Approximate calculation of multiple integrals 9780130438935, 0130438936

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APPROXI MATE CALCULATION

OF MULTIPLE INTEGRALS

A. H. Stroud II Department of Mathematics

Texas A cl M Univer:sity

Prentice-Hall, Inc. Englewood Cliffs, New Jersey

PREFACE

©

1971 by Prentice-Hall, Inc. Englewood Cliffs, N. J.

The theory of approximate calculation of single integrals is known in considerable detail. There are three books on this subject in English: V. I. Krylov, Approximate Calculation of Integrals, The Macmillan Company, 1962 (translated from first Russian edition, 1959, by A. H. Stroud);

All rights reserved. No part of this book may be reproduced in any form or by any means without permission in writing from the publisher.

A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas, Prentice-Hall, Inc., 1966; P. J. Davis and P. Rabinowitz. Numericallnlexralion, Blaisdell, 1967.

Current printing (last digit):

10 9 8 7. 6 S 4 3 2 13-043893-6 Library of Congress Catalog Card No. 77-1S9121

A second Russian edition of Krylov's book was published in 1967. The biggest change in this edition is the inclusion of four chapters on multiple 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 Romanian: - - - - - --- - - - --- . D. V. lonescu, Numerical Quadrature, Bucharest, 1957.

Printed in the United States of America

London Sydney CANADA, LTD., Toronto INDIA PRIVATE LIMITED, New Delhi JAPAN, INC., Tokyo

PRENTICE-HALL INTERNATIONAL, INC.,

PRENTICE-HALL OF AUSTRALIA, PTY. LTD., PRENTICE-HALL OF PRENTICE-HALL OF PRENTICE-HALL OF

The earliest interesting integration formulas for more than one variable were given in 1877 by J. C. Maxwell. However the Bibliography lists only 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. As such we have tried to include all of the important information presently known on this subject. However, in certain cases, which will be mentioned below, it has not been possible to do this. Some of our results are previously unpublished.

PREFACE

vi

The first part of the book mainly discusses theoretical questions: existence and construction of formulas and error estimates. The second part is mainly tables of formulas and computer programs. In Chapter 2 we discuss product formulas. The results of Sections 2.8 and 2.9 are new. Section 2.8 shows how a formula for a solid star-like n-dimensional region can be obtained if a formula is known for its surface. This generalizes the previously known result for the n-sphere. In Section 2.9 we derive a formula for a torus provided a formula is known for its cross section. Although most applications of this result are in three dimensions, the result applies in n dimensions, n ;;::: 3. Chapter 3 is devoted to nonproduct formulas. Sections 3.2 through 3.4 concern Newton-Cotes type formulas. In Section 3.2 the evaluation of a certain determinant related to the existence of these formulas is new. (We omit discussion of an algorithm, due to F. Stenger [3], for the construction of such formulas.) Section 3.3 gives Tchakaloff's result concerning the existence of formulas with all points inside the region and all coefficients positive. Section 3.4 gives one of the constructive proofs of Tchakaloff's result due to P. J. Davis. The most important topic in the remainder of Chapter 3 concerns the relation between the points in a formula and common zeros of sets of orthogonal polynomials in n variables. This investigation was started by Radon (see Section 3.12); additional results have recently been obtained by Stroud (Sections 3.7, 3.11, 3.13). Hopefully these results can be expanded into a still more complete theory. A lower bound for the number of points in a formula of odd degree given in Section 3.] 5 is new. From this it follows that a formula of degree 3 for the n-simplex must have at least n 2 points. (We have omitted the result of 1. P. Mysovskih [5] that a formula of degree 3 for the n-sphere and n-cube must have at least 2n points.) Chapter 4 discusses ways in which a formula for the m-cube can be used to obtain a formula for the n-cube, n > m, by methods other than product methods. Chapter 5 is a survey of topics on error estimates. It contains only the outlines of a few proofs. Sections 5.2 through 5.10 discuss theorems on representations of linear functionals given by

+

A. Sard, Linear Approximation, Amer. Math. Society, ]963, Chap. 4, and estimates for the error in integration formulas derived from these theorems. Sard's theory generalizes that of Peano for one variable. The complete theory is given by Sard and there is no need to reproduce it here. It has not been used for practical work in the past mainly for two reasons: (i) the lack of an easily readable account of the general ideas, and (ii) lack 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. This theory is due to R. E. Barnhill and is summarized in Sections 5.]]

PREFACE

vii

through 5.13. Essential to this theory are some fundamental theorems about linear functionals defined on a Hilbert space; statements of these theorems are included. Chapter 6 on quasi-Monte Carlo methods is also a survey. A comprehensive treatment of the so-called "number-theoretic methods" is given in N. M. Korobov, Number-Theoretic Methods in Approximate Analysis (in Russian), Moscow, ]963. We summarize some of Korobov's results and also results due to E. Hlawka, S. K. Zaremba, S. Haber and V. L. N. Sarma. In recent years S. L. Sobolev and other Russian authors have developed a theory of "formulas with a regular boundary layer." This is the only major topic we do not discuss. The reason we omit this is that we understand Sobolev is preparing a book also titled "Approximate Calculation of Multiple Integrals" which should give a complete discussion of this theory. Part of the review article by S. Haber [6] discusses Sobolev's work. Chapter 7 defines the regions for which formulas are tabulated in Chapter 8. We tabulate almost all formulas known to us which have some practical or theoretical importance. (There has not been space to include all the formulas tabulated by F. Stenger [I], [4].) Some of these formulas have not been published previously. With some formulas we also give constants needed for the error estimates discussed in Chap. 5. It is believed that none of these error constants has been published previously. Chapter 10 gives computer programs for a few selected formulas and programs for computing error constants. " I am indebted to James Brooking, now at Knolls Atomic Power Laboratory, Schenectady, New York, for his plotting routines which drew the 3-dimensional graphs of the Sard kernel functions in Chapter 5. Also I thank Bob Barnhill and Frank Stenger for comments on parts of the manuscript. I thank Mark Morgante who assisted in computingusome of the error C()Ilstants and Mrs. Liz Mciver, Mrs ..June Cassidy, Miss Joyce Staskiewicz and Mrs. Kristine Johnson for typing parts of the manuscript. Some of the new results contained in this book are results of research performed with support of the National Science Foundation under Grants GP-5675, GP-7364, GP-8954 and GP-13287. The computations were also supported in part from these grants. The remainder of the computations were supported by the State University of New York and were carried out on the IBM 7044 and the CDC 6400 at the Computing Center of the State University of New York at Buffalo. A. H. Stroud

CONTENTS

Part I THEORY Chapter 1

1 INTRODUCTION

3

1.1 1.2 1.3

The Approximations 3 Some Notation and Definitions 4 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 2.1 2.2 2.3 2.4

PRODUCT FORMULAS 23

Remarks 23 The n-Cube 25 General Cartesian Product Formulas 26 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

Chapter 3

-",:~'

3,1 3.1 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 l.U 3.11 3.13 3.14 3.15 3.16 3.17

Chapter 4 4.1 4.1 4.3 4.4

NONPRODUCT FORMULAS

53

Remarks 53 Existence of Formulas 54 Existence of Formulas with Positive Coefficients 58 Construction of Formulas with Positive Coefficients 63 Orthogonal Polynomials in n Variables 67 Some Particular Orthogonal Polynomials 70 Integration Formulas and Orthogonal Polynomials 75 Second Degree Formulas 79 Third Degree Formulas with 2n Points 88 Fifth Degree Formulas with n Z + n + 2 Points 92 Formulas of Degree 2m - 1 with m Z Points for Two Variables 96 Radon's Firth Degree Formulas 100 More on Integration Formulas and Orthogonal Polynomials 110 Some Formulas for the n-Simplex 114 A Lower Bound for the Number of Points in a Formula 118 Minimal Point Formulas 120 Richardson Extrapolation for the n-Cube 121

EXTENSIONS OF FORMULAS

127

Introduction 127 Extensions of an Arbitrary Formula for C m 129 Extensions of Symmetric and Fully Symmetric Formulas 133 Regions Other Than C m 136

Chapter 5

ERROR ESTIMATES

5.1 +-'----cSIr.I 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.U 5.13

Introduction 137 The SpaceuofFu-nclions B". 138 Estimates for I Elf) I 146 Examples 148 The Space of Functions B" ••, 157 Examples 159 General Spaces of Type B 168 Nonrectangular Regions in Two Dimensions 169 Error Constants for Some Product Formulas 172 Regions in Three Dimensions 176 The Hilbert Space £Z(~ p x 8 p) 178 Error Bounds for Analytic Functions 180 The Hypercircle Inequality and Integration Formulas Obtained From It 184 The Sarma-Eberlein Estimate of Goodness SE 188

5.14

137

Chapter 6

6.1 6.1 6.3 6.4 6.5 6.6 6.7

MONTE CARLO AND NUMBER-THEORETIC METHODS

193

Introduction 193 The Discrellancyof a Set of Points nnd the Error in a Quasi-Monte. Carlo Method 195 Number-Theoretic Methods 198 Stratified Sampling Methods 209 Qunsi-Monte Carlo Methods for Large n 210 Examples 212 Other Results 216

Part II TABLES

217

Chapter 7 THE REGIONS 7.1 7.1 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11

7.1Z 7.13 7.14 7.15 7.16 7.17 7.18 7.19

219

Introduction 219 C.-the n-Dimensional Cube 220 C:holl-the n-Dimensional Cubical Shell 220 S.-the n-Dimensional Sphere 220 S:boll-the n-Dimensional Spherical Shell 221 U.-the Surface of S. 221 G.-the n-Dimensional Octahedron 221 Tn-the n-Dimensional Simplex 222 E~' -Entire n-Dimensional Space with Weight Function exp (-xi - ... - .;) ~'~n,----~~~ E~-Entire n-Dimensional Space with Weight Function exp( -..; xi + ... + x!} 222 Hz-the 2-Dimensional Hexagon 223 ELP-the 2-Dimensional Ellipse with Weight Function [(x - c)Z + y2J-IIZ[(x + c)Z + yZJ-I/Z 223 PAR-First Parabolic Region 224 PARz-Second Parabolic Region 225 PAR3-Third Parabolic Region 225 CN: Cz-A 3-Dimensional Pyramid 225 CN: Sz-A 3-Dimensional Cone 226 TOR 3: Sz-A 3-Dimensional Torus with Circular Cross Section 226 TOR 3: Cz-A 3-Dimensional Torus with Square Cross Section 227

xl

xii

CONTENTS

CONTENTS

Chapter 8 TABLES OF FORMULAS 8.1 8.2 8.3 8.4 8.S 8.6

8.7 8.8 8.9 8.10 8.11 8.12

8.13 8.14 8.1S 8.16 8.17 8.18 8.19

Notation 228 C.-The n-Dimensional Cube 229 C:h ol1_The n-Dimensional Cubical Shell 266 S.-The II-Dimensional Sphere 267 S:bol1_The n-Dimensional Spherical Shell 293 U.-The Surface of S. 294 G.-The n-Dimensional Octahedron 303 T,.-The n-Dimensional Simplex 306 E~'-Entire n-Dimensional Space with Weight Function exp(-xf - ... - x;) 315 E~-Entire n-Dimensional Space with Weight Functionexp(-v'xt + ... +x~) 329 H z-The Two-Dimensional Hexagon 335 ELP-The Two-Dimensional Ellipse with Weight Function [(x -'c)l + yZ]-I/2[(x + c)l + yZ]-I/2 336 PAR-First Parabolic Region 338 PARz-Second Parabolic Region 338 PAR3-Third Parabolic Region 338 CN: Cz-A Three-Dimensional Pyramid 339 CN: Sz-A Three-Dimensional Cone 339 TOR3: Sz-A Three-Dimensional Torus with Circular Cross Section 340 TOR3: Cz-A Three-Dimensional Torus with Square Cross Section 340

Chapter 9

REGULAR POLYTOPES IN n DIMENSIONS 342

Chapter /0

COMPUTER PROGRAMS

10.1 10.2 10.3 10.4 10.S 10.6 10.7 10.8 10.9

228

346

Introduction 346 Iterated Integrals 346 Integrals over the Square, Circle and Sphere 350 Integrals over the n-Sphere, 2 ~ n ~ 8 353 Integrals over the Circumference of the Circle and the Surface of the Sphere 359 Integrals over Tori 362 Integrals over the Triangle, Tetrahedron and the n-Simplex, 2 =::;; n ~ 8 366 The Progressive Procedure (Richardson Extrapolation) for the n-Cube, 2 ~ n ~ 5 374 Computation of Error Constants 376

BIBLIOGRAPHY

390

AUTHOR INDEX SUBJECT INDEX

419 423

INDEX OF SYMBOLS

427

xiii

Z

«lI-« ::J

...J

U

-

-
0 on R •. For short we write

J(f)

=J[J(x., ... , x.»)

, Special cases of this are

== f·;: f w(x~~x.) dx • ... dx. J(x1'X!' : .. x:.) == f·;.· f w(x., ... ,x.)x1'X1' '" X:. dx. /(1)

'" dx. (1.2-1 )

Unless stated otherwise the exponents 11., in a monomial are assumed to be nonnegative integers. In Chapter 8 we write

V= /(1) We always assume that the region and weight function are such that the 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'. .

6

Part I

Tt;fEORY

Writing N constant.

=

O(nk) means that as n

-> 00

the ratio N/nk approaches a

.. ,x~) and P ",.,(x" .. . ,x~) will denote polynomials of exact

P",(XI"

degree m in Q",(x I'

XI' ••• ,X~. ••• ,

x~) and Q",. ,(x" ... , x.) will denote polynomials of degree

sm

in XI"" ,X~. If 1ft and"Y are sets then 100. Other methods need to be studied (but, obviously, one can expect serious numerical dif1iculties in any method proposed for constructing nonproduct formulas with several hundred points or more). If we do not eare if some of the B/ COllie out negattve then we can construet a formula of degree d using C:H points by solving a linear system as described in Sec. 3.2. If C: +d is large this also is not practical. In any case the resulting formulas are usually undesirable because of negative B/.

::: 0.S23S987~

using the 16-point Gauss- Legendre formula for both of the formulas (1.6-2) and (1.6-3). The approximate result is J. ::: 0.S2362038 Subroutine Q'f.,., U LT3 of Chapter 9 approximates the iterated integral in (1.6-S). One must provide three one-dimensional formulas. The program of

Problem. Given it region and weight function and an integer d> 3, find an effective procedure for constructing a formula of degree d.

(In Sec. 3.8 we show how to construct a formula of degree 2 using w(x .. ' .' . ,x.). Section 3.10 discusses the problem of constructin·g third-degree formulas.) There surely exist formulas of degree d with fewer than C:"''' points.

n

+ I points for arbitrary R. and

18

I

Part I

THEORY

INTRODUCTION

Ch,1

19

In Sec. 3.15 we show that a lower bound for the number of points in a formula of degree dis C:+k, where k = [dI2]. Except for d = 2 and a few low values of d for n = 2 it is not likely that this lower bound can be achieved. However, from formulas we know it is likely that there exist formulas of degree d using O(II(c1/11) points. In Sec. 3.8 we show that n + 1 is the minimum number of points for degree 2. For d = 3 the minimum number is not known in general; it is known that for C. and S., N = 2n is minimal (Sec. 3.9) and for T•• N = n 2 is minimal (Sec. 3.15). Therefore for d 2 3 and n > 2 the minimum number certainly depends on the region.

into irreducible factors P m, which are unique except for multiplicative constants and the order of the P m,' The set of curves whose equations are

Problem. For a given Rn. w(x., ... , x.) and d > 2 find the minimum number of points in a formula of degree d.

coincides with the curve whose equation is

+

In some cases it may turn out that a formula with the minimum number of points does not have all its points inside the region and all B, > O. For example, the third-degree formula T.: 3-1, which contains the minimum number of points, has one negative B,. (It is likely, but unproved, that Tn: 3-1 is the only third-degree formula for T. with n + 2 points.) Problem. For a given R., w(x., ... , x.) and d> 2 find the minimum number of points in a formula of degree d with all B/ > 0 and all points inside R •.

tion to Algebraic Geometry, Oxford University Press, London, 1949, pp.94-96. For a proof of Bezout's theorem and other properties of polynomials stated in this section see R. J. Walker, Algebraic Curves, Dover, New York, 1962. Let P m(x, y) be a polynomial of degree m in x and y. Pm has a factorization P.,(x, y) = P ml(X' y)P.,.(X, y) ... P m.(X, y) m = m. ml mk m/ > I ; = I, ... , k

+

Pml(X, y)

The above four problems are difficult and apparently we are not near a general solution to any of them. It would be useful if these problems could be solved for small d, say 4 < d < 7, for the more important regions. The following is probably solvable. Problem. For C. (or S., or Tn) find a fifth-degree formula with 0(n1) or O(nJ) points which has, for all large n, all B/ > 0 and all points inside the region.

= 0, ... ,Pm.(x, y) = p ..(x,y)

In Secs. 3.11 and 3.12 we show how to construct formulas by taking the points to be the common zeros of a set of orthogonal polynomials in two variables. For this we must know something about the common solutions of two polynomials p.(X,y)

=

0

where P 1ft and p. are polynomials of degrees m and s, respectively. The necessary information is given by Bezout's theorem and Max Noether's theorem, which we state below as Theorems 1.8-1 and 1.8-4, respectively. For a proof of Noether's theorem as we state it see J. G. Semple and L. Roth, Introduc-

0

(1.8-1) (1.8-2)

1. Let

amxm + am_.x m-.

+ ... + a.x + ao be considered as a polynomial in x and, in which no terms in , appear. Then Pm can always be factored as Pm

=

Pm = am(x - v.)(x -

and the curve Pm

=

Vl) ••.

(x - v m)

0 has m linear components x - v. = 0, ... , x - vm = 0

EXAMPLE

2. Let

PI = Xl + a.ox It is not difficult to show that the curve P l

1.8 Intersections of Plane Curves

=

0

The curves (1.8-1) are called components of the curve (1.8-2). If Pm(x, y) is irreducible then curve (1.8-2) consists of a single component. In all cases we assume that the coefficients of Pm and the P m, belong to the field of complex numbers. Thus it is possible, in particular cases, that there will be no real values of x and y which satisfy an equation such as (1.8-2). We give some examples. EXAMPLE

Problem. Find a method for constructing such formulas.

+ ... +

EXAMPLE

+ ao., + aDO = 0 is irreducible if and only if flo.

-::F- O.

3. Let

= (x - y + 1)2 = (Xl + 4x - , + l)(x - , + 1) Each of the curves PI = 0, P J = 0 consists of two components. The line x - , + 1 = 0 is a component of multiplicity 2 of PI = 0 and a simple component of P, = O. This line is a common component of the'curves PI = 0, P J = O. PI

PJ

It is clear that if two curves Pm(x,y) dO

p.(X,y)

=0

(1.8-3)

20

ParI I

THEORY

have a common component then the equations (1.8-3) have an infinite numbcr of common solutions. Theorem 1.8-1 (Bezout's Theorem). If the two curves (1.8-3) have no common component then these curves have exactly ms points of intersection. Of course some of the points of intersection of the curvt;s (1.8-3) can be complex. Also. some of these points can be multiple points of intersection. Suitable regard must be made for the multiplicity of a point of intersection in counting the total number. It is also possible that some points of intersection occur on the line "at infinity." Whenever we discuss plane curves in this book we shall write the curves in an affine system of coordinates. If two curves have points of intersection at infinity then it is sometimes convenient to write the curves in projective coordinates. This is discussed in the abovementioned book by R. J. Walker. The following is a weaker form of Bezout's theorem which we shall also use. Theorem 1.8-2. If two curves (1.8-3) have more than ms points of intersection then they have a common component. As an application of Theorem 1.8-2 we can prove Theorem 1.8-3. Theorem 1.8-3. There exist (m + I)(m + 2)/2 points ", do not all lie on a curve Qm(x, y) = O.

=(x,. y,) which

Proof. Take three points "I' "Z' "3 which do not lie on a line. Take a line QI.I(X, y) = 0 which does not pass through any of the points "I' "z, "3 and take on this line three points"., "" "6' The points" I' • • • • V 6 do not lie on a second-degree curve Qz = O. If they did, by Theorem 1.8-2, Qz and QI,I would have a common component and therefore QI.I would be a component of Qz. Then Qz would be the product of two linear factors. This is impossible since "I' "1' V3 do not lie on a line. Now take a line QI.Z = 0 which does not pass through any of the points VI'" • ,v6 and take on it four points V 7 , V 8 , V 9 , V IO • The points V I' ••• , V 10 do not lie on a third-degree curve Q 3 = O. If they did, by Theorem 1.8-2, QI,Z would be a component of Q3: Q3 = QZQI.Z' Since vI> ... , V6 do not lie on QI.1 = 0 then these six points must lie on Qz = O. We saw above that this is impossible. We can continue in this way, I )(m 2)/2 points which do not all lie on by induction, to find (m a curve Qm = O. This completes the proof.

+

+

There are generalizations of the above results for more than two variables. We shall use Bezout's theorem only for two variables.

INTRODUCTION

Ch.1

21

The other property of plane curves which we shall need is the following: Theorem 1.8-4 (Max Noether's Theorem). Assume Pm and 1', have no common component and that their common zeros (x" y,), i = I •... , ms. are distinct. If Pk(x. y) is a polynomial (which is not identically 7.cro) which is 7.cro nt nil these points then k :. min (III, .~) and thcre exist polynomials Qk-m, Qk-, so that P k = Qk-mPm

+ Qk-,P,

We shall use only this simple form of Noether's theorem; there are generalizations and other forms of this theorem. Note that either Qk-m or Qk-,' but not both, can be identically zero. 1.9 Convergence of a Sequence of Formulas

For a given R. and w(x i • tion formulas

•••

,x.) assume we have a sequence of integra-

J(f):-::: Here

J(f) ~,(f)

i N,

= f .~: f w(x

i • ••••

~,(f)

x.)f(x , •. .. ,x.) dx • ... dx.

NI

= t;.1 A~)f(vn =
2, N

=

M[(M - 1)·-1

=

{

+ (M -

1)·-1

~ [(M - I)' - I]

M-2

2n

For M odd and

+ .. , + (M -

+

I]

for M> 2 for M

>

I)

=

TABLE 2.5

The Number of Points, N, in Spherical Product Gauss and Spherical Product Lobatto Formulas of Degree 2M - ,

2 3 4 5 6 7

Lobalto

Gauss

Lobalto

7

7 19 55 163 487 1,459

16 52 160 484 1,456 4,372

IS 31 63 127 255

M= 5

M=4

M=3 n

39

Gauss

Lohalto

Gaus.r

21 85 341 1,365 5,461 21,843

21 101 501 2.501 12,501 62,501

16 64 256 1.024 4,096 16,384

n

M=6 Gaus.f Lohatto

M=7 Gau.f.f Lohalto

M=8 Lohalto Gallss

2 3 4 5 6

36 186 936 4,686 23,436

43 259 1,555 9,331 55,987

64 456 3,200 22,408 156,864

36 216 1,296 7,776 46,656

43 295 2,059 14,407 100,843

64 512 4,096 32,768 262,144

spherical product Lobatto formulas. For n = 2 these formulas are the same except for a rotation. The advantages of using Lobatto formulas in the spherical product were pointed out to the author by R. A. Sack (private communication). Formulas similar to those we have discussed for the II-sphere can be obtained for any integral of the form

2

3,

N = (M - I)' + (M - 1).:"1 (M-l)·+1 - I = M-2

r

PRODUCT FORMULAS

Ch. 2

+ .. , + (M -

I)

+

That these are the correct values for N in a spherical product Lobatto formula will be proven in Sec. 2.7, where the corresponding product formulas are discussed for the surface of S•. For M > 4 it is not easy to take advantage of the fact that some of the points coincide in spherical product Lobatto forfnulas. For M = 2 and M. = 3 these formulas have been constructed by special methods. For M = 2 the formula coincides with the 211-point third-degree formula S.: 3-1. For M = 3 it coincides with the fifth-degree formula S.: 5-4 which uses 2 + I - I points. Table 2.5 compares the number of points in spherical product Gauss and 0

Here R. is any II-dimensional region with centroid at the origin which has the same rotational symmetry as S. and w(r) is any weight function which depends only on r = (xt + .,. + x;) liZ • For example, we can obtain formulas for the integrals

... r~ Irl~e-"f(xl'"'' r~ ... r~ Irl~e-I'I!(xl"'" r~

x.) dX I x.) dX I

...

dx.

P>

-n

(2.6-14)

dx.

P>

-n

(2.6-15)

• ,.

To do this we need respectively, one-dimensional formulas

r~ j rj·-I +~e-"f(r) ~r:::::: #1 A"d(r,) L~ j rj·-I +~e-lrI!(r) dr::::::

t.

A,.,!(r,)

(2.6-16) (2.6-17)

40

Part I

THEORY

The spherical product of (2.6-16) and (2.6-6.k), k = I, ... , n - I. gives a formula for (2.6-14) IInalogolis to (2.6-9). Also, the spherical pnlduct or (2.6-16) and (2.6-13.k), k = I, ... , n - 1, gives a spherical product Lobatto formula for (2.6-14). Tables of Gauss formulas (2.6-16) and (2.6-17) for n + P = 2, 3, 4 are given by Stroud and Secrest [2; Tables 7 and 8].

2.7 Surface of the n-Sphere In this section we discuss the surface of the n-sphere, that is, the region xt

+ xf + ... + x; = 1 ( 2 . 7 - 1 )

which we denote by U", We are interested in approximations

f ... f f(x u.

l , · · ••

x,,) du

~ 1=:f;I B,f(v,. I , · · · ,

v,.,,)

(2.7-2)

=

f ...u. f xj'x;' .. , x:' du is the product of 1

(2.7-3)

d(J1

and the integrals (2.6-3.2), ... , (2.6-3.n-I). The Qnly dilference between (2.7-3) and (2.6-3.1) is that (2.7-3) is over twice the range as (2.6-3.1). Since, by (2.6-7), the angles (JI.I which we use in (2.6-4.1) are equally spaced and all the A 1.1 are equal it is not difficult to see that the formula we must use for (2.7-3) is

oI.' = (2i -

2M -

2M

(2.7-4)

2M

I;I AI.,P(cos 0 1." sin (JI,,) 7t

AI

.,= M 7t

l

=

1,

41

. •...• " - I • gIves (2.7-5)

(VI,I' VI,2,"" VI,,,) (iI' i 2, .. " i,,-I) I ~ik'-::M k ,. 1",.,1/

I

=

[where the VI. I, • . • , VI,,, are defined by (2.6-10) with rio = IJ which lie only on one-half of the su~face of U", The spherical product of (2.7-4) and (2.6-6.k), k = 2, ... ,n - I, gIves the same points (2.7-5) and also their negatives. In summary we can state the following. Theorem 2.7-1. Let Yk.,. A k ., be the points and coefficients in the M-point Gauss formulas ~2.6~6.k). k = I, ... , " - I. Then a formula of degree 2M - I for U IS gIven by the 2M"-1 points and coefficients (VI,I'

AI", '" i,,-I)

VI.2,· •• , VI.,,)

i

=(ii' i

(2.7-6)

A,,_I.,.c,

2 •••• ,

and let Y. be the surface of R •. For our purpose we need to assume that R n is starlike with respect to 4>. By this we mean that each ray which begins at 4> intersects Y" in exactly one point. We assume we are given a formula of degree d for Y.:

f·;: f !(xl, .. ·,xn)du== ;t. B,!(v"., ... , v,.• )

(2.8-1)

(We do not assume that the points in this formula lie on Y., but it is more desirable if they do.) Let r be a real number> 0 and define

rY"

Then

f··· rY N

= {rv:v

E

Y.J

f xr'xi' .. · x:'du = f··· fr' - '(rx,)", ... (rx.)"·dCT = f ... f xj'x;' ... x;' dCT Y.

r"-I +11

Y.

Therefore

f 'R:' f xj' ... x;' dx

I •

dx.

= =

r(f '" f o.

rYIi

xi' '" X;'dCT)dr

Ii r,,-I+· dr I··· I xi' o

Y.

X:'dCT (2.8-2)

44

ParI I

THEORY

As an

immediate consequence of (2.8-2) we

Theorem 2.8-1.

have

Theorem 2.8-1.

Ch. 2

PRODUCT FORMULAS

Theorem 2.8-2. With the assumptions of the previous paragraph, an integration formula of degree d for R:hen is given by

Suppose we are given a formula

rrvJ

f r·-I f(r) dr z. :f; A'/(r/)

(2.8-3)

l

o

/r I

r/v J

A/BJ

;

=1, ... , M,

j = I. ... , No

ArBJ

; = 1, ... , MI

j

= 1, ... ,No

(2.8-10)

where the vJ • B J are the points and coefficients in (2.8-1).

Proof.

of degree d. Then the points and coefficients

45

(2.8-4)

=

Let IX

... f R~h.1I

IXI

+ ... + IX.

be even and

f xi'x~' ... x«· dX "

I •••

dx

< d.

Then

"

are an integration formula of degree d for R•• where the vJ' BJ are the points and coefficients in (2.8-1).

If we define an n-dimensional cone in a slightly more general way than we have in Sec. 2.5 then R. can be considered as a cone with base Y•. Then Theorem 2.5-1 and Theorem 2.8-1 are different statements of the same result. A remark to this effect is made by Hammer, Marlowe, and Stroud [I]. It is easy to modify the result of Theorem 2.8-1 to construct integration formulas for the shell R~bell = {rv:v E Y.,O < PI < r < Pl < oo} (2.8-5)

(2.8-11)

which shows that (2.8-10) is exact for all monomials for which IX is even and 0 then R:hen does not contain the origin and it is usually not desirTABLE 2.6

whenever IX = IXI -+- ... + IX. is an odd integer < d. (Both the surface of the n-sphere and the surface of the n-cube have this property.) Consider an approximation

which is exact for /(r)

1= I

=

I, ... , M

j

=

-n

(J" ul sin (J" VI) A,BI j = I, ... ,N

Proof

It is straightforward to show that

III

1, ... , No

TOR:R,

x·y'z'dx dy dz

= Iu, xmy' da IIR, u Therefore, if Ot

2.9 n-Dimensional Tori

+p + )' s

Here we discuss formulas for regions which we call n-dimensional tori, n ~ 3. For n = 3 the most familiar such region is the torus defined by O O. XO > O. The parabolic coordinates of (xo,O) are (u, 0), u = ~ for Xo 2 O. or (0, v), v = -J -xu for Xu < O. The relation between Cartesian and parabolic coordinates is

(2.1 0-7)

y= 2uv -0

< u:::;;;o

(x 2

+ yl)l/l = u1 +

(2.11-1)

vl

52

Part I

THEORY

II

xSy" dx dy PAR

=

fa J:

4(u 1

+ V1)(U

(2.11-2) 1

-

v1)S(2uv)" dv du

As a consequence of(2.11-2) we have Theorem 2.U-l. If we are given an integration formula

fa J:

~

f(u, v) du dv

t.

of degree 2d + 2 for a rectangle then

II X,

=

g(x, y) dx dy PAR

uf - vf

Y,

=

~

2ul v,

B,/(u v,) "

f.

C,g(x Y/) t:1 " C/ = 4(u;

+ vl)B,

Chapter 3

NONPRODUCT FORMULAS

is a formula of degree d for PAR. Also

JJ

PAR

4(x~~ :~)I/l dx dy ~

t.

B,g(x y/) "

is a formula of degree d + 1.

3.1 Remarks In Chapter 2 we discussed formulas that could be constructed using combinaHons of one-variable formulas. This can be done for the common regions, but for most regions it cannot be done. This chapter discusses other methods for constructing formulas. This chapter is essentially a collection of special results. For a given region R. and a given degree d one would like a method for constructing a "good" formula of degree d for R •. No such general result is known. Probably the result closest to this which is known is that of P. J. Davis described in Sec. 3.4. Although Davis's method is "constructive" it seems likely that it will not be numerically practical for formulas with a large number of points-say, more than 100 or so. For one variable the construction of "good" formulas-that is, Gauss formulas-depends very much on the relation between integration formulas and orthogonal polynomials. Relatively little is known about the corresponding theory for several variables. Radon was the first to construct integration formulas in more than one variable using the theory of orthogonal polynomials (see Sec. 3.12). Recently the author has obtained some additional results about integration formulas and orthogonal polynomials (Secs. 3.7, 3.11, and 3.13). Whether or not these results are useful for the numerical construction of formulas remains to be studied. 53

54

Part I

THEORY

J.;.. J

(3.2-1)

N

:::::: ~ B,/(vl ... • • • • vl •N )

(0.1, 0.4), (0.4. 0.4). (0.1, 0.5)}.

of degree d with N = N(n, d) = (n + d)!/(n!d!) points. Such formulas always exist. They are the analogue of one-variable Newton-Cotes formulas. In the following discussion we shall use the well-known fact expressed in Theorem 3.2-1. Theorem 3.2-1. The number of monomials ::;:; d is N(n. d) = (n + d) !/(n !d!).

xT'x~' ..• x:'

of degree

In the proof of Theorem 1.8-3 we showed how to construct points in the plane (XI' YI) i = I •...• N N = N(2. d) (3.2-2)

=

0 of degree

< d. This means

that

0 mol

VIa

=

(3.2-6)

(m o' m o, m l )

The point VI is used in forming the first column of W~, VI is used in the second column, and so on. If we count the total number of occurrences, e" of m l in all the components of all the points (3.2-6) we find e l = 9. Consider one of the points (3.2-6) in which m, occurs, say V 8 • If we replace m l by mo in the first component of V8 then V8 coincides with V" columns 7 and 8 of Wi become identical, and det Wi becomes zero. This means that det Wi has a factor (ml - mol. Ifwe replace m, by mo in the third component of V8 (instead of in the first component) then V8 coincides with VZ' Thus det Wi has a second factor (m l - mol. Since there are nine occurrences of m .. det W~ must have a factor (ml - m o)9. Similarly, det Wi has factors (m z - moll and (ml - m l )3. Therefore

where K i!! po!!sibly n function of 111 0 .111,.111 2 , Next we show that K is a constant independent of 1110' m" m 2 • Note that det W~ is a homogeneous function of mo. mI' mI' This mean!! det W~ i!! n !!lIm of 101 term!!, where each term ha!! the form '1; III~'III~'III~' and k 0 I k, I k 1 i!! the same ror all terms. This is true since each of the terms is a product of 10 elements, with one element from each row of W~. We compute the degree of this homogeneous function. Let us write out the vectors (at .. at 2 • at]) which determine the rows of W~ side by side with the V,: (at,. at 1• at]) (0, O. 0)

(m p ,. m p ,' mp,) (111 0 .111 0 .111 0 )

(1,0,0) (2, 0, 0)

(ml' m o, mol

(0, 1,0) (1,1,0) (0, 2, 0) (0,0, 1) (1,0,1) (0.1,1)

(mo' m" mol

(m2' m o, mol (3.2-8)

(ml' m l , mol (mo, m 1, mol (mo' mo. mIl (m" m o, mIl (mo' m" mIl

(0. O. 2)

(mo. mo. m 2)

Let us write the total number of occurrences of a symbol z as a component of one of the vectors (3.2-8) as occ (z). The degree of the homogeneous function det W~ is the sum of all the components of all the vectors IZ; in symbols. degree det W~

=

2 occ (2)

+ occ (I)

(3.2-9)

But the degree of the right side of (3.2-5) is 2e 2

+ e, = 2 occ (m 1 ) + occ (m,)

(3.2-10)

Since occ (i) = occ (m,). ; = O. 1,2, Eqs. (3.2-9) and (3.2-10) show that K is a constant. Next we show that K = I. The product of the diagonal elements of Wf is m~m~mr and this term does not occur again among the 10! terms in det Wf. If we expand the right side of (3.2-7) the term Km~m~mr also occurs only once. Therefore K = 1. The only gap in generalizing the_above argument to arbitrary nand d is the proof that ed -

k

=occ

(m d _ k )

= nN(n -

I. k)

k = 0, I •... , d - 1

58

Part I

THEORY

We leave this to the reader. This completes the proof. (The author has not found this theorem in the literature.)

It was pointed out to the author by R. E. Barnhill that Biermann [I] was evidently the first to show (with different notation) that ~et W~ *- O. Another similar result is by Stenger [3]. In addition to showlIlg that the determinant is nonzero, Stenger gives an algorithm by which the linear system can be solved by solving a sequence of lower-order linear systems; this is a very surprising result. 3.3 Existence of Formulas with Positive Coefficients

For a given region and weight function the results of Sec. 3.2 show the existence of a formula of degree d with N = N(n, d) points. Here we shall show that if R is bounded and w(x I' ••• ,x.) is nonnegative on R. then there exists a fo~mula of degree d with at most N(n, d) points with the property that all the points belong to R. and all the coefficients are positive. This result is due to Tchakaloff [I]. For simplicity of notation we assume n = 2. The idea of the proof is the same for all n. We shall need a theorem about convex sets of points in EN due to H. Minkowski. We begin with some notation and definitions:

=

. (i) I[>N (0, 0, ... ,0) denotes the origin in EN' (ii) If u = (U I , ••• , UN), v = (VI"'" vN) are two points of EN' then (u, v) denotes the inner product (u, v)

=

U1V 1

+ ... + UNVN

Then -v'(u, u) is the distance between u and I[>N' (iii) I•.• denotes the straight line between I[>N and u.. 0•.• denotes the angle between the lines I•.• and I.... From analytIc geometry, _

(u, v)

cos 0•.• - -v'(u, u) -v'(v, v) Definition 1. A hyperplane .Yt'N-I of EN is an (N - I)-dimensional subspace of EN' (In general, we do not assume that I[>N E .Yt'N-I') Definition 2. A set of points .1f" of EN is called convex if v I E .1f", V1 E .1f" implies ..tVI (I - ..t)v1 E .1f" for every real number 0 0.

Ch. 3

NON PRODUCT FORMULAS

59

If .1f" is a convex cone then, from the definition, I[>N r=: .1f". It is not dillicult to see that if .1f" is not all of EN then I[>N is a boundary point of .1f" and, in an intuitive sense, I[>N is the vertex of .1f". Theorem 3.3-1 (Minkowskl's Theorem). Let .]f" be n Ilollcmpty closed convex set of points in EN and assume .1f" is not all of EN' If c = (c I ' • • • , cN ) is any point of EN which is not in .1f" then there exists a hyperplane .Yt'N-I which separates c from .1f". For a proof of Minkowski's theorem see either T. Botts, "Convex Sets," Amer. Math. Monthly, v. 49, 1942, pp. 527-535 (Theorem 2.1); or F. A. Valentine, Convex Sets, McGraw-Hili, New York, 1964 (Theorem 2.9). Theorem 3.3-2. If .1f" is a closed convex cone in EN which is not all of EN and if c E EN\.1f" then there exists a point D E EN for which (c, u)

0

for all v E .1f". Proof. By Theorem 3.3-1 t~ere exists a hyperplane .Yt'N'-I which separates c from .1f". Since I[>N is the vertex of .1f" we can take .Yt'N-I so that I[>N E .Yt'N-I' Let D be a point on the same side of .Yt'N-I as.1f" with the property that the line segment I•.• is perpendicular to .Yt'N-I' If v E .1f" then 0•.• < 90°. Therefore cos 0•.• > 0 and thus (v, u) > O. Also 0•.• > 90° and thus (c, u) < O. This completes the proof.

Theorem 3.3-3. Let " linear combination

D,

vI' D

• ,

=

vp be vectors of EN and suppose u is a

..tIV I

+ ... + Apvp

where AI' •.. , ..tp are positive. If p > N then D can be written as a linear combination, with nonnegative coefficients, of at most N of the v,, Proof. If p > N then the v, are linearly dependent so there exist real numbers }' ..... , }'p' not all zero, for which }'IV.

+ ... + }'pvp = 0

Without Joss of generality we can assume at least one of the }', is positive. Then JI. = max (}'lfA,) > O. Now we have I

u

~

'"

= elV I + ... + ep vp = (P...t,JI.- }',) -> 0 . I 1= , .•• ,p

e,

and at least one of the = O. Therefore u can be written as a linear combination, with nonnegative coefficients, of PI of the Vi •••• ' vP '

60

Par! !

THEORY

PI S, P - i. if PI < N the proof is complete. If PI argument. This completes the proof.

>

N we repeat the

ell.

3

NON PRODUCT FORMULAS

First we show that for each j the sequence {bl'}' ; = 1,2, ... , is bounded. Let w be a vector for which (v,, w) > 0 for al/ v, E yr(d). Then

Let.91 be an abstract set and assume that to every s E .91 there corresponds a unique vector v, E EN' The set of all such v, we denote by "I'"(.~). Definition 4. A set of vectors yr(.9I) of EN is called ajimdamental set of vectors if it has the following properties: (i) "1'"(.91) is bounded; that is there exists a real constant L for which ""'(v" v.) < L for all v, E yr(.9I). (ii) "1'"(.91) is closed and therefore compact. (iii) "1'"(.91) contains N linearly independent vectors. (iv) There exists a vector wEEN for which (v" w» 0 for all v. E yr(.9I). Note that if "1'"(.91) is a fundamental set of vectors and if (v" w) for all v, E "1'"(.91) then inf(v" w)

>0

>0

N

(u l, w)

=

bill'"" w)

>

blJ i~f(v" w)

>

0

0 we have

I

0< b -

0, Az > O. This completes the proof.

•

we could find a sequence {VII} of vectors of "I'"(~) for which

w) = 0

Since "1'"(.91) is compact, the sequence {v,.} has a convergent subsequence which converges to a vector VI t E "1'"(.91). It would follow that (v.. , w) = 0, which i's a contradiction. Definition 5. Given a fundamental set of vectors "I'"(.c1') of EN we define .Jf'N(~) to be the set of vectors U E EN which are a linear combination, with nonnegative coefficients, of at most N vectors of "1'"(.91). In other words U E .Jf'N('~) if and only if there exist VII E "I'"(~) and real numbers h, ~ 0, ; = I, ... , N, so that u=blv..

>

This implies that

inf(v" w) = 0

(VII'

f:1 bill',,,, w)

>

Since inf(v., w)

This is true for the following reason. If

lim 1--

61

+ ... +bNv'N

+

Definition 6. A vector C E EN is said to be positive with respect to the fundamental set "I'"(d) if for each wEEN which satisfies (v" w) > 0 for all v, E "1'"(.91) then it is also true that (c, w) ~ O. Theorem 3.3-5. If c is positive with respect to the fundamental set "1'"(.91) then c E .Jf'N(.9I). .

Proof. Let c

E EN\.Jf·N(~)'

By Theorem 3.3-2 there exists a vector

wEEN for which (c, w)


0, j = I, ... ,N. In Fig. 3-1 we take k = 3 so that each GL, has degree 5. The points in GL. are illustrated. 4. We choose d > 0 so that d

< min (Br, ... , B:> -

IIXN'IIM '

(3.4-1)

where M

=

max sup le,(x, y)1

I $.15,N

/I,

and if D is an N x N matrix with elements d then II D II is the matrix norm

"

66

Ch.3

Part I

THEORY

11011 = m,ax ~ Idlll

maxlA,1 ~ ,

0 on R z implies that (3.5-5) is positive for all Thus I z•z is positive-definite. To prove the existence and uniqueness of the basic orthogonal polynomials (3.5-4) it is not necessary to assume w(x p ••• , x.) ~ 0 on R •. It is sufficient to assume that I •.d_1 is nonsingular. For one variable it is well known that if P..(x) is orthogonal on an interval with respect to a nonnegative weight function to all Q.. _I(X) then all the IX.

70

Part I

THEORY

r

zeros of P". are real and distinct and lie inside the interval. We shall prove a weak analogue of this result for two variables. Let P".(x,y) be orthogonal to all Q",_I(X,y) for a given connected R z and nonnegative w(x,y). In Sec. 1.8 we pointed out that P'" has a unique factorization in the field of complex numbers. Pm also has a unique factorization in the field of real numbers; we write this as

= P!,(x, y)P!,(x, y) + ... + m" m, > I

P",(x, y)

m

= ml

... P!' + (n + 4)(n t + 6)

1 XlXl

n:t6

(3.5-7)

But P! • ... P!. changes sign on Rl at exactly the same places as does P".. Therefore Q1",- I does not change sign on R 1 • This contradicts (3.5-7). This completes the proof. 3.6 Some Particular Orthogonal Polynomials

We give here some orthogonal polynomials for the six regions C.' S., G., T., E~', E~. In addition to the basic orthogonal polynomials-defined in Sec. 3.5-there are also orthogonal polynomials known for S., T.,and E~t

If m l ,

••• ,

m. are nonnegative integers,

U '"h · .. ,In_ --:- K

a

am '"I

XI

•••

+ x + .,. + X Z I)" + ... + m. and is orthogonal to all poly~

(XlI (j X.' m

2

Z

" -

is a polynomial of degree m = m I nomials of lower degree over S•. K is a constant which we are free to choose to normalize the polynomial. These polynomials are discussed by Appell and Kampe de Feriet [I; pp. 257-281,328-330]. Tabie 3.4 gives these polynomials for n = 2 and m < 4. We have chosen the constants so that the coefficients are small integers.

.111 I

/4..

TABLE 3.4 Orthogonal Polynomials Kamp~ de F~riBt for S1

V O• O = 1 VI.O = X V 1.0 = 3x1

of

App~!!

Ch. :3

NON PRODUCT FORMULAS·

1m!!

TABLE 3.6 Basic Orthogonal Polynomials for Tn

pO.O ..... O = 1

+ y2

pl.O ..... O

- 1

VI.I == xy V3.0 = SX3 Vz I = 3x1y V.:o = 3Sx. V3.1 = Sx3y V2.l = Sx.

+ 3xyl - 3x + y3 - Y + 30xly2 + 3y. - 30Xl - 6y2 + 3 + 3xy3 - 3xy + 18x1y2 + Sy. - 6X2 - 6y1 + 1

= XI

1 nn

-

pl.O.O ..... O = xf -.

4 2 XI + (n + 2Hn + 3; ii+1

I XIXl - --(XI n 3

pl.I.O ..... O =

+

9

xl- iiTIxf

pl.O.O ..... O =

+ Xl) +

I

(n

+ 2)(n + 3)

18 + (iiT-4Hn + S)XI

I x i - ~XIXl 4 pl.I.O ..... O = xixl - - n+S n+;l

3. The n-octahedron G~. In Table 3.5 we give the basic orthogonal polynomials of degree ::;; 4. These polynomials have the same symmetry as those for the n-sphere.

2

pO.O •...• O = 1 =XI

pl.I.O ..... o =

XIXl

pl.O.O ..... o =

xl - (n

p2.I.O •...• o =

XiX2 -

pl.I.I.O ..... o

12 + 3)(n + 4)xl 2 (n + 3)(n + 4)X2

=XIXlXl

1 + 168 S)(n + 6)xl 192n(xt + xf + .. , + Xl) + 24(Sn2 + 2Sn + 18) S(n + S)(n + 6)(n 1 + Sn +-10) (n + 3)(n + 4)(n + S)(n + 6)(n 1 + . 12 pl.I.O ..... o = xlx2 - (n + S)(n + 6)X IX 2

+ S)~n +6)(xi + xi> 16(xt + xf + ... + x~) + (n + S)(n + 6)(n2 + Sn + 10) 4(nl - 3n - 30) + (n + 3)(n + 4)(n + S)(n + 6)(n 1 + Sn + 2 P2.1.I.O ..... o = xtX2Xl - (n + S)(n + 6)XlX3

Sn

+ 10)

n+S

10)

m=m l

••• x:"(l -

+ ... +m.

XI -

+

y - 1 6x l + 6xy + y2 ~ 6x - 2y + 1 3x 2 + 8xy + 3yl - 4x - 4y + 1 V,. O = 20X' + 30x2y + l2xy2 + yl - 30x2 - 24xy .- 3y2 + Ilx 3y - I Vl . 1 = 10x' + 36x2y + 27xyl + 4yl - 18xl - 36xy - 9yl 9x 6y - 1 V4.0 = 70x· + 140xly + 90x l yl + 20xy' y4 - 140xl _ 180x1y 90x 1 60xy 6y2 - 20x - 4y I - 60xyl - 4yl Vl. 1 = 3Sx· + l60x ly + 180x 2yl 64xyl + Sy4 - 80X' - 240x2y 6Ox 2 96xy 18y2 - 16x - 8y 1 - 144xy2 - 16yl U2• l = lSx4 120x'y 216x 1y l 120xy' lSy4 - 40x' _ 216x2y - 216xyl - 40yl 36x2 108xy 36y2 - 12x - 12y I

+

+

+

XIXlX,X4

= K ax"" ... am I1x"'· [XT'X~'

I 2x

+

+

4. The n-simplex Tn' In Table 3.6 we give the basic orthogonal polynomials of degree::;; 3. If m .. ... , m~ are nonnegative integers, U ,"., ...,...,.

V O• O = VI.O = V 2.0 = UI • I =

+

pl.l.O ..... o = xixf - (n

=

2

TABLE 3.7 Orthogonal Polynomials of AppBII Bnd Kampe dB Feriet for T1

p •• o.o •...• o = xf - S(n

pl.I.I.I.O ..... o

XI

is a polynomial of degree m and is orthogonal to all Q"'_I' K is a constant we are free to choose. These polynomials are discussed by Appell and Kampe de Feriet [l; pp. 99-113). Table 3.7 gives these polynomials for n = 2, m ::;; 4.

2 + 1)(n + 2)

- (n

4 (n+4)(n+S)

. 1 1 + (n + 4)(n + S)(x l + Xl + Xl) - (n + 3)(n + 4)(n + 5)

pl.O •...• O

= xi

+

+6 4)(n + S;

- (ii+3j(n

+~+~~+~~-~+~~+~~+~ pl.I.I.O ..... o =XIX2Xl - _I_(xlxl + XIXl +X2 Xl)

TABLE 3.5 Basic Orthogonal Polynomials for G.

pl.O.O •...• o

73

••• _

x~)m]

+

+ +

+ + + +

+

+

+

+

+

+ +

+

5. E~'-entire n-space with weight function exp ( - xf - ... - x~). The basic orthogonal polynomials are products of Hermite polynomials. That is, . p« .. «...... «.

=

H«,(xl)H«.(x 1 )

•••

where H",(u) is the orthogonal polynomial for exp(-u 1):

H •• (x~)

-00

~u

(3.6-1)


xz + 3(n + 3)(n + S)(2n + 11) Xz 2n + 7 p3.2.0 ..... 0 = xlxi - (n + 7)x1 - 3(n + 7)xlxi + • • • + x.Z)XI + 3(n + 3)(n2n++S)(2n + 11h l . + 2n12+ 7 (Z XI 7 pl.I.I.O..... o = xlxzx3 - 3(n + 7)XIX2Xl p2.2.1.0 ..... 0 = xixix3 - (n + 7)(xi + Xi)Xl + .. , + x.Z)X3 + (n + 3)(n2n++S)(2n + 11) X3 + 2n 4+ 7(Z XI 7 p4.1.0 ..... 0

= XtX2

- 6(n

pZ.I.I.I.O..... o = XfXZX1X4 - (n 1.1.1.1.0..... 0 = XIXZX3X4X,

+ 7)XZX3X4

= (x -

v.)(x - v z) ••• (x - vN)

Then N constants A., ... , AN can be found to make the formula

s:

w(x)/(x) dx

= ~ A,/(v,)

(3.7-1)

exact for all polynomials of degree S N + k (0 < k < N - 1) if and only if PN(x) is orthogonal to all polynomials of degree < k. A proof of this theorem (stated slightly differently) for the case k = N - I is given by Krylov [1; pp. 101-102]. The proof for arbitrary k, 0 < k < N - I is analogous. If k = N - I then the VI' A, in (3.7-1) are uniquely determined and the condition w(x) > 0 on [a, b] implies that all the v, are real and lie inside [a, b) and that all the AI are positive. In the remainder of this section we discuss the generalization of Theorem 3.7-1 to formulas (3.2-1). Given a set of orthogonal polynomials and their common zeros it is possible to determine if these zeros (or a subset of them) can be used as the points in an integration formula. This can be done by examining only the polynomials and their zeros. This result is Theorem 3.7-2. Conversely, given a formula of degree d with N < N(n, d) points it is possible to find a set of polynomials with certain orthogonality properties which have the points in the formula as common zeros. This result is Theorem 3.7-3. For simplicity of notation these theorems are stated for n = 2; they have a direct generalization to arbitrary n. We use the following notation:

16

Part I

THEORY

=

(i) M N(2, d) == (d + I)(d + 2)/2. (ii) P m.I(X, y) is a polynomial of degree exactly m. (iii) Qk(X, y), Qtf(x, y), and so on, are polynomials of degree


0, then there exist I linearly inde-

Pm.,(x,y)

s

=

1, ... ,1

(3.7-7)

, which satisfy the following: (I) each Pm,I is orthogonal to all Q"-m' and '(2) Pm,t(XI,y,) = .,' = Pm,I(XI,y,) = 0, i = I" .. , N, Proof. The fact that the polynomials (3.7-7) exist, are linearly independent, and have the (x" y,) as common zeros follows from the fact that I rows of the matrix (3.7-6) are linear combinations of the other rows. Since we are given that the formula has degree d it follows that

ff

R,

w(x, y)Pm,,(x, y)Q"-m(x, y) dx dy N

= ~ B,Pm"(x,, yl)Q"-m(x

"

y,)

=0

s

= I, ... , I

This proves orthogonality of each Pm,,' This completes the proof. We illustrate Theorems 3.7-2 and 3.7-3 with some examples.

Proof. Let LN be chosen to satisfy condition A4. We compute the B, in (3,7-4) by solving the linear system - X7'y~'

i

Q~,/J.

The following is a partial converse.

Theorem 3,7-2 (Stroud [12]), If Pm, .. ... , P m,I satisfy conditions A I, A2, A3, A4 above, then there exist constants B" i = I, ... , N so that

=

= 0,

Q~,/J(X" y,)

Therefore formu!a (3.7-4) is exact for formula (3.7-4) IS exact for LN

LN with the property that

+ ... + QtfPm,,= x·yll + I;b.-,,,.x·'yP'

w(x, y)Q,,(x, y) dx dy

= 0

This implies that (3.7-4) is also exact for x·y/J. This completes the proof.

about which we assume the following: AI. The polynomials (3.7-2) are linearly independent and each is orthogonal to all Qk(X, y). A2. Pm,t(XI,y,) = ... = Pm,I(XI,y,) = 0, where i = I, ... , N. A3. The matrix XM,N has rank N. In other words, for some set of indices LN the matrix XM,N is nonsingular. A4. For some LN for which XM,N is nonsingular the following is true. For each (tX, P) E L1\LN there exist polynomials i= 1, ... ,1 m:f(x, y) E

I(Q~,/J)

and

LN

We assume we are given a set of polynomials P m,t(x, y), .•. , P m,AX, y) 1> 2

and real numbers b.-,ll" (tX*, P*)

77

There remains to show that (3.7-4) is exact for each x·yP, (tX, P) E L1\LN. Let Q~,/J, (IX, P) E L1\LN denote the polynomial (3.7-3). From the left side of (3.7-3) it is clear that

L1

(ix) LN is a subset of L1 consisting of N elements. (x) XN,N is the N x N matrix whose rows are (x~yf, ... , x~y~)

NON PRODUCT FORMULAS

Ch. 3

rB (1

'··l(x"'yfl,) ...

(3.7-5)

EXAMPLE I. Let Rz be the square C z and w(x, y) = 1. Let the formula be the seven-point fifth-degree formula of Radon [I] l{iven as C z: 5-1 in Chapter 8:

Points . (0,0) (0, ±t) (±r. ±s)

.. _ ,rr

Coellicients 8 'T

2.

W"!J

~

_ ,It

!14

(3.7-8)

78

Part I

THEORY

Then M = 21, N = 7 and it can be verified that the matrix X7 • 7 whose rows are (xryf, .•• , x~y~) (tX,

p)

E

is nonsingular. By an application of Theorem 3.7-3 it can be shown that the seven points in this formula are the common zeros of the polynomials P, .• = x' -Ix P , . l = xyl - ~x (3.7-10) My

and each of these polynomials is orthogonal to all Ql' We can also illustrate Theorem 3.7-2 with this example. Assume we were somehow given the polynomials (3.7-10) which we can verify as being orthogonal to all Ql. We then determine that the common zeros of these polynomials are the points (3.7-8). Then we can compute the coefficients 8 ••.. , • 0 as the solution of a linear systcm with matrix (3.7-9). To vcriry that this gives a 7 fifth-degree formula we must show that these (Xit YI), 0 1are exact for all other monomials of degree ~ 5 which are not in the linear system. This means we must verify condition A4 for the monomials Xl,Xly,xyl,x4,Xly,xlyl,xyl,y·,X',X4y,Xlyl,Xlyl,xy4,y'

The following equations are this verification: P, .• = x' - Ix P , ., =xly +yl P , . l = xyl - ~x XP , • I yP, .• XP, . l yP , • l

Hy

= x· - IX2

= xly - Ixy = Xly2 - !Xl

= xyl - ~xy yP 3• l - XP 3• l = y. - Hy2 + ~x1 (Xl + VP3 •• = x' - 'J'rx y1P 3•• + IP 3. 3 = X'yl - ~x XyP 3•l + !P3. 3 = X2 y 3 + !y3 - Hy

(y2 + !)P 3.1 = xy. - ~x (y2 - ~)P3.3 - XyP 3. 1 = y' - Hyl + Hy XyP 3• 1 + iP 3. 1 = x4y + Iyl - My

EXAMPLE 2. Let R2 be the square C l and w(x, y) = 1. Let the formula be the 12-point seventh-degree formula C l : 7-2. By an application of Theorem 3.7-3 it can be shown that the points in this formula are the common zeros of the polynomials

p •. 1 = x. + Hxly1 p .... = x')' - ~x)'

+ y.

-Ix l -ly2

p •. 3

=

LN

NON PRODUCT FORMULAS

= ((O,O), (1,0), (0,

79

J), (2,0), (I, I), (0,2),

(3,0), (2, I), (1, 2), (0, 3), (4,0), (0,4)]

(3.7-9)

((O,O), (1,0), (0, I), (2,0), (I, I), (0, 2), (0, 3)}

P,., = xly + yl -

Ch. 3

+H

xy3 ~ Ix)'

These polynomials satisfy assumptions AI, A2, A3, A4 of Theorem 3.7-2 provided we take

'In Theorem 3.7-2 we assume that 'all the polynomials have the same degree m. This theorem can be modified so that the polynomials involved are not restricted to having the same degree. We illustrate this with the following example. EXAMPLE 3. Let Rl be the square C l and w(x, y) = I. Let the formula be the eight-point fifth-degree formula C 2: 5-3 . .It can be verified that the points in this formula are the common zeros of the three polynomials

= x' -

P, .• Pl •l

.. ,

p •.•

= x2yl

:xyl - twx y.1 - :Xly _ .T~y - ;Xl - ;y1

(3.7-11)

+ '/1

P3,I and P , .l are orthogonal to all Ql and P 4 •• is orthogonal to all Q •. The assumption of linear independence in A I is interpreted to mean that there are no polynomials Q .... Q •. l for which Q •.• P 3.• + Q •. lP , .l + P C• I which is true for the polynomials (3.7-11). Taking

== 0

Ls = {CO, 0), (1,0), (0, I), (2,0), (1, I), (0,2), (2, 1), (1, 2)}

we can verify that Xs. s is nonsingular. Also, the polynomials (3.7-11) satisfy A4 (with 1= 3) if in place of Pm ... P m. 1, P m. l we take P,. It P 3. 1, p •.•. The results of this section can be considered as part of a more complete theory which was more recently obtained by Stroud [17J. The more complete theory is given in Sec. 3.13. 3.8 Second-Degree Formulas

Let Rn and w(x •• ...• xn) be given. For a given degree d we would like to find the minimum number of points necessary for a formula of degree d. For d = 1 this problem is trivial. The formula with the single point (V I • I •

v •. l'

_ lex,,) -

/(1)

••• ,

k

vl • n)

=

(3.8-1)

1•...• n

with B I = 1(1) is the unique formula of first degree with one point. The point (3.8-1) is the centroid of Rn with respect to the weight function w(x., .••• xn)'

The first nontriviai case is d = 2. With an assumption of nondegeneracy of Rn and w(x •• ...• x.). which we state below. we can show that N = n 1 is the minimum number of points. We shall not assume in this section that

+

80

Part I

THEORY

w(x •• ... ,x.) ~ O. We follow the discussion of Stroud (2). Less general

results about these formulas were obtained by Georgiev (I I, [2), l3). l4), l5). [6) and by Thacher [I). In this section we shall use the notation 100 = 1(1) 1'0 = I(x,) II} = I(xlx}) i.j = I, ...• " (3.8-2) We assume only that R. and w(x .. ... , x.) have the property that the integrals (3.8-2) exist and are finite; that 100 > 0; and that the matrix -/

I •.•

=

110

1. 0 I ••

/ 10

III

00

1.0

I..

I zo

NON PRODUCT FORMULAS

Ch. 3

Given 1•. 1 we shall show how to find U and B. First we show how to lind a matrix T so that det'" = diagonal. For simplicity of notation we do this for n = 3. 0 we set Since 100 -I

100

and form

-[~

T,

1.2

1*

I ••

is nonsingular. 1•. 1 is a special case of the matrix 1•. 4 -. considered in Sec. 3.5. There it was shown that if w(x I' • • • • x.) > 0 on R •• then 1•. 1 is positivedefinite and hence nonsingular. If n + 1 points and coefficients V, = (V,.I' ... , v, .•) B, i = I, .. . , n + I (3.8-3)

I = 1,2,3

T,o=~

I••

/.1

and '('11.,.'1' is

"*

1.0

Iz. In

81

= T'I

=

T

1 3.1

liT

0 0

/ 00

0

0

n

TlO

T'0

0

0

J 0

0

o .

0

I~z It. It. I~·i IW It. IW IW . .

0

I [

IfIrl = 0 we must have some we form

TIO

o

"* 0 since det 1',1 "* O. Assuming It. "* 0

are to be a formula of degree 2 then they must be exact for the monomial integrals (3.8-2). These points and coefficients must then satisfy the system of nonlinear equations

+ ... + B.+.

BI

+Bl

Blvl.,v l .}

+ B"vl.,v".} + .,. + B.+lv.+I,IV.+I.} = I,}

o -

IW IW IW . .

i,j= 1, ... ,n

These equations can be written in matrix form as (3.8-4)

tJTBU = I•.• where 1 VI . I 1 v" .•

IW

VI .•

v" .• .

u=

I~I

"*

o

0

o

B=

+ 2hlt. + h"/~~) + IrIW i = 2,3

n "*

and choose h so that l\'tl O. If 0 we take h = 0 so that 11'.1 = Similarly we can find matrices T" T., T, so that 0 0 0 00 0 0 IW [/

1121

v.+ I .• BI 0

Bl

=

lIP = liT

= T~T~I(I)T3T. = ~

O. 0

f/OO

T'I(l'T , , 0

0

B .....

0

= 00 0

n·

I~"i

IW J IW IW.

0

0

0

IW

0

0 0

I~"i

0 0

0

IW

(3.8-5)

82

Part I

THEORY

where liZ;

NONpkUDUC i l-ukMULAS

Ch.3

* 0, Ig> * O. Defining

If we expand the right side of (3.8-9) we obtain lo,(loloo -I- "/10 -I- '2/Z0 1,,(lollO til .. 'l,('ollo 11}/21 ·1 , ',,(10/'0 -I- '1}1'I -I-

T=TITzT,T.T,

+ +

we have det T = 1 and

= TTI,.IT =

(UT)T8(UT)

D

where D is the diagonal matrix (3.8-5). Now we write

Z=UT~I:

VI •I

V I •2

V 1•1

V 1.1

V'.I

V'.l

V"I

V•. I {I.I

V •• 1

V •• 3

{I.l

{I.'~

1

{1.1

el.1

el.,

I

{3.1

e 3.2

{3.3

'1

-

83

V 3• 3

' 'I In

'll

113 lu

In

(3.8-6)

..

T: (XI' Xl' Xl)

-+

'2l32 ·1 · "/33)

*

From Theorem 3.8-] it follows that the points and coefficients v,, B" 1,2,3,4 will be a formula of degree 2 for R3 and w(x" Xl' X 3) if and only if the points and coefficients ~, ~ Tv B,• ; = I, 2, 3,4 are a formula " the ~,' the v, are determined by of degree 2 for TR, and w"'(z., Zl' %,). Given

;=

v,

1 { •. I { •• 1 { ,We can consider (3.8-6) as defining an affine transformation

-

(3.8-10)

+

+

'11

0 I .. 0 111 0 131

'Z.3

+ t2lz1 + ' 3l 3l ) + Izlu + 13l32)

By inspection we see that (3.8-10) is the element in the (i I)st row, (j ])st column of the matrix (3.8-5) and hence equals-by definitionzero. for i j. and lIP, for; = j. Q.E.D.

101

101

+ +

-I- ' 3/'0)

=

T-I/;,

We discuss how to find the /;" B,. With Z and D as defined above we have ZTBZ= D

(ZI' Zl' Z3)

(3.8-Il)

Since Z and D are both nonsingular we easily derive from (3.8-11)

where ZI

=

+'

101 +llrl 1IXZ j = 1,2,3

+ 131X,

But (3.8-I2) is equivalent to the system of equations

Theorem 3.8-1. Under the affine transformation T the region R, transformed into a region TR 3 , which has the properties

III

TRj

w*(z" Zz, Z3)Z,ZI dZ I dz z dZ 3 = 0

i

J

IIITA" W*(ZI' Zz, z,)z1 dZ I dz z dZ where we have set

Zo

J

=

Proof We give the proof for; of a multiple integral

*j

IIL.

I ~z

/W"'" I

+

I ~1

/W .. ,.l

'+

I ~2

lW"'"

]

I

;,j=],2,3

;*j

B,

(3.8-13)

+ /We, .•el.• + /We,.lel.2 + /We,.Jel.J = 0

Before we discuss the solution of (3.8-13) we note the following theorem.

=w(x"

;z,j;z, 1.

] +

loa

I loa (3.8-8)

= IW

Xl' x 3}

By the rule for transformation

IIITR.'W*(Z" Zz, Z3)Z, ZI dZ I dZl dZ 3 =

IS

i,I= 0, 1,2,3 == ], Ib'l1 loa' and W*(ZI' Z2' z,}

(3.8-]2)

(3.8-7)

(3.8-9)

W(X I, Xl' X 3) fi,./IJI dX I dx z dX3

Theorem 3.8-2. Let the number of 100 , II"', n~), IW which are positive be p, and the number which are negative be q, with p -I- q = 4. If all the /;" B" i = 1,2, 3,4 are to be real then p of the B,' must be positive and q of the B, must be negative. Proof. This theorem is a restatement of Sylvester's "law of inertia" for quadratic forms. (See, for example, F. R. Gantmacher, The Theory of Matrices, I, Chelsea, New York, 1959, pp. 296-298.) Q.E.D.

where

IJI = det~ = ox, fi ,.1 = (to,

det T

=

I

+ 'I/XI + ' lI Xl + ' 3,X3)(tOI + ,,~XI + I11X + IU X3) Z

A particular solution of (3.8-13) is given in Table 3.9. This solution is uniquely determined by the B,. The only restrictions which must be placed on the B, are

•

•

•

•

•

•

•

•

•

•

Ch. 3

NON PRODUCT FORMULAS

BI

+ B2 -I- B, + B. =

sign B,

= sign lW

85

100

i = 1,2,3

B. >0 To find additional solutions we make use of the following theorem. Theorem 3.8-3. (I) If ~/' B i " W*(ZI' Z1' z,), and

=

1,2,3,4 is any formula of degree 2 for

TR, and

.....

8

.. ;;

=

.

N

.::!. • "it

[~o

S:I S11

Su S1J

s~,]

o

S'I

Su Su

S:1

is any matrix which satisfies

SD- 18 T = D-I

(3.8-14)

then the points and coefficients ~

= (,:, I' ':.1' ,;,,)

,;,} = Slt'.1

B,

+ S1t,.1 + s,t,.,

(3.8-15)

are also a formula of degree 2 for TR" W*(ZI' Z1' z,). (2) If Q is an arbitrary 4 x 4 skew matrix (a real matrix which satisfies QT = -Q) with first row and column entirely zero such that neither D-I Q or D-I - Q is singular then

+

8

=

(D-I

+ QXD-I -

Q)-I

(3.8-16)

satisfies (3.8-14). (3) If for all Q, E S is nonsingular (where E is the 4 x 4 identity matrix) then all matrices S which satisfy (3.8-14) have the form of (3.8-16).

+

Proof Parts (2) and (3) are a special case of a result given by C. C. MacDuffee, The Theory of Matrices, Chelsea, New York, 1946, pp. 65-66. For part (I), if ZO-IZT = B-1 and SO- 1ST = 0 - 1 then (ZS)D-I(ZS)T = B-1. Therefore the the formula stated. Q.E.D.

--- --- --- --"

I

I

I

"

n

H

.defined by (3.8-15) gives

We comment on how the above results generalize to n dimensions. In general 1•. 1 will transform to a diagonal matrix D with elements

Ib'll, 1\'.', ... , I~)

i

84

~~

t.,

(3.8-17)

If p of the lit> are positive and q are negative, p + q = n + I, then to obtain a formula with all ~/' B, real, p of the B, must be positive and q must be negative. The only other restriction on the B, is that their sum be 1~'ll = 100 , If we choose the B, so that '

Part I

THEORY

86

sign H, = sign liP

i

=

NON PRODUCT FORMULAS

I, ... , n

>0

Bft+l

Ch.3

then a formula generalizing that given in Table 3.9 can be obtained. It is not difficult to write out the points in this formula by inspection of Table 3.9 and to verify that these points satisfy the general form of Eq. (3.8-13).

=

100

The n +

points

+

"*

Now we consider some specific regions R •. In all these cases we take w(XI' •.. ,x.) = 1. Figure 3-2 illustrates three formulas for the square C1 • Let R. be the n-cube C•. Then 1•. 1 is already diagonal so in the above discussion T is the identity and Points

•

Coefficients

r

00

+ r(V',lvJ.1 +- ... + v,.•vJ,.) = 0 II i, j

=

I, . .. , n

i

"* j

(3.8-18)

.

Let dist(v/• 0) denote the distance between V, and the origin and dist(v / vJ) the distance between v, and vr Then from (3.8-18), dist (v" 0) dist(v/ vJ)

= (vl,. + ... + vt.)l/l ~

[II I(~::B;' BI}J/l

= [(Vl,1 - VJ.I)l + ... + (v, .• = [III(B , + B'>Jl/l

VJ .• )l]l/l

(3.8-19)

B,B,

If we take the coefficients all equal then the formulas can be described as follows.

(-.)1. A) t

(-A.-A)

Proof. Taking B,

j

= ... =

B.+I

1, 1)

~,

-!)

-f,

~)

(-~,

0)

(1, -1)

,.j2(n

+ I),

t

Eq. (3.8-19) becomes

I)

(3.8-20)

Jf

If n = 2 or 3 then each simplex which satisfies (3.8-20) has all its vertices inside C•. For n >- 4 this is no longer true. For all n > 2 there is an n-simplex (3.8-20) which does have its vertices inside Cft' In fact, points V k with coordinates

3

2" ;?7

14

I

j)

=

2'/(n

These distances describe the vertices of the n-simplex as stated. Q.E.D.

4

."

V k • 1m - 1

(1,

=

=

dist(v/ 0)

•

; = 1, ... ,n

.!. + .!.(Vtl + ... + vl,.) =1.. 100 1'1 . B,

dist (v,, v,)

•

2· = '3

Theorem 3.8-5. A necessary and sufficient condition that the points v,, i = I, ... , n + I, are the points in a formula of degree 2 with equal coefficients for C. is that they are vertices of a regular n-simplex and lie on the surface of an n-sphere of radius Jn13 with center at the origin.

(A.O)

•

III

must satisfy

V,

Theorem 3.8-4. If 1ft .• is nonsingular then a formula of degree 2 for R. and w(x., ... , x.) cannot be obtained with fewer than n + I points. Proof. If such a formula with N < n + 1 points exists then the points and coefficients satisfy a matrix equation similar to (3.8-4), where U is an N by n + 1 matrix and B is N x N. Then UTBU is still an n + I by n 1 matrix and its rank is at most N. Therefore det (UTBU) = O. Since det I •.• 0 this is impossible. Q.E.D.

liP =

2·

87

~

IT 2mb 'V '3 cos n + 1

m =·1,2, ...

3

."

v

3

k,.

Fig. 3-2 Three second-degree threepoint formulas for the square with vertices (±J, ±J).

[ ; ] =

,[~J.

_ (_1)k

--:73

k

=

1, ... ,n + 1 (3.8-21)

if n is odd

the greatest integer
2 for most regions it is not known how to construct any third-degree formulas other than those described in Sec. 3.2. Also, the minimal number of points in a third-degree formula is not known in general for n > 2. (For n = 2, N = 4 is the minimal number; construction of such formulas is discussed in Sec. 3.11.) For a certain class of regions we can construct third-degree formulas with 2n points by a method similar to that used for second-degree formulas in Sec. 3.8. These formulas will be described here following the discussion of Stroud [2]. For the class of regions considered, Mysovskih [5] has shown that this method gives all third-degree formulas with 2n points and that 211 is the minimum number. We shall not prove Mysovskih's result. We assume that the region and weight function have the following properties (or can be transformed into a region with these properties): (i) I(x,) = I(x,x,x k ) = 0, where i,j, k = 1, ... , n. (ii) 1(1) 100 is finite.

=

where 1/,

= I(x,x,),

1.2

is nonsingular.

(3.9-2)

= I,. 00, n

UTBU

= il:'1

(3.9-3)

where I I

V I•2

VI .•

'V IVI •• I

V I •2

v2 ••

v•. 2

v,

U=

...

Iv

-B I

0

0

B2

0

10

o o

B= ,1

B.

By a method similar to that described in Sec. 3.8 we can find an n X n matrix T, with det T = I, so that ZTBZ = iTTI:'IT = where Z

~I'I

(3.9-1)

Equations (3.9-2) can be written

(iii) The n X n matrix of second-degree monomial integrals

1*".1-

89

=

UT and

iD

~(3.9-4)

90

ParI I

THEORY

el.1 el.l ez.1 ez.z

z=

NON PRODUCT FORMULAS

Ch. 3

with the only restrictions on the B, being (3.9-1) and sign B, = sign

III'

i = 1, ... ,n

We can obtain additional formulas in a manner analogous to Theorem 3.8-3. As an immediate consequence of (3.9-5) we have Theorem 3.9-2.

en.1

D~lf Lo

IW

en.n_ 0 0

0

.l~~}J

en.l 0

Theorem 3.9-2. If I:' I is positive-definite then /;" B" i = I, ... , " will be a solution of (3.9-5) if and only if the following hold:

'I

(i) The /;, are mutually orthogonal vectors. (ii) ~, is a distance .../lW/2B, from the origin. (iii) All B, are positive and their sum is 100/2 .

I

We consider some examples. Figure 3-3 gives three formulas for C z. Points

Then T defines an affine transformation

T: (XI' X z••..• xn) where

ZJ

=

tlJx l

--+

(z I' zz ••••• z.)

+ ... + tnJxn

j

=

Coefficients

•

•

(A, A) (A,-A)

•

•

(-,J1,,fi) (-,fi,-A)

1•...• n

which transforms R., w(x l , • • • • x.) onto TR n , W*(ZI' ... ,z.). From (3.9-4) we obtain ZD-IZ = iB-I, which is equivalent to 1

91

~

):z

m,"'·1 (3.9-5)

If the ~,. B, satisfy (3.9-5) then ±~" B" i = 1, ... ,n will be a third-degree formula for TR n, W*(ZI' •••• zn)' • Applying Sylvester's law of inertia for quadratic forms we obtain Theorem 3.9-1. Theorem 3.9-1. Let the number of Ill', i = I, ... , n, which are positive be p, and the number which are negative be q, with p + q = n. If all the ~" B, are to be real then p of the B, must be positive and q of them must be negative. A particular solution of (3.9-5) is

~ =( I

/!li..

0, ... , 0)

~2 =(o,~, ... ,o)

~n

=

~ (0, 0, . .. , 1/(ifj)

Fig. 3-3 Three third-degree fourpoint formulas for the square with vertices (± I, ± I).

"3

(-1,0)

!

•

2

4

(o,A)

"3

(O,-A)

"3

(1,

•

2

( 1, 0)

4

1

1)

J 1

(-1,-1)

J

(-A, -A)

!

( -A,-A)

5

"3

For the n-sphere S., if we choose all the coefficients equal the points will be any It mutually orthogonal vectors which lie on the surface of an n-sphere of radius ,.)n/(n + 2). One such formula is S.: 3-1. For the n-cube C.' if we choose all the coefficients equal the points will be any n mutually orthogonal vectors which lie on the surface of an n-sphere of radius .../1i{3. For all n > 2 there ~re such points, all of which lie inside C•. One such formula is Cn : 3-1. '

92

THEORY

Par' I

Ch. 3

NONPRODUCT FORMULAS

3.10 Fifth-Degree Formulas with nl + n + 2

- I(xt)

Points

In Secs. 3.8 and 3.9 we have used the theory of matrices to construct formulas of degrees 2 and 3. This method is not readily extended to construct formulas of higher degree. However, Stroud [8] has used such a method to construct a few fifth-degree formulas with n 1 n 2 points for the n-cube and other regions with similar symmetries. We describe these formulas here. The formulas which result are C.: 5-1, S.: 5-1, G.: 5-1, and E:': 5-1. For each of these regions formulas are obtained only for a few low values of n because it is only for these n that the formulas have all their points and coefficients real. If complex points and coefficients were permitted then these formulas would exist for most n. As we shall see it is not likely that such formulas exist, with real points and coelHcients, for any region for" large. We assume the region and weight function are such that

II

= I(x,xjx k) = I(x,xjxkx,xm ) = 0 i,j, k, I, m = I, ... , n

l(xtxD

l(xD

l(xix;)

I(xtx;)

I(xix;)

l(x.xD

=

f

1'/(X IX 2X;)

=I

or (3.10-2)

vt. vt.

r

I(xl)

l(xix,,_1 x.)

l(xD

I(xlx,x;)

l(xfxD

1(xfx 2 x,)

1(x.x1x._lx.)

1(x~x2x,)

l(x~xD

l(x.x,x._lx.)

l(x l x 2 x._.x.)

l(x.x,x._lx.)

l(x.x 2 )

I(x.x,)

.. ,

vir.•

V I • I V •• 2 V 2 • 1V 2 • 1

l(xU

,

l(xU

V M •I VM •1

0

0

B2

0

B=

and

140

I(x~xn = ... = l(x;_lx;)

=

122

VM •• _IV M ••

BI

1-1 = Z

1 = ![II 12] II I,

.J

= 120 =

= ... = I(x;) = ... = I(x!)

o o

0

'l

and that the integrals of all other monomials of degrees 2 and 4 are zero. We also write 1(1) = 100 , With these assumptions we can compute I-I. We find

O·

0

l(x._Ix.)

l(l)

U=l'

V~.I

I(X.X2) J(xlx,)

If w(x., ... ,x.> > 0 on R. then I is positive-definite. In such a case if we find a formula with all VI and BI real then all the BI will be positive. At this point we make additional assumptions about the monomial integrals of degrees 2 and 4. We assume

where

V!.1

l(xtx._lx.)

I, =

+ +

V 1• 1

I(x!)

I(x:x,)

I(X:X2)

12

i

(3.10-])

We define M = (n2 n 2)/2. We seek a formula with M points v, and their negatives with the coefficient, B of VI equal to the coefficient of -VI' Such a formula is automatically exact"for all the monomial integrals (3.10-1) of degrees I, 3, and 5. The condition that the formula be exact for all monomials of degrees 0, 2, and 4 can be written as the matrix equation

urDU

I(xtx;)

=

+ +

I(x,)

l(xtxD

93

du

o

o

IiI

o o

o

'0

IiI

o o

o o

o o o

BM •

121 -·

o

94

Parr I

THEORY

+ 2. j = M in (3.10-5) gives 4dn p' + [4d. + 4(n - 3)du + 6/inp2y2 + [(n - 4)d. + (n 2 - 5n + 8)du + i-(n - 4)(n - 5)/;:/ + 4(n - 4)li~PY' + 4d1p l + 2(n - 2)dlyl + do = 0 Taking i

where

d0 = ~ h I

d

-

no -/go/n

h,(l.o - I z;) d _ 100 [1.0 + (n - 2)/u) - (n - I)no • h,(l.o - 121 ) ho = 1'0 + (n - 1)/u hI = looho - nlio U

-

(3.10-3)

NON PRODUCT FORMULAS

Ch. 3

=

n

d,

t

,=1

vt, + (2du

.

+ liD t

~, - 1

, Pl.z, P l • l • (iv) There exist three first-degree polynomials

(xTy1, ... , xmyll)

,

0

< ex + P


a.

= (l,P 3.• = (l,P 3.1

=

-(I,P3.3

P 3.Z, P3.3 are linearly independent we must have

=(1,

hi =0 C. =0

This completes the proof.

a3 h3

=0 =0

C 3 =-(1,

106

Part I

THEORY

Ch.3

P,."

Theorem 3.12-4. The polynomials i = I, 2, 3, do not have a common component. Proof. First we show that the p,., do not have a common second-degree component PZ' Suppose P,., = PZPI.,' i = 1,2,3. Because the Pl., are linearly independent the p •. , are also linearly independent. Therefore we can find constants C.,C z, c, so that c.p •.• + czp •. z + ClP •. l = I

1

have nine common points or intersection. It is not ditlicult to see that two of these are on the line "at infinity" in projective coordinates. A point in the projective plane can be thought of as a triple (x, y, w). Two points (x .. Y., w.), (XZ'Yl' w l ) coincide if and only if

+ C2Pl .2 + C,P,.l = P z

x. :y. =X 1:Y2 x.:w. =x2:W Z Yt:w. =yz:w 2 The points for which W 0 (say W = 1) can be thought of as the "finite part" of the projective plane and the points for which w = 0 can be thought of as the line "at infinity." In projective coordinates the curves become y( N(n,k)

(3.15-3)

Recently Fritsch [2] has given an example of an n-dimensional region 2) which has a third-degree formula with n I points. All the points in"the formula are inside the region and all the coefficients are positive and equal. There are no other formulas known for which N = N(n, k). Theorem 3.15-3 is only the smallest possible improvement on (3.15-3) but it is the only result known for d odd. For simplicity we state this result for n = 2. Let (n

+

>

pk+I.O(X,

y). pk.I(X, y), ... , pO."+I(X. y)

+

be the k 2 basic orthogonal polynomials of degree k and w(x. y). Consider all possible polynomials of the form

(3.15-4)

+ I for

a given R z

(3.15-5) where

+ P, = k + I III + III = 113 + Il. (Il,. PI) '* (Ill' P3)

... -----'0 -

BN

=

where U is an N X N(n, k) matrix and I •. k is an N(n. k) X N(n, k) positive-definite matrix. An argument similar to the above shows this is impossible if N < N(n. k). Q.E.D.

i = 1,2,3,4

Il,

D=

o

119

If N < N(2, 2) = 6 then the rank of the matrices U. B. and hence of UTDU. is < 6. But I z.z is positive-definite and therefore has rank 6. Therefore (3.15-1) cannot be valid if N < 6. If nand d are arbitrary then the points and coefficients satisfy an equation

- xl

3.15 A Lower Bound for the Number of Points in a Formula

NONPRODUCT FORMULAS

PI

+ pz = P3 + P.

(Ill' PI)

(3.15-6)

'* (a., P.)

+

(Conditions (3.15-6) imply that the monomials of degree 2k 2 on the right side of (3.15-5) cancel, and therefore QZk+ I is indeed a polynomial of degree < 2k I, which is not identically zero.)

+

t Theorem 3.1.5-3. If at least one of the polynomials QZIc+ I defined by (3.15-5) and (3.15-6)-for all possible (XI' PI' i = 1,2,3, 4-satisfies

fL. w(x, y) QZk+ then a formula of degree 2k

+I

I(X, y) dx dy

"* 0

(3.15-7)

contains at least N(2, k)

Proof. Assume we had a formula of degree 2k

+I

points.

+I

with N(2, k) points or fewer. From Theorem 3.7-3 it would follow that all the points in the formula are zeros of each of the basic orthogonal polynomials (3.15-4). This implies that for all QZk+ I the integral (3.15-7) is zero. This contradicts the assumption that at least one of the integrals (3.15-7) is not zero. Q.E.D. This theorem was given by Radon [I] for fifth-degree formulas. In this case the statement of the theorem is that if at least one of the three integrals

ft. ft. fL.

W(X,y)(P3.0pO.3 - p2.lpl.Z) dx dv W(X,y)(P3.0pl.2 - p2.lp2.1) dx dy w(x,y)(PZ.lpO.3 - pl.2pI.2)dxdv

is not zero then a fifth-degree formula contains at least seven points. Theorem 3.15-3 has an obvious generalization for arbitrary nand d which we shall not state. For d = 3 we can therefore determine whether or not a third-degree formula can exist with n + I points. For example, for the n-simplex Tn' using the basic orthogonal polynomials of second degree given in Table 3.6, we find

f·;: f [P2.0 •.... 0pO.2.0..... O-

(PI.I.O..... O)Z] dX I ... dX n

I-n

(n

+ I)(n + 2)2(n + 3)2 "* 0

for n

NON PRODUCT FOFHv1UlAS

Ch. j

121

2. The (/I + 2)-point third-degree formula Tn: 3-1 has a minimum number of points. This was proved at the end of Sec. 3.15. 3. The 2n-point third-degree formulas of Sec. 3.9, for C n and similar regions, have a minimum number of points (see Mysovskih [5]). 4. For" > 2. Fritsch [2] givcs an example of an /I-dimensional region which has a third-degree formula with n + I points. These are minimum-point formulas. These are the only formulas known for all n > 2 for which number of points is known to be a minimum. For n = 2 we know the following: 5. If R2 satisfies

fL. w(x,y)[pz,opo.z -

(PI.I)Z] dx dy

"* 0

(3.16-1)

then, by Theorem 3.15-3, the minimum number of points is 4. The construction of such formulas is discussed in Sec. 3.11. 6. The six-point fourth-degree formulas S2: 4-1, E'z': 4-1, E'z: 4-1 contain a minimum number of points by Theorem 3.15-1. 7. The seven-point fifth-degree formulas of Radon [I], discussed in Sec: 3.12, contain a minimum number of points by Theorem 3.15-3. It is very likely, but has not yet been proved, that the following are also minimum-point formulas: 8. The 12-point seventh-degree formulas for n = 2, C2 : 7-1, C2 : 7-2, Sz: 7-1, E l ': 7-1, E'j: 7-\. 9. The 13-point fifth-degree formulas for n = 3, C,: 5-1, S2: 5-1, G 3 : " 5-1, E l ': 5-1, E'j: 5-\. '10. The 27-point seventh-degree formulas for n = 3, C 3 : 7-1, S,: 7-1, E'j': 7-1. E'j: 7-1. By parngraph~ 2, 3, and 4 it follow~ that for n ~ 2 and (/ > 3 the minimum number of points depends on the region.

>2

Therefore a third-degree formula for Tn must have at least " + 2 points. Formula Tn: 3-1 has this number of points. It is not known if Tn: 3-1 is the only third-degree formula for T. with n 2 points, but we believe it is.

+

3.16 Minimum-Point Formuias

Here we summarize all known formulas which have the minimum number of points for their degree I. The (n + I )-point second-degree formulas of Sec. 3.8 have the minimum number of points. This was proved in Theorem 3.8-4 which is a special case of Theorem 3.15-1.

3.17 Richardson Extrapolation for the n-Cube

Consider the trapezoidal formula for one variable

fl

f(x)dx =:::

~

f(-I)

+

h Zfe

k

N!!..;- I) + ~ f(1)

2 h= N-I

We shall denote the N terms on the right side of (3.17-1) by compute a sequence of such approximations

Ll:(f), Ll:(f), .... , L~~(f)

(3.17-1)

L~r(f).

If we

122

THEORY

Part I

then we can find a linear combination of the )1,.1

I:~:(f).

Lyness and McHugh show that the

I:1:(f) + ... + )I,.P I:~~(f)

was studied by Bulirsch [1]. He recommends using (3.17-4) in place of either (3.17-3) or (3.17-5). A similar process can also be used with the repeated midpoint formula and the repeated Simpson's formula. . An analogous procedure also can be carried out for the n-cube Cn' This will be discussed in the remainder of this section. We shall follow the discussion of Lyness and McHugh [I] but with somewhat different notation. They call this method a "progressive procedure." We shall describe the results and briefly how they are obtained. but proofs will not be given. We divide C. into M· equal subcubes. each of side length" = 21M and let v,. i = 1•...• M' denote the centers of these cubes. The approximation

J_II ... JI_I f(x I' .... x.)dx I... dx. :::: h' ,~ f(v,) M

> ... > lip (3.17-7) For each sequence (3.17-7) there exists constants )I P.' so that the approximation

hi' h z•.••• hp

JI ... JI -I

:::: )I p.1

hi

>

h2

f(xl •...• x.)dxl .. . dx.

-I

I:;::14(f)

+ ... + )I P.P I:r.ld(f) == I:mI4(f) p

is a formula of degree 2p - I. The For the sequence

)I

P.' are independent of the

=

I ..... p

_ )I

P.' -

)I P.'

123

are

(_1)'-'2;2P (p + i)! (p - i)!

(3.17-10)

Values of these constants are given in Table 3.11. TABLE 3.11

Values of T P.' for the Sequence hi

P

i

2

1 2

-0.33333 33333 33333 (1)0.13333 33333 33333

2 3

(-1)0.41666 66666 66667 -(1)0.10666 66666 66667 (1)0.20250 00000 00000

3

-(-2)0.21777 0.35555 -(1)0.26035 (1)0.32507

17177 55555 71428 93650

71778 55556 57143 79365

5 1 2 3 4 S

( - 3)0.11574 -(-1)0.67724 (1)0.14645 -(1)0.51791 (1)0.53822

07407 86172 08928 88712 88910

40741 48617 57143 52205 93474

6

1 2 3 4 5 6

-( - 5)0.33068 (-2)0.84656 -0.48816 (1)0.46233 -(2)0.12232 (I )0.90888

78306 08465 96428 50970 47479 31168

87831 60847 57143 01764 75790 83117

I 2 3 4 5 6 7

(-7)0.68893 -(-3)0.75249 0.10983 -(1)0.22416 (2)0.12742 -(2)0.25169 (2)0.15559

29805 99647 85302 76308 8169642857 24712 73582 16124 74781 07092 90709 44865 43228

7

= ;-1. ; =

P

)'P.'

1 2 3 4

4

(3.17-6)

can be called the M·-point product repealed midpoint formula. The sum on the right side of (3.17-6) will be denoted by I:l:'ld(f) We construct such a sum for a sequence of values of II.

=

NON PRODUCT FORMULAS

Ch. 3

(3.17-2)

which is an integration formula of degree 2p - I and will often be a much better approximation to the integral than any of the individual I:~:(f). This process is called Richardson extrapolation and the resulting formulas (3.17-2) are often called Romberg formulas. This computational process with h,=2 1 -' 1=1.2 •...• p (3.17-3) has been widely studied (see. e.g .• Bauer. Rutishauser. and Stiefel [I] and Wilf [1)). This process using the sequences hi = I. hk = 2- k / 2 • hUI = 3- 12 1 -(k/ZI (3.17-4) k = 2.4. 6•...• p h, = i-I i = 1.2•. ..• P (3.17-5)

h

t

1••••• p )'p.,

8

1 2 3 4 5 6 7 8

-(-8)0.10935 44413 - (-4)0.50166 56868 -(-1)0.17973 51866 0.74720 82375 -(1)0.8168052081 (2)0.32360 23405 -(2)0.50827 53227 (2)0.26906 06541

65023 50872 88312 78608 71672 16626 07878 64569

9

I 2 3 4 5 6 7 8 9

(-10)0.13669 30517 -(-5)0.26060 55516 ( - 2)0.22466 89833 -0.1839281815 (1)0.36464 51822 -(2)0.25888 18724 (2)0.77829 65878 -(3)0.10129 34227 (2)0.46887 18347

06279 10843 60390 57811 19497 13301 96438 44308 15659

10 2 3 4 5 6 7 8 9 10

-( -12)0.13807 37896 02302 (-6)0.10858 56465 04518 -(-3)0.22220 00934 33352 ( -I )0.35033 93934 43450 -(1)0.121548394073166 (2)0.14562 10532 32482 -(2)0.74177 51530 76970 (3)0.18007 71959 89882 - (3 }O.19988 74663 78781 (2)0.82206 35246 62433

(3.17-8) If we compute (3.17-8) for a sequence of values of P.

dimension n. (3.17-9)

I:mld(f)"

j'= 1.2.3, ...

(3.17-11)

this sequence will converge to the true value of the integral. whenever f is Riemann integrable. provided the sequence h" i = 1.2•.. , • grows at least

124

Part ,

THEORY

exponentially. (This is discussed for one variable by Bauer, Rutishauser, and Stiefel [IJ.) For the sequence (3.17-9) this is not truc and it is possible that the sequence (3.17-11) may be unstable and not converge. If n is moderately large-say n > 4-then a sequence other than (3.17-9) is difficult to use because the number of points in the L;::ld(f) will increase extremely rapidly. For p relatively small (say p < 7) and n = 2 or 3, a more rapidly increasing sequence of hi might be desirable, but this has not been studied. Table 3.12 gives three examples-one in each of the dimensions n = 3, TABLE 3.12

Comparison of the Progressive Procedure of Lyness and McHugh [1] with Product Gauss Formula

N

"/'tIc/Ilt'f GUII.fS

l~d~.r~,

I 2 3 4 5 6 7

I 8 27 64 125 216 343

(4

+ x + y + Z)-I dxdydz =

2.00000 00000 2.10562 77056 2.13033 50970 2.13961 15008 2.14403 79620 2.14648 10335 2.1479679460

000 277 017 066' 376 442 159

2.15214283259589

2.00000 00000 2.14083 69408 2.15125 90189 2.15206 79632 2.15213 60071 2.15214 21755 2.1521427669

000 369 589 627 023 233 347

2.00000 00000 000 2.1472995090016 2.15197 15948 128 2.15213 58746 756 2.15214 25167 156 2.15214 28170 465 2.1521428317 831

I ~d ~ d ~ d ~ I exp (I xtl + I x21 + IX31 + Ix41) dXI dX2 dX3 dX4 = I 2 3 4 5 6 7

I 16 81 256 625 1296 2401

16.00000 118.22489 113.45192 133.79488 129.576\1 136.91862 134.34704

00000 75829 18744 74694 77569 05526 10014

16.00000 00000 152.2998634439 104.30025 10405 181.55046 30816 82.33663 7\189 223.59136 26082 7.78566 26954

I HH~l H

~ exp (-XIX2 X3X3XS) dXI dX2 dX3 dX4 dxs =

I 2 3 4 5

t 32 243 1024 3125

0.96923 0.97016 0.97042 0.97052 0.97057

32344 7635 08327 9691 27635 2684 249764605 01372 4613

0.96923 0.97047 0.97065 0.97065 0.97065

32344 00322 25925 71518 71907

7635 3709 9500 8948 1301

139.47538 59223

16.00000 161.09428 118.54687 146.38683 \30.78799 142.70877 134.76825

00000 24716 96958 53451 06539 70752 65222

0.97065 71913 8839 0.96923 0.97065 0.97065 0.97065 0.97065

32344 57743 71906 71913 71913

7635 0404 1425 8682 8483

4, 5-using (3.17-8) with the sequence (3.17-9). The results using product Gauss formulas are also given. Here N = p" is the number of points in I;r,'d{f) and in the product Gauss formula; I;m'd(f)p uses alI the points used by I;;::'d{f), ... , I;~'d(f)

It is natural to compare I;m'd(f)p with the po-point product Gauss formula. Both of these formulas have degree 2p - I. In these examples the

1

NON PRODUCT FORMULAS

Ch.3

125

product Gauss rormula is always significantly better than l,;mld(f)p. In the rour-dimensional example the integrand is continuous but has discontinuous first partial derivatives; here the sequence (3.17-11) is obviously unstable. In this example it is also obvious that both the product repeated midpoint formula and the product Gauss formulas are much better than the progressive procedure; neither of these two methods can be unstable because, for each p. each of these two formulas is an n-dimensional Riemann sum (see Sec. 1.9). Although examples could surely be found for which the progressive procedure is better than the product Gauss formula the examples of Table 3.12 are not artifical. The product Gauss formula must. in general, be recommended over the progressive procedure. The progressive procedure can also be applied to other integration formulas. For eXllmple, in place of the product repeated midpoint formula (3.17-6) we could use the product trapezoidal rule. If we take the II-dimensional product of (3.17-1) with N = M + I, the formula which results is the (M + I)"-point product trapezoidal rule; it uses the vertices of the M' subcubes of side length h = 21M. We shall denote this (M + I)'-point formula by the symbol I;lr(f). which is the same as we used for one variable. There exists constants), p.I' which depend on the hi but not on n, so that the approximation

JI ... JI

f(x ..... , x,) dx , ... dx,

(3.17-12) :::::: )'p.1 I;1:(f) + ... + )'P.P I;1:(f) = I;tr(f)p is a formula of degree 2p - I. It turns out that the 'l' p.1 in (3.17-12) are identical with those in (3.17-8). Therefore. for the sequence (3.17-9). the )' p';l in Table 3.1I are those to be used in (3.17-12). Similar results apply for other formulas, for example the product repeated Simpson's formula. The 'l'p.1 depend on the mesh ratios "p and the degree of the formula. but are independent of II. If we have two different formulas of the same degree-such as L;:'ld{f) and I;1r (f) - the 'l' p.1 will be the same provided that each of the formulas is fully symmetric. (For the definition of a fully symmetric formula see Sec. 4.1.) For n = 1 the derivation of the Richardson extrapolation method for the repeated trapezoidal formula is based on the welJ known Euler-Maclaurin sum formula. The Euler-Maclaurin sum formula can be summarized by Theorem 3.17-1. . -I

-I

"1' "z' .. ..

Theorem 3.17-1. If I(x) has continuous derivatives of all orders then f l f(x)dx - h 4 c4

=

JI

-I

I;lr(f) - h 2c 2 flfl1l(X)dX P41(X) dx - h 6 c 6

fl

(3.17-13) f(6I(X) dx - ...

-I

where the Ca are constants which are independent offand h.

• ..

~Lt

.&.U

1"11r'_""i

Part I

Inll:V"l

For the derivation of (3.17-13) see Krylov (I; p. 212-216J. Expansions similar to (3.17-13)-but with different clk-are valid for other repeated formulas such as the repeated midpoint formula and the repeated Simpson's formula. Results similar to Theorem 3.17-1 are valid for C., n > 2. The derivation of these is given by Lyness [I]. The result for the product-repeated midpoint formula :Erld(f) can be stated as in Theorem 3.17-2. Theorem 3.17-2. If I(x .. ... ,x.) has continuous partial derivatives of all orders in C., then

II ... II -I

-I

I(x l , · · • ,x.) dX I

•••

dx.

=

-

t Chapter 4

EXTENSIONS OF FORMULAS

:Erld(f) - :E h 1SP1S .=1

(3.17-14) where PlS

=

I

:E c ..... •• '1+···.''''-' l 1

I

II a 1,Ia...l a Z

••• -1

'

-I

XI

1. dX I

XII ..

•••

dx.

Theel ..... 1 •• are constants which are independent of I and h. Now it is easy to show the existence of the Yp.l' We shall write (3.17-14) as (3.17-15) We write Eq. (3.17-15) for a sequence of values h = hi" .. , lip' Multiply the first of these equations by YI',I the second by )11'.1' and so on, and add. We obtain

:Emld(/)p

= /(/) + (1

(where (1 is a sum of terms which involve integrals of partial derivatives of lor order::::: 2p) provided we have I!,

'y' t:1

)II','

=

p

\"' )Ip"h,z'

t:1

I

=0

(3.1?-16)

s= I, ... ,p-I

The solution of this linear system gives the Yp," It is clear that this system is nonsingular and that the )I p,' depend only on the h, and not on n. For p moderately large, Lyness and McHugh [I] point out that the system (3.17-16) is ill-conditioned and difficulties are encountered if one tries to solve it numerically. They discuss how it can be solved in closed form. For the sequence (3.17-9) they obtain (3.17-10).

4.1 Introduction

In this chapter we discuss the following special problem: Suppose we have an integration formula of degree d for the m-cube C m with weight function = I. Is there some way we can get from this formula a formula of the same degree for C., n > m? " One way we can do this is by using product formulas as discussed in Chapter 2. That is, if we take the Cartesian product of a given formula of degree d for Coo with a formula of degree d for C._ '" we obtain a formula of degree d for C•. If the formula for Cm has N points and the formula for C.- m has M points, then the formula for C. will have NM points. In general we shall say a formula for C. is an extension of a formula for C m , m < n, if the formula for C. is constructed in some way which makes intrinsic use of the formula for Coo. In this chapter we discuss a few methods for extension of a formula for C", by methods other than product methods.

I)cllniUoll 4.t-l_ We:. s.tuaJl say that a [ormur

J"i- I

• "

fl f Xl' ..., . )ax,··· dx"1 ::::: .ff~::1 Ad -j

,.i " . . ,v,.... ) 4. I-J)

ror e mi .fJ'mmf!lric if the right side or (4. r-1 is nOl chllnged under any of the m I pr:rI1mtaHons or the vartablcs x I ' • • . • X"" !n oille words, (4.1 · 1) is symmetric pIO idea thal ir the formula cOllwillS the poin (V J• I' 1, l ' • .. ~ VJ. "'.)

v

lhen tile formula ,dso cOlli3ins lho point

4.1.2)

.. ~"'

.... 11~_,." .. .1

'''0

Inl:Vnl

Part !

(4.1 -3)

We note t1m!

where (PI'PZ"" ,Pm) is any permutation of (1,2, ... , m) and the coefficient of each of the points (4.1-3) is the same as the coefficient of (4.1-2). Also we define (V). I'

v). Z ••••

x::,-) =

I

Definition 4.1-2. We shall say that a formula (4.1-1) isfully symmetric if the right side of (4.1-1) is not changed under any of the 2m(m!) affine transformations of Cm onto itself. In other words (4.1-1) is fully symmetric provided that if it contains the point (4.1-2) then the formula also contains the point

(±v),P" ±v),P" ... , ±v).pJ

(4.1-4)

where (PI' Pz, ••• ,Pm) is any permutation of (l, 2, ... , m) and the m choices of sign in (4.1-4) can be made in any way we wish. Also, we define (v), I' V),Z,···, V),m)F8 to be the set which consists of all 2m(m!) points we have just described.

The first explicit discussion of extensions of formulas was given by Lyness [2), who discussed extensions of fully symmetric formulas. The method of Lyness was generalized to symmetric formulas by Stroud [14). These results will be discussed in Sec. 4.3. Stroud and Goit [I) have given a method of extension which can be applied to an arbitrary formula for Cm. This method will be described in Sec. 4.2; it is completely independent of the method of Sec. 4.3. In Sec. 4.4 we mention briefly a few simple results known concerning extensions of formulas for regions other than Cm. (The term "fully symmetric formula" was first used by Hammer and Stroud [2) in the sense we defined above. The term "symmetric formula" has been used in different ways by different authors. What Lyness [I) calls a symmetric formula (or rule) we call a fully symmetric formula. Similarly, what Stroud [6) calls a symmetric region R. would be more appropriate to call a fully symmetric region since the resulting formulas are fully symmetric with the above definition. What Stroud [8) calls a symmetric region could be called a centrally symmetric region and his formulas-which are unchanged under the transformation (XI' X z,"" x.) -+ (-XI' -Xl' •••• -x.)-could also be called centrally symmetric formulas.) In the next two sections we use the notation

fl ... fl x~'''· -I

-I

x:,- dX I

(lIl

1

+ I) . ~. (lim + I)

V

if all

m

III

are even

lif at least one

lI,

is odd

IC ...(X~I .•. x:."'x:''++l ... x:'''';;)

,V),,,,)8

= JC.(X~I

to be the set which consists of the m! points we have just described.

Ic.' 2. . The first kind of estimate is discussed in Sees. 5.2-5.10. This theory is due mainly to A. Sard and is a generalization of the Peano error estimates for one variable. The complete theory of these estimates is given by Sard (5; Chap. 4]. This theory gives estimates for the error, provided I(x, y) has a certain type of Taylor series expansion. Two examples of the type of Taylor series expansion whieh are permissible are given in Theorems 5.2-1 and 5.5-1. We state Sard's theory only in two special cases. These special cases concern spaces of functions Bp,q and Brp•f1 and, we believe, cover almost all practical applications. The theory is so intricate that even a complete statement of it is beyond our scope. For the theorems we do state we shall be content to give only an indication of the method of proof. We do things in this way to try to make it as clear and as easy to understand as possible. The reader who is interested in the gener~lizations and in complete proofs can refer to Sard [5; Chap. 4]. The second kind of estimate applies, in two dimensions. to a function of two complex variables, Z = x + iu, W = Y + iv, which is analytic for all (z, w) in a certain region 6'" x 6',. which contains RI • These estimates are'

138

Part I

THEORY

discussed in Secs. 5.11-5.13 and are due to Barnhill [I), (2). They are generalizations of estimates of P. J. Davis for one variable. Up to the present time these estimates have been studied only for the case w(x, y) = I. In principle the theory could be applied for other weight functions. In Sec. 5.14 we give a different kind of result, which is due to Sarma [I), [2), [3]. Here one is not interested in an upper bound for E[f) for a given.f. Instead one is interested in an "average" error over all functions which have a power series which converges for all points (x I ' • . • , x.) in C•. This information is given by the quantity SE' the square of which can be considered as the variance of the normalized error E*[fJ = 2--E[f). There are additional results known concerning the error in (5.1-1) which we shall not discuss. Results concerning the error in product Gauss formulas are given by Ahlin [I), Chawla [I), Stenger [2), and Stroud and Secrest [2; Sec. 4-5). Other papers dealing with error are by von Mises [2), Synge [I), Ermakov [2J, [5J, Hsu [6J, [7J, Levin (2), [4J, Lebedev [I), Martensen [I), Sobol, [3), Solodov [2), and Stancu (5). 5.2 The Space of Functions B""ff

Let Rl be a rectangle a