Numerical Analysis. Ian Jacques and Colin Judd. London New York CHAPMAN AND HALL. Department of Mathematics Coventry Lanchester Polytechnic

Transcription

1 Numerical Analysis

2 Numerical Analysis Ian Jacques and Colin Judd Department of Mathematics Coventry Lanchester Polytechnic London New York CHAPMAN AND HALL

3 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: DOl: / e-isbn-13: 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 QA297 Library of Congress Cataloging in Publication Data Jacques, Ian, Numerical analysis. Bibliography: p. Includes index. 1. Numerical analysis. I. Judd, Colin, II. Title QA297.J

4 Contents Preface vii Introduction I 1.1 Rounding errors and instability 2 2 Linear algebraic equations Gauss elimination Matrix decomposition methods Iterative methods 27 3 Non-linear algebraic equations Bracketing methods Fixed point iteration Newton's method Systems of non-linear equations 64 4 Eigenvalues and eigenvectors The power method Deflation Jacobi's method Sturm sequence iteration Givens' and Householder's methods The LR and QR methods Hessenberg form Methods of approximation theory Polynomial interpolation: Lagrange form Polynomial interpolation: divided difference form Polynomial interpolation: finite difference form Hermite interpolation Cubic spline interpolation Least squares approximation to discrete data Least squares approximation to continuous functions Numerical differentiation and integration Numerical differentiation Numerical integration: Newton-Cotes formulas 199

5 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

6 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