Numerical Analysis
Numerical Analysis Ian Jacques and Colin Judd Department of Mathematics Coventry Lanchester Polytechnic London New York CHAPMAN AND HALL
First published in 1987 by Chapman and Hall Ltd II New Fetter Lane, London EC4P 4EE Published in the USA by Chapman and Hall 29 West 35th Street, New York NY loooi 1987 I. Jacques and CJ. Judd Softcover reprint of the hardcover 1st edition 1987 ISBN-13: 978-94-010-7919-8 DOl: 10.1007/978-94-009-3157-2 e-isbn-13: 978-94-009-3157-2 This title is available in both hardbound and paperback editions. The paperback edition is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser. All rights reserved. No part of this book may be reprinted, or reproduced or utilized in any form or by any electronic, mechanical or other means, now known or hereafter invented, including photocopying and recording, or in any information storage and retrieval system, without permission in writing from the publisher. British Library Cataloguing in Publication Data Jacques, Ian Numerical analysis \. Numerical analysis I. Title II. Judd, Colin 519.4 QA297 Library of Congress Cataloging in Publication Data Jacques, Ian, 1957- Numerical analysis. Bibliography: p. Includes index. 1. Numerical analysis. I. Judd, Colin, 1952- II. Title QA297.J26 1987 519.4 86-26887
Contents Preface vii Introduction I 1.1 Rounding errors and instability 2 2 Linear algebraic equations 8 2.1 Gauss elimination 9 2.2 Matrix decomposition methods 16 2.3 Iterative methods 27 3 Non-linear algebraic equations 43 3.1 Bracketing methods 43 3.2 Fixed point iteration 50 3.3 Newton's method 57 3.4 Systems of non-linear equations 64 4 Eigenvalues and eigenvectors 71 4.1 The power method 72 4.2 Deflation 80 4.3 Jacobi's method 88 4.4 Sturm sequence iteration 97 4.5 Givens' and Householder's methods 102 4.6 The LR and QR methods 110 4.7 Hessenberg form 118 5 Methods of approximation theory 129 5.1 Polynomial interpolation: Lagrange form 130 5.2 Polynomial interpolation: divided difference form 138 5.3 Polynomial interpolation: finite difference form 145 5.4 Hermite interpolation 152 5.5 Cubic spline interpolation 159 5.6 Least squares approximation to discrete data 172 5.7 Least squares approximation to continuous functions 180 6 Numerical differentiation and integration 190 6.1 Numerical differentiation 191 6.2 Numerical integration: Newton-Cotes formulas 199
vi Contents 6.3 Quadrature rules in composite form 6.4 Romberg's method 6.5 Simpson's adaptive quadrature 6.6 Gaussian quadrature 7 Ordinary differential equations: initial value problems 7.1 Derivation of linear multistep methods 7.2 Analysis of linear multistep methods 7.3 Runge-Kutta methods 7.4 Systems and higher order equations 8 Ordinary differential equations: boundary value problems 8.1 The finite difference method 8.2 The shooting method Appendix References Solutions to exercises Index 209 215 221 227 233 233 244 254 258 265 266 271 277 279 281 323
Preface This book is primarily intended for undergraduates in mathematics, the physical sciences and engineering. It introduces students to most of the techniques forming the core component of courses in numerical analysis. The text is divided into eight chapters which are largely self-contained. However, with a subject as intricately woven as mathematics, there is inevitably some interdependence between them. The level of difficulty varies and, although emphasis is firmly placed on the methods themselves rather than their analysis, we have not hesitated to include theoretical material when we consider it to be sufficiently interesting. However, it should be possible to omit those parts that do seem daunting while still being able to follow the worked examples and to tackle the exercises accompanying each section. Familiarity with the basic results of analysis and linear algebra is assumed since these are normally taught in first courses on mathematical methods. For reference purposes a list of theorems used in the text is given in the appendix. The main purpose of the exercises is to give the student practice in using numerical methods, but they also include simple proofs and slight digressions intended to extend the material of the text. Solutions are provided for every question in each exercise. Many of the questions are suitable for hand calculation or are theoretical in nature, although some require the use of a computer. No attempt has been made to provide listings of programs or to discuss programming style and languages. Nowadays there is a massive amount of published software available and in our opinion use should be made of this. It is not essential for students to actually program a method as a prerequisite to understanding. We are indebted to Dr K.E. Barrett for his helpful comments on much of the text and to Mrs M. Schoales for her meticulous typing of the manuscript. IAN JACQUES COLIN JUDD