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Introduction to Maple Andre Heck Introduction to Maple With 84 illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest Andre Heck Expertise Center Computer Algebra Nederland (CAN) Kruislaan 413 1098 SJ Amsterdam The Netherlands Cover photograph courtesy of UNIPHOTO, Inc. Maple is a registered trademark of Waterloo Maple Software. Library of Congress Cataloging-in-Publication Data Heck, A. (Andre) Introduction to Maple / Andre Heck p. cm. Includes bibliographical references and index. ISBN-13: 978-1-4684-0521-7 e-ISBN-13: 978-1-4684-0519-4 DO I: 10.1 007/978-1-4684-0519-4 1. Maple (Computer file) 2. Algebra Data processing. I. Title. QA155.7.E4H43 1993 510' .285 ' 53--dc20 93-8631 Printed on acid-free paper. ©1993 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 1993 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is for bidden. The use of general descriptive names, trade names, trademarks, etc., in this pub lication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Bill Imbornoni; manufacturing supervised by Vincent Scelta. Photocomposed copy prepared from the author's 'lEX files. 987654321 Preface In symbolic computation on computers, also known as computer algebra, keyboard and display replace the traditional pencil and paper in doing mathematical computations. Interactive computer programs, which are called computer algebra systems, allow their users to compute not only with numbers, but also with symbols, formulae, equations, and so on. Many mathematical computations such as differentiation, integration, and series expansion of functions, and inversion of matrices with symbolic entries, can be carried out quickly, with emphasis on exactness of results, and without much human effort. Computer algebra systems are powerful tools for mathematicians, physicists, chemists, engineers, technicians, psychologists, sociologists, ... , in short, for anybody who needs to do mathematical computations. Com puter algebra systems are indispensable in modern pure and applied scien tific research and education. This book is a gentle introduction to one of the modern computer algebra systems, viz., Maple. Primary emphasis is on learning what can be done with Maple and how it can be used to solve (applied) mathematical problems. To this end, the book contains many examples and exercises, both elementary and more sophisticated. They stimulate you to use Maple and encourage you to find your way through the system. An advice: read this book in conjunction with the Maple system, try the examples, make variations of them, and try to solve the exercises. In this book, emphasis is on understanding the basic principles and ideas of Maple so that you can use it effectively to solve your mathematical problems. Factual knowledge or information about every built-in Maple facility can be obtained from the on-line help system or from the Maple documentation that comes along with the software. This book does not teach mathematics; it is understood that you know the theory behind the examples. By chosing a variety of problems and showing how Maple can be used to solve them, you should get an idea of the capabilities of the system. In this book, the usage of Maple as a programming language is not discussed at a higher level than that of defining simple procedures and VI Preface using simple language constructs. However, the Maple data structures are discussed in great detail because good understanding of them is necessary for manipulating and simplifying expressions effectively. This also forms a good starting point to acquaint you further with Maple as a programming language. About the Maple Version Used It is assumed that you use Maple V Release 2; it is available on many computer platforms, ranging from mainframes and workstations to desktop computers such as Macintosh, NeXT, Amiga, IBM PC, and compatibles. Most of the book should be system independent. About the Production of the Book This book was produced with Maple V Release 2 on a Silicon Graphics Indigo Server. The Maple version was customized by Waterloo Maple Soft ware to allow the capture in PostScript format of Maple output of separate commands. These PostScript results were embedded while typesetting the manuscript with TEX. In this way, "true Maple sessions" interleaved with comments, remarks, and explanations were produced. Therefore, you can be sure that you can reproduce the results on your terminal screen or on paper. Maple I/O has been typeset in Courier font so that you can easi ly distinguish it from other text fragments. Maple procedures have been typeset in bold face characters to distinguish them from ordinary words. The book was prepared in camera-ready format on the phototypesetter at CWI at a resolution of 1200 dots per inch. About the Origin of the Book In 1987 the author started to develop introductory Maple courses at the University of Nijmegen. Several revisions and updates of course material have appeared since then. The most important of these was the 1990 course book "Introductie in het gebruik van Maple" , which was joint work of Ernic Kamerich from the University of Nijmegen and the author. In this course book, the existing material was restructured, updated, extended, and many parts were rewritten. The present book is based on the 1990 course book, but the appearance of Maple V Release 2 has made many alterations and extensions in the text inevitable. Furthermore, many examples of practical usage of Maple have been included. Nevertheless, Ernic Kamerich's col laboration should be regarded as one of the most important steps towards readability and usability of this book. Acknowledgments Many people have contributed to this book. First of all, I would like to thank my friends and colleagues of the Symbolic Computation Group at Preface Vll the University of Waterloo and of Waterloo Maple Software for their sup port, encouragement, and willingness to answer my questions through out the past few years. I would like to thank Rudiger Gebauer from Springer Verlag for his interest in this book and his patience with me. I am greatly indebted to Darren Redfern and Bruce Barber for their careful and thorough reading of the book, and for improving my English and my style of writing. Michael Monagan's comments, suggestions, and criticism were invaluable. I would like to thank Ron Sommeling for the many discussions we had about Maple and the course material. Nancy Blachman and Bert Ruitenburg commented on an earlier draft of book. Jan Schipper's help in getting the manuscript in camera-ready format is acknowledged. Marc van Leeuwen's advice on using 'lEX was indispensable. And last, but not least, I wish to thank my colleagues of the CAN Foundation and at the CAN Expertise Center, notably Arjeh Cohen, Jan Sanders, and Leendert van Gastel, for reminding me of the fact that books must be published and for handling CAN affairs while I was working on the book. Despite all the help I got, I am sure that users of this book will come up with remarks, suggestions, corrections, etc. Please send them to CAN Expertise Center Attn. Andre Heck Kruislaan 413 1098 SJ Amsterdam The Netherlands or to the electronic mail address [email protected] Contents Preface ............................... v 1 Introduction to Computer Algebra. 1 1.1 What is Computer Algebra? ... 1 1.2 Computer Algebra Systems . . . . 2 1.3 Some Properties of Computer Algebra Systems 4 1.4 Advantages of Computer Algebra 11 1.5 Limitations of Computer Algebra 22 1.6 Maple .............. . 26 2 The First Steps: Calculus on Numbers .31 2.1 Getting Started ........ . · 31 2.2 Getting Help . . . . . . . . . . .34 2.3 Integers and Rational Numbers. · 38 2.4 Irrational Numbers and Floating-Point Numbers · 42 2.5 Algebraic Numbers .49 2.6 Complex Numbers. · 52 2.7 Exercises ..... · 55 3 Variables and Names .57 3.1 Assignment and Evaluation. 57 3.2 Unassignment .. 60 3.3 Full Evaluation .. .65 3.4 Names of Variables 69 3.5 Basic Data Types 72 3.6 Exercises ..... 74 x Contents 4 Getting Around with Maple .79 4.1 Input and Output .... 79 4.2 The Maple Library . . . 84 4.3 Reading and Writing Files 87 4.4 Formatted I/O ...... . 92 4.5 Code Generation .... . 97 4.6 Changing Maple to your own Taste 101 4.7 Exercises ............. . 104 5 Polynomials and Rational Functions 107 5.1 Univariate Polynomials . 107 5.2 Multivariate Polynomials 113 5.3 Rational Functions 115 5.4 Conversions 117 5.5 Exercises ..... . 119 6 Internal Data Representation and Substitution 121 6.1 Internal Representation of Polynomials 121 6.2 Generalized Rational Expressions 127 6.3 Substitution 129 6.4 Exercises ............. . 139 7 Manipulation of Polynomials and Rational Expressions 141 7.1 Expansion............. 141 7.2 Factorization............ 144 7.3 Canonical Form and Normal Form 147 7.4 Normalization 149 7.5 Collection 152 7.6 Sorting. 154 7.7 Exercises 155 8 Functions 157 8.1 Mathematical Functions 157 8.2 Arrow Operators 160 8.3 Maple Procedures 164 8.4 Recursive Procedure Definitions 166 8.5 unapply 171 8.6 Operations on Functions 173 8.7 Anonymous Functions . 174 8.8 Exercises. 174 Contents Xl 9 Differentiation 177 9.1 Symbolic Differentiation . 177 9.2 Automatic Differentiation. 186 9.3 Exercises ........ . 190 10 Integration and Summation 193 10.1 Indefinite Integration 193 10.2 Definite Integration .. 202 10.3 Numerical Integration . 208 10.4 Integral Transforms .. 208 10.5 Assisting Maple's Integrator 217 10.6 Summation. 221 10.7 Exercises .. 223 11 Truncated Series Expansions, Power Series, and Limits 227 11.1 Truncated Series Expansions 227 11.2 Power Series 237 11.3 Limits .. 241 11.4 Exercises. 243 12 Composite Data Types 245 12.1 Sequence 245 12.2 Set 248 12.3 List. 250 12.4 Array. 254 12.5 convert and map . 260 12.6 Exercises ..... 263 13 Simplification. 265 13.1 Automatic Simplification 265 13.2 expand. 268 13.3 combine 273 13.4 simplify 275 13.5 convert. 279 13.6 Trigonometric Simplification 281 13.7 Simplification w.r.t. Side Relations. 283 13.8 Exercises. 287 xii Contents 14 Graphics.................. 291 14.1 Some Basic Two-Dimensional Plots 291 14.2 Options of plot .......... . 296 14.3 The Structure of Two-Dimensional Graphics 302 14.4 Special Two-Dimensional Plots. 306 14.5 Plot Aliasing. . . . . . . ...... . 315 14.6 A Common Mistake ......... . 315 14.7 Some Basic Three-Dimensional Plots. 317 14.8 Options of plot3d .......... . 318 14.9 The Structure of Three-Dimensional Graphics. 322 14.10 Special Three-Dimensional Plots 325 14.11 Animation 332 14.12 Exercises ... 335 15 Solving Equations 339 15.1 Equations in One Unknown. 339 15.2 Abbreviations in solve. 341 15.3 Some Difficulties. . . . . . . 341 15.4 Systems of Equations ... 344 15.5 The Grobner Basis Method . 352 15.6 Numerical Solvers ... 359 15.7 Other Solvers in Maple 361 15.8 Exercises ..... . 367 16 Differential Equations . 369 16.1 First Glance at ODEs 369 16.2 Analytic Solutions . . 370 16.3 Taylor Series Method 380 16.4 Power Series Method 383 16.5 Numerical Solutions . 386 16.6 Perturbation Methods. 396 16.7 Liesymm. 413 16.8 Exercises ..... . 417 17 Linear Algebra: Basics 419 17.1 Basic Operations on Matrices . 419 17.2 Last Name Evaluation ... 423 17.3 The Linear Algebra Package 428 17.4 Exercises........... 433