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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. Do DESCRIBE(SYNTAX); for more details. 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).

## 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 MACSYMA 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. Do DESCRIBE(SYNTAX); for more details.

## 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. Do DESCRIBE(SYNTAX); for details.

## 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. Do DESCRIBE(SYNTAX); for more details.

## Definitions for Operators

operator: "!"
  The factorial operator, which is the product of all the integers from 1 up to its argument. Thus 5! = 1*2*3*4*5 = 120. The value of /the option FACTLIM (default: [-1]) gives the highest factorial which is automatically expanded. If it is -1 then all integers are expanded. See also the FACTORIAL, MINFACTORIAL, and FACTCOMB commands.

operator: "!!"
  Stands for double factorial which is defined as the product of all the consecutive odd (or even) integers from 1 (or 2) to the odd (or even) argument. Thus 8!! is 2*4*6*8 = 384.

operator: "#"
  The logical operator "Not equals".

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. Do APROPOS(DOT); for a list of the switches which affect the dot operator. DESCRIBE(switch-name); will explain them.

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 DESCRIBE(MACROS).

operator: ":="
  The function definition operator. E.g. F(X):=SIN(X) defines a function F.

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

Function: ABS (exp)
  returns the absolute value exp. If exp is complex, returns the complex modulus of exp.

special symbol: 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,

      if EXPR is E+D+C+B+A,
      then PART(EXPR,[2,5]);
      ==> D+A

  while

      PART(EXPR,ALLBUT(2,5))==>E+C+B

  It also works with the KILL command,

      KILL(ALLBUT(name1,...,namek))

  will do a KILL(ALL) except it will not KILL the names specified. Note: namei 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 (exp)
  returns the complex absolute value (the complex modulus) of exp.

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)
  largest integer <= X where X is numeric. FIX (as in FIXnum) is a synonym for this, so FIX(X); is precisely the same.

Function: EQUAL (expr1,expr2)
  used with an "IS", returns TRUE (or FALSE) if and only if expr1 and expr2 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) gives FALSE but IS(EQUAL(RAT(0),0)) gives TRUE. If a determination can't be made with EQUAL then a simplified but equivalent form is returned whereas = always causes either TRUE or FALSE to be returned. All variables occurring in exp are presumed to be real valued. EV(exp,PRED) is equivalent to IS(exp).

      (C1) IS(X**2 >= 2*X-1);
      (D1)                               TRUE
      (C2) ASSUME(A>1);
      (D2)                               DONE
      (C3) IS(LOG(LOG(A+1)+1)>0 AND A^2+1>2*A);
      (D3)                               TRUE

Function: EVAL
  causes an extra post-evaluation of exp to occur.

Function: EVENP (exp)
  is TRUE if exp is an even integer. FALSE is returned in all other cases.

Function: FIX (x)
  a synonym for ENTIER(X) - largest integer <= X where X is numeric.

Function: FULLMAP (fn, exp1, ...)
  is similar to MAP but it will keep mapping down all subexpressions until the main operators are no longer the same. The user should be aware that FULLMAP is used by the MACSYMA simplifier for certain matrix manipulations; thus, the user might see an error message concerning FULLMAP even though FULLMAP was not explicitly called by the user.

      (C1) A+B*C$
      (C2) FULLMAP(G,%);
      (D2)              G(B) G(C) + G(A)
      (C3)  MAP(G,D1);
      (D3)                G(B C) + G(A)

Function: FULLMAPL (fn, list1, ...)
  is similar to FULLMAP but it only maps onto lists and matrices

      (C1) FULLMAPL("+",[3,[4,5]],[[A,1],[0,-1.5]]);
      (D1)                      [[A + 3, 4], [4, 3.5]]

Function: IS (exp)
  attempts to determine whether exp (which must evaluate to a predicate) is provable from the facts in the current data base. IS returns TRUE if the predicate is true for all values of its variables consistent with the data base and returns FALSE if it is false for all such values. Otherwise, its action depends on the setting of the switch PREDERROR (default: TRUE). IS errs out if the value of PREDERROR is TRUE and returns UNKNOWN if PREDERROR is FALSE.

Function: ISQRT (X)
  takes one integer argument and returns the "integer SQRT" of its absolute value.

Function: MAX (X1, X2, ...)
  yields the maximum of its arguments (or returns a simplified form if some of its arguments are non-numeric).

Function: MIN (X1, X2, ...)
  yields the minimum of its arguments (or returns a simplified form if some of its arguments are non-numeric).

Function: MOD (poly)
  converts the polynomial poly to a modular representation with respect to the current modulus which is the value of the variable MODULUS. MOD(poly,m) specifies a MODULUS m to be used for converting poly, if it is desired to override the current global value of MODULUS. See DESCRIBE(MODULUS); .

Function: ODDP (exp)
  is TRUE if exp is an odd integer. FALSE is returned in all other cases.

operator: PRED
  (EVFLAG) causes predicates (expressions which evaluate to TRUE or FALSE) to be evaluated.

Function: RANDOM (X)
  returns a random integer between 0 and X-1. If no argument is given then a random integer between -2^(29) and 2^(29) -1 is returned. If X is FALSE then the random sequence is restarted from the beginning. Note that the range of the returned result when no argument is given differs in NIL MACSYMA from that of PDP-10 and Multics MACSYMA, which is -2^(35) to 2^(35) -1. This range is the range of the FIXNUM datatype of the underlying LISP.

Function: SIGN (exp)
  attempts to determine the sign of its specified expression 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)
  if X<0 then -1 else if X>0 then 1 else 0. If X is not numeric then a simplified but equivalent form is returned. For example, SIGNUM(-X) gives -SIGNUM(X).

Function: SORT (list,optional-predicate)
  sorts the list using a suitable optional-predicate of two arguments (such as "<" or ORDERLESSP). If the optional-predicate is not given, then MACSYMA's built-in ordering predicate is used.

Function: SQRT (X)
  the square root of X. It is represented internally by X^(1/2). Also see ROOTSCONTRACT. RADEXPAND[TRUE] - 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.

Variable: SQRTDISPFLAG
  default: [TRUE] - if FALSE causes SQRT to display with exponent 1/2.

Function: SUBLIS (list,expr)
  allows multiple substitutions into an expression in parallel. Sample syntax:

              SUBLIS([A=B,B=A],SIN(A)+COS(B));
               => SIN(B) + COS(A)

  The variable SUBLIS_APPLY_LAMBDA[TRUE] controls simplification after SUBLIS. For full documentation, see the file SHARE2;SUBLIS INFO.

Function: SUBLIST (L,F)
  returns the list of elements of the list L for which the function F returns TRUE. E.g., SUBLIST([1,2,3,4],EVENP); returns [2,4].

Variable: SUBLIS_APPLY_LAMBDA
  default:[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 "s 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. Note: SUBST is an alias for SUBSTITUTE. SUBST(eq1,exp) or SUBST([eq1,...,eqk],exp) are other permissible forms. The eqi are equations indicating substitutions to be made. For each equation, the right side will be substituted for the left in the expression exp. EXPTSUBST[FALSE] if TRUE permits substitutions like Y for %E**X in %E**(A*X) to take place. OPSUBST[TRUE] if 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.

      (C1) SUBST(A,X+Y,X+(X+Y)**2+Y);
                                       2
      (D1)                    Y + X + A
      (C2) SUBST(-%I,%I,A+B*%I);
      (D2)                             A - %I B

  (Note that C2 is one way of obtaining the complex conjugate of an analytic expression.) For further examples, do EXAMPLE(SUBST);

Function: SUBSTINPART (x, exp, n1, ...)
  is like SUBSTPART but works on the internal representation of exp.

      (C1) X.'DIFF(F(X),X,2);
                                         2
                                        d
      (D1)                         X . (--- F(X))
                                          2
                                        dX
      (C2) SUBSTINPART(D**2,%,2);
                                            2
      (D2)                             X . D
      (C3) SUBSTINPART(F1,F[1](X+1),0);
      (D3)                            F1(X + 1)
                    Additional Information
      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

      (C1) PART(X+Y+Z,[1,3]);
      (D1) 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[FALSE] is set to TRUE then END is returned when a
      selected part of an expression doesn't exist, otherwise an error
      message is given.
      (C1)  27*Y**3+54*X*Y**2+36*X**2*Y+Y+8*X**3+X+1;
                  3         2       2            3
      (D1)    27 Y  + 54 X Y  + 36 X  Y + Y + 8 X  + X + 1
      (C2)  PART(D1,2,[1,3]);
                        2
      (D2)          54 Y
      (C3)  SQRT(PIECE/54);
      (D3)            Y
      (C4)  SUBSTPART(FACTOR(PIECE),D1,[1,2,3,5]);
                                3
      (D4)           (3 Y + 2 X) + Y + X + 1
      (C5) 1/X+Y/X-1/Z;
                                       1   Y   1
      (D5)                           - - + - + -
                                       Z   X   X
      (C6) SUBSTPART(XTHRU(PIECE),%,[2,3]);
                                      Y + 1   1
      (D6)                            ----- - -
                                        X     Z

  Also, setting the option INFLAG to TRUE and calling PART/SUBSTPART is the same as calling INPART/SUBSTINPART.

Function: SUBSTPART (x, exp, n1, ..., nk)
  substitutes x for the subexpression picked out by the rest of the arguments as in PART. It returns the new value of exp. x may be some operator to be substituted for an operator of exp. In some cases it needs to be enclosed in "s (e.g. SUBSTPART("+",A*B,0); -> B + A ).

      (C1) 1/(X**2+2);
                                          1
      (D1)                              ------
                                         2
                                        X  + 2
      (C2) SUBSTPART(3/2,%,2,1,2);
                                          1
      (D2)                             --------
                                        3/2
                                       X    + 2
      (C3) A*X+F(B,Y);
      (D3)                          A X + F(B, Y)
      (C4) SUBSTPART("+",%,1,0);
      (D4)                         X + F(B, Y) + A

  Also, setting the option INFLAG to TRUE and calling PART/SUBSTPART is the same as calling INPART/SUBSTINPART.

Function: SUBVARP (exp)
  is TRUE if exp is a subscripted variable, for example A[I].

Function: SYMBOLP (exp)
  returns TRUE if "exp" is a "symbol" or "name", else FALSE. I.e., in effect, SYMBOLP(X):=ATOM(X) AND NOT NUMBERP(X)$ .

Function: UNORDER ()
  stops 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. Do DESCRIBE(ORDERGREAT); and DESCRIBE(ORDERLESS);, and also do EXAMPLE(UNORDER); for specifics.

Function: VECTORPOTENTIAL (givencurl)
  Returns the vector potential of a given curl vector, in the current coordinate system. POTENTIALZEROLOC has a similar role as for POTENTIAL, but the order of the left-hand sides of the equations must be a cyclic permutation of the coordinate variables.

Function: XTHRU (exp)
  combines all terms of exp (which should be a sum) over a common denominator without expanding products and exponentiated sums as RATSIMP does. XTHRU cancels common factors in the numerator and denominator of rational expressions but only if the factors are explicit. Sometimes it is better to use XTHRU before RATSIMPing an expression in order to cause explicit factors of the gcd of the numerator and denominator to be canceled thus simplifying the expression to be RATSIMPed.

      (C1) ((X+2)**20-2*Y)/(X+Y)**20+(X+Y)**-19-X/(X+Y)**20;
                                                      20
                           1           X       (X + 2)   - 2 Y
      (D1)             --------- - --------- + ---------------
                              19          20             20
                       (Y + X)     (Y + X)        (Y + X)
      (C2) XTHRU(%);
                                           20
                                    (X + 2)   - Y
      (D2)                          -------------
                                             20
                                      (Y + X)

Function: ZEROEQUIV (exp,var)
  tests whether the expression exp in the variable var is equivalent to zero. It returns either TRUE, FALSE, or DONTKNOW. 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) will return DONTKNOW because of the presence of an extra parameter. The restrictions are: (1) Do not use functions that MACSYMA does not know how to differentiate and evaluate. (2) If the expression has poles on the real line, there may be errors in the result (but this is unlikely to occur). (3) If the expression contains functions which are not solutions to first order differential equations (e.g. Bessel functions) there may be incorrect results. (4) The algorithm uses evaluation at randomly chosen points for carefully selected subexpressions. This is always a somewhat hazardous business, although the algorithm tries to minimize the potential for error.

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