label2"
, ..., respectively.
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) )$
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.
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.
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.
Default value: auto
This graphic option controls the way tic marks are drawn on the second x axis.
See xtics
for a complete description.
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.
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:
gr3d
: explicit
and parametric_surface
.
Example:
(%i1) draw3d(xu_grid = 10, yv_grid = 50, explicit(x^2+y^2,x,-3,3,y,-3,3) )$
See also yv_grid
.
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.
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))$
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
.
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
.
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
.
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
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
.
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
.
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
.
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
.
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
.
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)) $
Default value: true
This graphic option controls the way tic marks are drawn on the y axis.
See xtics
for a complete description.
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.
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.
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.
Default value: auto
This graphic option controls the way tic marks are drawn on the second y axis.
See xtics
for a complete description.
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.
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:
gr3d
: explicit
and parametric_surface
.
Example:
(%i1) draw3d(xu_grid = 10, yv_grid = 50, explicit(x^2+y^2,x,-3,3,y,-3,3) )$
See also xu_grid
.
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.
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
.
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
.
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
.
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
.
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
.
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
.
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])$
Default value: true
This graphic option controls the way tic marks are drawn on the z axis.
See xtics
for a complete description.
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.
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.
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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);
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))$
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);
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] )$
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]]))$
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]]));
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) )$
(%i2) draw2d(fill_color = brown, filled_func = true, explicit(x^2,x,-3,3) )$
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)$
See also filled_func
for filled functions.
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))$
(%i3) draw2d(palette = gray, image(im,0,0,30,30))$
(%i4) draw2d(palette = [15,20,-4], colorbox=false, image(im,0,0,30,30))$
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))$
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))$
(%i3) /* gray palette */ draw2d(palette = gray, image(im,0,0,30,30))$
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.
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" )$
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);
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]) )$
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]) )$
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]])) $
Plotting a triangle in 3D.
(%i1) draw3d( line_width = 2, mesh([[1,0,0],[0,1,0]], [[0,0,1],[0,0,1]])) $
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]])) $
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))$
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" )$
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) )$
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]))$
Drawing impulses.
(%i1) draw2d( points_joined = impulses, line_width = 2, color = red, points(makelist([random(20),random(50)],k,1,10)))$
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)) $
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)) $
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)) $
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)) $
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)) $
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))$
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) )$
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]]) )$
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]))$
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
.
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] )$
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));
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))$
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]))$
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
.
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) )$
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]))$
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)) )$
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Returns pixel from picture. Coordinates x and y range from 0 to
width-1
and height-1
, respectively.
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:
level
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)})
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:
rgb
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)})
Returns the negative of a (level or rgb) picture.
Returns true
in case of equal pictures, and false
otherwise.
Returns true
if the argument is a well formed image,
and false
otherwise.
Reads a file in xpm and returns a picture object.
Transforms an rgb picture into a level one by averaging the red, green and blue channels.
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)})
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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
.
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)$
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, %)$
(%i4) /* A list of countries */ make_poly_continent([Germany,Denmark,Poland])$ (%i5) apply(draw2d, %)$
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, %)$
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)$
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(%))$
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(%))$
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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]))$
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)$
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)$
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:
[spherical_projection,x,y,z,r]
: projects map boundaries on the sphere of
radius r centered at (x,y,z).
(%i1) load(worldmap)$ (%i2) draw3d(geomap(Australia), /* default projection */ geomap(Australia, [spherical_projection,2,2,2,3]))$
[cylindrical_projection,x,y,z,r,rc]
: re-projects spherical map boundaries on the cylinder of radius
rc and axis passing through the poles of the globe of radius r centered at (x,y,z).
(%i1) load(worldmap)$ (%i2) draw3d(geomap([America_coastlines,Eurasia_coastlines], [cylindrical_projection,2,2,2,3,4]))$
[conic_projection,x,y,z,r,alpha]
: re-projects spherical map boundaries on the cones of angle alpha,
with axis passing through the poles of the globe of radius r centered at (x,y,z). Both
the northern and southern cones are tangent to sphere.
(%i1) load(worldmap)$ (%i2) draw3d(geomap(World_coastlines, [conic_projection,0,0,0,1,90]))$
See also http://riotorto.users.sf.net/gnuplot/geomap for more elaborated examples.
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54.1 Introduction to drawdf | ||
54.2 Functions and Variables for drawdf |
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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.
Category: Differential equations · Category: Plotting · Category: Share packages · Category: Package drawdf · Category: Package draw
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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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55.1 The dynamics package | ||
55.2 Graphical analysis of discrete dynamical systems | ||
55.3 Visualization with VTK |
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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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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]);
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);
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]);
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]);
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]);
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]);
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]);
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]);
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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).
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"])$
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)$
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.
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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.
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).
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.
Default value: 500
The height, in pixels, of the graphics window. pixels must be a positive integer number.
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].
Default value: 10
The amount of time, in mili-seconds, between iterations among consecutive animation frames. time must be a real number.
Default value: 500
The width, in pixels, of the graphics window. pixels must be a positive integer number.
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.
Default value: Xmaxima: scene
name must be a string that will be written in the title of the
window created by scene
.
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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)$
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
.
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.
A sphere with default radius of 0.5 units and center at the origin.
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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
.
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.
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.
Default value: white
The color of the object. It accepts color names or hexadecimal
red-green-blue strings (see the color
option of plot2d).
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
.
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
.
Default value: 1
value must be a positive number which sets the height of a cone or a cylinder.
Default value: 1
The width of the lines, when option wireframe
is
used. value must be a positive number.
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.
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.
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.
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.
Only the vertices of the triangulation used to render the surface will
be shown. Example: [sphere, [points]]
Default value: 1
The size of the points, when option points
is
used. value must be a positive number.
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.
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.
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.
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.
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
.
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
.
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
.
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
.
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
.
Default value: 1
The height of a cube in the x direction. length must be a positive
number. See also ylength
and
zlength
.
Default value: 1
The height of a cube in the y direction. length must be a positive
number. See also xlength
and
zlength
.
Default value: 1
The height of a cube in z the direction. length must be a positive
number. See also xlength
and
ylength
.
Only the edges of the triangulation used to render the surface will be
shown. Example: [cube, [wireframe]]
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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.
56.1 Functions and Variables for engineering-format | ||
56.2 Known Bugs |
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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
.
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]
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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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.
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
57.1 Introduction to ezunits | ||
57.2 Introduction to physical_constants | ||
57.3 Functions and Variables for ezunits |
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
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.
y
is nondimensional.
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.
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
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
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
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.
y
is nondimensional.
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
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
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
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
Reverts the effect of declare_constvalue
. This function should be
loaded with load ("ezunits")
.
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
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
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
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
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
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
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
Reverts the effect of declare_dimensions
. This function should be
loaded with load ("ezunits")
.
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]
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
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]
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
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
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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58.1 Package f90 |
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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
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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59.1 Introduction to finance | ||
59.2 Functions and Variables for finance |
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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.
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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
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
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
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])$
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
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
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
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
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
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
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
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
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
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
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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60.1 Introduction to fractals | ||
60.2 Definitions for IFS fractals | ||
60.3 Definitions for complex fractals | ||
60.4 Definitions for Koch snowflakes | ||
60.5 Definitions for Peano maps |
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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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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
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])$
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])$
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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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])$
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
.
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
.
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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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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Continuous curves that cover an area. Warning: the number of points exponentially grows with n.
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)])$
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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61.1 Functions and Variables for ggf |
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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
.
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
.
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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62.1 Introduction to graphs | ||
62.2 Functions and Variables for graphs |
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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")
.
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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
Returns a copy of the graph g.
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
Returns the Clebsch graph.
Returns the complement of the graph g.
Returns the complete bipartite graph on n+m vertices.
Returns the complete graph on n vertices.
Returns the directed cycle on n vertices.
Returns the cycle on n vertices.
Returns the cuboctahedron graph.
Returns the n-dimensional cube.
Returns the dodecahedron graph.
Returns the empty graph on n vertices.
Returns the flower graph on 4n vertices.
Example:
(%i1) load ("graphs")$ (%i2) f5 : flower_snark(5)$ (%i3) chromatic_index(f5); (%o3) 4
Returns the graph represented by its adjacency matrix A.
Returns the Frucht graph.
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)$
Returns the union (sum) of graphs g1 and g2.
Returns the n x m grid.
Returns the great rhombicosidodecahedron graph.
Returns the great rhombicuboctahedron graph.
Returns the Grotzch graph.
Returns the Heawood graph.
Returns the icosahedron graph.
Returns the icosidodecahedron graph.
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
Returns the line graph of the graph g.
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]
Returns the mycielskian graph of the graph g.
Returns the graph with no vertices and no edges.
Returns the directed path on n vertices.
Returns the path on n vertices.
Returns the petersen graph P_{n,d}. The default values for
n and d are n=5
and d=2
.
Returns a random bipartite graph on a+b
vertices. Each edge is
present with probability p.
Returns a random directed graph on n vertices. Each arc is present with probability p.
Returns a random d-regular graph on n vertices. The default
value for d is d=3
.
Returns a random graph on n vertices. Each edge is present with probability p.
Returns a random graph on n vertices and random m edges.
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
Returns a random tournament on n vertices.
Returns a random tree on n vertices.
Returns the small rhombicosidodecahedron graph.
Returns the small rhombicuboctahedron graph.
Returns the snub cube graph.
Returns the snub dodecahedron graph.
Returns the truncated cube graph.
Returns the truncated dodecahedron graph.
Returns the truncated icosahedron graph.
Returns the truncated tetrahedron graph.
Returns the Tutte graph.
Returns the underlying graph of the directed graph g.
Returns the wheel graph on n+1 vertices.
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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 ]
Returns the average degree of vertices in the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) average_degree(grotzch_graph()); 40 (%o2) -- 11
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]]
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)$
Returns the chromatic index of the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) p : petersen_graph()$ (%i3) chromatic_index(p); (%o3) 4
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
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
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
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]]
Returns the diameter of the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) diameter(dodecahedron_graph()); (%o2) 5
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
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]
Returns the edge-connectivity of the graph gr.
See also min_edge_cut
.
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]]
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
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
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)
Returns the center of the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) g : grid_graph(5,5)$ (%i3) graph_center(g); (%o3) [12]
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]]
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]
Returns the number of edges in the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) p : petersen_graph()$ (%i3) graph_size(p); (%o3) 15
Returns the number of vertices in the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) p : petersen_graph()$ (%i3) graph_order(p); (%o3) 10
Returns the length of the shortest cycle in gr.
Example:
(%i1) load ("graphs")$ (%i2) g : heawood_graph()$ (%i3) girth(g); (%o3) 6
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))$
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))$
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]
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) []
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
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
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
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
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
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
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
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
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
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
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
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
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 ]
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]
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
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
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)$
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)$
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
Returns the minimum edge cut in the graph gr.
See also edge_connectivity
.
Returns the minimum vertex cover of the graph gr.
Returns the minimum vertex cut in the graph gr.
See also vertex_connectivity
.
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))$
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]
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
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) []
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]]
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]
Returns the radius of the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) radius(dodecahedron_graph()); (%o2) 5
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
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
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))$
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]]
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
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]
Returns the vertex connectivity of the graph g.
See also min_vertex_cut
.
Returns the degree of the vertex v in the graph 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]
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
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]
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]
Returns the list of vertices in the graph gr.
Example:
(%i1) load ("graphs")$ (%i2) vertices(complete_graph(4)); (%o2) [3, 2, 1, 0]
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]]]
Returns the Wiener index of the graph gr.
Example:
(%i2) wiener_index(dodecahedron_graph()); (%o2) 500
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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]
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
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
Adds all vertices in the list v_list to the graph 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
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
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
Removes the vertex v from the graph gr.
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Exports the graph into the file fl in the DIMACS format. Optional comments will be added to the top of the file.
Returns the graph from file fl in the DIMACS format.
Returns the graph encoded in the graph6 format in the string str.
Returns a string which encodes the graph gr in the graph6 format.
Exports graphs in the list gr_list to the file fl in the graph6 format.
Returns a list of graphs from the file fl in the graph6 format.
Returns the graph encoded in the sparse6 format in the string str.
Returns a string which encodes the graph gr in the sparse6 format.
Exports graphs in the list gr_list to the file fl in the sparse6 format.
Returns a list of graphs from the file fl in the sparse6 format.
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Draws the graph using the draw 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)$
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 )$
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 )$
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 )$
Example 5:
(%i1) load("graphs")$ (%i2) g: petersen_graph(20, 2); (%o2) GRAPH (%i3) draw_graph(g, redraw=true, program=planar_embedding); (%o3) done
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
Default value: spring_embedding
The default value for the program used to position vertices in
draw_graph
program.
Default value: false
If true then ids of the vertices are displayed.
Default value: false
If true then labels of the vertices are displayed.
Default value: center
Determines how to align the labels/ids of the vertices. Can
be left
, center
or right
.
Default value: false
If true then weights of the edges are displayed.
Default value: circle
Defines how vertices are displayed. See the point_type option for
the draw
package for possible values.
The size of vertices.
The color used for displaying vertices.
Default value: []
Display selected vertices in the using a different color.
Defines how vertices specified in show_vertices are displayed.
See the point_type option for the draw
package for possible
values.
The size of vertices in show_vertices.
The color used for displaying vertices in the show_vertices list.
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.
Specifies coloring of the vertices. The coloring col must be specified in the format as returned by vertex_coloring.
The color used for displaying edges.
The width of edges.
Defines how edges are displayed. See the line_type option for the
draw
package.
Display edges specified in the list e_list using a different color.
The color used for displaying edges in the show_edges list.
The width of edges in show_edges.
Defines how edges in show_edges are displayed. See the
line_type option for the draw
package.
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.
The coloring of edges. The coloring must be specified in the format as returned by the function edge_coloring.
Default value: false
If true
, vertex positions are recomputed even if the positions
have been saved from a previous drawing of the graph.
Default value: 15
The angle for the arrows displayed on arcs (in directed graphs).
Default value: 0.1
The length for the arrows displayed on arcs (in directed graphs).
Default value: 50
The number of iterations in the spring embedding graph drawing algorithm.
The terminal used for drawing (see the terminal option in the
draw
package).
The filename of the drawing if terminal is not screen.
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.
Specifies a list of vertices which will have positions fixed along a regular polygon.
Can be used when program=spring_embedding
.
Converts a list v_list of vertices to a list of edges of the path defined by v_list.
Converts a list v_list of vertices to a list of edges of the cycle defined by v_list.
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63.1 Introduction to grobner | ||
63.2 Functions and Variables for grobner |
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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.
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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 |
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lex
pure lexicographic, default order for monomial comparisons
grlex
total degree order, ties broken by lexicographic
grevlex
total degree, ties broken by reverse lexicographic
invlex
inverse lexicographic order
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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.
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.
Default value: false
Name of the default order for eliminated variables in
elimination-based functions. If not set, lex
will be used.
Default value: false
Name of the default order for kept variables in elimination-based functions. If not set, lex
will be used.
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.
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.
Default value: false
If set to true
, produce debugging and tracing output.
Default value: buchberger
Possible values:
buchberger
parallel_buchberger
gebauer_moeller
The name of the algorithm used to find the Groebner Bases.
Default value: false
If not false
, use top reduction only whenever possible. Top
reduction means that division algorithm stops after the first
reduction.
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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.
Adds two polynomials poly1 and poly2.
(%i1) poly_add(z+x^2*y,x-z,[x,y,z]); 2 (%o1) x y + x
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
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
Returns the syzygy polynomial (S-polynomial) of two polynomials poly1 and poly2.
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
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.
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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
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
poly_content
extracts the GCD of its coefficients
(%i1) poly_content(35*y+21*x,[x,y]); (%o1) 7
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.
Divide a polynomial poly1 by another polynomial poly2. Assumes that exact division with no remainder is possible. Returns the quotient.
poly_normal_form
finds the normal form of a polynomial poly with respect
to a set of polynomials polylist.
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.
poly_buchberger
performs the Buchberger algorithm on a list of
polynomials and returns the resulting Groebner basis.
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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 }.
poly_reduction
reduces a list of polynomials polylist, so that
each polynomial is fully reduced with respect to the other polynomials.
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.
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.
Returns a Groebner basis of the ideal span by the polynomials polylist. Affected by the global flags.
Returns a reduced Groebner basis of the ideal span by the polynomials polylist. Affected by the global flags.
poly_depends
tests whether a polynomial depends on a variable var.
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).
Returns the reduced Groebner basis of the colon ideal
I(polylist1):I(polylist2)
where polylist1 and polylist2 are two lists of polynomials.
poly_ideal_intersection
returns the intersection of two ideals.
Returns the lowest common multiple of poly1 and poly2.
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
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
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.
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.
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.
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.
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.
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
poly_saturation_extension
implements the famous Rabinowitz trick.
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64.1 Functions and Variables for impdiff |
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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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65.1 Introduction to interpol | ||
65.2 Functions and Variables for interpol |
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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'.
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Computes the polynomial interpolation by the Lagrangian method. Argument points must be either:
p:matrix([2,4],[5,6],[9,3])
,
p: [[2,4],[5,6],[9,3]]
,
p: [4,6,3]
, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
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
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
Computes the polynomial interpolation by the linear method. Argument points must be either:
p:matrix([2,4],[5,6],[9,3])
,
p: [[2,4],[5,6],[9,3]]
,
p: [4,6,3]
, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
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
Computes the polynomial interpolation by the cubic splines method. Argument points must be either:
p:matrix([2,4],[5,6],[9,3])
,
p: [[2,4],[5,6],[9,3]]
,
p: [4,6,3]
, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
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:
'd1
, default 'unknown
, is the first derivative at x_1; if it is 'unknown
, the second derivative at x_1 is made equal to 0 (natural cubic spline); if it is equal to a number, the second derivative is calculated based on this number.
'dn
, default 'unknown
, is the first derivative at x_n; if it is 'unknown
, the second derivative at x_n is made equal to 0 (natural cubic spline); if it is equal to a number, the second derivative is calculated based on this number.
'varname
, default 'x
, is the name of the independent variable.
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))$
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:
p:matrix([2,4],[5,6],[9,3])
,
p: [[2,4],[5,6],[9,3]]
,
p: [4,6,3]
, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.
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:
'varname
, default 'x
, is the name of the independent variable.
See also lagrange
, linearinterpol
, cspline
, minpack_lsquares
, and lbfgs
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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66.1 Introduction to lapack | ||
66.2 Functions and Variables for lapack |
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lapack
is a Common Lisp translation (via the program f2cl
) of the Fortran library LAPACK,
as obtained from the SLATEC project.
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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 ]
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
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
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 ]
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.
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 ]
Like dgeev
, but the matrix A is complex.
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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67.1 Introduction to lbfgs | ||
67.2 Functions and Variables for lbfgs |
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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
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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]
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.
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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68.1 Functions and Variables for lindstedt |
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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")
.
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69.1 Introduction to linearalgebra | ||
69.2 Functions and Variables for linearalgebra |
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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 ]
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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])
Return true if and only if M is a matrix and every entry of M is a matrix.
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.
If M is a matrix, swap columns i and j. If M doesn't have a column i or j, signal an error.
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 ()
.
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
.
Return the complex conjugate transpose of the matrix M. The function
ctranspose
uses matrix_element_transpose
to transpose each matrix
element.
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 ]
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.
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 ]
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
.
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.
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 ]
Return the n by n Hilbert matrix. When n isn't a positive integer, signal an error.
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
Invert a matrix M by using the LU factorization. The LU factorization is done using the ring rng.
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 ]
Return the Kronecker product of the matrices A and B.
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
.
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.
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 ]
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 ]
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.
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.
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
Return a two member list that gives the number of rows and columns, respectively of the matrix M.
If M is a block matrix, unblock the matrix to all levels. If M is a matrix, return M; otherwise, signal an error.
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))
.
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 ]
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 ()
.
If M is a matrix, return the dimension of the nullspace of M.
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.
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
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.
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.
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.
Return the rank of that matrix M. The rank is the dimension of the column space. Example:
(%i1) rank(matrix([1,2],[2,4])); (%o1) 1 (%i2) rank(matrix([1,b],[c,d])); Proviso: {d - b c # 0} (%o2) 2
If M is a matrix, swap rows i and j. If M doesn't have a row i or j, signal an error.
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 ]
Return a n by n matrix whose i-th row is
[1, x_i, x_i^2, ... x_i^(n-1)]
.
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
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.
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
70.1 Introduction to lsquares | ||
70.2 Functions and Variables for lsquares |
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
lsquares
is a collection of functions to implement the method of least squares
to estimate parameters for a model from numerical data.
[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
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]]
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
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]]
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
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
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
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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71.1 Introduction to minpack | ||
71.2 Functions and Variables for minpack |
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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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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]
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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72.1 Functions and Variables for makeOrders |
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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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73.1 Introduction to mnewton | ||
73.2 Functions and Variables for mnewton |
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mnewton
is an implementation of Newton's method for solving nonlinear
equations in one or more variables.
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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
.
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
.
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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74.1 Introduction to numericalio | ||
74.2 Functions and Variables for plain-text input and output | ||
74.3 Functions and Variables for binary input and output |
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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.
Category: File input · Category: File output · Category: Share packages · Category: Package numericalio
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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.
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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
.
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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
.
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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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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.
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.
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.
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.
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.
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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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.
Returns an input stream of 8-bit unsigned bytes to read the file named by file_name.
Returns an output stream of 8-bit unsigned bytes to write the file named by file_name.
Returns an output stream of 8-bit unsigned bytes to append the file named by file_name.
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
.
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
.
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
.
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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75.1 Introduction to ODEPACK | ||
75.2 Functions and Variables for odepack |
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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.
References: [1] Fortran Code is from http://netlib.org/odepack/
75.1.1 Getting Started |
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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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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
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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76.1 Introduction to operatingsystem | ||
76.2 Directory operations | ||
76.3 File operations | ||
76.4 Environment operations |
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Package operatingsystem
contains functions for operatingsystem-tasks, like file system operations.
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Change to directory dir
Create directory dir
remove directory dir
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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copies file file1 to file2
renames file file1 to file2
deletes file file1
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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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77.1 Functions and Variables for opsubst |
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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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78.1 Introduction to orthogonal polynomials | ||
78.2 Functions and Variables for orthogonal polynomials |
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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).
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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
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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
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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)
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orthopoly
To plot expressions that involve the orthogonal polynomials, you must do two things:
orthopoly_returns_intervals
to false
,
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)
The entire expression legendre_p (5, x)
is quoted; this is
different than just quoting the function name using 'legendre_p (5, x)
.
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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.
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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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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.
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.
The Chebyshev polynomial of the first kind of degree n.
Reference: Abramowitz and Stegun, equation 22.5.47, page 779.
The Chebyshev polynomial of the second kind of degree n.
Reference: Abramowitz and Stegun, equation 22.5.48, page 779.
The generalized Laguerre polynomial of degree n.
Reference: Abramowitz and Stegun, equation 22.5.54, page 780.
The Hermite polynomial of degree n.
Reference: Abramowitz and Stegun, equation 22.5.55, page 780.
Return true
if the input is an interval and return false if it isn't.
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.
The Laguerre polynomial of degree n.
Reference: Abramowitz and Stegun, equations 22.5.16 and 22.5.54, page 780.
The Legendre polynomial of the first kind of degree n.
Reference: Abramowitz and Stegun, equations 22.5.50 and 22.5.51, page 779.
The Legendre function of the second kind of degree n.
Reference: Abramowitz and Stegun, equations 8.5.3 and 8.1.8.
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);
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.
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.
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
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.
The spherical Bessel function of the first kind.
Reference: Abramowitz and Stegun, equations 10.1.8, page 437 and 10.1.15, page 439.
The spherical Bessel function of the second kind.
Reference: Abramowitz and Stegun, equations 10.1.9, page 437 and 10.1.15, page 439.
The spherical Hankel function of the first kind.
Reference: Abramowitz and Stegun, equation 10.1.36, page 439.
The spherical Hankel function of the second kind.
Reference: Abramowitz and Stegun, equation 10.1.17, page 439.
The spherical harmonic function.
Reference: Merzbacher 9.64.
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
.
The ultraspherical polynomial (also known as the Gegenbauer polynomial).
Reference: Abramowitz and Stegun, equation 22.5.46, page 779.
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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
.
79.1 Functions and Variables for ratpow |
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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
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
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)]]
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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80.1 Functions and Variables for romberg |
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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
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
.
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
.
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
.
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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81.1 Introduction to simplex | ||
81.2 Functions and Variables for simplex |
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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
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There are some tests in the directory share/simplex/Tests
.
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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));
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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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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
.
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
.
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
.
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
.
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
.
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.
After linear_program
returns,
pivot_count_sx
is the number of pivots in last computation.
pivot_max_sx
is the maximum number of pivots allowed by linear_program
.
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82.1 Introduction to simplification | ||
82.2 Package absimp | ||
82.3 Package facexp | ||
82.4 Package functs | ||
82.5 Package ineq | ||
82.6 Package rducon | ||
82.7 Package scifac | ||
82.8 Package sqdnst |
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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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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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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.
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
.
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
.
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.
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.
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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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)
.
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 ]
Returns the trace (sum of the diagonal elements) of matrix M.
To use this function write first load(functs)
.
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)
.
Returns true
if expr is nonzero and freeof (x, expr)
returns true
.
Returns false
otherwise.
To use this function write first load(functs)
.
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
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
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)
.
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)
.
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)
.
Returns the sum of the arithmetic series from 1 to n.
To use this function write first load(functs)
.
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)
.
Returns the Gaussian probability function
%e^(-x^2/2) / sqrt(2*%pi)
.
To use this function write first load(functs)
.
Returns the Gudermannian function
2*atan(%e^x)-%pi/2
.
To use this function write first load(functs)
.
Returns the inverse Gudermannian function
log (tan (%pi/4 + x/2))
.
To use this function write first load(functs)
.
Returns the versed sine 1 - cos (x)
.
To use this function write first load(functs)
.
Returns the coversed sine 1 - sin (x)
.
To use this function write first load(functs)
.
Returns the exsecant sec (x) - 1
.
To use this function write first load(functs)
.
Returns the haversine (1 - cos(x))/2
.
To use this function write first load(functs)
.
Returns the number of combinations of n objects taken r at a time.
To use this function write first load(functs)
.
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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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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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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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:
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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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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83.1 Introduction to solve_rec | ||
83.2 Functions and Variables for solve_rec |
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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)
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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
Default value: true
If simplify_products
is true
, solve_rec
will try to
simplify products in result.
See also: solve_rec
.
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
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
.
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
.
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
.
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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84.1 Introduction to stats | ||
84.2 Functions and Variables for inference_result | ||
84.3 Functions and Variables for stats | ||
84.4 Functions and Variables for special distributions |
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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'.
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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
.
Returns true
or false
, depending on whether obj is an
inference_result
object or not.
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]
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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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.
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:
'mean
, default 0
, is the mean value to be checked.
'alternative
, default 'twosided
, is the alternative hypothesis;
valid values are: 'twosided
, 'greater
and 'less
.
'dev
, default 'unknown
, this is the value of the standard deviation when it is
known; valid values are: 'unknown
or a positive expression.
'conflevel
, default 95/100
, confidence level for the confidence interval; it must
be an expression which takes a value in (0,1).
'asymptotic
, default false
, indicates whether it performs an exact t-test or
an asymptotic one based on the Central Limit Theorem;
valid values are true
and false
.
The output of function test_mean
is an inference_result
Maxima object
showing the following results:
'mean_estimate
: the sample mean.
'conf_level
: confidence level selected by the user.
'conf_interval
: confidence interval for the population mean.
'method
: inference procedure.
'hypotheses
: null and alternative hypotheses to be tested.
'statistic
: value of the sample statistic used for testing the null hypothesis.
'distribution
: distribution of the sample statistic, together with its parameter(s).
'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
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:
'alternative
, default 'twosided
, is the alternative hypothesis;
valid values are: 'twosided
, 'greater
and 'less
.
'dev1
, default 'unknown
, this is the value of the standard deviation
of the x1 sample when it is known; valid values are: 'unknown
or a positive expression.
'dev2
, default 'unknown
, this is the value of the standard deviation
of the x2 sample when it is known; valid values are: 'unknown
or a positive expression.
'varequal
, default false
, whether variances should be considered to be equal or not;
this option takes effect only when 'dev1
and/or 'dev2
are 'unknown
.
'conflevel
, default 95/100
, confidence level for the confidence interval; it must
be an expression which takes a value in (0,1).
'asymptotic
, default false
, indicates whether it performs an exact t-test or
an asymptotic one based on the Central Limit Theorem;
valid values are true
and false
.
The output of function test_means_difference
is an inference_result
Maxima object
showing the following results:
'diff_estimate
: the difference of means estimate.
'conf_level
: confidence level selected by the user.
'conf_interval
: confidence interval for the difference of means.
'method
: inference procedure.
'hypotheses
: null and alternative hypotheses to be tested.
'statistic
: value of the sample statistic used for testing the null hypothesis.
'distribution
: distribution of the sample statistic, together with its parameter(s).
'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
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:
'mean
, default 'unknown
, is the population's mean, when it is known.
'alternative
, default 'twosided
, is the alternative hypothesis;
valid values are: 'twosided
, 'greater
and 'less
.
'variance
, default 1
, this is the variance value (positive) to be checked.
'conflevel
, default 95/100
, confidence level for the confidence interval; it must
be an expression which takes a value in (0,1).
The output of function test_variance
is an inference_result
Maxima object
showing the following results:
'var_estimate
: the sample variance.
'conf_level
: confidence level selected by the user.
'conf_interval
: confidence interval for the population variance.
'method
: inference procedure.
'hypotheses
: null and alternative hypotheses to be tested.
'statistic
: value of the sample statistic used for testing the null hypothesis.
'distribution
: distribution of the sample statistic, together with its parameter.
'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
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:
'alternative
, default 'twosided
, is the alternative hypothesis;
valid values are: 'twosided
, 'greater
and 'less
.
'mean1
, default 'unknown
, when it is known, this is the mean of
the population from which x1 was taken.
'mean2
, default 'unknown
, when it is known, this is the mean of
the population from which x2 was taken.
'conflevel
, default 95/100
, confidence level for the confidence interval of the
ratio; it must be an expression which takes a value in (0,1).
The output of function test_variance_ratio
is an inference_result
Maxima object
showing the following results:
'ratio_estimate
: the sample variance ratio.
'conf_level
: confidence level selected by the user.
'conf_interval
: confidence interval for the variance ratio.
'method
: inference procedure.
'hypotheses
: null and alternative hypotheses to be tested.
'statistic
: value of the sample statistic used for testing the null hypothesis.
'distribution
: distribution of the sample statistic, together with its parameters.
'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
Inferences on a proportion. Argument x is the number of successes in n trials in a Bernoulli experiment with unknown probability.
Options:
'proportion
, default 1/2
, is the value of the proportion to be checked.
'alternative
, default 'twosided
, is the alternative hypothesis;
valid values are: 'twosided
, 'greater
and 'less
.
'conflevel
, default 95/100
, confidence level for the confidence interval; it must
be an expression which takes a value in (0,1).
'asymptotic
, default false
, indicates whether it performs an exact test
based on the binomial distribution, or an asymptotic one based on the Central Limit Theorem;
valid values are true
and false
.
'correct
, default true
, indicates whether Yates correction is applied or not.
The output of function test_proportion
is an inference_result
Maxima object
showing the following results:
'sample_proportion
: the sample proportion.
'conf_level
: confidence level selected by the user.
'conf_interval
: Wilson confidence interval for the proportion.
'method
: inference procedure.
'hypotheses
: null and alternative hypotheses to be tested.
'statistic
: value of the sample statistic used for testing the null hypothesis.
'distribution
: distribution of the sample statistic, together with its parameters.
'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
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:
'alternative
, default 'twosided
, is the alternative hypothesis;
valid values are: 'twosided
(p1 # p2
), 'greater
(p1 > p2
)
and 'less
(p1 < p2
).
'conflevel
, default 95/100
, confidence level for the confidence interval; it must
be an expression which takes a value in (0,1).
'correct
, default true
, indicates whether Yates correction is applied or not.
The output of function test_proportions_difference
is an inference_result
Maxima object
showing the following results:
'proportions
: list with the two sample proportions.
'conf_level
: confidence level selected by the user.
'conf_interval
: Confidence interval for the difference of proportions p1 - p2
.
'method
: inference procedure and warning message in case of any of the samples sizes
is less than 10.
'hypotheses
: null and alternative hypotheses to be tested.
'statistic
: value of the sample statistic used for testing the null hypothesis.
'distribution
: distribution of the sample statistic, together with its parameters.
'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
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:
'alternative
, default 'twosided
, is the alternative hypothesis;
valid values are: 'twosided
, 'greater
and 'less
.
'median
, default 0
, is the median value to be checked.
The output of function test_sign
is an inference_result
Maxima object
showing the following results:
'med_estimate
: the sample median.
'method
: inference procedure.
'hypotheses
: null and alternative hypotheses to be tested.
'statistic
: value of the sample statistic used for testing the null hypothesis.
'distribution
: distribution of the sample statistic, together with its parameter(s).
'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
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:
'median
, default 0
, is the median value to be checked.
'alternative
, default 'twosided
, is the alternative hypothesis;
valid values are: 'twosided
, 'greater
and 'less
.
The output of function test_signed_rank
is an inference_result
Maxima object
with the following results:
'med_estimate
: the sample median.
'method
: inference procedure.
'hypotheses
: null and alternative hypotheses to be tested.
'statistic
: value of the sample statistic used for testing the null hypothesis.
'distribution
: distribution of the sample statistic, together with its parameter(s).
'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
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:
'alternative
, default 'twosided
, is the alternative hypothesis;
valid values are: 'twosided
, 'greater
and 'less
.
The output of function test_rank_sum
is an inference_result
Maxima object
with the following results:
'method
: inference procedure.
'hypotheses
: null and alternative hypotheses to be tested.
'statistic
: value of the sample statistic used for testing the null hypothesis.
'distribution
: distribution of the sample statistic, together with its parameters.
'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
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:
'statistic
: value of the W statistic.
'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
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:
'conflevel
, default 95/100
, confidence level for the
confidence intervals; it must be an expression which takes a value
in (0,1).
The output of function linear_regression
is an
inference_result
Maxima object with the following results:
'b_estimation
: regression coefficients estimates.
'b_covariances
: covariance matrix of the regression
coefficients estimates.
b_conf_int
: confidence intervals of the regression coefficients.
b_statistics
: statistics for testing coefficient.
b_p_values
: p-values for coefficient tests.
b_distribution
: probability distribution for coefficient tests.
v_estimation
: unbiased variance estimator.
v_conf_int
: variance confidence interval.
v_distribution
: probability distribution for variance test.
residuals
: residuals.
adc
: adjusted determination coefficient.
aic
: Akaike's information criterion.
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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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
.
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
.
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
.
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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85.1 Functions and Variables for stirling |
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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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86.1 Introduction to String Processing | ||
86.2 Input and Output | ||
86.3 Characters | ||
86.4 String Processing | ||
86.5 Octets and Utilities for Cryptography |
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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)$
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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)$
Closes stream and returns true
if stream had been open.
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 .
Flushes stream where stream has to be an output stream to a file.
Example: See writebyte .
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.
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.
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 .
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)$
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)$
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()
.
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 .
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
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 .
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))$ |++++++++++++++++++++|
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
Removes and returns the first character in stream.
If the end of file is encountered readchar
returns false
.
Example: See make_string_input_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.
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
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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Characters are strings of length 1.
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
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.
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.
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
Returns true
if char_1 and char_2 are the same character.
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.
Returns true
if the code point of char_1 is greater than the
code point of 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.
Returns true
if obj is a Maxima-character.
See introduction for example.
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, ["$","£","€"]); (%o2) [24, 0A3, 20AC]
Warning: It is not possible to enter characters corresponding to code points larger than 16 bit in wxMaxima with SBCL on Windows when the external format has not been set to UTF-8. See adjust_external_format.
CMUCL doesn't process these characters as one character.
cint
then returns false
.
Converting a character to a code point via UTF-8-octets may serve as a workaround:
utf8_to_unicode(string_to_octets(character));
Returns true
if the code point of char_1 is less than the
code point of char_2.
Like clessp
but ignores case which is only possible for non-US-ASCII
characters when the underlying Lisp is able to recognize a character as an
alphabetic character. See remarks on alphacharp.
Returns true
if char is a graphic character but not a space character.
A graphic character is a character one can see, plus the space character.
(constituent
is defined by Paul Graham.
See Paul Graham, ANSI Common Lisp, 1996, page 67.)
(%i1) for n from 0 thru 255 do ( tmp: ascii(n), if constituent(tmp) then sprint(tmp) )$ ! " # % ' ( ) * + , - . / 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 { | } ~
Returns true
if char is a digit where only the corresponding
US-ASCII-character is regarded as a digit.
Returns true
if char is a lowercase character.
Note: See remarks on alphacharp.
The newline character (ASCII-character 10).
The space character.
The tab character.
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) [$, £, €]
Warning: In wxMaxima with SBCL on Windows it is not possible to convert code points larger than 16 bit to characters when the external format has not been set to UTF-8. See adjust_external_format for more information.
CMUCL doesn't process code points larger than 16 bit.
In these cases unicode
returns false
.
Converting a code point to a character via UTF-8 octets may serve as a workaround:
octets_to_string(unicode_to_utf8(code_point));
See octets_to_string, unicode_to_utf8.
In case the underlying Lisp provides full Unicode support the character might be
specified by its name. The following is possible in ECL, CLISP and SBCL,
where in SBCL on Windows the external format has to be set to UTF-8.
unicode(name)
is supported by CMUCL too but again limited to 16 bit
characters.
The string argument to unicode
is basically the same string returned by
printf
using the "~@c" specifier.
But as shown below the prefix "#\" must be omitted.
Underlines might be replaced by spaces and uppercase letters by lowercase ones.
Example (continued): Characters defined by names (Maxima with SBCL on GNU/Linux).
(%i3) printf(false, "~@c", unicode(0DF)); (%o3) #\LATIN_SMALL_LETTER_SHARP_S (%i4) unicode("LATIN_SMALL_LETTER_SHARP_S"); (%o4) ß (%i5) unicode("Latin small letter sharp s"); (%o5) ß
Returns a list containing the UTF-8 code corresponding to the Unicode code_point.
Examples: Converting Unicode code points to UTF-8 and vice versa.
(%i1) ibase: obase: 16.$ (%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]
Returns true
if char is an uppercase character.
Note: See remarks on alphacharp.
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.
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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Position indices in strings are 1-indexed like in Maxima lists. See example in charat.
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
Returns the list of all characters in string.
(%i1) charlist("Lisp"); (%o1) [L, i, s, p]
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.
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.
Returns a copy of string as a new string.
Like supcase but uppercase characters are converted to lowercase.
Returns true
if string_1 and string_2 contain the same
sequence of characters.
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.
sexplode
is an alias for function charlist
.
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.
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.
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
Returns the number of characters in string.
Returns a new string with a number of num characters char.
Example:
(%i1) smake(3,"w"); (%o1) www
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
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]
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.
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.
Like sremove
except that only the first substring that matches seq is removed.
Returns a string with all the characters of string in reverse order.
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
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.'
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.
Like subst
except that only the first substring that matches old is replaced.
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
Like strim
except that only the left end of string is trimmed.
Like strim
except that only the right end of string is trimmed.
Returns true
if obj is a string.
See introduction for example.
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
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
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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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.
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.
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
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.
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]
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]
Returns a number by concatenating the octets in the list of octets.
Example: See number_to_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]
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]
Convertes an object identifier (OID) to a list of octets.
Example: See octets_to_oid.
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.
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.
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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87.1 Functions and Variables for to_poly_solve |
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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.
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
prederror
does not alter the
simplification %and
expressions.
%and, %or
, and not
should be
fully parenthesized.
and
and or
are both
short-circuited. Thus and
isn't associative or commutative.
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.
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)
if
is nary, the operator %if
isn't
nary.
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.
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
prederror
does not alter the
simplification of %or
expressions.
%and, %or
, and not
; the binding powers of these
operators might not match your expectations.
and
and or
are both short-circuited.
Thus or
isn't associative or commutative.
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.
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.
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
funmake
(not compose_functions
)
signals an error:
(%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);
new_variable
.
(%i6) compose_functions([%g0]); (%o6) lambda([%g154], %g0(%g154)) (%i7) compose_functions([%g0]); (%o7) lambda([%g155], %g0(%g155))
Although the independent variables are different, Maxima is able to to deduce that these lambda forms are semantically equal:
(%i8) is(equal(%o6,%o7)); (%o8) 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.
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
float
is both an option variable (default
value false) and a function name.
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.
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
eliminate
, the function
elim
does not invoke solve
when the number of equations
equals the number of variables.
elim
works by applying resultants; the option
variable resultant
determines which algorithm Maxima
uses. Using sqfr
, Maxima factors each resultant and suppresses
multiple zeros.
elim
will triangularize a nonlinear set of polynomial
equations; the solution set of the triangularized set can be larger
than that solution set of the untriangularized set. Thus, the triangularized
equations can have spurious solutions.
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.
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.
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.
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)'
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
simpfuncs = l
, where l
is a list of functions.
Apply the composition of the members of l to each solution.
(%i1) to_poly_solve(x^2=%i,x); 1/4 1/4 (%o1) %union([x = - (- 1) ], [x = (- 1) ]) (%i2) to_poly_solve(x^2= %i,x, 'simpfuncs = ['rectform]); %i 1 %i 1 (%o2) %union([x = - ------- - -------], [x = ------- + -------]) sqrt(2) sqrt(2) sqrt(2) sqrt(2)
Sometimes additional simplification can revert a simplification; for example
(%i3) to_poly_solve(x^2=1,x); (%o3) %union([x = - 1], [x = 1]) (%i4) to_poly_solve(x^2= 1,x, 'simpfuncs = [polarform]); %i %pi (%o4) %union([x = 1], [x = %e ]
Maxima doesn't try to check that each member of the function list l
is
purely a simplification; thus
(%i5) to_poly_solve(x^2 = %i,x, 'simpfuncs = [lambda([s],s^2)]); (%o5) %union([x = %i])
To convert each solution to a double float, use simpfunc = ['dfloat]
:
(%i6) to_poly_solve(x^3 +x + 1 = 0,x, 'simpfuncs = ['dfloat]), algexact : true; (%o6) %union([x = - .6823278038280178], [x = .3411639019140089 - 1.161541399997251 %i], [x = 1.161541399997251 %i + .3411639019140089])
use_grobner = true
With this option, the function
poly_reduced_grobner
is applied to the equations before
attempting their solution. Primarily, this option provides a workaround
for weakness in the function algsys
. Here is an example of
such a workaround:
(%i7) to_poly_solve([x^2+y^2=2^2,(x-1)^2+(y-1)^2=2^2],[x,y], 'use_grobner = true); sqrt(7) - 1 sqrt(7) + 1 (%o7) %union([x = - -----------, y = -----------], 2 2 sqrt(7) + 1 sqrt(7) - 1 [x = -----------, y = - -----------]) 2 2 (%i8) to_poly_solve([x^2+y^2=2^2,(x-1)^2+(y-1)^2=2^2],[x,y]); (%o8) %union()
maxdepth = k
, where k
is a positive integer. This
function controls the maximum recursion depth for the solver. The
default value for maxdepth
is five. When the recursions depth is
exceeded, the solver signals an error:
(%i9) to_poly_solve(cos(x) = x,x, 'maxdepth = 2); Unable to solve Unable to solve (%o9) %solve([cos(x) = x], [x], maxdepth = 2)
parameters = l
, where l
is a list of symbols. The solver
attempts to return a solution that is valid for all members of the list
l
; for example:
(%i10) to_poly_solve(a * x = x, x); (%o10) %union([x = 0]) (%i11) to_poly_solve(a * x = x, x, 'parameters = [a]); (%o11) %union(%if(a - 1 = 0, [x = %c111], %union()), %if(a - 1 # 0, [x = 0], %union()))
In (%o2)
, the solver introduced a dummy variable; to re-index the
these dummy variables, use the function nicedummies
:
(%i12) nicedummies(%); (%o12) %union(%if(a - 1 = 0, [x = %c0], %union()), %if(a - 1 # 0, [x = 0], %union()))
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
fullratsubst
is
able to appropriately make substitutions, the solve variables can be nonsymbols:
(%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
to_poly_solve
consults the fact database to decide if the
argument to the absolute value is complex valued. When
(%i1) to_poly_solve(abs(x) = 6, x); (%o1) %union([x = - 6], [x = 6]) (%i2) declare(z,complex)$ (%i3) to_poly_solve(abs(z) = 6, z); (%o3) %union(%if((%c11 # 0) %and (%c11 conjugate(%c11) - 36 = 0), [z = %c11], %union()))
This is the only situation that the solver consults the fact database. If
a solve variable is declared to be an integer, for example, to_poly_solve
ignores this declaration.
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.
%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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88.1 Introduction to Units | ||
88.2 Functions and Variables for Units |
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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.
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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")
.
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")
.
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")
.
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.
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.
Default value: 2
This is the value supplied to metricexpandall
during the initial loading
of unit.
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89.1 Introduction to wrstcse | ||
89.2 Functions and Variables for wrstcse |
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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.
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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
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 ]
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]
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]
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]
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
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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90.1 Introduction to zeilberger | ||
90.2 Functions and Variables for zeilberger |
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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.
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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.
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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.
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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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Returns the hypergeometric anti-difference of F_k, if it exists.
Otherwise AntiDifference
returns no_hyp_antidifference
.
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
.
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
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.
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
.
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Default value: 5
MAX_ORD
is the maximum recurrence order attempted by Zeilberger
.
Default value: false
When simplified_output
is true
,
functions in the zeilberger
package attempt
further simplification of the solution.
Default value: linsolve
linear_solver
names the solver which is used to solve the system
of equations in Zeilberger's algorithm.
Default value: true
When warnings
is true
,
functions in the zeilberger
package print
warning messages during execution.
Default value: true
When Gosper_in_Zeilberger
is true
,
the Zeilberger
function calls Gosper
before calling parGosper
.
Otherwise, Zeilberger
goes immediately to parGosper
.
Default value: true
When trivial_solutions
is true
,
Zeilberger
returns solutions
which have certificate equal to zero, or all coefficients equal to zero.
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Default value: false
When mod_test
is true
,
parGosper
executes a
modular test for discarding systems with no solutions.
Default value: linsolve
modular_linear_solver
names the linear solver used by the modular test in parGosper
.
Default value: big_primes[10]
ev_point
is the value at which the variable n is evaluated
when executing the modular test in parGosper
.
Default value: big_primes[1]
mod_big_prime
is the modulus used by the modular test in parGosper
.
Default value: 4
mod_threshold
is the
greatest order for which the modular test in parGosper
is attempted.
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91.1 Error messages | ||
91.2 Warning messages |
This chapter provides detailed information about the meaning of some error messages or on how to recover from errors.
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[ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
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)
.
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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]
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Maxima supports several assignment operators. When trying to define a function
:=
has to be used.
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This message notifies about a classical division by zero error.
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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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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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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.
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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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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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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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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:
--dynamic-space-size <n>
allows to tell
sbcl to reserve n
megabytes for the heap. It is to note, though,
that sbcl has to handle several distinct types of memory and therefore
might be able to only reserve about half of the available physical
memory. Also note that 32-bit processes might only be able to access
2GB of physical memory.
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part()
was used to access the n
th item in something that has less than
n
items.
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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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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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91.2.1 Encountered undefined variable <x> in translation | ||
91.2.2 Rat: replaced <x> by <y> = <z> |
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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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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
.
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Introduction to Special Functions · airy_ai · airy_bi · airy_dai · airy_dbi
%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
array · arrayapply · arrayinfo · arraymake · arrays · arraysetapply · fillarray · listarray · make_array · rearray · remarray · subvar · translate_fast_arrays · use_fast_arrays
%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
bit_and · bit_length · bit_lsh · bit_not · bit_onep · bit_or · bit_rsh · bit_xor
Introduction to Numbers · cabs · carg · cbffac · conjugate · demoivre · exponentialize · imagpart · plog · polarform · polartorect · realpart · rectform · recttopolar · residue
% · %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
%e · %gamma · %i · %phi · %pi · constant · constantp · false · ind · inf · infinity · minf · true · und · zeroa · zerob
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
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
atomgrad · del · depends · derivabbrev · derivdegree · derivlist · derivsubst · diff · dscalar · express · gradef · gradefs · hessian · implicit_derivative · jacobian · vect_cross · wronskian
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
%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
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
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
elliptic_e · elliptic_ec · elliptic_eu · elliptic_f · elliptic_kc · elliptic_pi
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
%enumer · detout · eval · evflag · evfun · float · infeval · noeval · nouns · numer · pred · simp
' · " · : · :: · Nouns and Verbs · at · derivlist · ev · kill · refcheck · remvalue · values
expintegral_chi · expintegral_ci · expintegral_e · expintegral_e1 · expintegral_e_simplify · expintegral_ei · expintegral_li · expintegral_shi · expintegral_si · expintexpand · expintrep
%e_to_numlog · %edispflag · %emode · %enumer · exp · exptsubst · li · log · logabs · logarc · logconcoeffp · logcontract · logexpand · lognegint · logsimp · plog · polarform · taylor_logexpand
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
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
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
Introduction to Fourier series · Introduction to fast Fourier transform
apply · fullmap · fullmapl · funmake · macroexpand · macroexpand1 · macroexpansion · map · maperror · maplist · mapprint · maxapplydepth · maxapplyheight · outermap · scanmap
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% · %% · _ · __ · 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
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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
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[ · ] · 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
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
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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
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
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
!! · # · ' · " · ** · * · + · - · / · ^ · . · : · :: · ::= · := · < · <= · > · >= · = · @ · 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 · | · ~
Introduction to cobyla · Introduction to lbfgs · Introduction to minpack · Introduction to simplex · augmented_lagrangian_method
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
define_alt_display · info_display · mathml_display · multi_display_for_texinfo · reset_displays · set_alt_display · set_prompt · tex_display
Introduction to atensor · abasep · adim · af · aform · alg_type · asymbol · atensimp · av · init_atensor · sf
bit_and · bit_length · bit_lsh · bit_not · bit_onep · bit_or · bit_rsh · bit_xor
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
%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
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 · 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
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
JF · ModeMatrix · diag · dispJordan · jordan · mat_function · minimalPoly
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
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
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
eigen · eigenvalues · eivals · eigenvectors · eivects · gramschmidt · innerproduct · inprod · similaritytransform · simtran · ueivects · uniteigenvectors · unitvector · uvect
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
Package facexp · collectterms · facsum · facsum_combine · factorfacsum · nextlayerfactor
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
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
Introduction to Fourier series · absint · cosnpiflag · equalp · fourcos · fourexpand · fourier · fourint · fourintcos · fourintsin · foursimp · foursin · funp · remfun · sinnpiflag · totalfourier
fernfale · hilbertmap · julia_parameter · julia_set · julia_sin · mandelbrot_set · sierpinskiale · sierpinskimap · snowmap · treefale
Package functs · agd · arithmetic · arithsum · combination · covers · exsec · gaussprob · gcdivide · gd · geometric · geosum · harmonic · hav · nonzeroandfreeof · permutation · rational · rempart · tracematrix · vers · wronskian
GGFCFMAX · GGFINFINITY · ggf
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
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
dimacs_export · dimacs_import · graph6_decode · graph6_encode · graph6_export · graph6_import · sparse6_decode · sparse6_encode · sparse6_export · sparse6_import
add_edge · add_edges · add_vertex · add_vertices · connect_vertices · contract_edge · remove_edge
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
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
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
Introduction to interpol · charfun2 · cspline · lagrange · linearinterpol · ratinterpol
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 · | · ~
Introduction to lapack · dgeev · dgemm · dgeqrf · dgesv · dgesvd · dlange · zlange · zgeev · zheev
Introduction to lbfgs · lbfgs · lbfgs_ncorrections · lbfgs_nfeval_max
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 · 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 · rank · rowop · rowswap · toeplitz · vandermonde_matrix · zerofor · zeromatrixp
Introduction to lsquares · lsquares_estimates · lsquares_estimates_approximate · lsquares_estimates_exact · lsquares_mse · lsquares_residual_mse · lsquares_residuals · plsquares
Introduction to mnewton · mnewton · newtonepsilon · newtonmaxiter
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
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
Introduction to QUADPACK · quad_control · quad_qag · quad_qagi · quad_qagp · quad_qags · quad_qawc · quad_qawf · quad_qawo · quad_qaws
romberg · rombergabs · rombergit · rombergmin · rombergtol
Introduction to simplex · epsilon_lp · linear_program · maximize_lp · minimize_lp · nonegative_lp · nonnegative_lp · pivot_count_sx · pivot_max_sx · scale_lp
Introduction to solve_rec · product_use_gamma · reduce_order · simplify_products · simplify_sum · solve_rec · solve_rec_rat · summand_to_rec
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
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
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
%unitexpand · Introduction to Units · convert · metricexpandall · setunits · uforget · usersetunits
Vectors · scalefactors · vect_cross · vectorpotential · vectorsimp
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
Introduction to Units · Introduction to ezunits · Introduction to physical_constants
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
intopois · outofpois · poisdiff · poisexpt · poisint · poislim · poismap · poisplus · poissimp · poisson · poissubst · poistimes · poistrim · printpois
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
deftaylor · maxtayorder · pade · powerseries · revert · revert2 · taylor · taylor_logexpand · taylor_order_coefficients · taylor_simplifier · taylor_truncate_polynomials · taylordepth · taylorinfo · taylorp · taytorat · trunc · verbose
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
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
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
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
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
Introduction for Runtime Environment · batch · batchload · kill · load · loadfile · myoptions · nolabels · optionset · reset · save · stringout
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
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
%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
Package absimp · Package ineq · atensimp · foursimp · fullratsimp · hypergeometric_simp · logarc · radcan · ratsimp · rootscontract · scsimp · simplify_sum · trigexpand · trigrat · trigreduce · trigsimp · unknown · vectorsimp
Introduction to Rules and Patterns · define_opproperty · lassociative · linear · multiplicative · opproperties
%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
Introduction to String Processing · concat · sconcat · string
@ · defstruct · new · structures
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
Comments · Identifiers · Introduction to Strings · Introduction to operators · infix · matchfix · nary · nofix · postfix · prefix
get_tex_environment · set_tex_environment · get_tex_environment_default · set_tex_environment_default · tentex · tex · texput
Introduction to atensor · Introduction to ctensor · Introduction to itensor
absolute_real_time · decode_time · elapsed_real_time · elapsed_run_time · encode_time · parse_timedate · timedate
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
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
%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
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Applied Mathematics and Programming Division, K.U. Leuven
Applied Mathematics and Programming Division, K.U. Leuven
Institut für Mathematik, T.U. Wien
National Bureau of Standards, Washington, D.C., U.S.A
http://www.netlib.org/quadpack
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.
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ctensor
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