Table Of ContentThis is a volume in
COMPUTER SCIENCE AND APPLIED MATHEMATICS
A Series of Monographs and Textbooks
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COMPUTER SCIENCE AND SCIENTIFIC COMPUTING
Editor. WERNER RHEINBOLT
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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.
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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
