@@ -1,1713 +1,1713 @@
-\section{Algebraic Operators}
-
-\begin{Operator}{APPEND}
-\index{list}
-The \name{append} operator constructs a new \nameref{list} 
-from the elements of its two arguments (which must be lists).
-
-\begin{Syntax}
-\name{append}\(\meta{list},\meta{list}\)
-\end{Syntax}
-
-\meta{list} must be a list, though it may be the empty list (\name{\{\}}).
-Any arguments beyond the first two are ignored.
-
-\begin{Examples}
-alist := \{1,2,\{a,b\}\};      &      ALIST := \{1,2,\{A,B\}\} \\
-blist := \{3,4,5,sin(y)\};   &      BLIST := \{3,4,5,SIN(Y)\} \\
-append(alist,blist);       &      \{1,2,\{A,B\},3,4,5,SIN(Y)\} \\
-append(alist,\{\});          &      \{1,2,\{A,B\}\} \\
-append(list z,blist);      &      \{Z,3,4,5,SIN(Y)\}
-\end{Examples}
-
-\begin{Comments}
-The new list consists of the elements of the second list appended to the
-elements of the first list.  You can \name{append} new elements to the
-beginning or end of an existing list by putting the new element in a
-list (use curly braces or the operator \name{list}).  This is
-particularly helpful in an iterative loop.
-\end{Comments}
-\end{Operator}
-
-\begin{Operator}{ARBINT}
-\index{arbitrary value}
-The operator \name{arbint} is used to express arbitrary integer parts
-of an expression, e.g. in the result of \nameref{solve} when
-\nameref{allbranch} is on. 
-\begin{Examples}
-
-solve(log(sin(x+3)),x);    &
-\begin{multilineoutput}{6cm}
-\{X=2*ARBINT(1)*PI - ASIN(1) - 3,
- X=2*ARBINT(1)*PI + ASIN(1) + PI - 3\}
-\end{multilineoutput}
-\end{Examples}
-\end{Operator} 
-
-\begin{Operator}{ARBCOMPLEX}
-\index{arbitrary value}
-The operator \name{arbcomplex} is used to express arbitrary scalar parts
-of an expression, e.g. in the result of \nameref{solve} when
-the solution is parametric in one of the variable.
-\begin{Examples}
-
-solve({x+3=y-2z,y-3x=0},{x,y,z});    &
-\begin{multilineoutput}{6cm}
-\{X=\rfrac{2*ARBCOMPLEX(1) + 3}{2},
-  Y=\rfrac{3*ARBCOMPLEX(1) + 3}{2},
-  Z=ARBCOMPLEX(1)\}
-\end{multilineoutput}
-\end{Examples}
-\end{Operator} 
-
-\begin{Operator}{ARGLENGTH}
-\index{argument}
-The operator \name{arglength} returns the number of arguments of the top-level
-operator in its argument.
-
-\begin{Syntax}
-\name{arglength}\(\meta{expression}\)
-\end{Syntax}
-
-\meta{expression} can be any valid REDUCE algebraic expression.
-
-\begin{Examples}
-arglength(a + b + c + d);                             &      4 \\
-arglength(a/b/c);                                     &      2 \\
-arglength(log(sin(df(r**3*x,x))));                    &      1
-\end{Examples}
-
-\begin{Comments}
-In the first example, \name{+} is an n-ary operator, so the number of terms
-is returned.  In the second example, since \name{/} is a binary operator, the
-argument is actually (a/b)/c, so there are two terms at the top level.  In
-the last example, no matter how deeply the operators are nested, there is
-still only one argument at the top level.
-\end{Comments}
-\end{Operator}
-
-
-\begin{Operator}{COEFF}
-\index{coefficient}
-The \name{coeff} operator returns the coefficients of the powers of the
-specified variable in the given expression, in a \nameref{list}.
-
-\begin{Syntax}
-\name{coeff}\(\meta{expression}\name{,}\meta{variable}\)
-\end{Syntax}
-
-\meta{expression} is expected to be a polynomial expression, not a rational
-expression.  Rational expressions are accepted when the switch
-\nameref{ratarg} is on.  \meta{variable} must be a kernel.  The results are
-returned in a list.
-
-\begin{Examples}
-coeff((x+y)**3,x);           &       \{Y^{3} ,3*Y^{2} ,3*Y,1\} \\
-coeff((x+2)**4 + sin(x),x);  &       \{SIN(X) + 16,32,24,8,1\} \\
-high_pow;                    &       4 \\
-low_pow;                     &       0 \\
-ab := x**9 + sin(x)*x**7 + sqrt(y); 
-                             &       AB := SQRT(Y) + SIN(X)*X^{7}  + X^{9}\\
-coeff(ab,x);                 &       \{SQRT(Y),0,0,0,0,0,0,SIN(X),0,1\}
-\end{Examples}
-\begin{Comments}
-The variables \nameref{high\_pow} and \nameref{low\_pow} are set to the
-highest and lowest powers of the variable, respectively, appearing in the
-expression.
-
-The coefficients are put into a list, with the coefficient of the lowest
-(constant) term first.  You can use the usual list access methods
-(\name{first}, \name{second}, \name{third}, \name{rest}, \name{length}, and
-\name{part}) to extract them.  If a power does not appear in the
-expression, the corresponding element of the list is zero. Terms involving
-functions of the specified variable but not including powers of it (for
-example in the expression \name{x**4 + 3*x**2 + tan(x)}) are placed in the
-constant term.
-
-Since the \name{coeff} command deals with the expanded form of the expression,
-you may get unexpected results when \nameref{exp} is off, or when
-\nameref{factor} or \nameref{ifactor} are on.
-
-If you want only a specific coefficient rather than all of them, use the
-\nameref{coeffn} operator.
-
-\end{Comments}
-\end{Operator}
-
-
-\begin{Operator}{COEFFN}
-\index{coefficient}
-The \name{coeffn} operator takes three arguments: an expression, a kernel, and
-a non-negative integer.  It returns the coefficient of the kernel to that
-integer power, appearing in the expression.
-
-\begin{Syntax}
-\name{coeffn}\(\meta{expression},\meta{kernel},\meta{integer}\)
-\end{Syntax}
-
-\meta{expression} must be a polynomial, unless \nameref{ratarg} is on which
-allows rational expressions.  \meta{kernel} must be a kernel, and
-\meta{integer} must be a non-negative integer.
-
-\begin{Examples}
-
-ff := x**7 + sin(y)*x**5 + y**4 + x + 7;                &
-               FF := SIN(Y)*X^{5}  + X^{7}  + X + Y^{4}  + 7 \\
-coeffn(ff,x,5);              &        SIN(Y) \\
-coeffn(ff,z,3);              &        0 \\
-coeffn(ff,y,0);              &        SIN(Y)*X^{5}  + X^{7}  + X + 7 \\
-
-rr := 1/y**2+y**3+sin(y);    &        
-RR := \rfrac{SIN(Y)*Y^{2}  + Y^{5}  + 1}{Y^{2}} \\
-on ratarg; \\
-
-coeffn(rr,y,-2);             &        ***** -2 invalid as COEFFN index \\
-
-coeffn(rr,y,5);              &        \rfrac{1}{Y^{2}}\\
-                                        
-\end{Examples}
-
-\begin{Comments}
-If the given power of the kernel does not appear in the expression,
-\name{coeffn} returns 0.  Negative powers are never detected, even if
-they appear in the expression and \nameref{ratarg} are on.  \name{coeffn}
-with an integer argument of 0 returns any terms in the expression that
-do {\em not} contain the given kernel.
-\end{Comments}
-\end{Operator}
-
-
-\begin{Operator}{CONJ}
-\index{conjugate}\index{complex}
-
-\begin{Syntax}
-\name{conj}\(\meta{expression}\) or \name{conj} \meta{simple\_expression}
-\end{Syntax}
-This operator returns the complex conjugate of an expression, if that
-argument has an numerical value.  A non-numerical argument is returned as
-an expression in the operators \nameref{repart} and \nameref{impart}.
-
-\begin{Examples}
-conj(1+i); & 1-I \\
-conj(a+i*b); & REPART(A) - REPART(B)*I - IMPART(A)*I - IMPART(B)
-\end{Examples}
-
-\end{Operator}
-
-\begin{Operator}{CONTINUED_FRACTION}
-\index{approximation}\index{rational numbers}
-
-\begin{Syntax}
-\name{continued\_fraction}\(\meta{num}\) 
-or \name{continued\_fraction}\( \meta{num},\meta{size}\)
-\end{Syntax}
-This operator approximates the real number \meta{num}
-( \nameref{rational} number, \nameref{rounded} number)
-into a continued fraction. The result is a list of two elements: the
-first one is the rational value of the approximation, the second one
-is the list of terms of the continued fraction which represents the
-same value according to the definition \name{t0 +1/(t1 + 1/(t2 + ...))}.
-Precision: the second optional parameter \meta{size} is an upper bound
-for the absolute value of the result denominator. If omitted, the
-approximation is performed up to the current system precision.
-
-
-\begin{Examples}
-continued_fraction pi;
- & \{\rfrac{1146408},{364913},\{3,7,15,1,292,1,1,1,2,1\}\} \\
-continued_fraction(pi,100);
- & \{\rfrac{22},{7},\{3,7\}\} \\
-\end{Examples}
-
-\end{Operator}
-
-\begin{Operator}{DECOMPOSE}
-\index{decomposition}\index{polynomial}
-The \name{decompose} operator takes a multivariate polynomial as argument,
-and returns an expression and a \nameref{list} of 
-\nameref{equation}s from which the
-original polynomial can be found by composition.
-
-\begin{Syntax}
-\name{decompose}\(\meta{expression}\) or \name{decompose}
- \meta{simple\_expression}
-\end{Syntax}
-
-\begin{Examples}
-\begin{multilineinput}
-decompose(x^8-88*x^7+2924*x^6-43912*x^5+263431*x^4-
-          218900*x^3+65690*x^2-7700*x+234)
-\end{multilineinput}
-              & {U^{2} + 35*U + 234, U=V^{2} + 10*V, V=X^{2} - 22*X} \\
-     decompose(u^2+v^2+2u*v+1)  & {W^{2}  + 1, W=U + V}
-\end{Examples}
-
-\begin{Comments}
-Unlike factorization, this decomposition is not unique.  Further
-details can be found in V.S. Alagar, M.Tanh, \meta{Fast Polynomial
-Decomposition}, Proc. EUROCAL 1985, pp 150-153 (Springer) and J. von zur
-Gathen, \meta{Functional} 
-\meta{Decomposition of Polynomials: the Tame Case}, J.
-Symbolic Computation (1990) 9, 281-299.
-\end{Comments}
-\end{Operator}
-
-
-\begin{Operator}{DEG}
-\index{degree}\index{polynomial}
-The operator \name{deg} returns the highest degree of its variable argument
-found in its expression argument.
-
-\begin{Syntax}
-\name{deg}\(\meta{expression},\meta{kernel}\)
-\end{Syntax}
-
-\meta{expression} is expected to be a polynomial expression, not a rational
-expression.  Rational expressions are accepted when the switch
-\nameref{ratarg} is on.  \meta{variable} must be a \nameref{kernel}.  The
-results are returned in a list.
-
-\begin{Examples}
-
-deg((x+y)**5,x);            &        5 \\
-
-deg((a+b)*(c+2*d)**2,d);    &        2 \\
-
-deg(x**2 + cos(y),sin(x));   \\
-
-deg((x**2 + sin(x))**5,sin(x));                        &        5
-\end{Examples}
-\end{Operator}
-
-
-\begin{Operator}{DEN}
-\index{denominator}\index{rational expression}
-The \name{den} operator returns the denominator of its argument.
-
-\begin{Syntax}
-\name{den}\(\meta{expression}\)
-\end{Syntax}
-
-\meta{expression} is ordinarily a rational expression, but may be any valid
-scalar REDUCE expression.
-
-\begin{Examples}
-
-a := x**3 + 3*x**2 + 12*x;  &      A := X*(X^{2}  + 3*X + 12) \\
-
-b := 4*x*y + x*sin(x);      &      B := X*(SIN(X) + 4*Y) \\
-
-den(a/b);                   &      SIN(X) + 4*Y \\
-
-den(aa/4 + bb/5);           &      20 \\
-
-den(100/6);                 &      3 \\
-
-den(sin(x));                &      1
-\end{Examples}
-
-\begin{Comments}
-\name{den} returns the denominator of the expression after it has been
-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 does not have any
-other denominator, 1 is returned.
-
-Switch settings, such as \nameref{mcd} or \nameref{rational}, have an
-effect on the denominator of an expression.
-\end{Comments}
-\end{Operator}
-
-
-\begin{Operator}{DF}
-\index{derivative}\index{partial derivative}
-The \name{df} operator finds partial derivatives with respect to one or
-more variables.
-
-\begin{TEX}
-\begin{Syntax}
-  \name{df}\(\meta{expression}\name{,}\meta{var}
-	     \&optional\(\name{,}\meta{number}\)
-             \{\name{,}\meta{var}\&option\(\name{,}\meta{number}\)\}\optional\)
-\end{Syntax}
-\end{TEX}
-\begin{INFO}{
-\begin{Syntax}
-  \name{df}\(\meta{expression}\name{,}\meta{var}
-	     [\name{,}\meta{number}\]
-             \{\name{,}\meta{var} [ \name{,}\meta{number}] \} \)
-\end{Syntax}
-}\end{INFO}
-
-\meta{expression} can be any valid REDUCE algebraic expression.  \meta{var}
-must be a \nameref{kernel}, and is the differentiation variable.
-\meta{number} must be a non-negative integer.
-
-\begin{Examples}
-
-df(x**2,x);                 &         2*X \\
-
-df(x**2*y + sin(y),y);      &         COS(Y) + X^{2} \\
-
-df((x+y)**10,z);            &         0 \\
-
-
-df(1/x**2,x,2);             &        \rfrac{6}{X^{4}}\\
-
-df(x**4*y + sin(y),y,x,3);  &         24*X \\
-
-for all x let df(tan(x),x) = sec(x)**2; \\
-
-df(tan(3*x),x);             &         3*SEC(3*X)^{2}
-\end{Examples}
-\begin{Comments}
-An error message results if a non-kernel is entered as a differentiation
-operator.  If the optional number is omitted, it is assumed to be 1.
-See the declaration \nameref{depend} to establish dependencies for implicit
-differentiation.
-
-You can define your own differentiation rules, expanding REDUCE's
-capabilities, using the \nameref{let} command as shown in the last example
-above.  Note that once you add your own rule for differentiating a
-function, it supersedes REDUCE's normal handling of that function for the
-duration of the REDUCE session.  If you clear the rule
-(\nameref{clearrules}), you don't get back
-to the previous rule.
-\end{Comments}
-\end{Operator}
-
-\begin{Operator}{EXPAND\_CASES}
-\index{solve}
-When a \nameref{root\_of} form in a result of \nameref{solve} 
-has been converted to a \nameref{one\_of} form, \name{expand\_cases}
-can be used to convert this into form corresponding to the
-normal explicit results of \nameref{solve}. See \nameref{root\_of}.
-\end{Operator}
-
-\begin{Operator}{EXPREAD}
-\index{input}
-\begin{Syntax}
-\name{expread}\(\)
-\end{Syntax}
-
-\name{expread} reads one well-formed expression from the current input
-buffer and returns its value.
-
-\begin{Examples}
-expread(); a+b;  & A + B
-\end{Examples}
-
-\end{Operator}
-
-
-\begin{Operator}{FACTORIZE}
-\index{factorize}\index{polynomial}
-The \name{factorize} operator factors a given expression.
-\begin{Syntax}
-\name{factorize}\(\meta{expression}\)
-\end{Syntax}
-
-\meta{expression} should be a polynomial, otherwise an error will result.
-
-\begin{Examples}
-
-fff := factorize(x^3 - y^3);                          &
-                  \{X - Y,X^{2} + X*Y + Y^{2}\} \\
-fac1 := first fff;           &      FAC1 := X - Y \\
-factorize(x^15 - 1);        &
-\begin{multilineoutput}{5cm}
- \{X - 1,
-  X^{2} + X + 1,
-  X^{4} + X^{3} + X^{2} + X + 1,
-  X^{8} - X^{7} + X^{6} - X^{5} + X^{4} - X + 1\}
-\end{multilineoutput}\\
-lastone := part(ws,length ws);                          &
-        LASTONE := X^{8} - X^{7} + X^{6} - X^{5} + X^{4} - X + 1 \\
-setmod 2;                    &      1 \\
-on modular; \\
-factorize(x^15 - 1);        &
-\begin{multilineoutput}{5cm}
-\{X + 1,
- X^{2} + X + 1,
- X^{4} + X + 1,
- X^{4} + X^{3} + 1,
- X^{4} + X^{3} + X^{2} + X + 1\}
-\end{multilineoutput}
-\end{Examples}
-
-\begin{Comments}
-The \name{factorize} command returns the factors it finds as a \nameref{list}.
-You can therefore use the usual list access methods (\nameref{first},
-\nameref{second}, \nameref{third}, \nameref{rest}, \nameref{length} and
-\nameref{part}) to extract the factors.
-
-If the \meta{expression} given to \name{factorize} is an integer, it will be
-factored into its prime components.  To factor any integer factor of a
-non-numerical expression, the switch \nameref{ifactor} should be turned on.
-Its default is off. \nameref{ifactor} has effect only when factoring is
-explicitly done by \name{factorize}, not when factoring is automatically
-done with the \nameref{factor} switch.  If full factorization is not
-needed the switch \nameref{limitedfactors} allows you to reduce the
-computing time of calls to \name{factorize}.
-
-Factoring can be done in a modular domain by calling \name{factorize} when
-\nameref{modular} is on.  You can set the modulus with the \nameref{setmod}
-command.  The last example above shows factoring modulo 2.
-
-For general comments on factoring, see comments under the switch
-\nameref{factor}.
-\end{Comments}
-\end{Operator}
-
-
-\begin{Operator}{HYPOT}
-
-\begin{Syntax}
-hypot(\meta{expression},\meta{expression})
-\end{Syntax}
-
-If \name{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.
-
-\begin{Examples}
-hypot(3,4); & HYPOT(3,4) \\
-on rounded; \\
-ws; & 5.0 \\
-hypot(a,b); & HYPOT(A,B)
-\end{Examples}
-
-\end{Operator}
-
-
-\begin{Operator}{IMPART}
-\index{imaginary part}\index{complex}
-\begin{Syntax}
-\name{impart}\(\meta{expression}\) or \name{impart} \meta{simple\_expression}
-\end{Syntax}
-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 \nameref{repart} and \name{impart}.
-\begin{Examples}
-impart(1+i);   &  1 \\
-impart(a+i*b); &  REPART(B) + IMPART(A)
-\end{Examples}
-
-\end{Operator}
-
-
-\begin{Operator}{INT}
-\index{integration}
-The \name{int} operator performs analytic integration on a variety of
-functions.
-
-\begin{Syntax}
-\name{int}\(\meta{expression},\meta{kernel}\)
-\end{Syntax}
-
-\meta{expression} can be any scalar expression. involving polynomials, log
-functions, exponential functions, or tangent or arctangent expressions.
-\name{int} attempts expressions involving error functions, dilogarithms
-and other trigonometric expressions.  Integrals involving algebraic
-extensions (such as square roots) may not succeed. \meta{kernel} must be a
-REDUCE \nameref{kernel}.
-
-\begin{Examples}
-int(x**3 + 3,x);             &      \rfrac{X*(X^{3} + 12)}{4} \\\\
-int(sin(x)*exp(2*x),x);
-  &      - \rfrac{E^{2*X}*(COS(X) - 2*SIN(X))}{5} \\
-int(1/(x^2-2),x);
-  & \rfrac{SQRT(2)*(LOG( - SQRT(2) + X) - LOG(SQRT(2) + X))}{4} \\
-int(sin(x)/(4 + cos(x)**2),x);
-  &       - \rfrac{ATAN(\rfrac{COS(X)}{2})}{2} \\\\
-int(1/sqrt(x^2-x),x); & INT(\rfrac{SQRT(X)*SQRT(X - 1)}{X^{2}-X},X)
-\end{Examples}
-
-\begin{Comments}
-Note that REDUCE couldn't handle the last integral with its default
-integrator, since the integrand involves a square root.  However,
-the integral can be found using the \nameref{algint} package.
-Alternatively, you could add a rule using the \nameref{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
-\nameref{rounded}) and subtracting the lower from the higher.  Evaluation can
-be easily done by the \nameref{sub} operator.
-
-When \name{int} cannot find an integral it returns an expression
-involving formal \name{int} expressions unless the switch 
-\nameref{failhard} has been set. If not all of the expression
-can be integrated, the switch \nameref{nolnr} controls whether a partially
-integrated result should be returned or not.
-
-\end{Comments}
-\end{Operator}
-
-
-\begin{Operator}{INTERPOL}
-\index{interpolation}\index{polynomial}\index{approximation}
-\name{interpol} generates an interpolation polynomial.
-\begin{Syntax}
-   interpol(\meta{values},\meta{variable},\meta{points})
-\end{Syntax}
-
-\meta{values} and \meta{points} are \nameref{list}s of equal length and
-\meta{variable} is an algebraic expression (preferably a \nameref{kernel}).
-The interpolation polynomial is generated in the given variable of degree
-length(\meta{values})-1.  The unique polynomial \name{f} is defined by the
-property that for corresponding elements \name{v} of \meta{values} and
-\name{p} of \meta{points} the relation \name{f(p)=v} holds.
-
-\begin{Examples}
-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); & X^{3} - 1
-\end{Examples}
-
-\begin{Comments}
-The Aitken-Neville interpolation algorithm is used which guarantees a
-stable result even with rounded numbers and an ill-conditioned problem.
-\end{Comments}
-\end{Operator}
-
-
-\begin{Operator}{LCOF}
-\index{coefficient}\index{polynomial}
-The \name{lcof} operator returns the leading coefficient of a given expression
-with respect to a given variable.
-\begin{Syntax}
-\name{lcof}\(\meta{expression},\meta{kernel}\)
-\end{Syntax}
-
-\meta{expression} is ordinarily a polynomial. If \nameref{ratarg} is on,
-a rational expression may also be used, otherwise an error results.
-\meta{kernel} must be a \nameref{kernel}.
-
-\begin{Examples}
-lcof((x+2*y)**5,y);          &         32 \\
-lcof((x + y*sin(x))**2 + cos(x)*sin(x)**2,sin(x));
-			     &         COS(X)^{2} + Y \\
-lcof(x**2 + 3*x + 17,y);     &         X^{2} + 3*X + 17
-\end{Examples}
-
-\begin{Comments}
-If the kernel does not appear in the expression, \name{lcof} returns the
-expression. 
-\end{Comments}
-\end{Operator}
-
-
-\begin{Operator}{LENGTH}
-\index{list}
-The \name{length} operator returns the number of items in a \nameref{list}, the 
-number of
-terms in an expression, or the dimensions of an array or matrix.
-\begin{Syntax}
-\name{length}\(\meta{expr}\) or \name{length} \meta{expr}
-\end{Syntax}
-
-\meta{expr} can be a list structure, an array, a matrix, or a scalar expression.
-
-\begin{Examples}
-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);       &     EEX := \rfrac{X^{2} + 6*X + 9}{X - Y} \\
-length eex;                  &     5
-\end{Examples}
-
-\begin{Comments}
-An item in a list that is itself a list only counts as one item.  An error
-message will be printed if \name{length} is called on a matrix which has
-not had its dimensions set.  The \name{length} of an array includes the
-zeroth element of each dimension, showing the full number of elements
-allocated. (Declaring an array \IFTEX{$A$}{A} with \IFTEX{$n$}{n} elements
-allocates \IFTEX{$ A(0),A(1),\ldots,A(n)$}{A(0),A(1),...,A(n)}.) The
-\name{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.
-\end{Comments}
-\end{Operator}
-
-
-\begin{Operator}{LHS}
-\index{left-hand side}\index{equation}
-The \name{lhs} operator returns the left-hand side of an \nameref{equation},
-such as those
-returned in a list by \nameref{solve}.
-\begin{Syntax}
-\name{lhs}\(\meta{equation}\) or \name{lhs} \meta{equation}
-
-\end{Syntax}
-
-\meta{equation} must be an equation of the form \\
-\name{left-hand side} \name{=} \name{right-hand side}.
-
-\begin{Examples}
-polly := (x+3)*(x^4+2x+1); &
-POLLY := X^{5} + 3*X^{4} + 2*X^{2} + 7*X + 3 \\
-pollyroots := solve(polly,x); &
-\begin{multilineoutput}{1cm}
-POLLYROOTS := \{X=ROOT_OF(X_^{3} - X_^{2} + X_ + 1,X_),
-               X=-1,
-               X=-3\}
-\end{multilineoutput} \\
-variable := lhs first pollyroots; &
-VARIABLE := X
-\end{Examples}
-\end{Operator}
-
-
-\begin{Operator}{LIMIT}
-\index{limit}\index{l'Hopital's rule}
-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.        
-
-\begin{Syntax}
-\name{limit}\(\meta{expr},\meta{var},\meta{limpoint}\) or \\
-\name{limit!+}\(\meta{expr},\meta{var},\meta{limpoint}\) or \\
-\name{limit!-}\(\meta{expr},\meta{var},\meta{limpoint}\) 
-\end{Syntax}
-
-where \meta{expr} is an expression depending of the variable \meta{var}
-(a \nameref{kernel}) and \meta{limpoint} is the limit point. 
-If the limit depends upon the direction of approach to  the \meta{limpoint},
-the operators \name{limit!+} and \name{limit!-} may be used.
-
-\begin{Examples}
-limit(x*cot(x),x,0);&0\\
-limit((2x+5)/(3x-2),x,infinity);&\rfrac{2}{3}\\
-\end{Examples}
-
-\end{Operator}
-
-
-\begin{Operator}{LPOWER}
-\index{leading power}\index{polynomial}
-The \name{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.
-\begin{Syntax}
-\name{lpower}\(\meta{expression},\meta{kernel}\)
-\end{Syntax}
-
-\meta{expression} is ordinarily a polynomial. If \nameref{ratarg} is on,
-a rational expression may also be used, otherwise an error results.
-\meta{kernel} must be a \nameref{kernel}.
-
-\begin{Examples}
-lpower((x+2*y)**6,y);        &     Y^{6} \\
-lpower((x + cos(x))**8 + df(x**2,x),cos(x));
-			     &     COS(X)^{8} \\
-lpower(x**3 + 3*x,y);        &     1
-\end{Examples}
-\end{Operator}
-
-\begin{Operator}{LTERM}
-\index{leading term}\index{polynomial}
-The \name{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.
-\begin{Syntax}
-\name{lterm}\(\meta{expression},\meta{kernel}\)
-\end{Syntax}
-
-\meta{expression} is ordinarily a polynomial. If \nameref{ratarg} is on,
-a rational expression may also be used, otherwise an error results.
-\meta{kernel} must be a \nameref{kernel}.
-
-\begin{Examples}
-lterm((x+2*y)**6,y);        &     64*Y^{6} \\
-lterm((x + cos(x))**8 + df(x**2,x),cos(x));
-			    &     COS(X)^{8} \\
-lterm(x**3 + 3*x,y);        &     X^{3} + 3X
-\end{Examples}
-\end{Operator}
-
-
-\begin{Operator}{MAINVAR}
-\index{main variable}\index{polynomial}
-The \name{mainvar} operator returns the main variable (in the system's
-internal representation) of its argument.
-\begin{Syntax}
-\name{mainvar}\(\meta{expression}\)
-
-\end{Syntax}
-
-\meta{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
-\nameref{kernel}.
-
-\begin{Examples}
-test := (a + b + c)**2;      &
-     TEST := A^{2} + 2*A*B + 2*A*C + B^{2} + 2*B*C + C^{2} \\
-mainvar(test);               &        A \\
-korder c,b,a; \\
-mainvar(test);               &        C \\
-mainvar(2*cos(x)**2);        &        COS(X) \\
-mainvar(17);                 &        0
-\end{Examples}
-
-\begin{Comments}
-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 \name{mainvar} is concerned belong in the family \name{0}.  The
-canonical ordering can be changed by the declaration \nameref{korder}, as
-shown above.
-\end{Comments}
-\end{Operator}
-
-
-\begin{Operator}{MAP}
-\index{map}\index{composite structure}
-The \name{map} operator applies a uniform evaluation pattern
-to all members of a composite structure: a \nameref{matrix}, 
-a \nameref{list} or the arguments of an \nameref{operator} expression. 
-The evaluation pattern can be a
-unary procedure, an operator, or an algebraic expression with
-one free variable. 
-\begin{Syntax}
-  \name{map}\(\meta{function},\meta{object}\)
-\end{Syntax}
-\meta{object} is a list, a matrix or an operator expression.
-
-\meta{function} is 
-the name of an operator for a single argument: the operator
- is evaluated once with each element of \meta{object} as its single argument,
-
-or an algebraic expression with exactly one \nameref{free variable}, that is
-a variable preceded by the tilde symbol: the expression
- is evaluated for each element of \meta{object} where the element is
- substituted for the free variable,
-
-or a replacement \nameref{rule} of the form 
-\begin{Syntax}
-  \name{var} => \name{rep}
-\end{Syntax}
-  where \meta{var} is a variable (a \meta{kernel} without subscript)
- and \meta{rep} is an expression which contains \meta{var}.
-  Here \name{rep} is evaluated for each element of \meta{object} where 
-  the element is substituted for  \name{var}. \name{var} may be 
-  optionally preceded by a tilde.
-
-The rule form  for \meta{function} is needed when more than
-one free variable occurs.
-
-\begin{Examples}
-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))); &
-\begin{multilineoutput}{1cm}
-        [  3     6 ]
-        [ x     x  ]
-        [----  ----]
-        [ 3     6  ]
-        [          ]
-        [  5     6 ]
-        [ x     x  ]
-        [----  ----]
-	[ 5     6  ]
-\end{multilineoutput}\\
-map(~w*6, x^2/3 = y^3/2 -1); & 2*x^2=3*(y^3-2)\\
-\end{Examples}
-
-\begin{Comments}
-You can use \name{map} in nested expressions. It is not allowed to
-apply \name{map} for a non-composed object, e.g. an identifier or a number. 
-\end{Comments}
-\end{Operator}
-
-
-
-\begin{Command}{MKID}
-\index{identifier}
-The \name{mkid} command constructs an identifier, given a stem and an identifier
-or an integer.
-\begin{Syntax}
-\name{mkid}\(\meta{stem},\meta{leaf}\)
-\end{Syntax}
-
-\meta{stem} can be any valid REDUCE identifier that does not include escaped
-special characters. \meta{leaf} may be an integer, including one given by a
-local variable in a \nameref{for} loop, or any other legal group of
-characters.
-
-\begin{Examples}
-mkid(x,3);                  &               X3 \\
-factorize(x^15 - 1);        &  \begin{multilineoutput}{6cm}
-\{X - 1,
- X^{2} + X + 1,
- X^{4} + X^{3} + X^{2} + X + 1,
- X^{8} - X^{7} + X^{5} - X^{4} + X^{3}  - X + 1\}
-\end{multilineoutput}\\
-
-for i := 1:length ws do write set(mkid(f,i),part(ws,i));
-	  & \begin{multilineoutput}{6cm}
-X^{8} - X^{7} + X^{5} - X^{4} + X^{3} - X + 1
-X^{4} + X^{3} + X^{2} + X + 1
-X^{2} + X + 1
-X - 1
-\end{multilineoutput} \\
-\end{Examples}
-
-\begin{Comments}
-You can use \name{mkid} to construct identifiers from inside procedures.  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 \name{factorize} and \name{solve}.
-\end{Comments}
-\end{Command}
-
-
-\begin{Operator}{NPRIMITIVE}
-\index{primitive part}\index{polynomial}
-\begin{Syntax}
-\name{nprimitive}\(\meta{expression}\) or \name{nprimitive}
-   \meta{simple\_expression}
-\end{Syntax}
-This operator returns the numerically-primitive part of any scalar
-expression.  In other words, any overall integer factors in the expression
-are removed.
-
-\begin{Examples}
-nprimitive((2x+2y)^2); & X^{2} + 2*X*Y + Y^{2} \\
-nprimitive(3*a*b*c); & 3*A*B*C
-\end{Examples}
-\end{Operator}
-
-
-\begin{Operator}{NUM}
-\index{numerator}\index{rational expression}
-The \name{num} operator returns the numerator of its argument.
-\begin{Syntax}
-\name{num}\(\meta{expression}\) or \name{num} \meta{simple\_expression}
-\end{Syntax}
-
-\meta{expression} can be any valid REDUCE scalar expression.
-
-\begin{Examples}
-num(100/6);                 &         50 \\
-num(a/5 + b/6);             &         6*A + 5*B \\
-num(sin(x));                &         SIN(X)
-\end{Examples}
-
-\begin{Comments}
-\name{num} returns the numerator of the expression after it has been 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.  
-\end{Comments}
-\end{Operator}
-
-
-\begin{Operator}{ODESOLVE}
-\index{differential equation}\index{solve}
-The \name{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.
-
-\begin{Syntax}
-
-\name{odesolve}\(\meta{expr},\meta{var1},\meta{var2}\)
-
-\end{Syntax}
-
-\meta{expr} is  a single  scalar expression  such  that \meta{expr}=0  
-is the ordinary differential equation (ODE for short) to be  solved, or
-is an equivalent \nameref{equation}.
-
-\meta{var1} is the name of the dependent variable,
-\meta{var2} is the name of the independent variable.
-
-A differential in \meta{expr} is expressed using the \nameref{df}
-operator. Note that in most cases you must declare explicitly
-\meta{var1} to depend of \meta{var2} using a \nameref{depend}
-declaration -- otherwise the derivative might be evaluated to
-zero on input to \name{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 \name{arbconst} for the  arbitrary constants  in the  general
-solution.   The arguments of \name{arbconst} should  be new.
-A counter  \name{!!arbconst}  is used  to  arrange  this.
-
-\begin{Examples}
-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)\}
-\end{Examples}      
-
-\end{Operator}
-
-
-\begin{Type}{ONE\_OF}
-The operator \name{one\_of} is used to represent an indefinite choice
-of one element from a finite set of objects. 
-\begin{Examples}
-x=one_of{1,2,5}\\
-\explanation{this equation encodes that x can take one of the values
-1,2 or 5}\\
-\end{Examples}
-REDUCE generates a \name{one\_of} form in cases when an implicit
-\name{root\_of} expression could be converted to an explicit solution set.
-A \name{one\_of} form can be converted to a \name{solve} solution using
-\nameref{expand\_cases}.  See \nameref{root\_of}.
-\end{Type}
-
-\begin{Operator}{PART}
-\index{decomposition}
-The operator \name{part} permits the extraction of various parts or
-operators of expressions and \nameref{list}\name{s}.
-\begin{Syntax}
-\name{part}\(\meta{expression,integer}\{,\meta{integer}\}\optional\)
-\end{Syntax}
-
-\meta{expression} can be any valid REDUCE expression or a list, {\it
-integer} may be an expression that evaluates to a positive or negative
-integer or 0.  A positive integer \meta{n} picks up the {\it n} th term,
-counting from the first term toward the end.  A negative integer {\it n}
-picks up the {\it n} th term, counting from the back toward the front.  The
-integer 0 picks up the operator (which is \name{LIST} when the expression
-is a \ref{list}).
-
-\begin{Examples}
-part((x + y)**5,4);          &        10*X^{2}*Y^{3} \\
-part((x + y)**5,4,2);        &        X^{2} \\
-part((x + y)**5,4,2,1);      &        X \\
-part((x + y)**5,0);          &        PLUS \\
-part((x + y)**5,-5);         &        5*X *Y^{4} \\
-part((x + y)**5,4) := sin(x); &                          
-  X^{5} + 5*X^{4}*Y + 10*X^{3}*Y^{2} + SIN(X) + 5*X*Y^{4} + Y^{5} \\
-alist := \{x,y,\{aa,bb,cc\},x**2*sqrt(y)\};                 &
-			 ALIST := \{X,Y,\{AA,BB,CC\},SQRT(Y)*X^{2}\} \\
-part(alist,3,2);             &       BB \\
-part(alist,4,0);             &       TIMES
-\end{Examples}
-
-\begin{Comments}
-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, \IFTEX{$10x^2y^3$}{10x^2y^3},
-then the second term from that, \IFTEX{$x^2$}{x^2}, and finally the first
-component, \IFTEX{$x$}{x}.  If an integer's absolute value is too large for
-the appropriate expression, a message is given.
-
-\name{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 \name{part}.  \nameref{pri} must be on when
-\name{part} is used.
-
-When \name{part} is used on a polynomial expression that has minus signs, the
-\name{+} is always returned as the top-level operator.  The minus is found
-as a unary operator attached to the negative term.
-
-\name{part} can also be used to change the relevant part of the expression or
-list as shown in the sixth example line.  The \name{part} operator returns the
-changed expression, though original expression is not changed.  You can
-also use \name{part} to change the operator.
-\end{Comments}
-\end{Operator}
-
-\begin{Operator}{PF}
-\index{partial fraction}\index{rational expression}
-\begin{Syntax}
-pf(\meta{expression},\meta{variable})
-\end{Syntax}
-
-\name{pf} transforms \meta{expression} into a \nameref{list} of partial fraction
-s
-with respect to the main variable, \meta{variable}. \name{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.
-\begin{Examples}
-pf(2/((x+1)^2*(x+2)),x); &
-	\{\rfrac{2}{X + 2},\rfrac{-2}{X + 1},\rfrac{2}{X^{2} + 2*X + 1}\} \\
-off exp; \\
-pf(2/((x+1)^2*(x+2)),x);
- & \{\rfrac{2}{X + 2},\rfrac{- 2}{X + 1},\rfrac{2}{(X + 1)^{2}}\} \\
-for each j in ws sum j; & \rfrac{2}{( + 2)*(X + 1)^{2}}
-\end{Examples}
-
-\begin{Comments}
-If you want the denominators in factored form, turn \nameref{exp} off, as
-shown in the second example above.  As shown in the final example, the
-\nameref{for} \name{each} construct can be used to recombine the terms.
-Alternatively, one can use the operations on lists to extract any desired
-term.
-\end{Comments}
-
-\end{Operator}
-
-
-\begin{Operator}{PROD}
-\index{Gosper algorithm}\index{product}
-The operator  \name{prod} returns
-the indefinite or definite  product of a given expression.
-
-\begin{Syntax}
-
-\name{prod}\(\meta{expr},\meta{k}[,\meta{lolim} [,\meta{uplim} ]]\)
-
-\end{Syntax}
-
-where \meta{expr} is the expression to be multiplied, \meta{k} is the
-control variable (a \nameref{kernel}), and \meta{lolim} and \meta{uplim}
-uplim are the optional lower and upper limits. If \meta{uplim} is
-not supplied the upper limit is taken as \meta{k}. The
-Gosper algorithm is used.  If there is no closed form solution,
-the operator returns the input unchanged.
-
-\begin{Examples}
-prod(k/(k-2),k);&k*( - k + 1)\\
-\end{Examples}
-\end{Operator}
-
-
-\begin{Operator}{REDUCT}
-\index{reductum}\index{polynomial}
-The \name{reduct} operator returns the remainder of its expression after the
-leading term with respect to the kernel in the second argument is removed.
-\begin{Syntax}
-\name{reduct}\(\meta{expression},\meta{kernel}\)
-\end{Syntax}
-
-\meta{expression} is ordinarily a polynomial. If \nameref{ratarg} is on,
-a rational expression may also be used, otherwise an error results.
-\meta{kernel} must be a \nameref{kernel}.
-
-\begin{Examples}
-reduct((x+y)**3,x);          &       Y*(3*X^{2} + 3*X*Y + Y^{2}) \\
-reduct(x + sin(x)**3,sin(x));                           &       X \\
-reduct(x + sin(x)**3,y);                         &      0
-\end{Examples}
-
-\begin{Comments}
-If the expression does not contain the kernel, \name{reduct} returns 0.
-\end{Comments}
-\end{Operator}
-
-
-\begin{Operator}{REPART}
-\index{real part}\index{complex}
-\begin{Syntax}
-\name{repart}\(\meta{expression}\) or \name{repart} \meta{simple\_expression}
-\end{Syntax}
-
-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 \name{repart} and \nameref{impart}.
-\begin{Examples}
-repart(1+i);   & 1 \\
-repart(a+i*b); & REPART(A) - IMPART(B)
-\end{Examples}
-
-\end{Operator}
-
-
-\begin{Operator}{RESULTANT}
-\index{polynomial}
-The \name{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.
-\begin{Syntax}
-  \name{resultant}\(\meta{expression},\meta{expression},\meta{kernel}\)
-\end{Syntax}
-
-\meta{expression} must be a polynomial containing \meta{kernel} ;
-\meta{kernel} must be a \nameref{kernel}.
-
-\begin{Examples}
-resultant(x**2 + 2*x + 1,x+1,x);                        &      0 \\
-resultant(x**2 + 2*x + 1,x-3,x);                        &      16 \\
-\begin{multilineinput}
-resultant(z**3 + z**2 + 5*z + 5,
-          z**4 - 6*z**3 + 16*z**2 - 30*z + 55,
-          z);
-\end{multilineinput}    &      0 \\
-resultant(x**3*y + 4*x*y + 10,y**2 + 6*y + 4,y);       &
-           Y^{6} + 18*Y^{5} + 120*Y^{4} + 360*Y^{3} + 480*Y^{2} + 288*Y + 64
-\end{Examples}
-\begin{Comments}
-The resultant is the determinant of the Sylvester matrix, formed from the
-coefficients of the two polynomials in the following way:
-
-Given two polynomials: 
-
-\begin{TEX}
-\begin{displaymath}
-a_0x^n+a_1x^{n-1}+\cdots+a_n
-\end{displaymath}
-\end{TEX}
-\begin{INFO}
-{\begin{verbatim}
-    n       n-1 
- a x  + a1 x     + ... + an
-
-\end{verbatim}}
-\end{INFO}
-and
-\begin{TEX}
-\begin{displaymath}
-b_0x^n+b_1x^{n-1}+\cdots+b_n
-\end{displaymath}
-\end{TEX}
-\begin{INFO}
-{\begin{verbatim}
-    m       m-1 
- b x  + b1 x     + ... + bm
-
-\end{verbatim}}
-\end{INFO}
-form the  (m+n)x(m+n-1) Sylvester matrix by the following means:
-\begin{TEX}
-\begin{displaymath}
-  \left(\begin{array}{cccccccc}
-           0&\ldots&0&0&a_0&a_1&\ldots&a_n\\
-	   0&\ldots&0&a_0&a_1&\ldots&a_n&0\\
-	   \vdots&&&\vdots&&&\vdots\\
-	   a_0&a_1&\ldots&a_n&0&0&\ldots&0\\
-	   0&\ldots&0&0&b_0&b_1&\ldots&b_n\\
-	   \vdots&&&\vdots&&&\vdots\\
-	   b_0&b_1&\ldots&b_n&0&0&\ldots&0
-         \end{array}\right)
-\end{displaymath}
-\end{TEX}
-\begin{INFO}
-{\begin{verbatim}
-   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  
-
-\end{verbatim}}
-\end{INFO}
-
-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 \name{resultant} uses are those given in the article,
-``Computing in Algebraic Extensions,'' by R.  Loos, appearing in
-\meta{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:
-\begin{TEX}
-\begin{eqnarray*}
-  \mbox{resultant}(p(x),q(x),x)
-	&=& (-1)^{\deg p(x)*\deg q(x)}\cdot\mbox{resultant}(q(x),p(x),x),\\
-  \mbox{resultant}(a,p(x),x) &=& a^{\deg p(x)},\\
-  \mbox{resultant}(a,b,x) &=& 1
-\end{eqnarray*}
-where $p(x)$ and $q(x)$ are polynomials which have $x$ as a variable, and
-$a$ and $b$ are free of $x$.
-\end{TEX}
-\begin{INFO}
-{
-\begin{verbatim}
-  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
-\end{verbatim}
-where p(x) and q(x) are polynomials which have x as a variable, and
-a and b are free of x.
-}
-\end{INFO}
-
-Error messages are given if \name{resultant} is given a non-polynomial
-expression, or a non-kernel variable.
-\end{Comments}
-\end{Operator}
-
-
-\begin{Operator}{RHS}
-\index{right-hand side}\index{equation}
-The \name{rhs} operator returns the right-hand side of an \nameref{equation}, 
-such as those returned in a \nameref{list} by \nameref{solve}.
-\begin{Syntax}
-\name{rhs}\(\meta{equation}\) or \name{rhs} \meta {equation}
-\end{Syntax}
-
-\meta{equation} must be an equation of the form {\it left-hand side = right-hand
-side}.
-
-\begin{Examples}
-roots := solve(x**2 + 6*x*y + 5x + 3y**2,x);            &
-\begin{multilineoutput}{6cm}
-    ROOTS := \{X= - \rfrac{SQRT(24*Y^{2} + 60*Y + 25) + 6*Y + 5}{2},
-               X= \rfrac{SQRT(24*Y^{2} + 60*Y + 25) - 6*Y - 5}{2}\}
-\end{multilineoutput} \\
-root1 := rhs first roots;                               &
-     ROOT1 := - \rfrac{SQRT(24*Y^{2} + 60*Y + 25) + 6*Y + 5}{2} \\
-root2 := rhs second roots;                              &
-     ROOT2 := \rfrac{SQRT(24*Y^{2} + 60*Y + 25) - 6*Y - 5}{2}
-\end{Examples}
-
-\begin{Comments}
-An error message is given if \name{rhs} is applied to something other than an
-equation.
-\end{Comments}
-\end{Operator}
-
-\begin{Operator}{ROOT\_OF}
-\index{roots}\index{solve}
-When the operator \nameref{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 \name{root\_of}
-and a new local variable. An expression with a top level \name{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 \name{root\_of} construct is
-replaced by an operator \nameref{one\_of}. At this point it is
-of course possible to transform the result if the original \name{solve}
-operator expression into a standard \name{solve} solution. To
-effect this, the operator \nameref{expand\_cases} can be used.
-
-\begin{Examples}
-solve(a*x^7-x^2+1,x);&
-\{x=root\_of(a*x\_^7 - x\_^2 + 1,x\_)\}\\
-sub(a=0,ws);&
-\{x=one\_of(1,-1)\}\\
-expand_cases ws;&
-{x=1,x=-1}\\
-\end{Examples}
-The components of \name{root\_of} and \name{one\_of} expressions can be
-processed as usual with operators \nameref{arglength} and \nameref{part}.
-A higher power of a \name{root\_of} expression with a polynomial
-as first argument is simplified by using the polynomial as a side relation.
-\end{Operator}
-
-\begin{Operator}{SELECT}
-\index{map}\index{list}
-
-The \name{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 \nameref{free variable}.
-\begin{Syntax}
-  \name{select}\(\meta{function},\meta{object}\)
-\end{Syntax}
-\meta{object} is a \nameref{list}.
-
-\meta{function} is 
-the name of an operator for a single argument: the operator
- is evaluated once with each element of \meta{object} as its single argument,
-
-or an algebraic expression with exactly one \nameref{free variable}, that is
-a variable preceded by the tilde symbol: the expression
- is evaluated for each element of \meta{object} where the element is
- substituted for the free variable,
-
-or a replacement \nameref{rule} of the form 
-\begin{Syntax}
-  \name{var} => \name{rep}
-\end{Syntax}
-  where \meta{var} is a variable (a \meta{kernel} without subscript)
- and \meta{rep} is an expression which contains \meta{var}.
-  Here \name{rep} is evaluated for each element of \meta{object} where 
-  the element is substituted for  \name{var}. \name{var} may be 
-  optionally preceded by a tilde.
-
-The rule form  for \meta{function} is needed when more than
-one free variable occurs. The evaluation result of \meta{function} is
-interpreted as \nameref{boolean value} corresponding to the conventions of
-REDUCE. The result value is built with the leading operator of the
-input expression.
-\begin{Examples}
-  select( ~w>0 , {1,-1,2,-3,3}) &   \{1,2,3\} \\
-  q:=(part((x+y)^5,0):=list)\\
-  select(evenp deg(~w,y),q);& \{x^5 ,10*x^3 *y^2 ,5*x*y^4 \}\\
-  select(evenp deg(~w,x),2x^2+3x^3+4x^4);& 2x^2+4x^4\\
-\end{Examples}
-\end{Operator}
-
-\begin{Operator}{SHOWRULES}
-\index{rule}\index{output}
-\begin{Syntax}
-\name{showrules}\(\meta{expression}\) or
- \name{showrules} \meta{simple\_expression}
-\end{Syntax}
-
-\name{showrules} returns in \nameref{rule}\name{-list} form any 
-\nameref{operator} rules associated with its argument.
-
-\begin{Examples}
-showrules log; &
-\begin{multilineoutput}{6cm}
-\{LOG(E) => 1,
- LOG(1) => 0,
- LOG(E^{~X} ) => ~X,
- DF(LOG(~X),~X) => \rfrac{1}{~X}\}
-\end{multilineoutput}
-\end{Examples}
-
-Such rules can then be manipulated further as with any \nameref{list}.  For
-example
-\name{rhs first ws;} has the value {\em 1}.
-
-\begin{Comments}
-An operator may have properties that cannot be displayed in such a form,
-such as the fact it is an \ref{odd} function, or has a definition defined
-as a procedure.
-\end{Comments}
-\end{Operator}
-
-
-\begin{Operator}{SOLVE}
-\index{equation}\index{equation solving}
-\index{equation system}\index{root}\index{solve}
-The \name{solve} operator solves a single algebraic \nameref{equation} or a
-system of simultaneous equations.
-\begin{TEX}
-\begin{Syntax}
-% \name{solve}\(\meta{expression} \&option(, \meta{kernel})\) or
-% \name{solve}\(\name{\{}\meta{expression}\{,\meta{expression}\}
-%    \optional\name{\}}
-%           \&option(,\meta{kernel}\optional\)
-\name{solve}\(\meta{expression}\&option(, \meta{kernel})\) or \\
- \name{solve}\(\name{\{}\meta{expression}\&option( ,\meta{expression})
- \optional\name{\}}
-	  \&option(,\{\meta{kernel})\optional\name{\}\}}\)
-\end{Syntax}
-\end{TEX}
-\begin{INFO}
-{\begin{Syntax}
-\name{solve}\(\meta{expression} [ , \meta{kernel}]\) or
-
-\name{solve}\(\{\meta{expression},...\} [ ,\{ \meta{kernel} ,...\}] \)
-\end{Syntax} 
-}\end{INFO}
-
-If the number of equations equals the number of distinct kernels, the
-optional kernel argument(s) may be omitted.  \meta{expression} is either a
-scalar expression or an \nameref{equation}.  
-When more than one expression is given,
-the \nameref{list} of expressions is surrounded by curly braces.
-The optional list
-of \nameref{kernel}s follows, also in curly braces.
-
-\begin{Examples}
-sss := solve(x^2 + 7);      &
-\begin{multilineoutput}{6cm}
-Unknown: X
-SSS := \{X= - SQRT(7)*I,
-        X=SQRT(7)*I\}
-\end{multilineoutput}\\
-rhs first sss;               &       - SQRT(7)*I \\
-solve(sin(x^2*y),y);        &
-\begin{multilineoutput}{6cm}
-\{Y=\rfrac{2*ARBINT(1)*PI}{X^{2}}
- Y=\rfrac{PI*(2*ARBINT(1) + 1)}{X^{2}}\}
-\end{multilineoutput}\\
-off allbranch; \\
-solve(sin(x**2*y),y);        &      \{Y=0\} \\
-solve({3x + 5y = -4,2*x + y = -10},{x,y});
-   & \{\{X= - \rfrac{22}{7},Y=\rfrac{46}{7}\}\} \\
-solve({x + a*y + z,2x + 5},{x,y});
-   &       \{\{X= - \rfrac{5}{2},Y= - \rfrac{2*Z - 5}{2*A}\}\} \\
-% xval := rhs part(ws,1,1);    &       XVAL := - \rfrac{5}{2} \\
-ab := (x+2)^2*(x^6 + 17x + 1);
- &      AB := X^{8} + 4*X^{7} + 4*X^{6} + 17*X^{3} + 69*X^{2} + 72*X + 4 \\
-www := solve(ab,x);          &      \{X=ROOT_OF(X_^{6} + 17*X_ + 1),X=-2\} \\
-root_multiplicities;         &      \{1,2\}
-\end{Examples}
-\begin{Comments}
-Results of the \name{solve} operator are returned as \nameref{equation}\name{s} 
-in a \nameref{list}.
-You can use the usual list access methods (\nameref{first},
-\nameref{second}, \nameref{third}, \nameref{rest} and \nameref{part}) to
-extract the desired equation, and then use the operators \nameref{rhs} and
-\nameref{lhs} to access the right-hand or left-hand expression of the
-equation.  When \name{solve} is unable to solve an equation, it returns the
-unsolved part as the argument of \name{root_of}, with the variable renamed
-to avoid confusion, as shown in the last example above.
-
-For one equation, \name{solve} uses square-free factorization, roots of
-unity, and the known inverses of the \nameref{log}, \nameref{sin},
-\nameref{cos}, \nameref{acos}, \nameref{asin}, and
-exponentiation operators.  The quadratic, cubic and quartic formulas are
-used if necessary, but these are applied only when the switch 
-\nameref{fullroots} is set on; otherwise or when no closed form is available 
-the result is returned as
-\nameref{root\_of} expression. The switch \nameref{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 \nameref{root\_multiplicities}.  For systems of
-simultaneous linear equations, matrix inversion is used.  For nonlinear
-systems, the Groebner basis method is used.
-
-%If kernels are given for linear equations, and there are an unequal number
-%of kernels and equations, an error message is given.  If no kernels are
-%given, and there are more kernels in the equations than there are
-%equations, an error message is given.  
-Linear equation system solving is influenced by the switch \nameref{cramer}.
-%For nonlinear equations, it is
-%possible to have a consistent set in which the number of variables does not
-%match the number of equations.
-
-Singular systems can be solved when the switch \nameref{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 \nameref{requirements} constraints
-conditions for the system to become consistent.
-
-For a solvable linear and polynomial system with parameters 
-the variable \nameref{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 \nameref{varopt} is on (default), the system rearranges the 
-variable sequence for minimal computation time. Without \name{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 \nameref{arbcomplex} expressions 
-as long as the switch \nameref{arbvars} is on (default). Without
-\name{arbvars} no explicit equations are generated for free variables.
-\end{Comments}
-
-\begin{Related}
-\item[\nameref{allbranch} switch]
-\item[\nameref{arbvars} switch]
-\item[\nameref{assumptions} variable]
-\item[\nameref{fullroots} switch]
-\item[\nameref{requirements} variable]
-\item[\nameref{roots} operator]
-\item[\nameref{root\_of} operator]
-\item[\nameref{trigform} switch]
-\item[\nameref{varopt} switch]
-\end{Related}
-
-\end{Operator}
-
-\begin{Operator}{SORT}
-\index{sorting}
-The \name{sort} operator sorts the elements of a list according to
-an arbitrary comparison operator.
-\begin{Syntax}
-\name{sort}\(\meta{lst},\meta{comp}\)
-\end{Syntax}
-\meta{lst} is a \nameref{list} of algebraic expressions.
-\meta{comp} is a comparison operator which defines a partial
-ordering among the members of \meta{lst}.  \meta{comp} may be
-one of the builtin comparison operators like 
-\name{<}(\nameref{lessp}), \name{<=}(\nameref{leq})
-etc., or  \meta{comp}  may be the name of a comparison procedure.
-Such a procedure has two arguments, and it returns 
-\nameref{true} if the first argument
-ranges before the second one, and 0 or \nameref{nil} otherwise.
-The result of \name{sort} is a new list which contains the
-elements of \meta{lst} in a sequence corresponding to \meta{comp}.
-\begin{Examples}
- % Sort random integers\\
-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\}\\
- % Sort a set of polynomials, first for degree of x\\
- % and second for degree of y.\\
-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);&\{x^2,x*y,y^2\}
-\end{Examples}
-\end{Operator}
-
-\begin{Operator}{STRUCTR}
-\index{decomposition}
-The \name{structr} operator breaks its argument expression into named
-subexpressions.
-\begin{TEX}
-\begin{Syntax}
-\name{structr}\(\meta{expression} \&option(,\meta{identifier}
-                                  \&option(,\meta{identifier}))\)
-\end{Syntax}
-\end{TEX}
-\begin{Syntax}
-\name{structr}\(\meta{expression} [,\meta{identifier}[,\meta{identifier} ...]]\)
-\end{Syntax}
-\begin{INFO}
-\end{INFO}
-\meta{expression} may be any valid REDUCE scalar expression.
-\meta{identifier} may be any valid REDUCE \name{identifier}.  The first
-identifier
-is the stem for subexpression names, the second is the name to be assigned
-to the structured expression.
-
-\begin{Examples}
-structr(sqrt(x**2 + 2*x) + sin(x**2*z));     &
-\begin{multilineoutput}{6cm}
-ANS1 + ANS2
-    where
-        ANS2 := SIN(X^{2}*Z)
-        ANS1 := ((X + 2)*X)^{1/2}
-\end{multilineoutput}\\
-ans3;                        &      ANS3 \\
-on fort; \\
-structr((x+1)**5 + tan(x*y*z),var,aa);                  &
-\begin{multilineoutput}{6cm}
-VAR1=TAN(X*Y*Z)
-AA=VAR1+X**5+5.*X**4+10.*X**3+10.X**2+5.*X+1
-\end{multilineoutput}
-\end{Examples}
-\begin{Comments}
-The second argument to \name{structr} is optional.  If it is not given, the
-default stem \name{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 \nameref{savestructr} is
-on.
-
-If a third argument is given, the structured expression as a whole is named by
-this argument, when \nameref{fort} is on.  The expression is not stored
-under this
-name.  You can send these structured Fortran expressions to a file with the
-\name{out} command.
-\end{Comments}
-\end{Operator}
-
-
-\begin{Operator}{SUB}
-\index{substitution}
-The \name{sub} operator substitutes a new expression for a kernel in an
-expression.
-\begin{Syntax}
-%\name{sub}\(\meta{kernel}\name{=}\meta{expression}%
-%            \{,\meta{kernel}\name{=}\meta{expression}\}\optional,%
-%            \meta{expression}\)
-\name{sub}\(\meta{kernel}\name{=}\meta{expression}
-	    \{,\meta{kernel}\name{=}\meta{expression}\}\optional,
-	    \meta{expression}\) or \\
-\name{sub}\(\{\meta{kernel}\name{=}\meta{expression}\optional,
-	    \meta{kernel}\name{=}\name{expression}\},\meta{expression}\)
-\end{Syntax}
-
-\meta{kernel} must be a \nameref{kernel}, \meta{expression} can be any REDUCE
-scalar expression.
-
-\begin{Examples}
-sub(x=3,y=4,(x+y)**3);       &             343 \\
-x;                           &             X \\
-sub({cos=sin,sin=cos},cos a+sin b} & COS(B) + SIN(A)
-\end{Examples}
-\begin{Comments}
-Note in the second example that operators can be replaced using the
-\name{sub} operator.
-\end{Comments}
-\end{Operator}
-
-
-\begin{Operator}{SUM}
-\index{Gosper algorithm}\index{summation}
-The operator  \name{sum} returns
-the indefinite or definite  summation of a given expression.
-
-\begin{Syntax}
-
-\name{sum}\(\meta{expr},\meta{k}[,\meta{lolim} [,\meta{uplim} ]]\)
-
-\end{Syntax}
-
-where \meta{expr} is the expression to be added, \meta{k} is the
-control variable (a \nameref{kernel}), and \meta{lolim} and \meta{uplim}
-are the optional lower and upper limits. If \meta{uplim} is
-not supplied the upper limit is taken as \meta{k}. The Gosper
-algorithm is used. If  there is  no closed  form solution,  the operator 
-returns the input unchanged.
-
-\begin{Examples}
-sum(4n**3,n); &n^2 *(n^2  + 2*n + 1)\\
-sum(2a+2k*r,k,0,n-1);& n*(2*a + n*r - r)\\
-\end{Examples}
-\end{Operator}
-
-
-\begin{Operator}{WS}
-\index{work space}\index{interactive}
-The \name{ws} operator alone returns the last result; \name{ws} with a
-number argument returns the results of the REDUCE statement executed after
-that numbered prompt.
-\begin{Syntax}
-\name{ws} or \name{ws}\(\meta{number}\)
-\end{Syntax}
-
-
-\meta{number} must be an integer between 1 and the current REDUCE prompt number.
-
-\begin{Examples}
-
-\explanation{(In the following examples, unlike most others, the numbered
-prompt is shown.)} \\
-1: df(sin y,y);    & COS(Y) \\
-2: ws^2;           & COS(Y)^{2} \\
-3: df(ws 1,y);     & -SIN(Y)
-\end{Examples}
-
-\begin{Comments}
-\name{ws} and \name{ws}\name{(}\meta{number}\name{)} can be used anywhere 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 \name{ws;} returns the results of the differentiation.
-The current workspace (\name{ws}) can also be used inside files, though the
-numbered workspace contains only the \name{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 \nameref{input}.  The
-third stores results, when they are produced by statements, which are
-accessible by the \name{ws}\meta{ 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 \nameref{saveas} and \nameref{out} commands.
-% Or you could use the \name{forget} command to clear all history lists and
-% reset the prompt number to 1, which doesn't change any switch settings or
-% variable assignments.
-
-An error message is given if a reference number has not yet been used.
-\end{Comments}
-\end{Operator}
-
+\section{Algebraic Operators}
+
+\begin{Operator}{APPEND}
+\index{list}
+The \name{append} operator constructs a new \nameref{list} 
+from the elements of its two arguments (which must be lists).
+
+\begin{Syntax}
+\name{append}\(\meta{list},\meta{list}\)
+\end{Syntax}
+
+\meta{list} must be a list, though it may be the empty list (\name{\{\}}).
+Any arguments beyond the first two are ignored.
+
+\begin{Examples}
+alist := \{1,2,\{a,b\}\};      &      ALIST := \{1,2,\{A,B\}\} \\
+blist := \{3,4,5,sin(y)\};   &      BLIST := \{3,4,5,SIN(Y)\} \\
+append(alist,blist);       &      \{1,2,\{A,B\},3,4,5,SIN(Y)\} \\
+append(alist,\{\});          &      \{1,2,\{A,B\}\} \\
+append(list z,blist);      &      \{Z,3,4,5,SIN(Y)\}
+\end{Examples}
+
+\begin{Comments}
+The new list consists of the elements of the second list appended to the
+elements of the first list.  You can \name{append} new elements to the
+beginning or end of an existing list by putting the new element in a
+list (use curly braces or the operator \name{list}).  This is
+particularly helpful in an iterative loop.
+\end{Comments}
+\end{Operator}
+
+\begin{Operator}{ARBINT}
+\index{arbitrary value}
+The operator \name{arbint} is used to express arbitrary integer parts
+of an expression, e.g. in the result of \nameref{solve} when
+\nameref{allbranch} is on. 
+\begin{Examples}
+
+solve(log(sin(x+3)),x);    &
+\begin{multilineoutput}{6cm}
+\{X=2*ARBINT(1)*PI - ASIN(1) - 3,
+ X=2*ARBINT(1)*PI + ASIN(1) + PI - 3\}
+\end{multilineoutput}
+\end{Examples}
+\end{Operator} 
+
+\begin{Operator}{ARBCOMPLEX}
+\index{arbitrary value}
+The operator \name{arbcomplex} is used to express arbitrary scalar parts
+of an expression, e.g. in the result of \nameref{solve} when
+the solution is parametric in one of the variable.
+\begin{Examples}
+
+solve({x+3=y-2z,y-3x=0},{x,y,z});    &
+\begin{multilineoutput}{6cm}
+\{X=\rfrac{2*ARBCOMPLEX(1) + 3}{2},
+  Y=\rfrac{3*ARBCOMPLEX(1) + 3}{2},
+  Z=ARBCOMPLEX(1)\}
+\end{multilineoutput}
+\end{Examples}
+\end{Operator} 
+
+\begin{Operator}{ARGLENGTH}
+\index{argument}
+The operator \name{arglength} returns the number of arguments of the top-level
+operator in its argument.
+
+\begin{Syntax}
+\name{arglength}\(\meta{expression}\)
+\end{Syntax}
+
+\meta{expression} can be any valid REDUCE algebraic expression.
+
+\begin{Examples}
+arglength(a + b + c + d);                             &      4 \\
+arglength(a/b/c);                                     &      2 \\
+arglength(log(sin(df(r**3*x,x))));                    &      1
+\end{Examples}
+
+\begin{Comments}
+In the first example, \name{+} is an n-ary operator, so the number of terms
+is returned.  In the second example, since \name{/} is a binary operator, the
+argument is actually (a/b)/c, so there are two terms at the top level.  In
+the last example, no matter how deeply the operators are nested, there is
+still only one argument at the top level.
+\end{Comments}
+\end{Operator}
+
+
+\begin{Operator}{COEFF}
+\index{coefficient}
+The \name{coeff} operator returns the coefficients of the powers of the
+specified variable in the given expression, in a \nameref{list}.
+
+\begin{Syntax}
+\name{coeff}\(\meta{expression}\name{,}\meta{variable}\)
+\end{Syntax}
+
+\meta{expression} is expected to be a polynomial expression, not a rational
+expression.  Rational expressions are accepted when the switch
+\nameref{ratarg} is on.  \meta{variable} must be a kernel.  The results are
+returned in a list.
+
+\begin{Examples}
+coeff((x+y)**3,x);           &       \{Y^{3} ,3*Y^{2} ,3*Y,1\} \\
+coeff((x+2)**4 + sin(x),x);  &       \{SIN(X) + 16,32,24,8,1\} \\
+high_pow;                    &       4 \\
+low_pow;                     &       0 \\
+ab := x**9 + sin(x)*x**7 + sqrt(y); 
+                             &       AB := SQRT(Y) + SIN(X)*X^{7}  + X^{9}\\
+coeff(ab,x);                 &       \{SQRT(Y),0,0,0,0,0,0,SIN(X),0,1\}
+\end{Examples}
+\begin{Comments}
+The variables \nameref{high\_pow} and \nameref{low\_pow} are set to the
+highest and lowest powers of the variable, respectively, appearing in the
+expression.
+
+The coefficients are put into a list, with the coefficient of the lowest
+(constant) term first.  You can use the usual list access methods
+(\name{first}, \name{second}, \name{third}, \name{rest}, \name{length}, and
+\name{part}) to extract them.  If a power does not appear in the
+expression, the corresponding element of the list is zero. Terms involving
+functions of the specified variable but not including powers of it (for
+example in the expression \name{x**4 + 3*x**2 + tan(x)}) are placed in the
+constant term.
+
+Since the \name{coeff} command deals with the expanded form of the expression,
+you may get unexpected results when \nameref{exp} is off, or when
+\nameref{factor} or \nameref{ifactor} are on.
+
+If you want only a specific coefficient rather than all of them, use the
+\nameref{coeffn} operator.
+
+\end{Comments}
+\end{Operator}
+
+
+\begin{Operator}{COEFFN}
+\index{coefficient}
+The \name{coeffn} operator takes three arguments: an expression, a kernel, and
+a non-negative integer.  It returns the coefficient of the kernel to that
+integer power, appearing in the expression.
+
+\begin{Syntax}
+\name{coeffn}\(\meta{expression},\meta{kernel},\meta{integer}\)
+\end{Syntax}
+
+\meta{expression} must be a polynomial, unless \nameref{ratarg} is on which
+allows rational expressions.  \meta{kernel} must be a kernel, and
+\meta{integer} must be a non-negative integer.
+
+\begin{Examples}
+
+ff := x**7 + sin(y)*x**5 + y**4 + x + 7;                &
+               FF := SIN(Y)*X^{5}  + X^{7}  + X + Y^{4}  + 7 \\
+coeffn(ff,x,5);              &        SIN(Y) \\
+coeffn(ff,z,3);              &        0 \\
+coeffn(ff,y,0);              &        SIN(Y)*X^{5}  + X^{7}  + X + 7 \\
+
+rr := 1/y**2+y**3+sin(y);    &        
+RR := \rfrac{SIN(Y)*Y^{2}  + Y^{5}  + 1}{Y^{2}} \\
+on ratarg; \\
+
+coeffn(rr,y,-2);             &        ***** -2 invalid as COEFFN index \\
+
+coeffn(rr,y,5);              &        \rfrac{1}{Y^{2}}\\
+                                        
+\end{Examples}
+
+\begin{Comments}
+If the given power of the kernel does not appear in the expression,
+\name{coeffn} returns 0.  Negative powers are never detected, even if
+they appear in the expression and \nameref{ratarg} are on.  \name{coeffn}
+with an integer argument of 0 returns any terms in the expression that
+do {\em not} contain the given kernel.
+\end{Comments}
+\end{Operator}
+
+
+\begin{Operator}{CONJ}
+\index{conjugate}\index{complex}
+
+\begin{Syntax}
+\name{conj}\(\meta{expression}\) or \name{conj} \meta{simple\_expression}
+\end{Syntax}
+This operator returns the complex conjugate of an expression, if that
+argument has an numerical value.  A non-numerical argument is returned as
+an expression in the operators \nameref{repart} and \nameref{impart}.
+
+\begin{Examples}
+conj(1+i); & 1-I \\
+conj(a+i*b); & REPART(A) - REPART(B)*I - IMPART(A)*I - IMPART(B)
+\end{Examples}
+
+\end{Operator}
+
+\begin{Operator}{CONTINUED_FRACTION}
+\index{approximation}\index{rational numbers}
+
+\begin{Syntax}
+\name{continued\_fraction}\(\meta{num}\) 
+or \name{continued\_fraction}\( \meta{num},\meta{size}\)
+\end{Syntax}
+This operator approximates the real number \meta{num}
+( \nameref{rational} number, \nameref{rounded} number)
+into a continued fraction. The result is a list of two elements: the
+first one is the rational value of the approximation, the second one
+is the list of terms of the continued fraction which represents the
+same value according to the definition \name{t0 +1/(t1 + 1/(t2 + ...))}.
+Precision: the second optional parameter \meta{size} is an upper bound
+for the absolute value of the result denominator. If omitted, the
+approximation is performed up to the current system precision.
+
+
+\begin{Examples}
+continued_fraction pi;
+ & \{\rfrac{1146408},{364913},\{3,7,15,1,292,1,1,1,2,1\}\} \\
+continued_fraction(pi,100);
+ & \{\rfrac{22},{7},\{3,7\}\} \\
+\end{Examples}
+
+\end{Operator}
+
+\begin{Operator}{DECOMPOSE}
+\index{decomposition}\index{polynomial}
+The \name{decompose} operator takes a multivariate polynomial as argument,
+and returns an expression and a \nameref{list} of 
+\nameref{equation}s from which the
+original polynomial can be found by composition.
+
+\begin{Syntax}
+\name{decompose}\(\meta{expression}\) or \name{decompose}
+ \meta{simple\_expression}
+\end{Syntax}
+
+\begin{Examples}
+\begin{multilineinput}
+decompose(x^8-88*x^7+2924*x^6-43912*x^5+263431*x^4-
+          218900*x^3+65690*x^2-7700*x+234)
+\end{multilineinput}
+              & {U^{2} + 35*U + 234, U=V^{2} + 10*V, V=X^{2} - 22*X} \\
+     decompose(u^2+v^2+2u*v+1)  & {W^{2}  + 1, W=U + V}
+\end{Examples}
+
+\begin{Comments}
+Unlike factorization, this decomposition is not unique.  Further
+details can be found in V.S. Alagar, M.Tanh, \meta{Fast Polynomial
+Decomposition}, Proc. EUROCAL 1985, pp 150-153 (Springer) and J. von zur
+Gathen, \meta{Functional} 
+\meta{Decomposition of Polynomials: the Tame Case}, J.
+Symbolic Computation (1990) 9, 281-299.
+\end{Comments}
+\end{Operator}
+
+
+\begin{Operator}{DEG}
+\index{degree}\index{polynomial}
+The operator \name{deg} returns the highest degree of its variable argument
+found in its expression argument.
+
+\begin{Syntax}
+\name{deg}\(\meta{expression},\meta{kernel}\)
+\end{Syntax}
+
+\meta{expression} is expected to be a polynomial expression, not a rational
+expression.  Rational expressions are accepted when the switch
+\nameref{ratarg} is on.  \meta{variable} must be a \nameref{kernel}.  The
+results are returned in a list.
+
+\begin{Examples}
+
+deg((x+y)**5,x);            &        5 \\
+
+deg((a+b)*(c+2*d)**2,d);    &        2 \\
+
+deg(x**2 + cos(y),sin(x));   \\
+
+deg((x**2 + sin(x))**5,sin(x));                        &        5
+\end{Examples}
+\end{Operator}
+
+
+\begin{Operator}{DEN}
+\index{denominator}\index{rational expression}
+The \name{den} operator returns the denominator of its argument.
+
+\begin{Syntax}
+\name{den}\(\meta{expression}\)
+\end{Syntax}
+
+\meta{expression} is ordinarily a rational expression, but may be any valid
+scalar REDUCE expression.
+
+\begin{Examples}
+
+a := x**3 + 3*x**2 + 12*x;  &      A := X*(X^{2}  + 3*X + 12) \\
+
+b := 4*x*y + x*sin(x);      &      B := X*(SIN(X) + 4*Y) \\
+
+den(a/b);                   &      SIN(X) + 4*Y \\
+
+den(aa/4 + bb/5);           &      20 \\
+
+den(100/6);                 &      3 \\
+
+den(sin(x));                &      1
+\end{Examples}
+
+\begin{Comments}
+\name{den} returns the denominator of the expression after it has been
+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 does not have any
+other denominator, 1 is returned.
+
+Switch settings, such as \nameref{mcd} or \nameref{rational}, have an
+effect on the denominator of an expression.
+\end{Comments}
+\end{Operator}
+
+
+\begin{Operator}{DF}
+\index{derivative}\index{partial derivative}
+The \name{df} operator finds partial derivatives with respect to one or
+more variables.
+
+\begin{TEX}
+\begin{Syntax}
+  \name{df}\(\meta{expression}\name{,}\meta{var}
+	     \&optional\(\name{,}\meta{number}\)
+             \{\name{,}\meta{var}\&option\(\name{,}\meta{number}\)\}\optional\)
+\end{Syntax}
+\end{TEX}
+\begin{INFO}{
+\begin{Syntax}
+  \name{df}\(\meta{expression}\name{,}\meta{var}
+	     [\name{,}\meta{number}\]
+             \{\name{,}\meta{var} [ \name{,}\meta{number}] \} \)
+\end{Syntax}
+}\end{INFO}
+
+\meta{expression} can be any valid REDUCE algebraic expression.  \meta{var}
+must be a \nameref{kernel}, and is the differentiation variable.
+\meta{number} must be a non-negative integer.
+
+\begin{Examples}
+
+df(x**2,x);                 &         2*X \\
+
+df(x**2*y + sin(y),y);      &         COS(Y) + X^{2} \\
+
+df((x+y)**10,z);            &         0 \\
+
+
+df(1/x**2,x,2);             &        \rfrac{6}{X^{4}}\\
+
+df(x**4*y + sin(y),y,x,3);  &         24*X \\
+
+for all x let df(tan(x),x) = sec(x)**2; \\
+
+df(tan(3*x),x);             &         3*SEC(3*X)^{2}
+\end{Examples}
+\begin{Comments}
+An error message results if a non-kernel is entered as a differentiation
+operator.  If the optional number is omitted, it is assumed to be 1.
+See the declaration \nameref{depend} to establish dependencies for implicit
+differentiation.
+
+You can define your own differentiation rules, expanding REDUCE's
+capabilities, using the \nameref{let} command as shown in the last example
+above.  Note that once you add your own rule for differentiating a
+function, it supersedes REDUCE's normal handling of that function for the
+duration of the REDUCE session.  If you clear the rule
+(\nameref{clearrules}), you don't get back
+to the previous rule.
+\end{Comments}
+\end{Operator}
+
+\begin{Operator}{EXPAND\_CASES}
+\index{solve}
+When a \nameref{root\_of} form in a result of \nameref{solve} 
+has been converted to a \nameref{one\_of} form, \name{expand\_cases}
+can be used to convert this into form corresponding to the
+normal explicit results of \nameref{solve}. See \nameref{root\_of}.
+\end{Operator}
+
+\begin{Operator}{EXPREAD}
+\index{input}
+\begin{Syntax}
+\name{expread}\(\)
+\end{Syntax}
+
+\name{expread} reads one well-formed expression from the current input
+buffer and returns its value.
+
+\begin{Examples}
+expread(); a+b;  & A + B
+\end{Examples}
+
+\end{Operator}
+
+
+\begin{Operator}{FACTORIZE}
+\index{factorize}\index{polynomial}
+The \name{factorize} operator factors a given expression.
+\begin{Syntax}
+\name{factorize}\(\meta{expression}\)
+\end{Syntax}
+
+\meta{expression} should be a polynomial, otherwise an error will result.
+
+\begin{Examples}
+
+fff := factorize(x^3 - y^3);                          &
+                  \{X - Y,X^{2} + X*Y + Y^{2}\} \\
+fac1 := first fff;           &      FAC1 := X - Y \\
+factorize(x^15 - 1);        &
+\begin{multilineoutput}{5cm}
+ \{X - 1,
+  X^{2} + X + 1,
+  X^{4} + X^{3} + X^{2} + X + 1,
+  X^{8} - X^{7} + X^{6} - X^{5} + X^{4} - X + 1\}
+\end{multilineoutput}\\
+lastone := part(ws,length ws);                          &
+        LASTONE := X^{8} - X^{7} + X^{6} - X^{5} + X^{4} - X + 1 \\
+setmod 2;                    &      1 \\
+on modular; \\
+factorize(x^15 - 1);        &
+\begin{multilineoutput}{5cm}
+\{X + 1,
+ X^{2} + X + 1,
+ X^{4} + X + 1,
+ X^{4} + X^{3} + 1,
+ X^{4} + X^{3} + X^{2} + X + 1\}
+\end{multilineoutput}
+\end{Examples}
+
+\begin{Comments}
+The \name{factorize} command returns the factors it finds as a \nameref{list}.
+You can therefore use the usual list access methods (\nameref{first},
+\nameref{second}, \nameref{third}, \nameref{rest}, \nameref{length} and
+\nameref{part}) to extract the factors.
+
+If the \meta{expression} given to \name{factorize} is an integer, it will be
+factored into its prime components.  To factor any integer factor of a
+non-numerical expression, the switch \nameref{ifactor} should be turned on.
+Its default is off. \nameref{ifactor} has effect only when factoring is
+explicitly done by \name{factorize}, not when factoring is automatically
+done with the \nameref{factor} switch.  If full factorization is not
+needed the switch \nameref{limitedfactors} allows you to reduce the
+computing time of calls to \name{factorize}.
+
+Factoring can be done in a modular domain by calling \name{factorize} when
+\nameref{modular} is on.  You can set the modulus with the \nameref{setmod}
+command.  The last example above shows factoring modulo 2.
+
+For general comments on factoring, see comments under the switch
+\nameref{factor}.
+\end{Comments}
+\end{Operator}
+
+
+\begin{Operator}{HYPOT}
+
+\begin{Syntax}
+hypot(\meta{expression},\meta{expression})
+\end{Syntax}
+
+If \name{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.
+
+\begin{Examples}
+hypot(3,4); & HYPOT(3,4) \\
+on rounded; \\
+ws; & 5.0 \\
+hypot(a,b); & HYPOT(A,B)
+\end{Examples}
+
+\end{Operator}
+
+
+\begin{Operator}{IMPART}
+\index{imaginary part}\index{complex}
+\begin{Syntax}
+\name{impart}\(\meta{expression}\) or \name{impart} \meta{simple\_expression}
+\end{Syntax}
+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 \nameref{repart} and \name{impart}.
+\begin{Examples}
+impart(1+i);   &  1 \\
+impart(a+i*b); &  REPART(B) + IMPART(A)
+\end{Examples}
+
+\end{Operator}
+
+
+\begin{Operator}{INT}
+\index{integration}
+The \name{int} operator performs analytic integration on a variety of
+functions.
+
+\begin{Syntax}
+\name{int}\(\meta{expression},\meta{kernel}\)
+\end{Syntax}
+
+\meta{expression} can be any scalar expression. involving polynomials, log
+functions, exponential functions, or tangent or arctangent expressions.
+\name{int} attempts expressions involving error functions, dilogarithms
+and other trigonometric expressions.  Integrals involving algebraic
+extensions (such as square roots) may not succeed. \meta{kernel} must be a
+REDUCE \nameref{kernel}.
+
+\begin{Examples}
+int(x**3 + 3,x);             &      \rfrac{X*(X^{3} + 12)}{4} \\\\
+int(sin(x)*exp(2*x),x);
+  &      - \rfrac{E^{2*X}*(COS(X) - 2*SIN(X))}{5} \\
+int(1/(x^2-2),x);
+  & \rfrac{SQRT(2)*(LOG( - SQRT(2) + X) - LOG(SQRT(2) + X))}{4} \\
+int(sin(x)/(4 + cos(x)**2),x);
+  &       - \rfrac{ATAN(\rfrac{COS(X)}{2})}{2} \\\\
+int(1/sqrt(x^2-x),x); & INT(\rfrac{SQRT(X)*SQRT(X - 1)}{X^{2}-X},X)
+\end{Examples}
+
+\begin{Comments}
+Note that REDUCE couldn't handle the last integral with its default
+integrator, since the integrand involves a square root.  However,
+the integral can be found using the \nameref{algint} package.
+Alternatively, you could add a rule using the \nameref{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
+\nameref{rounded}) and subtracting the lower from the higher.  Evaluation can
+be easily done by the \nameref{sub} operator.
+
+When \name{int} cannot find an integral it returns an expression
+involving formal \name{int} expressions unless the switch 
+\nameref{failhard} has been set. If not all of the expression
+can be integrated, the switch \nameref{nolnr} controls whether a partially
+integrated result should be returned or not.
+
+\end{Comments}
+\end{Operator}
+
+
+\begin{Operator}{INTERPOL}
+\index{interpolation}\index{polynomial}\index{approximation}
+\name{interpol} generates an interpolation polynomial.
+\begin{Syntax}
+   interpol(\meta{values},\meta{variable},\meta{points})
+\end{Syntax}
+
+\meta{values} and \meta{points} are \nameref{list}s of equal length and
+\meta{variable} is an algebraic expression (preferably a \nameref{kernel}).
+The interpolation polynomial is generated in the given variable of degree
+length(\meta{values})-1.  The unique polynomial \name{f} is defined by the
+property that for corresponding elements \name{v} of \meta{values} and
+\name{p} of \meta{points} the relation \name{f(p)=v} holds.
+
+\begin{Examples}
+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); & X^{3} - 1
+\end{Examples}
+
+\begin{Comments}
+The Aitken-Neville interpolation algorithm is used which guarantees a
+stable result even with rounded numbers and an ill-conditioned problem.
+\end{Comments}
+\end{Operator}
+
+
+\begin{Operator}{LCOF}
+\index{coefficient}\index{polynomial}
+The \name{lcof} operator returns the leading coefficient of a given expression
+with respect to a given variable.
+\begin{Syntax}
+\name{lcof}\(\meta{expression},\meta{kernel}\)
+\end{Syntax}
+
+\meta{expression} is ordinarily a polynomial. If \nameref{ratarg} is on,
+a rational expression may also be used, otherwise an error results.
+\meta{kernel} must be a \nameref{kernel}.
+
+\begin{Examples}
+lcof((x+2*y)**5,y);          &         32 \\
+lcof((x + y*sin(x))**2 + cos(x)*sin(x)**2,sin(x));
+			     &         COS(X)^{2} + Y \\
+lcof(x**2 + 3*x + 17,y);     &         X^{2} + 3*X + 17
+\end{Examples}
+
+\begin{Comments}
+If the kernel does not appear in the expression, \name{lcof} returns the
+expression. 
+\end{Comments}
+\end{Operator}
+
+
+\begin{Operator}{LENGTH}
+\index{list}
+The \name{length} operator returns the number of items in a \nameref{list}, the 
+number of
+terms in an expression, or the dimensions of an array or matrix.
+\begin{Syntax}
+\name{length}\(\meta{expr}\) or \name{length} \meta{expr}
+\end{Syntax}
+
+\meta{expr} can be a list structure, an array, a matrix, or a scalar expression.
+
+\begin{Examples}
+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);       &     EEX := \rfrac{X^{2} + 6*X + 9}{X - Y} \\
+length eex;                  &     5
+\end{Examples}
+
+\begin{Comments}
+An item in a list that is itself a list only counts as one item.  An error
+message will be printed if \name{length} is called on a matrix which has
+not had its dimensions set.  The \name{length} of an array includes the
+zeroth element of each dimension, showing the full number of elements
+allocated. (Declaring an array \IFTEX{$A$}{A} with \IFTEX{$n$}{n} elements
+allocates \IFTEX{$ A(0),A(1),\ldots,A(n)$}{A(0),A(1),...,A(n)}.) The
+\name{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.
+\end{Comments}
+\end{Operator}
+
+
+\begin{Operator}{LHS}
+\index{left-hand side}\index{equation}
+The \name{lhs} operator returns the left-hand side of an \nameref{equation},
+such as those
+returned in a list by \nameref{solve}.
+\begin{Syntax}
+\name{lhs}\(\meta{equation}\) or \name{lhs} \meta{equation}
+
+\end{Syntax}
+
+\meta{equation} must be an equation of the form \\
+\name{left-hand side} \name{=} \name{right-hand side}.
+
+\begin{Examples}
+polly := (x+3)*(x^4+2x+1); &
+POLLY := X^{5} + 3*X^{4} + 2*X^{2} + 7*X + 3 \\
+pollyroots := solve(polly,x); &
+\begin{multilineoutput}{1cm}
+POLLYROOTS := \{X=ROOT_OF(X_^{3} - X_^{2} + X_ + 1,X_),
+               X=-1,
+               X=-3\}
+\end{multilineoutput} \\
+variable := lhs first pollyroots; &
+VARIABLE := X
+\end{Examples}
+\end{Operator}
+
+
+\begin{Operator}{LIMIT}
+\index{limit}\index{l'Hopital's rule}
+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.        
+
+\begin{Syntax}
+\name{limit}\(\meta{expr},\meta{var},\meta{limpoint}\) or \\
+\name{limit!+}\(\meta{expr},\meta{var},\meta{limpoint}\) or \\
+\name{limit!-}\(\meta{expr},\meta{var},\meta{limpoint}\) 
+\end{Syntax}
+
+where \meta{expr} is an expression depending of the variable \meta{var}
+(a \nameref{kernel}) and \meta{limpoint} is the limit point. 
+If the limit depends upon the direction of approach to  the \meta{limpoint},
+the operators \name{limit!+} and \name{limit!-} may be used.
+
+\begin{Examples}
+limit(x*cot(x),x,0);&0\\
+limit((2x+5)/(3x-2),x,infinity);&\rfrac{2}{3}\\
+\end{Examples}
+
+\end{Operator}
+
+
+\begin{Operator}{LPOWER}
+\index{leading power}\index{polynomial}
+The \name{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.
+\begin{Syntax}
+\name{lpower}\(\meta{expression},\meta{kernel}\)
+\end{Syntax}
+
+\meta{expression} is ordinarily a polynomial. If \nameref{ratarg} is on,
+a rational expression may also be used, otherwise an error results.
+\meta{kernel} must be a \nameref{kernel}.
+
+\begin{Examples}
+lpower((x+2*y)**6,y);        &     Y^{6} \\
+lpower((x + cos(x))**8 + df(x**2,x),cos(x));
+			     &     COS(X)^{8} \\
+lpower(x**3 + 3*x,y);        &     1
+\end{Examples}
+\end{Operator}
+
+\begin{Operator}{LTERM}
+\index{leading term}\index{polynomial}
+The \name{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.
+\begin{Syntax}
+\name{lterm}\(\meta{expression},\meta{kernel}\)
+\end{Syntax}
+
+\meta{expression} is ordinarily a polynomial. If \nameref{ratarg} is on,
+a rational expression may also be used, otherwise an error results.
+\meta{kernel} must be a \nameref{kernel}.
+
+\begin{Examples}
+lterm((x+2*y)**6,y);        &     64*Y^{6} \\
+lterm((x + cos(x))**8 + df(x**2,x),cos(x));
+			    &     COS(X)^{8} \\
+lterm(x**3 + 3*x,y);        &     X^{3} + 3X
+\end{Examples}
+\end{Operator}
+
+
+\begin{Operator}{MAINVAR}
+\index{main variable}\index{polynomial}
+The \name{mainvar} operator returns the main variable (in the system's
+internal representation) of its argument.
+\begin{Syntax}
+\name{mainvar}\(\meta{expression}\)
+
+\end{Syntax}
+
+\meta{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
+\nameref{kernel}.
+
+\begin{Examples}
+test := (a + b + c)**2;      &
+     TEST := A^{2} + 2*A*B + 2*A*C + B^{2} + 2*B*C + C^{2} \\
+mainvar(test);               &        A \\
+korder c,b,a; \\
+mainvar(test);               &        C \\
+mainvar(2*cos(x)**2);        &        COS(X) \\
+mainvar(17);                 &        0
+\end{Examples}
+
+\begin{Comments}
+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 \name{mainvar} is concerned belong in the family \name{0}.  The
+canonical ordering can be changed by the declaration \nameref{korder}, as
+shown above.
+\end{Comments}
+\end{Operator}
+
+
+\begin{Operator}{MAP}
+\index{map}\index{composite structure}
+The \name{map} operator applies a uniform evaluation pattern
+to all members of a composite structure: a \nameref{matrix}, 
+a \nameref{list} or the arguments of an \nameref{operator} expression. 
+The evaluation pattern can be a
+unary procedure, an operator, or an algebraic expression with
+one free variable. 
+\begin{Syntax}
+  \name{map}\(\meta{function},\meta{object}\)
+\end{Syntax}
+\meta{object} is a list, a matrix or an operator expression.
+
+\meta{function} is 
+the name of an operator for a single argument: the operator
+ is evaluated once with each element of \meta{object} as its single argument,
+
+or an algebraic expression with exactly one \nameref{free variable}, that is
+a variable preceded by the tilde symbol: the expression
+ is evaluated for each element of \meta{object} where the element is
+ substituted for the free variable,
+
+or a replacement \nameref{rule} of the form 
+\begin{Syntax}
+  \name{var} => \name{rep}
+\end{Syntax}
+  where \meta{var} is a variable (a \meta{kernel} without subscript)
+ and \meta{rep} is an expression which contains \meta{var}.
+  Here \name{rep} is evaluated for each element of \meta{object} where 
+  the element is substituted for  \name{var}. \name{var} may be 
+  optionally preceded by a tilde.
+
+The rule form  for \meta{function} is needed when more than
+one free variable occurs.
+
+\begin{Examples}
+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))); &
+\begin{multilineoutput}{1cm}
+        [  3     6 ]
+        [ x     x  ]
+        [----  ----]
+        [ 3     6  ]
+        [          ]
+        [  5     6 ]
+        [ x     x  ]
+        [----  ----]
+	[ 5     6  ]
+\end{multilineoutput}\\
+map(~w*6, x^2/3 = y^3/2 -1); & 2*x^2=3*(y^3-2)\\
+\end{Examples}
+
+\begin{Comments}
+You can use \name{map} in nested expressions. It is not allowed to
+apply \name{map} for a non-composed object, e.g. an identifier or a number. 
+\end{Comments}
+\end{Operator}
+
+
+
+\begin{Command}{MKID}
+\index{identifier}
+The \name{mkid} command constructs an identifier, given a stem and an identifier
+or an integer.
+\begin{Syntax}
+\name{mkid}\(\meta{stem},\meta{leaf}\)
+\end{Syntax}
+
+\meta{stem} can be any valid REDUCE identifier that does not include escaped
+special characters. \meta{leaf} may be an integer, including one given by a
+local variable in a \nameref{for} loop, or any other legal group of
+characters.
+
+\begin{Examples}
+mkid(x,3);                  &               X3 \\
+factorize(x^15 - 1);        &  \begin{multilineoutput}{6cm}
+\{X - 1,
+ X^{2} + X + 1,
+ X^{4} + X^{3} + X^{2} + X + 1,
+ X^{8} - X^{7} + X^{5} - X^{4} + X^{3}  - X + 1\}
+\end{multilineoutput}\\
+
+for i := 1:length ws do write set(mkid(f,i),part(ws,i));
+	  & \begin{multilineoutput}{6cm}
+X^{8} - X^{7} + X^{5} - X^{4} + X^{3} - X + 1
+X^{4} + X^{3} + X^{2} + X + 1
+X^{2} + X + 1
+X - 1
+\end{multilineoutput} \\
+\end{Examples}
+
+\begin{Comments}
+You can use \name{mkid} to construct identifiers from inside procedures.  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 \name{factorize} and \name{solve}.
+\end{Comments}
+\end{Command}
+
+
+\begin{Operator}{NPRIMITIVE}
+\index{primitive part}\index{polynomial}
+\begin{Syntax}
+\name{nprimitive}\(\meta{expression}\) or \name{nprimitive}
+   \meta{simple\_expression}
+\end{Syntax}
+This operator returns the numerically-primitive part of any scalar
+expression.  In other words, any overall integer factors in the expression
+are removed.
+
+\begin{Examples}
+nprimitive((2x+2y)^2); & X^{2} + 2*X*Y + Y^{2} \\
+nprimitive(3*a*b*c); & 3*A*B*C
+\end{Examples}
+\end{Operator}
+
+
+\begin{Operator}{NUM}
+\index{numerator}\index{rational expression}
+The \name{num} operator returns the numerator of its argument.
+\begin{Syntax}
+\name{num}\(\meta{expression}\) or \name{num} \meta{simple\_expression}
+\end{Syntax}
+
+\meta{expression} can be any valid REDUCE scalar expression.
+
+\begin{Examples}
+num(100/6);                 &         50 \\
+num(a/5 + b/6);             &         6*A + 5*B \\
+num(sin(x));                &         SIN(X)
+\end{Examples}
+
+\begin{Comments}
+\name{num} returns the numerator of the expression after it has been 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.  
+\end{Comments}
+\end{Operator}
+
+
+\begin{Operator}{ODESOLVE}
+\index{differential equation}\index{solve}
+The \name{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.
+
+\begin{Syntax}
+
+\name{odesolve}\(\meta{expr},\meta{var1},\meta{var2}\)
+
+\end{Syntax}
+
+\meta{expr} is  a single  scalar expression  such  that \meta{expr}=0  
+is the ordinary differential equation (ODE for short) to be  solved, or
+is an equivalent \nameref{equation}.
+
+\meta{var1} is the name of the dependent variable,
+\meta{var2} is the name of the independent variable.
+
+A differential in \meta{expr} is expressed using the \nameref{df}
+operator. Note that in most cases you must declare explicitly
+\meta{var1} to depend of \meta{var2} using a \nameref{depend}
+declaration -- otherwise the derivative might be evaluated to
+zero on input to \name{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 \name{arbconst} for the  arbitrary constants  in the  general
+solution.   The arguments of \name{arbconst} should  be new.
+A counter  \name{!!arbconst}  is used  to  arrange  this.
+
+\begin{Examples}
+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)\}
+\end{Examples}      
+
+\end{Operator}
+
+
+\begin{Type}{ONE\_OF}
+The operator \name{one\_of} is used to represent an indefinite choice
+of one element from a finite set of objects. 
+\begin{Examples}
+x=one_of{1,2,5}\\
+\explanation{this equation encodes that x can take one of the values
+1,2 or 5}\\
+\end{Examples}
+REDUCE generates a \name{one\_of} form in cases when an implicit
+\name{root\_of} expression could be converted to an explicit solution set.
+A \name{one\_of} form can be converted to a \name{solve} solution using
+\nameref{expand\_cases}.  See \nameref{root\_of}.
+\end{Type}
+
+\begin{Operator}{PART}
+\index{decomposition}
+The operator \name{part} permits the extraction of various parts or
+operators of expressions and \nameref{list}\name{s}.
+\begin{Syntax}
+\name{part}\(\meta{expression,integer}\{,\meta{integer}\}\optional\)
+\end{Syntax}
+
+\meta{expression} can be any valid REDUCE expression or a list, {\it
+integer} may be an expression that evaluates to a positive or negative
+integer or 0.  A positive integer \meta{n} picks up the {\it n} th term,
+counting from the first term toward the end.  A negative integer {\it n}
+picks up the {\it n} th term, counting from the back toward the front.  The
+integer 0 picks up the operator (which is \name{LIST} when the expression
+is a \ref{list}).
+
+\begin{Examples}
+part((x + y)**5,4);          &        10*X^{2}*Y^{3} \\
+part((x + y)**5,4,2);        &        X^{2} \\
+part((x + y)**5,4,2,1);      &        X \\
+part((x + y)**5,0);          &        PLUS \\
+part((x + y)**5,-5);         &        5*X *Y^{4} \\
+part((x + y)**5,4) := sin(x); &                          
+  X^{5} + 5*X^{4}*Y + 10*X^{3}*Y^{2} + SIN(X) + 5*X*Y^{4} + Y^{5} \\
+alist := \{x,y,\{aa,bb,cc\},x**2*sqrt(y)\};                 &
+			 ALIST := \{X,Y,\{AA,BB,CC\},SQRT(Y)*X^{2}\} \\
+part(alist,3,2);             &       BB \\
+part(alist,4,0);             &       TIMES
+\end{Examples}
+
+\begin{Comments}
+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, \IFTEX{$10x^2y^3$}{10x^2y^3},
+then the second term from that, \IFTEX{$x^2$}{x^2}, and finally the first
+component, \IFTEX{$x$}{x}.  If an integer's absolute value is too large for
+the appropriate expression, a message is given.
+
+\name{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 \name{part}.  \nameref{pri} must be on when
+\name{part} is used.
+
+When \name{part} is used on a polynomial expression that has minus signs, the
+\name{+} is always returned as the top-level operator.  The minus is found
+as a unary operator attached to the negative term.
+
+\name{part} can also be used to change the relevant part of the expression or
+list as shown in the sixth example line.  The \name{part} operator returns the
+changed expression, though original expression is not changed.  You can
+also use \name{part} to change the operator.
+\end{Comments}
+\end{Operator}
+
+\begin{Operator}{PF}
+\index{partial fraction}\index{rational expression}
+\begin{Syntax}
+pf(\meta{expression},\meta{variable})
+\end{Syntax}
+
+\name{pf} transforms \meta{expression} into a \nameref{list} of partial fraction
+s
+with respect to the main variable, \meta{variable}. \name{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.
+\begin{Examples}
+pf(2/((x+1)^2*(x+2)),x); &
+	\{\rfrac{2}{X + 2},\rfrac{-2}{X + 1},\rfrac{2}{X^{2} + 2*X + 1}\} \\
+off exp; \\
+pf(2/((x+1)^2*(x+2)),x);
+ & \{\rfrac{2}{X + 2},\rfrac{- 2}{X + 1},\rfrac{2}{(X + 1)^{2}}\} \\
+for each j in ws sum j; & \rfrac{2}{( + 2)*(X + 1)^{2}}
+\end{Examples}
+
+\begin{Comments}
+If you want the denominators in factored form, turn \nameref{exp} off, as
+shown in the second example above.  As shown in the final example, the
+\nameref{for} \name{each} construct can be used to recombine the terms.
+Alternatively, one can use the operations on lists to extract any desired
+term.
+\end{Comments}
+
+\end{Operator}
+
+
+\begin{Operator}{PROD}
+\index{Gosper algorithm}\index{product}
+The operator  \name{prod} returns
+the indefinite or definite  product of a given expression.
+
+\begin{Syntax}
+
+\name{prod}\(\meta{expr},\meta{k}[,\meta{lolim} [,\meta{uplim} ]]\)
+
+\end{Syntax}
+
+where \meta{expr} is the expression to be multiplied, \meta{k} is the
+control variable (a \nameref{kernel}), and \meta{lolim} and \meta{uplim}
+uplim are the optional lower and upper limits. If \meta{uplim} is
+not supplied the upper limit is taken as \meta{k}. The
+Gosper algorithm is used.  If there is no closed form solution,
+the operator returns the input unchanged.
+
+\begin{Examples}
+prod(k/(k-2),k);&k*( - k + 1)\\
+\end{Examples}
+\end{Operator}
+
+
+\begin{Operator}{REDUCT}
+\index{reductum}\index{polynomial}
+The \name{reduct} operator returns the remainder of its expression after the
+leading term with respect to the kernel in the second argument is removed.
+\begin{Syntax}
+\name{reduct}\(\meta{expression},\meta{kernel}\)
+\end{Syntax}
+
+\meta{expression} is ordinarily a polynomial. If \nameref{ratarg} is on,
+a rational expression may also be used, otherwise an error results.
+\meta{kernel} must be a \nameref{kernel}.
+
+\begin{Examples}
+reduct((x+y)**3,x);          &       Y*(3*X^{2} + 3*X*Y + Y^{2}) \\
+reduct(x + sin(x)**3,sin(x));                           &       X \\
+reduct(x + sin(x)**3,y);                         &      0
+\end{Examples}
+
+\begin{Comments}
+If the expression does not contain the kernel, \name{reduct} returns 0.
+\end{Comments}
+\end{Operator}
+
+
+\begin{Operator}{REPART}
+\index{real part}\index{complex}
+\begin{Syntax}
+\name{repart}\(\meta{expression}\) or \name{repart} \meta{simple\_expression}
+\end{Syntax}
+
+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 \name{repart} and \nameref{impart}.
+\begin{Examples}
+repart(1+i);   & 1 \\
+repart(a+i*b); & REPART(A) - IMPART(B)
+\end{Examples}
+
+\end{Operator}
+
+
+\begin{Operator}{RESULTANT}
+\index{polynomial}
+The \name{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.
+\begin{Syntax}
+  \name{resultant}\(\meta{expression},\meta{expression},\meta{kernel}\)
+\end{Syntax}
+
+\meta{expression} must be a polynomial containing \meta{kernel} ;
+\meta{kernel} must be a \nameref{kernel}.
+
+\begin{Examples}
+resultant(x**2 + 2*x + 1,x+1,x);                        &      0 \\
+resultant(x**2 + 2*x + 1,x-3,x);                        &      16 \\
+\begin{multilineinput}
+resultant(z**3 + z**2 + 5*z + 5,
+          z**4 - 6*z**3 + 16*z**2 - 30*z + 55,
+          z);
+\end{multilineinput}    &      0 \\
+resultant(x**3*y + 4*x*y + 10,y**2 + 6*y + 4,y);       &
+           Y^{6} + 18*Y^{5} + 120*Y^{4} + 360*Y^{3} + 480*Y^{2} + 288*Y + 64
+\end{Examples}
+\begin{Comments}
+The resultant is the determinant of the Sylvester matrix, formed from the
+coefficients of the two polynomials in the following way:
+
+Given two polynomials: 
+
+\begin{TEX}
+\begin{displaymath}
+a_0x^n+a_1x^{n-1}+\cdots+a_n
+\end{displaymath}
+\end{TEX}
+\begin{INFO}
+{\begin{verbatim}
+    n       n-1 
+ a x  + a1 x     + ... + an
+
+\end{verbatim}}
+\end{INFO}
+and
+\begin{TEX}
+\begin{displaymath}
+b_0x^n+b_1x^{n-1}+\cdots+b_n
+\end{displaymath}
+\end{TEX}
+\begin{INFO}
+{\begin{verbatim}
+    m       m-1 
+ b x  + b1 x     + ... + bm
+
+\end{verbatim}}
+\end{INFO}
+form the  (m+n)x(m+n-1) Sylvester matrix by the following means:
+\begin{TEX}
+\begin{displaymath}
+  \left(\begin{array}{cccccccc}
+           0&\ldots&0&0&a_0&a_1&\ldots&a_n\\
+	   0&\ldots&0&a_0&a_1&\ldots&a_n&0\\
+	   \vdots&&&\vdots&&&\vdots\\
+	   a_0&a_1&\ldots&a_n&0&0&\ldots&0\\
+	   0&\ldots&0&0&b_0&b_1&\ldots&b_n\\
+	   \vdots&&&\vdots&&&\vdots\\
+	   b_0&b_1&\ldots&b_n&0&0&\ldots&0
+         \end{array}\right)
+\end{displaymath}
+\end{TEX}
+\begin{INFO}
+{\begin{verbatim}
+   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  
+
+\end{verbatim}}
+\end{INFO}
+
+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 \name{resultant} uses are those given in the article,
+``Computing in Algebraic Extensions,'' by R.  Loos, appearing in
+\meta{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:
+\begin{TEX}
+\begin{eqnarray*}
+  \mbox{resultant}(p(x),q(x),x)
+	&=& (-1)^{\deg p(x)*\deg q(x)}\cdot\mbox{resultant}(q(x),p(x),x),\\
+  \mbox{resultant}(a,p(x),x) &=& a^{\deg p(x)},\\
+  \mbox{resultant}(a,b,x) &=& 1
+\end{eqnarray*}
+where $p(x)$ and $q(x)$ are polynomials which have $x$ as a variable, and
+$a$ and $b$ are free of $x$.
+\end{TEX}
+\begin{INFO}
+{
+\begin{verbatim}
+  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
+\end{verbatim}
+where p(x) and q(x) are polynomials which have x as a variable, and
+a and b are free of x.
+}
+\end{INFO}
+
+Error messages are given if \name{resultant} is given a non-polynomial
+expression, or a non-kernel variable.
+\end{Comments}
+\end{Operator}
+
+
+\begin{Operator}{RHS}
+\index{right-hand side}\index{equation}
+The \name{rhs} operator returns the right-hand side of an \nameref{equation}, 
+such as those returned in a \nameref{list} by \nameref{solve}.
+\begin{Syntax}
+\name{rhs}\(\meta{equation}\) or \name{rhs} \meta {equation}
+\end{Syntax}
+
+\meta{equation} must be an equation of the form {\it left-hand side = right-hand
+side}.
+
+\begin{Examples}
+roots := solve(x**2 + 6*x*y + 5x + 3y**2,x);            &
+\begin{multilineoutput}{6cm}
+    ROOTS := \{X= - \rfrac{SQRT(24*Y^{2} + 60*Y + 25) + 6*Y + 5}{2},
+               X= \rfrac{SQRT(24*Y^{2} + 60*Y + 25) - 6*Y - 5}{2}\}
+\end{multilineoutput} \\
+root1 := rhs first roots;                               &
+     ROOT1 := - \rfrac{SQRT(24*Y^{2} + 60*Y + 25) + 6*Y + 5}{2} \\
+root2 := rhs second roots;                              &
+     ROOT2 := \rfrac{SQRT(24*Y^{2} + 60*Y + 25) - 6*Y - 5}{2}
+\end{Examples}
+
+\begin{Comments}
+An error message is given if \name{rhs} is applied to something other than an
+equation.
+\end{Comments}
+\end{Operator}
+
+\begin{Operator}{ROOT\_OF}
+\index{roots}\index{solve}
+When the operator \nameref{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 \name{root\_of}
+and a new local variable. An expression with a top level \name{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 \name{root\_of} construct is
+replaced by an operator \nameref{one\_of}. At this point it is
+of course possible to transform the result if the original \name{solve}
+operator expression into a standard \name{solve} solution. To
+effect this, the operator \nameref{expand\_cases} can be used.
+
+\begin{Examples}
+solve(a*x^7-x^2+1,x);&
+\{x=root\_of(a*x\_^7 - x\_^2 + 1,x\_)\}\\
+sub(a=0,ws);&
+\{x=one\_of(1,-1)\}\\
+expand_cases ws;&
+{x=1,x=-1}\\
+\end{Examples}
+The components of \name{root\_of} and \name{one\_of} expressions can be
+processed as usual with operators \nameref{arglength} and \nameref{part}.
+A higher power of a \name{root\_of} expression with a polynomial
+as first argument is simplified by using the polynomial as a side relation.
+\end{Operator}
+
+\begin{Operator}{SELECT}
+\index{map}\index{list}
+
+The \name{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 \nameref{free variable}.
+\begin{Syntax}
+  \name{select}\(\meta{function},\meta{object}\)
+\end{Syntax}
+\meta{object} is a \nameref{list}.
+
+\meta{function} is 
+the name of an operator for a single argument: the operator
+ is evaluated once with each element of \meta{object} as its single argument,
+
+or an algebraic expression with exactly one \nameref{free variable}, that is
+a variable preceded by the tilde symbol: the expression
+ is evaluated for each element of \meta{object} where the element is
+ substituted for the free variable,
+
+or a replacement \nameref{rule} of the form 
+\begin{Syntax}
+  \name{var} => \name{rep}
+\end{Syntax}
+  where \meta{var} is a variable (a \meta{kernel} without subscript)
+ and \meta{rep} is an expression which contains \meta{var}.
+  Here \name{rep} is evaluated for each element of \meta{object} where 
+  the element is substituted for  \name{var}. \name{var} may be 
+  optionally preceded by a tilde.
+
+The rule form  for \meta{function} is needed when more than
+one free variable occurs. The evaluation result of \meta{function} is
+interpreted as \nameref{boolean value} corresponding to the conventions of
+REDUCE. The result value is built with the leading operator of the
+input expression.
+\begin{Examples}
+  select( ~w>0 , {1,-1,2,-3,3}) &   \{1,2,3\} \\
+  q:=(part((x+y)^5,0):=list)\\
+  select(evenp deg(~w,y),q);& \{x^5 ,10*x^3 *y^2 ,5*x*y^4 \}\\
+  select(evenp deg(~w,x),2x^2+3x^3+4x^4);& 2x^2+4x^4\\
+\end{Examples}
+\end{Operator}
+
+\begin{Operator}{SHOWRULES}
+\index{rule}\index{output}
+\begin{Syntax}
+\name{showrules}\(\meta{expression}\) or
+ \name{showrules} \meta{simple\_expression}
+\end{Syntax}
+
+\name{showrules} returns in \nameref{rule}\name{-list} form any 
+\nameref{operator} rules associated with its argument.
+
+\begin{Examples}
+showrules log; &
+\begin{multilineoutput}{6cm}
+\{LOG(E) => 1,
+ LOG(1) => 0,
+ LOG(E^{~X} ) => ~X,
+ DF(LOG(~X),~X) => \rfrac{1}{~X}\}
+\end{multilineoutput}
+\end{Examples}
+
+Such rules can then be manipulated further as with any \nameref{list}.  For
+example
+\name{rhs first ws;} has the value {\em 1}.
+
+\begin{Comments}
+An operator may have properties that cannot be displayed in such a form,
+such as the fact it is an \ref{odd} function, or has a definition defined
+as a procedure.
+\end{Comments}
+\end{Operator}
+
+
+\begin{Operator}{SOLVE}
+\index{equation}\index{equation solving}
+\index{equation system}\index{root}\index{solve}
+The \name{solve} operator solves a single algebraic \nameref{equation} or a
+system of simultaneous equations.
+\begin{TEX}
+\begin{Syntax}
+% \name{solve}\(\meta{expression} \&option(, \meta{kernel})\) or
+% \name{solve}\(\name{\{}\meta{expression}\{,\meta{expression}\}
+%    \optional\name{\}}
+%           \&option(,\meta{kernel}\optional\)
+\name{solve}\(\meta{expression}\&option(, \meta{kernel})\) or \\
+ \name{solve}\(\name{\{}\meta{expression}\&option( ,\meta{expression})
+ \optional\name{\}}
+	  \&option(,\{\meta{kernel})\optional\name{\}\}}\)
+\end{Syntax}
+\end{TEX}
+\begin{INFO}
+{\begin{Syntax}
+\name{solve}\(\meta{expression} [ , \meta{kernel}]\) or
+
+\name{solve}\(\{\meta{expression},...\} [ ,\{ \meta{kernel} ,...\}] \)
+\end{Syntax} 
+}\end{INFO}
+
+If the number of equations equals the number of distinct kernels, the
+optional kernel argument(s) may be omitted.  \meta{expression} is either a
+scalar expression or an \nameref{equation}.  
+When more than one expression is given,
+the \nameref{list} of expressions is surrounded by curly braces.
+The optional list
+of \nameref{kernel}s follows, also in curly braces.
+
+\begin{Examples}
+sss := solve(x^2 + 7);      &
+\begin{multilineoutput}{6cm}
+Unknown: X
+SSS := \{X= - SQRT(7)*I,
+        X=SQRT(7)*I\}
+\end{multilineoutput}\\
+rhs first sss;               &       - SQRT(7)*I \\
+solve(sin(x^2*y),y);        &
+\begin{multilineoutput}{6cm}
+\{Y=\rfrac{2*ARBINT(1)*PI}{X^{2}}
+ Y=\rfrac{PI*(2*ARBINT(1) + 1)}{X^{2}}\}
+\end{multilineoutput}\\
+off allbranch; \\
+solve(sin(x**2*y),y);        &      \{Y=0\} \\
+solve({3x + 5y = -4,2*x + y = -10},{x,y});
+   & \{\{X= - \rfrac{22}{7},Y=\rfrac{46}{7}\}\} \\
+solve({x + a*y + z,2x + 5},{x,y});
+   &       \{\{X= - \rfrac{5}{2},Y= - \rfrac{2*Z - 5}{2*A}\}\} \\
+% xval := rhs part(ws,1,1);    &       XVAL := - \rfrac{5}{2} \\
+ab := (x+2)^2*(x^6 + 17x + 1);
+ &      AB := X^{8} + 4*X^{7} + 4*X^{6} + 17*X^{3} + 69*X^{2} + 72*X + 4 \\
+www := solve(ab,x);          &      \{X=ROOT_OF(X_^{6} + 17*X_ + 1),X=-2\} \\
+root_multiplicities;         &      \{1,2\}
+\end{Examples}
+\begin{Comments}
+Results of the \name{solve} operator are returned as \nameref{equation}\name{s} 
+in a \nameref{list}.
+You can use the usual list access methods (\nameref{first},
+\nameref{second}, \nameref{third}, \nameref{rest} and \nameref{part}) to
+extract the desired equation, and then use the operators \nameref{rhs} and
+\nameref{lhs} to access the right-hand or left-hand expression of the
+equation.  When \name{solve} is unable to solve an equation, it returns the
+unsolved part as the argument of \name{root_of}, with the variable renamed
+to avoid confusion, as shown in the last example above.
+
+For one equation, \name{solve} uses square-free factorization, roots of
+unity, and the known inverses of the \nameref{log}, \nameref{sin},
+\nameref{cos}, \nameref{acos}, \nameref{asin}, and
+exponentiation operators.  The quadratic, cubic and quartic formulas are
+used if necessary, but these are applied only when the switch 
+\nameref{fullroots} is set on; otherwise or when no closed form is available 
+the result is returned as
+\nameref{root\_of} expression. The switch \nameref{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 \nameref{root\_multiplicities}.  For systems of
+simultaneous linear equations, matrix inversion is used.  For nonlinear
+systems, the Groebner basis method is used.
+
+%If kernels are given for linear equations, and there are an unequal number
+%of kernels and equations, an error message is given.  If no kernels are
+%given, and there are more kernels in the equations than there are
+%equations, an error message is given.  
+Linear equation system solving is influenced by the switch \nameref{cramer}.
+%For nonlinear equations, it is
+%possible to have a consistent set in which the number of variables does not
+%match the number of equations.
+
+Singular systems can be solved when the switch \nameref{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 \nameref{requirements} constraints
+conditions for the system to become consistent.
+
+For a solvable linear and polynomial system with parameters 
+the variable \nameref{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 \nameref{varopt} is on (default), the system rearranges the 
+variable sequence for minimal computation time. Without \name{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 \nameref{arbcomplex} expressions 
+as long as the switch \nameref{arbvars} is on (default). Without
+\name{arbvars} no explicit equations are generated for free variables.
+\end{Comments}
+
+\begin{Related}
+\item[\nameref{allbranch} switch]
+\item[\nameref{arbvars} switch]
+\item[\nameref{assumptions} variable]
+\item[\nameref{fullroots} switch]
+\item[\nameref{requirements} variable]
+\item[\nameref{roots} operator]
+\item[\nameref{root\_of} operator]
+\item[\nameref{trigform} switch]
+\item[\nameref{varopt} switch]
+\end{Related}
+
+\end{Operator}
+
+\begin{Operator}{SORT}
+\index{sorting}
+The \name{sort} operator sorts the elements of a list according to
+an arbitrary comparison operator.
+\begin{Syntax}
+\name{sort}\(\meta{lst},\meta{comp}\)
+\end{Syntax}
+\meta{lst} is a \nameref{list} of algebraic expressions.
+\meta{comp} is a comparison operator which defines a partial
+ordering among the members of \meta{lst}.  \meta{comp} may be
+one of the builtin comparison operators like 
+\name{<}(\nameref{lessp}), \name{<=}(\nameref{leq})
+etc., or  \meta{comp}  may be the name of a comparison procedure.
+Such a procedure has two arguments, and it returns 
+\nameref{true} if the first argument
+ranges before the second one, and 0 or \nameref{nil} otherwise.
+The result of \name{sort} is a new list which contains the
+elements of \meta{lst} in a sequence corresponding to \meta{comp}.
+\begin{Examples}
+ % Sort random integers\\
+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\}\\
+ % Sort a set of polynomials, first for degree of x\\
+ % and second for degree of y.\\
+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);&\{x^2,x*y,y^2\}
+\end{Examples}
+\end{Operator}
+
+\begin{Operator}{STRUCTR}
+\index{decomposition}
+The \name{structr} operator breaks its argument expression into named
+subexpressions.
+\begin{TEX}
+\begin{Syntax}
+\name{structr}\(\meta{expression} \&option(,\meta{identifier}
+                                  \&option(,\meta{identifier}))\)
+\end{Syntax}
+\end{TEX}
+\begin{Syntax}
+\name{structr}\(\meta{expression} [,\meta{identifier}[,\meta{identifier} ...]]\)
+\end{Syntax}
+\begin{INFO}
+\end{INFO}
+\meta{expression} may be any valid REDUCE scalar expression.
+\meta{identifier} may be any valid REDUCE \name{identifier}.  The first
+identifier
+is the stem for subexpression names, the second is the name to be assigned
+to the structured expression.
+
+\begin{Examples}
+structr(sqrt(x**2 + 2*x) + sin(x**2*z));     &
+\begin{multilineoutput}{6cm}
+ANS1 + ANS2
+    where
+        ANS2 := SIN(X^{2}*Z)
+        ANS1 := ((X + 2)*X)^{1/2}
+\end{multilineoutput}\\
+ans3;                        &      ANS3 \\
+on fort; \\
+structr((x+1)**5 + tan(x*y*z),var,aa);                  &
+\begin{multilineoutput}{6cm}
+VAR1=TAN(X*Y*Z)
+AA=VAR1+X**5+5.*X**4+10.*X**3+10.X**2+5.*X+1
+\end{multilineoutput}
+\end{Examples}
+\begin{Comments}
+The second argument to \name{structr} is optional.  If it is not given, the
+default stem \name{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 \nameref{savestructr} is
+on.
+
+If a third argument is given, the structured expression as a whole is named by
+this argument, when \nameref{fort} is on.  The expression is not stored
+under this
+name.  You can send these structured Fortran expressions to a file with the
+\name{out} command.
+\end{Comments}
+\end{Operator}
+
+
+\begin{Operator}{SUB}
+\index{substitution}
+The \name{sub} operator substitutes a new expression for a kernel in an
+expression.
+\begin{Syntax}
+%\name{sub}\(\meta{kernel}\name{=}\meta{expression}%
+%            \{,\meta{kernel}\name{=}\meta{expression}\}\optional,%
+%            \meta{expression}\)
+\name{sub}\(\meta{kernel}\name{=}\meta{expression}
+	    \{,\meta{kernel}\name{=}\meta{expression}\}\optional,
+	    \meta{expression}\) or \\
+\name{sub}\(\{\meta{kernel}\name{=}\meta{expression}\optional,
+	    \meta{kernel}\name{=}\name{expression}\},\meta{expression}\)
+\end{Syntax}
+
+\meta{kernel} must be a \nameref{kernel}, \meta{expression} can be any REDUCE
+scalar expression.
+
+\begin{Examples}
+sub(x=3,y=4,(x+y)**3);       &             343 \\
+x;                           &             X \\
+sub({cos=sin,sin=cos},cos a+sin b} & COS(B) + SIN(A)
+\end{Examples}
+\begin{Comments}
+Note in the second example that operators can be replaced using the
+\name{sub} operator.
+\end{Comments}
+\end{Operator}
+
+
+\begin{Operator}{SUM}
+\index{Gosper algorithm}\index{summation}
+The operator  \name{sum} returns
+the indefinite or definite  summation of a given expression.
+
+\begin{Syntax}
+
+\name{sum}\(\meta{expr},\meta{k}[,\meta{lolim} [,\meta{uplim} ]]\)
+
+\end{Syntax}
+
+where \meta{expr} is the expression to be added, \meta{k} is the
+control variable (a \nameref{kernel}), and \meta{lolim} and \meta{uplim}
+are the optional lower and upper limits. If \meta{uplim} is
+not supplied the upper limit is taken as \meta{k}. The Gosper
+algorithm is used. If  there is  no closed  form solution,  the operator 
+returns the input unchanged.
+
+\begin{Examples}
+sum(4n**3,n); &n^2 *(n^2  + 2*n + 1)\\
+sum(2a+2k*r,k,0,n-1);& n*(2*a + n*r - r)\\
+\end{Examples}
+\end{Operator}
+
+
+\begin{Operator}{WS}
+\index{work space}\index{interactive}
+The \name{ws} operator alone returns the last result; \name{ws} with a
+number argument returns the results of the REDUCE statement executed after
+that numbered prompt.
+\begin{Syntax}
+\name{ws} or \name{ws}\(\meta{number}\)
+\end{Syntax}
+
+
+\meta{number} must be an integer between 1 and the current REDUCE prompt number.
+
+\begin{Examples}
+
+\explanation{(In the following examples, unlike most others, the numbered
+prompt is shown.)} \\
+1: df(sin y,y);    & COS(Y) \\
+2: ws^2;           & COS(Y)^{2} \\
+3: df(ws 1,y);     & -SIN(Y)
+\end{Examples}
+
+\begin{Comments}
+\name{ws} and \name{ws}\name{(}\meta{number}\name{)} can be used anywhere 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 \name{ws;} returns the results of the differentiation.
+The current workspace (\name{ws}) can also be used inside files, though the
+numbered workspace contains only the \name{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 \nameref{input}.  The
+third stores results, when they are produced by statements, which are
+accessible by the \name{ws}\meta{ 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 \nameref{saveas} and \nameref{out} commands.
+% Or you could use the \name{forget} command to clear all history lists and
+% reset the prompt number to 1, which doesn't change any switch settings or
+% variable assignments.
+
+An error message is given if a reference number has not yet been used.
+\end{Comments}
+\end{Operator}
+