Maximas Database (Maxima 5.42.455.gc5fa8d39c Manual)

11 Maximas Database

11.1 Introduction to Maximas Database

11.2 Functions and Variables for Properties

Property: alphabetic

alphabetic is a property type recognized by declare. The expression declare(s, alphabetic) tells Maxima to recognize as alphabetic all of the characters in s, which must be a string.

See also declare.

Example:

(%i1) declare(n, integer, x, noninteger);
(%o1)                         done
(%i2) askinteger(n);
(%o2)                          yes
(%i3) askinteger(x);
(%o3)                          no

Categories: Declarations and inferences

Property: integervalued

declare(f, integervalued) tells Maxima to recognize f as an integer-valued function.

See also declare.

Example:

(%i1) exp(%i)^f(x);
                              %i f(x)
(%o1)                      (%e  )
(%i2) declare(f, integervalued);
(%o2)                         done
(%i3) exp(%i)^f(x);
                              %i f(x)
(%o3)                       %e

Categories: Declarations and inferences

Property: nonarray

The command declare(a, nonarray) tells Maxima to consider a not an array. This declaration prevents multiple evaluation, if a is a subscripted variable.

See also declare.

Example:

(%i1) a:'b$ b:'c$ c:'d$

(%i4) a[x];
(%o4)                          d
                                x
(%i5) declare(a, nonarray);
(%o5)                         done
(%i6) a[x];
(%o6)                          a
                                x

Categories: Expressions

Property: nonscalar

Makes atoms behave as does a list or matrix with respect to the dot operator.

See also declare.

Categories: Declarations and inferences Vectors Matrices

Function: nonscalarp (expr)

Returns true if expr is a non-scalar, i.e., it contains atoms declared as non-scalars, lists, or matrices.

See also the predicate function scalarp and declare.

Categories: Predicate functions Vectors Matrices

Property: posfun

declare (f, posfun) declares f to be a positive function. is (f(x) > 0) yields true.

See also declare.

Categories: Declarations and inferences Operators

Function: printprops
    printprops (a, i)
    printprops ([a_1, …, a_n], i)
    printprops (all, i)

Displays the property with the indicator i associated with the atom a. a may also be a list of atoms or the atom all in which case all of the atoms with the given property will be used. For example, printprops ([f, g], atvalue). printprops is for properties that cannot otherwise be displayed, i.e. for atvalue, atomgrad, gradef, and matchdeclare.

Categories: Declarations and inferences Display functions

Function: properties (a)

Returns a list of the names of all the properties associated with the atom a.

Categories: Declarations and inferences

System variable: props

Default value: []

props are atoms which have any property other than those explicitly mentioned in infolists, such as specified by atvalue, matchdeclare, etc., as well as properties specified in the declare function.

Categories: Declarations and inferences Global variables

Function: propvars (prop)

Returns a list of those atoms on the props list which have the property indicated by prop. Thus propvars (atvalue) returns a list of atoms which have atvalues.

Categories: Declarations and inferences

Function: put (atom, value, indicator)

Assigns value to the property (specified by indicator) of atom. indicator may be the name of any property, not just a system-defined property.

rem reverses the effect of put.

put evaluates its arguments. put returns value.

See also declare.

Categories: Declarations and inferences

Property: real

Property: imaginary

Property: complex

declare(a, real), declare(a, imaginary), or declare(a, complex) tells Maxima to recognize a as a real, pure imaginary, or complex variable.

See also declare.

Categories: Declarations and inferences

Function: rem (atom, indicator)

Removes the property indicated by indicator from atom. rem reverses the effect of put.

rem returns done if atom had an indicator property when rem was called, or false if it had no such property.

Categories: Declarations and inferences

Function: remove
    remove (a_1, p_1, …, a_n, p_n)
    remove ([a_1, …, a_m], [p_1, …, p_n], …)
    remove ("a", operator)
    remove (a, transfun)
    remove (all, p)

Removes properties associated with atoms.

remove (a_1, p_1, ..., a_n, p_n) removes property p_k from atom a_k.

remove ([a_1, ..., a_m], [p_1, ..., p_n], ...) removes properties p_1, ..., p_n from atoms a_1, …, a_m. There may be more than one pair of lists.

remove (all, p) removes the property p from all atoms which have it.

The removed properties may be system-defined properties such as function, macro, or mode_declare. remove does not remove properties defined by put.

A property may be transfun to remove the translated Lisp version of a function. After executing this, the Maxima version of the function is executed rather than the translated version.

remove ("a", operator) or, equivalently, remove ("a", op) removes from a the operator properties declared by prefix, infix, nary, postfix, matchfix, or nofix. Note that the name of the operator must be written as a quoted string.

remove always returns done whether or not an atom has a specified property. This behavior is unlike the more specific remove functions remvalue, remarray, remfunction, and remrule.

remove quotes its arguments.

Categories: Declarations and inferences

Property: scalar

declare(a, scalar) tells Maxima to consider a a scalar variable.

See also declare.

Categories: Declarations and inferences

Function: scalarp (expr)

Returns true if expr is a number, constant, or variable declared scalar with declare, or composed entirely of numbers, constants, and such variables, but not containing matrices or lists.

See also the predicate function nonscalarp.

Categories: Predicate functions Vectors Matrices

11.3 Functions and Variables for Facts

Function: activate (context_1, …, context_n)

Activates the contexts context_1, …, context_n. The facts in these contexts are then available to make deductions and retrieve information. The facts in these contexts are not listed by facts ().

The variable activecontexts is the list of contexts which are active by way of the activate function.

Categories: Declarations and inferences

System variable: activecontexts

Default value: []

activecontexts is a list of the contexts which are active by way of the activate function, as opposed to being active because they are subcontexts of the current context.

Categories: Declarations and inferences

Function: askinteger
    askinteger (expr, integer)
    askinteger (expr)
    askinteger (expr, even)
    askinteger (expr, odd)

askinteger (expr, integer) attempts to determine from the assume database whether expr is an integer. askinteger prompts the user if it cannot tell otherwise, and attempt to install the information in the database if possible. askinteger (expr) is equivalent to askinteger (expr, integer).

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

Categories: Declarations and inferences

Function: asksign (expr)

First attempts to determine whether the specified expression is positive, negative, or zero. If it cannot, it asks the user the necessary questions to complete its deduction. The user’s answer is recorded in the data base for the duration of the current computation. The return value of asksign is one of pos, neg, or zero.

Categories: Declarations and inferences

Function: assume (pred_1, …, pred_n)

Adds predicates pred_1, …, pred_n to the current context. If a predicate is inconsistent or redundant with the predicates in the current context, it is not added to the context. The context accumulates predicates from each call to assume.

assume returns a list whose elements are the predicates added to the context or the atoms redundant or inconsistent where applicable.

The predicates pred_1, …, pred_n can only be expressions with the relational operators < <= equal notequal >= and >. Predicates cannot be literal equality = or literal inequality # expressions, nor can they be predicate functions such as integerp.

Compound predicates of the form pred_1 and ... and pred_n are recognized, but not pred_1 or ... or pred_n. not pred_k is recognized if pred_k is a relational predicate. Expressions of the form not (pred_1 and pred_2) and not (pred_1 or pred_2) are not recognized.

Maxima’s deduction mechanism is not very strong; there are many obvious consequences which cannot be determined by is. This is a known weakness.

assume does not handle predicates with complex numbers. If a predicate contains a complex number assume returns inconsistent or redunant.

assume evaluates its arguments.

See also is, facts, forget, context, and declare.

Examples:

(%i1) assume (xx > 0, yy < -1, zz >= 0);
(%o1)              [xx > 0, yy < - 1, zz >= 0]
(%i2) assume (aa < bb and bb < cc);
(%o2)                  [bb > aa, cc > bb]
(%i3) facts ();
(%o3)     [xx > 0, - 1 > yy, zz >= 0, bb > aa, cc > bb]
(%i4) is (xx > yy);
(%o4)                         true
(%i5) is (yy < -yy);
(%o5)                         true
(%i6) is (sinh (bb - aa) > 0);
(%o6)                         true
(%i7) forget (bb > aa);
(%o7)                       [bb > aa]
(%i8) prederror : false;
(%o8)                         false
(%i9) is (sinh (bb - aa) > 0);
(%o9)                        unknown
(%i10) is (bb^2 < cc^2);
(%o10)                       unknown

Categories: Declarations and inferences

Option variable: assumescalar

Default value: true

assumescalar helps govern whether expressions expr for which nonscalarp (expr) is false are assumed to behave like scalars for certain transformations.

Let expr represent any expression other than a list or a matrix, and let [1, 2, 3] represent any list or matrix. Then expr . [1, 2, 3] yields [expr, 2 expr, 3 expr] if assumescalar is true, or scalarp (expr) is true, or constantp (expr) is true.

If assumescalar is true, such expressions will behave like scalars only for commutative operators, but not for noncommutative multiplication ..

When assumescalar is false, such expressions will behave like non-scalars.

When assumescalar is all, such expressions will behave like scalars for all the operators listed above.

Categories: Declarations and inferences

Option variable: assume_pos

Default value: false

When assume_pos is true and the sign of a parameter x cannot be determined from the current context or other considerations, sign and asksign (x) return true. This may forestall some automatically-generated asksign queries, such as may arise from integrate or other computations.

By default, a parameter is x such that symbolp (x) or subvarp (x). The class of expressions considered parameters can be modified to some extent via the variable assume_pos_pred.

sign and asksign attempt to deduce the sign of expressions from the sign of operands within the expression. For example, if a and b are both positive, then a + b is also positive.

However, there is no way to bypass all asksign queries. In particular, when the asksign argument is a difference x - y or a logarithm log(x), asksign always requests an input from the user, even when assume_pos is true and assume_pos_pred is a function which returns true for all arguments.

Categories: Declarations and inferences

Option variable: assume_pos_pred

Default value: false

When assume_pos_pred is assigned the name of a function or a lambda expression of one argument x, that function is called to determine whether x is considered a parameter for the purpose of assume_pos. assume_pos_pred is ignored when assume_pos is false.

The assume_pos_pred function is called by sign and asksign with an argument x which is either an atom, a subscripted variable, or a function call expression. If the assume_pos_pred function returns true, x is considered a parameter for the purpose of assume_pos.

By default, a parameter is x such that symbolp (x) or subvarp (x).

11.4 Functions and Variables for Predicates

Function: charfun (p)

Return 0 when the predicate p evaluates to false; return 1 when the predicate evaluates to true. When the predicate evaluates to something other than true or false (unknown), return a noun form.

Examples:

(%i1) charfun (x < 1);
(%o1)                    charfun(x < 1)
(%i2) subst (x = -1, %);
(%o2)                           1
(%i3) e : charfun ('"and" (-1 < x, x < 1))$
(%i4) [subst (x = -1, e), subst (x = 0, e), subst (x = 1, e)];
(%o4)                       [0, 1, 0]

Categories: Mathematical functions

Function: compare (x, y)

Return a comparison operator op (<, <=, >, >=, =, or #) such that is (x op y) evaluates to true; when either x or y depends on %i and x # y, return notcomparable; when there is no such operator or Maxima isn’t able to determine the operator, return unknown.

Examples:

(%i1) compare (1, 2);
(%o1)                           <
(%i2) compare (1, x);
(%o2)                        unknown
(%i3) compare (%i, %i);
(%o3)                           =
(%i4) compare (%i, %i + 1);
(%o4)                     notcomparable
(%i5) compare (1/x, 0);
(%o5)                           #
(%i6) compare (x, abs(x));
(%o6)                          <=

The function compare doesn’t try to determine whether the real domains of its arguments are nonempty; thus

(%i1) compare (acos (x^2 + 1), acos (x^2 + 1) + 1);
(%o1)                           <

The real domain of acos (x^2 + 1) is empty.

Categories: Declarations and inferences

Function: equal (a, b)

Represents equivalence, that is, equal value.

By itself, equal does not evaluate or simplify. The function is attempts to evaluate equal to a Boolean value. is(equal(a, b)) returns true (or false) if and only if a and b are equal (or not equal) for all possible values of their variables, as determined by evaluating ratsimp(a - b); if ratsimp returns 0, the two expressions are considered equivalent. Two expressions may be equivalent even if they are not syntactically equal (i.e., identical).

When is fails to reduce equal to true or false, the result is governed by the global flag prederror. When prederror is true, is complains with an error message. Otherwise, is returns unknown.

In addition to is, some other operators evaluate equal and notequal to true or false, namely if, and, or, and not.

The negation of equal is notequal.

Examples:

By itself, equal does not evaluate or simplify.

(%i1) equal (x^2 - 1, (x + 1) * (x - 1));
                        2
(%o1)            equal(x  - 1, (x - 1) (x + 1))
(%i2) equal (x, x + 1);
(%o2)                    equal(x, x + 1)
(%i3) equal (x, y);
(%o3)                      equal(x, y)

The function is attempts to evaluate equal to a Boolean value. is(equal(a, b)) returns true when ratsimp(a - b) returns 0. Two expressions may be equivalent even if they are not syntactically equal (i.e., identical).

(%i1) ratsimp (x^2 - 1 - (x + 1) * (x - 1));
(%o1)                           0
(%i2) is (equal (x^2 - 1, (x + 1) * (x - 1)));
(%o2)                         true
(%i3) is (x^2 - 1 = (x + 1) * (x - 1));
(%o3)                         false
(%i4) ratsimp (x - (x + 1));
(%o4)                          - 1
(%i5) is (equal (x, x + 1));
(%o5)                         false
(%i6) is (x = x + 1);
(%o6)                         false
(%i7) ratsimp (x - y);
(%o7)                         x - y
(%i8) is (equal (x, y));
(%o8)                        unknown
(%i9) is (x = y);
(%o9)                         false

When is fails to reduce equal to true or false, the result is governed by the global flag prederror.

(%i1) [aa : x^2 + 2*x + 1, bb : x^2 - 2*x - 1];
                    2             2
(%o1)             [x  + 2 x + 1, x  - 2 x - 1]
(%i2) ratsimp (aa - bb);
(%o2)                        4 x + 2
(%i3) prederror : true;
(%o3)                         true
(%i4) is (equal (aa, bb));
Maxima was unable to evaluate the predicate:
       2             2
equal(x  + 2 x + 1, x  - 2 x - 1)
 -- an error.  Quitting.  To debug this try debugmode(true);
(%i5) prederror : false;
(%o5)                         false
(%i6) is (equal (aa, bb));
(%o6)                        unknown

Some operators evaluate equal and notequal to true or false.

(%i1) if equal (y, y - 1) then FOO else BAR;
(%o1)                          BAR
(%i2) eq_1 : equal (x, x + 1);
(%o2)                    equal(x, x + 1)
(%i3) eq_2 : equal (y^2 + 2*y + 1, (y + 1)^2);
                         2                   2
(%o3)             equal(y  + 2 y + 1, (y + 1) )
(%i4) [eq_1 and eq_2, eq_1 or eq_2, not eq_1];
(%o4)                  [false, true, true]

Because not expr causes evaluation of expr, not equal(a, b) is equivalent to is(notequal(a, b)).

(%i1) [notequal (2*z, 2*z - 1), not equal (2*z, 2*z - 1)];
(%o1)            [notequal(2 z, 2 z - 1), true]
(%i2) is (notequal (2*z, 2*z - 1));
(%o2)                         true

Categories: Operators

Function: notequal (a, b)

Represents the negation of equal(a, b).

Examples:

(%i1) equal (a, b);
(%o1)                      equal(a, b)
(%i2) maybe (equal (a, b));
(%o2)                        unknown
(%i3) notequal (a, b);
(%o3)                    notequal(a, b)
(%i4) not equal (a, b);
(%o4)                    notequal(a, b)
(%i5) maybe (notequal (a, b));
(%o5)                        unknown
(%i6) assume (a > b);
(%o6)                        [a > b]
(%i7) equal (a, b);
(%o7)                      equal(a, b)
(%i8) maybe (equal (a, b));
(%o8)                         false
(%i9) notequal (a, b);
(%o9)                    notequal(a, b)
(%i10) maybe (notequal (a, b));
(%o10)                        true

Categories: Operators

Function: unknown (expr)

Returns true if and only if expr contains an operator or function not recognized by the Maxima simplifier.

Categories: Predicate functions Simplification functions

Function: zeroequiv (expr, v)

Tests whether the expression expr in the variable v is equivalent to zero, returning true, false, or dontknow.

zeroequiv has these restrictions:

Do not use functions that Maxima does not know how to differentiate and evaluate.
If the expression has poles on the real line, there may be errors in the result (but this is unlikely to occur).
If the expression contains functions which are not solutions to first order differential equations (e.g. Bessel functions) there may be incorrect results.
The algorithm uses evaluation at randomly chosen points for carefully selected subexpressions. This is always a somewhat hazardous business, although the algorithm tries to minimize the potential for error.

For example zeroequiv (sin(2 * x) - 2 * sin(x) * cos(x), x) returns true and zeroequiv (%e^x + x, x) returns false. On the other hand zeroequiv (log(a * b) - log(a) - log(b), a) returns dontknow because of the presence of an extra parameter b.

Categories: Predicate functions

• Introduction to Maximas Database:
• Functions and Variables for Properties:
• Functions and Variables for Facts:
• Functions and Variables for Predicates: