**EXPREAD** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

*expread*reads one well-formed expression from the current input
buffer and returns its value.

expread(); a+b; A + B

**FACTORIZE** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *factorize* operator factors a given expression into a list of
{factor,power} pairs.

*factorize*(<expression>)

<expression> should be a polynomial, otherwise an error will result.

fff := factorize(x^3 - y^3); 2 2 {{X + X*Y + Y ,1},{X - Y,1}} fac1 := first fff; 2 2 FAC1 := {{X + X*Y + Y ,1} factorize(x^15 - 1); 8 7 6 5 4 {{ X - X + X - X + X - X + 1,1}, 4 3 2 {X + X + X + X + 1,1}, 2 {X + X + 1,1}, {X - 1,1}} lastone := part(ws,length ws); LASTONE := {X - 1,1} setmod 2; 1 on modular; factorize(x^15 - 1); 4 3 2 {{X + X + X + X + 1,1}, 4 3 {X + X + 1,1}, 4 {X + X + 1,1}, 2 { X + X + 1,1}, {X + 1,1}}

The *factorize* command returns the factor,power pairs as a
list.
You can therefore use the usual list access methods (
first,
second,
third,
rest,
length and
part) to extract these pairs.

If the <expression> given to *factorize* is an integer, it will be
factored into its prime components. To factor any integer factor of a
non-numerical expression, the switch
ifactor should be turned on.
Its default is off.
ifactor has effect only when factoring is
explicitly done by *factorize*, not when factoring is automatically
done with the
factor switch. If full factorization is not
needed the switch
limitedfactors allows you to reduce the
computing time of calls to *factorize*.

Factoring can be done in a modular domain by calling *factorize* when
modular is on. You can set the modulus with t
he
setmod
command. The last example above shows factoring modulo 2.

For general comments on factoring, see comments under the switch factor.

**HYPOT** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

If *rounded* is on, and the two arguments evaluate to numbers, this
operator returns the square root of the sums of the squares of the
arguments in a manner that avoids intermediate overflow. In other cases,
an expression in the original operator is returned.

hypot(3,4); HYPOT(3,4) on rounded; ws; 5.0 hypot(a,b); HYPOT(A,B)

**IMPART** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

This operator returns the imaginary part of an expression, if that
argument has an numerical value. A non-numerical argument is returned as
an expression in the operators
repart and *impart*.

impart(1+i); 1 impart(a+i*b); REPART(B) + IMPART(A)

**INT** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *int* operator performs analytic integration on a variety of
functions.

<expression> can be any scalar expression. involving polynomials, log
functions, exponential functions, or tangent or arctangent expressions.
*int* attempts expressions involving error functions, dilogarithms
and other trigonometric expressions. Integrals involving algebraic
extensions (such as square roots) may not succeed. <kernel> must be a
REDUCE
kernel.

int(x**3 + 3,x); 3 X*(X + 12) ----------- 4 int(sin(x)*exp(2*x),x); 2*X E *(COS(X) - 2*SIN(X)) - ------------------------ 5 int(1/(x^2-2),x); SQRT(2)*(LOG( - SQRT(2) + X) - LOG(SQRT(2) + X)) ------------------------------------------------ 4 int(sin(x)/(4 + cos(x)**2),x); COS(X) ATAN(------) 2 - ------------ 2 int(1/sqrt(x^2-x),x); SQRT(X)*SQRT(X - 1) INT(-------------------,X) 2 X -X

Note that REDUCE couldn't handle the last integral with its defaul t integrator, since the integrand involves a square root. However, the integral can be found using the algint package. Alternatively, you could add a rule using the let statement to evaluate this integral.

The arbitrary constant of integration is not shown. Definite integrals can be found by evaluating the result at the limits of integration (use rounded) and subtracting the lower from the h igher. Evaluation can be easily done by the sub operator.

When *int* cannot find an integral it returns an expression
involving formal *int* expressions unless the switch
failhard has been set. If not all of the expr
ession
can be integrated, the switch
nolnr controls whether a partially
integrated result should be returned or not.

**INTERPOL** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

*interpol*generates an interpolation polynomial.

interpol(<values>,<variable>,<points>)

<values> and <points> are
lists of equal length and
<variable> is an algebraic expression (preferably a
kernel).
The interpolation polynomial is generated in the given variable of degree
length(<values>)-1. The unique polynomial *f* is defined by the
property that for corresponding elements *v* of <values> and
*p* of <points> the relation *f(p)=v* holds.

f := for i:=1:4 collect(i**3-1); F := 0,7,26,63 p := {1,2,3,4}; P := 1,2,3,4 interpol(f,x,p); 3 X - 1

The Aitken-Neville interpolation algorithm is used which guarantee s a stable result even with rounded numbers and an ill-conditioned problem.

**LCOF** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *lcof* operator returns the leading coefficient of a given expression
with respect to a given variable.

*lcof*(<expression>,<kernel>)

<expression> is ordinarily a polynomial. If ratarg is on, a rational expression may also be used, otherwise an error results. <kernel> must be a kernel.

lcof((x+2*y)**5,y); 32 lcof((x + y*sin(x))**2 + cos(x)*sin(x)**2,sin(x)); 2 COS(X) + Y lcof(x**2 + 3*x + 17,y); 2 X + 3*X + 17

If the kernel does not appear in the expression, *lcof* ret
urns the
expression.

**LENGTH** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *length* operator returns the number of items in a
list, the
number of
terms in an expression, or the dimensions of an array or matrix.

*length*(<expr>) or *length* <expr>

<expr> can be a list structure, an array, a matrix, or a scalar expression .

alist := {a,b,{ww,xx,yy,zz}}; ALIST := {A,B,{WW,XX,YY,ZZ}} length alist; 3 length third alist; 4 dlist := {d}; DLIST := {D} length rest dlist; 0 matrix mmm(4,5); length mmm; {4,5} array aaa(5,3,2); length aaa; {6,4,3} eex := (x+3)**2/(x-y); 2 X + 6*X + 9 EEX := ------------ X - Y length eex; 5

An item in a list that is itself a list only counts as one item. A
n error
message will be printed if *length* is called on a matrix which has
not had its dimensions set. The *length* of an array includes the
zeroth element of each dimension, showing the full number of elements
allocated. (Declaring an array A with n elements
allocates A(0),A(1),...,A(n).) The
*length* of an expression is the total number of additive terms
appearing in the numerator and denominator of the expression. Note that
subtraction of a term is represented internally as addition of a negative
term.

**LHS** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *lhs* operator returns the left-hand side of an
equation,
such as those
returned in a list by
solve.

*lhs*(<equation>) or *lhs* <equation>

<equation> must be an equation of the form

*left-hand side**=**right-hand side*.

polly := (x+3)*(x^4+2x+1); 5 4 2 POLLY := X + 3*X + 2*X + 7*X + 3 pollyroots := solve(polly,x); POLLYROOTS := {X=ROOT F(X3 - X2 + X + 1,X , O ) X=-1, X=-3} variable := lhs first pollyroots; VARIABLE := X

**LIMIT** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

LIMITS is a fast limit package for REDUCE for functions which are continuous except for computable poles and singularities, based on some earlier work by Ian Cohen and John P. Fitch. The Truncated Power Series package is used for non-critical points, at which the value of the function is the constant term in the expansion around that point. l'Hopital's rule is used in critical cases, with preprocessing of 1-1 forms and reformatting of product forms in order to apply l'Hopital's rule. A limited amount of bounded arithmetic is also employed where applicable.

*limit!+*(<expr>,<var>,<limpoint>) or

*limit!-*(<expr>,<var>,<limpoint>)

where <expr> is an expression depending of the variable <var>
(a
kernel) and <limpoint> is the limit poi
nt.
If the limit depends upon the direction of approach to the <limpoint>,
the operators *limit!+* and *limit!-* may be used.

limit(x*cot(x),x,0); 0 limit((2x+5)/(3x-2),x,infinity); 2 -- 3

**LPOWER** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *lpower* operator returns the leading power of an expression with
respect to a kernel. 1 is returned if the expression does not depend on
the kernel.

*lpower*(<expression>,<kernel>)

<expression> is ordinarily a polynomial. If ratarg is on, a rational expression may also be used, otherwise an error results. <kernel> must be a kernel.

lpower((x+2*y)**6,y); 6 Y lpower((x + cos(x))**8 + df(x**2,x),cos(x)); 8 COS(X) lpower(x**3 + 3*x,y); 1

**LTERM** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *lterm* operator returns the leading term of an expression with
respect to a kernel. The expression is returned if it does not depend on
the kernel.

*lterm*(<expression>,<kernel>)

<expression> is ordinarily a polynomial. If ratarg is on, a rational expression may also be used, otherwise an error results. <kernel> must be a kernel.

lterm((x+2*y)**6,y); 6 64*Y lterm((x + cos(x))**8 + df(x**2,x),cos(x)); 8 COS(X) lterm(x**3 + 3*x,y); 3 X + 3X

**MAINVAR** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *mainvar* operator returns the main variable (in the system's
internal representation) of its argument.

*mainvar*(<expression>)

<expression> is usually a polynomial, but may be any valid REDUCE scalar expression. In the case of a rational function, the main variable of the numerator is returned. The main variable returned is a kernel.

test := (a + b + c)**2; 2 2 2 TEST := A + 2*A*B + 2*A*C + B + 2*B*C + C mainvar(test); A korder c,b,a; mainvar(test); C mainvar(2*cos(x)**2); COS(X) mainvar(17); 0

The main variable is the first variable in the canonical ordering
of
kernels. Generally, alphabetically ordered functions come first, then
alphabetically ordered identifiers (variables). Numbers come last, and as
far as *mainvar* is concerned belong in the family *0*. The
canonical ordering can be changed by the declaration
korder, as
shown above.

**MAP** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *map* operator applies a uniform evaluation pattern
to all members of a composite structure: a
matrix,
a
list or the arguments of an
operator expression.
The evaluation pattern can be a
unary procedure, an operator, or an algebraic expression with
one free variable.

*map*(<function>,<object>)

<object> is a list, a matrix or an operator expression.

<function> is the name of an operator for a single argument: the operator is evaluated once with each element of <object> as its single argument,

or an algebraic expression with exactly one free variable, that is a variable preceded by the tilde symbol: the expression is evaluated for each element of <object> where the element is substituted for the free variable,

or a replacement rule of the form

*var*=> *rep*

where <var> is a variable (a <kernel> without subscript)
and <rep> is an expression which contains <var>.
Here *rep* is evaluated for each element of <object> where
the element is substituted for *var*. *var* may be
optionally preceded by a tilde.

The rule form for <function> is needed when more than one free variable occurs.

map(abs,{1,-2,a,-a}); 1,2,abs(a),abs(a) map(int(~w,x), mat((x^2,x^5),(x^4,x^5))); [ 3 6 ] [ x x ] [---- ----] [ 3 6 ] [ ] [ 5 6 ] [ x x ] [---- ----] [ 5 6 ] map(~w*6, x^2/3 = y^3/2 -1); 2 3 2*x =3*(y -2)

You can use *map* in nested expressions. It is not allowed
to
apply *map* for a non-composed object, e.g. an identifier or a number.

**MKID** _ _ _ _ _ _ _ _ _ _ _ _ **command**

The *mkid* command constructs an identifier, given a stem and an identifi
er
or an integer.

*mkid*(<stem>,<leaf>)

<stem> can be any valid REDUCE identifier that does not include escaped special characters. <leaf> may be an integer, including one given by a local variable in a for loop, or any other legal group of characters.

mkid(x,3); X3 factorize(x^15 - 1); {X - 1, 2 X + X + 1, 4 3 2 X + X + X + X + 1, 8 7 5 4 3 X - X + X - X + X - X + 1} for i := 1:length ws do write set(mkid(f,i),part(ws,i)); 8 7 5 4 3 X - X + X - X + X - X + 1 4 3 2 X + X + X + X + 1 2 X + X + 1 X - 1

You can use *mkid* to construct identifiers from inside pro
cedures. This
allows you to handle an unknown number of factors, or deal with variable
amounts of data. It is particularly helpful to attach identifiers to the
answers returned by *factorize* and *solve*.

**NPRIMITIVE** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

This operator returns the numerically-primitive part of any scalar expression. In other words, any overall integer factors in the expression are removed.

nprimitive((2x+2y)^2); 2 2 X + 2*X*Y + Y nprimitive(3*a*b*c); 3*A*B*C

**NUM** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *num* operator returns the numerator of its argument.

*num*(<expression>) or *num* <simple\_expression>

<expression> can be any valid REDUCE scalar expression.

num(100/6); 50 num(a/5 + b/6); 6*A + 5*B num(sin(x)); SIN(X)

*num*returns the numerator of the expression after it has b
een simplified
by REDUCE. As seen in the examples, this includes putting sums of rational
expressions over a common denominator, and reducing common factors where
possible. If the expression is not a rational expression, it is returned
unchanged.

**ODESOLVE** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *odesolve* package is a solver for ordinary differential
equations. At the present time it has still limited capabilities:

1. it can handle only a single scalar equation presented as an algebraic expression or equation, and

2. it can solve only first-order equations of simple types, linear equations with constant coefficients and Euler equations.

These solvable types are exactly those for which Lie symmetry techniques give no useful information.

<expr> is a single scalar expression such that <expr>=0 is the ordinary differential equation (ODE for short) to be solved, or is an equivalent equation.

<var1> is the name of the dependent variable, <var2> is the name of the independent variable.

A differential in <expr> is expressed using the
df
operator. Note that in most cases you must declare explicitly
<var1> to depend of <var2> using a
depend
declaration -- otherwise the derivative might be evaluated to
zero on input to *odesolve*.

The returned value is a list containing the equation giving the general
solution of the ODE (for simultaneous equations this will be a
list of equations eventually). It will contain occurrences of
the operator *arbconst* for the arbitrary constants in the general
solution. The arguments of *arbconst* should be new.
A counter *!!arbconst* is used to arrange this.

depend y,x; % A first-order linear equation, with an initial condition ode:=df(y,x) + y * sin x/cos x - 1/cos x$ odesolve(ode,y,x); {y=arbconst(1)*cos(x) + sin(x)}

**ONE\_OF** _ _ _ _ _ _ _ _ _ _ _ _ **type**

The operator *one_of* is used to represent an indefinite choice
of one element from a finite set of objects.

x=one_of{1,2,5}

this equation encodes that x can take one of the values 1,2 or 5

REDUCE generates a *one_of* form in cases when an implicit
*root_of* expression could be converted to an explicit solution set.
A *one_of* form can be converted to a *solve* solution using
expand_cases. See
root_of.

**PART** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The operator *part* permits the extraction of various parts or
operators of expressions and
list*s*.

*part*(<expression,integer>{,<integer>}*)

<expression> can be any valid REDUCE expression or a list,
integer may be an expression that evaluates to a positive or negative
integer or 0. A positive integer <n> picks up the n th term,
counting from the first term toward the end. A negative integer n
picks up the n th term, counting from the back toward the front. The
integer 0 picks up the operator (which is *LIST* when the expression
is a
list).

part((x + y)**5,4); 2 3 10*X *Y part((x + y)**5,4,2); 2 X part((x + y)**5,4,2,1); X part((x + y)**5,0); PLUS part((x + y)**5,-5); 4 5*X *Y part((x + y)**5,4) := sin(x); 5 4 3 2 4 5 X + 5*X *Y + 10*X *Y + SIN(X) + 5*X*Y + Y alist := {x,y,{aa,bb,cc},x**2*sqrt(y)}; 2 ALIST := {X,Y,{AA,BB,CC},SQRT(Y)*X } part(alist,3,2); BB part(alist,4,0); TIMES

Additional integer arguments after the first one examine the terms recursively, as shown above. In the third line, the fourth term is picked from the original polynomial, 10x^2y^3, then the second term from that, x^2, and finally the first component, x. If an integer's absolute value is too large for the appropriate expression, a message is given.

*part*works on the form of the expression as printed, or as it would
have been printed at that point of the calculation, bearing in mind the
current switch settings. It is important to realize that the switch settings
change the operation of *part*.
pri must be on when
*part* is used.

When *part* is used on a polynomial expression that has minus signs, the
*+* is always returned as the top-level operator. The minus is found
as a unary operator attached to the negative term.

*part*can also be used to change the relevant part of the expression or
list as shown in the sixth example line. The *part* operator returns the
changed expression, though original expression is not changed. You can
also use *part* to change the operator.

**PF** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

*pf*transforms <expression> into a
list of partial fraction
s
with respect to the main variable, <variable>. *pf* does a
complete partial fraction decomposition, and as the algorithms used are
fairly unsophisticated (factorization and the extended Euclidean
algorithm), the code may be unacceptably slow in complicated cases.

pf(2/((x+1)^2*(x+2)),x); 2 -2 2 {-----,-----,------------} X + 2 X + 1 2 X + 2*X + 1 off exp; pf(2/((x+1)^2*(x+2)),x); 2 - 2 2 {-----,-----,--------} X + 2 X + 1 2 (X + 1) for each j in ws sum j; 2 ---------------- 2 ( + 2)*(X + 1)

If you want the denominators in factored form, turn
exp off, as
shown in the second example above. As shown in the final example, the
for *each* construct can be used to re
combine the terms.
Alternatively, one can use the operations on lists to extract any desired
term.

**PROD** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The operator *prod* returns
the indefinite or definite product of a given expression.

where <expr> is the expression to be multiplied, <k> is the control variable (a kernel), and <lolim> and <uplim> uplim are the optional lower and upper limits. If <uplim> is not supplied the upper limit is taken as <k>. The Gosper algorithm is used. If there is no closed form solution, the operator returns the input unchanged.

prod(k/(k-2),k); k*( - k + 1)

**REDUCT** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *reduct* operator returns the remainder of its expression after the
leading term with respect to the kernel in the second argument is removed.

*reduct*(<expression>,<kernel>)

reduct((x+y)**3,x); 2 2 Y*(3*X + 3*X*Y + Y ) reduct(x + sin(x)**3,sin(x)); X reduct(x + sin(x)**3,y); 0

If the expression does not contain the kernel, *reduct* ret
urns 0.

**REPART** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

This operator returns the real part of an expression, if that argument has an
numerical value. A non-numerical argument is returned as an expression in
the operators *repart* and
impart.

repart(1+i); 1 repart(a+i*b); REPART(A) - IMPART(B)

**RESULTANT** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *resultant* operator computes the resultant of two polynomials with
respect to a given variable. If the resultant is 0, the polynomials have
a root in common.

*resultant*(<expression>,<expression>,<kernel>)

<expression> must be a polynomial containing <kernel> ; <kernel> must be a kernel.

resultant(x**2 + 2*x + 1,x+1,x); 0 resultant(x**2 + 2*x + 1,x-3,x); 16 resultant(z**3 + z**2 + 5*z + 5, z**4 - 6*z**3 + 16*z**2 - 30*z + 55, z); 0 resultant(x**3*y + 4*x*y + 10,y**2 + 6*y + 4,y); 6 5 4 3 2 Y + 18*Y + 120*Y + 360*Y + 480*Y + 288*Y + 64

The resultant is the determinant of the Sylvester matrix, formed f rom the coefficients of the two polynomials in the following way:

Given two polynomials:

n n-1 a x + a1 x + ... + an

and

m m-1 b x + b1 x + ... + bm

form the (m+n)x(m+n-1) Sylvester matrix by the following means:

0.......0 a a1 .......... an 0....0 a a1 .......... an 0 . . . . a0 a1 .......... an 0.......0 0.......0 b b1 .......... bm 0....0 b b1 .......... bm 0 . . . . b b1 .......... bm 0.......0

If the determinant of this matrix is 0, the two polynomials have a common root. Finding the resultant of large expressions is time-consuming, due to the time needed to find a large determinant.

The sign conventions *resultant* uses are those given in the article,
``Computing in Algebraic Extensions,'' by R. Loos, appearing in
<Computer Algebra--Symbolic and Algebraic Computation>, 2nd ed.,
edited by B. Buchberger, G.E. Collins and R. Loos, and published by
Springer-Verlag, 1983.
These are:

resultant(p(x),q(x),x) = (-1)^{deg p(x)*deg q(x)} * resultant(q(x),p(x),x), resultant(a,p(x),x) = a^{deg p(x)}, resultant(a,b,x) = 1

where p(x) and q(x) are polynomials which have x as a variable, an d a and b are free of x.

Error messages are given if *resultant* is given a non-polynomial
expression, or a non-kernel variable.

**RHS** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *rhs* operator returns the right-hand side of an
equation,
such as those returned in a
list by
solve.

*rhs*(<equation>) or *rhs* <equation>

<equation> must be an equation of the form left-hand side = right-hand side.

roots := solve(x**2 + 6*x*y + 5x + 3y**2,x); 2 SQRT(24*Y + 60*Y + 25) + 6*Y + 5 ROOTS := {X= - ---------------------------------, 2 2 SQRT(24*Y + 60*Y + 25) - 6*Y - 5 X= ---------------------------------} 2 root1 := rhs first roots; 2 SQRT(24*Y + 60*Y + 25) + 6*Y + 5 ROOT1 := - --------------------------------- 2 root2 := rhs second roots; 2 SQRT(24*Y + 60*Y + 25) - 6*Y - 5 ROOT2 := ---------------------------------- 2

An error message is given if *rhs* is applied to something
other than an
equation.

**ROOT\_OF** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

When the operator
solve is unable to find an explicit solution
or if that solution would be too complicated, the result is presented
as formal root expression using the internal operator *root_of*
and a new local variable. An expression with a top level *root_of*
is implicitly a list with an unknown number of elements since we
can't always know how many solutions an equation has. If a
substitution is made into such an expression, closed form solutions
can emerge. If this occurs, the *root_of* construct is
replaced by an operator
one_of. At this point it is
of course possible to transform the result if the original *solve*
operator expression into a standard *solve* solution. To
effect this, the operator
expand_cases can be used.

solve(a*x^7-x^2+1,x); 7 2 {x=root_of(a*x_ - x_ + 1,x_)} sub(a=0,ws); {x=one_of(1,-1)} expand_cases ws; x=1,x=-1

The components of *root_of* and *one_of* expressions
can be
processed as usual with operators
arglength and
part.
A higher power of a *root_of* expression with a polynomial
as first argument is simplified by using the polynomial as a side relation.

**SELECT** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *select* operator extracts from a list
or from the arguments of an n--ary operator elements corresponding
to a boolean predicate. The predicate pattern can be a
unary procedure, an operator or an algebraic expression with
one
free variable.

*select*(<function>,<object>)

<object> is a list.

<function> is the name of an operator for a single argument: the operator is evaluated once with each element of <object> as its single argument,

or an algebraic expression with exactly one free variable, that is a variable preceded by the tilde symbol: the expression is evaluated for each element of <object> where the element is substituted for the free variable,

or a replacement rule of the form

*var*=> *rep*

where <var> is a variable (a <kernel> without subscript)
and <rep> is an expression which contains <var>.
Here *rep* is evaluated for each element of <object> where
the element is substituted for *var*. *var* may be
optionally preceded by a tilde.

The rule form for <function> is needed when more than one free variable occurs. The evaluation result of <function> is interpreted as boolean value corresponding to the convention s of REDUCE. The result value is built with the leading operator of the input expression.

select( ~w>0 , {1,-1,2,-3,3}) {1,2,3} q:=(part((x+y)^5,0):=list) select(evenp deg(~w,y),q); 5 3 2 4 {x ,10*x *y ,5*x*y } select(evenp deg(~w,x),2x^2+3x^3+4x^4); 2 4 2x +4x

**SHOWRULES** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

*showrules*returns in
rule*-list* form any
operator rules associated with its argument.

showrules log; {LOG(E) => 1, LOG(1) => 0, ~X LOG(E ) => ~X, 1 DF(LOG(~X),~X) => --} ~X

Such rules can then be manipulated further as with any
list. For
example
*rhs first ws;* has the value 1.

An operator may have properties that cannot be displayed in such a form, such as the fact it is an odd function, or has a definition defined as a procedure.

**SOLVE** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *solve* operator solves a single algebraic
equation or a
system of simultaneous equations.

*solve*({<expression>,...} [ ,{ <kernel> ,...}] )

If the number of equations equals the number of distinct kernels, the optional kernel argument(s) may be omitted. <expression> is either a scalar expression or an equation. When more than one expression is given, the list of expressions is surrounded by curly br aces. The optional list of kernels follows, also in curly braces.

sss := solve(x^2 + 7); Unknown: X SSS := {X= - SQRT(7)*I, X=SQRT(7)*I} rhs first sss; - SQRT(7)*I solve(sin(x^2*y),y); 2*ARBINT(1)*PI {Y=--------------- 2 X PI*(2*ARBINT(1) + 1) Y=--------------------} 2 X off allbranch; solve(sin(x**2*y),y); {Y=0} solve({3x + 5y = -4,2*x + y = -10},{x,y}); 22 46 {{X= - --,Y=--}} 7 7 solve({x + a*y + z,2x + 5},{x,y}); 5 2*Z - 5 {{X= - -,Y= - -------}} 2 2*A ab := (x+2)^2*(x^6 + 17x + 1); 8 7 6 3 2 AB := X + 4*X + 4*X + 17*X + 69*X + 72*X + 4 www := solve(ab,x); {X=ROOT F(X6 + 17*X + 1),X=-2} O root_multiplicities; {1,2}

Results of the *solve* operator are returned as
equation*s*
in a
list.
You can use the usual list access methods (
first,
second,
third,
rest and
part) to
extract the desired equation, and then use the operators
rhs and
lhs to access the right-hand or left-hand exp
ression of the
equation. When *solve* is unable to solve an equation, it returns the
unsolved part as the argument of *root_of*, with the variable renamed
to avoid confusion, as shown in the last example above.

For one equation, *solve* uses square-free factorization, roots of
unity, and the known inverses of the
log,
sin,
cos,
acos,
asin, and
exponentiation operators. The quadratic, cubic and quartic formulas are
used if necessary, but these are applied only when the switch
fullroots is set on; otherwise or when no clo
sed form is available
the result is returned as
root_of expression. The switch
trigform
determines which type of cubic and quartic formula is used.
The multiplicity of each solution is given in a list as
the system variable
root_multiplicities. For systems of
simultaneous linear equations, matrix inversion is used. For nonlinear
systems, the Groebner basis method is used.

Linear equation system solving is influenced by the switch cramer.

Singular systems can be solved when the switch solvesingular is on, which is the default setting. An empty list is returned the system of equations is inconsistent. For a linear inconsistent system with parameters the variable requirements constraints conditions for the system to become consistent.

For a solvable linear and polynomial system with parameters the variable assumptions contains a list side relations for the parameters: the solution is valid only as long as none of these expressions is zero.

If the switch
varopt is on (default), the system rearranges
the
variable sequence for minimal computation time. Without *varopt*
the user supplied variable sequence is maintained.

If the solution has free variables (dimension of the solution is greater
than zero), these are represented by
arbcomplex expressions
as long as the switch
arbvars is on (default). Without
*arbvars* no explicit equations are generated for free variables.

_ _ _ allbranchswitch

_ _ _ arbvars switch

_ _ _ assumptions variable

_ _ _ fullroots switch

_ _ _ requirements variable

_ _ _ roots operator

_ _ _ root_of operator

_ _ _ trigform switch

_ _ _ varopt switch

**SORT** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *sort* operator sorts the elements of a list according to
an arbitrary comparison operator.

*sort*(<lst>,<comp>)

<lst> is a
list of algebraic expressions.
<comp> is a comparison operator which defines a partial
ordering among the members of <lst>. <comp> may be
one of the builtin comparison operators like
*<*(
lessp), *<=*(
leq)
etc., or <comp> may be the name of a comparison procedure.
Such a procedure has two arguments, and it returns
true if the first argument
ranges before the second one, and 0 or
nil otherwise.
The result of *sort* is a new list which contains the
elements of <lst> in a sequence corresponding to <comp>.

procedure ce(a,b); if evenp a and not evenp b then 1 else 0; for i:=1:10 collect random(50)$ sort(ws,>=); {41,38,33,30,28,25,20,17,8,5} sort(ws,<); {5,8,17,20,25,28,30,33,38,41} sort(ws,ce); {8,20,28,30,38,5,17,25,33,41} procedure cd(a,b); if deg(a,x)>deg(b,x) then 1 else if deg(a,x)<deg(b,x) then 0 else if deg(a,y)>deg(b,y) then 1 else 0; sort({x^2,y^2,x*y},cd); 2 2 {x ,x*y,y }

**STRUCTR** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *structr* operator breaks its argument expression into named
subexpressions.

<expression> may be any valid REDUCE scalar expression.
<identifier> may be any valid REDUCE *identifier*. The first
identifier
is the stem for subexpression names, the second is the name to be assigned
to the structured expression.

structr(sqrt(x**2 + 2*x) + sin(x**2*z)); ANS1 + ANS2 where 2 ANS2 := SIN(X *Z) 1/2 ANS1 := ((X + 2)*X) ans3; ANS3 on fort; structr((x+1)**5 + tan(x*y*z),var,aa); VAR1=TAN(X*Y*Z) AA=VAR1+X**5+5.*X**4+10.*X**3+10.X**2+5.*X+1

The second argument to *structr* is optional. If it is not
given, the
default stem *ANS* is used by REDUCE to construct names for the
subexpression. The names are only for display purposes: REDUCE does not
store the names and their values unless the switch
savestructr is
on.

If a third argument is given, the structured expression as a whole is named by
this argument, when
fort is on. The expression is not stored
under this
name. You can send these structured Fortran expressions to a file with the
*out* command.

**SUB** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *sub* operator substitutes a new expression for a kernel in an
expression.

*sub*(<kernel>*=*<expression>
{,<kernel>*=*<expression>}*,
<expression>) or

*sub*({<kernel>*=*<expression>*,
<kernel>*=**expression*},<expression>)

<kernel> must be a kernel, <expression> can be any REDUCE scalar expression.

sub(x=3,y=4,(x+y)**3); 343 x; X sub({cos=sin,sin=cos},cos a+sin b) COS(B) + SIN(A)

Note in the second example that operators can be replaced using th
e
*sub* operator.

**SUM** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The operator *sum* returns
the indefinite or definite summation of a given expression.

where <expr> is the expression to be added, <k> is the control variable (a kernel), and <lolim> and <uplim> are the optional lower and upper limits. If <uplim> is not supplied the upper limit is taken as <k>. The Gosper algorithm is used. If there is no closed form solution, the operator returns the input unchanged.

sum(4n**3,n); 2 2 n *(n + 2*n + 1) sum(2a+2k*r,k,0,n-1); n*(2*a + n*r - r)

**WS** _ _ _ _ _ _ _ _ _ _ _ _ **operator**

The *ws* operator alone returns the last result; *ws* with a
number argument returns the results of the REDUCE statement executed after
that numbered prompt.

*ws*or *ws*(<number>)

<number> must be an integer between 1 and the current REDUCE prompt number .

(In the following examples, unlike most others, the nu mbered prompt is shown.)

1: df(sin y,y); COS(Y) 2: ws^2; 2 COS(Y) 3: df(ws 1,y); -SIN(Y)

*ws*and *ws**(*<number>*)* can be used anywher
e the
expression they stand for can be used. Calling a number for which no
result was produced, such as a switch setting, will give an error message.

The current workspace always contains the results of the last REDUCE
command that produced an expression, even if several input statements
that do not produce expressions have intervened. For example, if you do
a differentiation, producing a result expression, then change several
switches, the operator *ws;* returns the results of the differentiation.
The current workspace (*ws*) can also be used inside files, though the
numbered workspace contains only the *in* command that input the file.

There are three history lists kept in your REDUCE session. The first
stores raw input, suitable for the statement editor. The second stores
parsed input, ready to execute and accessible by
input. The
third stores results, when they are produced by statements, which are
accessible by the *ws*< n> operator. If your session is very
long, storage space begins to fill up with these expressions, so it is a
good idea to end the session once in a while, saving needed expressions to
files with the
saveas and
out commands.

An error message is given if a reference number has not yet been used.

**Algebraic Operators**

**ALGEBRAIC** _ _ _ _ _ _ _ _ _ _ _ _ **command**

The *algebraic* command changes REDUCE's mode of operation to
algebraic. When *algebraic* is used as an operator (with an
argument inside parentheses) that argument is evaluated in algebraic
mode, but REDUCE's mode is not changed.

algebraic; symbolic; NIL algebraic(x**2); 2 X x**2; ***** The symbol X has no value.

REDUCE's symbolic mode does not know about most algebraic commands . Error messages in this mode may also depend on the particular Lisp used for the REDUCE implementation.

**ANTISYMMETRIC** _ _ _ _ _ _ _ _ _ _ _ _ **declaration**

When an operator is declared *antisymmetric*, its arguments are
reordered to conform to the internal ordering of the system. If an odd
number of argument interchanges are required to do this ordering,
the sign of the expression is changed.

<identifier> is an identifier that has been declared as an operator.

operator m,n; antisymmetric m,n; m(x,n(1,2)); - M( - N(2,1),X) operator p; antisymmetric p; p(a,b,c); P(A,B,C) p(b,a,c); - P(A,B,C)

If <identifier> has not been declared an operator, the flag
*antisymmetric* is still attached to it. When <identifier> is
subsequently used as an operator, the message *Declare* <identifier
>
*operator? (Y or N)* is printed. If the user replies *y*, the
antisymmetric property of the operator is used.

Note in the first example, identifiers are customarily ordered alphabetically, while numbers are ordered from largest to smallest. The operators may have any desired number of arguments (less than 128).

**ARRAY** _ _ _ _ _ _ _ _ _ _ _ _ **declaration**

The *array* declaration declares a list of identifiers to be of type
*array*, and sets all their entries to 0.

*array*<identifier>(<dimensions>)
{*,*<identifier>(<dimensions>)}*

<identifier> may be any valid REDUCE identifier. If the identifier was already an array, a warning message is given that the array has been redefined. <dimensions> are of form <integer>{,<integer>}*.

array a(2,5),b(3,3,3),c(200); array a(3,5); *** ARRAY A REDEFINED a(3,4); 0 length a; {4,6}

Arrays are always global, even if defined inside a procedure or bl ock statement. Their status as an array remains until the variable is reset by clear. Arrays may not have the same names as operators, procedures or scalar variables.

Array elements are referred to by the usual notation: *a(i,j)*
returns the jth element of the ith row. The
assignment operator
*:=* is used to put values into the array. Arrays as a whole
cannot be subject to assignment by
let or *:=* ; the
assignment operator *:=* is only valid for individual elements.

When you use
let on an array element, the contents of that
element become the argument to *let*. Thus, if the element
contains a number or some other expression that is not a valid argument
for this command, you get an error message. If the element contains an
identifier, the identifier has the substitution rule attached to it
globally. The same behavior occurs with
clear. If the array
element contains an identifier or simple_expression, it is cleared. Do
<not> use *clear* to try to set an array element to 0. Because
of the side effects of either *let* or *clear*, it is unwise
to apply either of these to array elements.

Array indices always start with 0, so that the declaration *array a(5)*
sets aside 6 units of space, indexed from 0 through 5, and initializes
them to 0. The
length command returns a list of the true num
ber of
elements in each dimension.

**CLEAR** _ _ _ _ _ _ _ _ _ _ _ _ **command**

The *clear* command is used to remove assignments or remove substitution
rules from any expression.

<let-type statement> *clear* <identifier>

<identifier> can be any *scalar*,
matrix,
or
array variable or
procedure name. <let-type statement> ca
n be any general
or specific
let statement (see below in Comments).

array a(2,3); a(2,2) := 15; A(2,2) := 15 clear a; a(2,2); Declare A operator? (Y or N) let x = y + z; sin(x); SIN(Y + Z) clear x; sin(x); SIN(X) let x**5 = 7; clear x; x**5; 7 clear x**5; x**5; 5 X

Although it is not a good idea, operators of the same name but tak
ing
different numbers of arguments can be defined. Using a *clear* statement
on any of these operators clears every one with the same name, even if the
number of arguments is different.

The *clear* command is used to ``forget" matrices, arrays, operators
and scalar variables, returning their identifiers to the pristine state
to be used for other purposes. When *clear* is applied to array
elements, the contents of the array element becomes the argument for
*clear*. Thus, you get an error message if the element contains a
number, or some other expression that is not a legal argument to
*clear*. If the element contains an identifier, it is cleared.
When clear is applied to matrix elements, an error message is returned
if the element evaluates to a number, otherwise there is no effect. Do
not try to use *clear* to set array or matrix elements to 0.
You will not be pleased with the results.

If you are trying to clear power or product substitution rules made with
either
let or
forall...*let*, you must
reproduce the rule, exactly as you typed it with the same arguments, up to
but not including the equal sign, using the word *clear* instead of
the word *let*. This is shown in the last example. Any other type of
*let* or *forall*...*let* substitution can be cleared
with just the variable or operator name.
match behaves the same as
let in this situation. There is a more compli
cated example under
forall.

**CLEARRULES** _ _ _ _ _ _ _ _ _ _ _ _ **command**

The operator *clearrules* is used to remove previously defined
rule lists from the system. <list> can
be an explicit rule
list, or evaluate to a rule list.

trig1 := {cos(~x)*cos(~y) => (cos(x+y)+cos(x-y))/2, cos(~x)*sin(~y) => (sin(x+y)-sin(x-y))/2, sin(~x)*sin(~y) => (cos(x-y)-cos(x+y))/2, cos(~x)^2 => (1+cos(2*x))/2, sin(~x)^2 => (1-cos(2*x))/2}$ let trig1; cos(a)*cos(b); COS(A - B) + COS(A + B) ----------------------- 2 clearrules trig1; cos(a)*cos(b); COS(A)*COS(B)

**DEFINE** _ _ _ _ _ _ _ _ _ _ _ _ **command**

The command *define* allows you to supply a new name for an identifier
or replace it by any valid REDUCE expression.

<identifier> is any valid REDUCE identifier, <substitution> can be a number, an identifier, an operator, a reserved word, or an expression.

define is= :=, xx=y+z; a is 10; A := 10 xx**2; 2 2 Y + 2*Y*Z + Z xx := 10; Y + Z := 10

The renaming is done at the input level, and therefore takes prece dence over any other replacement or substitution declared for the same identifier. It remains in effect until the end of the REDUCE session. Be careful with it, since you cannot easily undo it without ending the session.

**DEPEND** _ _ _ _ _ _ _ _ _ _ _ _ **declaration**

*depend*declares that its first argument depends on the rest of its
arguments.

<kernel> must be a legal variable name or a prefix operator (see kernel).

depend y,x; df(y**2,x); 2*DF(Y,X)*Y depend z,cos(x),y; df(sin(z),cos(x)); COS(Z)*DF(Z,COS(X)) df(z**2,x); 2*DF(Z,X)*Z nodepend z,y; df(z**2,x); 2*DF(Z,X)*Z cc := df(y**2,x); CC := 2*DF(Y,X)*Y y := tan x; Y := TAN(X); cc; 2 2*TAN(X)*(TAN(X) + 1)

Dependencies can be removed by using the declaration
nodepend.
The differentiation operator uses this information, as shown in the
examples above. Linear operators also use knowledge of dependencies
(see
linear). Note that dependencies can be nested
: Having
declared y to depend on x, and z
to depend on y, we
see that the chain rule was applied to the derivative of a function of
z with respect to x. If the explicit function of the
dependency is later entered into the system, terms with *DF(Y,X)*,
for example, are expanded when they are displayed again, as shown in the
last example. The boolean operator
freeof allows you to
check the dependency between two algebraic objects.

**EVEN** _ _ _ _ _ _ _ _ _ _ _ _ **declaration**

*even*<identifier>{,<identifier>}*

This declaration is used to declare an operator even in its first argument. Expressions involving an operator declared in this manner are transformed if the first argument contains a minus sign. Any other arguments are not affected.

even f; f(-a) F(A) f(-a,-b) F(A,-B)

**FACTOR** _ _ _ _ _ _ _ _ _ _ _ _ **declaration**

When a kernel is declared by *factor*, all terms involving fixed
powers of that kernel are printed as a product of the fixed powers and
the rest of the terms.

*factor*<kernel> {*,*<kernel>}*

<kernel> must be a
kernel or a
list of
*kernel*s.

a := (x + y + z)**2; 2 2 2 A := X + 2*X*Y + 2*X*Z + Y + 2*Y*Z + Z factor y; a; 2 2 2 Y + 2*Y*(X + Z) + X + 2*X*Z + Z factor sin(x); c := df(sin(x)**4*x**2*z,x); 4 3 2 C := 2*SIN(X) *X*Z + 4*SIN(X) *COS(X)*X *Z remfac sin(x); c; 3 2*SIN(X) *X*Z*(2*COS(X)*X + SIN(X))

Use the *factor* declaration to display variables of intere
st so that
you can see their powers more clearly, as shown in the example. Remove
this special treatment with the declaration
remfac. The
*factor* declaration is only effective when the switch
pri
is on.

The *factor* declaration is not a factoring command; to factor
expressions use the
factor switch or the
factorize command.

The *factor* declaration is helpful in such cases as Taylor polynomials
where the explicit powers of the variable are expected at the top level, not
buried in various factored forms.

Note that *factor* does not affect the order of its arguments. You
should also use
order if this is important.

**FORALL** _ _ _ _ _ _ _ _ _ _ _ _ **command**

The *forall* or (preferably) *for all* command is used as a
modifier for
let statements, indicating the universal appl
icability
of the rule, with possible qualifications.

*for all*<identifier>{,<identifier>}* *let*
<let statement>

or

*for all*<identifier>{,<identifier>}*
*such that* <condition> *let* <let statement>

<identifier> may be any valid REDUCE identifier, <let statement> can be an operator, a product or power, or a group or block statement. <condition> must be a logical or comparison operator returning true or false.

for all x let f(x) = sin(x**2); Declare F operator ? (Y or N) y f(a); 2 SIN(A ) operator pos; for all x such that x>=0 let pos(x) = sqrt(x + 1); pos(5); SQRT(6) pos(-5); POS(-5) clear pos; pos(5); Declare POS operator ? (Y or N) for all a such that numberp a let x**a = 1; x**4; 1 clear x**a; *** X**A not found for all a clear x**a; x**4; 1 for all a such that numberp a clear x**a; x**4; 4 X

Substitution rules defined by *for all* or *for
all*...*such that* commands that involve products or powers are
cleared by reproducing the command, with exactly the same variable names
used, up to but not including the equal sign, with
clear
replacing *let*, as shown in the last example. Other substitutions
involving variables or operator names can be cleared with just the name,
like any other variable.

The
match command can also be used in product and
power substitutions.
The syntax of its use and clearing is exactly like *let*. A *match
*
substitution only replaces the term if it is exactly like the pattern, for
example *match x**5 = 1* replaces only terms of *x**5* and not
terms of higher powers.

It is easier to declare your potential operator before defining the
*for all* rule, since the system will ask you to declare it an
operator anyway. Names of declared arrays or matrices or scalar
variables are invalid as operator names, to avoid ambiguity. Either
*for all*...*let* statements or procedures are often used to defin
e
operators. One difference is that procedures implement ``call by value"
meaning that assignments involving their formal parameters do not change
the calling variables that replace them. If you use assignment statements
on the formal parameters in a *for all*...*let* statement, the
effects are seen in the calling variables. Be careful not to redefine a
system operator unless you mean it: the statement *for all x let
sin(x)=0;* has exactly that effect, and the usual definition for sin(x) has
been lost for the remainder of the REDUCE session.

**INFIX** _ _ _ _ _ _ _ _ _ _ _ _ **declaration**

*infix*declares identifiers to be infix operators.

*infix*<identifier>{,<identifier>}*

<identifier> can be any valid REDUCE identifier, which has not already been declared an operator, array or matrix, and is not reserved by the system.

infix aa; for all x,y let aa(x,y) = cos(x)*cos(y) - sin(x)*sin(y); x aa y; COS(X)*COS(Y) - SIN(X)*SIN(Y) pi/3 aa pi/2; SQRT(3) - ------- 2 aa(pi,pi); 1

A
let statement must be used to attach function
ality to
the operator. Note that the operator is defined in prefix form in
the *let* statement.
After its definition, the operator may be used in either prefix or infix
mode. The above operator aa finds the cosine of the sum
of two angles by the formula

cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y).

Precedence may be attached to infix operators with the precedence declaration.

User-defined infix operators may be used in prefix form. If they are used in infix form, a space must be left on each side of the operator to avoid ambiguity. Infix operators are always binary.

**INTEGER** _ _ _ _ _ _ _ _ _ _ _ _ **declaration**

The *integer* declaration must be made immediately after a
begin (or other variable declaration such as
real
and
scalar) and declares local integer variables.
They are
initialized to 0.

*integer*<identifier>{,<identifier>}*

<identifier> may be any valid REDUCE identifier, except
*t* or *nil*.

Integer variables remain local, and do not share values with variables of
the same name outside the
begin...*end* block. When the
block is finished, the variables are removed. You may use the words
real or
scalar in the place of *integer*.
*integer* does not indicate typechecking by the
current REDUCE; it is only for your own information. Declaration
statements must immediately follow the *begin*, without a semicolon
between *begin* and the first variable declaration.

Any variables used inside *begin*...*end* blocks that were not
declared *scalar*, *real* or *integer* are global, and any
change made to them inside the block affects their global value. Any
array or
matrix declared inside a block is always glob
al.

**KORDER** _ _ _ _ _ _ _ _ _ _ _ _ **declaration**

The *korder* declaration changes the internal canonical ordering of
kernels.

*korder*<kernel>{*,*<kernel>}*

<kernel> must be a REDUCE
kernel or a
list of
*kernel*s.

The declaration *korder* changes the internal ordering, but not the print
ordering, so the effects cannot be seen on output. However, in some
calculations, the order of the variables can have significant effects on the
time and space demands of a calculation. If you are doing a demanding
calculation with several kernels, you can experiment with changing the
canonical ordering to improve behavior.

The first kernel in the argument list is given the highest priority, the
second gets the next highest, and so on. Kernels not named in a
*korder* ordering otherwise. A new *korder* declaration replaces
the previous one. To return to canonical ordering, use the command
*korder nil*.

To change the print ordering, use the declaration order.

**LET** _ _ _ _ _ _ _ _ _ _ _ _ **command**

The *let* command defines general or specific substitution rules.

*let*<identifier> *=* <expression>{,<identifier>
*=* <expression>}*

<identifier> can be any valid REDUCE identifier except an array, and in some cases can be an expression; <expression> can be any valid REDUCE expression.

let a = sin(x); b := a; B := SIN X; let c = a; exp(a); SIN(X) E a := x**2; 2 A := X exp(a); 2 X E exp(b); SIN(X) E exp(c); 2 X E let m + n = p; (m + n)**5; 5 P operator h; let h(u,v) = u - v; h(u,v); U - V h(x,y); H(X,Y) array q(10); let q(1) = 15; ***** Substitution for 0 not allowed

The *let* command is also used to activate a *rule sets
*.

*let*<list>{,<list>}+

<list> can be an explicit
rule *list*, or evaluate
to a rule list.

trig1 := {cos(~x)*cos(~y) => (cos(x+y)+cos(x-y))/2, cos(~x)*sin(~y) => (sin(x+y)-sin(x-y))/2, sin(~x)*sin(~y) => (cos(x-y)-cos(x+y))/2, cos(~x)^2 => (1+cos(2*x))/2, sin(~x)^2 => (1-cos(2*x))/2}$ let trig1; cos(a)*cos(b); COS(A - B) + COS(A + B) ------------------------ 2

A *let* command returns no value, though the substitution r
ule is
entered. Assignment rules made by
assign and *let*
rules are at the
same level, and cancel each other. There is a difference in their
operation, however, as shown in the first example: a *let* assignment
tracks the changes in what it is assigned to, while a *:=* assignment
is fixed at the value it originally had.

The use of expressions as left-hand sides of *let* statements is a
little complicated. The rules of operation are:

_ _ _ (i) Expressions of the form A*B = C do not change A, B or C, but set A*B to C.

_ _ _ (ii) Expressions of the form A+B = C substitute C - B for A, but do not change B or C.

_ _ _ (iii) Expressions of the form A-B = C substitute B + C for A, but do not change B or C.

_ _ _ (iv) Expressions of the form A/B = C substitute B*C for A, but do not change B or C.

_ _ _ (v)
Expressions of the form A**N = C substitute C for A**N in every expression of
a power of A to N or greater. An asymptotic command such as A**N = 0 sets
all terms involving A to powers greater than or equal to N to 0. Finite
fields may be generated by requiring modular arithmetic (the
modular
switch) and defining the primitive polynomial via a *let* statement.

*let*substitutions involving expressions are cleared by using
the
clear command with exactly the same expressio
n.

Note when a simple *let* statement is used to assign functionality to an
operator, it is valid only for the exact identifiers used. For the use of the
*let* command to attach more general functionality to an operator,
see
forall.

Arrays as a whole cannot be arguments to *let* statements, but
matrices as a whole can be legal arguments, provided both arguments are
matrices. However, it is important to note that the two matrices are then
linked. Any change to an element of one matrix changes the corresponding
value in the other. Unless you want this behavior, you should not use
*let* for matrices. The assignment operator
assign can be used
for non-tracking assignments, avoiding the side effects. Matrices are
redimensioned as needed in *let* statements.

When array or matrix elements are used as the left-hand side of *let*
statements, the contents of that element is used as the argument. When the
contents is a number or some other expression that is not a valid left-hand
side for *let*, you get an error message. If the contents is an
identifier or simple expression, the *let* rule is globally attached
to that identifier, and is in effect not only inside the array or matrix,
but everywhere. Because of such unwanted side effects, you should not
use *let* with array or matrix elements. The assignment operator
*:=* can be used to put values into array or matrix elements without
the side effects.

Local variables declared inside *begin*...*end* blocks cannot
be used as the left-hand side of *let* statements. However,
begin...*end* blocks themselves can be
used as the
right-hand side of *let* statements. The construction:

*for all*<vars>
*let*<operator>(<vars>)*=*<block>

is an alternative to the

*procedure*<name>(<vars>)*;*<block>

construction. One important difference between the two constructions is that
the <vars> as formal parameters to a procedure have their global values
protected against change by the procedure, while the <vars> of a
*let* statement are changed globally by its actions.

Be careful in using a construction such as *let x = x + 1* except inside
a controlled loop statement. The process of resubstitution continues until
a stack overflow message is given.

The *let* statement may be used to make global changes to variables from
inside procedures. If *x* is a formal parameter to a procedure, the
command *let x = *... makes the change to the calling variable.
For example, if a procedure was defined by

procedure f(x,y); let x = 15;

and the procedure was called as

f(a,b);

*a*would have its value changed to 15. Be careful when using *let*
statements inside procedures to avoid unwanted side effects.

It is also important to be careful when replacing *let* statements with
other *let* statements. The overlapping of these substitutions can be
unpredictable. Ordinarily the latest-entered rule is the first to be applied.
Sometimes the previous rule is superseded completely; other times it stays
around as a special case. The order of entering a set of related *let*
expressions is very important to their eventual behavior. The best
approach is to assume that the rules will be applied in an arbitrary order.