Computer algebra concepts and techniques lamagna 2019

Computer Algebra Concepts and Techniques COMPUTER ALGEBRA Concepts and Techniques Edmund A Lamagna Cover image: Doug Dunham, Butterflies (7,3) Tessellation inspired by the work of M C Escher Seven butterflies meet at left front wing tips and three meet at right rear wings There are eight colors; each group of seven butterflies meeting at left front wing tips is surrounded by a ring of butterflies of the eighth color The entire pattern is composed of interlocking rings of butterflies CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2019 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed on acid-free paper Version Date: 20181013 International Standard Book Number-13: 978-1-138-09314-0 (Hardback) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Library of Congress Cataloging-in-Publication Data Names: Lamagna, Edmund A., author Title: Computer algebra : concepts and techniques / Edmund A Lamagna Description: Boca Raton, Florida : CRC Press, [2019] | Includes bibliographical references Identifiers: LCCN 2018030134| ISBN 9781138093140 (hardback ; alk paper) | ISBN 9781315107011 (ebook) Subjects: LCSH: Algebra Data processing Classification: LCC QA155.7.E4 L36 2019 | DDC 512 dc23 LC record available at https://lccn.loc.gov/201803013 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com DEDICATION To my wonderful wife and my wondrous son, without whose love, support and patience this book would not have become a reality Contents Preface xi Acknowledgments xix Introduction 1.1 What is Computer Algebra? 1.2 Applications 1.2.1 Celestial Mechanics 1.2.2 Chemical Stoichiometry 1.2.3 Geometry 1.3 History 1.3.1 Early Systems (1961–1980) 1.3.2 More Recent Systems (1981–present) 1.4 Exercises 1 4 10 11 16 22 Computer Algebra Systems 2.1 Maple 2.1.1 Numbers 2.1.2 Polynomials and Rational Functions 2.1.3 Variables and Functions 2.1.4 Calculus 2.1.5 Equation Solving 2.1.6 Data Structures and Programming 2.2 Mathematica 2.2.1 Numbers 2.2.2 Polynomials and Rational Functions 2.2.3 Variables and Functions 2.2.4 Calculus 2.2.5 Equation Solving 2.2.6 Lists 2.2.7 Programming 2.3 Exercises 29 30 30 34 37 39 44 46 52 52 55 57 59 63 65 68 72 vii viii Contents Big Number Arithmetic 3.1 Representation of Numbers 3.2 Basic Arithmetic 3.2.1 Addition and Subtraction 3.2.2 Multiplication 3.2.3 Karatsuba’s Algorithm 3.2.4 Division 3.3 Greatest Common Divisor 3.3.1 Euclid’s Algorithm 3.3.2 Analysis of Euclid’s Algorithm 3.3.3 Extended Euclidean Algorithm 3.3.4 Binary GCD Algorithm 3.4 Rational Arithmetic 3.5 Exponentiation 3.6 Exercises 81 82 84 85 86 87 89 94 94 96 98 101 103 105 109 Polynomial Manipulation 4.1 Arithmetic on Polynomials 4.1.1 Dense and Sparse Representations 4.1.2 Addition and Multiplication 4.1.3 Karatsuba’s Algorithm 4.1.4 Sparse Polynomial Multiplication 4.1.5 Division 4.1.6 Pseudo-division 4.2 Polynomial GCDs 4.2.1 Adapting Euclid’s Algorithm 4.2.2 Polynomial Remainder Sequences 4.2.3 Extended Euclidean Algorithm 4.3 Multivariate Polynomials 4.3.1 Expanded and Recursive Forms 4.3.2 Data Structures for Polynomials 4.3.3 Multivariate GCD Computation 4.4 Exercises 115 116 116 117 119 121 123 124 127 127 132 136 137 138 138 142 144 Algebraic Simplification 5.1 Issues in Simplification 5.2 Normal and Canonical Forms 5.3 Simplification of Radicals 5.3.1 Algebraic Numbers and Functions 5.3.2 Rationalizing Denominators 5.3.3 Denesting Radicals 5.4 Transcendental Functions 5.4.1 Brown’s Normal Form Algorithm 5.4.2 Structure Theorem 5.4.3 Expanded and Combined Forms 149 150 153 157 157 161 164 169 170 173 177 Contents 5.5 ix 5.4.4 Euler’s Formula 5.4.5 Trigonometric Functions Exercises 180 183 187 Factorization 6.1 What is Factoring? 6.2 Rational Roots and Linear Factors 6.3 Schubert-Kronecker Factorization 6.4 Simplifying Polynomial Factorization 6.4.1 Monic Polynomials 6.4.2 Square-Free Decomposition 6.5 Roundabout Factorization 6.5.1 Factoring Polynomials mod m 6.5.2 Choosing a Modulus m 6.5.3 Roundabout Factorization Algorithm 6.6 Distinct Degree Factorization 6.6.1 Finding Distinct Degree Factors 6.6.2 Splitting Equal Degree Factors 6.7 Hensel Lifting 6.8 Exercises 193 193 195 196 199 199 200 204 204 207 209 210 210 212 216 220 Symbolic Integration 7.1 Antidifferentiation 7.2 Heuristic Methods 7.2.1 Derivative-Divides Method 7.2.2 Integration by Parts 7.2.3 Elementary Transformations 7.3 Rational Function Integration 7.4 Hermite’s Method 7.5 Horowitz’ Method 7.6 Rothstein-Trager Method 7.7 Liouville’s Principle 7.8 Risch Algorithm: Logarithmic Case 7.8.1 Polynomial Part 7.8.2 Logarithmic Part 7.8.3 Rational Part 7.9 Risch Algorithm: Exponential Case 7.9.1 Polynomials of Exponentials 7.9.2 Rational Functions of Exponentials 7.10 Exercises 225 226 228 229 230 231 232 235 240 243 249 251 251 255 257 258 258 261 262 Gră obner Bases 8.1 Solution of Polynomial Equations 8.2 Mathematical Applications 8.2.1 Linear Algebra 269 270 272 272 Bibliography 335 Karatsuba, Anatolii, and Yu Ofman 1962 “Multiplication of Many-Digital Numbers by Automatic Computers” Proceedings of the USSR Academy of Sciences 145 (2): 293–294 Knuth, Donald E 1997a The Art of Computer Programming, Volume 2: Seminumerical Algorithms Third edition Chap 4.3.1, Multiple-Precision Arithmetic: The Classical Algorithms; 4.5.1, Rational Arithmetic: Fractions; 4.5.2, The Greatest Common Divisor; 4.6.3, Evaluation of Powers Boston: Addison-Wesley Schă onhage, Arnold, and Volker Strassen 1971 Fast Multiplication of Large Numbers” Computing 7:281–292 Stein, Josef 1967 “Computational Problems Associated with Racah Algebra” Journal of Computational Physics (3): 397–405 Chapter 4, Polynomial Manipulation Aho, Alfred V., John E Hopcroft, and Jeffrey D Ullman 1974 The Design and Analysis of Computer Algorithms Chap 8, Integer and Polynomial Arithmetic Reading, Mass.: Addison-Wesley Brown, William S 1971a “On Euclid’s Algorithm and the Computation of Polynomial Greatest Common Divisors” Journal of the ACM 18 (4): 478– 504 Brown, William S., and Joseph F Traub 1971b “On Euclid’s Algorithm and the Theory of Subresultants” Journal of the ACM 18 (4): 505–514 Collins, George E 1967 “Subresultants and Reduced Polynomial Remainder Sequences” Journal of the ACM 14 (1): 128–142 Geddes, Keith O., Stephen R Czapor, and George Labahn 1992b Algorithms for Computer Algebra Chap 2.7, The Primitive Euclidean Algorithm; 2.8, Quotient Fields and Rational Functions; 3.7, Data Structures for Polynomials, Rational Functions, and Power Series; 7.1, Polynomial GCD Computation; 7.2, Polynomial Remainder Sequences Norwell, Mass.: Kluwer Karatsuba, Anatolii, and Yu Ofman 1962 “Multiplication of Many-Digital Numbers by Automatic Computers” Proceedings of the USSR Academy of Sciences 145 (2): 293–294 Knuth, Donald E 1997b The Art of Computer Programming, Volume 2: Seminumerical Algorithms Third edition Chap 4.6, Polynomial Arithmetic; 4.6.1, Division of Polynomials Boston: Addison-Wesley 336 Bibliography Chapter 5, Algebraic Simplification Berele, Allan, and Stefan Catoiu 2015 “Rationalizing Denominators” Mathematics Magazine 88 (2): 121–136 Bhalotra, Y., and Sarvadaman Chowla 1942 “Some Theorems Concerning Quintics Insoluble by Radicals” Mathematics Student 10:110–112 Borodin, Allan, et al 1985 “Decreasing the Nesting Depth of Expressions Involving Square Roots” Journal of Symbolic Computation (2): 169– 188 Brown, William S 1969 “Rational Exponential Expressions and a Conjecture Concerning π and e” American Mathematical Monthly 76 (1): 28–34 Cohen, Joel S 2002a Computer Algebra and Symbolic Computation: Elementary Algorithms Chap 7, Exponential and Trigonometric Transformations Natick, Mass.: A K Peters Collins, George E 1969 “Algorithmic Approaches to Symbolic Integration and Simplification” ACM SIGSAM Bulletin, no 12: 5–16 Jeffrey, David J., and Albert D Rich 1999 “Simplifying Square Roots of Square Roots by Denesting” Chap in Computer Algebra Systems: A Practical Guide, ed by Michael J Wester, 61–73 Chichester, UK: Wiley Landau, Susan 1994 “How to Tangle with a Nested Radical” The Mathematical Intelligencer 16 (2): 49–55 √ √ — 1998 “ + 3: Four Different Views” The Mathematical Intelligencer 20 (4): 55–60 Maxfield, John E., and Margaret W Maxfield 1971 Abstract Algebra and Solution by Radicals Philadelphia: W B Saunders Moses, Joel 1971a “Algebraic Simplification: A Guide for the Perplexed” Communications of the ACM 14 (8): 527–537 Mulholland, Jamie, and Michael Monagan 2001 “Algorithms for Trigonometric Polynomials” In ISSAC ’02: Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation, 245–252 Richardson, Daniel 1968 “Some Unsolved Problems Involving Elementary Functions of a Real Variable” Journal of Symbolic Logic 33 (4): 514–520 Risch, Robert 1979 “Algebraic Properties of the Elementary Functions of Analysis” American Journal of Mathematics 101 (4): 743–759 Rosenlicht, Maxwell 1976 “On Liouville’s Theory of Elementary Functions” Pacific Journal of Mathematics 65 (2): 485–492 Bibliography 337 Chapter 6, Factorization Akritas, Alkiviadis G 1989b “Elements of Computer Algebra with Applications” Chap 6, Factorization of Polynomials over the Integers New York: Wiley-Interscience Berlekamp, Elwyn R 1967 “Factoring Polynomials over Finite Fields” Bell System Technical Journal 46 (8): 1853–1859 Cantor, David G., and Hans Zassenhaus 1981 “A New Algorithm for Factoring Polynomials over Finite Fields” Mathematics of Computation 36 (154): 587–592 Davenport, James H., Yvon Siret, and Evelyne Tournier 1993a Computer Algebra: Systems and Algorithms for Symbolic Computation Second edition Chap 4.2.1, Factorization of Polynomials in One Variable; 4.2.2, Hensel’s Lemma London: Academic Press Geddes, Keith O., Stephen R Czapor, and George Labahn 1992c Algorithms for Computer Algebra Chap 6.5, The Univariate Hensel Lifting Algorithm; 8.2, Square-Free Factorization; 8.6, Distinct Degree Factorization; 8.7, Factoring Polynomials over the Rationals Norwell, Mass.: Kluwer Knuth, Donald E 1997c The Art of Computer Programming, Volume 2: Seminumerical Algorithms Third edition Chap 4.6.2, Factorization of Polynomials Boston: Addison-Wesley Lenstra, Arjen K., Hendrik W Lenstra, and L´aszl´o Lov´asz 1982 “Factoring Polynomials with Rational Coefficients” Mathematische Annalen 261 (4): 515–534 Mignotte, Maurice 1974 “An Inequality about Factors of a Polynomial” Mathematics of Computation 28 (128): 1153–1157 Murty, M Ram 2002 “Prime Numbers and Irreducible Polynomials” American Mathematical Monthly 109 (5): 452–458 Yun, David Y Y 1976 “On Square-Free Decomposition Algorithms” In SYMSAC ’76: Proceedings of the Third ACM Symposium on Symbolic and Algebraic Manipulation, 26–35 Zassenhaus, Hans 1969 “On Hensel Factorization” Journal of Number Theory (3): 291–311 338 Bibliography Chapter 7, Symbolic Integration Bronstein, Manuel 1998 Symbolic Integration Tutorial Tech rep INRIA Sophia Antipolis — 2006 Symbolic Integration I: Transcendental Functions Second edition Berlin: Springer Bronstein, Manuel, James H Davenport, and Barry M Trager 1998 Symbolic Integration is Algorithmic! Short course, Computers & Mathematics Conference, Massachusetts Institute of Technology Collins, George E 1969 “Algorithmic Approaches to Symbolic Integration and Simplification” ACM SIGSAM Bulletin, no 12: 5–16 Geddes, Keith O., Stephen R Czapor, and George Labahn 1992d Algorithms for Computer Algebra Chap 11.1, Rational Function Integration: Introduction Norwell, Mass.: Kluwer Hermite, Charles 1872 “Sur l’Int´egration des Fractions Rationelles” Annales ´ Scientifiques de l’Ecole Normale Sup´erieure (S´erie 2) 11:145–148 Horowitz, Ellis 1971 “Algorithms for Partial Fraction Decomposition and Rational Function Integration” In SYMSAC ’71: Proceedings of the Second ACM Symposium on Symbolic and Algebraic Manipulation, 441–457 Marchisotto, Elena A., and Gholam-Ali Zakeri 1994 “An Invitation to Integration in Finite Terms” College Mathematics Journal 25 (4): 295–308 Mead, D G 1961 “Integration” American Mathematical Monthly 68 (2): 152–156 Moses, Joel 1967 “Symbolic Integration” PhD thesis, Massachusetts Institute of Technology — 1971b “Symbolic Integration: The Stormy Decade” Communications of the ACM 14 (8): 548–560 Risch, Robert H 1969 “The Problem of Integration in Finite Terms” Transactions of the AMS 139:167–189 Ritt, Joseph F 1948 Integration in Finite Terms New York: Columbia University Press Rosenlicht, Maxwell 1968 “Liouville’s Theorem on Functions with Elementary Integrals” Pacific Journal of Mathematics 24 (1): 153–161 Rothstein, Michael 1976 “Aspects of Symbolic Integration and Simplification of Exponential and Primitive Functions” PhD thesis, University of Wisconsin, Madison Slagle, James R 1963 “A Heuristic Program that Solves Symbolic Integration Problems in Freshman Calculus” Journal of the ACM 10 (4): 507–520 Bibliography 339 Subramaniam, T N., and Donald E G Malm 1992 “How to Integrate Rational Functions” American Mathematical Monthly 99 (8): 762–772 Trager, Barry M 1976 “Algebraic Factoring and Rational Function Integration” In SYMSAC ’76: Proceedings of the Third ACM Symposium on Symbolic and Algebraic Manipulation, 219226 Chapter 8, Gră obner Bases Buchberger, Bruno 2006 “An Algorithm for Finding the Basis Elements of the Residue Class Ring of a Zero Dimensional Polynomial Ideal” Journal of Symbolic Computation 41 (3-4): 475–511 Buchberger, Bruno, and Manuel Kauers 2010 “Groebner Basis” Scholarpedia (10): 7763 http://www.scholarpedia.org/article/Groebner_basis — 2011 “Buchberger’s Algorithm” Scholarpedia (10): 7764 http://www scholarpedia.org/article/Buchberger’s_algorithm Cox, David A., John Little, and Donal O’Shea 2015 Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra Fourth edition Cham, Switzerland: Springer Mencinger, Matej 2013 “On Groebner Bases and Their Use in Solving Some Practical Problems” Universal Journal of Computational Mathematics (1): 5–14 Monagan, Michael B 2014 “Groebner Bases: What Are They and What Are They Useful For?” Maple Application Center http://www.maplesoft com/applications/view.aspx?SID=153693 Paprocki, Mateusz 2010 “Design and Implementation Issues of a Computer Algebra System in an Interpreted, Dynamically Typed Programming Language” MA thesis, University of Technology of Wroclaw, Poland http: //mattpap.github.io/masters-thesis/html/index.html Chapter 9, Mathematical Correctness Bernardin, Laurent 1999 “A Review of Symbolic Solvers” Chap in Computer Algebra Systems: A Practical Guide, ed by Michael J Wester, 101– 120 Chichester, UK: Wiley Bunch, Bryan 1982 Mathematical Fallacies and Paradoxes New York: Van Nostrand Reinhold 340 Bibliography Dummit, David S 1991 “Solving Solvable Quintics” Mathematics of Computation 57 (195): 387–401 Graham, Ronald L., Donald E Knuth, and Oren Patashnik 1994 Concrete Mathematics: A Foundation for Computer Science Second edition Upper Saddle River, N.J.: Addison-Wesley Jeffrey, David J 1994 “The Importance of Being Continuous” Mathematics Magazine 67 (4): 294–300 Jeffrey, David J., and Arthur C Norman 2004 “Not Seeing the Roots for the Branches: Multivalued Functions in Computer Algebra” ACM SIGSAM Bulletin 38 (3): 57–66 Jeffrey, David J., and Albert D Rich 1994 “The Evaluation of Trigonometric Integrals Avoiding Spurious Discontinuities” ACM Transactions on Mathematical Software 20 (1): 124–135 Kahan, William 1987 “Branch Cuts for Complex Elementary Functions, or Much Ado About Nothing’s Sign Bit” In The State of the Art in Numerical Analysis, ed by Arieh Iserles and Michael J D Powell, 165–212 Oxford, UK: Clarendon Press Stoutemyer, David R 1991 “Crimes and Misdemeanors in the Computer Algebra Trade” Notices of the AMS 38 (7): 778–785 Index Abel’s Theorem, 158, 243, 299, 318 Abel, Niels, 158 addition integer, 85–86 polynomial, 117–118 rational number, 104–105 Akritas, Alkiviadis, 204 Aldes, 13 algebraic computation, algebraic function, 157–158, 318–319 algebraic independence exponentials, 170, 175–176 logarithms, 174, 176 radicals, 159 algebraic manipulation, algebraic number, 157–158 dependent, 159 explicit, 161 implicit, 161 minimal polynomial, 159–160 representation, 159 Alpak, 12 Altran, 12 integer representation, 84 polynomial representation, 140, 156 Analitik, 16 antiderivative, 226 spurious discontinuity, 305–310 Apollo lunar program, asymptotic solution, 323 augmented matrix, 273 automatic simplification, 150 Axiom, 19–20 B´ezout’s identity, 98 Babbage, Charles, 10–11 Barton, David, 14 Berlekamp, Elwyn, 210 Bernardin, Laurent, 317, 321 Bhalotra, Y., 158 binary gcd algorithm, 101–103 Binet’s formula, 96 Bonadio, Allan, 21 Borodin, Allan, 164, 167 Bourne, Steve, 14 branch cuts, 310–316 fractional power transformations, 313–314 fractional powers, 310–313 logarithms, 314–315 trigonometric functions, 315–316 Bronstein, Manuel, 250 Brown’s normal form algorithm, 170–172 Brown, William, 12, 132, 135, 156, 170 Buchberger’s algorithm, 285 efficiency, 287–288 Buchberger’s Theorem, 283 Buchberger, Bruno, 270 Byron, Augusta Ada, 10–11 C/C++ programming language, 14, 17–21 Caley, 21 CAMAL, 14 cancellation, common factors, 300–301 Cannon, John, 21 canonical form, 150, 153–157 definition, 153 multivariate polynomial, 155–156 rational function, 156–157 trigonometric function, 184 univariate polynomial, 155 Cantor, David, 210, 212 Cantor-Zassenhaus factorization, 210–215 celestial mechanics, 4–7 Charybdis, 13 chemical stoichiometry, 7–8 Chou, Shang-Ching, 341 342 Chowla, Sarvadaman, 158 Cohen, A.M., 21 Cohen, Henri, 21 Collins, George, 12–13, 135, 156 combined form exponentials, 178–179 logarithms, 179 trigonometric functions, 185–186 complex number, 180 computer algebra definition, 1–2 history, 10–21 limitations, 299–300 computer algebra systems, 11–21 Altran, 12 Analitik, 16 Axiom, 19–20 Caley, 21 CAMAL, 14 Derive, 18–19 Form, 21 FORMAC, 12 GAP, 21 general purpose, Lie, 21 Macaulay, 21 Macsyma, 15 Magma, 21 Maple, 17 Mathematica, 18 MATHLAB, 13 muMATH, 16 MuPAD, 20 PARI/GAP, 21 PM, 12–13 REDUCE, 14 SAC, 12–13 SageMath, 20 SAINT, 11–12 Schoonschip, 16 Scratchpad, 15 Sheep, 16 SIN, 13–14 SMP, 18 special purpose, 2, 16, 21 Symbolic Mathematical Laboratory, 14 Texas Instruments calculators, 19 Theorist/Livemath, 21 Index Trigman, 16 content, polynomial, 128–129 Cox, David, 289 Davenport, James, 19 Delaunay, Charles, 5–7 DeMoivre’s formula, 183 denest radicals, 164–169 square roots, 165–169 Deprit, Andr´e, derivative-divides method, 229–230 Derive, 18–19 polynomial representation, 139 simplification, 152 differentiation, 226–227 Dirichlet, P.G Lejeune, 97 distinct degree factorization, 210–212 division general polynomial, 278–282 integer, 89–94 polynomial, 123–126 pseudo-division, 124–127 rational number, 104 Dummit, David, 318 elementary function, 225–226 elementary row operations, 273 Engelman, Carl, 13 equal degree factorization, 212–215 m = 2, 214–215 odd m, 212–214 equation solving, 316–324 algebraic functions, 318–319 exponential functions, 320–322 logarithmic functions, 320–322 polynomials, 317–318 trigonometric functions, 322–324 Euclid’s algorithm analysis, 96–98 integers, 94–98 monic polynomials, 130–132 polynomials, 127–132 primitive polynomials, 129–130 Euclidean remainder sequence, 133 Euler’s formula, 180–182, 320 Euler, Leonhard, 180 evaluation, 150 expanded form exponentials, 177–178 Index logarithms, 179 trigonometric functions, 184–185 explicit algebraic number, 161 exponential combined form, 178–179 exponential expanded form, 177–178 exponentials, integration of, 258–262 polynomial of exponentials, 258–261 rational function of exponentials, 261–262 exponentiation, 105–109 analysis, 108–109 binary powering, 106 factor method, 106–108 right-to-left binary powering, 106–107 extended Euclidean algorithm integers, 98–100 polynomials, 136–137, 162, 235, 245 extraneous root, 303–304 factorization, 193–220 “equivalent” monic polynomial, 199–200 Cantor-Zassenhaus, 210–215 definition, 193–194 distinct degree, 210–212 equal degree, m = 2, 214–215 equal degree, odd m, 212–214 Hensel lifting, 216–220 linear factor, 195 polynomial mod m, 204–207 repeated factor, 200–204 roundabout method, 204–210 Schubert-Kronecker, 196–198 square-free, 200–204 Fagin, Ronald, 164 Famous, computer algebra system, 152 fast Fourier transform, 89, 121 Fenichel, Robert, 152 Fitch, John, 14 Form, computer algebra system, 21 FORMAC, FORmula MAnipulation Compiler, 12 Formula Algol, 152 formula manipulation, Fortran, 12, 16 fractional powers, 310–314 343 Frick, Inge, 16 Fuchssteiner, Benno, 20 GAP, Groups, Algorithms and Programming, 21 Gauss, Carl Friedrich, 128 Geddes, Keith, 17, 229 general polynomial division, 278–282 geometry, 8–10 analytic, 9–10 theorem proving, 89 Glushkov, V.M., 16 Gonnet, Gaston, 1, 17 Gră obner basis construction, 282–288 Linear Combination Theorem, 282 reduced, 286–287 relation to input basis, 270271 Gră obner, Wolfgang, 270 graded lexicographical order, 277 Graham, Ronald, 302 Grayson, Daniel, 21 greatest common divisor (gcd/GCD) integer, 94103 multivariate polynomial, 142143 polynomial, Gră obner basis, 274–275 univariate polynomial, 127–136 Griesmer, James, 15 Gruntz, Dominik, Hearn, Anthony, 14 Henrard, Jacques, Hensel lifting, 216–220 linear, 220 quadratic, 220 Hensel, Kurt, 204 Hermite’s method, 235–239 Hermite, Charles, 234 Heron’s formula, 8–9 Hopcroft, John, 164 Horowitz’ method, 240–242 Horowitz, Ellis, 234 hyperbolic functions, 183 ideal, polynomial, 282 implicit algebraic number, 161 implicit equations, 275 improper integral, 304–305 344 indeterminate form, 302–303 integer arithmetic addition, 85 division, 89–94 exponentiation, 105–109 greatest common divisor, 94–103 least common multiple, 96 multiplication, 86–89 subtraction, 85–86 integer programming, 276 integer representation, 82–84 dynamic array, 82–84 fixed length array, 82–84 linked list, 82–84 integration, 225–262 xk dx, correctness, 301 antiderivative discontinuities, 305–310 derviative-divides, 229–230 elementary transformations, 231–232 exponential function, 258–262 Hermite’s method, 235–239 heuristic methods, 228–232 Horowitz’ method, 240–242 in finite terms (closed form), 249–251 integration by parts, 230–231 logarithmic function, 251–258 rational function, 232–248 Risch algorithm, 250–262 Rothstein-Trager method, 243–248 integration by parts, 230–231, 235 intermediate expression swell, 132, 151 inverse hyperbolic functions, 183 inverse trigonometric (arctrig) functions, 183 irreducible polynomial, 194 Jeffrey, David, 164–169, 309, 311 Jeffreys, William, 16 Jenks, Richard, 15 Kahan, William, 302 Kahrimanian, H.G., 12, 227 Karatsuba’s algorithm analysis, 88–89, 120–121 integers, 87–89 polynomials, 119–121 Index Karatsuba, Anatolii, 87, 121 Knuth, Donald, 302 Kronecker, Leopold, 196 Lam´e, Gabriel, 96 Lamagna, Edmund, 17 Landau, Edmund, 208 Landau, Susan, 164 Landau-Mignotte bound, 207–208, 216 Laplace’s Theorem, 233–234 Laplace, Pierre-Simon, 233 least common multiple (lcm/LCM) integer, 96 polynomial, Gră obner basis, 274–275 Lenstra, Arjen, 210 Lenstra, Hendrik, 210 LiE, 21 Liebniz, Gottfried, 249 linear algebra, 272–274 linear factor, 195 Liouville’s Theorem, 228, 249–250 exponential case, 258–259 Liouville, Joseph, 249 Lisp, 12–16, 19 Little, John, 289 logarithmic combined form, 179 logarithmic expanded form, 179 logarithmic transformations correctness, 314–315 logarithms, integration of, 251–258 polynomial of logarithms, 251–255 rational function of logarithms, 255258 Loos, Ră udiger, 13 Lov asz, L aszl o, 210 Macaulay, 21 Macsyma, 15 integer representation, 83 Magma, 21 Maple, 17, 30–52 arithmetic, 30–31 data structures, 46–48 differentiation, 39–40 equation solving, 44–46 numerical solution, 44–45 one variable, 44 system of equations, 45–46 Index exponential, 32 factoring, 193–194 floating point arithmetic, 3132 Gră obner basis, 270, 276, 284, 287 implicit algebraic number, 161 integer functions, 33 integer representation, 84 integration, 40, 228–229 limit, 41–42 list, 47–48 logarithm, 32 pattern matching, 152 polynomial expand, 34 factor, 34 greatest common divisor, 36–37 least common multiple, 36–37 polynomial representation, 140 programming, 49–52 rational function, 35–36 sequence, 46 series, 43 set, 47 summation, 40–41 trigonometric functions, 32 user-defined function, 38–39 variable, 37–38 Martin, William, 14, 15 Mathematica, 18, 52–72 arithmetic, 52–53 data structures, 65–68 differentiation, 59 equation solving, 63–65, 317 numerical solution, 64 one variable, 63 system of equations, 64–65 exponential, 54 floating point arithmetic, 53–54 implicit algebraic number, 161 integer functions, 54 integer representation, 84 integration, 59–60 limit, 61–62 list, 65–68 logarithm, 54 pattern matching, 152 polynomial expand, 55 factor, 55 345 greatest common divisor, 56–57 least common multiple, 56–57 polynomial representation, 140 programming, 68–72 rational function, 55–56 sequence, table iterator, 66 series, 62 set, 68 summation, 60 trigonometric functions, 53 user-defined function, 58 variable, 57–58 MATHLAB, 13, 15 McLaughlin, Don, Menabrea, Luigi, 10–11 Mignotte, Maurice, 208 Millen, Jonathan, 13 monic Euclidean algorithm, 130–132 monic polynomial, equivalent factors, 199–200 monic remainder sequence, 133 monomial, 277 graded lexicographical order, 277 order, 277–278 pure lexicographical order, 277 Morrison, Scott, 19 Moses, Joel, 13–15, 149, 152, 227 multiplication integer, 86–89 polynomial, 118–123 rational number, 103–104 sparse polynomial, 121–123 multivariate polynomial, 137–143 expanded/distributed form, 138 greatest common divisor, 142–143 recursive form, 138 representation, 138–141 muMATH, 16, 18 MuPAD, 20 Murty, M Ram, 198 nested radicals, 10 Neubă user, Joachim, 21 NEWTON, 17 Newton, Isaac, 196, 249 Nolan, J., 12, 227 normal form, 153–157 definition, 153 Normal/Canonical Form Theorem, 154 346 Norman, Arthur, 311 O’Shea, Donal, 289 Ofman, Yu., 87, 121 Ostrogradsky, Mikhail, 240 Ostrowski, Alexander, 250 parametric equations, 275 implicitization, 275 PARI/GP, 21 partial fraction decomposition, 157, 232, 235, 243, 251 Patashnik, Oren, 302 PM, Polynomial Manipulation, 12–13 polynomial representation, 156 polynomial factorization, 193–220 irreducible, 194 modular/mod m, 204 monic, 199 multivariate, 137–138 primitive, 199 square-free, 200 univariate, 116 polynomial arithmetic addition/subtraction, 117–118 long division, 123–126 multiplication, 118–123 multivariate gcd, 142–143 pseudo-division, 124–127 sparse multiplication, 121–122 univariate gcd, 127–136 polynomial equations, solution, 270–271 polynomial mod m factorization, 210–215 non-negative representation, 206–207 square-free, 209 symmetric representation, 206–207 polynomial remainder sequence, 132–136 definition, 134 Euclidean, 133 monic, 133 primitive, 133–134 rational, 132–133 reduced, 135 subresultant, 135 polynomial representation, 116–117, 138–141 Index dense, 116–117 descriptor block, 140–141 dynamic array, 139–141 linked list, 139–140 sparse, 116–117 primitive Euclidean algorithm, 129–130 primitive part, polynomial, 128–129 primitive polynomial, 199 primitive remainder sequence, 133–134 proper rational function, 233 pseudo-division, 124–127 quotient, 125–127 remainder, 125–127 pure lexicographical order, 277 Python programming language, 20 rational arithmetic, 103–105 rational exponential expression, 170 rational function integration, 232–248 logarithmic part, 243–248 Rothstein-Trager method, 243–248 rational part, 235–242 Hermite’s method, 235–239 Horowitz’ method, 240–242 rational remainder sequence, 132–133 rational root, 195 Rational Root Test, 195 rationalizing denominator, 161–164 REDUCE, 1415 integer representation, 83 polynomial representation, 139 reduced Gră obner basis, 286–287 reduced remainder sequence, 135 reduced row echelon form, 273 repeated factor, 200 Repeated Factors Theorem, 201–202 resultant, polynomials, 245 Rich, Albert, 16, 18, 164–169, 309 Richardson’s Theorem, 154 Richardson, Daniel, 154 Risch algorithm, 228, 250–262 Risch differential equation, 259 Risch, Robert, 174, 228, 250 Ritt, Joseph, 250 Rom, Arnold, Rosenlicht, Maxwell, 174, 250 Rothstein, Michael, 234, 244 Rothstein-Trager method, 243–248 Index Rothstein-Trager Theorem, 244–245 roundabout factorization, 204–210 roundabout factorization algorithm, 209–210 Roundabout Factorization Theorem, 205 S-polynomial, 282–283 SAC, Symbolic and Algebraic Computation, 12–13 polynomial representation, 156 SageMath, 20 SAINT, Symbolic Automatic INTegrator, 11–13, 227 Sammet, Jean, 12 Schă onhage, Arnold, 89 Schelter, William, 15 Schoonschip, 16 Schubert-Kronecker factorization, 196–198 Scratchpad, 15, 19 sets of measure zero, 300–301 Sheep, 16 simplifiers, classification, 152 catholic, 152 conservative, 152 pattern matching, 152 radical, 152 SIN, Symbolic INtegrator, 13–15, 227–228 Slagle, James, 11, 227 SMP, Symbolic Manipulation Program, 18 software bugs, 298–299 solution of polynomial equations, 270–271 Specht, Wilhelm, 208 spurious solution, 303–310 antiderivative discontinutity, 305–310 extraneous root, 303–304 improper integral, 304–305 Square Root Denesting Theorem, 166 Square Root of Cube Roots Denesting Theorem, 188–189 square-free polynomial, 200 polynomial mod m, 209 347 square-free decomposition, 200–204, 235, 251 Stein, Josef, 101 Stein, William, 20 Stillman, Michael, 21 stoichiometry, 7–8 Stoutemyer, David, 16, 18, 298, 312 Strassen, Volker, 89 Structure Theorem, 173–177 subresultant remainder sequence, 135 substitution/replacement problem, 152–153 subtraction integer, 85–86 polynomial, 117–118 rational number, 104–105 surd, 158 Sutor, Robert, 19 symbolic computation, 2, Symbolic Mathematical Laboratory, 14–15 Texas Instruments calculators, 19 Theorist/Livemath, 21 Tobey, Robert, 12 Tompa, Martin, 164 trace polynomial, 214 Trager, Barry, 19, 234, 244 transcendental function simplification, 169–187 trial quotient, 91–92 Trial Quotient Theorem, 92 Trigman, 16 trigonometric combined form, 185–186 trigonometric evaluation, 186–187 trigonometric expanded form, 184–185 trigonometric functions, 180, 182–187, 322–324 trigonometric identities, 182–183 trigonometric transformations correctness, 315–316 Veltman, Martinus, 16 Vermaserem, Jos, 21 von Schubert, Friedrich, 196 Watt, Stephen, 19 Weierstrass substitution, 307–309 Weierstrass, Karl, 307 348 Wolfram, Stephen, 18 Yun, David, 202 Zassenhaus, Hans, 210, 212, 220 zero-equivalence problem, 151–152, 160–161, 299 Index EDMUND ARTHUR LAMAGNA is a professor of computer science at the University of Rhode Island He earned a doctorate from Brown University, where he completed both his undergraduate and graduate studies while concentrating in computer science, applied mathematics, and engineering Ed’s professional interests lie at the intersection of computer science and mathematics In particular, he has contributed to the fields of computer algebra and to the design and analysis of algorithms In recent years, he has become interested in technical and societal aspects of cybersecurity, personal privacy, and election validation and security Throughout his career, Ed has also been involved in the development of innovative approaches for the teaching and learning of mathematics and computer science This work includes the use of technology to facilitate learning in college calculus, and the use of puzzles and games to develop analytical thinking skills .. .Computer Algebra Concepts and Techniques COMPUTER ALGEBRA Concepts and Techniques Edmund A Lamagna Cover image: Doug Dunham, Butterflies (7,3)... includes students, faculty, and professionals in scientific fields such as computer science, mathematics, engineering, and physics 1.1 What is Computer Algebra? Computer algebra is a branch of scientific... Encyclopedia of Computer Science (2000) by Gaston H Gonnet and Dominik W Gruntz defines computer algebra as “computation with variables and constants according to the rules of algebra, analysis, and other