SymbolicC++: An Introduction to Computer Algebra using Object-Oriented Programming

Transcription

1 SymbolicC++: An Introduction to Computer Algebra using Object-Oriented Programming

2 Springer-Verlag London Ltd.

3 Tan Kiat Shi, Willi-Hans Steeb and Yorick Hardy SymbolicC ++: An Introdurtion to Computer Algebra using Objert-Oriented Programming 2nd extended and revised edition i Springer

4 Tan KiatShi ILOG Co. Ltd., Tokyo, Japan Willi-Hans Steeb Y orick Hardy International School for Scientific Computing, Rand Africaans University, Johannesburg, South Africa ISBN Springer-Verlag London Berlin Heidelberg British Library Cataloguing in Publication Data Tan, Kiat Shi Syrnbolic C++ : an introduction to computer algebra using Object-oriented programming. - 2nd rev.ed. l. Object-oriented programming 2. C++ (Computer program language) 3. Algebra - Data processing 1. Title II. Steeb, W.-H. III. Hardy, Yorick 512'.00285'5133 ISBN ISBN (ebook) DOI / Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. Springer-Verlag London 2000 Originally published by Springer-Verlag London Limited in 2000 The use of registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may bemade. Typesetting: Camera ready by authors 34/ Printed on acid-free paper SPIN

5 Preface In this text we will show how object-oriented programming can be used to implement a symbolic algebra system and how the system is applied to different areas in mathematics and physics. In the most restrictive sense, computer algebra is used for the manipulation of scientific and engineering formulae. Usually, a mathematical formula described in the programming languages such as C, Fortran and Pascal can only be evaluated numerically, by assigning the respective values to each variable. However, the same formula may be treated as a mathematical object in a symbolic algebra system, which allows formal transformation, such as differentiation, integration and series expansion, on top of the numerical manipulations. This is therefore an indispensable tool for research and scientific computation. Object-oriented programming has created a new era for programming in computer science as it has been suggested as a possible solution to software development. Basically, object-oriented programming is an important approach to analyzing problems, designing systems and building solutions. By applying this method effectively, the software products become less error prone, easier to maintain, more reusable and extensible. The purpose of this book is to demonstrate how the features of object-oriented programming may be applied to the development of a computer algebra system. Among the many object-oriented programming languages available nowadays, we have selected C++ as our programming language. It is the most widely used object-oriented programming language, which has been successfully utilized by many programmers in various application areas. The design is based partly on acknowledged principles and partly on solid experience and feedback from actual use. Many experienced individuals and organizations in the industry and academia use C++. On top of the reasons stated above, we have selected C++ over other object-oriented languages because of its efficiency in execution speed and its utilization of pointers and templates. Chapter 1 introduces the general notion of Computer Algebra. We discuss the essential properties and requirements of a computer algebra system. Some pitfalls and limitations are also listed for reference. Finally, we present a computer algebra system - SymbolicC++ This new system has many advantages over existing computer algebra systems. A brief summary of features anel function ali ties is given. v

6 VI PREFACE Chapter 2 presents the general mathematics for a computer algebra system. We describe how fundamental mathematical quantities are built up to form more complex mathematical structures. Chapter 3 gives a brief introduction to some computer algebra systems available in the market place, such as Reduce, Maple, Axiom, Mathematica and MuPAD. The basic operations are described for each system. Examples are used to demonstrate the features of these systems. The basic concepts of object-oriented programming, such as objects, classes, abstract data types, message passing, inheritance, polymorphism and so on are introduced in Chapter 4. Examples are given to assist the readers in understanding these important concepts. The object-oriented language Java is discussed in detail and two classes are developed. Finally, the chapter briefly describes three other object-oriented programming languages - Eiffel, Smalltalk and Oberon. In Chapter 5, we introduce the language tools in C++ Along with the description, we construct the String data type, which serves as a carriage for introducing the facilities available in C++. Some other examples are also given. The Standard Template Library is also introduced together with a large number of examples. Finally we describe recursion. By the end of this chapter, all the features needed in the book will have been introduced. Chapter 6 gives a collection of useful classes for computer algebra. We investigate very long integers, rational numbers, complex numbers, quaternions, exact derivatives, vectors, matrices, arrays, strings, bit vectors, finite sets and polynomials. They are the building blocks of mathematics as described in Chapter 2. The internal structures and external interfaces of these classes are described in great detail. In Chapter 7, we describe how a mathematical expression can be constructed using object-oriented techniques. We introduce the new computer algebra system SymbolicC++ and describe its internal representations and public interfaces. Several examples are also presented to demonstrate the functionalities of the system. In Chapter 8, we apply the classes developed in Chapters 6 and 7 to problems in mathematics and physics. Applications are categorized according to classes. Several classes may be used simultaneously to solve a particular problem. Many interesting problems are presented, such as ghost solutions, Pade approximant, Lie series techniques, Picard's method, Mandelbrot set, etc. In Chapter 9, we discuss how the programming language Lisp can be used to implement a computer algebra system. We implement an algebraic simplification and differentiation program. Furthermore, we develop a Lisp system using the objectoriented language C++.

7 PREFACE VII The header files of the classes (abstract data type) introduced in Chapters 6 and 7 are listed in Chapter 10, while in Chapter 11, we show how a PVM (Parallel Virtual Machine) can be used with abstract data types. In Chapter 12, we describe some error handling techniques. We introduce the concept of exception handling. Examples are also given for demonstration purposes. Finally, in Chapter 13 we show how Gnuplot and PostScript can be used to draw figures. The level of presentation is such that one can study the subject early on in ones education in science. There is a balance between practical programming and the underlying language. The book is ideally suited for use in lectures on symbolic computation and object-oriented programming. The beginner will also benefit from the book. The reference list gives a collection of textbooks useful in the study of the computer language C++ [6], [13], [23], [29], [31], [37], [45], [55]. For data structures we refer to Budd (1994) [9]. For applications in science we refer to Steeb et al (1993) [48], Steeb (1994) [49] and Steeb (1999) [52]. The C++ programs have been tested with all newer C++ compilers which comply to the C++ Standard and include an implementation of the Standard Template Library. Without doubt, this book can be extended. If you have comments or suggestions, we would be pleased to have them. The addresses of the authors are: Willi-Hans Steeb: whs~na.rau.ac.za Tan Kiat Shi: Yorick Hardy: steeb_wh~yahoo.com ktan~ilog.co.jp yorickhardy~yahoo.com SymbolicC++ was developed by the International School for Scientific Computing. The Web pages of the International School for Scientific Computing are The second web page also provides the header files for SymbolicC++

8 Contents 1 Introduction 1.1 What is Computer Algebra? Properties of Computer Algebra Systems 1.3 Pitfalls in Computer Algebra Systems. 1.4 Design of a Computer Algebra System 2 Mathematics for Computer Algebra 2.1 Rings and Fields 2.2 Integers Rational Numbers. 2.4 Real Numbers Complex Numbers 2.6 Vectors and Matrices 2.7 Quaternions Polynomials Differentiation Integration Commutativity and Noncommutativity 2.12 Tensor and Kronecker Product Exterior Product Computer Algebra Systems 3.1 Introduction Reduce Basic Operations Example Maple Basic Operations Example Axiom Basic Operations Example Mathematica Basic Operations IX

9 x CONTENTS Example MuPAD Basic Operations Example... 4 Object-Oriented Programming 4.1 Objects, Classes and Abstract Data Types Objects Abstract Data Types (ADT) Classes Message Passing. 4.3 Inheritance Polymorphism Inclusion Polymorphism Operation Polymorphism Parametric Polymorphism 4.5 Object-Oriented Languages C Java Other Object-Oriented Languages. 4.6 Summary Basic Tools in C Pointers and References 5.2 Classes Constructors and Destructor. 5.4 Copy Constructor and Assignment Operator 5.5 Type Conversion Operator Overloading. 5.7 Class Templates Function Templates. 5.9 Friendship Inheritance Virtual Functions 5.12 Wrapper Class Standard Template Library Introduction The Namespace Concept The Vector Class The List Class The Stack Class The Queue Class The Deque Class The Bit Set Class

10 CONTENTS XI The Set Class The Map Class The Algorithm Class Recursion Summary Classes for Computer Algebra The Very long Integer Class Abstraction Data Fields Constructors Operators Type Conversion Operators Private Member Functions Other Functions Streams The Rational Number Class Abstraction Template Class Data Fields Constructors Operators Type Conversion Operators Private Member Functions Other Functions Streams The Complex Number Class Abstraction Template Class Data Fields Constructors Operators Type Conversion Operators Other Functions The Quaternion Class Abstraction Template Class Data Fields Constructors Operators Other Functions Streams The Derive Class Abstraction 225

11 XII Data Fields Constructors Operators Member Functions Possible Improvements 6.6 The Vector Class Abstraction Templates Data Fields Constructors Operators Member Functions and Norms Streams The Matrix Class Abstraction Data Fields Constructors Operators Member Functions and Norms. 6.8 Array Classes Abstraction Data Fields Constructors Operators Member Functions 6.9 The String Class Abstraction Data Fields Constructors Operators Member Functions Type Conversion Operator Possible Improvements 6.10 Bit Vectors Abstraction Data Fields Constructors Member Functions Private Member Functions 6.11 The Linked List Class Abstraction The List Class The Link Class The List Iterator CONTENTS

12 CONTENTS 6.12 The Polynomial Class Abstraction Template Class Data Fields Constructors Operators Type Conversion Operators Private Member Functions Other Functions Streams OExample The Set Class Abstraction Template Class Data Fields Constructors Operators Member Functions Streams Example Summary... 7 The Symbolic Class 7.1 Object-Oriented Design The Expression Tree Polymorphism of the Expression Tree. 7.2 Data Fields and Types of Symbol 7.3 Constructors. 7.4 Operators Functions Simplification Canonical Forms Simplification Rules and Member Functions 7.7 Commutativity Symbolic and Numeric Interface 7.9 Summary Applications 8.1 Bit Vector Class Prime Numbers 8.2 Verylong Class Big Prime Numbers Inverse Map and Denumerable Set Godel Numbering... XIII

13 XIV 8.3 Verylong and Rational Classes Logistic Map Contracting Mapping Theorem Ghost Solutions Iterated Function Systems Verylong, Rational and Derive Classes Logistic Map and Ljapunov Exponent. 8.5 Verylong, Rational and Complex Classes Mandelbrot Set 8.6 Symbolic Class Polynomials Cumulant Expansion Exterior Product Symbolic Class and Symbolic Differentiation First Integrals Spherical Harmonics Nambu Mechanics Taylor Expansion of Differential Equations Commutator of Two Vector Fields Lie Derivative and Killing Vector Field 8.8 Matrix Class Hilbert-Schmidt Norm Lax Pair and Hamilton System Pade Approximant Array and Symbolic Classes Pseudospherical Surfaces and Soliton Equations 8.10 Polynomial and Symbolic Classes Picard's Method Lie Series Techniques Spectra of Small Spin Clusters Nonlinear Maps and Chaotic Behaviour Numerical-Symbolic Application Summary Lisp and Computer Algebra 9.1 Introduction Basic Functions of Lisp Examples from Symbolic Computation Polynomials Simplifications Differentiation Lisp System based on C++ CONTENTS

14 CONTENTS XV 10 Program Listing Verylong Class Rational Class Complex Class Quaternion Class Derive Class Vector Class Matrix Class Array Class String Class O.10Bit Vector Class Linked List Class Polynomial Class 546 1O.13Set Class O.14Symbolic Class PVM and Abstract Data Types Error Handling Techniques Error State Exception Handling Gnuplot and PostScript 659 Bibliography 663 Index 667