Sequence Transformations and Their Applications

SEQUENCE TRANSFORMATIONS AND THEIR APPLICATIONS Jet Wimp DEPARTMENT OF MATHEMATICAL SCIENCES DREXEL UNIVERSITY PHILADELPHIA, PENNSYLVANIA

@

1981

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COPYRIGHT © 1981, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

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Library of Congress Cataloging in Publication Data Wimp, Jet. sequence transformations and their applications. (Mathematics in science and engineering) Bibliography: p. Includes index. 1. Sequences (Mathematics) 2. Transformations (Mathematics) 3. Numerical analysis. I. Title. II. Series. QA292.W54 515'.24 80-68564 ISBN 0-12-757940-0

PRINTED IN THE UNITED STATES OF AMERICA

81 82 83 84

987654321

Preface

In this book we shall be concerned with the practical aspects of sequence transformations. In particular, we shall discuss transformations T mapping sequences in a Banach space 81 (often, but not always, the complex field) into sequences in 81. Certain practical requirements are ordinarily made of T: that its domain f» contain an abundance of" interesting" sequences and for S E f» also as + e E ~, e being any constant sequence; further, we shall usually require that T satisfy the following requirements: (i) T is homogeneous: T(as) = aT(s) for any scalar a; (ii) T is translative: T(s + e) = T(s) + T(e) for any constant sequence e; (iii) T is regular for s: if s converges, then T(s) converges to the same limit. Often more than (iii) is required, namely, (iii')

T is accelerative for s: T(s) converges more rapidly than s.

This requirement sometimes takes the form that lim II{T(s)}n - sll [s, - sliP

=

f3 < I

n~OCJ

for some indexp ~ I, where {T(s)}n and Sn are the nth components of T(s) and s, respectively, and s is the limit of s. Historically, most of the work done in this area up to 1950 focused on transformations that are also linear: T(s + t) = T(s) + T(t). Such transformations have a very simple structure, namely, the components of T(s) ix

x

Preface

can be characterized by weighted scalar means of the components of s (at least when :!4 is separable), and such transformations have beautiful theoretical properties. [The classical work in this area is the book "Divergent Series" (Hardy, 1956), and more modern developments are discussed in Cooke (1950), Zeller (1958), Petersen (1966), and Peyerimhoff (1969).] However, linear methods are distinctly limited in their usefulness primarily because the class of sequences for which the methods are regular is too large. In defense of this somewhat paradoxical statement, I only remark that experience indicates the size of the domain of regularity of a transformation and its efficiency(i.e., the sup of p values in the foregoing equation) seem to be inversely related. Furthermore, linear transformations whose domains of regularity are all convergent sequences (called regular transformations) generally accelerate convergence at most linearly, i.e., p = 1, 0 < f3 < 1. Obviously, for safety's sake, when one uses a nonregular method, one wants a criterion for deciding when s belongs to its domain of regularity. This, however, is not the problem it might seem to be. Linear regular transformations are discussed (at length, in fact) in this book, but primarily those transformations whose application can be effected through a certain simple computational procedure called a lozenge method. As the reader will find, the subject touches virtually every area of analysis, including interpolation and approximation, Pade approximation, special functions, continued fractions, and optimization methods, to name a few; and the proofs of the theorems draw their techniques from all these disciplines. Incidentally, I have included a proof only if it is either short or conceptually important for the discussion at hand. It was simply not feasible to include very detailed and computational proofs, e.g., estimates for the Lebesgue constants for various transformations (Section 2.4), or inequalities satisfied by the iterates in the e-algorithm, or long proofs whose flavor was totally that of another discipline-results on Pade theory, for instance, or results requiring the theory of Hilbert subspaces. In such cases, I have always indicated where the proof can be found. The techniques given will, I hope, be useful in any practical problem that requires the evaluation of the limit of a sequence: the summation of series, numerical quadrature, the solution of systems of equations. Particularly welcome should be the discussion of methods to accelerate the convergence of sequences arising from Monte Carlo statistical experiments. Since the convergence of Monte Carlo computations is so poor, O(n -1/2), n being the number of trials, techniques for enhancing convergence are highly desirable. A closely related subject is the iterative solution of (operator) equations. In fact, any sequence transformation can be used to define such an iterative method (cf. Chapter 5). However, this is not the subject proper of this book,

Preface

xi

there being available already several excellent works in this area. I have, in fact, restricted myself mostly to material which has not appeared in book form in English. Some of the material is available in French [any numerical analyst will have on his shelf C. Brezinski's two important volumes (Brezinski, 1977, 1978)], but much of the material has never appeared in book form, some has not appeared in published papers [the thesis work of Higgins (1976) and Germain-Bonne (1978) for instance], and much is new altogether. I have not usually opted for abstraction. In most instances the transformations can be generalized from complex sequences to Banach-spacevalued sequences, and often I have indicated how this can be done and have established appropriate convergence results. But where abstraction can confuse rather than elucidate, I have left well enough alone. For instance, I believe that the theory of Pade approximants, at least for my purposes, is most firmly at home in classical function theory. My notation may at times seem idiosyncratic, but it is one I have found necessary to diminish clutter and bring some focus to the development. Before the reader gets into the book, I strongly advise him to read the section on notation. Otherwise, certain unfamiliar conventions-for instance, xnR: Yn, which I have found most useful-may well render the material completely opaque. The notation for special functions is, by and large, as in the Bateman manuscript volumes. Ad hoc notation is explained in Notation or as needed. I have provided many numerical examples, but these are illustrative only, not exhaustive. The reader interested in further numerical examples and applications should consult C. Brezinski's (1978) book, and, for a comparison of methods, the survey of Smith and Ford (1979). The problem of rounding is always an annoying one in a book dealing with numerical methods. Generally speaking, all numbers free from decimal points or occurring in definitions may be considered exact. Others, particularly those occurring in tables, have been rounded to the number of places given. However, I should be surprised if I have been consistent.

Acknowledgments

Several people have contributed to this book. John Quigg has read and commented on some of the material. Bob Higgins, my former student, has provided most of the theory in Chapters 12 and 13. Steve Yankovich and Stanley Dunn have contributed their programming and analytical skills for the preparation of numerical examples. Drexel University has been generous in its support and encouragement. I am grateful to Alison Chandler, whose combined typing and mathematical skills led to such a beautifully prepared manuscript, and to Don Johnson and Harold Schwalm, Jr., who assisted in the proofreading. Finally, I consider myself fortunate to be working in a field where friends are so easily made. My colleagues have proved to be warm and enthusiastic. I have enjoyed thoroughly meeting and exchanging ideas with Bernard Germain-Bonne and Florent Cordellier. I am particularly indebted to correspondence and discussions with Claude Brezinski. He has generously provided me with unpublished results (Chapter 10). Some of the ideas in the book originated in a lengthy afternoon discussion with Claude and other colleagues. That meeting demonstrated to me the delights of the mutual, as opposed to solitary, quest.

XIII

Notation

Spaces

.,({

metric or pseudometric space

1/

linear space

fff

topological vector space over real or complex field

fJI

Banach space

-*

dual space

B(81, fJI') space of all bounded linear mappings of one Banach space into

another

IITII

n

= sUPllxll:511I T (x) ll, TEB,xEfJI cone in fff (n contains a nonzero vector and if x

E

n, A.X E n, A. > 0).

for any matrix A = [aiJ, 1 :s; i :s; n, 1 :s; j :s; m, first subscript of aij denotes row position, the second column position, of the element

Real and Complex Numbers

space of ordered complex p-tuples,

p

complex numbers space of ordered real p-tuples, p > 1 xv

> 1

XVI

Notation

fJIl

real numbers

fJIlO

nonnegative real numbers

fJIl+

positive reals

J

integers

JO

nonnegative integers

J+

positive integers

m, n, k, r, i, j

generally denote integers

d(A, B)

=

D(A, B)

=

infxEA,YEB

[x -

SUPXEA,YEB

yl

[x -

yl

= {z[lz - al < p}

Np(a)

oNp(a)

= {z[lz - al = p}

Np(a)

= {z[lz - al : : ; p}

NiO)

=

Np

N1 = N

the unit circle

Sequences boldface letters denote sequences, s, t, etc, for any space d, d s denotes the space of sequences with elements in d; s = {s.} E ds, Sn E sf de

space of convergent sequences

d

space of null sequences, e.g., d a metrizable t.v.s.

N

e., «.. «;

fJIl TM, fJIlTQ' 'CE=(r) special real and complex sequence spaces (see Sections 1.4, 1.5, 2.2) related sequences (the space d must be such that the definitions make sense)

n

a: r:

S =

~

1;

lim

Sn

a o = So,

so

Notation

h:

h.

=

L' L"

r.+ dr.

=

(s.+ I

.1 s : {.1 s }. = .1 s. , k

k

-

xvii

s)/(s. - s)

k ~ 1

k

indicates first term of sum is to be halved indicates first and last term of (finite) sum are to be halved

• f(k/n), T,,(f) = -1 L"

n

/1 k=O

sequence relationships: two sequences x, y

~

1

(trapezoidal sum)

let R be a binary relationship between members of

x.R :Y.

means x.Rv; holds for an infinite number of values of n

x.Ry.

means x.Ry. holds for alln sufficiently large this notation is used only when the sequence variable is n

Example

IA.kl s.

1 means: for some no, IA.kl ~ 1, 0

s ks

/1,

n > no

Functions

'I'

class of real nondecreasing bounded functions on [0, 00) having infinitely many points of increase

'1'* subclass of 'I' such that

LOOt' dt/J


0,

(EX 1)n:

Jo

=

t + 1

8 = 25 8

= 1.25 In 5

s divergent

s divergent, but generated by

0.5963473611

generated by 8 n+ 1 = 20[s,;

+ 2sn +

IOr

1

80

;

= 1;

8 = 1.368808107

(IT 2)n:

generated by

Sn+1 =

(20 - 28'; - 8~)/1O;

80

= 1;

s divergent (LUB)n:

ak

=

( _1) k + 1 '

8 = 1.131971754

= greatest integer contained in

Notation

xix

Numerics

Generally, in tables n SF representing a number is a rounded value; for instance, n=3 n = 3.1 tt = 3.14 n = 3.142, '" . For rational numbers, it may occasionally be important to know that the given value is exact. If that is the case, we write ~

= 1.5

(exact).

In definitions, all numbers are exact, e.g., s, = (1.18t, or it is indicated by ... that the number has been truncated.

Chapter 1

Sequences and Series

1.1. Order Symbols and Asymptotic Scales, Continuous Variables Let 1/ and 1/' (see Notation) be equipped with pseudometrics d and d', respectively; let n be a cone in 1/ and ¢, IjJ E /T(1/, 1/').

¢=

O(IjJ)

n

in

(1)

means for some M > 0 there is an R(M) > 0 such that d'(¢, O)jd'(IjJ, 0) < M,

Further,

¢ = o(ljJ)

d(x,O) > R.

XEn,

III

n

(2)

(3)

means for any e > 0 there is an R(e) > 0 such that d'(¢, O)jd'(IjJ, 0) < e,

d(x,O) > R.

XEn,

(4)

If ¢, IjJ depend parametrically on a E .sf and (2) holds for all a E .sf, then we shall write "¢ = O(IjJ) in n uniformly in .sf," and similarly for (3). F or the foregoing definitions to apply, the implicit assumption is made that denominators are never zero; for example, there must be some R such that d'(l/J, O) oF 0, X E n, d(x, 0) > R. Thus anytime an order symbol is used, an implicit statement is being made about the zeros of d'(IjJ, 0). The concept of asymptotic equivalence is often useful. This is written in and means both ¢ - IjJ

n

= o(ljJ) and IjJ - ¢ = o(¢) in n.

(5)

2

I. Sequences and Series

Now let cj» E :Y sC'f/, y'), where Y and 'Or' are linear spaces with pseudometrics d and d', cj» is called an asymptotic scale in Q if, for every k ;;::: 0, in

(6)

Q,

and if this holds uniformly in k or uniformly in some parameter space, we speak of a uniform asymptotic scale (properly qualified). See Erdelyi (1956) for many examples. Letf E :Y(Y, y'), A E res and cj» be an asymptotic scale in Q. The statement in

(7)

Q

is to be read "f has the right-hand side as an asymptotic expansion in Q with respect to the scale cj»" and means, for every k ;;::: 0,

f -

k

L A r4>r = O(4)k) r;O

in

Q.

(8)

Often cj» is understood from context, so "with respect to the scale cj»" may be deleted from the definition. Note that 0, 0, and -- are transitive and ~ is symmetric. Clearly no asymptotic scale can contain the zero vector or two identical vectors. If d' is a metric induced by a norm I ·11, the asymptotic expansion (7) is unique (but not otherwise). This is a simple consequence of the fact that 4> = 0(1]), l/J = 0(1]) then imply 4> + l/J = 0(1]). Thus assume another expansion (7) with coefficients A' holds. Setting k = in (8) and its analog and subtracting the two gives

°

(A o - A~)4>o = 0(4)0), or lAo -

A~

I
0.

1.2. Integer Variables In discussions of sequences, the relevant variable x in 4> or l/J takes values in JO. We write 4>n or l/Jn for 4> or l/J, respectively, or, when there is a possibility of confusion with the index of an asymptotic scale, 4>(n) or l/J(n). 1.1(1) is then written

and means that for some M > 0, there is an N > d'(4)n, O)jd'Cl/Jn' 0) < M

for

°

such that

n> N.

(1)

1.3. Sequences and Transformations in Abstract Spaces

3

A similar modification is made of 1.1(3). An additional complexity occurs when ¢ and t/J depend on a p-tuple with elements in say, n = (ml' m2' ... , m p ) . It is usually important to know exactly how the elements mj become infinite, and it is hardly ever sufficient to say, for instance, that ml + m 2 + ... + mp > N. In fact, the concept of a path in n-space becomes important (see Section 1.3).

r:

1.3. Sequences and Transformations in Abstract Spaces

In this book we shall be concerned with two kinds of sequence transformations. The first is the transformation ofagiven sequences E d sinto a sequence S E.r4's with, generally, a formula given to compute sn in terms of elements of s. (In some situations there is no explicit formula.) The other case is where the given sequence s is mapped into a countable set of sequences S(k), k ~ 0, with a formula given (called a lozenge algorithm) for filling out the array {S~k)}, n, k ~ 0. The whole point is to compare the convergence of the transformed sequence(s) with that of the original sequence. The most useful concepts are formulated in the definitions that follow. Definition 1. (i) (ii) (iii)

Let s, tEAte, a metric space.

t converges as s means d(sn, s) = O(d(t n, t)) and d(t n, t) = O(d(sn's)), t converges more rapidly than s means d(tn' t) = o(d(sn, s)). The convergence of sis pth order if, for some p E

r,

d(sn+ I' s)

= O(d(sn, s)") (I)

and

d(sn' s)"

=

O(d(sn+ I' s)).

It is easy to show that p, if it exists, is unique.

Definition 2.

Let T E !!T(d, JIt s) where d

c

JIt e and T(s)

=

s.

(i) Tis regular for d ifs Ed=> S E Ate and s = s. (ii) T is accelerative for d (or accelerates d) if T is regular for d and S converges more rapidly than s, s e ss, Definition 3. SEd.

Let T E 5"(d, Jlts)whered

c

Jlt s . T sumsdifT(s) E JIt c-

P = {(im,Jm)lim, JmEjO} IS called a path if io 00 along certain paths but not along others. The following definitions contain the key ideas. Let P be a path and <jJ(n, k), t/!(n, k) E §'(P, 1/') where 1/ is equipped with a pseudometric d. <jJ

= O(t/!) in P

(2)

means for some M > 0 there is an N > 0 such that d(<jJ, O)/d(t/!, 0) < M for E P, n + k > N. A similar interpretation is made of o.

(n, k)

5

104. Properties of Complex Sequences

Definition 4. Let vIt be a metric space, T and let 7k(s) = S(k), k ~ 0.

E

:Ysed, vIt s) where d

c

vIt c

(i) T is called regularfor d on P if sEd = d(s~kl, s) = 0(1) in P. (ii) T is called accelerative for d on P if T is regular for d on P and if d(s~k),

s)/d(sn' s) = 0(1)

in

P,

s e se.

(3)

If, in the foregoing definitions, d == vIt c- we shall omit the wordsror d and say simply that .r is regular, etc. We now discuss certain computational aspects of the foregoing definitions. Usually To = I, the identity transformation, so s(O) = S and an algorithm that is computationally feasible for filling out the array {S~k)} will start with the values s~O) = s; and assign one and only one value to each (n, k) position in the array. There seems to be no easy characterization of those algorithms that are feasible in this sense. However, several important ones have been discovered recently. Among these are formulas of the kind s~O) =

Sn,

n, k

~

0,

(4)

called a deltoid; and S~k+ I)

=

H(S~k;/), S~k~ I' S~k»,

n, k ~ 0,

S~-I)

= 0,

s~O)

= Sn, n

~ 0;

S~k+l)

=

H'(S~kL, S~k-l), S~k),

n, k

S~-lI

= 0,

s~O)

= Sn' n

~

~

0,

(5)

0, (6)

called rhomboids. There is as yet no general theory for constructing such algorithms. Those that are known have been derived using ad hoc arguments from diverse areas of analysis: Lagrangian interpolation, the theory of orthogonal polynomials, and the transformation theory of continued fractions. Much work remains to be done in this area. For transformations in vector spaces, there are several important concepts that involve the linearity of the underlying space. Definition 5. Let T E :Y(d, 11s) where d c; 11s- T is linear if, for all x, y E d and c l , C2 E '??, T(c1x + e2Y) = C 1 T(x) + C2 T(y); otherwise, Tis nonlinear. T is homogeneous if T(cx) = cT(x) for xEd, c E '?? T is translative if T(d + x) = d + T(x), where d is a constant sequence (dn == d) whenever d + x, XEd. 1.4. Properties of Complex Sequences When the metric space of the previous sections is the complex field, its sequence space possesses elegant properties. Some of these have been long

6

1. Sequences and Series

known, and others are surprisingly recent. This section contains a discussion of some of these results. Definition 1.

Let

S E ~c

and

rn+dr n = (sn+ I

s)/(sn - s) = p

-

+ 0(1).

(1)

(i) If 0 < Ipi < I, s converges linearly and we write s E ~l' (ii) If p = 1, s converges logarithmically and we write S E ~l"

Theorem 1.

Let Ip I i= 0, 1. Then , Sn+1 I1m

n-e co

Remark.

Sn -

S

S

=p

1iff

I'IHl a n +~I = p. n-e oo an

(2)

For the divergent case Ipi> 1, S can be any number.

Proof The validity of either limit implies an i=. O. Assume, without loss of generality, an =f 0 for any n; otherwise delete the finite number of ans that are zero and relabel the members of a and s. =: We have [a n+I

+ (s,

- s)]/(sn - s) ~ p

(3)

or

an+1

~

an = (p - 1)(sn_1 - s).

(p - 1)(sn - s),

(4)

Dividing the former by the latter shows (4')

Note that for this part of the theorem p can be zero. =: We do only the convergent case 0 < Ipl < 1. The other is similar. Since I an converges,

s, -

S

=

00

I

k=n+ 1

ai,

(5)

We can write EE~N'

(6)

Let gn = sup j'2:n

lejl.

(7)

1.4. Properties of Complex Sequences

Then g E

~N'

7

Taking products in (6) gives

= aopn

an

n

n-I

j=O

(1

+ G),

(8)

empty products interpreted as 1. Define

(9)

Thus

.

11m F" ~

n-e co

I Lr pkl k=O

1

>

Ipl,+1

I I

1_

P

-lpr+ 1

- II - pi

(1 + I) . 11 - pi I - Ipl

(10)

For r sufficiently large, the right-hand side is >0. Thus lim F; > 0 and IIFn is bounded. Now S

n+ 1 Sn -

L

-s

'=.

S

au

k=n+1

ak+

/00L

1

k=n+1

a,

=

L00

k=n+1

a k P(1

+ Gk) /00 L

=. p + Un'

k=n+1

ak

(11) (12)

(The foregoing operations are valid since it will turn out that s, of. s.) Thus jUnl~·

00

L

k=n+1

~.Cgn+1

lakGkl/lan+IJFn+1

00

L

k=O

Iplk(1

+ gn+d

Cg n+ 1

(13)

l-p(l+gn+I)'

which actually shows a bit more, namely, Sn+1 Sn -

S S

=

P

+ o(suplak+1 k>n

pak

_

II).•

(14)

8

I. Sequences and Series

Corollary. Proof

Cfl a :=; Cfl/.

This is true since (15)

limlanl 1 / n slim lan+1/anl . •

Another useful result has to do with the order of growth of partial products. Theorem 2.

If

», = for some t

E Cfl N, Gj

n-l

Il (1 + G),

j=O

n ~ 1,

Vo = 1

i= -1,j ~ 0, then there is an t*

E CflN

(16)

such that (17)

Proof

We have

n-1

Un

= ell (1 + Gj)'

(18)

j=no

(19) but the quantity in square brackets is the Cesaro means of a null sequence, and hence the nth term of a null sequence, say, .0, s diverge to

+ 00, and b; =

n

L akbk = o(sn)'

k=O

0(1). Then (46)

Proof n

L

k=O

f.lnkbk = 0(1),

(47)

22

l. Sequences and Series

where (48) by the Toeplitz limit theorem, Theorem 2.1(3). Note b may be a complex sequence. •

Theorem 5.

Case I.

Let

S E ~s,

an >.0 and h; = an/l1an with I1h n = 0(1).

Sa; . O. Then s diverges and (50)

Proof

Case I. h; - ho

n-I

= k~O I1hk =

0

(n-I ) k~O = o(n), 1

(51)

by the lemma (with s = {n}). This means

1)r 1 = 0(1),

[n(an+ dan -

(52)

or, since an+ dan ,(n + f3t ,=0

±

se r

(f3 - aY-,'(r - (}):-, (s - r).(n + f3)

k

=. L c:(n + f3)o-S + O(nO- k- I ), s=o

and from this the theorem follows immediately.

•

+

O(nO- k- l )

(64)

Chapter 2

Linear Transformations

2.1. Toeplitz's Theorem in a Banach Space The most famous result dealing with the regularity of linear transformations is the Toeplitz limit theorem. In its classical guise, this concerns the convergence of transformations of (6's where the (n + 1)th member of the transformed sequence is a weighted mean of the first n + 1 members of the original sequence: n

s, = L PnkSk' k=O

(1)

The theory of this transformation is covered quite adequately in the existing literature (Knopp, 1947, Hardy, 1956; Petersen, 1966, Peyerimhoff, 1969). For what follows, we shall need an abstract version of the theorem. This, in a way, is fortunate, since the proof is cleaner than the proof of n, and so the matrix

° ° °

fl OO

U

=

[flnkJ

= [

fllO fl20

flll

(2)

fl21

is lower triangular. If rows sum to 1, n

L flnk

k=O

= 1,

(3)

then U, or the transformation defined by U, T(s) fin

=

n

L flnkSk' k=O

n

= s,

= 0, 1,2, ...

(4)

is called a triangle. We now restate the Toeplitz limit theorem in a form suitable for U in Eq. (2). Theorem 1.

(i) (ii) (iii)

U is regular iff

Li:=o Iflnkl ~ M; LI:=o flnk = 1 + 0(1); flnk = 0(1), k fixed.

(ii) and (iii) are obvious. Condition (i) of Theorem 2.1(2) produces ~ M for Isjl ~ 1, but for n fixed, there is an s for which this maximum is attained, namely, Sj = sgn flnj' The smallest M that will do is, in fact, the norm of T, and Proof

ID=o flnjsjl

IITII = sup M n , n

u, =

n

L Iflnkl· k=O

•

(5)

Of course, if U is a triangle, condition (ii) can be deleted. A method U satisfying (i)-(iii) is called a Toeplitz method. If flnk ?: 0, U is called positive. Complex Toeplitz methods are very useful and, when applied to the right sequences, can greatly enhance convergence. Because of their numerical stability, positive methods are the most frequently used ones.

28

2. Linear Transformations

Real positive Toeplitz triangles (even triangles that are "nearly" positive) have an important limit-preserving property; i.e., negative elements appear in only a finite number of columns of V iff (6)

jar all real bounded sequences s [see Cooke (1955, p. 160)].

One cannot expect too much from any linear summability method. The improvement in convergence is, in general, no greater than exponential; in other words, (s, - s)j(sn - s) = O(t), 0 < y < 1, and one cannot find a method, at least a positive triangle, that is accelerative for all convergent sequences. To see this, let V be such a method and s a monotone decreasing null sequence. Then

f

k=O

JinkSk > ~

f

JinkSn

k=O

= 1.

(7)

~

Pennacchi (1968) has shown that no method of the form p

s, = L JijSn-p+j,

(8)

j=O

where the Jij are independent of n, can be accelerative for all sequences. (The foregoing is a band Toeplitz process with constant diagonals.) A minor modification of his proof permits the generalization that no band Toeplitz process can be accelerative for all Cf/c. Whether any Toeplitz method can be accelerative for all Cf/c is an important open question. There are many triangles that sum divergent bounded sequences, but it is a consequence of the Banach-Steinhaus theorem that no regular triangle can sum all bounded sequences (Schur, 1921). The polynomial A Pn( )

~ Jink Ak = On (A1 -_ AAnk) ' = Lk=O

k=l

nk

n

~

0,

(9)

is called the nth characteristic polynomial of the triangle U. The regularity and accelerative properties of V are intimately connected with the location of the complex zeros Ank of Pn(A). A useful function, called the measure of V, is (10) For all AEN, (11)

2.2. Complex Toeplitz Methods

and K

K

= 1.

29

is called the modulus of numerical stability of U. When U is regular,

Let '{J Em(r) C

'{Js denote

sn

=

S

the space of all exponential sequences of the form

+ cdi + czYz + '" + cmY;:',

(12)

where cj =1= 0 is complex and Yj E I",a nonempty compact subset ofthe complex plane not containing O. We assume the Yj are distinct. As the following theorem shows, the properties of the measure of U determine whether or not U is regular and accelerative for this important class of sequences.

Theorem 2. (i) (ii)

AE r.

Let 0"(..1.) Let 0"(..1.)

Proof

Let U be a triangle with measure 0"(..1.), .s4 = =1= =1=

'{J Em(r).

I, A E r. Then U sums .s4 with s = s iff 0"(..1.) < 1, A E r. A, AEre N. Then U is accelerative for .s4 iff 0"(..1.) < IAI,

The basic inequality from which these statements follow is (0"(..1.) - e)" O. To show sufficiency in (ii), for instance, choose an s so that the above holds for Z = Yl, Yz,.··, Ym' and let y* = suplv.]. Next pick 1 > b > 0 so that (14)

O"(Yr) ~ IYrl(l - b).

One has 8n 15" -

S \ 5

~.

CI8"-sl C~ Icrl[IYrl(l-b)+e]" *n n!

+

T)

crr(r)

00

r~lr(r+T+n)r(r-n)

(6)

results by using the known formula for 2F1(I). Thus

I

Icr+n+ll(n + r)! n - n! r = 0 (n + r)n+ r + 1 r!

If I < ~

:::; Ipln+l

~~~Ivrl r(~(~

1) (n

+

1, 2n

+

1;

Ipl),

(7)

being Tricomi's -function, since (n + T)n+r+ 1 is increasing in T. However, each term in the Taylor series for is decreasing in n. Letting n = 0 gives an upper bound, and the theorem results. • The foregoing also shows that U is, ultimately, exact (r n =.0) for sequences of the form

s, = Since the weights method is

/lnk

S

+

m

I

C r.

r=l(n+T)r

(8)

alternate in sign, the numerical stability of this

(9) or K(U)

= 3 + J8 = 5.828,

even worse than the Salzer method. The method is regular for another important class of sequences.

(10)

40

2. Linear Transformations

Theorem 2.

{(sn - s)}

Proof

Let d be the class of

S E ((j s

= r E ~TM' U is regular for d. For some

whose remainder sequences

t/J E '1', (11)

and thus t, = (_I)n

f p~r-l,

0)(1 - 2t) dt/J,

t)

= (_I)n+l f/~r-I'O)(t)d¢,

1¢(t) = t/J ( -2- .

(12)

The following facts are in Erdelyi (1953, vol. 2,10.14,10.18):

tE[-I,l]; p~r-I,O)(COS

8)

~

K(8)n- 1/ 2 cos[(n

(13)

+ r/2)8 + C],

(14)

(14) holding uniformly on compact subsets of (0, n), K (>0) being integrable on such subsets. Pick b, 0 < b < 1, and write

Irnl s

r.

d¢

+

C'n-

1/ 2

f_-I

bK(8)d¢

+

f-/¢,

o = arccos t.

(15)

Now pick b to make the first and last integrals < e/3; the second will be O. (1)

[See Lotockii (1953), Vuckovic (1958), and for applications and other references Cowling and King (1962/1963) and Agnew (1957).J Theorem 2.2(3) as it stands provides no information about the regularity of U, but, starting with Eq. 2.2(24) the proof is easily modified:

IJlnk I -< R k nn

j~l

= CR k

(a

+j

- 1 + I/R) ( a +') , ]

R > 1

r(n + a + I/R) r(n + a + 1)

= RkO(n(l/R)-l) = 0(1)

III

n.

(2)

Since U is a positive triangle, this is all that is required to show that U is regular.

2.3.9. Romberg Weights The Romberg weights are a triangle, not a positive one, that bears a close relationship to an extrapolation procedure attributed to Richardson and also

2.3. Important Triangles

45

to a method attributed to Romberg for improving the accuracy of integration by the trapezoidal rule. Both procedures are treated at length in most books on numerical analysis; see, for instance, Isaacson and Keller (1966), Bauer et al. (1963), or the articles of Bulirsch and Stoer (1966, 1967). Take a > 1, Po = 1, and

PiA) =

JJ A _ o n

1_ a

?

k'

n ~ 1.

(1)

/lnk then is the kth symmetric function of (17- 1,17- 2 , ••• , a- n); but it is not necessary, nor even desirable, to compute /lnk to use U. It is much more convenient to use a lozenge algorithm (see Chapter 3). We have shown that U is regular. Furthermore

(2) and the product on the right is convergent as n -+ 00. It cannot converge to zero unless one of the factors is zero. Thus U accelerates an s in the space of exponential sequences ~Em(r) iff the y, of largest modulus [see 2.1(12)] is equal to a- k for some k ~ 1. Since (an + I - l)Pn+ l(A) = (Aa n+ 1 - I)P n(A), (3) one gets, by equating powers of A, the recursion formula (a n +

1

-

1)/ln+l.k

=

an+1/ln,k_l - Iln,k'

(Ilnk = 0,

k < 0, k > n).

n, k ~ 0

(4)

2.3.10. Hiqqins Weights

Higgins weights are designed for sequences having the following behavior. (1) cf. Eq. 2.3.3(10). In contrast to the summation formulas of Section 2.3.3, the method to be derived here is regular. Let

y E rt.

(2)

By the same steepest descent argument used in Section 2.3.4,

vv,,/n! ~ j2n/n(zo + l)e Y(Zo+ll( - zo)- n, The Toeplitz limit theorem shows U is regular.

Zo = -0.278464543.

(3)

46

2. Linear Transformations

We now demonstrate two theorems about this process. According to Theorem 1.6(6), (1) may be rewritten 5"

~ 5 + (-

or

~(m)

f (n +c~ yr:

c'

+ (-1)" L

(4)

r= 1

~(m)

( r y + (n + "yr+ r=1 n + y m

5" =.5

1)"

m 2 1,

l'

(5)

a bounded sequence. [(5) holds also when (4) is convergent.] Note that

I"

k=O

(_I)"+k ll

(k

r"k

+ y)1

= 0,

1 :-s; j :-s; n,

(6)

and this provides the following theorem. Theorem 1.

Let (1) hold with y > 0. Then

Is" - 51 :-s; Proof

Left to the reader.

m < n.

supl~~m)l/ym+1,

"

(7)

•

When the series on the right-hand side of (4) is absolutely convergent for n = 0, much more can be said. Theorem 2. Let the series on the right-hand side of (4) converge absolutely, y > 0. Then

(8)

where M is independent of nand k is the smallest integer such that Proof

ci of.

0.

The integral representation 2.3.3(12) gives (9)

Therefore, - I -< W 1 ~ I c~ +" I < 1 ()" 1r; L. - r - - W C Y Y , " r= 1 y"

Choosing k to be the smallest integer with sequence asymptotically by

Ir"1

c(y)

= ~ ler I. L. r= 1

r'

Y

(10)

c~

of. 0, we can describe the error

~ 1e~I/nk

(11)

and combining this with (10) proves the theorem.

•

2.3. Important Triangles

47

The computational gain on using the transformation (2) on series of the kind (l) is spectacular, the convergence being accelerated more than exponentially, actually, by a factor Anlnn. It is rare indeed that a linear method performs this well. Of course since the method is a positive triangle, it is numerically stable (K = 1). The analysis of this procedure illustrates very well the gulfthat exists between the acceleration of sequences that oscillate about their limits and those that do not (logarithmically convergent sequences being cases of the latter). For the former, there exists a profusion of highly efficient summation procedures, while for the latter, the suitable methods are much less efficient and invariably nonregular. For sequences that neither oscillate about their limits nor approach their limits monotonically, almost all known methods fail. (Recent numerical evidence, however, indicates that the implicit procedures of Chapter 9 hold some promise for such sequences.) 2.3.11. Inverse Methods

Some interest attaches to the inverse of a Toeplitz method V, i.e., the triangle U* = [Il~k] defined by

t; =

n

L Ilnk Sk, k=O

Sn

=

n

L ll~kSk k=O

(1)

for all sequences s, s. The characteristic polynomials P~(A) of U* usually cannot be found explicitly. In one case, however, this can be accomplished, that is, for the nonregular methods discussed in Section 2.3.5. If (2) and application of the formula in Erdelyi et al. (1953, vol. 2, 10.20(3» gives

where

P~(A) = (r + 2A dldA) v,.(A) n+t

(3)

(4) Furthermore,

IAI < 1 IAI> 1.

(5)

48

2. Linear Transformations

For litl < 1, (5) follows from (4) by dominated convergence and taking a termwise limit. For Iitl > 1, consider the integral

f

u« /3, it) =

(1 - t)n+a(1

= (1 + it)n+ P

f

+ itt)n+P dt

zn+>(1 - yz)n+ P dz, IY

= (1 + it)"+p[f +

y = it/(I + it)

{~J

= (1 + it)2n+a+ P+lit- n->-IB(n + a + 1, n + /3 + 1) - (1/it)I(/3, a; Ilit)

(6)

The use of Stirling's formula and the relation (n

+a+

I)I(a,

/3; it) =

n -a -l/3 21\ -it) F ( n- +

(7)

shows (5) for Iitl > 1. A quick computation shows U* is regular, but it is a poor method to use on exponential sequences since S will converge more slowly than itn for all litl < 1. A considerable amount of research has been done on inverse methods. The paper by Wilansky and Zeller (1957) contains some important results and a number of useful references. 2.4. Toeplitz Methods Applied to Series of Variable Terms; Fourier Series and Lebesgue Constants

Often it is important to discern the effect of U on a series of variable terms: f(z) = s(z) = sn(z) = sn(z) =

co

L !k(z),

k:;O n

L fk(Z),

k=O n

n

L J.1nk Sk(Z) = k=O L Vndk(Z),

k=O

(1)

2.4. Toeplitz Methods Applied to Series of Variahle Terms

49

A straightforward application of Cauchy's integral formula shows that for U to sum a Taylor series about the origin anywhere within its circle of convergence, it is sufficient that Pn(A) -> 0 uniformly for all IAI ~ 1 - 15, for every o < 15 < 1. Obviously this is a weaker condition than regularity. Necessary and sufficient conditions are presented later [Theorem 4.3.1(1)]. Applications to Fourier series present somewhat different problems. Let f E L( - n, n) and let ak' bk be the Fourier coefficients generated by f, ak

= ~1

I

n _"

f(x) cos kx dx,

bk

= ~1

I

n _"

f(x) sin kx dx.

(2)

Let sn(x)

= !ao +

n

L (a

k=l

k

cos kx

+ b, sin kx).

(3)

Assume that U is a real triangle and that six) is the result of applying the summability method U to sn(x). The convergence of s; can be related to the constants (4) called the Lebesgue constants for U. The standard theorem establishing the connection is due to Hardy and Rogosinki (1956). to

Theorem 1. Let U be regular with Ln(U) bounded. Then sn(x) converges (5)

wherever this exists. If f is continuous on a compact set K c [ - n, n], then sn(x) converges uniformly to f(x) on K. Conversely, if Ln(U) is unbounded there is an f E C[ - n, n] for which six) -f> f(x) at some point x E [ - z, n]. Proof

See Hardy and Rogosinski (1956, pp. 58ff.).

•

An important related result is due to Nikolskii (1948).

Theorem 2. (i) (ii)

limn~oo sn(x) = f(x)

limn~ 00 /lnk

at every Lebesgue point of f iff

= 0 and

Ln(U) is bounded.

Proof The proof is established by an appeal to results of Banach on weak convergence in Banach spaces [see Nikolskii (1948)]. •

50

2. Linear Transformations

As a philosophical consequence of such theorems, much research has centered on describing the asymptotic properties of L n( U) for various summation methods. Concerning the Hausdorff transformation Ilnk =

f

(~)

x\1 - x)n-k d

(6)

(see Section 2.3.3), Lorch and Newman (1961), improving the earlier work of Livingston (1954), have found the following result. Theorem 3.

Let U for (6) be regular. Then Ln(U) = C cP In n

+ o(ln n),

(7)

where (8) the sum extending over all the discontinuities ~k (at most countable) of , and ~(f(x)) represents the mean value of the almost periodic function f(x). Furthermore,

f1 IdI

4

0::;; C cP ::;; 2 n

(9)

0

and CcP = 0 iff is continuous.

e;

Theorem 4. Let E E be monotone. Then there exists an increasing absolutely continuous for which

(10)

LiU) i= O(E n In n).

This result establishes that the error term o(ln n) in (7) is the best possible and cannot be improved even for an increasing absolutely continuous . For the Cesaro method

L (U)

=

n


0 such that for each k > 0 lim

nal Vnk - II >

0,

(18)

then (19) implies f is constant almost everywhere (a.e.).

Proof

Let

h = -1

I" f(x)e- lkX. dx.

2n _"

Then (20) and so (21) There exists a subsequence {nj}, nj -+ 00, such that both lim j_ oo njjvnj,k - 11 = Ck > 0 and also, by Holder's inequality, limj_oo njllsn/x) - f(x) I 1 = O. But this implies Ck I fk I = 0 for each k > O. Since a function in X is uniquely characterized by its Fourier coefficients, f must be a constant. • This shows that the approximation in norm of sn(x) to f(x) by Toeplitz methods satisfying (18) cannot be improved beyond the critical order n a: no matter how smooth f is. Saturation theory deals with the optimal order of approximation to functions E E c X by a triangle U. For instance, consider the Cesaro means, Vnk = (n + 1 - k)/(n + 1). One cannot have IISix)f(x)11 = 0(n- 1 ) for f E C[ -n, nJ no matter how smooth f is, since IV nk - 11 = k/(n + 1) and a = 1 in the previous theorem. For all nonconstant functions in C[ - n, n J, sn(x) approximates f with an order at most O(n- 1). In fact, this order is actually attained since for f = eix , Ilsn(x) - f(x)11 = I/(n + 1). One says the Cesaro triangle is saturated in C[ - n, nJ with order 0(n- 1 ) . One problem is to characterize those elements in X for which the optimal order is attained. In some cases this can be done. Define -r

~ f(x)

=

f"

1 _/(x - t) cot "2t dt, 2n

the integral being a principal value integral.

(22)

53

2.5 Toeplitz Methods and Rational Approximations; The Pade Table

Theorem 6. Let six) be the Cesaro means of the Fourier series for f(x), X = C[ -n, n]. Then

Ilsn(x) - f(x)11 = O(n- 1 ) iff](x)EC[-n,n],esssuPIJ'(x)1
- 1.

(8)

The characteristic polynomial for the method defined by (2) and (3) is then (9) so P; has its zeros on the ray connecting 0 and y. An argument based on Eq. 2.3.6(3) and Theorem 2.2(5) shows that U is regular iff y is real and y < O. In this case sn(z, y) ~ s(z) for all z interior to the circle of convergence of (1). Also, the rational approximation sn(z) will converge for all z real, negative, and interior to the circle of convergence of (1). For many important functions, however, this appraisal of convergence is far too weak. These are the functions that have a representation as Stieltjes integrals

s(z) =

J

OO

dljt

--,

o 1 - zt

Ijt E tJl*,

z rt= Supp Ijt.

(10)

2.5 Toeplitz Methods and Rational Approximations; The Pade Table

Theorem 1. Define

Let the representation (5) hold and t/J have compact support. a=sup{tltESUppt/J}.

Let YE~, l1Y¢ [0, IJ, zalyER, Then

°s zal» s 1, a> - t, /3 > -1.

1FnCz, y)1 s KnCz), Kn(z)

55

:'=::

(11)

(12)

zt

2 sup - o sr s« I 1 - zt

q = max(a,

/3, -1),

I fiall/y -

1 la/2+1/4Iyl-/l/2-1/4 nQ+l/2 , q!ly-1/2 + 1/Y _ 11 2n+ v

n ---+

J

(13)

00.

Proof

Fn(z, y) =

_R~a,/l)(I/y)-l LX) [ztl(1

-

zt)JR~a,/l)(zt/y) dt/J.

(14)

The proof will require the following well-known estimate (Szego, 1959, p. 194). For wi (0, 1),

R~a,/l)(w)

:'=::

_1_ (w _ 1)-a/2-1/4 w-/lI2-1/4(w l / 2

2JM

+ ~-=-i)2n+v,

(15)

branch cuts for (w - l )" and w" being taken along ( - 00, IJ and ( - 00, OJ, respectively, This result holds uniformly on compact subsets of ~ - [0, 1]. Using the fact that R~a,/l)(x) can be bounded algebraically (Erdelyi et aI., 1953, vol. II, 10.18(12)) completes the proof. •

Corollary. Under the conditions stated above, sn(z) converges exponentially to s(z). Further, the rational approximations sn(z, az) converge uniformlytos(z)oneverycompactsubsetSof~ - Dla, w),alsoexponentially;i.e., ZES, (16) 1"

= sUPI(az)-1/2 ZES

for some M and

Example.

e. Note 1"

+ Jl/az -

11- 2,

< 1.

Let

s(z) = F(1,

/3; v; z),

v=a+/3+1.

(17)

56

2. Linear Transformations

Then a = 1, (18)

and (19)

On expanding (1 - zt)-l in powers of t one finds the first n terms, that is, the coefficients of 1, z, ... , zn- 1 vanish by virtue of the orthogonality properties of R~IZ·/l)(t). Thus

= O(z2n+l);

[znBiz)]s(z) - [znAn(z)]

(20)

i.e., in this case the rational approximation yields the [n/n] entry in the Pade table for F(1, fJ; (f. + fJ + 1; z), These rational approximations, by virtue of the theorem, converge uniformly On compact subsets of C& - [1, 00). For an extensive discussion of the construction and properties of Pade approximants, see Chapter 6. Using R~IZ·/l)(t) =

a useful formula for

(

l)n

~~

dn

(1 - t)-lZt-/l dt" [(1 - t)lZ+nt/l+ n],

rn can be derived by integration by parts:

rn(z) = .~n 0.

(2)

(Without loss of generality we may assume P -1 = 0, P 1 = 1.) Suppose a generating function for the set Pn(x) exists: co

L znpix).

g(z, x) =

(3)

n~O

Let tJ denote the operator tJ

= z dldz, Then

tJg(z, x)

=

co

L nznpn(x).

(4)

n~O

If the coefficients An' Bn, C, are rational in n the substitution of (3) into (2), multiplying by the lowest common multiple of An' Bn, C n and using the properties of the ~ operator, produces an ordinary linear differential equation for g (in the variable z). If a fundamental set of solutions for the related homogeneous equation can be determined, then the equation for g can, in principle, be solved. Once g is found, Pollaczek shows that X(z) = z-t S(Z-1) can be found, where x(z)

=

f~, z- t

(5)

and then, by using the inversion formula for the Hilbert transform (Shohat and Tamarkin, 1943, p. xiv) one can determine e :

1 = lim - -2' £-0

£>0

XI

it 0

[X(t + is) - X(t - is)J dt.

(6)

The appropriate weights for computing the [n - 11nJ Pade approximants to s(z) = Z-1 X(Z-1) cannot, in general, be given in closed form but can be

61

2.6. Other Orthogonal Methods; Pollaczek Polynomials and PaM Approximants

computed conveniently from (2):

o s k s n, n ~ 1, k i

< 0, j < 0, or j > i; 0"00

=

~

0;

(7)

1.

For the class of transcendental functions to be discussed, this is tantamount to having a closed form expression for the [n - lin] Pade approximant. The necessity for solving linear equations is avoided (cf. Secti~n 6.5) and, in fact, the s-algorithm for generating Pade approximants is tedious. The most general case considered by Pollaczek was for An' B n, en bilinear functions of n having a common denominator. Through a suitable normalization the recursion relationship can be written

+ c)Pn - 2[(n - I + A + a + c)z + b]Pn- 1 (8) + (n + 2..1. + c - 2)P n - 2 = 0, so An = 2(n - I + A + a + c)/(n + c), etc. We shall assume a, b, c, and A are real and a > Ib I, 2..1. + c > 0, C ~ 0, although often an appeal to continuity (n

will enable some of these conditions to be relaxed. In what follows p-=l", z ¢ ( - I, I), will denote that branch of the function that is positive for z positive and> 1. Let

B(z) = az

+b

p-=l"

= - i( at +

() B +t

, b)

t E ( - 1, 1),

' v~ I - t2

_ i(at

( ) B_t

+ b)

=t+

UL(t)

~' 2

vI - t

iJ1=7,

(9)

tE(-I,I).

We wish to solve the integral equation x(z) =

where

n2 X(z) =

2

-

.

2Ar(c

+

f:oo dl/JI(z -

l)r(c

(A + c

+ 2..1.)w(z)F

+ B(z))F

t),

z¢ Supp

(1 c-+A..1.+1 + B(z), c + 11 + B(z)

(1 - A+ B(z) c I c

l/J

+ A + B(:)

w(zf

)

(10)

W(Z)2

) (11)

62

2. LinearTransformations

Pollaczek's work guarantees that a solution l/J E '1'* exists; it will be given by the inversion formula (6). It turns out that l/J is differentiable. Writing dl/J = p dt, we have

pet) = i2 1 -

2 Ar(c

x [ (A H+(c)

+

l)r(c

+ 2..1)

w+(t)H +(c + 1) + c + B+(t))H +(c)

1 - ..1+ B+(t), c

= F ( ..1+ C + B+(t)

I

- (A

w+(t)

w_(t)H _(c + 1) ] + c + B_(t))H _(c)'

(12)

2) ,

etc. The computations are rather complicated, but straightforward. First, use the fact that oi : = W.;:l, and then Eq. 2.10(2) of Erdelyi et al. (1953) on H _(c + 1) and H _(c). Next use Eqs. 2.8(25) and 2.8(26). The result is

(13) _ Us -

2-2c-2).--2B+

w+

F

(2 - 2..12 _- cc,_1-A _AB+- B+ Iw+. 2)

The Wronskian of this pair of solutions of the hypergeometric equation is easily determined by the standard techniques, so that finally exp[(2 arccos t - n)(at + b)/j!=t2] x (1 - t 2)" - 1/21rCA + c + B+(t)W

pet) = 0,

tE(-I,I)

(14)

ItI > 1.

Pollaczek has shown that the polynomials defined by (8) are orthogonal with respect to this weight function. From this basic result a number of other Hilbert transforms and the recursion relationship for the corresponding orthogonal polynomials may be

2.7. Other Methods for Generating Toeplitz Transformations

obtained.Ifz ~ ez and s ~ 0, then

+ cos ¢,t

~

et + cos ¢,a

~

sin ¢/s,b

~

63

-sin ¢ cos ¢/s,

l' e- Zi (0) is not much better than vertical convergence. Numerical evidence seems to bear this out. As an example take f = (x + 0.05)- 1. The results of applying the algorithm are given in Table I. Note S~2) is nearly as accurate as any other entry in the table yet easier to compute and less subject to roundoff error than entries on its right. It is a curious contrast that in most other deltoid algorithms diagonal convergence is more rapid than vertical convergence. 3.2. General Deltoids The Romberg integration scheme leads one to analyze the more general deltoid scheme defined by (1)

Let S E ({;e. The idea is to determine when the above transformation is regular for any P. If S~k) ---> S is to hold, an induction argument shows (2)

72

3. Linear Lozenge Methods

From here on assume this holds. As before, write k (k) _

Sn

for some constants

flkm'

(3)

"

L. {tkmSn+m

-

m=O

Substituting (3) in (1) shows

k+ 1

k

k

I flk+ l.mSn+m = akm=O I flkmSn+m+ + b,m=O I flkmSn+m' m=O 1

(4)

and this will hold for all possible sequences iff (flkm

Multiplying by

Am

= 0, m < 0, m > k).

and summing from m = 0 to k P k+

1(..1.) =

(akA

(5)

+ 1 gives

+ bk)Pk(A)

(6)

(7)

Furthermore, in the notation of Chrystal (1959, vol. I, p. 431), flkm = ( -1)k+mg\_mO·),

(8)

where ,o/lr(A) is the product of AI, ... , Ak taken r at a time. An application of Theorem 2.2(4) furnishes the next result. Theorem 1. Let ak ri. [0, 1]. Then the transformation defined by (1) is regular for all paths P iff L (1 - a; 1) converges. It often happens, of course, that S~k) goes to S along some P much more rapidly than Sn goes to S as n -+ 00. Then the scheme defined by (1) is computationally desirable, as is the case for Romberg integration. This algorithm can be derived heuristically on the assumption that the given sequence s; behaves as Sn

=

S

+

k

I

r= 1

crA~,

(9)

(see Chapter 10). Consequently, one would expect the algorithm to be exact (s~k) == s), when s has such a representation. This turns out to be true and does not even require convergence. Theorem 2. Let s have the above representation with a j i= 0 for some complex constants c., Then S~k) == S, n ~ O. Proof

Trivial.

•

3.3. Deltoids Obtained by Extrapolation

73

3.3. Deltoids Obtained by Extrapolation Other deltoid formulas are generated by the Neville-Aitken formalization of the Lagrangian interpolation polyuomial [see Householder (1953, pp.202ff.)]. Let x be an arbitrary sequence of distinct numbers, and suppose there is a function f(x) such that f(x) = Sj,j 2 0, s being the sequence to be transformed. Consider the algorithm given by the following computational scheme: n, k 2 0,

s~O)

=

sn,

n 2 O.

(1)

One sees immediately, by referring to the appendix, that S~k) is the value at z = 0 of the Lagrangian polynomial of degree k in z that interpolates to Sj at Xj' n -:::;, j -:::;, n + k. Now f(x n ) = s., so if X n , for instance, is decreasing and f is reasonably behaved, f(O) = s, and for nand/or k large S~k) will closely approximate s. Laurent (1964) has shown necessary and sufficient conditions for diagonal (k ---+ (0) regularity of the algorithm. A minor additional effort enables us to show the same conditions are equivalent to regularity for any path. Theorem 1. Let x be monotone decreasing to zero. Then the algorithm (1) is regular for all paths iff Xn/X n+ 1 2 o: > 1, n 2 O.

Proof

By the formulas in the appendix, we can write k (k)

Sn

_"

-

1... Ilkm (n) Sn+m'

Ilkm(n)

m=O

=

k

f1 i=O,i*m X n +

i -

Xn+m

(2)

Note I Ilkm = 1. =: Assume that on the contrary where

EE

9f; and contains a subsequence converging to zero. Then k- I

Ilnk

=

f1 i=O Xn+i -

Xn+i Xn+k

> -----Xn+k-I -

Xn+k

(3)

(4)

since each term in the product is greater than 1. Thus Ilnk

> (1

+ En+k-I)/f;n+k-I

(5)

and, taking n fixed, we can pick k values -> 00 such that En + k is a member of the aforementioned subsequence of E. Thus Ilkn contains a positively divergent subsequence. Therefore, by the Toeplitz limit theorem, the transformation

74

3. Linear Lozenge Methods

defined by (1) is not regular (in k). In fact, this shows that (1) is not regular for any diagonal path. =: Taking products of xn/x n+ 1 ~ r:J. from n = j to k - 1 and reciprocating gives

k > j.

(6)

Write IJlkml

= A· B,

Then

(7)

n

A = m-I( 1 _ 1

=0

X

n

+m

Xn+m - i - I

)-1

(8)

But x, + .J», + m _ 1 _ 1 :s: o: - i - I , so A :s: d, where d is the limiting value of the convergent monotone increasing sequence m (1 - rx-i-I)-I [see Knopp (1947, p. 219, Theorem 3)]. Also,

n

B

__

j

:s:

n --_._--

k-m-l

Xn+m -

~O

n

X n+ m+ i+ 1

k-m-I

rx

n

Xn+m +i+ 1

-i-I

.

1 -rx - I

i=O

k-m-I

1


0 as k -> 00, m ~ 0 fixed, an application of a result soon to be given, Theorem 5.2(1), completes the proof. • Example 1. Let Xi = o', 0 < (J < 1. This choice yields a special case of the Romberg-Richardson algorithm 3.1(8). The algorithm is regular for all paths. Example 2.

Let

Xi

= 1/(i + 1). Then

Jlkm

(_1)m+k

= ---- k!

(n

+m+

(k)

1)k m '

(11)

3.3. Deltoids Obtained by Extrapolation

75

and (k+ Sn

1) _

-

(n

+ k + 2)S~k~ 1 k + 1

(n

+

l)s~k)

n, k

;:0:

0,

s~o)

= Sn, n > 0, (12)

is a deltoid formalization of Salzer's weight scheme for y = 1, Eq. 2.3.4(1). Clearly, the hypotheses of the theorem are violated. In fact, it is easy to show the algorithm (12) is regular on no vertical or diagonal path. Nevertheless, used on appropriate sequences, the technique is very valuable. Let s, = (GAM)n so that an ~'l/(n

+

1)

+ In[n/(n +

1)],

n

>

ao = 1.

1,

(13)

Then s, -> Y = 0.5772156649, Euler's constant. Suppose only ten terms of s are available. How weIl can one do in computing y? Table II lists the tenth ascending diagonal of S~k), each term of which requires so, ... , S9'

Example 3.

Let

Xi =

1/(1

+ i?

Then

_ 2( _l)m+k(n

ftk

m

+ m + 1)2k+ 1 (k)

---------~

k!(2n+m+2)k+1

m

(14)

and S(k+

I)

(k) = ( n + k + 2) 2 Sn+

(k

n

+

1)(2n

1 -

(

n

+ 1)2Sn

(k)

+ k + 3)

This algorithm is appropriate for sequences behaving as

co/n2

+ c l/n 4 + ... ; Table II

k

S~~k

0 1 2 3 4 5 6 7 8 9

0.626 0.578 0.577219 0.577214 0.577215590 0.577215682 0.577215669 0.577215665 0.5772156643 0.5772156644

(15)

76

3. Linear Lozenge Methods Table III k

S~~-k

0 5 10 15 20 25

0.6932 0.6931469 0.69314705 0.693147089 0.693147099 0.693147100

e.g., the sequence of iterates 1;, in the trapezoidal rule 3.1(13). Table III gives some entries on the 26th ascending diagonal for f(x) = (x + 1)-1 (In 2 = 0.6931471805.) It is easy to show that for the Toeplitz array U corresponding to (these weights yield the diagonal entries s~)) one has K(U) ;:::: e2 = 7.389,

~km(O)

(16)

worse than that for the methods given in Sections 2.3.4 or 2.3.5.Such numerical instability dictates great caution in the use of (15). There are several ways of looking at the acceleration properties of lozenge algorithms. One is to compare rapidity of convergence along different paths. Very little work has been done in this area. However, an interesting condition for horizontal acceleration in the previous algorithm is due to Brezinski (1972).

Theorem 2. Let x be monotone decreasing to zero and x.fx; + 1 ;:::: rx > 1. Then for S E CfS s , s~k+ 1) converges more rapidly than S~k), n ..... 00, k fixed, iff (17) Proof

Left to the reader.

•

The algorithm of this section can be derived formally from the assumption that S behaves as s; =

k

S

+ L crx~.

(18)

r::::. 1

Not surprisingly, the algorithm is exact for such sequences, even when x depends on s.

Theorem 3.

Let s have the foregoing representation with C m • Then S~k) == S, n ;:::: O.

i #- i. for some complex constants

Proof

Trivial.

•

Xi

#- Xj'

77

3.4. Example: Quadrature Based on Cardinal Interpolation

3.4. Example: Quadrature Based on Cardinal Interpolation A class of quadrature formulas derived from a general Hermite cardinal interpolation formula provides an excellent example of the summation process of Section 3.3. It has long been known that the approximation of a doubly infinite integral by a trapezoidal sum 1=

f:a)j(X)dX~hff(mh)

(1)

gives surprisingly good results in many cases; i.e., the series on the right approaches rapidly the value of the integral as h ---> O. For instance, ifj = «>' and h = 1, the sum has the value 1.77264, to be compared to = 1.77245. This agreement is nothing short of phenomenal, considering how few values ofjare required to define the sum, and indicates that something profound is gomg on. In many instances, however, there are knotty computational problems associated with (1). It may happen that the right-hand side is indeed a good approximation to the integral, but converges very slowly; in fact, those small values of h that give a good approximation produce a slowly convergent series. An example is

fi

1=

1 fa)

~

dx

_a)

h

1+~2 ~ ~

f

a)

I 1 + (mh)2'

(2)

One would like a procedure to calculate I based on as few evaluations of the sum as possible. One approach is to truncate the sum at N, which depends on h, and to try to find, given h and a suitable class of functions f, the values of N that produce optimal accuracy. This approach is the basis of the so-called tanh rule. However the iterates in that rule are not suitable for the application of the present summation procedure. A more general procedure, the BL protocol, is required; the subject is discussed in Section 11.3. Note that any procedure to compute I is adaptable to the computation of finite integrals; for instance, the substitution x = tanh t gives an integral over (-1, 1). (Some writers have conjectured that this change of variable is, in some sense, the best choice; again see the discussion in Section 11.3.) The quadrature formulas to which Section 3.3 is to be applied are generalizations of (1). Let.f: 9f ---> rc and h > O. The series ~(.f)(z)

=

sin

L f(mh) -W- - , a)

-a)

Wm

m

Wm

tt

= h (z

is called the cardinal interpolation series of the function Obviously, m e J, ~(.f)(mh) = f(mh),

- mh),

(3)

f with respect to

h.

(4)

78

3. Linear Lozenge Methods

This formula and its remainder have been thoroughly investigated [e.g., Kress (1971); McNamee et al. (1971)]. Here a more general interpolation series is required. Define the p + 1 entire functions tiz), 0 :s; q :s; p, by

I

[1tzJr

_ zq [sin(1tz/h)JP+ 2«p-q)/2) tq(z) - , /h I a; h q.

1tZ

r= 0

r even

(IX) indicating largest integer contained in IX. a, of the Taylor series,

[sin z

~-J P + l

Lemma.

Let 0 :s; r, q

(5)

'

00

I

a.e",

,=0

==

a,(p) are the coefficients

Izi < n.

(6)

reven

s

p. Then

(7) Proof

Since tiz) = zq/q!

+ zp+ IUq(Z),

where "« is entire, (7) is immediate. Now let f : ;Jl -->

rc, /

s

(8)

p,

• P

I I 00

-

q

h > O. The series

E CP(~),

Tp,h(f)(Z) =

o :s;

00

q= 0

pq)(mh)tiz - mh)

(9)

is called the pth cardinal interpolation series of/with respect to h. Clearly ::q Tp,h(f)(z)lz=mh = pq)(mh),

o :s; q :s; p,

m

E

J.

(10)

For functions / analytic and bounded in a strip [- ia, ia] x ~, a remainder formula is easy to derive [see Kress (1972)]. Its exact form is not important for our purposes. It suffices to say that for all x the remainder is O(e -7t(p+ I la/h), It is the integration of (9) that provides the desired quadrature formula: Ip,h(f) = h I

00

_. 00

P

I

q=O

q even

bqpq)(mh), (11)

79

3.4. Example: Quadrature Based on Cardinal Interpolation

A simple recursion formula exists for the computation of bq (Kress, 1972).

If the first several such formulas are recorded, it turns out that I Oh is given by (1), I z p+l,h = Izp,h' and

14 h

=

I Oh

5h3 + 16 Z n

h5 L j"(mh) + 644 -00 n 00

(12)

L j""(mh). 00

-00

A remainder formula can be computed directly from that for (9). Theorem. Letfbe analytic in [-ia, ia] x !Jll,f(x uniformly for - a ~ y ~ a, and

+ iy)

~

Oasx

~

± 00 (13)

Then I =

f':'

00

f(x) dx exists and (14)

This is a generalization of a result (p = 1) first given, apparently, by Luke (1969, vol. 2, p. 217). Now for p,f, N > 1 fixed (and thus a, which may be taken as the distance from!Jll to the nearest singularity of f), let h = N/(n + 1) and define O~n~N-1.

(15)

Equations (14) and 3.3(18) suggest taking x, =

e-na(p+Z)n/N

(16)

in the summation formula 3.3(1). Thus one can compute the S~k) array for ~ N - 1. Equation (16) turns out to be a very happy choice, Take as an example f(x) = l/n(x Z + 1), p = 0, and consider formula (2). Then a = I. The lozenge formula is

o~ n + k

O~n+k~N-1.

(17)

Table IV gives the results for N = 4. Thus we have I to almost eight significant figures with only four evaluations ofthe sum in (2).

80

3. Linear Lozenge Methods Table IV n

s;

0

1.524868619

s~l)

1.0903314 [ I 2

1.018129443

3

1.003741873

\(3)

S~2)

0.976293939 0.999181169 0.999966081

·n

1.000214887

0.999999598

1.000001532

3.5. General Rhombus Lozenges This section shows how a lozenge algorithm can be developed for the orthogonal triangles discussed in Section 2.3.6. Theorem 1. Let {Pk(X)} be a system of polynomials orthogonal on [ -1, IJ with respect to Ij; E '¥ with PH I(X) = (Akx

+ Bk)Pk(X) -

CkPk-l(X),

k ?: 0, P-l == O.

(1)

Then the sequence transformation defined by n, k ?: 0,

where

ak = (B k + Ak)O"k/O"H 1, bk = 2A kO"k/aO"k+ l ' Ck = -CkO"k-J!O"k+l, O"k = O"k(a) = Pk(2/a + I),

= 0,

S~-l)

s~O)

=

Sn,

(2)

(3)

a> 0,

is regular for any path P. Proof

First, note that O"k satisfies

= [(2/a + 1)A k + Bk]O"k - CkO"k-l· This shows that ak + bk + Ck = 1. Thus for some constants Ilkm, O"k+ 1

(4)

k S(k) n -

'\'

(5)

S

1-J J1km n + m ,

m=O

and putting Sn == 1 in (2) shows that [llkmJ is a triangle. Proceeding as before, we find that Pk(A) satisfies

P- 1 == 0,

(6)

3.5. General Rhombus Lozenges

81

and this is precisely the recursion relationship satisfied by Pk(2)./a + 1)/ Pk(2/a + 1), and since the two agree when k = 0 and k = 1, identically

Pk(J.) = Pk(2)./a + 1). Pk(2/a + 1)

(7)

Theorem 2.2(5) then asserts that U is regular. The rest of the proof is as in Theorem 3.2(1). • The computational scheme for the algorithm is as follows:

0 So

0

~( I) '0

S(2) , 0

SI

0 S

0

~\"~S(2) /1

2

S~I)

3) Sb

S(3)

, 1

S~2)

S3

In the algorithm it is much more efficient to compute t~k)

= (J kS~k)

(8)

and then divide t~k) by (Jk (which itself satisfies a simple recursion relation) to get S~k). The algorithm becomes t~k+ 1) t~-I) S~k)

(Jk+ 1

= (B k + == 0,

Ak)t~k)

t~O)

= t~~)/(Jk; = [(2/a +

=

2A + ~k t~k~ 1 a

-

Ckt~k-l),

n, k ~ 0, (9)

Sn;

1)A k + Bk](Jk - Ck(Jk~ I'

k ~ 1.

As an example consider the Chebyshev polynomials T,,(x) with a = 1. These satisfy (10) Thus (11)

82

3. Linear Lozenge Methods

Even simpler is the algorithm given by the Chebyshev polynomials of the second kind Uk(x): Uk+l(x)=2xU k ( x ) - U k -

1( X ) ,

i

:».

U_

1

=, 0 ;

U o = l , (12)

and so (13) For I;. G"k(l) = {I, 3,17,99,577,3363,19601,114243, ...},

(14)

G"k(1) = {I, 6,35,204,1189,6930,40391,235416, ...}.

(15)

and for Uk Both satisfy k

~

1.

(16)

Applying the I;. algorithm to the sequence s, = LI:=o (_I)k with the computational scheme yields Table V for t~k). This sequence is divergent, and thus S~k) cannot sum s along all paths P (s must converge if that is to happen). However, it is easy to show that S~k) -> t as k -> 00. This, in fact, is the traditional "sum" assigned to the sequence [see Knopp (1947, Chap. XIII) for a historical discussion]. The sequence {s~)} obtained from the preceding table is one of dramatic precision: {I, 0.333, 0.529, 0.495, 0.501, 0.49985, 0.500025, 0.499996,...}.

(17)

3.5. General Rhombus Lozenges

~B

It is likely that no other linear method is more efficient than 7k for sequences that alternate around their limits. Two-dimensional algorithms can also be developed for the nonregular class of triangles of Section 2.3.5. The case r = 1 (Legendre polynomials) is particularly simple: (k+1) _ S" -

+

( 2k

1)[2S(k) - S(k)] - kS(k "+1"" (k + 1)

1)

,

n, k > 0,

(18)

In general, this algorithm will be regular only along vertical paths. However for an important class of sequences it is regular along any path. Theorem 2. Let r regular along any P.

E

9l T M • Then the transformation defined by (18) is

Proof (k)

_

r" -

(k)

s"

L J.1kmr"+m k

_

s -

-

so for any p. Furthermore, it is a Hilbert space (11)

Theorem 1. The optimization problem for ;if has a unique solution given by

VnO = Co

+

(1- A)-I;

sksn-I;

(12)

where ( 13)

In particular, when mk = k, VnO = Vnl and Ilellx = An + 1/(1 - ,.1.)3.

= ... = vn. n- I = I,

V,m

= (1 - A)-I,

This approximation, which works" best" for all .Yf, is, then, a o - A)-I. Thus

+ an-I + an(1

s;

= Sn + anA/(1 - A),

+ a I + ... (14)

and the transformation offers little improvement over ordinary convergence. Things are not much better for /~c, the space of convergent real sequences, as will be shown. Let A be an infinite real matrix [au], i,j 2 0, and denote by An the (n + 1) x (n + 1) truncate of A, and by s, E .~n+ I the n + I truncate (so,···, sn) ofs E ~s· A is assumed to satisfy the following three hypotheses: (1) (2) (3)

An is positive definite; sUPn,ilanil S M; Iimn~oo «; = aOk> k 2 O.

4.2. Optimal Approximations in I' and .4Ic

Now define ,1t

==

,1t(A) =

{SE.~sIS~PSnAn-lsJ

<XJ}'

93

( 15)

Germain-Bonne (1978) has established the following. Theorem 2.

,1t'(A) is a Hilbert space with inner product

(s, t) = lim snA; ltJ.

(16)

Furthermore, Yr(A) is a Hilbertian subspace of both /00 and case the representer ~(k) having the property for all is the (k perty

+

SE

s

= (s, s)

for all

SE

In either (17)

,yt'(A)

I)th row of A. In the latter case the representer

.gfc.

~

£'(A)

having the pro(18)

is the first row of A. Now let e = ~ -

n

L

k=O

(19)

J.1.nkS(k).

The problem is to determine J.1.nk to minimize lIell.#". It can be shown that the problem has a solution, and the resulting Toeplitz method is regular for

Yf(A).

As an example, let d > I and take

A =

Then An is positive definite and

"'1

1 d ... 1 1I I d · .. .

r

·· ·

..

(20)

.

n+d-l -1 -1

···-1

-1

-1

(I)

(21 )

-1 One has (22)

94

4. Optimal Methods and Methods Based on Power Series

and the space Yl' is the space of sequences in f!Ilc for which lim s, A nor, those satisfying

I

00

k=1

(Sk -

SO)2


N. Then choose ko ~ N to make the contribution of the second and third terms to the integral < e12, and then n large enough so the first term contributes < e12. (Note vnk -+ 1, k fixed.) =: This requires a much more subtle argument. We shall only sketch the proof. The reader unable to fillin details should consult Gordon's paper (1975). (ii) is obvious. To establish the necessity of (i), assume it is false. This means given e > 0, IVnk I ~ (1 + e)k holds for two nand k sequences with the n sequence unbounded. But, clearly, the k sequence must also be unbounded. Now we construct two sequences inductively, {nr } , {k r } , as follows: (i) (ii) (iii)

Let no = ko = 0; assume nr - 1 , kr - 1 have been chosen; pick n~ > nr - 1 so that

Ivnd < 1 + s,

(4)

(this is possible because U is a triangle); (iv)

choose n, >

n;., k; > k

r- 1

so that

'±:./l." i ! =

IVn"kd

! J=kr-

(v)

~ (l + e)k~;

(5)

choose k, > n..

We now construct a sequence s E Cf}". Choose 15 E (0, I) so that b( 1

+ e) =

p

> 1.

(6)

Let ao

= 1,

with empty sums interpreted as

°

and sgn

°=

(7)

0.

96

4. Optimal Methods and Methods Based on Power Series

Then n s· n a·v· I . I = I .~ Is_nrI = II.~ J Iln r,] J nr,] J-O J-O r

r

2 £5k~(I

>

+ e)k~

kr

L: lajllvnr)

-

- 1

j=O

l; - L £5j(l + e) 2 kr -

I

j=O

l~

Thus s contains an unbounded sequence.

I

+e

- --' --> 1-£5

CfJ.

(8)

•

It is easy to show that the regularity of U for rc" implies Pn(A.) = 0(1) for each A. EN. However, regularity for rc" does not imply regularity. Take

Ilnk =

{

n

0,

0~k~n-2

-n, I,

k=n-l

+

(9)

k = n.

Then (10) but U is not regular. A Taylor sequence s is a sequence such that ISn II/n = 0(1). Clearly, the space of Taylor sequences contains rc". Bajsanski and Karamata (1960) have shown that U (not necessarily a triangle) takes a Taylor sequence into a Taylor sequence iff, given e > 0, there is an M such that IIlnk I < ekM", n, k 2 O. For additional material on such sequences, see the thesis by Heller referenced in Bajsanski and Karamata (1960). The transformation we wish to study here is defined by formal power series. Let S E rc sand G(z)

=

00

L ak_Izk.

(11)

k=1

Let z = f(w) =

L h-l W \

k=l

f(1) = 1.

(12)

Define the sequence b by 00

L bk -

k=l

Iw

k

= G(f(w»

(13)

97

4.3. Methods Based on Power Series

Finally define T(s)

= s by n

~

o.

(14)

(If j~ + II + ... + In -# 0, we can use this to define sn for all n.) Further exploring the properties of U requires its matrix representation. Let

[I( w)]m =

00

'\'

1... j'k-I,m wk ,

m~l.

k=m

(15)

Define n

Pnk

=

I ./j,k+ j=k

Os k S n.

I'

(16)

Thus .fie, I = .fie, fJno = 1 + 0(1). Substituting the series (15) into (13) and interchanging the order of summation gives

bn =

n

I ak.fn.k+l k=O

(17)

so n

s; = Ko l I

j=O

hj

= r~;;ol

n

I

k=O

akr~nk (18)

Define

k = n k < n,

(19)

An application of Theorem 1 then furnishes immediately the next result.

Theorem 2. (i)

given

U is regular for L:

> 0,

IfJnkl

I

= 1.

(21)

Proof O. But then G(f(w» is analytic for Iw 1 < 1 + J, for some J > O. In particular (22) Thus V is regular for s. =: Assume (21) is not satisfied. Then there is a point w* such thatf(w*) = z* where Iw*1 < 1 and Iz*1 > 1 since f(1) = 1 and by the continuity of few). Let (23)

The corresponding sequence s is in '"5". Since G(f(w» has a pole at w = w* the radius of convergence of this series is Iw* I < 1. The series, therefore, cannot converge at w = 1, so limn~oo s; does not exist. • Corollary. Let feu) have a radius of convergence> 1. Then the two conditions sUPlwl=llf(w)1 = 1 and Ifink I

n

0,

---> 00.

99 and

(25)

Further, it is easily established by induction that Il"k ~ O. Thus

" " Illnk = I Illnkl = 1

k=O

k=O

and the Toeplitz limit theorem can be invoked.

•

4.3.2. Applications

Example 1. Let f(w) = _ 1 [ w 1 + q 1 - qw/(1

+ q)

]

,

q

~

O.

(1)

This gives, for all practical purposes, the Euler (E, q) method, although this formulation differs slightly from the standard one, the weights here having row sum of 1. Then k~O

(2)

n

~

O.

Theorem 4 provides the well-known result that U is regular.

Example 2. Consider the case where f is a polynomial of degree 1,2, or 3. Usually condition 4.3.1(21) is easier to check than Eq. 4.3.1(20). If f(w) = w - w 2 for example, condition (21) shows that U is not regular for 'l?" since f( -1) = -2. For first-degree polynomials, the only 'l? ,,-regular methods arise from f(w) = w, the identity transformation. For second-degree polynomials, (21) requires that f(w) = w(w + a)/(l + a) with a real and positive. Again, U is regular. For third-degree polynomials, both regularity and C(}" regularity arise. The results are as follows: first f(w) = w(w 2

+ zw + [3)/(1 + II + [3).

(3)

100

4. Optimal Methods and Methods Based on Power Series

For real

IX,

(i) (ii) (iii)

p, (21) holds iff at least one of the following holds:

IX,PZO;

- 1 < p < 0, -1, IX S

P
Pn+2, ... ,Pn+k-l)].

r~k)/rn

(17)

Note that (14) shows that G is regular for d p : Clearly g's continuity and value (15) at pe is necessary and sufficient for G to accelerate d o: • Note that if G fails to be regular for d some neighborhood of p.

Example 1. defined by

p'

then 9 must be unbounded in

Anticipating a little, we now discuss the Aitken (F-process, (18)

Note s; is always defined for n sufficiently large since an #. an+ l' SEdp : Here k = 2 and g(x) = 1/(1 - x). The theorem confirms the well-known fact that Aitken's D2-process accelerates all sequences in d p for each P, i.e., accelerates '{}/. Smith and Ford (1979) have given a useful one-way generalization of the previous theorem in which the functions 9 are allowed to depend on n as well as k. We again assume G == Gn satisfies (ii) and (iii).

Theorem 5. Let k ::2: 2 and the functions gn be continuous in a common neighborhood K of pe and converge uniformly in K as n -+ 00 to a function 9 with the property g(pe) = 1/(1 - p).

Then G as defined by (1) is accelerative for d Proof

p .

By uniform convergence gn(Pn+l,···,Pn+k-l) = 1/(1- p)

+ '1n,

where" is a null sequence. Thus r(k)

I :n

I = 1 + (p -

( 1+ ) '1n = (p -

1 + Dn) 1 _ P

()a null sequence, and this gives the theorem.

1)'1n

+ (1

(19)

D ~ p) + Dn'1n, (20)

•

5.2. Path Regularity for Certain Lozenges

105

Example 2. Let y be a fixed sequence EC(}s. Define the generalized Aitken i5 2 -process by sn

=

+ an+l(1

Sn

- (an+2Yn/an+1Yn+l»-1.

(21)

(It is assumed that the denominator in Eq. (21) is nonzero-more about this

later.) Again k = 2, but

(22) Thus if Yn/Yn+ 1 = 1 + 0(1), (21) is accelerative for C(}/. [Then, of course (21) is defined for n large enough. If the denominator of the right-hand side vanishes for any n*, it is customary to put sn* = sn*'] • 5.2. Path Regularity for Certain Lozenges When each element S~k) in a lozenge can be written as a weighted sum of s; +k' then certain simple conditions ensure the path regularity of S~k) even though the algorithm may be nonlinear. To be precise, let

Sn' ... ,

S(k) n

=

"IIrkm n+m' m=O k

n. k 2 0,

S

L.

(1)

where Ilkm == Ilkm(n, s). Theorem 1. For s E

C(}C

let

(i) L~=o Illkml :::; M, M independent ofn; (ii) L~=o Ilkm = 1; and (iii) Ilkm = 0(1), k ---+ 00, uniformly in n, along P. Then the algorithm defined by (1) is regular for s along P. Proof

By (ii) we can write, for s E c(}c, k

rn(k) -_

=

(k)

Sn

-

_"

S -

ko

L.

m=O k

L + m=ko+ L m=O

rn+mllkm

(2)

1

Applying (i) gives Ir~k)1 :::; sup j"2:.n

Irjl

ko

L IIlkm I +

m=O

sup IrjlM.

j>n+ko

(3)

106

S. Nonlinear Lozenges; Iteration Sequences

Either k -. 00 or n ---+ 00 on P (or both). If k ---+ 00 pick k o to make the second term < t:/2 for all n and then the first term will be less than t:/2 for k large. If n ---+ 00 simply take the direct limit. This shows r~k) ---+ 0 on P, or S~k) ---+ S. • This theorem means, in effect, that an analog ofthe Toeplitz limit theorem holds one way for certain nonlinear algorithms. This provides us with a one-way generalization of the linear deltoid obtained by extrapolation, Theorem 3.3(1).

Theorem 2.

Let Xn+k+ 1

Xn -

n, k

~

n

0,

~

0,

(4)

where x (which may depend on s) is monotone decreasing to zero. Then if ex > 1, n ~ 0, the above algorithm is regular for all paths.

Xn/x n+ 1 ~

Proof

Left to the reader.

•

5.3. Iteration Sequences

5.3.1. The GBW Transformation The algorithm to be studied in this section is a case of algorithm 5.2(4) with X n = Llsn = an + 1 •

Equation 3.3(2) gives (k ) Sn

= ~ ~

m=O

n k

Sn+m

i=O

te m

(

an + i + 1

an + i + l

-

n, k

~

n

O.

~

an + m + l

)

.

0,

(1)

(2)

If a, # a., i # j, then S~k) is defined. This restriction on s will be held in force throughout. The transformation (1) was discovered independently by Wimp (1970) and Germain-Bonne (1973). Ek(sn) == S~k) is homogeneous, translative, and exact when s has the form

(3)

5.3. Iteration Sequences

107

for some cj not all zero, Theorem 3.2(3). This rather obscure nonlinear difference equation offers little clue to the behavior of s itself, and one would expect the exactness problem for this algorithm to be rather intractable, certainly in comparison with the easy analyses of the preceding chapters. It turns out, however, that E; is a very natural transformation to use for a class of sequences of fundamental importance in numerical analysis.

Definition. s E ~s is an iteration sequence at S if there is a noneonstant function 4>(z) analytic in a region d with s e se, 4>(s) = s, 14>'(s)1< 1, and Sn+ 1 = 4>(sn)'

n :?: 0,

Sn E

d.

(4)

4> is analytic in a disk centered at S and

It is easy to show that if

= ro is sufficiently small, then s, lies in the disk for all nand s,

So -

s. As the reader may know, these sequences arise in the attempt to find solutions of scalar equations of the form z = 4>(z), or, in abstract spaces, solutions of operator equations x = Ax. For the latter, the abstract Brezinski-Havie process with f~) = (LlsnY is the process to study (see Chapter to). Here we discuss only the scalar case. It is easy to show the class of convergent iteration sequences is a subset of C(J, provided 0 < 14>'(a)I < 1 and 1S - So 1is sufficiently small for then (sn+ 1 - s)/(sn - s) -+ 4>'(a). In practice, however, Sn does not usually converge either because insufficient information is available to enable one to choose So close enough to S or because the function 4>(z) lacks the property 14>'(s) 1< 1. For computational aspects of such sequences, see, for instance, Isaacson and Keller (1966) or Householder (1953), and, for a discussion in abstract spaces, Kantorovich and Akilov (1964). The first result is a representation theorem. S

-+

Theorem 1. Let s satisfy (3) with 11 + l/c l l < 1.Then for ro sufficiently small, s is an iteration sequence at s. Conversely if s is an iteration sequence at S and the function inverse to 4>(z + s) - (z + s) at z = 0 is a polynomial of degree .

Rewriting (3) as rn =

Cl

Llrn + ... + ck(Llrnt

(5)

and reversing this series gives (6)

or (7)

108

5. Nonlinear Lozenges; Iteration Sequences

Now let

¢(z) = S + (1

+

l/c t)(z - s)

+ c~(z

- s)Z +

....

(8)

Then ¢ has all the required properties. (s.)

=

=

s; -

1,

So =

1.5.

(22)

Here (X = (1 + )5)/2 = 1.618033989. None of the usual sequence transformations works very well in summing this very rapidly divergent sequence. In fact, nearly all of them produce divergent sequences, an exception being the GBW transformation. Yet the Steffensen function produces a rapidly convergent sequence, s. 4>( 4>(s.)) - 4>2(S.) S.+1 = 4>(4)(sn)) - 24>(sn) +

(23)

s;

s = {1.5, 1.6429, 1.6189, 1.6180344, ...}.That the original GBW transformation produces only mild convergence while the iteration function constructed by analogy to the GBW transformation produces very rapid convergence is not so paradoxical as it may seem. This is because the iteration function cJ)k uses much more precise information about the sequence (22) than the GBW sequence transformation, namely, the exact form of the iteration function 4>. The Overholt procedure also produces an iteration function. Theorem 3. Let 4> be an iteration function of first order at not a root of 1. Then the function 'Pk(4)) defined recursively by

k

~

0, 'P o(4» = Proof

(X

with 4>'(rx)

4>, is an iteration function of order k + 1 at least.

Left to the reader.

•

For k = 0, this method also yields the Steffensen iteration function. The third iteration procedure arises from the formal elimination of the constants c., 1 S r S k, from the system of equations y= 4>i

+

k

"f.Cr(4)r+i - 4>r+i-1)'

r=l

o sj s k.

(25)

116

5. Nonlinear Lozenges; Iteration Sequences

Some readers may recognize this as the same formal procedure that leads to the Schmidt sequence transformation, to be discussed at length in Chapter 6. One interprets (25) as k + 1 equations in the k unknowns c., As such, for consistency, the determinant of the augmented matrix of the system must vanish. This produces a determinantal equation that may be solved for Y, which can be relabeled 1 k, and

cP-x cPz - cP

cPk - cPk-1 cPk+ 1 - cPk

cPk+1 - cPk cP-X cPz - cP

cPZk - cPZk-1 cPk - cPk-1 cPk+1 - cPk

cPk+ 1

cPZk - cPZk- 1

x

cP 1~

To analyze

)~

=

cPk I 1

-

cPk

(26)

it is convenient to rewrite it

r, = a + ILi(i-1)(cPj~ l)lk+ dILi(i+ l)(cPj~ l)k, k + 1 in the numerator determinant and 1 S

i.e., 1 s i, j s denominator determinant, and

Lik+ l(cP) = Lik(cPj+ 1) - Lik(cjJ),

(27) i, j skin the

k,j Z 0,

LiO(cPo) = LiO(x) = x - a.

(28)

We now need the following lemma.

Lemma. Let 1 s i,j

where

bi(x) =

00

LC

r=O

i_

s

m,

(29)

1,r Xr,

(30)

Let 11

(31)

--+ O.

Then D m = ICi-1,j-1IVm(X1,XZ,oo.,xm)

+ O(l1m(m -

l)/ 2+ 1) .

(32)

Proof In what follows it will be convenient to let Dm be a generic notation not necessarily involving the same bi(x) wherever the symbol appears.

5.3. Iteration Sequences

1I 7

Proof is by induction. Assume (32) true for 1 .:::;; m .:::;; N - 1. By the explicit formula for Vm [Appendix 1(4)], this implies l.:::;;m':::;;N-I; 1 .:::;; i,j N-l

c.(x·) l)

= "c. i....J

d;(x)

=

r=O

l-

s

= b·(x.) - d.(x.) J

1 ,r x~J

I

I)

(33)

N,

(34)

,

L C;-I.r X j. 00

r=N

Then

Ic;(x)I = IC;-I,j-lllxt11 = IC;-I,j-ll VN( X 1 , · · · , x N ) ,

1 .:::;; i.t

« N.

(35)

The remainder R N may be written RN

=

N

L

L

r= 1 (Ul.U2.···.Ur)ENSr

(X U'X U 2

' "

x ur t TN ( U 1,

U 2,···,

ur ) ,

(36)

where nSk is the set of combinations of (1, 2, ... , n) taken k at a time and TN is a determinant of order N containing di(x uJ)/x~J in the columns uJ' , j = 1,2, ... ,r, and C;(Xk) in the kth column if k # Uj' By Laplace's expansion (Aitken, 1956, p. 78) TN ( U 1 , u z,"" ur ) may be expanded by minors chosen from the columns u j and their cofactors whose elements are chosen from the remaining N - r columns. These latter are determinants of the form D N - " and each may be estimated as I] ~ 0 by (33). Thus (36) may be written RN

=

L O[r(NI](N-r)(N-r-1)/2] N

r= 1

=

L O[I]N(N- 1l/2+r(r+ l l/2] N

r=

=

O(I]N(N- 1l/2 + I).

(36')

1

This establishes the lemma for m = N. Since the result is true for m = 1, the proof is complete. • We now return to the analysis of Yk . A(k)(X)

=

A(k)(X)

L bklx 00

may be written a)'.

r= 1

(37)

Let ¢(x)

=

a

+

00

L Cr(X -

r=1

a)'.

(38)

I 18

5. Nonlinear Lozenges; Iteration Sequences

Then the bkr can be computed recursively from the c; For instance, b ll = C I - I, b 21 = (c I - 1)2, b 3 1 = (C I - 1)3,

b'j = Cj, b22 = Cz(CI - 1)(cI + 2), b 32 = Cz(C I - 1?(ci + 3c I

(39)

+ 3),

Theorem 4. Let Ibi + l, j lk' Ibi • j + l i b 1 ::::; i,j ::::; k, be nonzero and let C I not be a root of 1. Then Y k is an iteration function of order k + 1.

Proof For the numerator determinant in (27) we use the lemma with the identifications m

= k + I, 1'/ = x -

IX,

(40)

and for the denominator determinant m = k,

1'/

=

x -

IX.

(41)

Since Ibi-I.jlk+ I = Ibi.j+ l i b the use of (5) gives )'" =

IX

+

Ibi.j+dk

Ibi+l,jik

C~(k+2)/2 [J(e{ -

and this proves the theorem.

1)(x - 1X)k+I[1

j=1

+ O(x

- IX)] (43)

•

This process also yields the Steffensen iteration function when k = 1. 5.3.5. Iteration Functions in Abstract Spaces

Iteration functions suitable for the solution of operator equations can be derived in a straightforward manner. Let cjJ: ~ -+!!4. Let cjJ* be any element from the dual of ~ and consider I

cjJk+1 - cjJk :

Sn+k- 1

Sn

S

=

~(rn' rn)/~(l, rn)·

(11)

== s, we must have for some aj not all zero

If S~k)

aor n + a l l\r n +

.,. + ak l\rn+k-I

= 0,

n ;::: 0,

(12)

or, rewriting, (13) Differencing gives Co

°

l\r n +

C1

+ ... + Ck l\rn+k = 0, 1f'k(1, rn) < k + 1, so ~ ==

l\rn+ 1

°

(14)

and if Co = or Ck = 0, rank and S~k) is not defined. This means the minimal order of the difference equation (13) is k. By Lemma 2, r« can contain no algebraic terms. Thus r, E :ff k • =: Wk(l, rn ) -# and

corn

°

+ clrn+ 1 + ... + Ckrn+k =

(15)

0,

Rewriting gives c~rn

+ c;

l\r n +

so Rank jf"ir n, rn) < k

+

... + c~ l\rn+k-l = 0, I and ~(rn' rn) == 0.

(16) •

126

6. The Schmidt Transformation; Thee-Algorithm

Example (k = I, the Aitken c5 2 -process). and only when s; = S + cAn, C #- 0, A #- 0, 1.

el is defined and exact when

Example (k = 2). e2 is defined and exact iff either s, = S + n, C lC2 #- 0, ..1[, ..1 2 #- 0, I, A[ #- ..1 2 , or s; = S + (Cln + C2)A A #- 0, 1. c2A~,

Cl A~ Cl

+

#- 0,

6.4. The Effect of ek on Certain Series The work of this section will illustrate the remarks ofthe previous section. It turns out that ek works well on (renders more rapidly convergent) sequences that behave as exponential sequences and does poorly on sequences that contain terms algebraic in n. By elementary row and column manipulations,

(1) and k~1.

(2)

Since lv,. can be written (3)

(4) Now let (5)

lv,.(I, Anpn) = ~_I(An(A - 1)2p: , An(A - 1)2p:>, = (A - 1)2k Ak(n+k-1)lv,._I(P:' P:),

P: = nO(co + c't/n + ...).

By Lemma 1.7(2) and the representation (1),

(6)

127

6.4. The Effect of e, on Certain Series

0(0 - 1)(0 - k

x

+

O(0 - 1) ... (0 - k

=

1)n8 -

1)(8-k)A k

ct+1n(k+

O(D - 1)··· (D - 2k

k

D (-I)j( -D)j' k

D

D(O - 1)··· (0 - k

+

1)

0- k

(D - k) ... (0 - 2k

+

1)

Dj

= 1 ( -

D(-D)jjL k

j= 1

Theorem 1. Let s, have the asymptotic expansion 00 c 8 Sn '" S + .-1.nn "L. -.-!-r' Co #- O. r=O

+

k

l)n 8- 2k

1)jj! and

l¥,.(Pn, Pn) ~ ct+ 1n(k+l)(8-kl

We combine these results as a theorem.

l)n 8 -

(7)

j= 1

and differencing rows gives A k =

+

n

(8)

(9)

Then for .-1. #- 1, 0 #- 0, 1, ... , k - 1, (k) _

Sn

-

S

+

co~n+2kn8-2kk!( -O)k [ (.-1. _ 1)2k 1

+0

(~)J n '

(10)

and thus, for fixed k, ek accelerates convergence on vertical paths of all convergent sequences of the form (9). For .-1. = 1, 0 #- 0, 1, ... , k - 1, (k) _

Sn

-

S

(I)J

8 cok! n [ O)k 1 + 0;;

+ 0-

,

(11)

and so ek does not accelerate convergence.

For the effect of ek on exponential sequences of the form S + L Ck .-1.;, see Theorem 6.8(4). We have already shown that el' which is Aitken's (F-process, is accelerative for rtf/ (see Section 5.1, Example I); ek is not accelerative for rtf/ when k > 1. The following argument, due to Smith and Ford (1979), shows what happens when k = 2. In the notation of Chapter 5, (12) e2(Sn) = s, + an+lg(Pn+b Pn+2, Pn+3)' (The k in that chapter is here 4.) According to Theorem 5.1(4), eisn) accelerates d p iff g is continuous at (p, P, p) and has the value 1/(1 - p) there.

128

6. The Schmidt Transformation; The t: -Algorithm

But _

(

gXI,X2,X3 ) -

If Xl = p, X2 =

(1

+ X2)(X 2 1

X2(X I -

X, X3

)(

X3 -

-

Xl

)

xd -

(

Xz( X3 - Xl) )( X2 - Xl X IX2-

1)

(13)

= y, the denominator is zero on the curve

X(p - 1)(y - p)

= (x - p)(xp - 1)

(14)

and the numerator on the curve X(y - p)

= (1 + x)(x - p).

(15)

The only x-values common to these curves must satisfy p - 1

= (xp - 1)/(1 + x),

or

X

=

p.

(16)

Therefore, y = p as well. We conclude that 9 is unbounded arbitrarily close to (p, p, p), so e2 does not accelerate d P' e b in fact, is not regular for d p when k> 1. For totally monotone sequences, an interesting convexity property holds for ek(s.), To establish it requires an inequality due to Bergstrom. Lemma. Let A, B be positive definite matrices and let Ai, B, denote the submatrices obtained by deleting the ith rows and ith columns. Then

IAI/IAd + IBI/IBd Proof

::s;

IA + BI/IA i + Bil·

See Beckenbach and Bellman (1961, p. 67).

Theorem 2.

(17)

•

Let s, t E 9f!TM' Then ek(s.)

+

ek(t.) ::s; ekeS.

+ t.).

(18)

Proof The case in which either s or t is the zero sequence is obvious. Assume neither is the zero sequence. Then neither of the distribution functions (see Section 1.5) for these functions can be equivalent to the zero function, and it is easy to show that »-k is positive definite in either case. The rest of the proof follows immediately from Eq. 6.4(1). •

A similar statement is possible when s, t

E 9f!TO'

6.5. Power Series and ek ; The PaM Table When ek is applied to the partial sums of a power series the result, obviously, is a rational approximation. This section investigates the nature of this approximation.

6.5. Power Series and ek ; The Pade Table

Definition.

129

Let s(z) be analytic at 0, 00

L akz\

s(z) =

(1)

k=O

Let A be a polynomial of degree ~ p, B ( =1= 0) a polynomial of degree ~ q. If

=

s(z)B - A

O(ZP+q+1),

z

-->

0 in

(2)

C(j,

then the rational form A/B is called the p, q Pade approximant to s(z) and written [p/qJ. • When [p/qJ exists, it is unique when written in lowest terms. To show this, assume otherwise, i.e., that

s(z)B* - A* = O(zp+q+ 1).

(3)

AB* - A*B = O(zp+q+ 1).

(4)

Then

But the left-hand side is a polynomial of degree not exceeding p O(zp+q+ 1). It must therefore vanish identically. Thus

+ q, yet

A/B = A*/B*.

is

(5)

Now let

A =

p

L r:tjZ j,

j=O

B=

q

L f3jX j.

(6)

j=O

Equation (2) requires

(ao

+ a1z + .. ·)(f3o + = O(zp+q+ 1).

f31 Z +

... +

f3 qzq) - (r:t o + r:t 1z +

... + r:tpz P)

(7)

Then f3j must satisfy

ap+1f3o

+ ap!31 + ... + ap-1+1!3q =

0, (8) a:

j

= 0.

This is a homogeneous system of q equations in q + 1 unknowns and so always possesses nontrivial solutions. The r:t j may then be computed from

= aof3o, r:t 1 = a1f3o + a of3h

()(o

(9)

130

6. The Schmidt Transformation; Thee-Algorithm

°

A nontrivial solution with f30 i= will exist iff H~~q+ lea) i= 0. Thus if all the determinants H~q~ q+ 1 (a) i= 0, p, q z 0, there will exist for each p, q an A and B of degrees s; p, :::; q, respectively, satisfying (2). This does not guarantee that A will be of exact degree p (although B will be of exact degree q since 0:0 i= 0, f3q i= 0). However, if the like determinants formed with the reciprocal series

-

1

s(z)

=

ao

L bkzk k;O

(10)

are nonvanishing, then A will also be of maximum degree. Such a series, or such a set of'Pade approximants, Wall calls normal, although this terminology is not universal [cf. Henrici (1977, vol. 2, Chap. 12)]. With the approximants [p/qJ one can associate the following geometrical configuration known as the Pade table for s(z): [%J [O/IJ [0/2J

[1/0J [1/IJ [1/2J

[2/0J [2/1J [2/2J

The elements in the first row are simply the partial sums of the power series for s, [P/OJ = L);o a.z', those in the first column the partial sums of the series for l/s. [P/qJ can always be obtained by brute force, i.e., by the above method of undetermined coefficients. However, this way of computing the Pade elements for a function tends to be very unstable numerically. Luke has found that in many cases double-precision arithmetic is not adequate for even moderate values of n (10 or so). Note, however, that the coefficient matrix of the system (8) is of Toeplitz type and thus can be inverted easily by using algorithms due to Trench (1964, 1965). (Toeplitz matrices are those having northwest-tosoutheast diagonals constant.) Trench's algorithms require O(n 2 ) operations, as compared with the usual methods, which characteristically require O(n 3 ) operations. In fact, the Trench algorithms can be used to compute ek(sn) systematically. [In Section 10.6 we show how a Trench algorithm can be used to compute a transformation that includes ek(sn) as a special case.J To my knowledge, however, no work has yet been done on assessing the computational stability of Trench's methods. [For an excellent general discussion of methods for inverting Toeplitz and Hankel matrices, see Cornyn (1974).J Closed-form expressions for the Pade elements are known for only a few special functions (other than rational functions) and then only for diagonal approximants [p/pJ or off-diagonal approximants, [pip - IJ or [p - l/p]. The functions are special cases or confluent limits of the Gaussian hypergeometric function F(l, b; c; z); see Luke's books (1969) for details. For certain

6.5. Power Series and ek; The Pade Table

131

other functions (see Section 2.5.3) the [p - 1, pJ Pade may be computed systematically without solving equations, but since closed-form expressions for the Taylor coefficients of these functions are not known, the Pade can hardly be said to be closed form. The following is due to Shanks (1955). Theorem 1.

Let s,

= LJ=o a.z'. Then

S~k)

= [n + k/k].

Note that it is possible for Wk(I) to vanish identically. However, as Shank shows, this can happen only if at the same time Wk(sn) == O. In such cases one defines ek(Sn) = ek_ 1 (Sn) = ... = ek_ ,(sn), where r is the smallest integer such that Wk-,(sn) 1= O. With this understanding, the theorem still holds. A number of theorems on Pade approximants can be translated directly into statements about the effect of ek on partial sums of power series [see Gilewicz (1978, Section 6.3.1)]. (There are interesting results available on convergence in measure also. Here we shall be concerned only with uniform convergence.) Theorem 2 (de Montessus de Balloire, 1902). Let s(z) be meromorphic in N R and have exactly k poles (each counted as often as its multiplicity) in N R' Then S~k)(Z) = s(z) + 0(1), n -.. 00, (11) uniformly on every compact subset of N R with the poles removed.

Proof

See de Montessus de Balloire (1902).

•

The most far-reaching result to date on the convergence of subsequences of S~k)(Z) is due to Bearden (1968). The proof will not be given here. Theorem 3. Let s(z) be analytic at zero and meromorphic in a domain f0 containing zero. Then given any compact subset :ft of f0 with the poles removed there is some subsequence S~~i)(Z) that converges uniformly to s(z) in :ft. These theorems cannot be strengthened in any obvious way. For instance, regarding Theorem 2, Perron (1929) has given an example of a function analytic in a disk N o for which s~l)(z) (Aitken's (j2_process) diverges over a set dense in N p : Several computational algorithms have been developed to relate the various components of the Pade table, e.g., schemes due to Wynn (the ealgorithm, Section 6.2), Baker, Longman, and Pindor. For a good survey of these, see Gilewicz (1978, Section 7.3). Some of these schemes are computationally more efficient than the s-algorithm [see Baker and Gammel (1970, Chap. 1)].

132

6. The Schmidt Transformation; The c-Algortthm

The previous results have touched on convergence only along vertical paths. Convergence along other paths is a much more difficult matter. The theorems to date-such as those of Chisholm-Gilewicz and Zinn-Justinrequire troublesome hypotheses. One of these results will be given later. As an example of what can happen, consider the diagonal path [k/k], i.e., the path corresponding to s~)(z). If s is given by a formal power series 0 a.z', then it is known that the Pade approximant [k/k] = s~>Cz) is the (2k + I)th approximant to the (formal) continued fraction corresponding to s (when such exists); see Henrici (1977, vol. 2, Chap 12). It would be tempting to assert that for s(z)analytic at 0, s~)(z) -+ s(z) in some neighborhood ofO. However, this is not, in general, the case, as shown by an example in Perron (1957, vol. 2, p. 158). Nor indeed, must a.z' converge for s~)(z) to converge. Apparently, additional conditions must be placed on the coefficients aj to ensure the convergence of the Pade table. In a special but very important case, namely, when the a j are the moments of some function t/J, many of these convergence questions can be answered satisfactorily. We shall tabulate below the most far-reaching results to date. (Since proofs in Pade theory tend to be very computational and to rely heavily on function theory, they have been omitted.) In what follows, let t/J E 'P* and also Supp t/J = [0, CfJ). (When the latter is not true, the results can be refined in an obvious way.) Let

I;;,

I

s(z) =

Sn(Z)

=

dt/J o 1 - zt

5 00

n

I

z ¢ (0,

--,

IX),

(12)

a.z',

j~O

j

~

-1,

and let {pV)(z)}r~o be the sequence of orthogonal polynomials associated with dt/Jj' Theorem 4. S~k) is the ratio of two polynomials of exact degree n k, respectively. The denominator polynomial B is given by

+ k and (13)

Thus all the poles of S~k)(Z) lie in [0, CfJ).

6.5. Power Series and e.; The Pade Table

133

Theorem 5 S~k)(Z) =

La zr + zn+ L k

n

1

r=O r

a(n)

kr r= 1 1 - ZXk~)'

(14)

where the as and the xs are the weights and abscissas, respectively, of the Gaussian quadrature process based on dljJn' Theorem 6. (i)

Let z be real negative.

(_l)n+ I[S~k+ I)(Z) -

S~k)(Z)]

::2: 0,

(ii) (_l)n+l[s~k+I)(z) - S~k~Z p, then, for (iv)

~

°::; z

< 1/p,

S~k~2(Z)::; S~k+1)(Z)::; S~k~I(Z)::; S~k)(Z)::; S~k+I)(Z)::;

Theorem 7. - [0, (0). If

S~k)(Z)

s(z).

converges along any path P to a function analytic in

L la 00

j= 1

j

l- I/(2j + l ) =

(15)

00,

all paths produce a common limit. Further, if (1) converges in some N R'

R > 0, S~k)(Z) converges along any path P to s(z) in ~ - [R, 00).

In all cases above, convergence is uniform on compact subsets of the indicated region. Theorem 8 z

Theorem 9.

if (0,

00].

(16)

Let (17)

for some R > 0. Then s~)(z) converges to s(z) as k subsets of ~ - [0, OC!).

-+ 00

uniformly on compact

For Theorem 4,see Wall (1948, p. 388)and Allen et al. (1975); for Theorems 5,8, and 9 see Allen et al. (1975); for Theorems 6 and 7 see Baker and Gammel (1970). [Actually, Theorem 9 follows directly from Theorem 5 by an application ofVspensky's result (l928)on quadrature formulas for infinite intervals.]

134

6. The Schmidt Transformation; The s-Afgorlthm

Example.

If I)(

> -I,

(18)

then pjj>(z)

= LV+ a+ 1)(z).

(19)

We shall require the formula

~~z;}: \{I(a + 1, a + b + 2; -~}

lOOe-lta(1 - zt)b dt =

[argt -z)1
-1.

tt,

In the present case

foo

-I _ e-1/Zr(1)( + 1) a+ l e t (-z) -liz

S(Z) -

(20)

-a-I dt.

(21)

[The integral is the incomplete Gamma function F( -I)(, -1/z).J Using the Rodrigues formula for the Laguerre polynomial in (16) and integrating k times by parts gives r(k)(z)n

-

(-I)"k!r(c) \{I(a,c;-I/z)e-1/Z + 1)( _zy+l (a, c; -1/z) ,

r(1)(

a = n

+ + k + 2,

c= n

I)(

(22)

+ + 2. I)(

Now \{I(a, c; w)

(a, c; w)

~

2na c -

exp( -4~) r(a)r(c)

1

a

-+ 00,

[arg w] < tt

(23)

[see Slater (1960, p. 80)]. Thus r~k)(z) ~ [2n( -In( -zY+ IJ exp( -1/z) exp( -4J -k/z),

k

-+ 00,

[argf -z)1 < n.

Table I k

slkl ·0

2 4 6 8 10 \2 14

0.615 0.598802 0.596816 0.596459999 0.596378884 0.596357234 0.596350734

(24)

6.5. Power Series and ek ; The PaM Table

135

This agrees, for n = 0, with a result given by Luke (1969, vol. 2, p. 200). [The formula (5.10) of Allen et al. (1975) seems to be wrong.] For a numerical example, take z = 0, Z = -1, Then ek in diagonal modes sums the highly divergent sequence s, = O! -I! +2! -3! + .. , + (-I)nn! = (FAC)n

(25)

to the value

s

=

foo l e -

t

+t

o

dt

= 0.5963473611.

(26)

is not regular for s, in vertical modes. Let us tabulate some elements on the leading diagonal (see Table I). According to Eq. (24),

ek

S~)

-

S

k

~ 2rc exp(1 - 4Jk),

-> 00.

Table II

z = 0.85 s;

s~1)

S~2)

1 1.85 2.37 2.60 2.67 2.695 2.6979 2.6983 2.698328 2.698338

3.2 2.79 2.71 2.70 2.6985 2.698346 2.698331 2.69833043 2.6983303937

2.67 2.695 2.6980 2.69831 2.698329 2.69833035 2.6983303913 2.6983303925

z = 1.05 Sn

s~1)

I 2.05 3.36 4.82 6.99 10.38 16.18 27.09 49.80

-5.66 -2.35 -0.54 0.86 2.23 3.85 6.05

,(2)

•n

11.13 -36.51 -5.88 -2.11 -0.18

(27)

136

6. The Schmidt Transformation; Thee-Algorithm

ei, particularly when computed by means of the s-algorithm, is one of the few simple computational tools available for the numerical analytic continuation of an analytic function. What happens if one tries to continue a function beyond a natural boundary? Brezinski (1978) presents the example of (28)

for which aN is a natural boundary. The results are given in Table II. The behavior of the algorithm seems to reflect the functional realities. A series whose coefficients are given by (12) is called a Stieltjes series.lfthe Taylor series for s(z) is not a Stieltjes series, it is still possible to say something about the convergence of S~k} on general paths, but restrictions, which may in practice be unverifiable, must be placed on the zeros of ~(sn), ~(1). Let (1) hold near 0 with s; its partial sums. The following result is due to Chisholm

(1966).

Theorem 10. Let s(z) be meromorphic in N R' Let P be a path and let the zeros of ~(sn), ~(1) have no accumulation point in NIT for (J < Rand (n, k) E P. Then S~k}(Z) converges to s(z) uniformly in some nonempty region ~, 0 c ~ C NIT'

6.6. Geometrical Significance of the Schmidt Transformation

As Tucker (1973) has pointed out, ek has an elegant geometrical interpretation. Let s E f!lls and let (1)

be a sequence of points in f!llk + 1. Denote by !fl the line through (0, 0, ... , 0) and (1, 1, ... , 1). Assume a unique plane can be passed through Pn' Pn+ b ... , Pn+k' Its equation is Xl

X2

X k+ 1

s,

Sn+ 1

Sn+k

Sn+k

Sn+k+1

Sn+2k

= O.

(2)

If s converges, then lim P« = s(1, 1, ... , 1).

(3)

6.6. Geometrical Significance of the Schmidt Transformation

137

which lies on }P. If the plane intersectsze', say at (tn, tn' ... , t n), then it is reasonable to expect that t; will be closer to s than any of the components of p., n ::;; i ::;; n + k. Now,

(4)

Solving for

tn

and performing obvious determinant manipulations shows t n = ek(Sn)'

(5)

Figure 1 illustrates k = 1 (Aitken's 15 2-process).

(e I (sn) ,e I (sn)) I I

I I

I

I I I I I I I

.1

I

~---I----I----+--------'------'~

5

Fig. 1

XI

138

6. The Schmidt Transformation; The s-Algorithm

6.7. The s-Algorithm The s-algorithm, due to Wynn (1956b), is an economic computational procedure for calculating ek(s.) without the necessity of evaluating determinants. It is a lozenge algorithm, actually a rhomboid of the kind 1.3(5). Not only does the s-algorithm make the application of the Schmidt algorithm much more practical, but it helps to clarify some of the convergence properties of the latter, particularly in its application to monotone sequences.

Theorem 1. Let n, k :2 0, c:~\ = 0,

(I)

(It is assumed that all quantities are defined.) Then n, m :2 0,

(2)

n, m :2 0.

(3)

and Proof The proof is quite computational and depends on an expansion of the ratio of determinants due to Schweins. The proof is sketched in Appendix

2. •

The computational scheme for the s-algorithm is as follows.

0(3)

'-'-1

As a numerical example, take the iterative sequence considered by Wynn, S.+l

=

i(s; + 2),

n > 0,

So

= 0.

(4)

139

6.7. The s-Algorlthm

Table III" 11

£(n)

Sri

o

e~)

'I

E~ n)

€~)

0.0 2

0.5

2

0.5625

3

60.23529.41

0.5791015625

4

0.5838396549

5

0.5852171856 a

0.5714285714

16

211.05540 725.9366

The true root of x 2

-

4x

0.5851063830 0.5857319781 0.5857818504

89.1111108

1658.713

0.5857434871 0.5857857313

20262.3

+ 2 is 0.5857864375

This sequence converges (very slowly) to the smaller zero of the Laguerre polynomial Lz(x) = x 2 - 4x + 2, i.e., x = 0.5857864375. Table III shows the effect of the e-algorithm on Sn' It is apparent the convergence of e~~) is much superior to that of Sk' Note the odd entries, e~l+ l' diverge as n ---+ 00. This is generally the case. Tables IV and V show the effect of the algorithm on some other sequences. 4n ) satisfies e~~ = H~m+ 1)(s)/H~m)(Ll2s),

e~n~+l e~~+2 -

=

(5)

H~m)(Ll3s)/mm+1)(Lls),

e~~ = _[H~m+1)(Lls)]2/H~m+1)(Ll2s)H~m)(Ll2s).

The first two properties are obvious and the third is deducible using a determinantal expansion of the kind given in the appendix. The two following theorems show the effect of 4n ) on sequences in ~TM and ~TO' Theorem 2.

Let

SE

~TM'

Then

(i) 0::; e~~+2 ::; e~~, n, m ;;::: 0; (ii) e~~+ 1 ::; e~~-l ::; 0, n, m ;;::: 0; and (iii) ekn + 1) ::; 4n>, n, k ;;::: O.

Further the s-algonthm is regular along any such that rE~TM and limn~oo e~~+l = -00.

En, 2m] path for sequences

Proof The three inequalities are fairly straightforward, requiring the use of Theorem 1.6(6).They are left as an exercise. The rest of the proofis trickier. Assume without loss of generality that S = O. Inequalities (iii) and (i) show

140

6. The Schmidt Transformation; The c -Algorithm Table IV" (LN 2),

E'O)

.,

(PI 2 ) ,

ckO)

(EX 3),

riO)

0 2 4 6 8 10 12 14

1 0.833333 0.783333 0.759524 0.745635 0.736544 0.730134 0.725372

1 0.7 0.693333 0.693152 0.693147332 0.693147185 0.693147180688 0.693147180564

I 1.361 1.464 1.512 1.540 1.558 1.571 1.580

1 1.45 1.552 1.590 1.609 1.620 1.626 1.630

1 7.72 25 60.832 128.196 247.903 452.973 795.351

1 -2 25 25 25 25 25 25

a

f,~~

m

.,

k

In 2 = 0.693147180560. 11';6 1.644934.

is bounded and positive decreasing in n, and hence convergent. Putting

= 0, 1,2, ... , in (i) shows vertical regularity.

Inequality (i) shows that E~~ is positive decreasing in m and bounded, and hence convergent. Let (6)

m-->oo

so that (7) Now,

l.

-

(8)

since E~:-n E~~+I $; 0 by (iii). Taking limits shows t« $; t n + But taking limits in (iii) with k = 2m shows t n+ 1 $; tn' Hence t; is a constant, and letting n ~ XJ in (7) shows tn = O. This gives horizontal regularity. It is an easy exercise to show that (iii) guarantees regularity for any path.

•

Table V"

k

.ro:

(FAe),

c,

2

2 4

0.666 0.615 0.602 0.598 0.597 0.596817

20 620

6

8

10 12 u

/:,,0J = 0.596353077, S'~ [e-'/(I +

I)J dt

= 0.596347

6.8. The Stability of the s-Algorithm

Theorem 3. (i) (ii)

°S

(iii)

Let

SE

.0lTO • Then

e~2';:~ 2 S e~2';:l,

e~2';:+ I) S e~2';::

(-lte~~+ 1 S (-lte~n~_1 (-It(e~:

1) -

( - 1t( e~:-;t

-

e~~)

so,

n e~:~ \ -

Ite 0, there is an c > 0 such that for Iltll < c,

Il

f (X + t) -

,f ;, fU)(x)t U) II s elltll J=oJ·

m

•

(3)

For details and examples, see Dieudonne (1969). Now consider the iterative scheme for determining a fixed point s of the function f(x), (4)

6.11. Fixed Points of Differentiable Functions

147

If f is differentiable in some neighborhood of s and all members of s are sufficiently close to s, one may write (5) so generally the convergence of the process is poor and first order [see Definition l.3(liii)]. However, the vector e-aigorithm, as defined by Eq. 6.10(2) may be used to derive a quadratically (second-order) convergent process in the case that f!J 1 = f!J 2 = f!IlP and f is twice continuously differentiable on d. In this case, of course, 1'(x) is a real p x p matrix. Let s = f(s) and Qm(f'(s)) be the set of vectors x for which m is the degree of the minimal polynomial of x (with respect to 1'(s)). Define s as above and s by Sn

= S + [f'(s)Jn(so - s).

Theorem. Let 1 not be an eigenvalue of 1'(s) and let t:F>' exist for all So sufficiently close to s with So - s E Qm(f'(s)). Put t:io~

= G(so, Sl>

.•• ,

(6)

W>' i + j

~ 2m,

S2m) = H(so).

(7)

0,

(8)

Then the computational procedure

t i + 1 = H(tJ,

i

~

is, for to sufficiently close to s and to - s E Qm(f'(s)), quadratically convergent to s.

Proof

See Gekeler (1972).

Gekeler assumes that f is analytic (possesses a p-tuple series absolutely convergent in a neighborhood of s), but this seems not to be required. The author gives some interesting numerical examples. The s-algorithm has an obvious generalization to sequences of square matrices. For rectangular matrices, the generalized matrix inverse of Moore (1920) and Penrose (1955) is applicable, and many writers have investigated generalizations of the e-algorithm applied to sequences of such matrices, particularly as they arise in the solution of linear systems of equations; see Pyle (1967), Wynn (1966, 1967), and Greville (1968). For a numerical example of the theorem let f: f!Il4 --+ f!Il4 be defined by

f(x) = s

+

A(x - s)

+ Q(x

- s)

(9)

where and (11)

148

6. The Schmidt Transformation; The s-Algorithm

where D is the diagonal matrix (0.9, 0.8, 0.7, 0.6) and U is the orthogonal matrix 1

U

1

1

=:2

[

~

1 1

-1 -1

1 -1 1 -1

-~j

-1 .

(12)

1

It is easy to show that

f'(s)

= A.

(13)

The mapping f is quadratic and f(x)

=

s + f (s)(x - s) f

!,,(s)

+ -2- (x

- SPl.

(14)

s turns out to be (1,1,1, 1)T. With the initial vector to = (2,2,2, 2)T, the method described in the theorem produces iterates t, for which (15)

Gekeler, who gives other examples, states that the method seems to produce the best results when the Jacobian matrix of the system s = f(s) is symmetric.

Chapter 7

Aitken's £52-Process and Related Methods

7.1. Aitken's (j2-Process The most famous example of the Schmidt algorithm is a method usually attributed to Aitken (1926) but that is, in fact, much older. The method, which is discussed in most books on numerical analysis, results on taking k = 1 in ei . We use the following notation: A(s) = S, where (1) Pn+l

= 1.

This definition guarantees that S always exists. The following results are obvious: (i) A is defined and regular for all convergent sequences having the property (j > 0, n > 0, (2) (thus A is regular when applied to the partial sums of convergent real alternating series); (ii) if A is not regular for s E C(}C, then some subsequence of p has limit 1. Thus A is regular for of the form

C(}" which

was established earlier. If s is a sequence Co

# 0,

Re

e < 0,

(3)

then A is regular for s but does not accelerate s; see Theorem 6.4(1). (This is a logarithmically convergent sequence.) A, however, is not regular. 149

150

7. Aitken's iF-Process and Related Methods

Example 1

(Lubkin, 1952). 8=

so

80

Let s be defined by the partial sums of

1 +!-!-i+t+i-···.

= 1. 8

81

(4)

= 1 + !.

(5)

= n/4 + ! In 2.

We can write (6) and

_ 82m = 82m _

82m+1

=

( -l)m(2m

+ 3)

+ (2m + 2)(4m + 5)'

(7)

+ 4) + 3)"

(8)

82m+1

+ (-1)

m

(2m

(2m

Thus A is not regular for s. Note that s contains essentially three distinct convergent subsequences. One of these, 82m' converges to 8. This is no coincidence. Theorem 1 verges to 8.

(Tucker).

Let s

E '(jc.

Then some subsequence of

Proof Suppose no subsequence of s converges to Pn i=. 1. Now

8.

s con-

This means an i=. 0, (9)

Thus the assumption holds iff no subsequence of means

Vn

for some

converges to zero. This B > 0

(10)

or

But then. by Theorem 1.5(1), s diverges. a contradiction. Corollary 1. Corollary 2. s diverges.

If sand S E '(j s- then

8

•

= 8.

If s is such that s is properly divergent (i.e., Is; I ~ 00). then

Proof If s were convergent some subsequence of s would be convergent. a contradiction. •

7.1. Aitken's iF-Process

151

Tucker (1967, 1969), has obtained several sets of conditions that ensure that A accelerates convergence. These conditions, generally speaking, amount to restricting s to rt'l or else are reformulations of the condition (Llsn)2/(rnLl2 sn) ~ 1. Brezinski's result, Theorem 1.7(5), results in a criterion for certain real sequences. Theorem 2. Let a be ultimately positive and monotone decreasing, and Ll(an/Llan) = 0(1). Then s converges and A accelerates the convergence of s. Proof

Direct application of Theorem 1.7(5).

•

There are two useful results describing the effect of A on the partial sums of power series. Let sn(z) be the effect of A on sn(z) = and s(z) =

n

L akzk

(12)

k=O

L akzk, 00

k=O

[z] < c5.

(13)

We know (see Section 6.5) that the analyticity of s(z) in N R does not guarantee the convergence of sn(z). However, the following results, whose proofs are omitted, provide some information. Theorem 3 (Tucker, 1969). Let IPnl S. P < 1 and let Sn(1) converge more rapidly than sn(1). Then sn(z) converges more rapidly than sn(z) [to s(z)] for each z such that 0 < [z] < lip. Theorem 4 (Beardon, 1968). Let s(z) be analytic and bounded in N R' Then there is some subsequence oSnlz) that converges to s(z) uniformly on every compact subset of N R'

Aitken's c5 2 -process also accelerates convergence of hyperlinearly convergent series, i.e., series of the form Pn = 0(1), Pn = rn+i/r n. However, comparatively speaking, the method does not work as well on these sequences as on linearly convergent ones. Reich (1970) observes that it is more logical to compare oSn with Sn+2 rather than Sn' since the computation of oSn involves s, + 2 . For linearly convergent sequences, one still has rnlrn + 2 -+ O. The 2 examples s, = n:" or 2 -n show this is not true for hyperlinear sequences. Conditions for A to accelerate convergence of the infinite product (1 + an) (i.e., accelerate convergence of the sequence of partial products) have been investigated by Tucker. One result is that if IPn I s. P < 1 and an =1= - 1, n ~ 0, then A accelerates the convergence of (1 + an)iff Llpn -+ O. A sums divergent exponential sequences in certain cases.

n

n

152

7. Aitken's il'-Process and Related Methods

Theorem 5.

Let s be real, divergent, and bounded, and let Pn = - 1 + 0(1),

Pn eventually monotone. Then A sums s. See Goldsmith (1965).

Proof

Example 2.

A sums the series

•

L (-It (to !).

There are ways of modifying the b 2-process when the original is ineffective, for instance, in such problems as determining by the power method eigenvalues of a matrix which are close together. Iguchi (1975, 1976) discusses means of doing this and gives many examples. 7.2. The Lubkin W-Transform This is a transformation introduced by Lubkin (1952) that is sort of an iteration of the Aitken b2-process. The formula is

The work of Chapter 5 shows immediately that the W -transform is accelerative for rtf,. Further, for any P, W is regular in a sequence space slightly larger than d p : To make this precise, consider D; = 1 - 2Pn + 1 + PnPn+ 1 and let Pn = P - bn, Ibnl :-:; A, A > O. Then D; = I(p - 1)2 - p(bn + bn+ 1) + 2bn+ 1 + bnbn+ 11

> Ip Then

o, >

11 2

-

2A(lpl

+

I) - ,1,2.

(2)

0 for A :-:; ,1,*, (3)

This proves the following result.

Theorem 1. The W -transform is defined and regular for all convergent sequences s having the property

0< Ipi < 1, ,1,* as above.

(4)

7.2. The Lubkin W-Transform

Theorem 2.

s: is defined and s: = s, n 1, iff s, has the form (ja + b + 1) s, = + K n . b ' n 1, Ja + ~

n

S

where K #- 0, a#- 1,ja Proof

153

j=l

+ b #- 0,

-1, 1,j

~

~

(5)

n~1.

(6)

1.

Notice that s: may be written

The requirement s: == s means Sn satisfies a first-order linear difference equation that may be solved by the usual techniques. Cordellier (1977) is responsible for the clever observation (6). Setting a = shows W is exact for exponential sequences, convergent or not. •

°

There is a close connection between accelerativeness for the Aitken (jz-process and W. Theorem 3. Let A (resp. W) accelerate s. Then W (resp. A) accelerates s iff

~ (1 _2an+z + an+ z). (1_an+z)Z an+ an+ an 1

1

(7)

Proof Immediate by the use of Theorem 1.5(4). Note in accordance with my convention [see Definition 7.1(1)] A or W may be undefined for a (finite) number of values of n. • For more results of the W-transform, see Lubkin (1952) and Tucker (1967, 1969). To close the discussion, observe that for a large class of logarithmically convergent sequences, the W-transform is accelerative, whereas the A-process is not. Theorem 4.

Let

an ~ norco Then W accelerates

Sn'

+ ct/n + cz/n z + .. '],

S: -

Proof

S

_ 2no+ 1 1) [1

(e +

+ O(n- 1 )].

Left to the reader. [Note that the denominator of (1) is e(e

+ O(n- 3 ) .]

•

(8)

In fact

-- = x n- z

Re e < -1.

(9)

+ 1)

154

7. Aitken's Y-Process and Related Methods

7.3. Related Algorithms A number of variants and generalizations of the c5 2 -process have been given. In Aitken's process one assumes s, converges as (1) Samuelson (1945) assumed sn+ 1

-

S

(2)

~ A(Sn - S)2.

Replacing n by n + 1 and eliminating A from the two equations produces a quadratic equation for S qua sn' Ostrowski (1966) assumed more generally

m

~

2,

(3)

and proposed the scheme sn = Sn+l

+ (!a n+l!m+l/Iannsgn(sn+l

- s).

(4)

Of course, there can be a problem in determining the appropriate sign above. Jones's method (1976) includes the c5 2 -method and takes s; = Sn - L1sn/(d - 1),

(5)

where d is a root of (6) It is an easy matter to show the procedure is exact (s, = s) when s; satisfies (7) for some A E C(l, m ~ 1. The selection of the correct root of (5) is not really a difficult matter-Jones has a discussion of this. The procedure is intended to be used on sequences which converge or diverge hyperlinearly, for instance, if one takes m = 2 in (5), (6) will sum the sequence Sn+ 1 =

s; -

1,

So

= 2,

(8)

to its "correct" value, (1 + )5)/2. Note, however, the method is not accelerative for C(l, since (6) with ~sn + Ii~sn replaced by p has a root = p iff m = 1. All the above methods, however, have severe, perhaps fatal, computational deficiencies. If s, converges linearly, one is better off using a column of the a-algorithm to sum s. If s converges hyperlineariy, why use an acceleration method at all? It is my experience that one picks up in s; at most an extra significant figure or so over those present in s, + 2, which is used to compute sn' Finally, if s diverges hyperlinearly, severe loss of significance problems are

7.3. Related Algorithms

155

encountered. If m > 2, it is unlikely these can be overcome even on the largest computers. Iguchi (1975, 1976) discusses a generalization of the c:5 2 -process based on sn

(m

--+ 00

=

gives the

Sn+2

c:5

2

+ (Sn+2

-process).

-

Sn)

I

m

k;l

(

n

a +2 an + l

)2k,

(9)

Chapter 8

Lozenge Algorithms and the Theory of Continued Fractions

8.1. Background

In Chapter 6 it was shown how the Schmidt algorithm, when applied to the partial sums of a power series, produced the upper half of the Pade table. Since the diagonal Pade elements are the (2n + 1, 2n + 1) approximants of the continued-fraction representation of the function defined by the power series, it seems clear some formal connection must exist between the Schmidt transformation, i.e., the s-algorithm, and the theory of continued fractions. In fact, the s-algorithm is just one of several computational formats relating various elements of the Pade table. This chapter shows how two algorithms, the "I-algorithm and the calgorithm, can be derived from the theory of continued fractions. The theory is both elegant and satisfying because it establishes a deep connection between an algorithm derived purely algebraically and certain important ideas in function theory. The analysis in this chapter will depend heavily on material by Wall (1948) and Henrici (1977, Vol 2, Chapter 12). 8.2. The Quotient-Difference Algorithm; The "I-Algorithm

This section considers a procedure due to Rutishauser, who developed it and explored its application in a series of books and papers [see e.g., Rutishauser (1954, 1957)]. We shall not deal extensively with the properties of the quotient-difference (q-d) algorithm here, but use it primarily as a tool for obtaining the other lozenge algorithms, the '1- and s-algorithms. 156

157

8.2. The Quotient Difference Algorithm; The I]-Algorithm

A formal (not necessarily convergent) power series U = ao + a1z + azz z + ...

(1)

and a formal continued fraction of the kind (2)

are said to correspond to each other if the nth approximant Piz)/Qn(z) of

K, with

Po = 0,

. ..

(3)

and

Qo = 1, (4)

if expanded in powers of z, satisfies

(5)

It is not clear that such a correspondence need exist. But the following theorem states when this happens.

Theorem 1. For U, there is at most one corresponding K. There is exactly one such K if and only if the Hankel determinants satisfy

¥ 0,

n

~

k

0,

~

1. (6)

an+Zk-Z Proof

See Henrici (Vol. 2, p. 518). •

The q-d algorithm provides a systematic way of obtaining {qn} and {en} from {an}' We assume the condition H~k) ¥ 0 of the previous theorem holds, but for the present the development is purely formal and no assumptions are made about convergence. The even part of K is Z

K,

=

J

Z

1 qlelz qzezz ao [ 1 - q1z- 1 - z(qz + e 1)- 1 - Z(q3 +ez)~ ....

Its approximants are PZn/Qzn' The odd part of K is

«; =

[1+

aO

J.

z qlZ qZelZ .. . 1 - Z(ql + el)- 1 - z(qz + ez)-

(7)

(8)

158

8. Lozenge Algorithms and the Theory of Continued Fractions

Its approximants are P2n+ I/Q2n+ I' Now consider a sequence of functions {U;(z)} that have continued fraction developments (2) with corresponding coefficients a~, {q~i)}, {e~)}, and so Equating (7) for k (k)

ao [

1

+

+

+ elk)~

(k+ I)

= a k + ao

+ ak + I Z + ak + 2 Z2 + ....

(9)

I with (8) for k gives

q (k)Z I

1 - z(q~)

Uk = ak

q(k)e(k)z2 2

1 - z(q~)

I

+ e~l)-

I

]

...

q(k+1)e(k+ll z2 1

z [ 1 _ zqlk+ll_ I _ z(q~+1)

]

+ elk+ 1)- ....

1

(10)

For the sake of the formal development, assume these fractions terminate. Then a uniqueness argument [see Wall (1948, Chapter IX)] can be invoked to show they are equal coefficient by coefficient. The result is q~k)

+ e~k) =

e~k)q~k~l

=

e~k_\1)

+ q~k+ I),

q~k+l)e~k+I),

k :;::: 0, n

e:

1;

(11)

k :;::: 0,

e:

1.

(12)

n

To obtain starting values, observe that a~) = ai ; so

qlk l = ak+ dak

and

e~)

= 0, k > O.

(13)

Equation (9) shows U

=

ao

+ alz + a2z2 + ... + aNz N + /V+ 1U N+ I ,

(14)

for any N. But taking N sufficiently large shows (15) The q-d scheme may be arranged as follows:

8.2. The Quotient Difference Algorithm; The 'I-Algorithm

159

Table I

1

0

-2

1

2

0

!

0

6

i

1 -n

1

4:

0

1

1

-6

3

1

-6 1

1

TO

-TO

20

1

-20

1

5

0

The quantities in each formula constitute the four corners of a lozenge or rhombus, and one moves out in the table using first (11), then (12), and then repeating, as indicated in the above array. As an example, take U = e', The results are given in Table I. Thus Z 1 z i,z i,z /oz e = 1=-1+1=-1+1=-1+'" (16)

tz

which is, apart from an equivalence transformation, the known continued fraction for e', In this case, the quantities q~k), e~k) may be written in closed form q~k) = (n + k - l)j(k + 2n - 2)(k + 2n - 1), k Z 0, n > 1; (17) e~) = -nj(k + 2n - 1)(k + 2n), k Z 0, n Z 1. One way of making the q-d formulas easier to use is to label each quantity by its direction from the center of the lozenge: E represents east, etc. Then (11) and (12) become E = WSjN. (18) E = W + S - N, It can be shown that q~) = HLn~

IHkn-l)jHLn)HLn;/),

e~k) =

HLn+ I)Hkn+-II)/Hkn)Hkn~ I'

(19)

Thus the q-d recursion relations induce a recurrence relation sometimes attributed to Aitken (1931) but, in fact, known to Hadamard (1892). Of course the s-algorithm also makes a statement, by means of Eq. 6.8(5), about Hankel determinants. Theorem 2.

Define

[H~k)(6.a)]2

H~O) =

1. Then

- H~k)(6.2a)H~k)(a)

n Z 0, H~k)(a)2 -

+ H~k+ 1)(6.a)mk k Z 1;

I )(6.2a)

l(a) + H~k_\I)(a)H~k;II)(a) n, k z 1.

H~k~ l(a)H~k~

= 0,

= 0,

(20)

(21)

160

8. Lozenge Algorithms and the Theory of Continued Fractions

Remark. (21) can be shown independently of the q-d algorithm by using Sylvester's expansion; see Section A.3 of the Appendix.

Bauer [Bauer (l959, 1965); Bauer et al. (1963)) seems to be the first to trace the connection between the q-d algorithm and the s-algorithm. The basic idea is to convert the continued fraction K, which is equivalent to the formal power series V, into a Euler continued fraction K'

=~

Pl P2 P3 1- 1 + Pl - 1 + P2- 1 + P3-

(22)

Under appropriate conditions [see Wall (1948 p. 17, Theorem 2.1)) this continued fraction is equivalent to the infinite series V'

= aO(1 + J/IP2 "'Pr)

(23)

in the sense that the nth numerator of (22) is equal to the sum of the first n terms of (23) and the nth denominator is 1. The IJ-algorithm establishes a correspondence between the terms of the above series and the coefficients aj of V. The s-algorithm results on interpreting the IJ-algorithm for sequences. In what follows all convergence considerations are disregarded, since these are thoroughly discussed in Chapter 6. The required conversion of the continued fraction K ' to K depends on shameless algebraic trickery. Recall that the denominators of K satisfy Q2m(Z) = -qm zQ2m-iz)

+ Q2m-l(Z),

Q2m+ l(Z) = Q2m(Z) - emzQ2m- l(Z),

m:::::-:1.

(24)

m:::::-:1.

(25)

1,

(26)

Let A be an arbitrary complex parameter. Write qm =

Q2m-l(A) - Q2m(A) ;·Q2m-2(A)

m:::::-:

em =

Q2m(A) - Q2m+ 1(A) AQ2m-l(A)

m:::::-:1.

(27)

Defining gm(A) = Qm + 1(A)/Qm(A),

m

> 0,

go(A) == I,

(28)

> 0.

(29)

(note go(A) = 1), we also have gm(A)gm+ 1(,1.) = Qm+2(,1.)/Qm(A),

m

8.2. The Quotient Difference Algorithm; The II-Algorithm

161

Using the formula (2.2) in Wall shows

( A) = zQm-l(Z)[Qm(.1) - Qm+l(.1)] Pm z, .1Qm+ l(Z)Qm-l(.1) ,

(30)

m~1.

Then

K(z) = K'(z) = ~

Pl(Z, A) pz(z, A) 1- 1 + Pl(Z, .1)- 1 + P2(Z, .1)-

(31)

Now let .1= z:

K(.1) = K'(.1) = ~

Pl P2 1- 1 + Pl - 1 + P2-

(32)

where

Pm == Pm(A, A) = [1 - gm(.1)]/gm(A),

m ~ 1.

(33)

m ~ 1,

(34)

1.

(35)

Also, from (26) and (27),

qm = [g2m-z(.1)j.1][1 - g2m-l(.1)], em = [g2m-l(A)/.1] [ 1 - g2m(.1)],

m

~

Assume, as with the q-d algorithm, that both K and K' are used with two different functions, and U k + 1 , associated with quantities g~>, q~>, and e~l. Applying the q-d algorithm to K and using (34) and (35) with all quantities superscripted by k gives for n = 1 in (11)

v,

(g~ + 1)/.1)(1

- g\k+ 1l)

= (g~) /.1)(1

- gt l)

+ (g\k l/.1)( 1 -

g~l)

(36)

or (37) For n

= 1, (12) gives

which when combined with (37) gives (1 - g\k+l»(1 -

g(km'l

g~+l»

= (1 -

g~l)(l

_

(39)

g~kl).

Continuing this process gives a lozenge algorithm for the computation of (40)

k

~

0, m

~

1, (41)

162

8. Lozenge Algorithms and the Theory of Continued Fractions

with starting values g~)

= 1,

(42)

To derive the 17-algorithm, let .,(k)

'1m

= a k Ak

m flp(k) r= 1

r

(43)

,

with (44) Then, since p~)

= Pm' we have from (32) and (23) U(A) = 171>°)

+ 17\°) + 17~0) + .,.,

(45)

and so we have defined a series transformation of U(A). Of course, the 17~) satisfy lozenge relationships. For instance, let (46) Substituting (43) in the above, factoring 17~~-1 from the numerator and 17~:!1 from the denominator, and pairing off factors by 2s using (44) and the g~) recursion relationships yields

(47) and this provides the first of two relationships. The iterates in the n-alqorithm are defined by

(48)

1 .,(k)

'12m

1

+~ = '12m+l

1 .,(k+ I) '12m-l

1

+ '12m .,(k+ 1)'

k,m 20,

(49)

with starting values

(50) The derivation of the second relationship above is straightforward, By the 17-algorithm we mean the summation of the sequence defined by

(51) in terms of

(52)

8.2. The Quotient Difference Algorithm; The I}-Algorithm

163

The computational scheme is as follows: ao = 17bO)

= 17b1)

al/l

17~O)

1711) a2/l 2 = 17l?)

YJ~O) 17~I)

17~O)

1712)

17~1 )

a3/l 3 = YJb3)

17~2)

1713 ) a4/l4

= 17b4)

Symbolically the 17-algorithm may be written N + E = W + S => E = W + S - N,

~+~=~+~=>E= (~+~_~)-l NEWS

(53)

WSN'

the formulas being applied alternately. Often it is convenient to take /l = 1 in the algorithm. As an example, consider the divergent series 0!-1!+2!-3!+4!-···,

(54)

The 17 table is as follows: 1

1

-1

2

3"

2

6

I

-6

24

24 -20 -120

TI 3

5

4

91 6

6

3

-2

-6

2

-21

4

-91

10 8

-35

-s4

-2TI

164

8. Lozenge Algorithms and the Theory of Continued Fractions

The original series is therefore transformed into the series 1-

!

+

i - l1

+

41 9 -

+ ... ,

2~1

(55)

whose first six terms provide the sum 0.5882352 (cf. Example 6.5). The s-algorithm results from interpreting the 1J-algorithm as a sequence rather than a series transformation. Let

=

e~~

k-1

I

r=O

+

1Jt)

2m-1

I

r=O

(56)

1J~k).

Then the starting value e~)

=

k-1

I

r=O

1Jt)

k-1

I

=

QrA

(57)

r

r=O

is the kth partial sum of the original series and 2m-1

e(O)

2m

= '" L..,;

(58)

"(0)

nr

r=O

is the 2mth partial sum of the transformed series. We find that e~: 1)

-

e~~

=

1Jbk)

+

=

IJ~~

+

= 1J~~ +

2m-1

L

r=O

2m-1

1J~k+ 1)

L

-

r=O

2m-1

I

r=O

1J~k)

2m

1J~k+ 1)

-

L lJ~k)

r=1

k-1

L (1J~\+1) + lJ~r-r.-V

r=O

-

lJ~k~+1

-

1J~~+2)

(59)

or (60) and so the odd partial sums of the transformed series are given by 1) e(2m

=

2m

'" L..t

r=O

1](0) r .

(61)

Now let (62) and for convenience, let e~\ = O.

8.2. The Quotient Difference Algorithm; the x-Algorithm

165

One can show, as above, that

B~:;~ - B~~+ 1

=

1/1J~~+ 1, (63)

B~~+1 - B~:!~ = 1/1J~~, B~~ + 2

-

B~: 1)

= IJ~~ + i -

Applying to these formulas the iteration rules for the IJ-algorithm shows that the following hold for both even and odd subscripts:

n, k

~

0,

(64)

with B~\

= 0,

B~)

=

k~l

I

r;O

arAr.

(65)

Other Lozenge Algorithms and Nonlinear Methods

Chapter 9

9.1. A Multiparametere-Algorithm

The s-algorithm may be considered one of a class of lozenge algorithms that depend on an arbitrary fixed sequence y E '(/5. The s-algorithm results on choosing y = {c}. For any sequence W E '(/5' define the linear operator R: '(/5 --> '(/5 by {R(w)}n = R(wn) = ~(Wn/Yn).

(1)

R may be iterated by means of the rule

(2)

so that R 2(w n) = M~(wn/Yn)/Yn}, etc. Note that if Yn = c, then Rk(wn) =

~kwn/ck.

Now define RWn

Wn+ m RWn+ m

Rmwn Yn RWn

Rmwn+ m Yn+m RWn+ m

Rmwn

Rmwn+ m

Wn

e\f~

ein~+l

= fm(wn) =

= Ilfm(Rw n).

(3)

(4) 166

9.1. A Multiparameten-Algorithm

167

4n ) satisfies

Theorem 1.

f.1n~1

= f.1n~+/) + Yn(e1n+ l ) - 81n») - I, e~)l = 0, eg') = w-lv«. n

n, k ~ 0.

~

0,

(5)

Proof The proof is the same as that for the e-algorithrn and is left to the reader. •

The most useful case occurs on choosing (6)

and defining (7) Then eg') = Sn, n ~ O. The transformation ek is translative and homogeneous. Exactness theorems for this transformation are, of course, more difficult than those for the s-algorithm because of the nature of the operator R. However, some information is available; see Brezinski (1977, pp. l l lff.). Example.



°

fl(n)···J,.(n) (2)

where

Cn

=

[~

•

1

...



P, let O.

(14)

on P iff S~k) converges to S on P in the

(i) constant, P any path; (ii) not necessarily constant, P a path (n, k) with n Proof

For n

-> 00.

+ k large, 11 = const the algorithm can be derived in the same way the Schmidt transformation was derived. One assumes that Sn behaves as Sn

=S+

k

L

(1)

crfr(n).

r~l

Taking ¢ of Eq. (1), replacing n by m, n :-: : ; m :-: : ; n + k, and considering those equations and the above as k + 2 equations in the k + 1 unknowns (¢, S), Cb C z , " " C k produces the requirement S

0

f~(n)

(¢,sn)

I

(¢, ftc(n»

s, -

(¢,

Sn+k)

(¢, fl(n

+ k»

(¢, ftc(n

= 0,

(2)

+ k»

but this is clearly equivalent to Eq. 10.1(2) when S = s~kl, cI> = const. Conversely, when s; has the form (I), then E k will be exact, S~k) == S, provided the algorithm is defined. For the study of this algorithm, there are two modes of regularity or accelerativeness to consider. One pertains to weak convergence, i.e., convergence in the seminorm I(¢, .) I.The other is the usual strong convergence, convergence in the norm. The regularity result below, though based on pretty specific properties off,., is often applicable. Theorem 1.

Let s E fJB c and

lim(¢, f,.(n

+

1»/(¢, f,.(n»

= b,

# 1,

1 :-: : ; r :-: : ; k,

(3)

where b, # bj , i # j. Define

(4) (which we assume exists for n sufficiently large) and denote by Am the proposition "lIfm(n)II/(¢, fm(n»

= 0(1)."

(5)

Then along any vertical path.

In

(i) if '7n is bounded, then E; is regular for s in the seminorm I(¢,.) I; (ii) if '7n ----> b, for some j, then Ek accelerates s in seminorm; (iii) if '7n is bounded and Am holds, 1 :-: : ; m :-: : ; k, then E, is regular for s norm;

10.2. The Case I/J Constant

if'1n ---> b, for some j, Am holds, 1 :s; m :s; k, m

(iv)

=1=

181

j, and (6)

then E k accelerates s in norm.

Proof ship r(k) n

~ ~

All these statements are immediate consequences of the relation-



k-l

_ S + '" 1...

Sn -

° °

m=O

r m Tn(m) ,

== s,

n ~

0, iff

(2)

where T~m), S m S k - 1, is a basis of solutions of the (scalar) equation = satisfying

;J!lk

TU) = m

J mJ'.

Os m,j S k - 1.

(3)

Proof

(4)

10.3. The Topological Schmidt Transformation

183

Taking ¢ of both sides gives

o = ~~k) =

(5)

Jtv,.(~n' d~n)/Jtv,.(I, d~n)'

By Theorem 6.3, the definition and exactness of ~~k) imply ~n E x:k : Let {r~O), r~l), ... , r~k-l)} be a basis of gIIk where gIIk(~n) = 0, satisfying (2). Then :


(9)

j=O

and finally O~i~j~k+l O~j 0, (8)

to = a = 0,

p = 1.

Here G is known explicitly, but, surprisingly, not more than ten or so tabular values are required to determine I to almost five places despite the fact that 9 is singular at zero. Thus we may assume, for the example11(0) = 0, that 11 values of G are known and tabulate the 11th ascending diagonal of S~k) (see Table I). Example 2 g(x) = -e-X(x G(t)

+

x> 0,

1)lx 2 ,

= e-tlt - lie,

I =

J'"

(9)

g(x) dx = e-

I

= 0.367879441,

to = a = p = 1.

The sixth ascending diagonal is tabulated in Table II. In this example double precision (16 significant figures) was used, and Sb1 5 ) is accurate to 16 significant figures. This indicates the method has great numerical stability, at least when applied to monotonic integrands. Table II k 0 I 2

5-'

k

S~~k

-0.367466316 -0.367863093 -0.367878981

3 4 5

- 0.367879339 -0.367879477 - 0.367879363

Sl')

203

11.1. Introduction; The G-Transform Table III

Example 3.

k

S\k~ _ k

k

S\k~ -k

0 2 4 6

1.04471 0.99818 1.00015 0,99968

8 10 12

0.99996 0.99967 0,99996

This has an oscillatory integrand, corresponding to

I = Then

f

OO

sin x - x cos x

o

x

G(t)

=

2

dx = 1.

(10)

1 - (sin t}/t.

(11)

Some elements on the 13th ascending diagonal are tabulated in Table III. The error in #5) is 1 X 10- 6 . Obviously, the algorithm was not designed for integrands that decay algebraically or logarithmically. For J~ x-l(ln x)" 2 dx, as another example, Sb1 5 ) = 1.262, while the true value of the integral is l/ln 2 = 1.443. We now look at the exactness problem for this algorithm. Theorem 1. For some complex constant do, d 1 , ••• , db let A. E f:(}c be a sequence of roots with negative real parts of the exponential polynomial H(A) = do

k-l

+ AI

r=O

(12)

dr+ le"rp •

Then if (13)

where Pm(t) is a polynomial of degree less than the multiplicity of Am' infinite sums being allowed subject to convergence conditions, the transformation (5) is exact for each t; i.e., Ilk) == I, t > a, provided the denominator of (5) does not vanish. Proof

Define !£(f) = dof(t)

k-l

+ I

r=O

dr+d'(t

+ rp).

(14)

If g satisfies the equation !£(g) = 0, then, by integration between t and

do[G(t) - 1]

k-l

+ I

r=O

dr+lg(t

+ rp) =

0,

t > a,

00,

(15)

204

II. The Brezinski-Havie Protocol and Numerical Quadrature

°

so the numerator of the determinantal expression of I~k) - I will vanish. Let ,10 be a root of H(A) of multiplicity m. We need only show 2(ti e AOI) = for O:5:j:5:m-l. We can write eAIH(A)

d

k-l

= co eAI + "c _ 1... r+ 1 dt r=O

eA(I+rp )

(16)

O:5:j:5:m-l,

(17)

'

so

which was to be shown.

•

Corollary 1 (k = 1). Let f E L(O, iff g(t) = Me-at, M -# 0, Re a > 0. Corollary 2.

00).

Then 1~IJ is defined and exact

For some complex constants db ... , db d,

+ dk -# 0, let A be a sequence of roots with negative real parts of

°

k- 1

"d 1... r+ 1 e Lrp .

r=O

Then

IlkJ, k

~

+ ... (18)

1, is defined and exact for

=

g(t)

L: Pm(t)e

Am l

(19)

,

where Pm is as in Theorem 1.

Proof Completion of the proof, which requires Heymann's theorem to guarantee the nonvanishing of the denominator of Ilk!, is left to the reader; see Section 6.3. • The following result on accelerativeness is easy to demonstrate.

Theorem 2. Let D(t) denote the denominator of I:k ) and Mr(t) the rth cofactor of the first column of D. Let Mr/D be bounded. Let I exist, g be bounded, and 1 :5: r :5: k.

(20)

Then lim {(Wl - l)/[G(t) - I]} = 0. t~oo

Proof

Left to the reader.

•

(21)

1l.2 The Computation or Fourier Coefficients

205

Let us take as an example the important case k = 1, III _

~

-

G(t

+ p)

- G(t)g(t + p)jg(t) . 1 - g(t + P)jg(t)

(22)

If g(t + p)jg(t) = A + 0(1), 0 < A < 1, the hypotheses of Theorem 2 are satisfied ~ in fact, in this case the conditions are necessary and sufficient for the accelerativeness of Ipl; see Gray and Atchison (1967). The algorithm is most suitable for integrands that behave exponentially. Obviously iff = o(t-a), the conditions of theorem are not satisfied; in fact, for k = 1, one has (23)

An algorithm suitable for cases in which f behaves algebraically can be obtained by making an exponential substitution in (2)-(6). This amounts to taking in the BH protocol f,(n)

= topn+r-lg(topn+r-I), to

~ a ~

1, p > 1.

(24)

However, these equations offer no clear computational advantage over (7), since tabular values of G for very large t are required. An exactness theorem analogous to Theorem 1 is easily established for the new algorithm. Details are left to the reader. Theorem 2 remains unchanged. For the important case k = 1, these results show the algorithm is exact for functions f(t) = Mt- a, M #- 0, Re IX > 1, and accelerative if f(t) = O(t- a), Re IX > 1. The papers by Gray, Atchison, and Clark detail many other properties of the k = 1 algorithm. 11.2. The Computation of Fourier Coefficients Suppose it is required to compute the Fourier coefficients I(m)

=

L

f(x) cos(2nmx) dx,

(1)

and that a sequence s of values of the trapezoidal sums (2)

is known. Further, assume that Romberg integration (Section 3.1) has been applied to Sn to produce a value of 1(0) accurate to as many figures as are required of 1(m).

206

II. The Brezinski-Havie Protocol and Numerical Quadrature

The BH protocol, combined with a method due to Lyness (1970, 1971) can be used to attack this problem. To be accurate, we should speak of a "class" of methods, since Lyness's theory has a great deal of flexibility, which allows one to take advantage of additional data, i.e., a knowledge of the derivatives off Here only the simplest form of his algorithm will be used. (It seems a pity that Lyness's work, uncomplicated and beautifully ingenious, has received almost no attention from the authors of books on numerical analysis.) Supposefhas the Fourier series development f(y)

=

1(0)

+

2JI f

f(x) cos[2nk(x - y)] dx.

Let y assume the values jim and sum from j = be expressed

2

°to m -

(3)

1. The result may

00

I

k=l

l(km) = rm ,

(4)

[For details, see Luke (1969, Vol. II, p. 215).] Now, the Mobius inversion formula (Hardy and Wright, 1959, p. 237) states that, subject to certain convergence conditions, the sum m

~

(5)

1,

may be inverted to yield 00

r; = I

k= 1

ilk G k·m,

m

~

1,

(6)

where ilk is the Mobius function, ilk

=

f~

1

(-1)'

k = 1 if k has a square factor if k is the product of r prime numbers.

(The first ten values of ilk are + 1, -1, -1,0, -1, applied this formula to the sum (4) to obtain l(m) =

1

(7)

+ 1, -1,0,0, + 1). Lyness

00

2 k;/krk.m.

(8)

This is the series from which we wish to compute l(m). We show how the BH protocol can be applied to the partial sums of this series. Let 1 n+ 1 lim) = -2 I ilkrk'm, k=l

n

~

0,

(9)

11.3. The tanh Rule

207

and define R; =: I n(m) - l(m) =

1

2

L 00

k;n+2

(10)

Ilkrk'm'

From the fact that (11) fj(n) =

L 00

k;n+ 2

Ilk

k2 j

'

(12)

However, (13)

so I

fj(n) = (2j) -

n

+

1

k~1

Ilk k2j

'

n ::::: 0, j

>

1,

(14)

and to complete the BH protocol one takes 1 n+ 1 s, = I n(m) = -2 L Ilkrk'm, k;l

rn =

T" =

T" -

1(0),

~n in f(~), n k;O

1(0)

=

f

(15) f(x) dx.

[The numbers (2j) are extensively tabulated; see e.g., Abramowitz and Stegun (1964).J One would expect, based on the representation 10.4(1), that rin ) = Oin" 2k- 2), n -> 00. (This has not been proved, of course.) The original series, Eq. (8), converges only as n- 2 . Iffhas derivatives, i.e., if the values of c., c 2 , ••• , c 2 r + I' are known, these may be used in an obvious way to make the process even more efficient, with T" minus the first several terms in the series (11) taken for T". 11.3. The tanh Rule The basis of the tanh rule is the approximation of a doubly infinite integral by means of a trapezoidal approximating sum. Thus the quadrature process is similar to the methods based on cardinal interpolation. However, there is an important difference, one that changes completely the nature ofthe

208

11. The Brezinski-Hiivie Protocol and Numerical Quadrature

error term: The infinite sum is truncated at ± N(h). The problem is, how should N be chosen to obtain optimal results? Following Schwartz (1969), we make a change of variable in the finite integral J~ 1 g(x) dx. Let ljJ be a reasonably smooth function that is monotone and maps ( -1, 1) into ( - 00, (0).

~ hrt_f'(rh)g(ljJ(rh)).

flg(X)dX = f:oog(ljJ(t))ljJ'(t)dt

(1)

How should ljJ and h == hen) be chosen? Schwartz suggested ljJ(t) = tanh(!t) (hence the name "tanh rule") and h = nj2FJ. For integrands 9 in Hardy class H 2 , Haber (1977) has computed the asymptotic form of the error norm and has shown that for the above choice of ljJ, the choice of h is optimal. [The functions in the Hardy class H 2 are iO 2 functions analytic in N for which I f(re ) 1 dO is bounded as r ---+ 1.] Let 9 E H 2 and define

gJr

( )= h Sn

9

It can be shown that S that

i

s(g) =

r= -n

-

f

1 g(x)

(2)

dx,

g(tanh(nh/2)) 2 cosh 2(nh/2) ,

h=

nj2FJ.

(3)

s; is a bounded linear functional on H 2 • Haber found (4)

Note that this seems to be considerably inferior to the bound obtained for the trapezoidal rule in Section 3.4. However, there the sum is not truncated and the class of functions is smaller. Haber's computations seem to indicate that a good choice for the BH protocol is Jj(n) = e-(Jr/J].lJri/(n

+ l)U- 1 )/ 2 .

(5)

The function g(x) = (l - x 2 y is in H 2 provided Re a> I = rca + l)fi/rca + ~) and

-i.

Then

n 2: 1 (6)

n 2: 1,

and

So

= O.

11.3. The tanh Rule Table IV BH Protocol Applied to 1

7

Ci

k

2

4 6 8

8

12

=

Sk

-t 1_ 13 =

(tanh rule)

2.611931003 2.586166070 2.586239244 2.586715520 2.586937436 2.587032111

2.587109559 s~)

2.266890051 2.563060233 2.586139159 2.587082817 2.587108878 2.587109544

2

=

IX

=

Sk

fO o

-

209

x 2 )' dx

i, 1_ L4

(tanh rule)

2.440806880 2.399070105 2.396475368 2.396260717 2.396257569 2.396267876

=

2.396280467 )'(kl

'0

2.048670072 2.371528094 2.395295728 2.396255106 2.396279761 2.396280440

Table IV displays Sk versus s~), i.e., vertical versus diagonal, convergence for the choice (5) and the cases (X = -t and (X = -t. Clearly, the BH protocol is a powerful tool to use in conjunction with the tanh rule.

Chapter 12

Probabilistic Methods

12.1. Introduction

Historically, the construction of summability methods has been based on the philosophy and techniques of classical analysis. Actually, the problem of accelerating the convergence of a sequence is more at home in a probabilistic setting. A formulation in terms of prediction theory or recursion filtering, for instance, immediately suggests the minimization of the expectation {E(lrnl)} of the transformed error sequence if the original sequence is interpreted as a sequence of random variables.t By assuming certain distribution functions for the {sn} and performing this minimization, one is led naturally to a class of methods for transforming sequences. Of course, the methods will depend on the parameters of the chosen distributions. If these parameters are unknown, any well-known estimation technique can be applied. Each estimation technique provides a different summation method. Although the construction of summation methods has not traditionally been based on probabilistic techniques, the methods themselves have been put to extensive probabilistic use. For example, Chow and Teicher (1971) represent the strong law as a trivial special case of the following Toeplitz summability. Let {X n}:'= 1 be independent identically distributed random variables with finite first moment. Suppose (1) «.> 0, n > 0, t Good sources for the theory of probability and stochastic processes needed in this chapter are Papoulis (1965) and Miller (1974). 210

12.2. Derivation of the Methods

211

and 11 ;::::

0,

(2)

diverges. Define the transformed sequence {1;,} by

1;, =

n

s;; 1 L a.x;

11 ;::::

j=O

O.

(3)

If 1;, - C, -+ 0 almost surely for some centering constants {Cn}, then {X n} is called an-summable with probability 1. Note that the strong law is obtained by using C, = EX,

11;:::: 0,

(4)

the common mean of the underlying distribution, and

11;:::: O.

(5)

The summation methods to be derived here are nonlinear and nonregular. They are simple to use. They are useful for summing classical series and also for summing "statistical" series whose terms are realizations of random sequences. Numerical examples of both kinds of applications are included here. The advantages the methods hold for statistical applications are clear: For series defined by complicated experiments in which obtaining data is difficult and expensive, the use of the proper summation method based on an appropriate probabilistic assumption can result in practical advantages. Finally, we shall show that for one large and important class of sequences, the methods are regular, namely, the sequence space of partial sums of alternating series whose terms in absolute value are monotone decreasing. No other nonregular method has been shown to be regular for this sequence space. 12.2. Derivation of the Methods To motivate our derivation, suppose that the series Lk='O a k is a realization of the following" experiment": Let {xdk'= 1 be a sequence of independent random variables with and where

Ipl < 1 and q
O.

-

(6)

Thus ultimately {en - l)a n } is monotone decreasing. Therefore lim nan + 1 = lim nan

n--+ co

n- 00

exists. Now from (7)

we can conclude

ak!ak -

1

:OS::.l - YI(k - 1),

1 n ak 1[NL-+ ak Pn=-I-:os::In nk=l ak-l

n

k=l

ak-l

k=N+l

(8)

Y)J

(1 - - , k- 1

(9)

or

Pn

:os::

[n - yin n

+ M(n)]ln,

(10)

where {M(n)} is a bounded sequence. Here we have used the fact that n

1

I k_ k=N+l

+ 0(1).

(11)

1 - P« 2 [y In n - M(n)]ln,

(12)

1 = In n

Thus

218

12. Probabilistic Methods

and for n large enough, the right-hand side is positive. Thus

an

--

(14)

00

This gives the result for Method I. Now

IPnln~.

[1 + M(n) ~ yin nT

=. exp [ n In ( 1 + =. ex p[n(M(n)

M(n) - yin n

~

yin n

n)J '

+ e~n»)

l

(15)

by a Taylor's series argument, where {sen)} is a null sequence. Thus

nlPnln {3n 2 - i' -- IAII > IAII, 2 ~ j

~ k,

u(r)

E

Y} (29)

Note Ye is a generalization of CC Ek(N) [see Eq. 2.2(12)]. Theorem 4.

Proof

Let

Method II is regular for Ye; Method I accelerates Ye. uUl E

Y. By an application of Theorem 1.4(2), (30)

220

12. Probabilistic Methods

Now, an + 1

k

L A~+lu~)[1 + o(1)J

=

r=1

-

k

k

r=1

r=1

L A~u~) = L A~u;;l[(Ar -

Thus .

hm

n-e co

rUn An (r) 1 .fn<TI -_ .hm

I

lUn

n-+oo

Uo I(I) Iexp [( n In I-, I + C (r)

A

Uo

(r) n

r

1\..1

(I)

Cn

)J

_ -

1)

0,

+ 0(1)].

(31)

2 :s; r :s; k, (32)

by virtue of the fact that In IArlAl I < O. Thus

lim an + dA~u~1)

= Al - 1.

(33)

Also, (34) Dividing these two limits gives lim an+ dan

= AI'

(35)

Since p is the Cesaro mean sequence of the sequence {anla n- d, it also converges to AI' and so (1) gives

sn =

s

+

±

An - l u (rn -)

r= 2 r

I

[(1 + ~) + 1 _ Al

O(1)J

+

AnI - 1U (nI- ) I 0(1) .

(36)

Thus (37) n~

00

From (32), (38) Dividing these limits shows that

= 0,

(39)

~ = ~ [1 + 0(1)J,

(40)

lim [(Sn - s)/(sn - s)J and this is the desired result. For the result for Method II, note that

I - P«

1 - Al

and this, used in (25), implies the method is regular for s.

•

12.3. Properties of the Methods

Note that all these methods map the partial sums of So

=

s; = 11(1 - x),

1,

I:=

n 2: 1.

0

221

x" into (41)

This property is shared by other nonlinear transformation, for example, the Shanks ef transformation (Shanks, 1955). In fact, a transformation related to Method I was mentioned in passing in Shanks (1955, pp. 25-26) under the name "geometric extrapolation." This transformation is defined by

sn =

Sn -

lim.(anlan-l)sn-l

n 2: 1.

-"---"-'-"-'-~"--"~

I - limn(anla n- 1 )

(42)

Method III is, for certain classes of sequences, the most effective method of all. We now make this more precise. Lemma.

Let

(- 00,0].

WE ~

be bounded and belong to a bounded subset S of

(43) it follows that

lim s,

Proof

Let sup

ZES

= s.

I-zI l-z

= d

(44)

< 1.

(45)

Note also that

\1-zl- 1 :::;; 1,

Z

E

S.

(46)

Since (47) one can write

Choose N such that WnE S, n > N, and ISn - s] rk is bounded, Irk \ < C

= Irnl < s, n> N. Since

222

12. Probabilistic Methods

or (50) Now (51) for fixed k, so lim

n-e co

(n)d. k

(52)

= 0,

taking lim sup term by term gives

rrm If.1 < s,

(53)

or, since c: was arbitrary, or

Theorem 5.

lim f.

= O. •

(54)

Method III is regular for all alternating series

for which sup bk/bk- 1 = M
S(k l, but here it is more useful to think of {Snk} as a rectangular, rather than a triangular, array.) We shall assume that

°

lim Snk =

k-

fln,

n 2 0,

00

k 20. Definition. regular if

(3)

The transformation defined by (1) is called horizontally lim

k-oo

Snk

=

fln'

227

n 20,

(4)

228

13. Multiple Sequences

and vertically regular if k

~

(5)

0.

The material in the remainder of this section is due to Higgins (1976). Theorem 1. The transformation defined by (1) is horizontally regular iff (i) given n

0,

~

n

k

L L IlliY I ~ R n

;=0 j=O

for some positive number R; independent of k; (ii) given n ~ 0, lim

°

k

L 1l~1 =

bnr ;

k-oo j=O

(iii) given n

0, j

~

~

0,

n,

~ i ~

lim lliY = 0.

k-oo

Proof p/2, convergence being obtained regardless of the order in p-space in which the terms are added up.

Proof The most elegant demonstration uses the theory of theta functions. This proof is given in Section 13.2.2. •

For a discussion of the physical context in which such sums arise, see the classic treatise by Born and Huang (1954). We shall take an approach with these sums that is fundamentally different from the procedures used previously in this book to accelerate the convergence of series or sequences. The techniques given here will not be general, but will very much depend on the specific character off This is, of course, very much in contrast to the previous work-for instance, the fact that the remainder sequence possessed an asymptotic series of Poincare type-where only the general form of the sequence or series was of interest. The present kind of endeavor might be called the analytic approach to sequence transformations. The arguments used will depend on known properties of mathematical functions, such as theta functions, and on the application of a powerful formula from classical analysis, the Poisson summation formula.

13.2. Crystal Lattice Sums

233

13.2.1. Exact Methods Definition.

Let f be locally L(O, 00) and let the integral .A(f; s) = {"" x·- 1f(x) dx

(1)

converge for Re s = to, Re s = t 1, to < t t- .A is called the Mellin transform

off

Clearly the integral converges for IX

< Re s < {3

(2)

where IX = inf to and {3 = sup t l' (2) is called the strip of absolute convergence of (1). The Mellin inversion theorem states that iffis of bounded variation in a neighborhood of x E (0, 00), then, for any IX < C < {3, f(x+)

+

f(x-) _ _ 1 l' fC+iR «cj, ) . 1m JI't, S X 2m R~"" c-iR

'---'-----'--=--'-----'- -

2

-r

s

d

S,

(3)

Usually.A may be continued analytically into a larger region q; of the complex s plane, for instance, f(s)

-. = a

f"" x

.-1

0

e

-ax

d

X,

Re a> 0.

(4)

°

Here IX = 0, {3 = 00, q; = Cfi - {a, -1, -2, ... }. The theta functions for x > are defined as follows.

(5)

For some of the many beautiful properties of these now almost forgotten functions, consult Whittaker and Watson (1962), Hancock (1909, Vol. I), or Bellman's more recent book (1961), which is compulsively readable. A good collection of formulas is in Abramowitz and Stegun (1964). The following notation is standard:

B/O, q) = BlOlr),

(6)

234

13. Multiple Sequences

Thus 8i(0Iix/n) corresponds to taking q = e- x in 8i(0, q). Formulas such as the following can be found in Hancock (1909, Chapter XVIII):

(7)

Similar formulas exist for OJ, etc.; see Hancock or Jacobi (1829), who gives a list of 47 such relationships. It can be shown that 8/0Iix/n) has an algebraic singularity at x = 0; hence Mellin transforms of Oz, 8 3 - 1, 84 - 1, etc., have a half-plane of convergence. The Mellin transforms of theta functions generally involve meromorphic functions such as Riemann's zeta function, defined for Re s > 1 by (s)

=

1

L s' n 00

(8)

n: 1

We shall need the formulas (1 - 2 1 - ' )( s)

=

00

(_1)"-1

n: 1

n

L

"

Re s > 1; (9)

Re s > 1. Another useful function is Re s > 1,

(10)

which satisfies the relationship L(1 - s)

= (2/n)'r(s) sin(ns/2)L(s).

(11)

Obviously, L(s) can be expressed in terms of the generalized zeta function 00

(s, a)

1

= n:O L (n + a)S'

-a ¢ JO,

Re s > 1.

(12)

13.2. Crystal Lattice Sums

235

The Mellin transforms of powers of the theta functions can be found from such formulas as (7). For instance,

A[8~(01~)J = 4AL~o(-lte-(n+1/2)(2k+l)XJ (-It

00

= 4r(s)

n.~o (n + t)S(2k + I)' 2s

00

= 4r(s)L(s)n~o (2n + I)S = 4(2

S -

1)r(sK(s)L(s). (13)

Mellin transforms of products of theta functions can be found by using the

Landen transformations,

8iO, q)83(0, q)

= ¥1~(0, ql/2),

8iO, q)8iO, q) = te-"i/48~(0, i q I / 2),

(14)

83(0, q)8iO, q) = 8i(0, q2),

and the formula

A{f(ax); s} = a-sA{f(x); s}.

(15)

Table I gives some of the Mellin transforms that can be found this way. Table I Mellin Transforms Involving OJ

=

Ii/Ol ixln)

f(x)

O2

2(2 2 '

04

2(2 1 - 2 ' - 1)[(s)(2s) 4(2' - 1)r(s)(s)L(s)

03 O~ O~

O~

(0 3 (0 4

- I

l)r(s)(2s)

1)2 1)2

_

_

0 304 -

-

I 1)(04

0 20 304 0'2

OJ - I

01-

4r(s)(s)L(s) 1)[(s)(s)L(s) 4r(s)[L(s)(s) - «2s)] 4(1 - 2 1 - 2')[(S)[(2s) - L(s)(s)] 2>+ [(2' - 1)[(s)(s)L(s) 22-'(2 1 - ' - l)f(s)(s)L(s) -2 2 - ' [(s )[r ' ( 2s ) + (I - 2 1-')(s)L(s)] -2'+ [r(s)L(2s - I ) 16(1 - 2'-')(1 - 2-')[(s)(s)(s - I)

4(2 1 - ,

- I

020 3

(0 3

-

2[(s)(2s)

I

O~O~ O~O~ - I 8~0~

-

I)

8(1 - 2 2 - 2')f(s)(s)(s - I) - 8(1 - 21-»(1 - 2 2 -')f(S)(5)(S - I) 2>+ 2f(s)L(s)L(s - I) _2 3-'(1 - 2 2 - ' )(1 - 2[-')r(s)(s)(s - I) 2 2 +' (1 - 2 1 - 5 )(1 - 2-')f(s)(s)(s - I)

236

13. Multiple Sequences

To see how these formulas can be used to obtain closed-form expressions for lattice sums, consider

~, = L,

1 r(s)

--

-00

= - 1

r(s)

foo x

e

s-1 -(m 2+m 2+m 2+m 2)x

0

I

2

foo xS-1[e~(Olix/n) 0

3

4

dX

1] dx

= 8(1 - 22 - 2 S)((S)((s - 1).

(16)

[Later it is shown that this sum converges for Re s > p/2 = 2. Since ((s - 1) has a pole at s = 2, the result is sharp.] As another example, consider

= L(s)((s)

(17)

- ((2s).

A short table (Table II) lists two-dimensional sums determined by Glasser. Table II cc

S =

L' J(m,n)

J

S

(m 1 + n 2 ) - ' (_l)m+n(m 1 + n 2 ) - ' (_I)n+ '(m 2 + n 2 ) - ' [(2m + 1)2 + (2n + 1)2r'. m, n Z 0

4«s)L(s) -4(1 - 2'-2')(s)L(s) 22-'(1 - 2'-')(s)L(s) 2-'(1 - r')(s)L(s) 2(1 - 2-' + 2'-1')(s)L(s)

(m 2

+ 4n 1 ) - '

13.2. Crystal Lattice Sums

237

Certain other related sums have been obtained, i.e.,

L' (m + mn + n ) - S = 6(s)g(s), 00

2

2

g(s)

=

-00

I

00

n=O

[(3n

+ 1)-S - (3n + 2)-S] (18)

(Fletcher et al., 1962, p. 95), and (19) whose derivation is rather complicated (Glasser, 1973b). Obviously, the following case can be expressed by a single sum:

L (ml

mj?l

+

m2

+ ... +

mp )

-r

s

=

L (k + Pk 00

k=O

1) (k 1 y' +P

(20)

The difficulty in computing odd-dimensional sums by the use of theta functions is that most of the known theta function identities involve an even number of theta functions. Glasser (1937b) uses a number-theoretic approach to obtain additional sums, and the theory of basic hypergeometric series (Glasser, 1975) can be used to deduce the five-dimensional sum

L

ml?:O;m2,"';m5~

(m 1m2 + m1m3 + m3m4 + m4ml + m2mS)-S

1

= (S)(S - 2) - (2(S - 1). (21)

(The region of convergence of this sum cannot be deduced from the theorem of Section 13.2.) 13.2.2. Approximate Methods: The Poisson Summation Formula

Many approximation techniques have been developed to deal with lattice sums, beginning, perhaps, with Born's and Huang's approach, which uses values of the incomplete gamma function. That approach is not very adaptable to general values of s. Other approaches (van der Hoff and Benson, 1953; Benson and Schreiber, 1955; Hautot, 1974) use methods that convert the sum to a multidimensional sum involving the modified Bessel functions K v • This might, at first glance, seem to be compounding the problems. However, the transformed sums converge with extraordinary rapidity, and often the contributions at just a few lattice points serve to give six- or eightplace accuracy. Several approaches are possible, including one (Hautot, 1974)using Schlornilch series. My own preference is to begin with the following striking result, which can be found in any book on Fourier methods [e.g., Butzer and Nessel (1971, p. 202)].

238

13. Multiple Sequences

Let f

Theorem.

E

L( -

F(x) =

00,

(0),

Loooo e-iX~f(t)

dt,

X E

(1)

!Jll.

Then, iff is of bounded variation, 2n

L 00

k=-oo

f(x

+ 2kn) =

lim

n

X E

n-cok=-n

where, at points of discontinuity, f(a)

Proof

L eikxF(k),

= -t[f(a+) +

See Butzer and Nessel (1971).

211,

(2)

f(a-)].

•

There follows a list of formulas that will subsequently be of use. For the computation of the integrals involved, consult Erdelyi et al. (1954, Vol. I). f(t) = e- at2 cos bt, a E g~+, b e .OJ;

f

e- a(x+2kn)2 cos[b(x

+ 2kn)J

= _1_

- 00

f

2J"1W -

00

eikxe-(k2+b2)/4a COSh(bk). (S-l) 2a

f(t) = Itl±IlK(altJ),

f

eikX(k 2 + a 2)+11-1/2 =

-00

a E!Jll+;

2J1r + 1/2)

(S-2)

(2a)±Ilr(±/l 00

x

L

[x

-00

+ 2knl±IlKialx + 2knl).

(By analytic continuation and use ofthe well-known asymptotic properties of K Il , one finds that these sums are convergent and equal when Re( ± /l) > 0.) f(t) = Jt 2 + a 2 - l e - b-/tT +a' , a, b E ,OJ+; n

L 00

L 00

[(x + 2kn)2 + a2r1/2e-bv'{.x+2kn)2+a2 =

-00

-00

eikxK o(aJb 2 + k 2); (S-3)

a, b E .0/1.+ ;

L 00

~ e- b-/(;:;: 2kn)'+-a2 ab_ oo

L eikxJb 2 + k 2- 1 00

=

-00

x K 1(aJb 2 + k 2);

(S-4)

a, b E ]1+;

}br.3 a ±I'b 1/H Il L 00

-00

L 00

=

-00

[(x

+

2kn)2

+ a 2J±Il/2-1/4K±I1_ 1/z(bJ(x+2kn)2+ a2)

eikx(k2 + b 2)+11/2K,,(aJb 2

+ e),

/l E

t.{j.

(S-5)

239

13.2. Crystal Lattice Sums

We are now in a position to complete the proof of the theorem in Section 13.2. Let (3) (Without loss of generality we may assume that A = 0.) Then

S

=

9=

1

r(s)

Jo 9 dx, (00

Xs- 1

L

(4)

f(M)e-IIMII2x.

(5)

Imjl

o.

k, (j --> «[n, and substituting the result in (15) proves the lemma.

-->

•

--!-,

For A ~ 1, Re v >

Lemma 2.

I

K.(A)eA r(Re v + 1) AV ~ jr(v + -!-)I KRe.(l) =

I Proof

(16)

A(

(17)

CV •

This follows immediately from the integral

Kv(z)e= -- = ZV

- r

v

2

+ -1)-1 2

Re v>

-1-,

foo e 0

-zt[t (I + -t)]V-I/2 dt

Re z >

2

'

o. •

(18)

Lemma 3 (19) Proof

(20) so (mi

and the lemma follows.

+ ... +

m;)1/2 ~ (l/p)(m l

+ ... + mp )

(21)

•

A straightforward application of all these results shows that for s ~

1< 2s+p+I/2rr2S-I/2cs_I/z{N + 1)2S-1 exp{-[2rr(p

R

I

N

r(s)

-

x {I - exp[ -2rr(N

+

2s + p+ 1/2 rr2s-I/2 r(s)

K s-

+

1)2}

1)/p]}I-P

x {I - exp[(2s - 1)/(N >::::

- 1)/p](N

-!-

+

1 / 2 ( 1)

1) - 2rr(p - l)(N

+

exp{ -[2rr(p - 1)/p](N

l)/p]}-I

+

1)2}, N

--> 00.

(22) For instance, if N = 2, the truncated sum will contain 26 terms if p = 3. The exponential term above is 4.2 x 10- 1 7 . If only seven terms are taken (N = I), the exponential term is still only 5.3 x 10- 8 .

242

13. Multiple Sequences

The case s = 1 of (9) is particularly important. It gives eix(m, + ... +mpl

v- 2 2=n -oomt+···+mp 00

'\' 00

L.

x

L.

eix(m'+"'+m p'

l )

-00

(ml ... ·.mp-,l"O

e-j;;'T+"'+~--;lx+2mp"l

00

cos kx

Jmi + ... + m;-t

k=t

k2

+ 2L

(23)

a rapidly convergent series of exponentials. Obviously the forgoing procedure is easily modified to account for sums with denominator 11M - Ails, A = (at, ... , a p ) . For many special cases, see Hautot's paper. 13.2.3. Laguerre Quadrature

This is an elementary but very accurate method for hand computations. It can be applied for certain functionsfwhen s - 1 - 1P is a value {3 for which the abscissas and weights for the Laguerre quadrature formula for xfJe- x have been tabulated, e.g., {3 = 0, -t, -1-. -1, etc (Concus et al., 1963.) This is illustrated for f == 1.

(1) h(x)

=

exx P/ 2[03(0Iix/ny - 1].

The integral on the right is easily evaluated by Laguerre quadrature, since the series for 0 3 converges with great rapidity. For example, let P = 2, s = l

(2)

Laguerre quadrature with just three abcissas yields S = 9.0352, while the true value is 9.0336.

Appendix

A.I. Lagrangian Interpolation

Let x, yEC(;'s, and denote by p~k)(Z) the polynomial of degree k that at assumes the values Yn, Yn+l'···' Yn+k' respectively. (It is assumed the x j are distinct.) Then X n, Xn-b .•• , Xn+k

(k)( ) _ " k

~

Z -

~h+m

m=O

Il k

Xn+i

Z -

(

i=O X n + m i*m

)

Xn+i

.

(1)

It is easily shown that p~k) satisfies the recursion relationship (k+1)_ Pn -

(

(k)

) X n-ZPn+l-

Xn -

(

) (k) X n+k+l- ZPn

Xn+k+ 1

,

n,

k>O

-

,

(0)_ Pn - Yn,

n 2:: 0, (2)

by putting z = Xi' n :s; i :s; n + k + 1. Another useful expression for p~k) comes from expanding the determinant

Yn+ 1

1 1 1

Yn+k

1

p~k)

Yn

Z

Z2

Zk

2

x kn

X n+l

xn X~+ 1

X~+l =0.

Xn+k

X~+k

X~+k

Xn

243

(3)

244

Appendix

Let

Uj E~,

Vm( u 1,

and denote the Vandermonde determinant Vm by

Uz,""

um)

=

Ul

ui

ui

Uz

u~

ui

Um

Z Um

n n

m-l

=

m

i=O j=i+ 1

m Um

(Uj -

uJ

(4)

Expanding the determinants (3) by minors of the first column and using (4) shows that the determinantal expression is the same as the sum (1).

A.2. The Formula for the s-Algorlthm The proof of Eqs. 6.7(1)-6. 7(3) depends on two determinantal identities. It will be very useful to use Aitken's shorthand notation for determinants, writing only diagonal elements. For instance,

al a3a4a 7 b 1b 3b4b 7 d 1d 3d4d 7 el e3e4e 7

'

(1)

and so forth. The two identities are the obvious generalizations to n x n determinants of

which relates determinants with different first rows, and

!albzC3d41IalbzC3esl-lalbzC3dsllalbzC3e41 = lalbzc3d4esllalbzc31, (3)

which is an expression of the cross product of determinants whose last rows and columns differ in a certain way [see Aitken (1956, p. 108, No.2; p. 49, No.8)]. First, Eq. 6.8(1) is true when k = 1 for

(4)

A.2. The Formula for the ,,-Algorithm

Next consider the case k = 2m, m

~

245

1. Let

1

(5)

and -1

Q'n

=

L\sn+m

(6)

1

L\sn+ Zm

We must show these are the same. Rearranging the elements of the first gives

Qn

=

L\ZSn+ 1

L\ZSn+m

L\Zsn

L\ZSn+m

L\ ZSn+Zm_1

L\ ZSn+m_1

L\sn + 1

L\sn+m

L\sn

L\ ZSn+1

L\ZSn+m

L\Zsn

L\ZSn+m

L\ ZSn+Zm_1

L\ ZSn+m_1

L\ZSn+ 1

L\ZSn+m

L\ ZSn+m_1

L\zSn+zm_z

L\sn + 1

L\sn+m

L\ZSn+ 1

L\ZSn+m

L\ ZSn+m_1

L\ZSn+Zm_Z

(7)

246

Appendix

and using the determinantal identity (2) above one gets Eq. (8).

Qn =

I

I

~Sn+l

~sn+m

~sn

~2Sn+l

~2Sn+m

~2Sn

~2Sn+m_l

~2Sn+2m_l

~2Sn+m_l

~2Sn+ 1

~2Sn+m

~2Sn+m

~2Sn+2m_l

~Sn+l

~Sn+m

Ss;

~Sn+ 1

~Sn+m

~2Sn+ 1

~2Sn+m

~2Sn

~2Sn+ 1

~2Sn+m

~2Sn+m

~2Sn+2m-l

~2Sn+m_l

~2Sn+m_l

~2Sn+2m_2

~Sn

~Sn+m

~Sn+l

~Sn+m+ 1

~Sn+m-l

~Sn+2m-l

~Sn+m

~Sn+2m

~Sn

~Sn+m

~Sn+ 1

~Sn+m+ 1

~Sn+m

~Sn+2m

~Sn+l

~Sn+m

~Sn+m

~sn+2m-l

(8)

The second quantity [Eq. (6)] may be written

Q'

n

Dn

= (-It

~sn

~sn+m

~Sn+l

~Sn+m+l

~Sn+m-l

~sn+ 2m-l

~sn+m

~Sn+2m

Dn

=

~Sn+ 1

~Sn+m+l

~Sn+l

Ss;

~Sn+m

~Sn+ 2m

~Sn+m

~Sn+m-l

Sn+ 1

Sn+m-l

(9)

~Sn+l

~Sn+m

~Sn

~Sn+l

~Sn+m

~Sn+m

~Sn+2m-l

~Sn+m-l

~Sn+m

~Sn+2m

Sn+ 1

Sn+m

s;

I

247

A.3. Sylvester's Expansion Theorem

On D; we use the second identity to find

Dn

=

ASn + 1

Asn + m + 1

ASn + m

ASn + 2m

ASn AS n + m 1

1

ASn + 1

ASn + m

ASn + m

AS n + 2 m -

(10) 1

Elementary determinant manipulations show that the first factor above is ( - l)k times the first factor in the denominator of Qn. Thus Qn = Q~. The proof for k = 2m + 1 is similar. A.3. Sylvester's Expansion Theorem Let A be an n x n determinant, n 2 3, with elements aij and denote the minor of element aij by Mij. Let

D=

(1)

Then (2)

[see Muir (1960, p. 132)].

Bibliography

Abramowitz, M., and Stegun, 1. A. (eds.) (1964). "Handbook of Mathematical Functions." National Bureau of Standards Applied Mathematics Series # 55, Washington, D.C. Agnew, R. P. (1952). Proc. Amer. Math. Soc. 3, 550-556. Agnew, R. P. (1957). Michigan Math. J. 4, 105-128. Aitken, A. C. (1926). Proc. Roy. Soc. Edinburgh Sect. A 46, 289-305. Aitken, A. C. (1931). Proc, Roy. Soc. Edinburgh Sect. A 51, 80-90. Aitken, A. C. (1956). "Determinants and Matrices." Oliver & Boyd, London. Allen, G. D., Chui, C. K., Madych, W. R., Narcowich, F. J., and Smith, P. W. (1975). J. Approx. Theory 14, 302-316. Atchison, T. A.. and Gray, H. L. (1968). SIAM J. Numer. Anal. 5, 451-459. Bajsansk i. B.. and Kararnata, J. (1960). Acad. Serhe Sci. Publ. Inst. Math. 14, 109-114. Baker. G. A .. Jr., and Gammel. J. L. (eds.) (1970). "The Pade Approximant in Theoretical Physics." Academic Press, New York. Banach, S. (1932). "Theorie des Operations Lineaires." Chelsea, New York. Baranger, J. (! 970). C.R. Acad. Sci. Paris Ser. A 271, 149-152. Bauer, F. L. (1959). In" On Numerical Approximation" (R. E. Langer ed.). Univ. of Wisconsin Press, Madison, Wisconsin. Bauer, F. L. (1965). In "Approximation of Functions "(H. Garabedian, ed.). American Elsevier, New York. Bauer, F. L. et al. (1963). Proc. Symp. Appl. Math. Amer. Math. Soc. 15. Beardon, A. F. (1968). J. Math. Anal. Appl. 21, 344-346. Beckenbach, E. F., and Bellman, R. (1961). "Inequalities." Springer-Verlag, Berlin and New York. Bellman, R. (1961). " A Brief Introduction to Theta Functions." Holt, New York. Bellman, R. (1970). "Introduction to Matrix Analysis," 2nd ed. McGraw-Hill, New York. Bellman, R., and Cooke, K. L. (1963). "Differential-Difference Equations." Academic Press, New York. Benson, G. c., and Schreiber, H. P. (1955). Canad. J. Phys. 33, 529-540. Birkoff, G. D. (1930). Acta Math. 54, 205-246. Birkhoff, G. D., and Trjitzinsky, W. J. (1932). Acta Math. 60, 1-89. Born, M., and Huang, K. (1954). "Dynamical Theory of Crystal Lattices," Oxford Univ. Press, London and New York. 249

250

Bibliography

Brezinski, C. (1970). C.R. Acad. Sci. Paris Ser. A 270,1252-1253. Brezinski, C. (1972). RAIRO RI, 61-66. Brezinski, C. (1975). Calcolo 12, 317-360. Brezinski, C. (1976). J. Comput. Appl. Math. 2,113-123. Brezinski, C. (1977). "'Acceleration de la Convergence en Analyse Nurnerique." SpringerVerlag, Berlin and New York. Brezinski, C. (1978). "Algorithmes d'Acceleration de la Convergence: Etude Numerique." Editions Technip, Paris. Bulirsch, R., and Stoer, J. (1966). Numer . Math. 8,1-13. Bulirsch, R., and Stoer, J. (1967). Numer. Math. 9, 271-278. Butzer, P. L., and Nessel, R. J. (1971). "Fourier Analysis and Approximation." Academic Press, New York. Chisolm, J. S. R. (1966). J. Math. Phys. 7, 39. Chow, Y. S., and Teicher, H. (1971). Ann. Math. Statist. 42, 401-404. Chrystal, G. (1959). "Algebra." Chelsea, New York. Concus, P., Cassatt, D., Jaehnig, G., and Melby, E. (1963). Math. Camp. 17, 245-256. Cooke, R. G. (1955). "Infinite Matrices and Sequence Spaces." Dover, New York. Cordellier, F. (1977). C.R. Acad. Sci. Paris Ser. A 284, 389-392. Cornyn, J. J., Jr. (1974). Direct Methods for Solving Systems of Linear Equations Involving Toeplitz or Hankel Matrices, NRL Memorandum Rep. 2920. Naval Research Laboratory, Washington, D.C. Cowling, V. F., and King, J. P. (1962/1963). J. Analyse Math. 10, 139-152. Davis, P. (1963). "Interpolation and Approximation." Ginn (Blaisdell), Waltham, Massachusetts. Dieudonne. J. (1969). "Foundations of Modern Analysis." Academic Press, New York. Erdelyi, A. (1956). "Asymptotic Expansions." Dover, New York. Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricorni, F. G. (1953). "Higher Transcendental Functions," Vols. 1,2, and 3. McGraw-Hill, New York. Erdelyi, A .. Magnus. W., Oberhettinger, F., and Tricomi. F. G. (1954). "Tables of Integral Transforms," Vols. I and 2. McGraw-Hill. New York. Esser, H. (1975). Computing 14, 367-369. Fletcher, A., Miller, J. C. P., Rosenhead, L., and Comrie, L. J. (1962). "'An Index of Mathematical Tables." Vol. I. Oxford Univ. Press (Blackwell), London and New York. Freud, G. (1966). "Orthogonal Polynomials." Pergamon, Oxford. Gantrnacher, F. R. (1959). "The Theory of Matrices," Vols. I and 2, Chelsea, New York. Garreau, G. A. (1952). Nederl. Akad. Wetensch. Proc. Ser. A 14,237-244. Gekeler, E. (1972). Math. Camp. 26, 427-435. Germain-Bonne, B. (1973). RAIRO RI, 84-90. Germain-Bonne, B. (1978). Thesis, Univ. des Sciences et Techniques de Lille. Gilewicz, J. (1978). "Approximants de Pade," Lecture Notes in Mathematics #667. SpringerVerlag, Berlin and New York. Glasser, M. L. (l973a). 1. Malh. Phvs. 14,409-414. Glasser, M. L. (l973b). J. Math. Phys. 14,701-703. Glasser, M. L. (1974).1. Math. Phys. 15, 188-189. Glasser, M. L. (1975). J. Math. Phys. 16, 1237-1238. Goldsmith, D. L. (1965). Amer. Math. Monthly 72,523-525. Golomb, M. (1943). Bull. Amer. Math. Soc. 49, 581-592. Gordon, P. (1975). SIAM 1. Math. Anal. 6, 860-867. Gray, H. L., and Atchison, T. A. (1967). SIAM J. Numer. Anal. 4, 363-371. Gray, H. L.. and Atchison, T. A. (I 968a). J. Res. Nat. Bur. Standards 72B, 29-31. Gray, H. L., and Atchison, T. A. (l968b). Math. Camp. 22, 595-606. Gray, H. L., and Clark, W. D. (1969). J. Res. Nat. Bur. Standards 73B, 251-273.

Bibliography

251

Greville, T. N. E. (1968). Univ. of Wisconsin Math. Res. Center Rep. #877. Haber, S. (1977). SIAM J. Numer. Anal. 14,668-685. Hadamard, J. (1892). J. Math. Pures. Appl. 8, 101-186. Hancock, H. (1909). "Lectures on the Theory of Elliptic Functions," Vol. I. Dover, New York. Hardy, G. H. (1956). " Divergent Series." Oxford Univ. Press, London and New York. Hardy, G. H., and Rogosinski, W. W. (1956). "Fourier Series." Cambridge Univ. Press, London and New York. Hardy, G. H., and Wright, E. M. (1954). "An Introduction to the Theory of Numbers." Oxford Univ. Press, London and New York. Hautot, A. (1974). J. Math. Phys. 15,1722-1727. Havie, T. (1979). BIT 19,204-213. Henrici, P. (1977). "Applied and Computational Complex Analysis," Vols. 1 and 2. Wiley (lnterscience), New York. Higgins, R. L. (1976). Thesis, Drexel Univ. Householder, A. S. (1953). "Principles of Numerical Analysis." McGraw-Hill, New York. Iguchi, K. (1975). Inform. Process. Japan 15, 36-40. Iguchi, K. (1976). Inform. Process. Japan 16, 89-93. Isaacson, E., and Keller, H. B. (1966). "Analysis of Numerical Methods." Wiley, New York. Jacobi, C. G. I. (1829). "Fundamenta Nova Theoriae Functionum Ellipticarum." Konigsberg. Jacobi, C. G. I. (1846). J. Reine Angew. Math. 30,127-156. Jakimouski, A. (1959). Michigan Math. J. 6, 277-290. Jameson, G. J. O. (1974). "Topology and Normed Spaces." Chapman & Hall, London. Jones, B. (1970). J. Inst. Math. Appl. 17,27-36. Kantorovich, L. V., and Akilov, G. P. (1964). "Functional Analysis in Normed Spaces" (transl. by D. E. Brown and A. P. Robertson). Pergamon, Oxford. King, R. F. (1979). SIAM J. Numer. Anal. 16,719-725. Knopp, K. (1947). "Theory and Application ofInfinite Series." Hafner, New York. Kress, R. (1971). Computing 6, 274-288. Kress, R. (1972). Math. Camp. 26, 925-933. Krylov, V. I. (1962). "Approximate Calculation of Integrals." Macmillan, New York. Kummer, E. E. (1837). J. Reine Angew. Math. 16,206-214. Lambert, J. D., and Shaw, B. (1965). Math. Camp. 19,456-462. Lambert, J. D., and Shaw, B. (1966). Math. Camp. 20, 11-20. Laurent, P. J. (1964). Thesis, Grenoble. Levin, D. (1973). Internat. J. Comput. Math. B3, 371-388. Livingston, A. E. (1954). Duke Math. J. 21, 309-314. Lorch, L., and Newman, D. J. (1961). Canad. J. Math. 13,283-298. Lorch, L., and Newman, D. J. (1962). Comm. Pure Appl. Math. 15, 109-118. Lotockii, A. V. (1953). Iranor. Gas. Ped. Inst. u: Zap. Fiz-Mat. Nauki 4,61-91. Lubkin, S. (1952). J. Res. Nat. Bur. Standards Sect. B 48,228-254. Luke, Y. L. (1969). "The Special Functions and Their Approximations," Vols. I and 2. Academic, New York. Luke, Y. L. (1979). On a Summability Method, notes, Univ. of Missouri, Kansas City, Missouri. Luke, Y. L., Fair, W., and Wimp, J. (1975). Camp. Math. Appl. 1,3-12. Lyness, J. N. (1970). Math. Comp., 24, 101-135. Lyness, J. N. (1971). Math. Comp., 25,59-78. McLeod, J. B. (1971). Computing 7,17-24. McNamee, J., Stenger, F., and Whitney, E. L. (1971). Math. Camp. 25, 141-154. Miller, K. S. (1974). "Complex Stochastic Processes." Addison-Wesley, Reading, Massachusetts. Milne-Thomson, L. M. (1960). "The Calculus of Finite Differences." Macmillan, London.

252

Bibliography

de Montessus de Balloire, R. (1902). Bull. Soc. Math. France 30, 28~36. Moore, E. H. (1920). Bull. Amer. Math. Soc. 26, 394-395. Muir, T. (1960). "'A Treatise on the Theory of Determinants." Dover, New York. Nikolskii, S. M. (1948). Izt: Akad. Nauk SSSR Ser. Mat. 12,259-278. Olevskil. A. M. (1975). "'Fourier Series with Respect to General Orthogonal Systems," SpringerVerlag, Berlin and New York. Olver. F. W. J. (1974)... Asymptotics and Special Functions." Academic Press. New York. Ortega, J. M., and Rheinboldt, W. C. (1970). "Iterative Solution of Nonlinear Equations in Several Variables." Academic Press, New York. Ostrowski, A. M. (1966). "'Solutions of Equations and Systems of Equations," Academic Press, New York. Ostrowski, A. M. (1973). "Solutions of Equations in Euclidean and Banach Spaces." Academic Press, New York. Overholt, K. J. (1965). BITS, 122-132. Papoulis, A. (1965). "' Probability, Random Variables, and Stochastic Processes." McGraw-Hili, New York. Pennacchi, R. (1968). Calcolo 5,37-50. Penrose, R. (1955). Proc, Cambridge Philos. Soc. 51,406-413. Perron, O. (1929). "Die Lehre von den Kettenbriichen." Chelsea, New York. Perron, O. (1957). "Die Lehre von den Kettenbruchen," 3rd ed., Vols. I and 2. Teubner, Stuttgart. Petersen, G. M. (1966). "Regular Matrix Transformations." McGraw-Hili, New York. Peyerimhoff, A. (1969). "'Lecture Notes of'Summability." Lecture Notes in Mathematics 11 107. Springer-Verlag, Berlin and New York. Pollaczek, F. (1956). "Sur une Generalisation des Polynomes de Jacobi." Gauthiers-Villars, Paris. Pyle, L. D. (1967). Number. uo». 10,86--102. Rainville, E. D. (1960). "Special Functions," Macmillan, New York. Reich, S. (1970). Amer. Math. Monthly 77,283-284. Richtrneyer, R. D. (1957). "Difference Methods for Initial Value Problems." Wiley (lnterscience), New York. Rutishauser, H. (1954). Z. Anqew. Math. Phys. 5, 233-251. Rutishauser, H. (1957). "Der Quotienten-Differenzen Algorithmus." Birkhauser- Verlag, Basel. Salzer, H. E. (1955). J. Math. Phys. 33, 356-359. Salzer, H. E. (1956). MTAC 10,149-156. Salzer, H. E., and Kimbro, G. M. (1961). Math. Camp. 15,23-29. Samuelson, P. A. (1945). J. Math. Phys. 24,131-134. Scheid, F. (1968). "Numerical Analysis, Schaum's Outline Series." McGraw-Hili, New York. Schmidt, J. R. (1941). Philos. Mag. 32, 369-383. Schur, I. (\921). J. Reine Anqew. Math. 151,79-111. Schwartz, C. (1969). J. Comput. Phys. 4, 19-29. Schwartz, L. (\961/1962). In "Serninaire Bourbaki," fasc. 3. Benjamin, New York. Shanks, D. (1955). J. Math. Phys. 34, 1-42. Shaw, B. (1967). J. Assoc. Comput. Math. 14, 143-154. Shohat, J. A., and Tomarkin, J. D. (1943). "The Problems of Moments," American Mathematical Society, Providence, Rhode Island. Shoop, R. A. (1979). Pacific J. Math. 80, 255-262. Slater, L. J. (1960). "Confluent Hypergeometric Functions." Cambridge Univ. Press, London and New York. Smith, A. C. (1978). Utilitas Math. 13,249-269. Smith, D. A., and Ford, W. F. (1979). SIAM J. Numer. Anal. 16,223-240.

Bibliography

253

Szego, G. (1959)... Orthogonal Polynomials." American Mathematical Society, Providence, Rhode Island. Titchmarsh, E. C. (1939). "The Theory of Functions." Oxford Univ. Press, London and New York. Todd, J. (ed.) (1962). "Survey of Numerical Analysis." McGraw-Hill, New York. Traub, J. F. (1964)... Iterative Methods for the Solution of Equations." Prentice-Hall, Englewood Cliffs, New Jersey. Trench, W. F. (1964). SIAM 1. Appl. Math. 12,515-522. Trench, W. F. (1965). SIAM J. Appl. Math. 13, 1102·1107. Tucker, R. R. (1967). Pacific 1. Math. 22, 349..359. Tucker, R. R. (1969). Pacific 1. Math. 28, 455-463. Tucker, R. R. (1973). Faculty Rei'. Bull. N.C. A and T State o-«, 65, 60·63. Uspensky, J. V. (1928). Trans. Amer. Math. Soc. 30, 542-559. van der Hoff, B. M. E., and Benson, G. C. (1953). Canad. 1. Phys. 31,1087-1091. Vuckovic, V. (1958). Acad. Serbe. Sci. Pubi.Tnst . Math. 12, 125-136. Walls, H. S. (1948)... Analytic Theory of Continued Fractions." Chelsea, New York. Wasow, W. (1965). "Asymptotic Expansions For Ordinary Differential Equations." Wiley, New York. Whittaker, E. T., and Watson, G. N. (1962). "A Course of Modern Analysis." Cambridge Univ. Press, London and New York. Wilansky, A., and Zeller, K. (1957). J. London Math. Soc. 32, 397-408. Wimp, J. (1970). SIAM J. Numer. Anal. 7, 329-334. Wimp, J. (1972). Math. Compo 26, 251-254. Wimp, J. (I 974a). Computing 13,195-203. Wimp, J. (1974b). J. Approx. Theory 10,185-198. Wimp, J. (l974c). Numer. Math. 23,1-17. Wimp, J. (1975). Acceleration methods, In "Encyclopedia ofComputer Science and Technology," Vol. 1. Dekker, New York. Wright, E. M. (1955). J. Reine Anqew. Math. 194,66-87. Wynn, P. (I 956a). J. Math. Phys. 35, 318-320. Wynn, P. (I956b). Math. Tables Aids Comput. 10,91-96. Wynn, P. (l956c). Proc. Cambridge Philos. Soc. 52,663-671. Wynn, P. (1959). Numer. Math. 1, 142-149. Wynn, P. (1961). Nieuw Arch. Wisk 9,117-119. Wynn, P. (1962). Math. Camp. 16, 301-322. Wynn, P. (1963). Nordisk Tidskr.Lnformar-Behandl. 3,175-195. Wynn, P. (1966). SIAM 1. Numer. Anal. 3, 91-122. Wynn, P. (1966). Univ. of Wisconsin Math. Res. Center Rep. #626. Wynn, P. (1967). Univ. of Wisconsin Math. Res. Center Rep. # 750. Wynn. P. (1972). C. R. Acad. Sci. Paris Ser. A 275, 1065-1068. Zeller, K. (1952). Math. Z. 56, 18-20. Zeller, K. (1958)... Theorie du Lirnitierungsverfahren." Springer- Verlag, Berlin and New York. Zemansky, M. (1949). C.R. Acad. Sci. Paris 228,1838-1840.

Index

A 2-process,

Aitken 5 104, 149-152 applied to power series, 151 generalized, 105, 154, 167, 184 B

Birkhoff-Poincare scales, 15-23

c Continued fractions, 156-165 Convergence equivalence to, 33 hyperlinear, 154 linear, 6 logarithmic, 6

Exponential polynomials, 203 Extrapolation, deltoids obtained by, 73-76

F Fixed points of differentiable functions, 146-148 Fourier coefficients, computation of, 205-207 Fourier series, summation of, 48-53 G G-transform, 200-205

H Hankel determinants, 14, 157 Heat conduction, equation for, 100 Hilbertian subspace, 9~94

D

Deltoid,S, 71-80 Difference equations, analytic theory, 16

E s-algorithm, 138-148 generalization of, 144-146 stability of, 141-142 Equivalence, asymptotic, I Euler's constant, 75

Implicit summation, 171-174 Interpolation, Neville-Aitken formula for, 73 Iteration functions abstract spaces, 118-119 construction of, 112-118 L

Laguerre quadrature, 91 Lebesgue constants, 48-53 255

256

Index

Lozenge algorithms, 3-5 linear, 67-76 nonlinear, 101-106 M

Means, see Transformation Method, see Transformation Modulus of numerical stability, see Numerical stability N

Numerical analysis, rational formulas for, 142-144 Numerical stability, modulus of, 29

s Saturation, 51-53 Scale, asymptotic, 1-2 Sequences complex, properties of, 5-12 Laplace moment, 84-90 iteration, 106-108 linearly convergent, 6 logarithmically convergent, 6 Taylor, 96 totally monotone, 12-14 totally oscillatory, 12-14 Stieltjes integrals, quadrature formulas for, see Quadrature Summation methods, see Transformation Sums, lattice, 232-242

o Order symbols, 1-2

T

p Pade approximants, see Rational approximations Path,3 Poisson summation formula, 238 Pollaczek polynomials, 59-63 Polynomials, orthogonal, 40-44, 80-83 Products, partial, growth of, 8

Q Quadrature, numerical, 69-71 based on BH protocol, 200-209 based on cardinal interpolation, 77-80 based on G-transform, 200-205 based on Romberg integration, 67-71 based on tanh rule, 207-209 Quotient-difference algorithm, 156-159 R

Rational approximations, 53-59 gamma function, 58 Gaussian hypergeometric function, 56-57 Pade, 54-57, 128-136 for Stieltjes integrals, 132-136 Rhomboid,S, 80-83 Richardson extrapolatin, 67-71 Romberg integration, see Quadrature

T-matrix, Abel, 66 Taylor formula, generalized, 146-148 Transformation accelerative, 3 Brezinski-Havie, 175-209 quadrature by, 200-209 e-algorithm. 120-148 multiparameter, 166-167 'TJ-algorithm for, 160 GWB, 106-108 homogeneous,S implicit summation, 171-174 Levin t and u, 189-198 linear,S Lubkin, 152-153 multiple sequences, 227-231 nonlinear,S Overholt, 108-110 probabilistic, 210-226 p-algorithm for, 168-169 regular, 3 Schmidt, 120-147 geometric interpretation of, 136-137 topological, 182-185 a-algorithm for, 169-171 Toeplitz, 24-26 applied to series of variable terms, 48-53 band,28 based on power series, 94-100

Index

characteristic polynomials for, 28 Chebyshev weights, 43--44 Euler (E, q)method, 99 Euler means, 34 (f,-'Yk) means, 51 Hausdorff, 34 Higgins weights, 45--46 Lotockil, 44 measure of, 28 nonregular, 38--40 optimal, 90-94 orthogonal, 40-43, 80-83 positive, 27

257

Richardson procedure, 67-71 generalized, 181 Romberg weights, 44--45 generalized, 181 rational approximations obtained with, 54 Riesz means, 65 Salzer means, 35-38 weighted means, 33 translative,S W, 152-153 Trench algorithm, 198-199 Triangle, 27