Computer Algebra: Symbolic and Algebraic Computation

282 Pages·1982·6.13 MB·English

by  Prof. Dr. R. Loos (auth.)

Computing Supplementum 4
Computer Algebra
SylfJbolic and
Algebraic COlfJpufafion
Edited by B. Buchberger, G. E. Collins, and R. Loos
in cooperation with R. Albrecht
Springer-Verlag Wien GmbH
Prof. Dr. Bruno Buchberger
Institut fur Mathematik
Johannes Kepler Universităt Linz, Austria
Praf. Dr. George Edwin Collins
Computer Science Department
University of Wisconsin-Madison
Madison, Wis., U.S.A.
Prof. Dr. Riidiger Loos
Institut fUr Informatik I
Universităt Karlsruhe
Federal Republic of Germany
This work is subject to copyright.
AII rights are reserved, whether the whole Of part of the material is concerned, specifically those of
translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine Of
similar means, and storage in data banks.
© 1982 by Springer-Verlag Wien
Originally published by Springer-VerlagjWien in 1982
With 5 Figures
Library ofCongress Cataloging in Publication Data. Main entry under title: Computer algebra. (Computing. Supplementum; 4.) Includes
index. 1. Algebra -Data processing -Addresses, essays, lectures. 2. Machine theory -Addresses, essays, lectures. 1. Buchberger, Bruno. Il.
Collins, G. E. (George Edwin), 1928 - . III. Loos. R. (Rildiger), 1938 - . IV. Series: Computing (Springer. Verlag). Supplementum; 4.
V. Title: Symbolic and algebraic computation. QAI55.7.E4C65. 1982.512'.028'54.82·10577
ISSN 0344-8029
ISBN 978-3-211-81684-4 ISBN 978-3-7091-3406-1 (eBook)
DOI 10.1007/978-3-7091-3406-1
Preface
The journal Computing has established a series of supplement volumes the fourth
of which appears this year. Its purpose is to provide a coherent presentation of a
new topic in a single volume. The previous subjects were Computer Arithmetic
1977, Fundamentals of Numerical Computation 1980, and Parallel Processes and
Related Automata 1981; the topic of this 1982 Supplementum to Computing is
Computer Algebra. This subject, which emerged in the early nineteen sixties, has
also been referred to as "symbolic and algebraic computation" or "formula
manipulation".
Algebraic algorithms have been receiving increasing interest as a result of the
recognition of the central role of algorithms in computer science. They can be easily
specified in a formal and rigorous way and provide solutions to problems known
and studied for a long time. Whereas traditional algebra is concerned with
constructive methods, computer algebra is furthermore interested in efficiency, in
implementation, and in hardware and software aspects of the algorithms. It
develops that in deciding effectiveness and determining efficiency of algebraic
methods many other tools - recursion theory, logic, analysis and combinatorics,
for example - are necessary. In the beginning of the use of computers for symbolic
algebra it soon became apparent that the straightforward textbook methods were
often very inefficient. Instead of turning to numerical approximation methods,
computer algebra studies systematically the sources of the inefficiency and searches
for alternative algebraic methods to improve or even replace the algorithms.
The remarkable results of computer algebra so far are scattered over conference
proceedings, journal papers and technical reports. Until now no comprehensive
treatment has been undertaken. The present volume is a first attempt at removing
this gap. In sixteen survey articles the most important theoretical results, algorithms
and software methods of computer algebra are covered, together with systematic
references to literature. In addition, some new results are presented. Thus the
volume should be a valuable source for obtaining a first impression of computer
algebra, as well as for preparing a computer algebra course or for complementary
reading.
The preparation of some papers contained in this volume has been supported by
grants from the Austrian "Fonds zur Forderung der wissenschaftlichen For
schung" (Project No. 3877), the Austrian Ministry of Science and Research
(Department 12, Dr. S. Hollinger), the United States National Science Foundation
(Grand MCS-8009357) and the Deutsche Forschungsgemeinschaft (Lo-231-2). The
work on the volume was greatly facilitated by the opportunity for the editors to stay
VI Preface
as VIsItors at the University of Delaware (Department of Computer and
Information Sciences), at the General Electric Company (Research and
Development Center, Schenectady) and at Rensselaer Polytechnic Institute
(Mathematical Sciences Department), respectively. Our thanks go to all these
institutions. The patient and experienced guidance and collaboration of the
Springer-Verlag Wien during all the stages of production are warmly appreciated.
The editors of the Cooperative editor of
Supplementum Computing
B. Buchberger R. Albrecht
G. Collins
R. Loos
Contents
Loos, R.: Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . 1
Buchberger, B., Loos, R.: Algebraic Simplification . . . . . . . . . . 11
Neubiiser, J.: Computing with Groups and Their Character Tables. 45
Norman, A. C.: Integration in Finite Terms. . . . . . . . . . . . . . 57
Lafon, J. c.: Summation in Finite Terms . . . . . . . . . . . . . . . 71
Collins, G. E.: Quantifier Elimination for Real Closed Fields: A Guide to the
Literature . . . . . . . . . . . . . . . . . . . . . . 79
Collins, G. E., Loos, R.: Real Zeros of Polynomials . . . 83
Kaltofen, E.: Factorization of Polynomials. . . . . . . . . 95
Loos, R.: Generalized Polynomial Remainder Sequences . 115
Lauer, M.: Computing by Homomorphic Images. . . . . 139
Norman, A. C.: Computing in Transcendental Extensions 169
Loos, R.: Computing in Algebraic Extensions. . . . . . . 173
Collins, G. E., Mignotte, M., Winkler, F.: Arithmetic in Basic Algebraic
Domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
van Hulzen, J. A., Calmet, J.: Computer Algebra Systems. . . 221
Calmet, J., van Hu1zen, J. A.: Computer Algebra Applications 245
Mignotte, M.: Some Useful Bounds. 259
Author and Subject Index . . . . . . . . . . . . . . . . . . . . . 265
Computing, Suppl. 4, 1-10 (1982)
© by Springer-Verlag 1982
Introduction
R. Loos, Karlsruhe
Abstract
In this introduction we first give a working definition of computer algebra. We then describe the
organization of research activities in this field. Finally the overall structure and the intention of the
present volume on computer algebra is explained. Some technical information (basic references,
notation etc.) about the volume is given.
1. What Is Computer Algebra?
1.1 An Attempt at a Definition
Computer algebra is that part of computer science which designs, analyzes,
implements and applies algebraic algorithms. We may consider this as a working
definition for the scope of the following chapters but we do not claim that this is the
only possible one. The definition is based on the view that the study of algorithms is
at the heart of computer science [I, 6, 13]. Compared with other algorithms,
algebraic algorithms have simple formal specifications, have proofs of correctness
and asymptotic time bounds which can be established on the base of a well
developed mathematical theory. Furthermore, algebraic objects can be represented
exactly in the memory of a computer, so that algebraic computations can be
performed without loss of precision and significance. Usually, algebraic algorithms
are implemented in software systems allowing input and output in the symbolic
notation of algebra. For all these reasons, computer algebra has attracted an
increasing interest in computer science, mathematics, and application areas.
Negatively, one could say that computer algebra treats those subjects which are too
computational to appear in an algebra text and which are too algebraic to be
presented in a usual computer science text book. Another negative description says
that computer algebra deals with non-numeric computations, a description which
had only some content in the very early days of computing. Somewhat in between is
Knuth's definition of seminumerical algorithms [12]. The chapter of this name in
his series of books is the best introduction to parts of our subject.
1.2 The Relation to Algebra
It is not easy to separate computer algebra from mathematical disciplines like
algebra, analysis (calculus) and numerical analysis. Algebra has a long tradition of
constructive tools, going back to Euclid's algorithm (at least), to Kronecker's [14]
concept of constructions "in endlich vielen Schritten", to H. Weyl's [17]
I Computing, Suppl. 4
2 R. Loos
intuitionistic insistence on constructions, to Grete Hermann's [11] actual bounds
for the number of steps in ideal theory algorithms. Even the influential book of van
der Waerden on Modern Algebra [16] contains a bulk of constructive algebraic
methods - at least in its early editions. Birkhoff [2, 3] emphasizes the role of
"modern (i.e. symbolic) algebra" in computing. Many of the concepts and results of
his universal algebra [4] belong to the basis of a rigorous theory of simplification
(see this volume). Many-sorted term algebras correspond most precisely to the
name symbolic algebra which is sometimes used instead of computer algebra.
Already the Arabian word algebra means to shuffle a term from one side to the
other side of an equation and Hasse [10] defines algebra as the theory of solving
equations built up from the four arithmetical operations. Surprisingly, he points
out already in 1957 that "the process (the general formal rules, formalisms,
algorithms) of solving the equations, but not the objects" are at focus in algebra.
Therefore, from this point of view there is very little difference between algebra and
computer algebra as we described it. The same point of view is elaborated by
Buchberger and Lichtenberger [7]. However, we would like to maintain formally a
difference even if the same subject and the same algebraic algorithm is considered:
in algebra the general rules underlying an algorithm are the primary object of study
whereas in computer science the tool itself, the algorithm in all its aspects, including
its efficiency and implementation, is the central theme. A little synopsis of
constructive proofs presented in a sample of algebra books reveals quickly that very
different views are possible and that, for example, the books of Lang represent the
other end of the spectrum of views on algebra. We refer the reader to [3J and the
following chapters to find his own point of view.
1.3 The Relation to Analysis
The reader may be surprised to see in a text on computer algebra algorithms for
integration, for computing with elementary transcendental functions or even for
solving differential equations. There are several reasons for this inclusion. First: the
algorithms just mentioned operate on terms and formulas and they produce
symbolic output. Second: a closer look at these algorithms shows that all of them
achieve their solution by some kind of algebraization of the problem. We mention
only the family of Liouville type theorems. Third: there are algebraic counterparts
of analytical concepts. The derivative of a polynomial can be defined in a pure
combinatorial manner. The Newton iterations for computing zeros can be replaced
by p-adic expansions. Both cases do not involve limit considerations. Finally, there
are methods to represent exactly entities defined as limits and having infinite
numerical representations, which leads us to the next section.
1.4 The Relation to Numerical Analysis
Many computer algebra computations produce formula output which, in turn, is
used as input to numerical procedures, e.g. for integration of rational functions in
several dimensions, the first and perhaps the second integration is done symboli
cally and then the remaining ones numerically. Numerical procedures use finite
precision arithmetic and are based on the theory of approximation. A numerical
root finder working with finite precision, for example, cannot always isolate the
Introduction 3
roots, but only isolate clusters of roots the diameter of which depends on the
precision and many other parameters. In principle, it is possible and desirable to
specify numerical algorithms with the same formal rigor as algebraic algorithms;
however, the amount of detail required is much higher and resembles less
transparently the specification of the mathematical problem. On the other side, an
algebraic algorithm pays for the exactness with - in general much - higher
execution times and storage requirements compared to its numerical counterpart.
However, there are many instances of problems where an approximation does not
make much sense, for example for decision procedures for elementary algebra, for
counting real roots of polynomials or even for the degree of a polynomial in one
indeterminate, defined as the exponent of the highest non-vanishing term. Since 0
numerically is defined only within the given precision, consequently the degree is
also only defined within this precision - a questionable concept of degree. Having
seen that Sturm sequences with floating point coefficients count real zeros, giving
sometimes more zeros in the subintervals than in the entire interval, the price for a
reliable answer seems acceptable. To overstate the point, an algebraic algorithm
will never trade the mathematical specification of the output for efficiency to get it
approximately.
A consequence of this consideration is that the analytic and empirical analysis of
the efficieny of algebraic algorithms is necessary for a responsible use of time and
memory. This includes the analysis of the growth of the outputs; they may be so
large that any content of information is lost.
Let us consider the relation between numerical and algebraic algorithm more
closely in the case of real algebraic numbers. In the numerical case the irrational
number rx may be represented by two floating point numbers rxm and [; > 0 such that
rxm - C < a < am + c. In the algebraic case d is the zero of a positive, primitive
squarefree integral polynomial A(x) of positive degree. In order to indicate which
zero of A(x) is rx, one may use an interval I, with rational endpoints rand s such that
r < rx < s. In addition, one requires that a is the only zero of A with this property.
Conceptually, a is represented by A, I, and a bisection algorithm which for any
inputs A, Ia = (r, s) computes m = (r + s)/2 and selects (r, m) if A(r)A(m) < 0 or
(m, s) if A(m)A(s) < 0 or (m, m) if A(m) = O. In this way Ir - sl can be made
arbitrarily small at any desired point in an algebraic computation with a, whereas in
the finite precision representation the interval length will grow very quickly.
An idea reconciling the efficiency of numerical arithmetic with the reliability of
algebraic computations is k-precision arithmetic where k can dynamically be
adjusted such that the algebraic specifications are still met. This has to be carefully
checked for every algorithm implemented.
To use integers of arbitrary large size "is natural since computers are now fast
enough that any restrictions imposed by the word length of the computer are
artificial and unnecessary" (Collins [8]).
2. The Organization of Research Activities in Computer Algebra
From the early beginnings of computer algebra, research activities in this field have
been coordinated by SIGSAM (the Special Interest Group on Symbolic and
I'
4 R. Laos
Algebraic Manipulation) of ACM (Association for Computing Machinery).
SIGSAM publishes a Bulletin that reports regularly on all events and developments
in computer algebra. Information about membership in SIGSAM and subscription
to the Bulletin may be obtained from
Association for Computing Machinery, 1133 Avenue of the Americas, New
York, NY 10036, U.S.A.
Furthermore, many other organizations in mathematics and computer science have
helped to develop computer algebra by including special sessions on this subject
into their meetings and conferences.
The European research activities in computer algebra are not centrally organized so
far. There are several independent research groups at various universities and
research institutes. The following professional organizations loosely coordinate the
activities:
NIGSAM (Nordic Interest Group for Symbolic and Algebraic Manipulation):
Address: Y. Sundblad, Department of Numerical Analysis and Computer
Science, KTH, S-10044 Stockholm, Sweden.
The group publishes the "NIGSAM-News".
SAM-AFCET (Le Group "Calcul Formel" d'Association Fran<;aise pour la
Cybernetique Economique et Technique):
Address: M. Bergman, IREM, Faculte de Science de Luminy, Case 901,
F-13288 Marseille Cede x 9, France; J. Calmet, IMAG, Laboratoire
d'Informatique et de Mathematiques Appliquees de Grenoble, BP 53, F -38041
Grenoble Cedex, France.
The group publishes the CALSYF-Bulletin.
SEAS-SMS (SHARE European Association, Symbolic Mathematical Computa
tion Group):
Address: J. H. Davenport, Cambridge University, Computer Laboratory,
Corn Exchange Street, Cambridge CB2 3QG, England.
Recently, the European groups have established a steering committee. All persons
and groups interested in computer algebra are invited to contact this committee.
The coordinator is
H. van Hulzen, Twente University of Technology, Department of Applied
Mathematics, P.O. Box 217, 7500 AE Enschede, Netherlands.
SIGSAM, partly in collaboration with the European professional organizations,
has organized a series of symposia whose proceedings form a significant portion of
what constitutes computer algebra:
SYMSAM 1966:
ACM Symposium on Symbolic and Algebraic Manipulation, Washington,
D.C., March 29 - 31, 1966. Proceedings (Floyd, R. W., ed.): Commun. ACM
9,547 -643 (1966).