xref: /dports/math/reduce/Reduce-svn5758-src/doc/misc/intro.tex

\documentstyle{article}
\author{}
\date{}
\title{REDUCE: Overview}
\begin{document}
\sloppy
\maketitle

\section*{Introduction}
The first version of {\small REDUCE} was developed and published by
Anthony C.  Hearn about 25 years ago.  The starting point was a class of
formal computations for problems in high energy physics (Feynman diagrams,
cross sections etc.), which are hard and time consuming if done by hand.
Although the facilities of the current {\small REDUCE} are much more
advanced than those of the early versions, the direction towards big
formal computations in applied mathematics, physics and engineering has
been stable over the years, but with a much broader set of applications.

Like symbolic computation in general, {\small REDUCE} has profited by the
increasing power of computer architectures and by the information exchange
made available by recent network developments.  Spearheaded by A.C.
Hearn, several groups in different countries take part in the {\small
REDUCE} development, and the contributions of users have significantly
widened the application field.

Today {\small REDUCE} can be used with a variety of hardware platforms
from the {\small DOS}-based personal computer up to the Cray
supercomputer.  However, the primary vehicle is the class of advanced UNIX
workstations.

Although {\small REDUCE} is a mature program system, it is extended and
updated on a continuous basis.  Since the establishment of the {\small
REDUCE} Network Library in 1989, users take part in the development, thus
reducing the incompatibilities encountered with new system releases.

In July 1995 version 3.6 of {\small REDUCE} was released.  Information
regarding the available implementations can be obtained by electronic
mail: submit `send info-package' to `reduce-netlib@rand.org',
`reduce-netlib@can.nl' or
`reduce-netlib@pi.cc.u-tokyo.ac.jp'.
The same information is available from an Internet gopher server with
the address info.rand.org.  The network library files are in a ``REDUCE
Library'' directory under the directory ``Publicly Available Software''.
The relevant URL is gopher://info.rand.org/11/software/reduce .  Finally,
a World Wide Web {\REDUCE} server with URL http://www.rrz.uni-koeln.de/REDUCE/
is also supported.  In addition to general information about {\REDUCE}, this
server has pointers to the network library, the demonstration versions,
examples of {\REDUCE} programming, a set of manuals, and the {\REDUCE} online
help system.

\section{Problem solving}

The primary domain of {\small REDUCE} is the solution of large scale
formal problems in mathematics, science and engineering. {\small REDUCE}
offers a number of powerful operators which often give an immediate answer
to a given problem, e.g. solving a linear equation system or computing a
determinant (with symbolic entries, of course).  More typical however are
relatively complicated applications where only the combination of several
evaluation steps leads to the desired result.  Consequently the
development of {\small REDUCE} primarily is oriented towards a collection
of powerful tools, which enable problem solving by combination.

In some cases even complete new algorithmic bases will be required for
problem solving. {\small REDUCE} supports this by various interfaces to
all levels of symbolic evaluation, and the modules of {\small REDUCE} and
of the {\small REDUCE} Network Library demonstrate by example how this
technique is to be used.
\section{Data Types, Structures}

\subsection{Elementary Expressions}

The central object of {\small REDUCE} is the formal expression, which is
built with respect to the common mathematical rules.  Elementary items are

\begin{itemize}
\item numbers (integers, rationals, rounded fractionals, real or
complex); the domain can be selected dynamically,
\item symbols (names with or without indices)
\item functional expressions (names followed by a parameter list)
\item operator symbols {\bf +,-,*,/,**}
\item parentheses for precedence control.
\end{itemize}

A symbol here can play the role of an unknown in the mathematical sense,
as well as a placeholder for a value.  An expression can be assigned to a
symbol as a value such that later all references to the symbol are
replaced by the assigned value.


Examples of elementary expressions:

\begin{verbatim}
    3.1415928      % fraction
    a              % simple variable
    (x+y)**2 / 2   % quadratic expression
    log(u)+log(v)  % function
\end{verbatim}

\subsection{Aggregates}

There are data structures that collect a number of formal expressions:

\begin{itemize}
\item An {\tt equation} is an object where the operator {\bf =} takes
highest precedence, with two slots for expressions, the {\bf lhs} and
{\bf rhs}:
\begin{verbatim}
     p=u**2
\end{verbatim}

\item A {\tt list} is a linear sequence of expressions, where each of the
members is elementary or itself an aggregate.  There are operations for
construction, join, decomposition and reordering of lists:
\begin{verbatim}
    {2,3,5,7,11,13,17,19}
\end{verbatim}

\item An {\tt array} is a rectangular multidimensional structure; the
elements are identified by integer indices.  Elements always have a value,
which defaults to zero.

\begin{verbatim}
 array primes(10);
 primes(0):=2;
 for i:=1: 10 do
  primes(i):=nextprime(primes(i-1));
\end{verbatim}

\item A {\tt matrix} is a named structure of rows and columns, whose
elements are identified by two positive integers.  For matrices with
compatible dimensions and for matrices and scalars there are operations
corresponding to the laws of linear algebra.
E.g. using the derivative operator {\bf df} to construct
a Jacobian
\begin{verbatim}
   matrix jac(n,n);
   for i:=1:n do for j:=1:n do
    jac(i,j):=df(f(i),x(j));
\end{verbatim}

\end{itemize}

\section{Programming Paradigms}

For specifying symbolic tasks and algorithms {\small REDUCE} offers a set
of different programming paradigms:

 \subsection{Algebraic Desk Calculator}

Using {\small REDUCE} as a desk calculator for symbolic and numeric
expressions is the simplest approach.  Formulas can be entered, combined,
stored and processed by a set of powerful operators like differentiation,
integration, polynomial GCD, factorization etc.  Any formula will be
processed immediately with the objective of finding its most complete
simplification, and the result will be presented on the screen as soon as
available.

Example: Taylor polynomial for {\bf x*sin(x)}

\begin{verbatim}
for i:=0:5 sum
  sub(x=0,df(x*sin(x),x,i)) * x**i
            / factorial(i);

    1   4    2
 - ---*X  + X
    6
\end{verbatim}


\subsection{Imperative Algebraic Programming}

Evaluation of a single formula with the immediate output of the result is
a special case of a statement of the {\small REDUCE} programming language,
which, from a syntactical standpoint, is part of the ALGOL family.  This
programming language allows the user to code complicated evaluation
sequences such as conditionals, groups, blocks, iterations controlled by
counters or list structures, and the definition of complete parameterized
procedures with local variables.

Example: definition of a procedure for expanding a function to a Taylor
polynomial:

\begin{verbatim}
procedure tay(u,x,n);
  begin scalar ser,fac;
    ser:=sub(x=0,u);fac:=1;
    for i:=1:n do
    <<u:=df(u,x); fac:=fac*i;
      ser:=ser+sub(x=0,u)*x**i/fac >>;
    return(ser);
  end;
\end{verbatim}

A call to this procedure:
\begin{verbatim}
tay(x*sin(x),x,5);
\end{verbatim}
yields
\begin{verbatim}
    1   4    2
 - ---*X  + X
    6
\end{verbatim}

Example: a recursive program for collecting a basis of Legendre
polynomials from the recurrence relation:

\begin{verbatim}
    P{n+1,x) = ((2n+1)*x*P(n,x) - n*P(n-1,x))/(n+1)
\end{verbatim}


The infix operator "." adds a new element to the head of a list.

\begin{verbatim}
procedure Legendre_basis(m,x);
  % Start with basis of order 1
   Legendre_basis_aux(m,x,1,{x,1});

procedure Legendre_basis_aux(m,x,n,ls);
    % ls contains polynomials n, n-1, n-2 ...
  if n>=m then ls     % done
  else Legendre_basis_aux(m,x,n+1,
  (((2n+1)*x*first ls - n*second ls)/(n+1))
       . ls);
\end{verbatim}

A call to this procedure
\begin{verbatim}
Legendre_basis(3,z);
\end{verbatim}
yields
\begin{verbatim}
  5   3    3
{---*Z  - ---*Z,
  2        2

  3   2    1
 ---*Z  - ---,
  2        2

 Z, 1}
\end{verbatim}

\subsection{Rule Oriented Programming}

In {\small REDUCE}, global algebraic relations can be formulated with
rules.  A rule links an algebraic search pattern to a replacement pattern,
sometimes controlled by additional conditions.  Rules can be activated
(and deactivated) globally, or they can be invoked with a limited scope
for single evaluations.  So the user has an arbitrary precise control over
the algebraic simplification.

Example:  Expanding trigonometric functions for combined arguments; the
tilde symbol represents an implicit for--all.

\begin{verbatim}
Sin_Cos_rules:=
{sin(~x+~y)=>sin(x)*cos(y) + cos(x)*sin(y),
 cos(~x+~y)=>cos(x)*cos(y) - sin(x)*sin(y)};
\end{verbatim}

Global activation is achieved by
\begin{verbatim}
let Sin_Cos_rules;
\end{verbatim}

Note: {\small REDUCE} has no predefined "knowledge" about these
relations for trigonometric functions, as they can be used as production
rules in either form depending on whether expansion or collection is
required; only the user can define which mode is adequate for his problem.

Using rules, a complete calculus can be implemented; the rule syntax here
is very close to the mathematical notation for multistep cases.

Example: Definition of Hermite polynomials:

\begin{verbatim}
operator Hermite;
Hermite_rules:=
{Hermite(0,~x) => 1,
 Hermite(1,~x) => 2*x,
 Hermite(~n,~x) => 2*x*Hermite(n-1,x)
          -2*(n-1)*Hermite(n-2,x)
              when n>1};

let Hermite_rules;
\end{verbatim}

Generation of a Hermite polynomial:

\begin{verbatim}
Hermite(4,z);

    4       2
16*Z  - 48*Z  + 12
\end{verbatim}

\subsection{Symbolic Imperative Programming}

The paradigms described so far give access to the {\small REDUCE}
facilities at the top level.  They enable a compact programming close to
the application problem.  No knowledge about the internal data structures
is necessary, since {\small REDUCE} converts data automatically to the
formats needed locally for each evaluation step.  On the other hand, such
frequent conversions are time consuming and so for very large problems it
might be desirable to keep intermediate results in the internal form in
order to avoid the conversion overhead.  Here the ``symbolic'' mode of
{\small REDUCE} can be used, which allows the access to internal data
structures and procedures directly with the same syntax as in top level
programming.

Of course, this level of programming requires some knowledge about {\small
LISP} and about internal {\small REDUCE} structures.  However, it enables
the implementation of algorithms with the highest possible efficiency.

\section{Algebraic Evaluation}

The evaluation of expressions is the heart of {\small REDUCE}.  Because of
its great complexity, it is only briefly touched on here.  One central
problem in automatic formula manipulation is the detection of identity
between objects, e.g. the confirmation
     {\bf a + b  =   b + a }
under the assumption of commutative addition.

It is well known that this problem is equivalent to the problem of
recognizing that an expression is zero, in other words to the existence of
an algorithm for the transformation of a formula into an equivalent
canonical normal form.  Unfortunately there is no universal canonical
form; only for subcases, for example polynomials, rationals, and ideals,
are canonical forms known.  Therefore {\small REDUCE} evaluation is based
on a canonical form for rational functions (i.e., quotients of
multivariate polynomials), where symbols or function expressions play the
role of variables ({\small REDUCE}: kernels). {\small REDUCE} attempts to
tranform as many functions as possible into the canonical form by applying
additional heuristic rules.

A coarse sketch of evaluation is as follows:

\begin{itemize}
\item a symbol with an assigned value is
replaced by the value,
\item a call for a known procedure is
replaced by the value produced by the procedure invocation,
\item matching rules are applied,
\item polynomials are expanded recursively using a lexicographic
order of variables (kernels): a multivariate polynomial is a
polynomial in its highest variable with decreasing exponents,
where the coefficients are polynomials in the remaining
variables,
\item a rational function is converted into a form with common
denominator (i.e., a quotient of two polynomials).
\end{itemize}

This is, of course, a highly recursive process, which is applied until no
more transformations are possible.
\section{Approximations}

In the domain of symbolic computation, mostly exact arithmetic is used,
especially with algorithms from the classical Computer Algebra.  That
aspect is supported by {\small REDUCE} with arbitrarily long integer
arithmetic and, built on top of that, rational and modular (p-adic)
numbers.

The values of transcendental functions with general numeric arguments do
not fall into these domains, even if symbols like  {\bf pi}, {\bf e}, {\bf i}
are attached.  Nevertheless symbolic computation can be used for fields beyond
classical algebra, for example in the domain of analytic approximations in
numerical mathematics.

\subsection{Power Series}

Power series are a valuable tool for the formal approximation of
functions, e.g. in the area of differential equations. {\small REDUCE}
supports several types of power series, among them univariate Taylor
series with variable order and multivariate Taylor series with fixed
order.

\subsection{Rounded Numbers}

For several decades, floating point numbers have been recognized as a
useful tool for numerical computations, although they do not possess most
of the algebraic properties of numbers.  In {\small REDUCE} they are
incorporated as "rounded numbers" which, when compared to classical
floating point numbers (e.g. in the IEEE view) they offer interesting
additional properties:

\begin{itemize}
\item  the mantissa length can be selected
arbitrarily (i.e., selected as a number of decimal digits),
\item there is no limit for the exponent and so no
upper or lower limit for the magnitude of a number.
\end{itemize}

Technically, this arithmetic is implemented by an embedding of the
standard (hardware) floating point operations in a software package, which
tries to execute as much as possible in fast hardware and which converts
to software emulation as soon as the hardware limits are passed.  Based on
this number domain, attractive algorithms can be implemented, which start
with coarse approximations and then refine the overall precision in an
adaptive style when approaching the desired solution.

\subsection{Interface for Numerical Programs}

A field of growing importance for symbolic computation is the use of
algorithms of mixed symbolic-numeric type, when for example a symbolic
calculation carries out formal transformations on an equation system for
control or conditioning of a numerical solver.  Examples are the automatic
programming of Jacobians for ODE solvers, or the reduction of the order of
a system by exploiting formal symmetries.  By the cooperation of symbolic
and numeric components, {\small REDUCE} offers several facilities for the
generation of partial or complete programs in languages such as FORTRAN or
C.  As automatically generated programs tend to flood the target
compilers, {\small REDUCE} also provides for the optimization of the
numeric code.
\section{I/O}

In interactive mode, {\small REDUCE} normally prints results in a two
dimensional ``mathematical'' form, where exponents are raised, quotients
are printed with denominator below numerator, and matrices are represented
as rectangular blocks.  The output can be influenced by a variety of
switches, e.g. for reordering or collecting of terms.

For special purposes, additional output forms are available:

\begin{itemize}
\item linear form: the data can be re-used for later
input in {\small REDUCE} or another system,
\item foreign syntax: the expressions are printed in
the syntax of FORTRAN, C or another programming language
for the direct insertion in numeric codes,
\item TeX: indirect formatting as input for the TeX
layout program to be inserted into a publication.
\end{itemize}

Examples for {\bf q:=(x+y)**3}:

natural (default) output:
\begin{verbatim}
      3      2          2    3
Q := X  + 3*X *Y + 3*X*Y  + Y
\end{verbatim}
for later re--use:
\begin{verbatim}
Q := X**3 + 3*X**2*Y + 3*X*Y**2 + Y**3$
\end{verbatim}
as contribution to a FORTRAN source:
\begin{verbatim}
      Q=X**3+3.*X**2*Y+3.*X*Y**2+Y**3
\end{verbatim}
for a LaTeX document:
\begin{verbatim}
\begin{displaymath}
q=x^{3}+3 x^{2} y+3 x y^{2}+y^{3}
\end{displaymath}
\end{verbatim}

In addition to direct terminal access, I/O can also be redirected to
files.
\section{Open System}

In contrast to most other symbolic math systems, {\small REDUCE}
traditionally is completely open:

\begin{itemize}
\item {\small REDUCE} is written in a language {\small RLISP}, which
incorporates the functionality of {\small LISP} in a user friendly syntax.
At the same time {\small RLISP} is the language of application.

\item Traditionally {\small REDUCE} is delivered with all sources.  So the
algorithmic basis is visible to any user.  Even the {\small REDUCE}
translator (compiling {\small RLISP} to {\small LISP}) is delivered as
source code.

\item Any internal {\small REDUCE} function and data structure can be
accessed by the user directly (in symbolic style programming).  Most of
the {\small REDUCE} implementations contain a {\small LISP} compiler, such
that the user can produce very efficient modules. {\small REDUCE} can be
integrated into other ({\small LISP}-) packages as an algebraic engine.

\item {\small REDUCE} inherits automatically from {\small LISP} the
facility of dynamic loading of modules, of incremental compilation and
dynamic function redefinition.  Even the kernel of {\small REDUCE} is open
for local modification.  Obviously this is a dangerous feature where
system integrity is concerned, but, on the other hand, an innovative user
finds a rich testbed here.
\end{itemize}

One effect of the liberality of {\small REDUCE} is the large number of
application packages written by users.  Many of these packages now are now
included in {\small REDUCE} or in the {\small REDUCE} Network Library.
\renewcommand{\thesection}{Appendix \Alph{section}}
\setcounter{section}{0}

\section{REDUCE Packages}

State: late 1993.

\begin{itemize}
\item ALGINT integration for functions involving roots
(James H. Davenport)

\item ARNUM  algebraic numbers (Eberhard Schr\"ufer)

\item ASSIST  useful utilities for various applications (Hubert Caprasse)

\item AVECTOR vector algebra (David Harper)

\item CALI  computational commutative algebra (Hans-Gert Graebe)

\item CAMAL  calculations in celestial mechanics (John Fitch)

\item CHANGEVAR transformation of variables in differential equations
(G. \"{U}\c{c}oluk)

\item COMPACT condensing of expressions with polynomial side relations
        (Anthony C. Hearn)

\item CRACK solving overdetermined systems of PDEs or ODEs
(Andreas Brand, Thomas Wolf)

\item CVIT Dirac gamma matrices (V.Ilyin, A.Kryukov, A.Rodionov,
         A.Taranov)

\item DESIR differential equations and singularities
(C. Dicrescenzo, F. Richard-Jung, E. Tournier)

\item EXCALC  calculus for differential geometry (Eberhard Schr\"ufer)

\item FIDE code generation for finite difference schemes
(Richard Liska)

\item GENTRAN  code generation in FORTRAN, RATFOR, C (Barbara Gates)

\item GNUPLOT  display of functions and surfaces (Herbert Melenk)

\item GROEBNER computation in multivariate polynomial ideals
         (Herbert Melenk, H.Michael M\"oller, Winfried Neun)

\item HEPHYS  high energy physics (Anthony C. Hearn)

\item IDEALS  arithmetic for polynomial ideals (Herbert Melenk)

\item INVSYS involutive polynomial systems (Alexey Zharkov)

\item LAPLACE  Laplace and inverse Laplace transform (C. Kazasov et al.)

\item LIE  functions for the classification of real n-dimensional Lie
algebras (Carsten, Franziska Sch\"obel)

\item LIMITS  finding limits (Stanley L. Kameny)

\item LININEQ  linear inequalities and linear programming (Herbert Melenk)

\item NUMERIC  solving numerical problems using rounded mode (Herbert Melenk)

\item ODESOLVE ordinary differential equations (Malcolm MacCallum et al.)

\item ORTHOVEC calculus for scalar and vector quantities
         (J.W. Eastwood)

\item PHYSOP additional support for non-commuting quantities (Mathias Warns)

\item PM  general algebraic pattern matcher (Kevin McIsaac)

\item REACTEQN  manipulation of chemical reaction systems (Herbert Melenk)

\item RLFI, TRI  TeX and LaTeX output (Richard Liska, Ladislav Drska,
Werner Antweiler)

\item ROOTS  roots of polynomials (Stanley L. Kameny)

\item SCOPE  optimization of numerical programs (J. A. van Hulzen)

\item SPDE symmetry analysis for partial differential equations
         (Fritz Schwarz)

\item SPECFN  special functions (Chris Cannam et al.)

\item SPECFN2  special special functions (Victor Adamchik,
Winfried Neun)

\item SUM sum and product of series (Fuji Kako)

\item SYMMETRY  symmetry-adapted bases and block diagonal forms of
symmetric matrices (Karin Gatermann)

\item TAYLOR multivariate Taylor series (Rainer Sch\"opf)

\item TPS  univariate Taylor series with indefinite order
         (Alan Barnes, Julian Padget)

\item WU  Wu algorithm for polynomial systems (Russell Bradford)
\end{itemize}
\section{Examples}

Polynomials, rational functions:

\begin{verbatim}
coeff(X**3 + 3*X**2*Y + 3*X*Y**2 + Y**3,x);

  3    2
{Y ,3*Y ,3*Y,1}

gcd(X**2 + 4*X + 3,X**2 - 2*X - 3);

X + 1

resultant(X**2 + 4*X + 3,X**2 - 2*X - 3,x);

0

decompose(x**6+6x**4+x**3+9x**2+3x-5);

  2            3
{U  + U - 5,U=X  + 3*X}

factorize(x**6+6x**4+x**3+9x**2+3x);

    2      3
{X,X  + 3,X  + 3*X + 1}

roots(x**6+6x**4+x**3+9x**2+3x-5);

{X=0.543562, X=-0.775167, X=-0.27178+1.79488*I,
 X=-0.27178-1.79488*I, X=0.38758+1.8576*I,
 X=0.38758-1.8576*I}

interpol({0,7,26,63},z,{1,2,3,4});

 3
Z  - 1

\end{verbatim}
partial fraction decomposition:
\begin{verbatim}
pf(2/((x+1)^2*(x+2)),x);

    2      - 2       2
{-------,-------,----------}
  X + 2   X + 1          2
                  (X + 1)

\end{verbatim}
Matrices:
\begin{verbatim}
 m:=mat((1,x),(2,y));

     [1  X]
M := [    ]
     [2  Y]


 1/m;

[   - Y         X    ]
[---------  ---------]
[ 2*X - Y    2*X - Y ]
[                    ]
[    2         - 1   ]
[---------  ---------]
[ 2*X - Y    2*X - Y ]

 det m;

 - 2*X + Y

\end{verbatim}
Ordinary differential equations:
\begin{verbatim}
odesolve(df(y,x)=y+x**2+2,y,x);

    X                2
{Y=E *ARBCONST(1) - X  - 2*X - 4}

\end{verbatim}
Linear system (hidden):
\begin{verbatim}
 solve({(a*x+y)/(z-1)-3,y+b+z,x-y},
       {x,y,z});

      - 3*(B + 1)
{{X=--------------,
        A + 4

      - 3*(B + 1)
  Y=--------------,
        A + 4

      - A*B - B + 3
  Z=----------------}}
         A + 4

\end{verbatim}
Transcendental equations:
\begin{verbatim}
 solve(a**(2*x)-3*a**x+2,x);

    2*ARBINT(2)*I*PI
{X=------------------,
         LOG(A)

    LOG(2) + 2*ARBINT(3)*I*PI
 X=---------------------------}
             LOG(A)

\end{verbatim}
Polynomial systems:
\begin{verbatim}

solve(
 { a*c1 - b*c1**2 - g*c1*c2 + e*c3,
   -g*c1*c2 + (e+t)*c3 -k*c2,
   g*c1*c2 + k*c2 - (e+t) * c3},
  {c3,c2,c1});

{{C1=ARBCOMPLEX(2),

   C1*(-C1*B*E-C1*B*T+A*E+A*T)
C2=---------------------------,
            C1*G*T-E*K

          2
   C1*(-C1 *B*G+C1*A*G-C1*B*K+A*K)
C3=-------------------------------}}
            C1*G*T-E*K


\end{verbatim}
Structural analysis:
\begin{verbatim}
compact(s*(1-(sin x**2))
       +c*(1-(cos x)**2)
       +(sin x)**2+(cos x)**2,
   {cos x^2+sin x^2=1});

      2           2
SIN(X) *C + COS(X) *S + 1

\end{verbatim}
Calculus:
\begin{verbatim}

 df(exp(x**2)/x,x,2);

     2
    X      4    2
 2*E  *(2*X  - X  + 1)
-----------------------
           3
          X

 int(x^3*exp(2x),x);

  2*X     3      2
 E   *(4*X  - 6*X  + 6*X - 3)
------------------------------
              8

 limit(x*sin(1/x),x,infinity);

1
\end{verbatim}
Series:
\begin{verbatim}
on rounded; taylor(sin(x+1),x,0,4);

0.8414709848079 + 0.5403023058681*X
                                       3
- 0.420735492404*X - 0.09005038431136*X
                    4
+ 0.03506129103366*X  + ...

sum(n,n);

 N*(N + 1)
-----------
     2

prod(n/(n+2),n);

      1
--------------
  2
 N  + 3*N + 2
\end{verbatim}
Complex numbers:
\begin{verbatim}

 w:=(x+3*i)**2;

      2
W := X  + (6*I)*X - 9
\end{verbatim}
Rounded numbers:
\begin{verbatim}
precision 25; pi**2;

9.869 60440 10893 58618 83449 1
\end{verbatim}
Modular numbers:
\begin{verbatim}
on modular; setmod 17; (x-1)**2;

 2
X  + 15*X + 1

factorize ws;

{X + 16,X + 16}
\end{verbatim}
\begin{thebibliography}{9}

\item F. Brackx, D. Constales: Computer Algebra with
{\small LISP} and {\small REDUCE}, Kluwer, 1991

\item J.H. Davenport, Y. Siret, E. Tournier: Computer Algebra,
second printing, Academic Press, London, 1989

\item Anthony C. Hearn: {\small REDUCE} User's Manual, Version 3.6,
The Rand Corporation, Santa Monica (CA), 1995

\item F. W. Hehl, V. Winkelmann, H. Meyer:
{\small REDUCE}, ein Kompaktkurs \"uber die Anwendung von Computer-Algebra,
(in German), Springer 1993

\item Malcolm MacCallum, Francis Wright:
Algebraic Computing with {\small REDUCE},
Oxford University Press, 1991

\item Norman MacDonald:  {\small REDUCE} for physicists,
Institute of Physics Publishing, Bristol, UK, 1994

\item Gerhard Rayna, {\small REDUCE}, Software for Algebraic Computation,
Springer, New York, 1987

\item D.Stauffer, F.W.Hehl, N.Ito, V.Winkelmann, J.G.Zabolitzky:
 Computer Simulation and Computer Algebra, Lectures for Beginners,
 third enlarged edition, Springer 1993

\bibitem W.-H. Steeb. D. Lewien: Algorithms and Computation with
{\small REDUCE}, BI Wissenschaftsverlag, 1992

\item J. Ueberberg: Einf\"uhrung in die Computeralgebra
mit {\small REDUCE}(in German), BI Wissenschaftsverlag, 1992

\item {\small REDUCE} Network Library, Bibliography,
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\end{thebibliography}
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