Approximate Calculation of Integrals (Dover Books on Mathematics)

386 Pages·2006·5.39 MB·English

by  V. I. Krylov (Author) Arthur H. Stroud (Translator)

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APPROXIMATE
CALCULATION
OF INTEGRALS
Vladimir Ivanovich Krylov
Translated by
Arthur H. Stroud
DOVER PUBLICATIONS, INC.
Mineola, New York
Bibliographical Note
This Dover edition, first published in 2005, is an unabridged republication of
the Arthur H. Stroud translation, originally published by The Macmillan
Company, New York, in 1962. The text has been translated from the Russian
book Priblizhennoe Vychislenie Integralov, Gos. Izd. Fiz.-Mat. Lit., Moscow,
1959.
Library of Congress Cataloging-in-Publication Data
Krylov, V. I. (Vladimir Ivanovich), 1902-
[Priblizhennoe vychislenie integralov. English]
Approximate calculation of integrals / Vladimir Ivanovich Krylov ; trans-
lated by Arthur H. Stroud.
p. cm.
Originally published: New York : Macmillan Co., 1962.
Includes index.
ISBN 0-486-44579-8 (pbk.)
1. Integrals. 2. Approximation theory. I. Title.
QA311.K713 2005
515'.43-dc22
2005051789
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501
PREFACE
The author attempts in this book to introduce the reader to the princi-
pal ideas and results of the contemporary theory of approximate integra-
tion and to provide a useful reference for practical computations.
In this book we consider only the problem of approximate integration of
functions of a single variable. We almost completely ignore the more
difficult problem of approximate integration of functions of more than one
variable, a problem about which much less is known. Only in one place
do we mention double and triple integrals in connection with their reduc-
tion to single integrals.
But even for single integrals the author has omitted many interesting
considerations. Problems not touched upon are, for example, methods of
integration of rapidly oscillating functions, the calculation of contour
integrals of analytic functions, the application of random methods, and
others. The book is devoted for the most part to methods of mechanical
quadrature where the integral is approximated by a linear combination of
a finite number of values of the integrand.
The contents of the book are divided into three parts. The first part
presents concepts and theorems that are met with in the theory of quadra-
ture, but are at least partially outside of the programs of higher academic
institutions.
The second part is devoted to the problem of calculation of definite
integrals. Here we consider, in essence, three basic topics: the theory
of the construction of mechanical quadrature formulas for sufficiently
smooth integrand functions, the problem of increasing the precision of
quadratures, and the convergence of the quadrature process.
In the third part of the book we study methods for the calculation of
indefinite integrals. Here we confine ourselves for the most part to a
study of methods for constructing computational formulas. In addition
we indicate stability criterions and the convergence of the computational
process.
My colleagues in this work, M. K. Gavurin and I. P. Mysovskich, ex-
amined a large part of the manuscript and I am very thankful for their
remarks and advice.
Academy of Sciences of the
Byelorussian Socialist Soviet Republic V. I. KRYLOV
v
TRANSLATOR'S PREFACE
This book provides a systematic introduction to the subject of approxi-
mate integration, an important branch of numerical analysis. Such an
introduction was not available previously. The manner in which the book
is written makes it ideally suited as a text for a graduate seminar course
on this subject.
A more exact title for this book would be Approximate Integration of
Functions of One Variable. As in many aspects of the theory of func-
tions the theory developed here for functions of one variable is very
difficult to extend to functions of more than one variable, and the cor-
responding results are mostly unknown. Several years from now, after
methods for integration of functions of more than one variable have been
investigated more thoroughly, a book entirely devoted to this subject will
be needed.
As a source of reference for other topics concerning approximate inte-
gration see "A Bibliography on Approximate Integration," Mathematics
of Computation (vol. 15, 1961, pp. 52-80), which was compiled by the
translator. This is a reasonably complete bibliography, particularly for
papers published during the past several decades.
The only significant change in this translation from the original is the
inclusion in the appendices of slightly more extensive tables of Gaus-
sian quadrature formulas. The formulas in Appendix A for constant
weight function are taken from a memorandum by H. J. Gawlik and are
published with the permission of the Controller of Her Britannic Maj-
esty's Stationery Office, and the British Crown copyright is reserved.
I wish to thank Dr. V. I. Krylov for the assistance he provided in
furnishing a list of corrections to the original edition. I am also in-
debted to Professor G. E. Forsythe for the interest he expressed on be-
half of the Association for Computing Machinery in having this book pub-
lished in the present monograph series. Finally I am indebted to James
T. Day for his interest in this book and for his assistance in reading
parts of the manuscript.
University of Wisconsin
Madison, Wisconsin A. H. STROUD
vi
CONTENTS
Preface v
Translator's Preface vi
PART ONE. PRELIMINARY INFORMATION
Chapter 1. Bernoulli Numbers and Bernoulli Polynomials 3
1.1. Bernoulli numbers 3
1.2. Bernoulli polynomials 6
1.3. Periodic functions related to Bernoulli polynomials 13
1.4. Expansion of an arbitrary function in Bernoulli
polynomials 15
Chapter 2. Orthogonal Polynomials
2.1. General theorems about orthogonal polynomials 18
2.2. Jacobi and Legendre polynomials 23
2.3. Chebyshev polynomials 26
2.4. Chebyshev-Hermite polynomials 33
2.5. Chebyshev-Laguerre polynomials 34
Chapter 3. Interpolation of Functions 37
3.1. Finite differences and divided differences 37
3.2. The interpolating polynomial and its remainder 42
3.3. Interpolation with multiple nodes 45
Chapter 4. Linear Normed Spaces. Linear Operators 50
4.1. Linear normed spaces 50
4.2. Linear operators 54
4.3. Convergence of a sequence of linear operators 59
Vii
viii Contents
PART TWO. APPROXIMATE CALCULATION OF DEFINITE
INTEGRALS
Chapter 5. Quadrature Sums and Problems Related to Them. The
Remainder in Approximate Quadrature 65
5.1. Quadrature sums 65
5.2. Remarks on the approximate integration of periodic
functions 73
5.3. The remainder in approximate quadrature and its
representation 74
Chapter 6. Interpolatory Quadratures 79
6.1. Interpolatory quadrature formulas and their remainder
terms 79
6.2. Newton-Cotes formulas 82
6.3. Certain of the simplest Newton-Cotes formulas 92
Chapter 7. Quadratures of the Highest Algebraic Degree of
Precision 100
7.1. General theorems 100
7.2. Constant weight function 107
7.3. Integrals of the form fb (b - Z) (x - a)p f(x) dx and
their application to the calculation of multiple integrals 111
7.4. The integral ,f 7x' f(,) dx 129
oa
7.5. Integrals of the form j °° xa e : f (x) dx 130
Chapter 8. Quadrature Formulas with Least Estimate of the
Remainder 133
8.1. Minimization of the remainder of quadrature formulas 133
8.2. Minimization of the remainder in the class L,,(r) 134
8.3. Minimization of the remainder in the class C, 149
8.4. The problem of minimizing the estimate of the remainder
for quadrature with fixed nodes 153
Chapter 9. Quadrature Formulas Containing Preassigned Nodes 160
9.1. General theorems 160
9.2. Formulas of special form 166
9.3. Remarks on integrals with weight functions that change
sign 174
Contents ix
Chapter 10. Quadrature Formulas with Equal Coefficients 179
10.1. Determining the nodes 179
10.2. Uniqueness of the quadrature formulas of the highest
algebraic degree of precision with equal coefficients 183
10.3. Integrals with a constant weight function 187
Chapter 11. Increasing the Precision of Quadrature Formulas 200
11.1. Two approaches to the problem 200
11.2. Weakening the singularity of the integrand 202
11.3. Euler's method for expanding the remainder 206
11.4. Increasing the precision when the integral representa-
tion of the remainder contains a short principle sub-
interval 229
Chapter 12. Convergence of the Quadrature Process 242
12.1. Introduction 242
12.2. Convergence of interpolatory quadrature formulas for
analytic functions 243
12.3. Convergence of the general quadrature process 264
PART THREE. APPROXIMATE CALCULATION OF INDEFINITE
INTEGRALS
Chapter 13. Introduction 277
13.1. Preliminary remarks 277
13.2. The error of the computation 281
13.3. Convergence and stability of the computational
process 288
Chapter 14. Integration of Functions Given in Tabular Form 298
14.1. One method for solving the problem 298
14.2. The remainder 302
Chapter 15. Calculation of Indefinite Integrals Using a Small
Number of Values of the Integrand 303
15.1. General aspects of the problem 303
15.2. Formulas of special form 309
Chapter 16. Methods Which Use Several Previous Values of the
Integral 320
16.1. Introduction 320
16.2. Conditions under which the highest degree of precision
is achieved 323