Methods of Numerical Integration PDF
Preview Methods of Numerical Integration
This is a volume in COMPUTER SCIENCE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks This series has been renamed COMPUTER SCIENCE AND SCIENTIFIC COMPUTING Editor. WERNER RHEINBOLT A complete list of titles in this series is available from the publisher upon request. METHODS OF NUMERICAL INTEGRATION SECOND EDITION Philip J. Davis APPLIED MATHEMATICS DIVISION BROWN UNIVERSITY PROVIDENCE, RHODE ISLAND Philip Rabinowitz DEPARTMENT OF APPLIED MATHEMATICS THE WEIZMANN INSTITUTE OF SCIENCE REHOVOT, ISRAEL ACADEMIC PRESS, INC. Boston San Diego New York London Sydney Tokyo Toronto This book is printed on acid-free paper. @ COPYRIGHT © 1984, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. A Division ofHarcourt Brace & Company 525 Β Street, Suite 1900, San Diego, CA 92101-4495 United Kingdom Edition published by ACADEMIC PRESS LIMITED 24-28 Oval Road, London NW1 7DX Library of Congress Cataloging in Publication Data Davis, Philip J, Date Methods of numerical integration. (Computer science and applied mathematics) Includes bibliographies and index. 1. Numerical integration. I. Rabinowitz, Philip. II. Title. III. Series. QA299.3.D28 1983 515'.624 83-13522 ISBN 0-12-206360-0 (alk. paper) PRINTED IN THE UNITED STATES OF AMERICA 94 95 96 97 98 99 QW 7 6 5 4 To Ε. R. and H. F. D. Preface to First Edition The theory and application of integrals is one of the great and central themes of mathematics. For applications, one often requires numerical values. The prob- lem of numerical integration is therefore a basic problem of numerical analysis. It is often an elementary component in a much more complex computation. Numerical integration is, paradoxically, both simple and exceedingly difficult. It is simple in that it can often be successfully resolved by the simplest of methods. It is difficult in two respects: first, in that it may require an inordinate amount of computing time, verging in some unfavorable situations toward im- possibility; second, in that one can be led through it to some of the deepest portions of pure and applied analysis. And not only to analysis but, in fact, to diverse areas of mathematics; for inasmuch as the integral has successfully invaded many areas, it is inevitable that from time to time these areas turn about and contribute their special methods and insights to the problem of how integrals may be computed expeditiously. The problem of numerical integration is also open-ended; no finite collection of techniques is likely to cover all the possibili- ties that arise and to which an extra bit of ingenuity or of special knowledge may be of great assistance. This book presents what we think are the major methods of numerical integra- tion. We have tried to produce a balanced work that is both useful to the programmer and stimulating to the theoretician. There are portions of the book where deep results of analysis are derived or are alluded to; yet, it has been our hope that most of the final results have been expressed in a way that is accessible to those with a background only in calculus. xi xii PREFACE TO FIRST EDITION The past generation, under the impact of the electronic computer, has wit- nessed an enormous productivity in the field of numerical integration. It would not misrepresent the situation to say that a learned journal devoted solely to this topic could have been established. The published work has run the range from subtle computer programs to highly theoretical questions of interest primarily to analysts, both classical and functional. In surveying and compiling material for this book we found ourselves very much in the position of Tristram Shandy who required a year to write up one day of his life. We wish to thank many friends and colleagues who have provided us with suggestions. Particular thanks go to Dr. Harvey Silverman who helped prepare the material on the Fast Fourier Transform. Thanks also go to the Weizmann Institute of Science for granting P. J. Davis a Weizmann fellowship during the Spring term of 1970 and to the Office of Naval Research for continued support. Preface to Second Edition Even though a journal of numerical integration has not yet appeared on the scene, it is not because of lack of material. Since the publication of the first edition of our book, itself a revision of previous work, there has been consider- able activity in the field of numerical integration touching upon almost every section of this book. In addition to the many papers appearing in the ever- increasing number of journals in numerical analysis, several books on numerical integration have appeared, as well as many internal reports and a number of Ph.D. theses, notably those of Genz, Haegemans, Mantel, de Doncker, Malik, and Neumann. In addition, two conferences have been held in Oberwolfach and workshops in Los Alamos and Kyoto. Furthermore, numerical integration sub- routines have been included in the IMSL and NAG libraries, and a package of such routines, QUADPACK, has just made its debut. In preparing this revision, we have tried to remain faithful to the aims of the first edition and have essentially attempted to update the material in view of the fruitful work done in the past eight years. Since we wished to keep the book to a reasonable size, we had to be quite selective and our choice has tended more to the practical rather than the theoretical aspects of the field. Thus, we have considerably expanded Chapter 5, Approximate Integration in Two or More Dimensions, and Chapter 6, Automatic Integration. On the other hand, we have not added much on the subject of optimal quadratures, even though much has been written on this subject. In addition, the work of the Soviet school is not well represented here since much of it is inaccessible to those of us who cannot read Russian mathematics. xiii xiv PREFACE TO SECOND EDITION In this edition, we have attempted to correct all the errors in the first edition and wish to thank all those who pointed out these errors, in particular, Professors M. Mori and W. Gautschi, for their lists of errata. We also extend our thanks to our many colleagues, especially James Lyness and Uri Ascher, with whom we have had conversations or correspondence for their contributions to this revision. Of course, we alone are responsible for its contents. We hope it will be as well received as our previous efforts. Chapter 1 Introduction LI Why Numerical Integration? Nurnerical integration is the study of how the numerical value of an integral can be found. The beginnings of this subject are to be sought in antiquity. A fine example of ancient numerical integration, but one that is entirely in the spirit of the present volume, is the Greek quadrature of the circle by means of inscribed and circumscribed regular polygons. This process led Archimedes to an upper and lower bound for the value of π. Over the centuries, particularly since the sixteenth century, many methods of numerical integration have been devised.t These include the use of the fun- damental theorem of integral calculus, infinite series, functional relation- ships, differential equations, and integral transforms. Finally, and this is of prime importance in this volume, there is the method of approximate integra- tion, wherein an integral is approximated by a linear combination:): of the t For a brief history of the older portions of our subject, see Moors [1] and Runge and Willers [1]. % Nonlinear combinations occur occasionally, e.g., in extrapolation by rational functions, the epsilon algorithm, and other nonlinear acceleration techniques as well as in inner product integration rules. 2 1 INTRODUCTION values of the integrand f(x)dx % wf(x) + wf(x) + ··· + w /(x ), x x 2 2 n n - oo < a < < -r-oo. (1.1.1) In equation (1.1.1), x , x , x„ are π points or abscissas usually chosen so x 2 as to lie in the interval of integration, and the numbers Wj, w ,..., w„ are η 2 " weights " accompanying these points. Occasionally, values of the deriva- tives of the integrand appear on the right-hand side of (1.1.1), which is frequently called a rule of approximate integration. The terms mechanical or approximate quadrature are also employed for this type of numerical process. With an abundance of quite general and sophisticated methods for obtain- ing values of integrals, one may properly ask why such primitive approxima- tions as those provided by (1.1.1) should be developed and utilized. The answer is very simple: The mathematically sophisticated methods do not always work, and even if they do work, it may not be advantageous to use them. Take, for example, the method embodied in the fundamental theorem of integral calculus. With this method \bf(x)dx = F(b)-F(a), (1.1.2) where F(x) is an indefinite integral (an antiderivative) of f(x). If the indefinite integral is readily available and sufficiently simple, (1.1.2) can .provide a most expeditious computation. But, as is well known, the process of integration often leads to new transcendental functions. Thus, the simple integration J dx/x leads to the logarithm, which is not an algebraic function, whereas the integration \e~xl dx leads to a function that cannot be ex- pressed in finite terms by combinations of algebraic, logarithmic, or expon- ential operations. Even if the indefinite integral is an elementary function and can be obtained without undue expenditure of labor, it may be sufficiently complicated for one to pause before applying (1.1.2). Take, for example, rx dt 1 x2 + xjl + 1 J nT? - V 2 g x 2 - x y2 + l 0 + —^7=|arctan-—^ h arctan —Jï-—I. (1.1.3) V 2! y/2-x v/2 + x! [In a previous version of this book, formula (1.1.3) was given incorrectly. The present formula is (hopefully) correct. The probability of finding an
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