Approximate calculation of multiple integrals

221 Pages·1971·41.363 MB·English

by  A. H. Stroud

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PREFACE
© 1971 by Prentice-Hall, Inc. The theory of approximate calculation of single integrals is known in
Englewood Cliffs, N. J. considerable detail. There are three books on this subject in English:
V. I. Krylov, Approximate Calculation of Integrals, The Macmillan Com
pany, 1962 (translated from first Russian edition, 1959, by A. H.
Stroud);
All rights reserved. No part of this book A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas, Prentice-Hall,
may be reproduced in any form or by any means Inc., 1966;
without permission in writing from the publisher.
P. J. Davis and P. Rabinowitz. Numericallnlexralion, Blaisdell, 1967.
A second Russian edition of Krylov's book was published in 1967. The
Current printing (last digit):
biggest change in this edition is the inclusion of four chapters on multiple
10 9 8 7. 6 S 4 3 2 integrals. these chapters were written by I. P. Mysovskih. In addition there
are a number of Russian books by V. I. Krylov and several co-authors
devoted mainly to tables of one variable formulas. There is also a book in
13-043893-6 Romanian: - - ------ - - - ---..
Library of Congress Catalog Card No. 77-1S9121
D. V. lonescu, Numerical Quadrature, Bucharest, 1957.
The earliest interesting integration formulas for more than one variable
were given in 1877 by J. C. Maxwell. However the Bibliography lists only
Printed in the United States of America about 15 papers before 1945. Therefore most of what is known about this
subject has been discovered fairly recently; in fact, a significant amount has
been published since work was begun on this book four years ago. Much
remains to be done.
This book is meant to be a research monograph and reference work.
PRENTICE-HALL INTERNATIONAL, INC., London As such we have tried to include all of the important information presently
PRENTICE-HALL OF AUSTRALIA, PTY. LTD., Sydney known on this subject. However, in certain cases, which will be mentioned
PRENTICE-HALL OF CANADA, LTD., Toronto
below, it has not been possible to do this. Some of our results are previously
PRENTICE-HALL OF INDIA PRIVATE LIMITED, New Delhi
PRENTICE-HALL OF JAPAN, INC., Tokyo unpublished.
vi PREFACE PREFACE vii
The first part of the book mainly discusses theoretical questions: through 5.13. Essential to this theory are some fundamental theorems about
existence and construction of formulas and error estimates. The second part linear functionals defined on a Hilbert space; statements of these theorems
is mainly tables of formulas and computer programs. are included.
In Chapter 2 we discuss product formulas. The results of Sections 2.8 Chapter 6 on quasi-Monte Carlo methods is also a survey. A com
and 2.9 are new. Section 2.8 shows how a formula for a solid star-like prehensive treatment of the so-called "number-theoretic methods" is given
n-dimensional region can be obtained if a formula is known for its surface. in
This generalizes the previously known result for the n-sphere. In Section 2.9
N. M. Korobov, Number-Theoretic Methods in Approximate Analysis (in
we derive a formula for a torus provided a formula is known for its cross
Russian), Moscow, ]963.
section. Although most applications of this result are in three dimensions,
the result applies in n dimensions, n ;;::: 3. We summarize some of Korobov's results and also results due to E. Hlawka,
Chapter 3 is devoted to nonproduct formulas. Sections 3.2 through 3.4 S. K. Zaremba, S. Haber and V. L. N. Sarma.
concern Newton-Cotes type formulas. In Section 3.2 the evaluation of a In recent years S. L. Sobolev and other Russian authors have developed
certain determinant related to the existence of these formulas is new. (We a theory of "formulas with a regular boundary layer." This is the only major
omit discussion of an algorithm, due to F. Stenger [3], for the construction topic we do not discuss. The reason we omit this is that we understand
of such formulas.) Section 3.3 gives Tchakaloff's result concerning the Sobolev is preparing a book also titled "Approximate Calculation of Mul
existence of formulas with all points inside the region and all coefficients tiple Integrals" which should give a complete discussion of this theory. Part
positive. Section 3.4 gives one of the constructive proofs of Tchakaloff's of the review article by S. Haber [6] discusses Sobolev's work.
result due to P. J. Davis. The most important topic in the remainder of Chapter 7 defines the regions for which formulas are tabulated in
Chapter 3 concerns the relation between the points in a formula and common Chapter 8. We tabulate almost all formulas known to us which have some
zeros of sets of orthogonal polynomials in n variables. This investigation was practical or theoretical importance. (There has not been space to include
started by Radon (see Section 3.12); additional results have recently been all the formulas tabulated by F. Stenger [I], [4].) Some of these formulas
obtained by Stroud (Sections 3.7, 3.11, 3.13). Hopefully these results can be have not been published previously. With some formulas we also give con
expanded into a still more complete theory. A lower bound for the number stants needed for the error estimates discussed in Chap. 5. It is believed that
of points in a formula of odd degree given in Section 3.] 5 is new. From this none of these error constants has been published previously.
it follows that a formula of degree 3 for the n-simplex must have at least Chapter 10 gives computer programs for a few selected formulas and
+
n 2 points. (We have omitted the result of 1. P. Mysovskih [5] that a programs for computing error constants.
formula of degree 3 for the n-sphere and n-cube must have at least 2n points.) " I am indebted to James Brooking, now at Knolls Atomic Power
Chapter 4 discusses ways in which a formula for the m-cube can be Laboratory, Schenectady, New York, for his plotting routines which drew
used to obtain a formula for the n-cube, n > m, by methods other than the 3-dimensional graphs of the Sard kernel functions in Chapter 5. Also
product methods. I thank Bob Barnhill and Frank Stenger for comments on parts of the
Chapter 5 is a survey of topics on error estimates. It contains only the manuscript. I thank Mark Morgante who assisted in computingusome of
outlines of a few proofs. Sections 5.2 through 5.10 discuss theorems on the error C()Ilstants and Mrs. Liz Mciver, Mrs ..J une Cassidy, Miss Joyce
representations of linear functionals given by Staskiewicz and Mrs. Kristine Johnson for typing parts of the manuscript.
Some of the new results contained in this book are results of research
A. Sard, Linear Approximation, Amer. Math. Society, ]963, Chap. 4,
performed with support of the National Science Foundation under Grants
and estimates for the error in integration formulas derived from these GP-5675, GP-7364, GP-8954 and GP-13287. The computations were also
theorems. Sard's theory generalizes that of Peano for one variable. The com supported in part from these grants. The remainder of the computations
plete theory is given by Sard and there is no need to reproduce it here. It were supported by the State University of New York and were carried out on
has not been used for practical work in the past mainly for two reasons: the IBM 7044 and the CDC 6400 at the Computing Center of the State
(i) the lack of an easily readable account of the general ideas, and (ii) lack University of New York at Buffalo.
of tabulated constants for specific formulas. We have tried to provide these
two things. There is a second type of error estimate for analytic functions. A. H. Stroud
This theory is due to R. E. Barnhill and is summarized in Sections 5.]]
CONTENTS
Part I
THEORY 1
Chapter 1 INTRODUCTION 3
1.1 The Approximations 3
1.2 Some Notation and Definitions 4
1.3 Contrasts Between One Variable and More Than
One Variable 6
1.4 Affine Transformations of Formulas 7
I.S Desirable Properties of a Formula; Choice of a
Formula 12
1.6 Regions for Which Formulas Are not Known 14
1.7 Unsolved Problems 17
1.8 Intersections of Plane Curves 18
1.9 Convergence of a Sequence of Formulas 21
Chapter 2 PRODUCT FORMULAS 23
2.1 Remarks 23
2.2 The n-Cube 25
2.3 General Cartesian Product Formulas 26
2.4 The n-Simplex 28
2.S n-Dimensional Cones 31
2.6 The n-Sphere 32
2.7 Surface of the n-Sphere 40
2.8 Formulas for a Region from Formulas for Its
Surface 43
2.9 II-Dimensional Tori 46
2.10 The Ellipse 49
2.11 Regions Bounded by Orthogonal Parabolas 51
Ix
x CONTENTS :ONTENTS xl
Chapter 3 NONPRODUCT FORMULAS 53 Chapter 6 MONTE CARLO AND
NUMBER-THEORETIC METHODS 193
3,1 Remarks 53
3.1 Existence of Formulas 54
3.3 Existence of Formulas with Positive Coefficients 58 6.1 Introduction 193
3.4 Construction of Formulas with Positive Coefficients 63 6.1 The Discrellancyof a Set of Points nnd the Error
3.5 Orthogonal Polynomials in n Variables 67 in a Quasi-Monte. Carlo Method 195
3.6 Some Particular Orthogonal Polynomials 70 6.3 Number-Theoretic Methods 198
3.7 Integration Formulas and Orthogonal Polynomials 75 6.4 Stratified Sampling Methods 209
3.8 Second Degree Formulas 79 6.5 Qunsi-Monte Carlo Methods for Large n 210
-",- 3.9 Third Degree Formulas with 2n Points 88 6.6 Examples 212
3.10 Fifth Degree Formulas with nZ + n + 2 Points 92 6.7 Other Results 216
:~'
l.U Formulas of Degree 2m - 1 with mZ Points for
Two Variables 96
3.11 Radon's Firth Degree Formulas 100
3.13 More on Integration Formulas and Orthogonal
Polynomials 110
3.14 Some Formulas for the n-Simplex 114
3.15 A Lower Bound for the Number of Points in a Part II
Formula 118
3.16 Minimal Point Formulas 120 TABLES 217
3.17 Richardson Extrapolation for the n-Cube 121
Chapter 4 EXTENSIONS OF FORMULAS 127 Chapter 7 THE REGIONS 219
4.1 Introduction 127
7.1 Introduction 219
4.1 Extensions of an Arbitrary Formula for C 129
m 7.1 C.-the n-Dimensional Cube 220
4.3 Extensions of Symmetric and Fully Symmetric
7.3 C:holl-the n-Dimensional Cubical Shell 220
Formulas 133
7.4 S.-the n-Dimensional Sphere 220
4.4 Regions Other Than Cm 136 7.5 S:boll-the n-Dimensional Spherical Shell 221
7.6 U.-the Surface of S. 221
7.7 G.-the n-Dimensional Octahedron 221
Chapter 5 ERROR ESTIMATES 137
7.8 Tn-the n-Dimensional Simplex 222
7.9 E~' -Entire n-Dimensional Space with Weight
5.1 Introduction 137
+-'----cSIr.I The SpaceuofFu-nclions B". 138 Function exp (-xi - ... - .;) ~'~n,----~~~
5.3 Estimates for I Elf) I 146 7.10 FE~u-nEcntitoinr eex np-(D -.im.;ex nis i+on a..l .S p+a cxe! }w ith2 2W2 eight
5.4 Examples 148
7.11 Hz-the 2-Dimensional Hexagon 223
5.5 The Space of Functions B" •• , 157
7.1Z ELP-the 2-Dimensional Ellipse with Weight
5.6 Examples 159 Function [(x - c)Z + y2J-IIZ[(x + c)Z + yZJ-I/Z 223
5.7 General Spaces of Type B 168
7.13 PAR-First Parabolic Region 224
5.8 Nonrectangular Regions in Two Dimensions 169
7.14 PARz-Second Parabolic Region 225
5.9 Error Constants for Some Product Formulas 172
7.15 PAR3-Third Parabolic Region 225
5.10 Regions in Three Dimensions 176
7.16 CN: Cz-A 3-Dimensional Pyramid 225
5.11 The Hilbert Space £Z(~ p x 8 p) 178
7.17 CN: Sz-A 3-Dimensional Cone 226
5.U Error Bounds for Analytic Functions 180
7.18 TOR 3 : Sz-A 3-Dimensional Torus with Circular
5.13 The Hypercircle Inequality and Integration Formulas
Cross Section 226
Obtained From It 184
7.19 TOR 3 : Cz-A 3-Dimensional Torus with Square
5.14 The Sarma-Eberlein Estimate of Goodness SE 188 Cross Section 227
xii CONTENTS CONTENTS xiii
Chapter 8 TABLES OF FORMULAS 228 AUTHOR INDEX 419
8.1 Notation 228
8.2 C.-The n-Dimensional Cube 229 SUBJECT INDEX 423
8.3 C:hol1_The n-Dimensional Cubical Shell 266
8.4 S.-The II-Dimensional Sphere 267
8.S S:bol1_The n-Dimensional Spherical Shell 293 INDEX OF SYMBOLS 427
8.6 U.-The Surface of S. 294
8.7 G.-The n-Dimensional Octahedron 303
8.8 T,.-The n-Dimensional Simplex 306
8.9 E~'-Entire n-Dimensional Space with Weight
Function exp(-xf - ... - x;) 315
8.10 E~-Entire n-Dimensional Space with Weight
+ ...
Functionexp(-v'xt +x~) 329
8.11 H z-The Two-Dimensional Hexagon 335
8.12 ELP-The Two-Dimensional Ellipse with Weight
+ + +
Function [(x -'c)l yZ]-I/2[(x c)l yZ]-I/2 336
8.13 PAR-First Parabolic Region 338
8.14 PARz-Second Parabolic Region 338
8.1S PAR3-Third Parabolic Region 338
8.16 CN: Cz-A Three-Dimensional Pyramid 339
8.17 CN: Sz-A Three-Dimensional Cone 339
8.18 TOR3: Sz-A Three-Dimensional Torus with Circular
Cross Section 340
8.19 TOR3: Cz-A Three-Dimensional Torus with Square
Cross Section 340
Chapter 9 REGULAR POLYTOPES
IN n DIMENSIONS 342
Chapter /0 COMPUTER PROGRAMS 346
10.1 Introduction 346
10.2 Iterated Integrals 346
10.3 Integrals over the Square, Circle and Sphere 350
10.4 Integrals over the n-Sphere, 2 ~ n ~ 8 353
10.S Integrals over the Circumference of the Circle and
the Surface of the Sphere 359
10.6 Integrals over Tori 362
10.7 Integrals over the Triangle, Tetrahedron and the
n-Simplex, 2 =::;; n ~ 8 366
10.8 The Progressive Procedure (Richardson
Extrapolation) for the n-Cube, 2 ~ n ~ 5 374
10.9 Computation of Error Constants 376
BIBLIOGRAPHY 390
-«l
-

«
0 <1: ...J U ::J ...J I- Z
I
.
L

::J ~ a.-...J z I-IJJ (j 0::: <:( .:J
JIwoer

- t
:
,
Chapter I
NTRODUCTION
1.1 The Approximations
The purpose of this book is to discuss methods for approximate calcula
tion of multiple integrals. Most of the approximations we consider have the
form
J .. J
.~ w(x ,x.)f(x, •...• x.) dx, ... dx.
i • •••
(I.I-I)
:::::::: LN
Bt!(v"I"'" v".)
Is I
Here R. is a given region in n-dimensional Euclidean space E. and w(x
= =l= •
. ',' • x.) is a given weight function-often w(x x.) I. The v,
l •••••
(v,.!' •.. .• v, .•) are points which lie in E. and are called the points (or nodes)
of the formula; the B, are constants which do not depend on f(x x.)
I' ••••
and are called the coefficients of tlte formula. Throughollt this book we shall
>-
assume w(xl ••••• x.) 0 on R. unless stated otherwise.
Most of the formulas we discuss are exact (o~ ~ y~rlltlll class o[ pol~ -
·ri1farsln x~.-. -:-: ,-i ..- We say-that-formula (I~ I-I> has degree d (or degree of
-<
exactlless eI) if it is exact for all polynomials in XI' •• • • x. of degree d
+
and there is at least one polynomial of degree d I for which it is not exact.
In other words, (1.1-1) has degree d if it is exact for any linear combination
of monomials
xi'xi' ... x:- (1.1-2)
(l .. (lz ••••• (l. nonnegative integers
o <IXI + IXz + '" + IX. < d
+
and there is at least one monomial of degree d I for which (I.I-I) is not
exact.
In Part I of this book we discuss the theory of these formulas. In Chapter
3
4 THEORY Par' I Ch. 1
INTRODUCTION 5
8 of Part Ii we tabulate the most important known formulas of the form of defined in the Riemann sense, and with some of its simpler properties. We
(1.1-1). Formulas are given for the following regions: . also nsslIme ral11ilinrity with relnted ideas, such as a Riemann sum in II
C. the unit n-cube, dimensions and an II-fold iterated integral:
C:hOIl the n-cubical shell, "" f""("') f f"'··'( ..
f ,·····x.o.>
S. the solid unit n-sphere, . . . f(x., ... ,x.) dx• ... dx.
,. ".(.Y,) ' .... (~I ......' ,. I)
S~holl the lI-sJ"lhericnl shell,
v. All vectors are delined by lowercase boldface letters, for example. V,
the surface of the unit n-sphere,
x, v" The components of a vector are written as follows:
G. the n-octahedron (or n-crosspolytope),
=
T. the n-simplex, x (x., x2, ••• , x.)
EE~" eennttiirree nn--ssppaaccee wwiitthh wweeigighht tf ufunnctcitoionn e xepx(p -( ,-.xj~ x t- + ...... -+ xx!!)).. v, = (v, .• , v,. 2' ••• , V, •• )
All matrices are denoted by boldface capital letters, for example, X, A, C,
TOR: S2 the three-dimensional torus with circular cross section, and
I. The symbol XT denotes the transpose of X. The symbol X-I denotes the
TOR: C the three-dimensional torus with square cross section.
2 inverse or X. Given a matrix X we shall not use any systematic notation for
Precise definitions of these regions are given in Chapter 7. Formulas the elements of X.
are also given for a few other special two- and three-dimensional regions,
E. denotes real n-dimensional Euclidean space.
which are also defined in Chapter 7.
R. denotes a region in E• .
The approximations we give for integrals over U• • the surface of the n is the dimension of E., usually n > 2.
unit II-sphere, have a slightly different form than (1.1-1). We write these in
Usually N denotes the number of points in a formula (1.1-1) and the
the form
coefficients in the formula will be denoted by B ..... , BN (sometimes by
(1.1-3) A., ... , AN)'
A weight function is denoted by w(x., ... , x.). Usually we assume
The integral on the left side of (1.1-3) is an (II - I )-fold integral ;f(x., ... , x;,) w(x., ... , x.) > 0 on R •.
is assumed defined in terms of II-dimensional Cartesian coordinates, which
For short we write
are restricted to lie on U.: =
xf + xl + .. , + x: = I J(f) J[J(x., ... , x.»)
,
The points V, which appear on the right side of (1.1-3) are also usually given
in terms of Cartesian coordinates and also lie on U• .
Special cases of this are
Each formula given in Chapter 8 has an individual label. auch as C2: 2-4 f·;: f
or S.: 5-1. The system of labeling is explained at the beginning of Chapter 8. == w(x~~x.)
/(1) dx • ... dx.
We give one formula, Ca: 2-1, which ~ses~arti~derivatives of the f·;.· f
integrand. - x:.) ==
J(x1'X!' : .. w(x., ... ,x.)x1'X1' '" X:. dx. '" dx.
(1.2-1 )
1.2 Notation and Definitions
Unless stated otherwise the exponents 11., in a monomial
As mentioned in Sec. 1.1, definitions for the regions we consider are
given in Chapter 7 and the system for labeling the formulas tabulated in
are assumed to be nonnegative integers. In Chapter 8 we write
Chapter 8 is explained in Sec. 8.1. The main purpose of this section is to define
the symbols and basic notation to be used. V= /(1)
We assume the reader is familiar with the concept of a multiple definite We always assume that the region and weight function are such that the
integral, monomial integrals (1.2-1) are finite. -
The symbol (.I'J denotes the largest integer which is less than or equal to
the positive real number .1'. .