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Operators

nary

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 syntax.

nofix

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 syntax.

operator

See operators.

postfix

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 syntax.

prefix

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 syntax.

Definitions for Operators

Operator: "!"

The factorial operator. For any complex number x (including integer, rational, and real numbers) 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 floating point number 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)). If x is anything else, x! is not simplified.

The variables factlim, minfactorial, and factcomb control the simplification of expressions containing factorials.

The functions gamma, bffac, and cbffac are varieties of the gamma function. makegamma substitutes gamma for factorials and related functions.

See also binomial.

The factorial of an integer, half-integer, or floating point argument is simplified unless the operand is greater than factlim.

(%i1) factlim: 10$
(%i2) [0!, (7/2)!, 4.77!, 8!, 20!];
          105 sqrt(%pi)
(%o2) [1, -------------, 81.44668037931193, 40320, 20!]
               16

The factorial of a complex number, known constant, or general expression is not simplified. Even so it may be possible 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) [(%i + 1)!, 7.188082728976031, 4.260820476357003, 

                                          1.227580202486819]

The factorial of an unbound symbol is not simplified.

(%i1) kill (foo)$
(%i2) foo!;
(%o2)                       foo!

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, -------------, 81.44668037931193, 40320, 20!]
               16

Operator: "!!"

The double factorial operator.

For an integer, float, or rational number n, n!! evaluates to the product n (n-2) (n-4) (n-6) ... (n - 2 (k-1)) where k is equal to entier (n/2), that is, the largest integer less than or equal to n/2. Note that this definition does not coincide with other published definitions for arguments which are not integers.

For an even (or odd) integer n, n!! evaluates to the product of all the consecutive even (or odd) integers from 2 (or 1) through n inclusive.

For an argument n which is not an integer, float, or rational, n!! yields a noun form genfact (n, n/2, 2).

Operator: "#"

Represents the negation of syntactic equality =.

Note that because of the rules for evaluation of predicate expressions (in particular because not expr causes evaluation of expr), not a = b is not equivalent to a # b in some cases.

Examples:

(%i1) a = b;
(%o1)                         a = b
(%i2) is (a = b);
(%o2)                         false
(%i3) a # b;
(%o3)                         a # b
(%i4) not a = b;
(%o4)                         true
(%i5) is (a # b);
(%o5)                         true
(%i6) is (not a = b);
(%o6)                         true

Operator: "."

The 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.

Operator: ":": The assignment operator. E.g. A:3 sets the variable A to 3.

Operator: "::": Assignment operator. :: assigns the value of the expression on its right to the value of the quantity on its left, which must evaluate to an atomic variable or subscripted variable.

Operator: "::=": The "::=" is used instead of ":=" to indicate that what follows is a macro definition, rather than an ordinary functional definition. See macros.

Operator: ":=": The function definition operator. E.g. f(x):=sin(x) defines a function f.

Operator: "="

denotes an equation to Maxima. To the pattern matcher in Maxima it denotes a total relation that holds between two expressions if and only if the expressions are syntactically identical.

The negation of = is represented by #. Note that because of the rules for evaluation of predicate expressions (in particular because not expr causes evaluation of expr), not a = b is not equivalent to a # b in some cases.

Operator: and

The logical conjunction operator. and is an n-ary infix operator; its operands are Boolean expressions, and its result is a Boolean value.

and forces evaluation (like is) of one or more operands, and may force evaluation of all operands.

Operands are evaluated in the order in which they appear. and evaluates only as many of its operands as necessary to determine the result. If any operand is false, the result is false and no further operands are evaluated.

The global flag prederror governs the behavior of and when an evaluated operand cannot be determined to be true or false. and prints an error message when prederror is true. Otherwise, and returns unknown.

and is not commutative: a and b might not be equal to b and a due to the treatment of indeterminate operands.

Operator: or

The logical disjunction operator. or is an n-ary infix operator; its operands are Boolean expressions, and its result is a Boolean value.

or forces evaluation (like is) of one or more operands, and may force evaluation of all operands.

Operands are evaluated in the order in which they appear. or evaluates only as many of its operands as necessary to determine the result. If any operand is true, the result is true and no further operands are evaluated.

The global flag prederror governs the behavior of or when an evaluated operand cannot be determined to be true or false. or prints an error message when prederror is true. Otherwise, or returns unknown.

or is not commutative: a or b might not be equal to b or a due to the treatment of indeterminate operands.

Operator: not

The logical negation operator. not is a prefix operator; its operand is a Boolean expression, and its result is a Boolean value.

not forces evaluation (like is) of its operand.

The global flag prederror governs the behavior of not when its operand cannot be determined to be true or false. not prints an error message when prederror is true. Otherwise, not returns unknown.

Function: abs (expr): Returns the absolute value expr. If expr is complex, returns the complex modulus of expr.

Keyword: additive

If declare(f,additive) has been executed, then:

(1) If f is univariate, whenever the simplifier encounters f applied to a sum, f will be distributed over that sum. I.e. f(y+x) will simplify to f(y)+f(x).

(2) If f is a function of 2 or more arguments, additivity is defined as additivity in the first argument to f, as in the case of sum or integrate, i.e. f(h(x)+g(x),x) will simplify to f(h(x),x)+f(g(x),x). This simplification does not occur when f is applied to expressions of the form sum(x[i],i,lower-limit,upper-limit).

Keyword: allbut

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

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

while

(%i3) part (expr, allbut (2, 5));
(%o3)                       e + c + b

It also works with the kill command,

kill (allbut (name_1, ..., name_k))

will do a kill (all) except it will not kill the names specified. Note: name_i means a name such as function name such as u, f, foo, or g, not an infolist such as functions.

Declaration: antisymmetric: If declare(h,antisymmetric) is done, this tells the simplifier that h is antisymmetric. E.g. h(x,z,y) will simplify to - h(x, y, z). That is, it will give (-1)^n times the result given by symmetric or commutative, where n is the number of interchanges of two arguments necessary to convert it to that form.

Function: cabs (expr): Returns the complex absolute value (the complex modulus) of expr.

Declaration: commutative: If declare(h,commutative) is done, this tells the simplifier that h is a commutative function. E.g. h(x,z,y) will simplify to h(x, y, z). This is the same as symmetric.

Function: entier (x): Returns the largest integer less than or equal to x where x is numeric. fix (as in fixnum) is a synonym for this, so fix(x) is precisely the same.

Function: equal (expr_1, expr_2)

Used with an is, returns true (or false) if and only if expr_1 and expr_2 are equal (or not equal) for all possible values of their variables (as determined by ratsimp). Thus is (equal ((x + 1)^2, x^2 + 2*x + 1)) returns true whereas if x is unbound is ((x + 1)^2 = x^2 + 2*x + 1) returns false. Note also that is(rat(0)=0) yields false but is (equal (rat(0), 0)) yields true.

If a determination can't be made, then is (equal (a, b)) returns a simplified but equivalent expression, whereas is (a=b) always returns either true or false.

All variables occurring in expr_1 and expr_2 are presumed to be real valued.

The negation of equal is notequal. Note that because of the rules for evaluation of predicate expressions (in particular because not expr causes evaluation of expr), notequal is not equivalent to not equal in some cases.

ev (expr, pred) is equivalent to is (expr).

(%i1) is (x^2 >= 2*x - 1);
(%o1)                         true
(%i2) assume (a > 1);
(%o2)                        [a > 1]
(%i3) is (log (log (a+1) + 1) > 0 and a^2 + 1 > 2*a);
(%o3)                         true

Function: notequal (expr_1, expr_2)

Represents the negation of equal (expr_1, expr_2).

Note that because of the rules for evaluation of predicate expressions (in particular because not expr causes evaluation of expr), notequal is not equivalent to not equal in some cases.

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);
`macsyma' was unable to evaluate the predicate:
equal(a, b)
 -- an error.  Quitting.  To debug this try debugmode(true);
(%i5) maybe (notequal (a, b));
(%o5)                        unknown
(%i6) maybe (not equal (a, b));
(%o6)                        unknown
(%i7) assume (a > b);
(%o7)                        [a > b]
(%i8) equal (a, b);
(%o8)                      equal(a, b)
(%i9) maybe (equal (a, b));
(%o9)                         false
(%i10) notequal (a, b);
(%o10)                   notequal(a, b)
(%i11) not equal (a, b);
(%o11)                        true
(%i12) maybe (notequal (a, b));
(%o12)                        true
(%i13) maybe (not equal (a, b));
(%o13)                        true

Operator: eval: As an argument in a call to ev (expr), eval causes an extra evaluation of expr. See ev.

Function: evenp (expr): Returns true if expr is an even integer. false is returned in all other cases.

Function: fix (x): A synonym for entier (x).

Function: fullmap (f, expr_1, ...)

Similar to map, but fullmap keeps mapping down all subexpressions until the main operators are no longer the same.

fullmap is used by the Maxima simplifier for certain matrix manipulations; thus, Maxima sometimes generates an error message concerning fullmap even though fullmap was not explicitly called by the user.

(%i1) a + b*c$
(%i2) fullmap (g, %);
(%o2)                   g(b) g(c) + g(a)
(%i3) map (g, %th(2));
(%o3)                     g(b c) + g(a)

Function: fullmapl (f, list_1, ...)

Similar to fullmap, but fullmapl only maps onto lists and matrices.

(%i1) fullmapl ("+", [3, [4, 5]], [[a, 1], [0, -1.5]]);
(%o1)                [[a + 3, 4], [4, 3.5]]

Function: is (expr)

Attempts to determine whether the predicate expr is provable from the facts in the assume database.

If the predicate is provably true or false, is returns true or false, respectively. Otherwise, the return value is controlled by the global flag prederror. When prederror is false, is returns unknown for a predicate which cannot be proven nor disproven, and reports an error otherwise.

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);
`macsyma' was unable to evaluate the predicate:
a > 0
 -- an error.  Quitting.  To debug this try debugmode(true);
(%i4) prederror: false$
(%i5) is (a > 0);
(%o5)                        unknown

Function: maybe (expr)

Attempts to determine whether the predicate expr is provable from the facts in the assume database.

If the predicate is provably true or false, maybe returns true or false, respectively. Otherwise, maybe returns unknown.

maybe is functionally equivalent to is with prederror: false, but the result is computed without actually assigning a value to prederror.

See also assume, facts, and is.

Examples:

(%i1) maybe (x > 0);
(%o1)                        unknown
(%i2) assume (x > 1);
(%o2)                        [x > 1]
(%i3) maybe (x > 0);
(%o3)                         true

Function: isqrt (x): Returns the "integer square root" of the absolute value of x, which is an integer.

Function: max (x_1, x_2, ...): Returns the maximum of its arguments (or returns a simplified form if some of its arguments are non-numeric).

Function: min (x_1, x_2, ...): Returns the minimum of its arguments (or returns a simplified form if some of its arguments are non-numeric).

Function: mod (p)

Function: mod (p, m)

Converts the polynomial p to a modular representation with respect to the current modulus which is the value of the variable modulus.

mod (p, m) specifies a modulus m to be used instead of the current value of modulus.

See modulus.

Function: oddp (expr): is true if expr is an odd integer. false is returned in all other cases.

Operator: pred: As an argument in a call to ev (expr), pred causes predicates (expressions which evaluate to true or false) to be evaluated. See ev.

Function: make_random_state (n)

Function: make_random_state (s)

Function: make_random_state (true)

Function: make_random_state (false)

A random state object represents the state of the random number generator. The state comprises 627 32-bit words.

make_random_state (n) returns a new random state object created from an integer seed value equal to n modulo 2^32. n may be negative.

make_random_state (s) returns a copy of the random state s.

make_random_state (true) returns a new random state object, using the current computer clock time as the seed.

make_random_state (false) returns a copy of the current state of the random number generator.

Function: set_random_state (s)

Copies s to the random number generator state.

set_random_state always returns done.

Function: random (x)

Returns a pseudorandom number. If x is an integer, random (x) returns an integer from 0 through x - 1 inclusive. If x is a floating point number, random (x) returns a nonnegative floating point number less than x. random complains with an error if x is neither an integer nor a float, or if x is not positive.

The functions make_random_state and set_random_state maintain the state of the random number generator.

The Maxima random number generator is an implementation of the Mersenne twister MT 19937.

Examples:

(%i1) s1: make_random_state (654321)$
(%i2) set_random_state (s1);
(%o2)                         done
(%i3) random (1000);
(%o3)                          768
(%i4) random (9573684);
(%o4)                        7657880
(%i5) random (2^75);
(%o5)                11804491615036831636390
(%i6) s2: make_random_state (false)$
(%i7) random (1.0);
(%o7)                   .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)                  .2310127244107132
(%i12) random (10.0);
(%o12)                  4.394553645870825
(%i13) random (100.0);
(%o13)                  32.28666704056853

Function: sign (expr): Attempts to determine the sign of expr on the basis of the facts in the current data base. It returns one of the following answers: pos (positive), neg (negative), zero, pz (positive or zero), nz (negative or zero), pn (positive or negative), or pnz (positive, negative, or zero, i.e. nothing known).

Function: signum (x)

For numeric x, returns 0 if x is 0, otherwise returns -1 or +1 as x is less than or greater than 0, respectively.

If x is not numeric then a simplified but equivalent form is returned. For example, signum(-x) gives -signum(x).

Function: sort (list, p)

Function: sort (list)

Sorts list according to a predicate p of two arguments, such as "<" or orderlessp.

sort (list) sorts list according to Maxima's built-in ordering.

list may contain numeric or nonnumeric items, or both.

Function: sqrt (x)

The square root of x. It is represented internally by x^(1/2). See also rootscontract.

radexpand if true will cause nth roots of factors of a product which are powers of n to be pulled outside of the radical, e.g. sqrt(16*x^2) will become 4*x only if radexpand is true.

Option variable: sqrtdispflag

Default value: true

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

Function: sublis (list, expr)

Makes multiple parallel substitutions into an expression.

The variable sublis_apply_lambda controls simplification after sublis.

Example:

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

Function: sublist (list, p)

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

Example:

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

Option variable: sublis_apply_lambda: Default value: true - controls whether lambda's substituted are applied in simplification after sublis is used or whether you have to do an ev to get things to apply. true means do the application.

Function: subst (a, b, c)

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

subst is an alias for substitute.

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.

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.

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

For further examples, do example (subst).

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

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

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

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

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

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

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

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

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

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

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

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

Function: subvarp (expr): Returns true if expr is a subscripted variable, for example a[i].

Function: symbolp (expr)

Returns true if expr is a symbol, else false. In effect, symbolp(x) is equivalent to the predicate atom(x) and not numberp(x).