Nonlinear Methods in Numerical Analysis (North-Holland Mathematics Studies)

NORTH-HOLLAND MATHEMATICS STUDIES Studies in Computational Mathematics (1) Editors:

C. Brezinski University of Lille Villeneuvedxscq, France

L. Wuytack University of Antwerp Wilrijk, Belgium

NORTH-HOLLAND -AMSTERDAM

NEW YORK

0

OXFORD *TOKYO

136

NONLINEAR METHODS IN NUMERICAL ANALYSIS

Annie CUYT Luc WUYTACK University ofAntwerp Belgium

1987

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD *TOKYO

‘.c Elsevier Science Publishers B.V.,

1987

Allrights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, niechanical, photocopying, recording or otherwise, without the priorpermission of the copyright owner.

ISBN: 0 444 70189 3

Publishers:

ELSEVIER SCIENCE PUBLISHERS B.V. P.O. BOX 1991 1000 BZ AMSTERDAM THE NETHERLANDS

Sole distributors for the U.S.A. and Canada:

ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52VANDERBILT AVENUE NEW YORK, N.Y. 10017 U.S.A.

Library of Congress CataioginginPublifation Data

Cuyt, Annie, 1956Nonlinear methods in nurrerical analysis. (North-Holland mathematics studies ; 1 3 6 ) (Studies I n ccnputational mathemtics ; 1) Bibliography: p Includes index. 1. m r i c a l analysis. I. Wytack, L. (Luc), 194311. Title. 111. Series. IV. Series: Studies in ccnputational mathematics : 1.

.

QA297.C89 1987 ISBN 0-444-70189-3

519.4

(U.S.)

PRINTED IN THE NETHERLANDS

86-32932

To Annelies Van Soom from her mother. Annie Cuyt

PREFACE. Most textbooks on Numerical Analysis discuss linear techniques for the solution of various numerical problems. 01ily a small number of books introduce and illustrate nonlinear methods. This book accumulates several nonlinear techniques mainly resulting from the use of Pad6 approximants and rational interpolants. First these types of rational approximants are introducrd and afterwards methods based on their use are developed for the solution of standard problems in numerical mathematics : convergence acceleration, initial value problems, boundary value problems, quadrature, nonlinear equations, partial differential equations and integral equations. The problems are allowed to be univariate or multivariate. The treatment of the univariate theory results from a course given by the second author a t the University of Leuvcm and completed by the first author with many new theorems and numerical rcsiilts. The discussion of the multivariate theory is based on research work by thc first author. The text as it stands is now used for a graduate course in Numerical Analysis at the University of Antwerp. The book brings together many results of research work carried out at the University of Antwerp during the past few yrars. We particularly mention results of Guido Claessens, Albert Wambecq, Paul Van der Cruyssen and Brigitte Verdonk. Let us now give a survey of the conterits of this book and a motivation for the problems treated. Since continued fractions play an important role, Chapter I is an introduction to this topic. We mention somc basic properties, evaluation algorithms and convergence theorems. From the section dealing with convergence we can already learn that in certain situations nonlinear approximations are more powerful than linear approximations. The rervnt notion of branched continued fraction is introduced in the multivariate section and will be used for the construction of multivariate rational interpolants.

In Chapter I1 Pad6 approximants arc’ treated. They are local rational approximants for a given function. The problems of existence, unicity and computation are treated in detail. Also the convergence of sequences of Pad6 approxirnants and the continuity of the Pad6 operator which associates with a function its Pad6 approximant of a certain order, are considered. Again a special section is devoted to the multivariate case. We do not discuss the relationship between Pad6 approximants and orthogonal polynomials or the moment problem.

Preface

In Chapter 111 rational interpolants are defined. Their function values fit those of a given function at some given points. Many results of the previous chapter remain valid for this more general case where the interpolation conditions are spread over several points. In between the rational interpolation case and the Pad6 approximation case lies the theory of rational Hermite interpolation where each interpolation point can be assigned more than one interpolation condition. Some results on the convergence of sequences of rational Hermite interpolants are mentioned and multivariate rational interpolants are introduced in two different ways. The previous types of rational approximants are used in Chapter IV to develop several numerical methods for the solution of classical problems such as convergence acceleration, nonlinear equations, ordinary differential equations, numerical quadrature, partial differential equations and integral equations. Many numerical examples illustrate the different techniques and we see that the nonlinear methods are very useful in situations where we are faced with singularities. However, one must be careful in applying the nonlinear methods due to the fact that denominators in the formula can get small. We tried t o make the text as self-contained as possible. Each chapter also contains a problem section and a section with remarks that indicate extensions of the discussed theory. References to the literature are given at the end of each chapter in alphabetical order. In the text we refer to them within square brackets. Formulas and equations are numbered as (a.b.), where a indicates the chapter number and b the number of the formula in that chapter.

In preparing the text the authors did benefit from discussions with many colleagues and friends. We mention in particular Claude Brezinski (Lille) , Marcel de Bruin (Amsterdam), William Gragg (Lexington), Peter Graves-Morris (Canterbury), Louis Rall (Madison), Nico Temme (Amsterdam) , Helmut Werner (Bonn). We also thank Drs. A. Sevenster from North Holland Publishing Co who encouraged us to write this book and Mrs. F. Schoeters and Mrs. R. Vanmechelen who typed the manuscript. Ant werp

Annie Cuyt Luc Wuytack

1

.

CHAPTER I: Continued Fractions

$1.Notations and definitions

. . . . . . . . . . . . . . . . . . . . .

$2. Fundamental properties . . . . . . . . . . . . . 2.1. Recurrence relations for P,, and Qn . . . . 2.2. Euler-Minding series . . . . . . . . . . 2.3. Equivalence transformations . . . . . . . 2.4. Contraction of a continued fraction . . . . 2.5. Even and odd part . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

3

. . . . . . . . . . . . . . . . . . . . . . . . .

4

$3.Methods t o construct continued fractions . . . . . . . . . . . . . 3.1. Successive substitution . . . . . . . . . . . . . . . . . 3.2. Equivalent continued fractions . . . . . . . . . . . . . . 3.3. The method of Viscovatov . . . . . . . . . . . . . . . . 3.4. Corresponding and associated continued fractions . . . . . . . . . . . . . . . . 3.5. Thiele interpolating continued fractions

12 12 15 16 17 19

. 4 .5 .6 .7 . . . . . . . . .9

$4. Convergence of continued fractions . . . . . . . . . 4.1. Convergence criteria . . . . . . . . . . . . . 4.2. Convergence of continued fraction expansions . . 4.3. Convergence of corresponding cf for Stieltjes series

. . . . . . 20 . . . . . . 20

. . . . . . 25

. . . . . . 29

$5. Algorithms t o evaluate continued fractions . . . . . . . . . . . . 31 5.1. The backward algorithm . . . . . . . . . . . . . . . . 31 5.2. Forward algorithms . . . . . . . . . . . . . . . . . . . 3 2 5.3. Modifying factors . . . . . . . . . . . . . . . . . . . . 34 56 . Branched continued fractions . . . . . . . . . . . . . . . . . . 41 6.1. Definition of branched continued fractions . . . . . . . . . 41 6.2. A generalization of the Euler-Minding series . . . . . . . . 42 6.3. Some recurrence relations . . . . . . . . . . . . . . . . . 47 6.4. A multivariate Viscovatov algorithm . . . . . . . . . . . . 49 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

. . . . . . . . . . . . . . . . . . . . . . . . . . .

56

. . . . . . . . . . . . . . . . . . . . . . . . . .

58

Remarks References

2

“J’ai e u 1 ’honneur de prksenter d 1 ’Acadkmie e n 1802 un mkmoire 6oua le titre : Essai d’une me‘thode ge‘nkrale pour rkduire toutea sortea de skriea e n fraction8 continues. Aprka ce temps ayant eu occaaion de penaer encore ci cette matikre, j’ai f a i t de nouvelles rkjlezions qui peuvent aervir d perfectionner et simplifier la mkthode dont il a’agit. Ge sont ces rkfleziona que j e prksente maintenant d la sociktk savante. ”

B . VISCOVATOV - “De la me‘thode gknkrale pour rkduire toutea sortea de quantit4.9 e n fraction8 continues” (1805).

3

I.1. Notatdom and defindtdona

81. Notations and deflnitions. A continued fraction is a n expression of the form

bo+---

61

--

bl+

-____ b2 ___--___

+ b3+

02

...

a3

' * * +

bi

a;

+ . ..

where the ai and bi are real (or complex) numbers or functions and are respectively called partial numerators and partial denominators. Instead of the expression above we will most of t h e times use the following compact notations:

or

b o + f $ j? = l

(1.1.)

The truncation n=0,1,2,.

is called the nth convergent of the continued fraction (1.1.). If lim

n-+m

G, = C

exists and is finite, then the continucd fraction is said t o be convergent and C is called the value of the continued fraction. Clearly C, is a rational expression (1.2.)

+

where P, and &, are polynomials of a certain degree in the 2n 1 partial numerators and denominators 60, a l l61 I . . ., a,, 6,. The polynomials P, and Q n are respectively called the nth numerator and nth denominator of the continued fraction (1.1.).

4

1.2.1. Recurrence relation8 for

Pn and Qn

§2. Fundamental properties.

2.1. Recurrence relation8 for P, and Qn.

The nth numerators and denominators satisfy the same three-term recurrence relation, but with differentstarting values. This relation is given in the following theorem.

Theorem 1.1.

(1.3.)

Proof The proof is performed by induction. 0bviously

and so the formulas (1.3.) are valid for n = 1. Let us now suppose the validity of (1.3.) for n 5 k. We will prove i t for n = k + 1. We have, using (1.2.),

Consequently, by using (1.3.) for

p1

= k,

5

I. 2.2. Euler-Minding seriea

I

2.2. Euler-Minding 8 eries. It is easy now t o give an expression for the difference of two consecutive convergents of a continued fraction.

Theorem 1.2. If

Q n Qn-1

# 0, then cn- cn-l= ( - I ) n + l

al

a 2 . . .an

Q n Qn-i

(1.4.)

Proof One can show, by induction arid using the recurrence relations (1.3.), th a t for n > l

P,

Qn-l -

Qn

= (-l)n+lal a 2 . . .a,

From this (1.4.) follows immediately. This theorem can now be used to give a more explicit formula for Cn.

I

I. 2.5. Equivalence tranaformationa

6

Theorem 1.3. If

Qi

# 0 for 1 5 i 5 n, then Cn = bo

+

n i= 1

a l . , .ai (-l)i+l -

(1.5.)

Qi-1 Q i

Proof We have, by means of formula (1.4.),

=(-l)n+l

a1. . .a, Q n Qn-1

+ (-1)n

(11.

. .an-l

~

Qn-1 Qn-2

+

,

..+

(11 Qi

Qo

+ bo

I

The expression (1.5.) is the nth partial sum of the series

c 00

bo f

i= 1

(-1)i+1

631..

.a;

Qi-1 Q i

which is called the Euler-Minding series. Thus we have associated a series with a continued fraction such that the nth partial sum of the series equals the nth convergent of the continued fraction. This interrelation between series and continued fractions can be used to apply well-known results for series t o the theory of continued fractions. 2.3. Equivalence tranaformationa. Let p i into

# 0 for i 2 0. The transformation th at alters the continued fraction (1.1.) (1.6.)

is called an equivalence transformation. Clearly (1.1.) and (1.6.) have the same convergents. By performing equivalence transformations a continued fraction can be rewritten in a prescribed form. For instance, if a; # 0 for i 3 1, then (1.1.) can be rewritten as

I.2.4. Contraction of a continued fraction

7

(1.7.) by choosing P1 =

1 -

a1

and 1

pi = ; ; for i 2 2. t

Pt-1

Hence one can limit th e study of continued fractions to continued fractions of the form (1.7.). Such a continued fraction is called a reduced continued fraction [lo p. 4801. 2.4. Contraction of a continued fraction.

Let us consider the following problem. Suppose we are given a sequence (Cn},E- of subsequently differcnt elements and we want t o construct a continued fraction of which C, is the nth convergent.

Theorem 1.4. If C ,

# G,-l

for n

2 1, then the continued fraction bo

+ r=l

with

bi =

c;- c;-z - c;-z

Ci-1

has the elements of the sequence {Cn},en as convergents.

8

1.2.4. Contraction of a continued fraction

Prod We write

with P, = C , and &, = 1. Since the partial numerators and denominators of the continued fraction

bo

+

M

I

i= 1

with convergents C, must satisfy the relations (1.3.1,we get the following system of equations in the unknowns bo, ai and b;:

for

k2 2 :

b; Ci-1 + ai Gi-2 = Ci bi + a; = 1

A solution of this system is given by

(1 3.) I

If also C , # Cn-2 for n >_ 2, then by means of an equivalence transformation the continued fraction with partial numerators and denominators given by (1.8.), can be written as

1.2.5. Even and odd part

9

The formulas (1.8.) and (1.9.) can be used t o compute a contraction of a continued fraction, i,e. a continued fraction constructed in such a way t h a t its convergents form a subsequence of the sequence of convergents of the given continued fraction. We shall now illustrate this. 2.5. Even and o d d part. Consider a continued fraction with convergents { C n } n E ~ . The even part of this continued fraction is a continued fraction with converwhile , the odd part is a continued fraction with convergents gents { C Z ~ } , ~ N { C 2 n + l } n E ~Theorem . 1.4. enablcs us t o construct those even and odd parts. We now derive a formula for the even part and give an analogous formula for the odd part without proof. Consider the continued fraction

with convergents

The partial numerators and dcnomiriators of the even part as expressed in the partial numerators and denominators a ; and b i , are computed as follows. Let

use the formulas (1.8.) and perform an equivalence transformation with

We then get the following continued fraction

+

2

i- 2

with

1.2.5. Even and odd part

10

and

which is equal to

Analogous to formula (1.4.) one can prove t h a t

and that

(see problems (1) and (2) a t the end of this chapter). Finally we get for the even part

bo +

~ a 1 b2 _

lbi bz

+z 00

+ a2

I_

4 2 i - 2

l r n i - 2

I

__ _a 2 a_3 b4_ ~ ~ -

/ ( b z b3

+ a 3 ) b4 + b2 a 4

a2i--1

hi--1

b2i--4

+ a2i--1)

b2i

-t b2i-2 a2!

(1.1Oa.)

11

I.2.5. Even and odd part

In the same way one can prove that the odd part is

We illustrate this procedure with the following example. To compute C = 6 with r real positive, we first write r = b2 positive. So C2- b2 = a or

Hence

i= 1

If we take

t

= 102, we get 415- = 10 +

c-

i= 1

The even part of this continued fraction is

and the odd part is

2!

+a

with a

12

I.S.1. Succeaas've subatitutdon

83. Methods to construct continued fractions. We are now going t o describe some methods t h a t can be used t o write a given number or function as a continued fraction. Other techniques can for example be found in [lo pp. 487-5001 and [12 pp. 78-1501. Some convergence problems of such continued fractions are treated in the next section. 3 . j . Successive subatitution.

Let f be a given number or function. Write To = j and compute TI, T2,.. .,Tn+l such that

where b o , a; and b; are chosen freely and can be functions of the argument of f. In this way we get

By continuing this method of successive substitution we get an expression of the form (1.1.). It is important t o check whether f is really the value of this continued fraction or for which arguments of f this is true. Such problems are treated further on. We shall illustrate this method by calculating a continued fraction expansion for e z . Since

we choose bo = 1.

1.8.1. Successive substitution

13

So we get

TI= e z - l = z

b X + ( X

-

2

1

22 ++ ... 12

which suggests us t o write

with

Some easy computations show that

T2= X

1

2 1-(;

+...)

2-(;

2-2

So we can write

with

+...)

+

( y + .) ..

14

I.3.1. Successive substitution

T3

22 3

+ . . .)

= --(1

If we continue in this way, we find th at for a; = (i - 1). and i+l

which again suggests the choice

k=l

ai+l = i s

(d-

and

b,+l

= (i

bi = i

- s,

+ 1) - x, such th a t

1

1)xI + ~ (n1)s - _ _ _ In - 2 + Tn+l

i - 2

or by continuing the process [12 p.1301

eZ=l+

Another examp

I

2

__

11-x

i- 2

(1.1 1.)

is given by the construction of a continued fract.an for

C=

fi)= q T T x g

with s and y real positive. Proceeding as in the previous section we can write

C2= I ( x , Y) = b2(x, Y) + 4 5 , Y)

with b(s, y) = x + y and a(x, y) = sy. Then

and consequently

1.3.2. Equavalent continued fructiona

15

3.2. Equivalent continued fractionrr.

A series m

i=O

and a continued fraction bO f i=l

are called equivalent if for every n

2 0 the nth partial sum n i=O

of the series equals the nthconvergent

Remember that the Euler-Minding series and the continued fraction (1.1.) are equivalent, 'hmsforrning a given series into an equivalent continued fraction can also be done by means of formula (1.9.), with n

Gn =

C

d;

i=O

We obtain (1.12.)

In the following example Consider

czod ; is a power series.

16

I.S.S. The method of Vimovatov

Then after an equivalence transformation (1.12.) is

1+

1$

+cIT-

- ~ - _ _ (i - 2)! i! +’Tx 1 (%- I)! t.

00

i=2

If we perform another equivalence transformation, we get

(1.13 .) Remark t h a t a n equivalent continued fraction will converge if and only if the given power series converges. For our example this means that (1.13.) converges on the whole complex plane. Hence, by substitution of x by -2, (2

- l)z/

i=2

converges for all x. Consequently the continued fraction in the righthand side of (1.11.) converges for arbitrary 2.

3.3. The method of Viscovotou. This method is used t o develop a continued fraction expansion for functions given as the quotient of two power series [27]. Let

Then

I.3.4. Correeponding and uaaociated continued fraction8

17

This procedure can be repeated and if we define dk,i

for k

= dk--l,O

dk-2,i+l

- dk-2,O d k - l , i + l

> 2 and d 2 0, we finally get. (1.14.)

In case do; = 0 for i 2 1 then f ( x ) is a power series itself and we shall prove in the next section t h a t the method of Viscovatov can be used t o compute a corresponding continued fraction. If f(x) is the quotient of two polynomials and thus a rational function, this method can be used t o write f in the form of a continued fraction (see also problem (8)).

3.4. Cotreaponding and associated continued fraction8

A continued fraction

for which the Taylor series development of the nth convergent C,(z) origin matches a given power series

around the

M

i- 0

up t o and including the term of degree n is called corresponding t o this power series. In other words, for a corresponding continued fraction, if

then for every n we have e i = ci for i = 0 , . . ., n. A lot of methods t o construct corresponding continued fractions are treated in chapter 11.

18

I.3.4. Corresponding and aaaocdated continued fraction8

A continued fraction

for which the Taylor series development of the nth convergent C,(z) matches a given power series

c co

c; x i

i= 0

up to and including the term o degree 2n is called associatec A corresponding continued fraction can be turned into an associated one by calculating the even part. Let us now again consider the algorithm of Viscovatov. For the continued fraction (1.14.) we define

fo = f given by (1.14.) fl

= dl0 - doofo

fk

= &,02 fk-2

- dk--l,O

Then by induction it is easy to see th at the form

k

fk-1 fk(X)

= 2 , 3 , 4 , .. .

can be developed into a series of

i=O

One can also prove by induction th at for the kth convergent

pk _ - dl01 &k Id00 ~

ZOzl +... + I-ddl0 + -d.3051 d20

+

~

the relation

holds. Hence if f(s) is given by the series expansion f ( 2 ) = co

+ c1z +

C2Z2

+ ...

dk,Oz /dk--l,O

I

I.8.5. Thiele interpolating contanued fraction8

19

the algorithm of Viscovatov when applied t o (f(z)- C O ) / Z with d l ; = c l + ; for i 2 0, generates a continued fraction of the form

of which the Taylor series development of the kth convergent matches the power series (f(z) - cO)/z up t o and including the term of degree k - 1. In this way we obtain for f(z) the corresponding continued fraction co

+

( p -+ id20 d30zl +...

c l z l 4- dzozi

because the Taylor series development of the kth convergent matches the power series f ( z ) up to and including the term of degree k.

3.5. Thiele interpolating continued fract io n8. This technique is dealt with completely in chapter 111. It uses interpolation data and reciprocal differences. Thicle typo continued fractions will be constructed both for univariate and multivariate functions. In the univariate case continued fractions of the form

b"

+

2

i=l

/ad. - GI! 6;

will be used while we shall need branched continued fractions similar t o those in (1.29.) for the multivariate case.

I.d.1. Convergence criteria

20

54. Convergence of continued fractions.

4.1. Convergence criteria. The following result is a classical convergence criterion for reduced continued fractions, and is due to Seidel [21]. It dates from 1846.

Theorem 1.5. If

bi

> 0 for i 2

1, then the continued fraction

bo

converges, if and only if the series

+

c

i= 1

czl

bi

diverges.

Proof The Euler-Minding series for

It is an alternating series because a; = 1 and bi > 0 imply t h a t &i > 0 for i 2 1. The nth denominator & n is bounded below by 0 = min(1, b l ) . This can be proved by induction from the recurrence relation for Qn. If we put r n = Q n Qn-l then the rn are monotonically increasing because

Consequently

1.4.1.Convergence criteria

21

czl

Thus if bi diverges, the sequence {rn}nEmtends t o infinity. This implies, by a theorem of Leibniz on alternating series, t h a t the EulerMinding series

c:,

converges. On the other hand, if b , converges then we can prove t h a t rn is bounded above. To do so, we first prove that Qn

< (1 + 6 1 )(I + 62). . .(I + bn)

This is obvious for n = 1 . Assume now that it is also true for n the recurrence relations,

Consequently

Q~ because ez If we put

< ebl

e b 2 . . .ebn

> 1 + z for z > 0. M

i= 1

and u = e+

then

5 k, then using

1.4.1. Convergence criteria

22

This implies that the terms of the Euler-Minding series do not converge t o sero. I As an example, consider the continued fraction

An equivalence transformation rewrites it as I

I

11 -I- -11 l o + -(10 120

I

I

+

11 110

+

11 120

+...

Clearly this continued fraction converges. Note that the convergents satisfy (yn - l o = -

2

20

+ (Cn-* - 10)

such that for the value G of the continued fraction

c = 10 +

2 10+c

or

G2 = 102

m.

Since all convergents are positive we get c = The next theorem is valid for continued fractions of the form (1.1.)with bo = 0, also if the partial numerators or denominators are complex numbers. It was proved by Pringsheim [18] in 1899.

Theorem 1.8. The continued fraction

converges if jb;l 2 1u;I IC,( < 1 if n 2 1.

+ 1 > 1 for d 2 1. For the nth convergent G, we have

1.4.1. C o n v e r g e n c e c r i t e r i a

23

Proof First we prove the upper bound on G,. Let =

An(.)

an

+z

-__

b,

Then

Thus

lc,~=

0 82 0

.,.

8n(0)1

For the nth denominator Q n we have

r with 0 < r < 00, then

The Stieltjes transform of g ( t ) is defined as

(1.17.)

A proof of the existence of this transform is given in [lo p. 5781. The series (1.16.) has convergence ridius 1/r and can be regarded as a formal power series expansion of F ( z ) which is analytic in the cut complex plane @\[$,m) [I0 p. 5811.

Theorem 1.10. The corresponding continued fraction for f(z)given by (1.16.) converges t o F ( z ) given by (1.17.) for all z in C \ [ t , co). The convergence is uniform OIL every closed and bounded subset of the cut complex plane. The proof which was originally given by Markov can be found in [17 p. 2021. A simple example of a Stieltjes series is

Here g ( t ) = t for 0

5 t 5 1 and

g ( t ) = 1 for

t 2 1.

30

1.4.3. Convergence of Corresponding continued fractions

The Stieltjes transform is 1 F ( z ) = -- ln(1- z ) z

As a consequence of theorem (1.10.) we get that the corresponding continued fraction for f converges to F for all z in C\[l, cm).

1.5.1. The backward algorithm

31

§5. Algorithms to evaluate continued fractions. If we want t o know an approximation for the value of a continued fraction, we must compute one or more convergents C,. The recurrence relations (1.3.) for the nth numerator and denominator provide a means t o calculate the ntA convergent since

This algorithm is called forward because i t is possible t o compute Cn+l from the knowledge of Cn with little extra work. We shall now discuss some other algorithms used for the computation of convergents.

5.1. The backward algorithm. The nth convergent C , can easily be calculated as follows: Put

rn+l,n = 0 and compute

Then

A drawback of this method is that it must fully be repeated for each convergent we want t o compute. It is impossible t o calculate Cnfl starting from Cn. But the algorithm appears to be niinierically stable in a lot of cases [2].

I. 5.2. Forward algor%'thms

32

5.2. Forward algorithms. The following theorem can be found in [14].

Theorem 1.11. The nth convergent of the continued fraction

i=l

is the first unknown

XI,,

li

of the tridiagonal system of linear equations

...

bl

-1

0

a2

b2

- 1

0

a3

b3

...

0

*.

...

-1

0

0

a,

=I

0

a1

0

(1.18.)

0

b,

(see also problem (5)). Consequently, algorithms for the solution of a linear tridiagonal system, and especially for the computation of t h e first unknown, are also algorithms for the calculation of C,. If backward Gaussian elimination is used to solve (1.18.), then the coefficient matrix of (1.18.) is transformed into a lower triangular matrix and the computation of ~ 1 =,r l , ~n is precisely the backward algorithm. It is also easy to see t h a t the computation of C , via the recurrence relations (1.3.) is equivalent with solving (1.18.) by means of the so-called shooting method. If we choose XI,, = z?; = 0 then according to (1.18,) z f L = --a1 and (0) Xk+l,,

If we choose

~

1 =,

(1) Zk+l,n

(0) - akxk-i,,

+ b k x k(0) ,r,

zit!,~ = 1 then

$1

k

= 2 , . .. ,? -I1

= b l - a1 and

- a k Z k( -11) , n + b k x k( 1, n)

k

= 21 . . . 1

-

The last equation of the linear system (1.18.) is after substitution of t h e first (n- 1) equations in it, merely a linear function g(xl,n). The zi,, we are looking for is a root of g(zl,,), in other words the intersection point of the z-axis with the straight line y = g(z).

I.5.2. Forward algortthma

33

so

or

After comparison of the starting values for the recursive formulas (1.3-)1when applied to

28

i=l

with those for the recu sive cornpiitation of

since the recurrence relations are identical.

2 k( 0, n)

34

1.5.3. Modifying factore

Another forward algorithm for the computation of C , is obtained when (1.18.) is solved by forward Gaussian elimination and backsubstitution. The resulting formulas are

ri,n = b;

+ t i -ail . n

e' = 2 , . . .,n

Finally C , = bo + ~ 1 , ~ . More algorithms for the calculation of C, can be found in the literature. We refer among others to [25] and [6].

5.3. Modifying factor8. Even efficient ways t o calculate C, do not guarantee that, C, is a good approximation for the value G = lim C , of the continued fraction. Since the nth n-yw convergent results from truncating

we shall call the chopped off part

the nth tail of the continued fraction. Clearly n-1

I

(1.19.)

and

35

I. 5.5. Modifying factore

Let again for n 2 1 &(5)

an = -

b,

+z

Then

and

Hence, in order to estimate G , it may be better to replace the tail T,,by a value different from zero. In many cases the tails do not even converge to zero. Suppose r,, is such an approximation for T,,.We shall then calI

I’,

= bo

+ a1

o 92 o

. . . o s,(m)

the nth modified convergent with 7, the nth modifying factor. The next theorems illustrate in which cases modifying factors are really worthwile, in the sense that

First of all we study the behaviour of the tails [17 p. 931.

Theorem 1.12. If the continued fraction (1.1.) is such that lim a; = a and lim b; = b with

+

t+m

c+m

a,b E G and if the quadratic equation x2 bx - a = 0 has two roots 5 2 with 1x11 < 1221 then lim T,,= z1 ,403

x1

and

36

1.5.8. Modifying factors

This behaviour of the tails suggests to choose

In order to study the effect of this modifying factor we rewrite the expression IC - rnl/1C - CnI as follows, Using the three-term recurrence relation for (1.19.) we find

Analogously

This leads to

with

where

ho = 03

If the continued fraction (1.1.) satisfies the conditions of theorem 1.12. then

1.5.3.Modifying factors we shall denote

D

+

37

- 1~11 d , = inax lam - a \ =

Ib

511

m>n

en = inax Ib, - bl m>n

En =

dnD D2 - 2 d ,

+ 0, = b + b,,

a, = a

b,

Tn = 51 + En B, = h,

+ 51

Using these notations we can formulate the next theorem [24].

Theorem 1.13. Let the continued fraction

bo be such that Jim t-m

ai =

+

c-1 4 i= 1

6;

a and lim b; = b with a , b E C and let s-

00

51

be the

strictly smallest root of the quadratic equation x 2 + b x - a = 0 with

If also

(1.20c.) then

38

I. 5.3.Modifying factors

Proof Let us first show that in (1.20b.) we have Ib d, 5 D a / 3 and this implies

since

0 5 Hence or

16 +

+ 511 2

En. We know that

4- l ~ l 5l 1

Ib + 511

( D 2- 2 d n ) l b + 5 1 1 2 Dd, J b + 5 1 ) 2 En

In order to bound I(C - r n ) / ( C - Cn)\ we shall now calculate an upper bound for I(Tn - zl)/Tn\. Since lim Tn =

n-+m

and

51

Iiin b , = b

n-co

we have lim

n-m

En = 0

and and hence for fixed n We can then write for fixed k > max(m,ra)

1.5.9. Modifying factors

and

because

Using this upper bound for J f k - 1 -t Bk-11 we can also prove it to be an upper bound for \ & 2 + & 2 \ . Repeating this procedure as long as l a k l I d, and J B k - i l 5 e n , finally assures for k - I = n IFn

NOWsince dn _< la1/2 = I(b

+z

+ Pnl 5 E n

l ) I /~2 ~ and 2dn _< 2 D 2 / 3 we have

Next we shall compute an upper bound for Ihn/(h, + x1)1 which is the second factor in I(C - rn)/(C- Cn)l. Note already that

39

40

I. 5.3. Modifying

factora

By induction it follows that

bec,ause

This gives us

Using the estimatfesfor I(Tn - 21)/Tn\ and Jhn/(hn finish the proof. We have

+ XI)\

it is now easy t o

1.6.1. Definition of branched continued fraction8

41

56. Branched continued f'ractions.

6.1. Definition of branc hed continued fractions. If the denominators bi in the continued fraction

are themselves infinite expressions, then it is called a branched continued fraction. The b; are called the branches and we need a multi-index t o indicate a convergent. Consider for instance thc expression ai

1

The (n,mo, nl,...,nnfth convergent is then the subexpression

i= 1

We will use branched continued fractions to construct a multivariate Viscovatov algorithm for the computation of multivariate continued fraction expansions of the form

where k is the number of variables we are dealing with. Input of such an algorithm is a multivariate power series. Vice versa, given a branched coritiniied fraction, we can also construct an EulerMinding series of which the successive partial sums equal a given sequence of convergent s .

42

Z.6.2. A generalization of the Euler-Minding series

6.2. A generalization of the Euler-Minding aeries.

Let us consider continued fractions

( 1.21a.) for i = 0 , 1 , 2 , . . .. If C c ) denotes the nth convergent of (1.21a.) then according to (1.4.)

where we have written

We will now generalize (1.4.) for the branched continued fraction (1.21b.) Let us denote by Pn/Qnthe subexpression

(1.22.)

So P,/Qn is the (n,n,n - 1 , . . . , l , O ) t h convergent of (1.20.). Another subexpression we shall need is

1.6.2. A generalization of the Euler-Minding aerie8

43

( 1.23.) which is in fact the k t h convergent of Pn/Q,. These subconvergents can be ordered in a table

where we proceed in a certain row from one value to the next one by using (1.3.) for (1.22.) :

with RPj = 1 = Sin), R P ) = Cia) and Sl;' = 0. If we want t o develop a formula analogous t o (1.5.) for the branched continued fraction (1.21.) we must compute a n expression for t h e difference

(1.25.) Remark t h a t in comparison with P n - l / Q n - l the expression Pn/Qn contains an e x t r a term in each of the involved convergents of B;. Also Bn is not taken into account in P n - l / Q n - l . In order t o compute (1.25.) we must b e able to proceed from one row in the table of subconvergents t o the next row. The following t.heorem is a means to calculate the differences - RF-') and Sin) - sin-') Rk

44

1.6.2.A generalization of the Eder-Minding series

Theorem 1.14.

F o r n 2 2 a n d k = 1, ..., n - 1

and

Proof We shall perform the proof only for Rp' - Rf-') because it is completely analogous for sin) . Choose k and n and write down the recurrence relation (1.24.) for row n and row n - 1 in the table of subconvergents:

sin-')

R P ) = cn-kRk-l (k) ( n ) + a k R k(-n2 )

By subtracting we get

1.6.2. A generalization of the Euler-Minding series

45

The first three starting starting values are easy t o check and for R p ) -

I

again (1.4.) is used.

From t h e above theorem we see t h a t up t o an additional correction term the values R f ) - R P - l ) a nd Sin)- Sin-') also satisfy a three-term recurrence relation. By means of this result we can write for the numerator of ( 1 . 2 5 . ) :

because Rn-2 (n-1) / S L i l ) and Rn-l (n-1) /SLq1) are consecutive convergents of th e finite continued fraction n-1

t

1.6.2.A generalization of the Euler-Minding series

46

h this way

(1.26.)

We remark that (1.26.) reduces t o (1.4.) if the continued fraction (1.21.) is not branched because then R r ) = R p ) and Sikj = Sin) for all n 2 k. Consequently the classical Euler-Minding series will t u r n out to be a special case of the Euler-Minding series for branched continued fractions.

Theorem 1.15.

For n 1 2 the convergent C n , n , n - l , . . . , lof , ~ the branched continued fraction (1.21.) can be written as

I

&

Qi

i=2

Qi-1

Qi

47

1.6.8. Some recurrence relataono

Proof The result is obvious if we write

and insert (1.26.) for P i / Q i

I

- P,-1/&,-1.

As a result of the previous theorcm we can associate with the branched continued fraction (1.21.) the series

I

Qi

J

Qi

Qi- I

of which the successive partial sums equal the successive convergents Cn,n,n-i ,...,1,0 of (1-21.). 6.8. Some recurrence relations.

In order to formulate a multivariate Viscovatov algorithm we first show that a corresponding continued fraction can be obtained from a system of recurrence relations [ 161. Consider the problem of constructing a continued fraction expansion of the form

f(x) =

,f$ +

4 +

I 1

...

(1.27.)

for a given series expansion

Remark that (1.27.) coincides with (1.14.) after an equivalence transformat ion. Instead of using Viscovatov's algorithm, the coefficients a; can also

I . 6.3. Some recurrence relata'ons

48

be deduced from the following set of recurrence relations. Define fo = f

given by

(1.27.)

- fo

I1

=

fk

= akxfk--2 - f k - 1

a1

As mentioned in section 3.4.

fk(2)

k = 2,3,4,. . .

(1.28a.) (1.28b.)

can h e developed in a series of the form 00

i= 1

Equating coefficients in relation (1.28a.)

i= 1

i=l

we find

and for k 2 2 by means of (1.28b.)

we get

Using these formulas all the coefficients in the continued fraction (1.27.) can be computed. Hence we can also construct a continued fraction expansion of the form i= 1

1

I.S.4. A multivciriate Vaacovatov algorithm for the power series f(x) = c r ) reasoning t o (f(x) - co(0))/x.

49

+ c(l(l) z + c p ) z2+. . . by applying the previous

6.4. A multivariate Viacovatov algorithm

Let us now apply this reasoning t80the following problem [15]. We restrict ourselves t o the bivariate case only t o avoid notational difficulties. Given a double power series

try to End a branched continued fraction of the form

We define

fo = f fl

given by

= a l l - (1

fk =

(1.29.)

+ g 1 + h1)fo

a k k z y f k - 2 - ( 1 $. g k -k

A series expansion for

fk(z,

hk)fk-l

y) is then of the form

k

= 2 , 3 , 4 , .. .

(1.30a.) (1.30b.)

50

I.6.4. A multivariate Viscovatov algorithm

while gk(z) and h k ( y ) can be written as (1.31.f i= 1 00

(1.32 .)

Equating coefficients in formula { 1.30a.)

we obtain

i--1

and doing the same with (1.30b.)

we find for k 2 2

I.6.4.A multivariate Viscovatov algorithm

The coefficients a k + i , k and a k , k + i are computed by applying (1.27.) and (1.28.) to t,he series (1.31.) and (1.32.) As a consequence one can obtain a coutinued fraction expansion of the form

for a double power series

by applying the previous reasoning t o the power series

and compute the coefficients n;o and a o i in

51

52

I.6.4. A multivariate Viscovatov algorithm

by applyicg (1.27.) and (1.28.) to the series

i= 1

c

and

co

c$'yj

j= 1

To illustrate this technique we consider 'the following example. Take 1 1 + -12z 2 + z y + -g21 2 + 61 3 + -&+ --5y2+ 2 2 1 1 + 241 4 + 61 3y + -412 y 2 + -xy3 + -y4 + . . . 6 24 1 1 1 1 = 1 + z(l + + 6-z2 + . . .) + y ( l f -I/2 + -I/2 + . . .) 2 6 1 1 1 + q ( 1 + 12 + -9 + -61x 2 + -sy 1- -r/2 + . . .) 2 4 6

=l+z+y

--5

--5

1

-y3 6

--2

--2

--5

Since the problem is completely symmetric it is sufficient to calculate the coefficients aiO, d:") and el!,"' with i 5 j. Then

Using the above formulas we obtain

I.S.4. A multivariate Viscouatou algorithm and

53

1

a22 = 2

p) = 1 2 36

d y ) = 16.

*.-

Applying (1.27.) and (1.28.) to

gl(s) -- -

- -1+

-1x + o x 2 12 hl(?/) - - - - +1 i p +1 O y Y 2 X

2

+ ...

2

+...

and

we finally get for

(ez+Y

- e2

- eY

+ l ) / z y the branched continued fraction I

1 ~

11 +

( +

and for

xY14

I1+

.)1+

.. .)'

...

..

.) + (

+

..

.) + (

IF+ .. .) +

+ ..

er+g

1+

($+

I*+

XY

\I+

Iq+ ,*+

. .) + (,*+

I++

...) +

(,-+ + II+

(p+

4

lq+

4

+ (lqJ+

+ ...

I

I. Problems

54

Problems. (1)

Prove that

(3)

Prove formula (l.lOb,) for the odd part of a continued fraction.

(4)

The convergents of the continued fraction

with b;

> 0 for t 2 0 satisfy

a) c can},,^ is a monotonically increasing sequence. b) { C 2 n + l } n E is ~ a monotonically decreasing sequence. c) for n and rn arbitrary: Czrn+l > C2n

(5)

a) The nth numerator and denominator of the continued fraction

satisfy

Pn =

55

I . Problems

Qn=

...

0

63

...

0

*.

-1

0

an

1

0

bl

-

a2

b2

- 1

0

a3

0

...

bn

b) Also c) Prove theorem 1.11. a) Construct a continued fraction with convergents c n

+

= (1

%)(I + 71)..

+ 7n)

where rk(l ’yk) # 0 for k 2 0. b) Use it t o give a continued fraction expansion €or

-_ sin(nx) TX

- (1

(

- x)(l+ x ) 1 - ;)(I

+ ;)(I - ;)(1+

;). .

If are the convergents of a given continued fraction, construct a continued fraction with convergents

where a E R. This procedure is called the extension of a continued fraction.

(10)

How is the method of Viscovatov to be adapted if

d20

Give a continued fraction representation for gence.

and discuss its conver-

Prove formula (1.26.) using (1.24.) n - 1 times.

= O?

I. Remark5

50

Remarks. (1)

Facts about the history of continued fractions can be found in [5]. This history goes back t o Euclids algorithm t o compute the greatest common divisor of two integers (300 B.C.) but the first conscious use of continued fractions dates from the 16th century.

(2)

The notion of nth numerator and denominator satisfying a three-term recurrence relation can be generalized to compute solutions of a ( k 2) term recurrence relation with k + 1 initial data:

+

The (k + 1)-tuple of elements

is then called a generalieed continued Fraction [26].

(3)

More general forms of coutinued fractions where a i and bi are no longer real or complex numbers, are possible. We refer to the works of Fair is], Hayden [9], Roach [19], Wynn [30, 311 and Zemanian [32]. A lot of references on the theory of continued fractions can also be found in the bibliographies edited by Brezinski [4].

(4)

If a continued fraction

with nth convergent C,, converges to a finite limit C , then C - C , is called the nth truncation error. An extensive analysis of truncation errors is given in [I1 pp. 297-3281.

(5)

Another type of bivariate continued fraction expansions can for instance be found in [22]. They are of the form

I. Remarks

57

where the continued fractions B!J'(xy) are given by

i = 1,2, j = 1,2

and obtained by inverting power series. More types of branched continued fractions are given in [3] and [13].

58

I. References

References.

[

I ] Abramowite M. and Stegun I. Handbook of Mathematical functions. Dover publications, New York, 1968.

[

21 Blanch G . Numerical evaluation of continued fractions. SLAM Rev. 6, 1964, 383-421.

[

31 Bodnarc'uk P. and Skorobogatko W . (in Russian) Branched continued fractions and their applications. Naukowaja Dumka, Kiev, 1974.

[

41 Brezinski C.

[

History of continued fractions and Pad6 approximants. 51 Brezinski C. Springer, Heidelberg, 1986.

[

61 Cuyt A. and Van der Cruyssen P . Rounding error analysis for forward continued lraction algorithms. Comput. Math. Appl. 11, 1985, 541-564.

[

A bibliography on Pad6 approximation and related subjects. Publications ANO, Universiti: de Lille, France.

71 de Bruin M. and van Rossum €I.

Pad6 Approximation and its applications. Lecture Notes in Mathematics 888, Springer , Berlin, 1981.

[

81 Fair W. Noncommutative continued fractions. SIAM J. Math. Anal. 2, 1971, 226-232.

[

Continued fractions in Banach spaces. Rocky Mountain J. 91 Hayden 2’. Math. 4, 1974, 357-370.

[ 101 Henrici P.

Applied and computational complex analysis: vol. 2. John Wiley, New York, 1976.

[ 111 Jones W. and Thron W .

Continued fractions: analytic theory and applications. Encyclopedia of Mathematics and its applicalions: vol. 11, Addison-Wesley, Reading, 1980.

[ 121 Khovanskii A. generalizations t o Groningen, 1963.

[ 131 Kuchminskaya K.

The application of continued fractions and their problems in approximation theory. Noordhoff,

(in Russian) Corresponding and associated branched continued fractions for double power series. Dokl. Akad. Nauk Ukrain.SSR Ser. A 7, 1978, 614-617.

I. References

\

59

141 MikloSko J . Investigation of algorithms for numerical computation of continued fractions. USSR Cornputational Math. and Math. Phys. 16, 1976, 1-12.

[ 151 Murphy J . and O’Donohoc M.

A two-variable generalization of the Stieltjes-type continued fraction. J . Comput. Appl. Math. 4, 1978, 181190.

[ 161 Murphy J . and O’Donohoe M.

Some properties of continued fractions with applications in Markov processes. J . Inst. Math. Appl. 16, 1975, 5771.

[ 171 Perron 0.

Die Lehre von den Kettenbruchen 11. Teubner, Stuttgart,

1977.

[ 181 Pringsheim A.

Uber die Konvergenz unendlicher Kettenbriiche. S.-B.Bayer. Akad. Wiss. Math.-Nat. KI. 28, 1899, 295-324.

[ 191 Roach F .

Continued fractions over an inner product space. AMS Proceedings 24, 1970, 576-582.

[ 201 Sauer R. and Szabd F .

Mathematische Hilfsmittel des Ingenieurs 111.

Springer, Berlin, 1968.

[ 211 Seidel L.

Untersuchungen iiber die Konvergenz und Divergenz der Kettenbriiche. Habilschrift, Munchen, 1846.

[ 221 Siemaszko W.

Branched continued fractions for double power series. J . Comput. Appl. Math. 6, 1980, 121-125.

[ 231 Stieltjes T .

Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse 8, 1894, 1-22 and 9, 1894, 1-47.

[ 241 Thron W. and Waadeland H .

Accelerating Convergence of Limit Periodic Continued Fractions K ( a n / l ) . Numer. Math. 34, 1980, 155-170.

[ 251 Van der Cruyssen P.

A continued fraction algorithm. Numer. Math.

37, 1981, 149-156.

[ 261 Van der Cruyssen P.

Linear Difference Equations and Generalized Continued Fractions. Computing 22, 1979, 269-278.

[ 271 Viscovatov B.

De la mCthode g6n6rale pour reduire toutes sortes de quantitCs en fractions continues. MCm. h a d . Impkriale Sci. St-Petersburg 1, 1803-1806, 226-247.

I. References

60

[ 281 Wall H.

Analytic theory of continued fractions. Chelsea, Bronx, 1973.

[ 291 Worpitzky J.

Untersuchungen uber die Entwickelung der monodronen und monogenen Funktionen durch Ket,tenbriiche. Friedrichs-Gymnasium und Realschule Jahresbericht, 1865, 3-39.

[ 30) Wynn P.

Continued Fractions whose coefficients obey a noncommutative law of multiplication. Arch. Rational Mech. Anal. 12, 1963, 273-312.

[ 311 Wynn P.

Vector continued fractions. Linear Algebra 1 , 1968, 357-395.

Continued fractions of operator-valued analytic func[ 321 Zernanian A. tions. J. Approx. Theory 11, 1974, 319-326.

61

.

CRAPTER II: Pad6 Approximants 51. Notations and definitions

. . . . . . . . . . . . . . . . . . . .

52 . Fundamental properties . . . . . . . . . 2.1. Properties of the Pad4 approximant 2.2. Block structure of the I’ad6 table . 2.3. Normality . . . . . . . . . . . .

. . . .

§3. Methods t o compute Pad6 approximants 3.1. Corresponding continued fractions 3.2. The qd-algorithm . . . . . . . 3.3. The algorithm of Gragg . . . . 3.4. Determinant formulas . . . . . 3.5. The method of Viscovatov . . . 3.6. Recursive algorithms . . . . . . 3.7. The 0. Since (a) must be valid:

with t Hence

> 0.

for all k and C satisfying m' 5 k 5 m! + t and n’ 5 C 5 n’ + t . This again contradicts the normality of rm,n(z). To prove t h a t (a) and (b) guarantee the normality of rm,n(z) we proceed as follows. Suppose rm,n(z)= rk,r(z) for certain k and C with k > m or C > n. For an integer B t h a t satisfies

w[(f

40

-Po)

57

2

k

+C+1

we find, by using (b), t h a t s>k-m This contradicts theorem 2.3.

or a > C - n

I

Normality of a Pad6 approximant can also be guaranteed by the nonvanishing of certain determinants.

74

II. 2.8. Normality

We introduce the notation Cm

... ...

Cm+n-1

...

Cm

Cm-1

Cm+l

Cm-n Cm+l-n

Cm

with det D,,o = 1. The following result c a n be proved [40 p. 2431. Theorem 2.6. The Pad4 approximant

For Stieltjes series this theorem and the following lemma [30 p. 605)lead to a remarkable result.

Lemma 2.1. Let p(t) be a real-valued, bounded, nondecreasing function defined on ( 0 , ~ ) and let the integrals

1

0)

ci =

t'dg(t)

exist for all e' 2 0. If g ( t ) has at least k points of increase then for all rn >_ 0 and for n = 0 , . . . , k we have

If g ( t ) has an infinite number of points of increase then for all m ,n 2 0

II. 2.9. N o r rnality

Clearly for Stieltjes series

with g ( t ) having infinitely many points of increase, the Iatter is true and hence we can conclude the following.

Theorem 2.7. Let f be a Stieltjes series and let g be a real-valued, bounded, nondecreasing function having infinitely many p0int.s of increase. Then for all m , n 2 0 the Pad6 approximant rrn,-, for

is normal.

75

II.3.1.Corresponding continued fractions

76

§3. Methods to compute Pad6 spproximents.

In the sequel of the text we suppose t h a t every Pad4 approximant in the Pad6 table itself satisfies the condition (2.2.). By theorem 2.3. this is the case if for instance min(m - m',n - n') = 0 for all m and n. A siirvey of algorithms for computing Pad6 approximants is given in [50] and P11.

3.1. Corresponding continued fractions.

The following theorem sball be used to compute the difference of neighbouring Pad4 approximants in the table.

Theorem 2.8.

If

and

P2

rm+k,n+C = -

42

with k, t

2 0 then a polynomial u(z) exists with

Proof

For the expression pi42 - p2q1 we can write

This completes the proof.

118.1. Corresponding continued fractions

77

Let us now consider the following sequence of elements on a descending staircase in the Pad4 table Tk

= {fk,ol

rk+l,ol rk+l,l) rk+2,i1..

.} for k 2 0

and the following continued fraction

Theorem 2.9. if every three consecutive elements in T k are different, then a continued fraction of the form (2.4.) exists with d k + i # 0 for i 2 1 and such t h a t the nth convergent equals the (n + l)thelement of Tk. Proof Put rk+;,j

=

Pi+ j

--

Q i+i

f o r i = j , j + l a n d j = 0 , 1 , 2 ,.... A continued fraction of which the nth convergent equals

is according t o theorem 1.4., given by

Here we have already used the fact t h a t QO = & I = 1. By theorem 2.8. we find that

78

II.3.1. Corresponding continued fractaone

i = 3 , 2 , ...

for certain nonzero numbers a; and b;. So the continued fraction (2.5a.) is

with

and

c

k+l P l =rk+l,O

=

CiXi

i=O

By performing an equivalence transformation we finally get a continued fraction of the form (2.4.). I In this way we are able to construct corresponding continued fractions for functions j analytic in the origin.

Theorem 2.10.

element of To ( n 2 O), then If the nth convergent of (2.4.) equals the ( n + (2.4.) is the corresponding continued fraction t o the power series (2.2.).

Proof Let Pn/Qn be the nth convergent of (2.4.). Then w ( j Qn - Pn) 2 n

because Pn/&, is also the ( n +

+ 1 and Qn(0) = 1

element of

To.

11.8.2. The qd-dgorithm

Hence

(f-2) (i)

(O)=O

79

j = 0 ,... , n

because Q , ( x ) is nontrivial in a neighbourhood of the origin. So the Taylor series development of the nth convergent matches the given power series u p t o and including the term of degree n. In other words, (2.4.) is the corresponding continued fraction t o (2.1.). I

By continued fractions of the form (2.4.) one can only compute Pad4 approximants below the main diagonal in the Pad6 table, For the right upper half of the table one can use the reciprocal covariance property of Pad6 approximants, given in problem (4) a t the end of this chapter. We now t u r n t o the problem of the calculation of the coefficients d k + i in (2.4.) for k > 1 starting from the coefficicnts c o , c l , c 2 , . . ., and not from the knowledge of the Pad6 approximants. 3.2. The qd-algorithm. Consider the following continued fraction

If the coefficients qPt1) and er")

are computed as in theorem 2.9. then the convergents of gk equal the elements of 1 6 . If we calculate the even part of g k ( X ) we get

80

11.3.2. The qd-algorithm

If we calculate the odd part of

g k - ~ ( z ) we

get

The even part of gk(z) and the odd part of g k - l ( Z ) are two continued fractions which have the same convergents r k , o , r k + l , l , r k + 2 , 2 , . . . and which also have the same form. Hence the partial numerators and denominators must be equal, and we obtain (431 for k 2 1 and e 2 1

(2.6a.) (2.Ab.)

The numbers q y ) and e p ) are usually arranged in a table, where the superscript (k) indicates a diagonal and the subscript t! indicates a column. This table is called the qd-table. Table 2.9.

11.3.2. The gd-algorithm

81

The formulas (2.6.) can now also bc memorized as follows: e p ' is calculated such that in the following rhombus the sum of the two elements on the upper diagonal equals the sum of the two elements on the lower diagonal

and qcl is computed such t h a t in the next rhombus the product of the two elements on the upper diagonal equals the product of the two elements on the lower diagonal

e

1"’

*

Since the qd-algorithm computes the coefficients in (2.4.), it can be used t o compute the Pad6 approximants below the main diagonal in the Pad6 table. To calculate t h e Pad4 approximarits in the right upper half of the table, the qd-algorithm itself can be extended above the diagonal and the following results can be proved [32 pp. 615-6171.

82

II.3.2. The qd-algorithm

TGble 2.4.

...

... ... ...

1

-(z) = wo

f

and for k

2

+ w15 + 20222 + . . .

1

If the elements in the extended qd-table are all calculated by the use of (2.6.) using the above starting values, then the continued fraction

supplies the Pad6 approximants on the staircase

II.3.3. The algorithm of Gragg

83

To illustrate the qd-scheme we will again calculate some Pad6 approximants for the function exp(z). Table 2.3. looks like 0

1

2

0 1 3

_ -1 6

-

0

_ -1

12

1

-

0

4

1 5

_ -1 2ti

1

6

3 20

1 _-1.0

...

-.

0

...

From this we get

It is obvious t h a t difficulties can arise if the division in (2.6b.) cannot be performed by the fact t h a t e p ) = 0. This is the case if the Pad6 table is not normal for consecutive elements i n T k can then be equal. Reformulations of the qd-algorithm in this case are giver) in [I31 and [30]. 9.9. The algorithm of Gragg.

Let us now consider ascending staircases in the Pad6 table. Take

and consider the continued fractioii

118.8. The algorithm of Gragg

84

In an analogous way as for T k one can compute the coefficients f yl and *I_"’ such t h a t the athconvergent of this continued fraction equals the (n l)*h element of s k . Remark that fk(s) is not an infinite expression since s k is a finite sequence. If we compute the odd part of f k + l and the even part of fk, we again get continued fractions of the same form t h a t have the same convrrgents. This reasoning provides us with formulas for and "I [29]: for k 2 1 $4) = 0

+

where

1 -(z) = wo

f

and for k

+ w1z + w222 + . . .

2 1 and 1 5 !t 5 k - 1

These quantities are arranged in a table as follows. The superscript denotes now an upward sloping diagonal.

Table 2.5.

II. 9.4 Determinant for mu1QB I

For the computation of gt( k ) and et(k)..

8p)

and

+

85

fy' wc have similar rhombus rules as for the f(kj

t

and

3.4. Det e rminant formulas . One cau also solve the system of equations (2.3b.) and thus get explicit formulas for the Pad6 approximant. For

we write

II.3.4. Determinant formulas

86

Theorem 2.11.

If the Pad6 approximant of order ( m , n )for f is given by

and if D = det Dm,,, # 0, then

1

Po(%) = 5

and

Proof

L

Since D # 0 the homogeneous system (2.3b.) has a unique solution for the choice 60 = 1. Thus the following homogeneous system has a nontrivial solution:

c,+lbo

+ sbi + z2b2 + . . * +x"bn = 0 + c w b l + * . .+ cm+l-,,bn = 0

c,+,bo

+

(1 - qo(x))bo

Cm+n-Ibl

+ . . . + cmbn

=O

This implies that the determinant of the coefficient matrix of this system is zero:

=o

II.3.4. Determinant for mu1a8

87

and it proves the formula for qo(z). If we take a look at f ( z ) q o ( x ) we have

Because the polynomial po(z) contains all the terms of degree less than or equal t o rn of the series f(z)qo(z), we got the determinant expression for po(z) given above. I The determinant formula for qo(z) can also very easily be obtained by solving (2.3b.) using Cramer's rule after choosing bo = Dm+. The determinant expressions are of course only useful for srnall values of m and n because the calculation of a determinant involves a lot of additions and multiplications. They merely exhibit closed form formulas for the solution. From the proof of theorem 2.11. we can also deduce t h a t

(f QO

-Po)(.)

=

;1

cm+l

where 00

p k ( z )=

ci

zi = f(z) - F k ( z )

i=k+ 1

This gives a n explicit formula for the error ( f - r m,= ) ( z ) in terms of the coefficients c i in f(z). For Stieltjes series

with convergence radius

$, it is possible using the error formula, to indicate

88

II. 8.4. Determinant f~ rrnula8

within a finite set of Pad6 approximants which one is the most accurate on (-03, +[ . The most interesting result is the following one.

Theorem 2.12.

Let f be a Stieltjes series. For the Pad6 approximants in the set

and for those in

{rm,n I O

we have

i m + n I 2k + I}

This means that when k increases, the best PadC approximants for a Stieltjes series among the elements in successive triangles r0,o r1,o

ro,1

...

rO,k

are the elements on the descending staircase To,in other words they are the successive convergents of the corresponding continued fraction for the Stieltjes series. For other subsets of the PadC table similar results exist because, when f is a Stieltjes series, the errors ( f - r m , n ) ( z ) are linked by inequalities throughout the entire table. The interested reader is referred to [8].

11.3.5. The method of Viscovatov

89

3.5. The method of Viscovatov.

By the method of Viscovatov described in section 3.3. of chapter I, the recursive generation of staircase sequences Tk of Pad6 approximants is absolutely straightforward in the case of a normal Pad4 table. We have proved in secttion 3.4. of chapter I that the constructed continued fraction is corresponding. Hence it generates the elements on To. If the rnrthod of Viscovatov is applied to k i=O

for the construction of a corresponding continued fraction

then the elements on Tk are obt,airied from

However in order t o obtain the normaliz,cd

the normalization qo(0)= 1 has t o be built into the algorithm via a n equivalence transformation. We reformulate it as follows. For

i=O

we put

and for j

22

90

11.9.6. Recursive algorithms

Then

3.6. Recursive dgorithms.

It is also possible t o calculate Pad4 approximants on ascending staircases by means of a recursive computation scheme. To this end we formulate the following recurrence relations for which we again assume normality of the Pad6 table. First we introduce the notation m

i= 0

Theorem 2.13. If the Pad6 approximants

are normal, then

11.3.6.Recursive algorithm

91

Furthermore

because of the assumption of normality and because of the uniqueness of the Pad6 approximant. I This theorem enables us to calculate ~ 3 1 9 3when p1 /qI and p2 192 are given. We denote this by rm,n

--*

Theorem 2.14. If

then

Computationally this means rm-1,n

t Combining theorem 2.13. and 2.14. we can compute the elements on an ascending staircase in the Pad6 table, starting with the first column. Other algorithms exist for the computation of Pad6 approximants in a row, column or diagonal of the Pad6 table, instead of on staircases. We do not mention them here, but we refer t o [l], [36] and [41].

113.7. The r-algorithm

92

3.7. The t-algorithm. Consider again a continued fraction M

I

with convergents

Using theorem 1.2. of the previous chapter, we know t h a t

For the continued fraction gmwn(x) given by (2.5b.) we get for k = 2n

and for gm-,-l(z)

with k = 'Ln + 1 we have

1 -+ rm,n+l

- rm>n

__ _ _1 _ ~-_ r m , n - rm-1,n

QXn

P2ni-2 Q2n

- P2n Q2n3-2

Consequently the elements in a normal Pad4 table satisfy the relationship

II.3.7. The e-algorithm

93

where we have defined

The identity (2.8.) is a star identity which relates

and is often written as

( N - c)-*+ ( S - c)-I = ( E - c)-1+ ( W -

c)-I

If we introduce the following new notation for our Pad4 approximants

we obtain a table of €-values where again the subscript indicates a column and the superscript indicates a diagonal.

Table 2.6.

...

(3)

€0

...

The ci"') are the partial sums F,(Z) of the Taylor series f(z).

94

II.3.7. The c-algorithm

Remark the fact that only even column-indices occur. The table can be completed with odd-numbered columns in the following way. We define elements 1 m = 0 , 1 , .. . (m-n-1) - (m-n) €gn+l - €2,-1 - ~ _ _ (2.9.) n = 0 , 1 , .. . (m-n) - €::-n-i)

+

€2n

with

Table 2.7.

II.3.7. The c-algorithm

95

from which we can easily conclude by induction on n that

or (2.10.) The relations (2.9.) and (2.10.) are a means t o calculate all the etements in table 2.7. and hence also t o calculate all the Pad6 approximants in table 2.6. This algorithm is very handy when one needs the value of a Pad6 approximant for a given x and one does not want t o compute the coefficients of the Pad6 approximant explicitly. The c-algorithm was introduced in 1956 by Wynn [51]. To illustrate the procedure we calculate part of the completed €-table for f(z)= ez with z = 1. Compare the obtained values with e = 2.718281828.. .

Table 2.8.

I(z) = exp(4 x=1

1. 00 0 0 0 0 1.oooooo

3.000000 2.750000

2. 500000 6. 00 0 0 0 0

2.722222

2.666667 24. 0 0 0 0 0

- 2.000000 - 30.00000 - 264.0000

2.718750

2. 708333

2 .5 0 0 0 0 0

1.000000

2. 00 0 0 0 0

3 .0 0 0 0 0 0

2.000000

00

2. 000000

1.500000

1.000000

1. 00 0 0 0 0

0

0

0

0

2.727273

2 .6 6 6 6 6 7 1 0 .0 0 0 0 0

2 .7 1 8 7 5 0

2 .7 1 4 2 8 6 2 4 3 .0 0 0 0

2 .7 1 8 3 1 0

2 .7 1 7 0 4 0 30 1 2 .0 0 0 2 .7 1 8 2 5 4

-2280.000

120 . 000 0 2.718333

2. 716667 720.0000 2. 7 18056

Again computational difficulties can occur when the Pad6 table is not normal. Reformulations of the €-algorithm in this case can be found in [15] and [52].

I I . 4 . f . Numerical ezamples

96

§4. Convergence of Pad6 approximants.

Let us consider a sequence S = {ro, r l , r2, . . .} of elements from the Pad6 table for a given function f ( z ) . We want t o investigate the existence of a function F ( s ) with ri(z) = F'(z)

lim

i-cc

and t h e properties of t h a t function F ( z ) . In general the convergence of S will depend on the properties of f . Before stating some general convergence results we give the following numerical examples. One can already remark tha t t he poles of the elements in S will play an important role. A lot of information on the convergence of Pad6 approximants can also be found in [4].

4.1. Numerical ezamples. For f(x) = ez and r i ( z ) = rm,,.,(z)with m + n = i, we know [40 p. 2461 that lim

i-00

for all

ri(z) = ez

2

in C

We illustrate this with t he following numerical results. Table 2.9.

f(z) = e" z = l e = 2.718281828.. .

_-

2 3 4

1.000000 2.000000 2.500000 2.666667 2.708333

1

2

3

4

00

2.000000 2.660667 2,714286 2.727949 2.718254

3.000000 2.727273 2.7 18750 2.7 18310 2.718284

2.666867 2.716981 2.718232 2.718280 2.718282

3.000000 2.750000 2.722222 2.718750

Next we consider t he case t h a t f is a rational function. For 2+

f(z) = 21-

10

ZZ.4.2. Convergence of columna in the Pad6 table

97

the Taylor series expansion 10 + converges for 1x1

< 1. If

+ l o x 2 + z3 + 1oe4 + . . .

r;(z) = r i , l ( z )then i-1

ri(z) =

C

+

ckzk

k=O

CiZ' Ci+I

1--2

For i even the pole of r;(z) is 10 and for i odd the pole of r i ( z ) is &,. In these points the sequence r ; ( e )docs not converge t o f(z).

For

In( 1 + z) = 1 -- z + - - z3 + z4 ... Z 2 3 4 5 the Taylor series expansion converges for IzI < 1 while the diagonal Pad6 a g proximants r;,i(z) converge t o f for all z in C\(-oo, -I]. The following results illustrate this.

f(z) =

~

Table 2.10.

f(2)

=

In(] --

+ z) 2

f(1) = 0.69314718..

f ( 2 ) = 0.54930614..

i 0 1 2 3 4

i d 1) ____r~.

.

1.000000 0.700000 0.693333 0.693152 0.693147

r ;, i ( 2) 1.000000 0.571429 0.550725 0.549403 0.549313

4.2. Convergence of columna in the Pad6 table. First we take ti(.) = r i , o ( z ) ,the partial sums of the Taylor series expansion for f(z). The following result is obvious.

98

Theorem 2.15.

II.4.9. Convergence of the diagonal elements

I

with ! t > 0, then = {ri,l)}iEN If f is analytic in B(0,r) = {Z 121 < r) (I converges uniformly t o f on every closed and bounded subset of B(0, r). Next take rj(z) = ri,l(z)Jthe Pad4 approximants of order (i, 1) for f . It is possible to construct functions f t h a t are analytic in the whole complex plane but for which the poles of the r;,l are a dense subset of C [40 p. 1581. So in general S will not converge. But the following theorem can be proved [6].

Theorem 2.16. If f is analytic in B(0,r), Lhen a subsequence of {r,,l}aN exists which converges uniformly t o f on every closed and bounded subset of R ( 0 , r).

In [3] a similar result was proved for S = (r;,2(z)}iEm. For meromorphic functions f it is also possible t o prove the convergence of certain columns in the Pad4 table [22]. Theorem 2.17.

If f is analytic in B(0, r) except in the poles w1 . .

wk of f with total multiplicity converges uniformly to f on every closed and bounded subset of B ( O , r ) \ { ~ l , . . - , w k } .

n, then {r+};€N

4.9. Convergence of the diagonal elements.

In some cases a certain kind of convergence can be proved. It is called convergence in measure [39]. Theorem 2.18. Let f be meromorphic and G a closed and bounded subset of C. For every S in R$? there exists an integer k such t h a t for i > k we have lri,i(z) - f(x)I

=

c

dij

yi

i-t j>mn+rn+n+l

Proof

If we put

then the conditions

(2.15a.)

where Ck = 0 if k

< 0. With this notation we have

c m

P(Z,

and

Y) =

k-0

Adz, I/)

108

11.6.1. Definition of multivariate Pad6 lapprozinaants

If we solve the homogeneous system for the & ( s , y), choosing Cm(W)

.

Gm+l-,(Z,V)

v ) .. .

CmIZ, v )

*.

B o ( z , y )= Cm+n-l(Z,

and using Cramer's rule, and if we substitute this solution in the system of equations defining the A ~ ( zy), , then we get precisely p(z, y) and q(z, V ) given by the determinant expressions (2.11 .) and (2.12.). m

So p(z,y) and q(z,y) given by (2.11) and (2.12.) constitute a solution of (2.14.).

Now the expressions (2.11.) and (2.12.) may be identically zero, but one can prove that problem (2.14.) always has at least one nontrivial solution [16]. For the definition of the ( m , n ) multivariate Pad6 approximant we first need the following property. Theorem 2.22.

If p1 , q1 and p2, 42 both satisfy the condition (2.14.) then

Proof

We proceed as in the univariate case. Write p1 92 - p2 q1 as

We know that

and consequently

11.8.2. Block structure of the multivariute Pad6 table

Now

3 ( ~ 1 q 2- P Z Q I )

5

2m.n

109

+m +n I

SO p i q 2 - p2q1 is identically zero.

The multivariate Pad6 approximant of order (m, n) for j ( z , y) is now defined as the irreducible form

of a rational function p ( z , y)/q(z, y) where p and q satisfy (2.14.). For these rm,n a lot of properties of univarjate Pad6 approximants remain valid.

6.2. Block structure of the multivariate P a d 6 table. From the previous section we easily conclude the following. Theorem 2.23.

For every rn and n a unique multivariate Pad6 approximant of order (m,n) for f(z, y) exists. Let us now first take a look at an example. Consider

f(z,y) = 1 +

=1

2

+ sin(zy)

0.1 - 1/ m

00

i= I

i- 0

(2i + I)!

+ 10Z -1- l 0 l z y + 1 0 0 o x y ~+ . . .

Take m = 1 = n. Then we have t o look for p(z,y) and q ( z , y ) of the form p(z,y) = a102 -tnoly

+ a20z2 + a l l z y + ao2v 2

110

II.6.2. Block atructure of the multiuariate Pad4 table

such that

According t o theorem 2.21. a solution is given by

p(sJ) =

I

+

1 10s l0lxy

= 102

A1

+ 1OOs2 - l0lsy

= 10s - l0lzy

1 1+1oz 1 0s l 0 l z y p(z,y) = 1ooosy2 l 0 l s y

1 l0lsy

1 102 q(s,y) = 1ooosy2 l 0 l s y

0 1 10s

1 1 10s

11.6.2. Block attucture of the multivariate Pad6 table

We obtain

111

+ 10z2 - 20.2211 + 10y2 = 2 - 1.0ly - I0.lzy + lOy2 4- 2.012y2 f;

- 1.01y

__ _.

r112(2,y)

In general the following results can be proved about the order and degree of numerator and denominator of rm,n(zJy). Since po/qo is the irreducible form of a rational function p / q with p and of the form (2.13.), we may write

and because of ( 2 . 1 5 ~ )also with

So we can define mI = a p o - wqo n' = aqo

-

wqo

0bviously

This definition of mr and n' is an extension of the univariate definition, because in the univariate case wqo = 0.

11.6.2. Block structure of the multivariate Pad6 table

11 2

Theorem 2.24. If the Pad6 approximant of order ( m , n )for f ( z , y ) is given by

then an integer 8 with 0 bivariate polynomial

5 5 5 min(m - m',n - d > and

a homogeneous

exist such that p(z,y) = S(z,y)po(z,y) and q(z,y) = S(z,y)qo(z,y) satisfy (2.14.). Proof Since

is computed from a solution of (2.14.), we may consider nontrivial polynomials pl(z, y) and q l ( z , y) and write

with

p1

and 41 satisfying (2.14.) and with

c

tij zi yJ

+ wqo

2 mn

aT

T(z,y) =

i+j=wT

a bivariate polynomial.

Clearly wql = w T

and thus

w T = mn - wqo with

5

2 0.

+8

11.6.2. Block structure of the multivariate Pad4 table Also

wT

5I~T

with

which implies

O

585

min(wc - m',n - n’)

Write i+ j=mra--wgc+s

Because

the proof is completed. Theorem 2.25.

If the Pad4 approximant of order

(m, 92) for

f m , n ( Z 1 Y)

j(z,yf is given by

Po

= --(x, Y) Qo

and mr = 8 p o - wqo and nf = ayo - wqol then: (a) w ( f q o -PO) = wqo (b) for k and fk,L(XI

(c) m

5 rn’

+ mr + nr+ t + 1 with t 2 0

!. satisfying m’ 5 k !?- n’) and

5k5

mr

+t

and n’

5 t 5 n’

-t t we define

For p ( z ) = p o ( z ) t ( z ) and q ( z ) = qo(z) t ( z ) we know from (a) that

Since k

+ ! 5 m' + n' + t +

8

we have

(f q - p ) ( z ; )= 0 i = 0, . . ., k + 1

134

111.2.2. The table of rational interpolants

and thus

because

Since p o / q o is the irreducible form of p / q , where p and q satisfy (jq-p)(z;)=O

i = O ,...,m+ n

it is possible, according t o theorem 3.3., t o find an integer 0 5 R 5 mdn(rn- m , n - n ) such t h a t

The upper bound on 8 implies the upper bounds m respectively. This finishes the proof.

8

with

+ t and n + t on m and n I

Remark that condition (b) in theorem 3.4. is only necessary t o assure t h a t the integer t in (a) is as large as possible. It is important t o emphasize t h a t the table of rational interpolants has the block structure described above, only when the interpolation points (20,.. ., + )z, are ordered in such a fashion t h a t the points xi in which (I4 - P ) ( Z i ) # 0 are the last points in the interpolation set. For chosen m and n such a numbering can always be arranged and then we know that all the rational interpolants lying in a square emanating rrom rrnf,=!and with rmt+t,n~+tas its furthermost corner are equal to p o / q o . Let us take a look at the following example. Let 2; = i for i = 0 , . . . , 5 with j ( z 0 ) = I, f ( x 1 ) = 1, f(x2) = 5 / 3 , j ( z 3 ) = 5/2, f(z4) = 17/5, f(z.5) = 13/5. Then

III.2,S.Normality

Po(4

r3,2(x) = -~~

4015)

-

135

1 + x2 1+ z

and (f qo - pa)(x;) = 0 for d = 0 , . . ., 4, while (f qo - po)(xs) # 0. This implies that r2,1 = r2,2 = r3,l = r3,2. If t h e interpolation points are not numbered such t h a t the conditions of theorem 8.4. are satisfied then it is possible for 2 dements in t h e table of rational interpolants t o be equal without having a square of equal elements. A disturbance in the numbering of the interpolation points in theorem 3.4. implies a disturbance in the block-structure. Let us illlistrate this. For 2; = i when i = 0 , . . . , 3 arid f ( x 0 ) = 2, f ( x 1 ) = 312, f ( ~ 2 )= 4/5, f(z3) = 112 we find

q , o ( z ) = r2,1(x) = 2 - 2 X

but r?,o(z)

=2 -

2 x - --x 1 5 10

2

and

One can also check t h a t the rational interpolation problem of order (k,!) for f as formulated in 13.1.) has a solut,iou if the integers k and f are such that

and if the conditions of theorem 3.4. are satisfied. 2.3. Normality. Again we call an entry of the table normal if it occurs only once in t h a t table. A necessary condition for the normality of the rational interpolant rm,n(x) is formulated in the following theorem.

138

KII.2.9. No rmaltty

Theorem 3.5.

If rm,,, = @

Qo

is normal and if (f qo - p~)(z;)= 0 for z' = 0 , . . ., m'

+ n’, then

m = m' and n = n’ ( j q o -PO)(%;)

f 0 for i = m + n + 1, m + n + 2

(IQo - po)(z;)

= o for i = 0,.. .,m l +

we know that rm,n= r m r , n i . Hence, if rm,n is normal, we have m = m' and n = n’. Now suppose that (f qo - p ~ ) ( z , + ~ ~ +=~ 0. ) Then

This contradicts the normality of r,,,+. If (I90 - ~ 0 ) ( ~ m + n + 2= ) 0 then

satisfy

(1q - p)(z;) = 0

for i = 0 , . , ., m

+n+2

Thus rm," = rm+l,n+l which is again a contradiction with the normality of rm,n-

I

111.2.9. Normality

137

Conclusion (b) in theorem 3.5. does not imply tha t ( f q o - p o ) ( z i ) f 0 for i

2m

+n + 1

That the conditions (a) and (b) are not sufficient to guarantee normality of r m , n ( z ) is illustrated as follows. I d z; = i for i = 0 , 1 , 2 , . . . and f(z0) = 0, f(z1) = I, f(z2) = 3, f(z3) = 4, f(zi) = i for i = 4 , 5 , 6 , .. .. For rn = 1 and n = 0 we find r,,+(z) = z with (a) and (t,) in theorem 3.5. satisfied. But rnL,n is not normal because r m , n = r k , t for k 2 3 and e 2 2. However, it is possible t o formulate a sufficient condition for the normality of rm.=.

Theorem 3.6.

If rm,n = YO

with m = m', n = n' and (fq0 - po)(z;) = 0 for a t most m + n + 1 points from the sequence {s;}~,]N, then T ~ is normal. , ~ Proof Let us suppose that rm,nis not normal, in other words that rm,n = r h , t for k >_ m, t? 2 n and with k t- !. > m + n. According t o theorem 3.3. an integer 8 exists with 0 5 8 5 min(k - m, 1 - ti) and a points {yl, . . . , y e } exist such that, the polynomials

and

satisfy

Hence ( f q o - p o ) ( Z i ) = 0 for a t least k + t? + 1 - 8 points in (z;)~&. Since 6 is bounded above by k - m and t? - n, we conclude th a t k + L + 1 - 8 > m + n 1 which contradicts the fact th at ( j q o - p o ) ( z ; ) = 0 for at most rn + n + 1 points from {zi);Em I

+

III.3.1. Interpolating continued fractions

138

$3. Methods to compute rational interpolants.

In the sequel of this chapter we suppose t h a t every rational interpolant r,,=(z) itself satisfies the interpolation conditions (3.1.). This is for instance satisfied if min(m - m', n - n’) = 0. 9.1. Interpolating continued fractions.

Theorem 3.7.

If

and rm+k,n+C

with k,C

2. 0 then

a polynomial

du

U(Z)

P2

=42

exists with

5 max(k - 1, t - I)

and

where

Proof As we assumed, the rational functions ?-m,n and rrn+k,=+( both satisfy [3.1.): for d = 0 , . . .,rn + n

(fql

-p l ) ( s i ) =0

(fq2

- p 2 ) ( ~ ;= ) 0 for i

= 0 , . . .,m

+ k + n + t?

Consequently (P1q2

- P 2 Q l ) ( Z i ) = KfQ2 - P 2 ) q l ] ( Z i ) - [ ( f q l

-p1)q2](2;)

=0

111.3.2. InvcrRp difference8

139

+

i = O , . . . , m n and thus a polynomial ~ ( z ) exists such that ) t~(z)B,+,+l(z). It is easy t o see t h a t a u 5 max(k - 1 , 1 - 1) (PlQZ' - p z q l ) ( ~ = since a ( p 1 q z - pzqi) 5 max(m + n + k , m n L). I

for

+ +

If we consider the staircase of rational interpolants ~ k = { ~ k , 0 , r k + 1 , 0 ~ r k + l , i ~ r k + 2 , i I for ~ ~ ~ }k

2 0

it is possible t o compute coefficients d;(i 2 0) such that the convcrgents of the continued fraction do + d l ( 2 - 2 0 ) dk(2 - 20). . .(z - X k - 1 )

+ + I

.

.

are precisely the subsequent elements of

Tk.

Theorem 3.8.

If every three consecutive elements in Tk are different, then a continued fraction of the form (3.3.)exists with d k + , # 0 for i 2 1 and such t h a t the nthconvergent equals the (n + 1lth element of Tk.

The proof is left t o the reader because it is completely analogous to the one given for theorem 2.9. We shall now describe methods t h a t can be used to calculate those coefficients d;(i 2 0).

3.2.Inverse d#e re nces . Inverse differences for a function f givcn in G are defined as follows:

po[z] = f(z) for every

for every

2

in G

.

Z O , Z ~ , . .,zk

in G

140

111.3.2. Inverse daflerencea

w e call p k [ Z 0 7 . . .,zk] the kth inverse difference of f in the points 20,. . ., Z k . Usually inverse differences depend on the numbering of the points 2 0 , . . .,Z k although they are independent of the order of the last two points. If we want t o calculate an interpolating continued fraction of the form

(3.4.)

we have to compute the inverse differences in table 3.1.

Table 3.1. Po I201

Theorem 3.9.

If di = p;[so,. . ,,xi] in the continued fraction (3.4.), then the Cn of (3.4.) satisfies

ltth

Cn(z;) = f(zi) for i = 0 , . . .,n if C , ( z i ) is defined. Proof

From the definition of inverse differences we know t h a t for n

2 1:

convergent

III.8.2. Inverse d i f e r e n c e s

141

With d; = cpi[xo,.. . l z;] it follows that C,, satisfies the imposed interpolation 1 conditions. The continued fraction po[zo]

2-20 2-z1 + ( c P l [ ~ O , X l+l-JP2[2UJ21,221I+ ... --

is called a Thiele interpolating continued frsction. To illustrate this technique we give the following exarnple. Consider the data: z i = i for i = 0, . . .,3, f ( ~=)1, f ( z l ) = 3, /(Q) = 2 and f ( x 3 ) == 4 . We get

3

1/2

4

1

4

3/10

The rational function

1

I

2 2-11 I+ + ___ + 2-21

I1/2

213

13/10

= -5. x 2 - 5 x - 6 42 - 6

= r(z)

142

III.3.2. Inueree differences

indeed satisfies r(zi) = f(zi)for 4 = 0 , . . ., 3. In the previous example difficulties occured neither for the computation of the inverse differences nor for t h e evaluation of r(z;). We shall illustrate the existence of such computational difficulties by means of some examples. Consider again the data: 2 0 = 0, z1 = I, z2 = 2 with f ( s 0 ) = 0, f(z1) = 3 = f ( ~ 2 ) . Then the table of inverse differences looks like 0 3

113

3

213

Hence

3

I

I

is not defined for x = 20, and thus we cannot guarantee t h e satisfaction of the interpolation condition r(x0) = f ( 2 0 ) . If we cousider the data: zi = i for i = 0 , . . . , 4 with ~ ( Z O= ) 1, f(z1) = 0, f ( z 2 ) = 2, f(23) = -2 and f ( q ) = 5 then p2[z0,21,23] is not finite. This does not imply the nonexistence of t he ratioual interpolant in question. A simple permutation of t h e interpolation d a t a enables us t o continue the computations. For zo = 0, z1 = 2, 22 = 1, z3 = 3 and z 4 = 4 we get

I

2

2

0

-1

113

-2

-1

-113

-3

5

z

-2

-917

The rational function

7/12

III. 8.9. Reciprocal differencea

143

satisfies r(zi) = f(z;) for d = 0,.. .,4. In order t o avoid this dependence upon the numbering of the data we will introduce reciprocal differences. 9.9. Reciprocal diferences.

Reciprocal differences for a function f given in G are defined as follows :

po[z]= f ( z ) for every z in G

w e call P k ( Z 0 , . . ., z k ] the kth reciprocal difference of the function f in the points 20,.. ., Z k . There is a close relationship between inverse and reciprocal differences as stated in the next property.

Theorem 3.10.

For k

2 2 and for all

. ., Z k in (7:

20,.

Proof

The relations above are an immediate consequence of the definitions. This theorem is helpful for the proof of the following important property. Theorem 3.11. pk(201..

.,Z k ] does not depend upon the numbering

of the points

X O , . . ., Z k .

I

144

III.3.9. Reciprocal diffe T ence 8

Proof We consider the continued fraction (3.4.)and calculate the kth convergent by means of the recurrence relations (1.3.):

i = 1, ..., k

For even k = Zj,this convergent is of the form

+ a l z + . . . + 0 j Z’ bo + 6 1 2 + . . . + b j z3

a0

____I____

and for odd k = 2 j - 1, it is of the form

+ . . . + a j zi+ biz + . . . + b3-1. zi-’

ao -t- a12

_ _ I _ -

bo

In both cases we calculate the coefficients of the terms of highest degree in numerator and denominator, using the recurrence relations for the kfh convergent and the previous theorem : for k even we get

and for k odd

Since Pk[%o,. . ., zk] appears to be a quotient of coeflicients in the rational interpolant, i t is independent of the ordering of the 20, .. ., z k because the rational interpolant itself is independent of th at ordering. I

111.3.3. Reciprocal differences

145

The interpolating continued fraction of the form (3.4.) can now also be calculated as follows: compute a table o f rcciprocal differences and put do = p o [ z o ] , d l = p 1 [zo, zl]and for i 2 2: d, =; p t [ z o , . . ., z z ]- p r - 2 [ z ~ , .. ., zi-21. U p t o now we have only constructcd rational interpolants lying on the descending staircase To. To calculate a ration:tl interpolant on T k with k > 0 one proceeds as follows. Obviously it is possiblr to construct a continued fraction of the form

(3.5.) whose convergtwts are the e l e m r n ~ sof ?i, Clearly CO,. . ., c k + l are the divrdtd difrerences f[zoj,. . f[z0, . , z k + l ] since rk,O and r k + l , O , the first two coriwrgents, are the polynomial interpolants for j of degree k and k + 1 respectivc3ly If we want t o calculate for instance r k + i t we need the (2!Jth convergent of ( 3 5.) In order to compute the coefkienl,s d k + * for = 2, . . .,2e we write .j

To define

8

we proceed as follows. Thc conditions rk+l.l(z;) = j ( z ; )

imply t h a t

So

A(Z)

8

for i = 0,.. ., k

+ 2t?

must satisfy

is the ( Z t -

convergcrit of the continued fraction

III.3.4.A generalization

146

of the qd-algorithm

Hence 8(z) belongs t o t h e descending staircase To in the table of rational interpolaats for the function -.q

f-P

As soon as t h e coefficients C O , . . ., c k + l are known, the function q / ( f - p) can be constructed and inverse or reciprocal diiferences for it can be computed. The coeficients dk+i with i 2 2 are preciscly those inverse differences. So finally the computation of an clement i n T k for f is reduced to the computation of an c l e m mt in 7’0 for 4_ _ I - P 3.4. A generalization of the qd-algorzthm.

Consider continued fractions of the form gk(2)

= co

+

c k

i= 1

c;(.

- 2 " ) ( 2 - el). . .(z

- Zi-1)

(3.6.)

Theorem 3.12.

If every three consecutive elements in Tk are different, then a continued fraction of the form (3.6.) exists with c k + J # 0, q?+') # 0, e y + ' ) # 0, 1 + q y + ' ) ( z o - z k + 2 ; - , ) # 0, 1 + e[,k+')(zo- ~ k + ~ # ; ) 0 for a 2 1 and such element of Tk. t h a t the nth convergent equals the ( n

+

IIZ.3.4.A generalization

of

the qd-algorithm

147

Proof

For t h e elements in Tk we put rk+i,j

=

Pk+i,j --__

qk+i,j

for i = j , j + l a n d j = 0 , 1 , 2 , . .

and for the convergents of gk(Z) we put, prl C, = -- for n = 0 , 1 , 2 , . . . with

&O

= &I = 1

Q R

lJsing theorem I .4. a continued fraction with nth convergent equal to

I:,

(2

r,

( n 2 0)

is, alter a n equivalence transforIr\ixLron,

(3.7.)

For

we find by means of theorem 3.7. (,hat f o r

. .(.

C k + 1 ( 5 - Z ).

+

with

Ck+1

1------r

# 0 since rk,o # r k + l , o .

-

1,

Zk)l

2

1,

’2-2

ai(z - z k + i ) /

148

III.S.4.A generalization of the qd-algorithm

For (3.7.) it is even true t h a t

we find t h a t (3.7.) can be written as (3.6.).

I

To calculate the coefficients qLF+') and in (3.6.) one can use the following recurrence relations. Compute the even part of the continued fraction g k ( 2 ) and the odd part of the continued fraction gk-l(z). These contractions have the same convergents r k , O , r k + l , 1 , r k + 2 , 2 , . . . and they also have the same form. In this way one can check [2] that: for k 2 I

III.3.5. A generalization of the algorithm of Gragy

149

These coefficients are usually ordered as in the next table

Table 3.2.

Again the superscript denotes a diagonal i n the table and the subscript a column. Another qd-like algorithm exists for continued fractions of another form than the one given in (3.6.). Although it is computationally more efficient, it has less interesting properties and so we do not mention it here but refer t o [3].

3.5. A generalization of the algorithm of Gragg. The previous algorithm generalized the qd-algorithm and calculated elements on descending staircases. We can also generalize the algorithm of Gragg and calculate rational interpolants on ascending staircases [3]. To this end we assume normality of the table of rational interpolants. Consider for k 2 1 t h e staircase

and continued fractions of the form fk(2) = c g

+

c k

i= 1

C;(.

- 2 0 ) . . .(z - . & I )

-

Ck(2

- 20).

. . ( z - Zk-1)j 1 (3.8.)

150

111.8.5.A generalization of the algorithm of Gragg

Similar to theorem 3.12. one can prove that there exist coefficients f?) # 0 and 8:b) # 0 such t ha t the successive convergents of f k are the elements of s k , as soon as three consecutive elements of s k are different from each other. Making use of the relations existing between neighbouring staircases sk and S k + l we get the following recurrence relations: for k 2 1

f (1k ) - Ck-1 ck

(3.9.)

The coefficients

fr)

and

can be arranged in a two-dimensional table. Table 3.3.

f‘,“’

...

f I”

f I“’

...

Each upward sloping diagonal contains the coefficients which are necessary t o construct the continued fraction (3.8.). It is easy t o see th a t the formulas (3.9.) reduce to the corresponding algorithm of Gragg for the calculation of Pad6 approximants in case all the interpolation points coincide with the origin.

III.S.6. The generalized c-algorithm

151

3.6. The generalized c-algorithm. Let us again consider two neighbouring staircases Sm+nand Sm+n+l.Each of them can be represented by a continued fraction of the form (3.8.). The successive convergents of the continued fraction constructed from Sm+ncan be obtained by means of the forward recurrence relations (1.3.). If we write [4] rm,n=

Pm,n -~ Qm,n

then

(3.10.)

and (3.11.)

Consequently, using (3.10.),

Using (3.10.) and (3.11.) we get

152

111.3.6. The generalized

E-

algorithm

Combining these two relations, we obtain

Performing analogous operations on Sm+n+lwe obtain

Using this result it is possible to set up the following generalized E- algorithm [4], in the same way as the c-algorithm for Pad6 approximants was constructed from the star identity (2.8.) :

(")=o

6- 1

(-n-l)

€2 n

$)

=O

= rm,o(z)

m=O,l,

...

n=0,1,

...

m = 0 , 1 , . ..

111.3.7.Stoer ’5 recursive m e t h o d

153

9.7. Stoer’s recursive m e t h o d .

The use of recursive methods is especially interesting when one needs the function value of an interpolant and not the interpolant itself. Several recursive algorithms were constructed for the rational interpolation problem, one of which is the generalized ealgorithm. Other algorithms can be found in [lo, 16, 221. We shall restrict ourselves here t o t h o presentation of the algorithm described by Stoer. Let m p!&(z) =

c

ai

xi

i= 0

be defined by

i= 0

in other words, they solve the interpolation problem (3.2.) starting a t z, and let a& and b!$n indicate the coefficients of degree m and n in the polynomiais p!& and qk!n respectively. The following relations describe the successive calculation of the rational interpolants lying on the main descending staircase

with and

154

111.3.7.Stoer's recursive method

Proof We will perform the proof only for the first set of relations, because the second (i) and part is completely analogous. In case one wants to proceed from pn,n-l px:A)l to p?)- the degree of the numerator may not be raised. The coefiicient of the term of degree n + 1 in the right hand side of (3.12.) is indeed

To check the interpolation conditions in xi for i = j,. . ., j + 274 we divide the set of interpolation points into three subsets: (a) ( f q ( , i - pk),Jx,)

= -(xi - x

G+l)

3+2n) an,n-l

(b) (fq(,', - px',)(x;) = 0 for i = j + 1 , . . ., j

(d

(i)

(fqn,n--l - p n , n - - l ) ( Z i ) = 0

+ 2n - 1 since

Again these relations can easily be adapted for the calculation of rational interpolants on other descending staircases. To calculate the interpolants in

one starts with

1119.7.Stoer’8 recursive method

155

where the c; are divided differences of f. To calculate the interpolants in

one starts with

where the wi are divided differences of l / f . As for Pad4 approximants, one can also give explicit determinantal formulas for the numerator and denominator of rm,Jz). We will postpone this representation until the next section.

156

III.4.1. Definition of rational Hermite interpolants

54. Rational Hermite interpolation.

4.1. Definition of rational Hermite interpolante. Let the points ( Z , } ~ ~heNdistinct and let the numbers s;(i 2 0) belong to IN. Assume t ha t the derivatives f(')(z;) of the function f evaluated a t the point 2; are given for f = 0, . . ., 8 ; - 1. Consider fixed integers j , k, m and n with

i m+n+l=Ca; i- 0

+

k

The rational Hermite interpolation problem of order (m, n) for f consists in finding polynomials m

and

n

q(z) =

C

b; zi

with p / q irreducible and satisfying

In this interpolation problem 8i interpolation points coincide with z;, so 8; interpolation conditions must be fulfilled in 2;. Therefore this type of interpolation problem is also often referred t o a5 the osculatory rational interpolation problem 1211. In case 8; = 1 for all d 2 0 then the problem is identical t o the rational interpolation problem defined at the beginning of this chapter. In case all the interpolation conditions must be satisfied in one single point 20 then the osculatory rational interpolation problem is identical to the Pad6 approximation problem defined in the previous chapter.

III..&1. Definition of rational Hermite interpolants

157

Instead of considering problem (3.13.) we can look at the linear system of equations

(fQ- P ) ( " ( . i )

=0 (14 - P ) ( L ) ( z i + l= ) 0

for -! = 0,. . .,8 ; - 1 with i = 0 , . . . , j for C = O , ..., k - 1

(3.14.)

and this related problem always has a nontrivial solution for p ( z ) and q(z), since it is a homogeneous system of m + n 1 equations in rn + n + 2 unknowns. Again distinct solutions have the same irreducible form p o l 4 0 and we shall call

+

PO

rm,n = --

40

where 90 is normalized such that qO(z0) = 1, the rational Hermite interpolant of order (rn,n) for f. The rational Hermite interpolation problem can be reformulated as a NewtonPad6 approximation problem. We introduce the following notations: y~ = zo for -! = 0 , . . ., 80 - 1 y,qi)+l = z i for C = 0 , . . ., 8 ; c i i = 0 for i > j c i j = f[yi,. . .,t/i] for i 5 j

- 1 with d ( i ) = 8 0

+ 81 + . . . + 8i-1 (i 2. 1)

with possible coalescence of points in the divided difference f [ y i , . . .,g j ] . If we put

with

Bj(4 =

n

i

(2

- YL-1)

then formally

This series is called the Newton series for f. Problem (3.14.) is then equivalent with the computation of polynomials

158

111.4.1.Definit i o n of ratio no1 He rmit e interpol ant8

and

such that

Problem (3.15.) is called the Newton-Pad6 approximation problem of order (n, n) for f . To determine solutions p and q of (3.15.) the divided differences di

= (f q - p ) ( y o , .

..,yi]

for i = 0 , . . .,m+

must be calculated and put equal t o zero. The following lemma, which is a generalization of the Leibniz rule for differentiating a product of functions, is a useful tool.

Lemma 3.1.

For the proof we refer t o 119). Using lemma 3.1. it is now possible t o write down the linear systems of equations that must be satisfied by the coefficients a; and b; in p and q : COO COI

bo = QO

60

combo

+ c11 61 =

+

Clmbl

+ ...+

(3.16a.) cmmb,

= a,

(3.16b.)

III.d.2. The table of rational Hermite interpolant8

159

Since the problems (3.14.) and (3.15.) are equivalent, the rational function rm,= can as well be called the Newton-Pad6 approximant of order ( m ,n ) to I. In the same way as for the rational interpolation problem the following theorem can be proved. Theorem 3.14.

The rational Hermite interpolation problem (3.13.) has a solution if and only if the rational Hermite interpolant rm,== po/qo satisfies (3.14.). 4.2. The table of rational Hermite interpolants. Once again we will order the interpolants rm,nin a table with double entry: r0,o

r0,1

f0,2

...

r3.0

For a detailed study of the structure of the rational Hermite interpolation table we refer to [3].We will only summarize some results. They are based on the following property. Theorem 3.15.

If the rank of the linear system (3.18b.) is n - t then (up t o a normalization) a unique solution P(z) and q(z) of (3.16.) exists with

ap 5

m-t

aqj

The proof is left as an exercise (see problem (5)). Again one can see that in case all the interpolation points coincide with one single point, these determinant formulas reduce to the ones given in the chapter on Pad6 approximants since the divided differences reduce to Taylor coefficients.

4.4. Continued fraction repreeentation. If one considers staircases

in the table of rational Hermite interpolants, one can again construct continued fractions of which the successive convergents equal the elements of Tk.It is easy to see that these continued fractions are of the form

+ - dk.+2(Z \

- Yk+l)I I

&+dZ

+/

- Yk+2)l

+...

The coefficients 6,. . .,dh+l are divided differences (with coalescence of points) and the other coefficients can still be obtained using the generalized qdalgorithm. The generalization of Gragg's algorithm and the generalized calgorithm also remain valid for the calculation of rational Hermite interpolants.

164

III.4.5. Thiele's continued fraction ezpansion

4.5. Thiele '8 continued fraction ezpansion. From theorem 3.9. we write formally

(3.17.)

We consider now the limiting case z ; -+ xo for i

p,(z) =

21 .

C P ~ ( ~ O ,* J. zjl

z; lim -+ 2 ...,j

a=o,

Then (3.17.) becomes I

I

I

This is a continued fraction expansion of around 5 0 . Formula (3.18.) is obtained from (3.17.) in the same way as we set u p a Taylor series development from Newton's interpolation formula. Since (3.18.)is formal, one has t o check for which values of z the righthand side really converges t o f(z). We can calculate the p j ( z ) using Thiele's method [17] :

and with

III.4.5. Thiele '8 continued fraction ezpunaion

165

we have

Now p j - l [ z o , . . . , z j - l ] does not depend upon the ordering of the points 20

,...,xj-1. s o

Consequently (3.19a.)

To calculate

one uses the reIationship

(3.19b.)

We apply the formulas (3.19.) to construct a continued fraction expansion of f ( z ) = ez around the origin: P o ( % ) = e"

tpl(z) = e-"

iPr(z0)

=1

pl(z) = e-"

pz(z) = -2e"

'pZ(Z0)

= -2

pZ(z) = -ez

p3(z) = - 3 ~ "

'p3(ZO)

= -3

p4(z) = 2e"

'p4(ZO)

=

= 5e-"

'P5(z0)

=

p3(z)

= -2e-=

P 4 ( 4 = e2

p5(2)

188

So we get

111.4.5. Thiele’e continued fraction ezpanaion

111.5. Convergence of rational Hermite interpolanta

167

55. Convergence of rational Hermite interpolants. The theorems of chapter I can be used to investigate the convergence of interpolating continued fractions. We shall now mention some results for the convergence of columns in the table of rational Hermite interpolants. r2,o, ] . . .}, in other words The first theorem deals with the first column { r o , o ,q , ~ it is a convergence theorem for interpolating polynomials. For given complex points (20,. , z j } we define the lemniscate

.

~ ( z o , ..,z,, t) = { z E C

1

~ (z zo)(z - 2 1 ) . . .(z - z,)l = r}

Broadly speaking, the convergence of an arbitrary series of interpolation does not depend on the entire sequence of interpolation points y; (as defined in the Newton-Pad6 approximation problem) but merely on its asymptotic character, as can be seen in the next theorem.

Theorem 3.18. Let the sequence of interpolation points (yo, y1 ,~ sequence

>it

Vk(j+l)+i

. .) be

2 , .

asymptotic t o the

= zi

for i = 0,.. . , j . If the function j(z) is analytic throughout the interior of the lemniscate B(z0 , .. ., z,, r) then the rm,0 converge to f on the interior of B(z0 , . . .,z j , r). The convergence is uniform on every closed and bounded subset interior to B(z0,. . . ] z j , r).

For the proof we refer to [20 p. 611 and 19 pp. 90-911. Let us now turn t o the case of a meromorphic function f with poles 2 0 1 , . . ., wn (counted with their multiplicity). For the rational Hermite interpolant of order (m,n) we write Pm,n

rm,n = __

4m,n

and for the minimal solption of the Newton-Pad4 approximation problem of order (m,n) we write fS,,,(z) and qm,n(z).

168

111.5. Convergence of rational Hermite inderpolanta

Let the table of minimal solutions for the Newton-Pad4 approximation problem be normal. According to [6] we then have &j,,+ = n. Let wim) ( (i = 1 , . . .,n) be the zeros of qm,n for rn = 0 , 1 , 2 , . . - and let p i = I(wi - Z O ) ( W ; - 2 1 ) .. .(wi - z j ) l with 0 < p1 5 p 2 5 . . . 5 pn 5 at < r for a positive constant a.

Theorem 3.19. If the sequence of interpolation points (yo, yl, 112, . . .} is asymptotic t o the sequence { Z O , Z ~ ,...,zj,zo,z1, . . .,zj,20, 21,. . .,z j , . . .}, if f is meromorphic in the interior of B(z0,. . ., zg’, r) with poles w 1 , . . , , w , counted with their multiplicity and if the table of minimal solutions for the Newton-PadB approximation problem is normal, then

+

= w , o(am>

i = 1, ..., n

and

uniformly in every closed and bounded subset of the interior of B(z0, . . .,z j , r) not containing the points ~ 1 , ...,W n . The proof is given in [4].

III. 6.1. Interpolating branched continued fraction8

169

86. Multivariate rational interpolants. We have seen t h at univariate rational interpolants can be obtained in various equivalent ways: one can calculate the explicit solution of the system of interpolatory conditions, start a recursive algorithm, calculate the convergent of a continued fraction or solve t he Newton-Pad6 approximation problem. We will generalize t h e last two techniques for niiiltivariate functions. These generalizations are written down for t he case of two variables, because the situation with more than two variables is only notationally more difficult. More details can be found in [7] and [8]. 6.1, Interpolating branched continued fractions.

Given two sequences { Z O , Z ~ , Z ? , . . .} and {yo,y1,y2 ,...} of distinct points we will interpolate the bivariate function f ( z , y ) at the points in (zo, X I , ~ 2 , . ..} X { y o , y l , y 2 , . . .}. 'TO this end we use branched continued fractions symmetric in t h e variables z a n d y and we define bivariate inverse differences as follows:

170

MI. 6.1. Interpolating branched continued fractions

Theorem 3.20. (3.20.)

with

Proof From theorem 3.9. we know that

Let us introduce the function gO(z, y) by

where

By calculating inverse differences

(0)

for go we obtain

III.6.l. Interpolating branched continued fractions

1

where ho(z,y) =

&I

yo, y]. So already

By computing inverse differences R (j 0, k) for ho we get

It is easy to see that

fl

- I/n

171

172

III.6.I . Interpolating branched continued fractione

From this we find by induction that

So we can write

where

If we introduce inverse differences which provides us with a function

C:l for hl

g 1 we can repeat the whole reasoning and inverse differences ?rj,k: (1 1

III. 6.1.Interpolating branched continued fraction8

173

In the same way as for ho we find

Finally we obtain the desired interpolatory continued fraction.

I

To obtain rational interpolants we are going to consider convergents of the branched continued fraction (3.20.). To indicate which convergent we compute we need a multi-index

The Fith convergent is then given by

with

For these rational functions the following interpolation property can be proved.

III. 6.1. Inte t p o lat ing branched c o ntinue d fr actt o )a8

174

Theorem 3.21. The convergent CK(Z,y) of (3.20.) satisfies C d Z t l , YC,) = I(ZC, t YC,)

for

( e l , &)

belonging to

Proof Let C = min (&, Cz). From theorem 3.20. we know that

where

Now f(%, yt,) = CE(ZC,, yt,) if and only if the following conditions are satisfied Cln

Q0

5 i 5 L : L1 I

mi,

and Lz 5 mi,

This is precisely guaranteed by saying ( t i , &) E I .

I

;...-

111.6.2. General order Newton-Pad6 approzimant8

175

For instance, if moz 2 m l , 2 ... 2 mnz and mOy 2 m l y 2 ... 2 mny the . as we can boundary of the set I is given by n = ( n , m o z J m o y.,.,mnr,mny), tell from the next picture which is drawn for n = 2.

m1y

mZy

-. I:.

0

-*-.

-. . 4

*

4

0

0

0

~

m2x

m1x mOx

Figure 3.8.

We illustrate this technique with a simple numerical example. Let the following d ata be given: z; = i for i = 0 , 1 , 2 , . . . and yg = j for j = O,1,2,. .. with f ( z i , Y i )= (i + j)'. Take A = ( 1 , 2 , 2 , 1 , 1 ) . Then we have to compute

The resulting convergent is

6.2. General order Newton-Pad6 approzimants.

Consider two sequences of real points { Z ; ) ~ ~ N and { Y , } ~ ~where N coalescent points get consecutive numbers. For a bivariate function f ( 2 , I/)we define the following divided differences

176

111.6.2. General order Newton-Padk approzimanta

(3.21a.) or equivalently

One can easily prove that (3.21a.) and (3.21b.) give the same result. When the interpolation points z;,. . ., zi+ri-l and yj, . . .lgj+sj-l coincide, then one must bear in mind that

We consider the following set of basis functions for the real-valued polynomials in two variables:

This basis function is a bivariate polynomial of degree i

+j .

III.6.2. General order Newton-Pad6 approzimants

177

With c k ; , J j = f [ z k , . . .,z i ] [ y c , .. ., yj] we can then write in a purely formal manner [l pp. 160-1641

C

f(z,1/> =

coi,oj

B i i ( z ,YI

(i,j)EINa

The following lemmas about products of basis functions B i j ( z ,y) and about bivariate divided differences of products of functions will play an important role in the sequel of the text.

Lemma 3.2. For k

+ L 2 i + j the product B i i ( ~ , y&(z,y) ) p=o

=

u-0

Proof We write B i j ( ~v ,) = Bio(2,v ) B o j ( z ,Y). Since Bio(z,y) is a polynomial in z of degree i we can write

and

with the convention th at an empty product is equal to 1. Consequently

u=o

p=o

which gives the desired formula if we put, ,A

= a,&.

178

111.6.2. General order Newton-Pedt apptoziments

A figure in IN2 will clarify the meaning of this lemma. If we multiply B;j(z,v) by &(Z, g) and k + l >_ a + J' then the only occuring Bpu(z,v ) in the product are those with ( p , v ) lying in the shaded rectangle.

I

k

i

k+i

Figure 3.3.

Lemma 3.3.

The proof is by induction and analogous t o the proof of th e univariate case. The definition of multivariate Newton-Padh approximants which we shall give is a very general one. It includes the univariate definition and the multivariate Pad6 approximants from the previous chapter as a special case as we shall see at the end of this section. With any finite subset D of IN2 we associate a polynomial

Given the double Newton series

111.6.2. General order Newton-Pad4 approzimanta

179

with c o i , ~ = , f[z0, . . ., z ; ] [ y o , .. ., y j ] ] we choose three subsets N , D and E of IN2 and construct an “ / D I E Newton-Pad4 approximant to f ( z , y ) as follows:

C

p(z,y) =

(i,j)EN

q(z,y)=

q j ~ ; (z,y) j b;j Rij (2, y )

( i , j W

(f Q - P)(z, y ) =

C

( N from “numeratorn)

(3.22a.)

(Dfrom “denominator”)

(3.22 b.)

d i j Bij (2,

Y)

( E from “equatiom”)

(3.22c.)

(i,j)EN2\E

We select N , D and E such that

D has n + 1 elements, numbered ( i o , j o ) , . . ., (in,in)

N C E E satisfies the rectangle rule, i.e. if (i,j) E E then (k,L) E E for k 5 iand L 5 j E\N has at least n elements.

Clearly the coeBcients d;, in

(f q - P N Z , Y ) =

C

dij

Bij(z,y)

(i,i)ENa

are dij

=(~~-P)[zo,...,z~~[Yo,...,Y~~

So the conditions ( 3 . 2 2 ~are ) equivalent with

(.f q - P)[zo,. . ., ~ i ] [ y o ,. ..,i/i] = 0 for (i,j) in E

(3.23.)

The system of equations (3.23.) can be divided into a nonhomogeneous and a homogeneous part:

( I q ) [ z o , . Z i ] [ y ~ ,..., Y j ] = P [ ~ o ., .,. Z i ] [ ~ o ,...,y j ] (f q)[zo, . . ., z ; ] [ y o ,. . ., y j ] = O for (i,j ) in E\N a,

Let’s take a look a t the conditions (3.23b.).

for ( i , j ) in N

(3.23a.) (3.23b.)

111.6.2.General order Newton-Pad6 approximants

180

Suppose that E is such t h a t exactly n of the homogeneous equations (3.23b.) are linearly independent. We number the respective n elements in E\N with ( h l ,k l ) , . . ., (hR?k,) and define the set = { ( h l i I),

.‘

. J (hnl

‘n)}

c E\N

( H from “homogeneous equations”)

By means of lemma 3.3. we have

Since the only nontrivial q [ z o , . . zP][yo,. . .,yu] are the ones with ( p , v) in D we can write .J

Remember that f [ z C c J . ,. z i ] [ y U ,..,yj] . = 0 if p > i or v > j. So the homogeneous system of n equations in n + 1 unknowns looks like

because

D

= {(iO, ~ O ) J * . * J (in?jn))

As we suppose the rank of the coefficient matrix t o be maximal, a solution q(z, y) is given by

111.6.2. General order N e w t o n - P a d 6 approximanta

181

By the conditions (3.23a.) and lemma 3.3. we find

Consequently a determinant representation for p(z y) is given by

(3.25b.)

If for all k, L 2 0 we have Q(ZkJYc)# 0 then

$(z,g) can be written as

with e i i = +[so,.. ., z i ] [ y o , .. yj]. Hence by the use of lemma 3.2. and since E satisfies the inclusion property .)

The following theorem describes which interpolation conditions are now satisfied by P / 9 .

182

111.6.2. General order Newton-Pad4 approzimanta

Theorem 3.22.

If

q ( Z k , yt)

# 0 for

( k , I!) in E then

where

If

r k = 1 = 6~

this reduces t o

Proof Given rk and 6~ for fixed (zk,yc), consider the following situation for the interpolation points, with respect to E

(3.26.)

I

I

, I

k Figure 3.4.

I

I 1

k+fk.1

I

*

111.6.2. Gener af order Nerut o n - f ad6 approzimant 8

and define

T

Using these d e h i t i o n s we rewrite I as

with

Because q ( z t , yt)

# 0 for ( k , l ) in R we have

183

IIl.S.2. General order Newton-Pad4 approximonte

184

To check the interpolation conditions we write

apf” R ‘J. . = p t - u ( B ~Boj) o

-

ax@ dy”

If we cover N2\E with three regions

because B;a(z,y) contains a factor (x - zk)pE+l, and

au

B~~

ay”

l(zkrYl)

= 0 for (i,j ) in B and

(I(,

v) in I

because Boj(x, y) contains a factor (y - Y $ ‘ E + ~ . Analogously

a’ B~~ axp

l(x*iYl)

= 0 for (i,j)in

C and

( p , v ) in I2

The most general situation for the interpolation points with respect to E is slightly more complicated but completely analogous to the one given in (3.26.). We illustrate this remark by means of the following figure:

111.6.2. General order Newton-Pad6 approzirnants

185

I

I

I

I

The proof in this case is performed in the same way as above.

I

We will now obtain the determinant representation given in theorem 3.17. for univariate Newton-Pad6 approximants, from the determinant representations (3.25a.) and (3.25b.). Consider the Newton interpolating series for f ( x , 0) and choose

+

If the points {(k,O) I m + 1 5 k 5 m n } supply linearly independent equations, then the determinant representations for p(z, 0) and q(z, 0) are

186

111.6.2.General order Newton-Pad6 approzimunts

We c a n also obtain the multivariate Pad6 approximants defined in the previous chapter as a special case. One only has t o choose

D = { ( i , j )I m n 5 i + j 5 mn+n} N={(i,j)Imn~%’+jImsa+m} b’= { ( i , j ) 1 mn 5 i t j _< mn + m t n }

because when all the interpolation points coincide with the origin, then

Bij

( 2 , ~= ) Z ’

yi

Let us now illustrate the multivariate setting by calculating a Newton-Pad4 approximant for 2 f ( z , ? / )= 1 + ___ sin(xy) 0.1 - y

+

with y j = ( j- 1)fi

i = 0 , 1 , 2 ' ... j = 0,1,2,. . .

The Newton interpolating series looks like 10 f(Z,YI = 1 + x+ zb + d3 O.l+fi O.l+&

Choose

111.6.2. General order Newton-Pad4 approzimants

Writing down the system of equations (3.23b.), it is easy t o check that

LI

= {(2? ')I

('1

2))

The determinant formulas for p(z, V ) and q(z, v ) yield 1 q ( z , f l )=

-

V-tJ;;

c02,01

c12,11

c01,02

L'11,02

c02,11

c01,12

100 1 0.01 - T (1 - 0.1 fi(Y

+

with

1

1

Finally we obtain

- 0.1+2-y 0.1 - y

+ d4)

187

IH. Prob lema

188

Problems. Let

o rm,= = P~40

be the rational interpolant of order (m,n) for f ( z ) with m' = apo and fir = r3g.o. Prove th at there exist at least m' + nr + 1 points {yl , .. .,ye} from the points ( 2 0 , . . ., z ,+,} such th a t rm,,(vi) = f(vi) for i = 1 , . . ., 5 . Formulate and prove the reciprocal and homografic covariance of rational inter pol ants. Prove theorem 3.8. Which interpolation conditions are satisfied by the nth convergent of the continued fraction (3.4) if a) d, = 0 b) d, = (x, Prove theorem 3.17 Prove the following result for the error function (f - rm,,)(z). Let I be ~ . an interval containing all the interpolation points 5 0 , . . ., z ~ + Then 'dz E I ,

3y, E I :

Compute rZ,z(z) satisfying rZ,z(si) = f(q) for i = 0 , . . ., 4 and r1,3(z) satisfying rl,s(zi) = f(zi) for i = 0 , . . ., 4 with z; = d (i = 0 , . . .,4) and f(z0) = 4, f ( z 1 ) = 2, f ( z 2 )= 1, f(z3) = -1, f ( Z 4 ) = -4If f ( z ) is a rational function

with at 5 m and 8 s 5 n then k with 0 5 k 5 2 max(m, n).

'pk[Zo,.

. .,X k - 1 ,

21

is constant for a certain

111. P r o ble m8

(9)

If for some k, p k [ Z O , . . tion.

(10)

Compute

r2,1(2)

(i = 0 , . . ., 3) and

of

.)

zk-1,2]

189

is constant then f(z) is a rational func-

r 2 , 1 ( z ; ) = f(z;) for i = 0 , . . . , 3 with z i = i = I , f(z1)= 3, f(z2) = 2, f(z3) = 4 J by means

satisfying f(z0)

a) the generalized qd-algorithm

b) the algorithm of Gragg (11)

Construct a continued fraction expansion using Thiele's method for f ( z ) = l n ( l + z) around 3: = 0.

(12)

Calculate the inverse differences for gl(z,y) and 7r!.li for h l ( ~y), and perform one more step in the proof of theorem 3.20. in order to obtain the contribution b---zl)(Y - Y l ) B 2 ( 2 , Y)

(!:i

with B 2 ( Z 1 Y ) = Pz[% 2 1 , Z2l[Yo,

in the continued fraction (3.20.).

Y l , Y21

III. Remarka

190

Remarks. (1)

Instead of polynomials

m

i=O

and

n i-0

one could also use linear combinations m

i= 0

and

of basis functions (g;}iEN, which we call generalized polynomials, and study the generalized rational interpolation problem

A unique solution of this interpolation problem exists provided ( g i ( Z ) ) ; ~ N satisfies the Haar condition, i.e. for every k 2 0 and for every set of distinct points (20,. . .,zk} the generalieed Vandermonde determinant

Examples of such interpolation problems can be found in [lS]. A recursive algorithm for the calculation of these generalized rational interpolants is given in [HI.

III. Re m ar ka

(2)

19 1

The Newton-Pad6 approximation problem (3.15.) is a linear problem in t ha t sense t hat rm,ncan he considered as the root of the linear equation

where p and q are determined by the following interpolation conditions (q f - p ) ( z j ) = 0

+

j = 0 , . . .,m n

Instead of such linear equations one can also consider algebraic equations

where the polynomials pi of degree m;are determined by

More generally we can consider for different functions fo(z), . . ., fk(Z) the interpolation conditions

An extensive study of this type of problems is made in [14] and [lo]. (3)

Rational interpolants have also been defined for vector valued functions [ l l , 251 using generalized vector inverses. For other definitions of multivariate rational interpolants we refer to [17] and [S]: Siemaszko uses nonsymmetric branched continued fractions while in [8] Stoer's recursive scheme for the calculation of univariate rational interpolants is generalized t o the multivariate case.

111.Reference8

192

References. Computing methods I. Addison Wesley,

[

11 Berezin J. and Zhidkov N . New York, 1965.

[

A generalization of the qd-algorithm. J. Comput. Appl. 21 Clsessens G. Math. 7, 1981, 237-247.

[

A new algorithm for osculatory rational interpolation. 31 Clsessens G . Numer. Math. 27, 1976, 77-83.

[

41 Claessens G.

[

A useful identity for the rational Hermite interpolation 51 Claessens G. table. Numer. Math. 29, 1978, 227-231.

[

61 Claessens G. On the Newton-Pad4 approximation problem. J. Approx. Theory 22, 1978, 150-260.

[

71 Cuyt A. and Verdonk B. General order Newton-Pad4 approximants for multivariate functions. Numer. Math. 43, 1984, 293-307.

[

81 Cuyt A. and Verdonk B. Computing 34, 1985, 41-61.

[

91 Davis Ph.

Some aspects of the rational Hermite interpolation table and its applications. Ph. D., University of Antwerp, 1976.

Multivariate rational interpolation.

Intcrpolation and approximation. Blaisdell, New York, 1965.

Contribution B l’approximation de fonctions de la variable complexe au sens Hermite-Pad6 et de Hardy. Ph. D., University of Grenoble, 1980.

[ 101 Della Dora J .

[ 111 Graves-Morris P. and Jenkins C . Generalised inverse vector valued rational interpolation. In [22], 144-156. ( 121 Larkin F.

Some techniques for rational interpolation. Comput. J. 10, 1967, 178-187.

An algorithm for generalized rational inter[ 131 Loi S. and Me Innes A. polation. BIT 23, 1983, 105-117.

[ 141 Liibbe W.

Ueber ein allgemeines Interpolationsprobiem und lineare Identitaten zwischen benachbarten Losungssystemen. Ph. D., University of Hannover, 1983.

III. Reference8

193

151 MuhIbach G . The general Neville-Aitken algorithm and some applications. Numer. Math. 31, 3978, 97-110.

[ 161 Saleer H .

Note on osculatory rational interpolation. Math. Comp. 16, 1962, 486-491.

[ 171 Siemaszko W.

Thielp-type branched continued fractions for twovariable functions. J. Corn put. Appl. Math. 9, 1983, 137-153.

[ 181 Stoer J.

Ueber zwei Algorithmen zur Interpolation mit rationalen Funktionen. Numer. Math 3, 1961, 285-304.

[ 191 Thiele T.

Interpolationsrechnung. Teubner, Leipzig, 1909.

[ 201 Walsh J.

Interpolation and approximation by rational functions in the complex domain. Amer. Math. SOC.,Providence Rhode Island, 1909.

[ 211 Warner D.

Hermite interpolation with rational functions. Ph. D., University of California, 1974.

[ 221 Werner H . and Biinger H .

Pad6 approximation and its applications. Lecture Notes in Mathematics 1071, Springer, Berlin, 1984.

[ 231 Wuytack L. On some aspects of the rational interpolation problem. SIAM. J. Numer. Anal. 1 1 , 1974, 52-80. [ 241 Wuytack L.

On the osculatory rational interpolation problem. Math. Comp. 29, 1975, 837 - 843.

[ 251 Wynn P.

Continued fractions whose coefficients obey a noncommutative law of multplication. Arch. Rational Mech. Anal. 12, 1963, 273-31 2. Ueber einen Interpolations-Algorithmusund gewisse andere Formeln, die in der Theorie der Interpolation durch rationale Funktionen bestehen. Numer. Math. 2, 1960, 151-182.

[ 261 Wynn P .

195

.

CBAPTER N:Applications

5 1. Convergence 1.1. 1.2. 1.3. 1.4. 1.5.

The The The The The

acceleration . . . . . . . . . . . . . . . . . . . . univariate t-algorithm . . . . . . . . . . . . . . . qd-algorithm . . . . . . . . . . . . . . . . . . . . algorithm of Bulirsch-Stoer . . . . . . . . . . . . . p-algorithm . . . . . . . . . . . . . . . . . . . . . multivariate €-algorithm . . . . . . . . . . . . . .

197

. 197 205

. 209 213

. 216

$2. Nonlinear equations . . . . . . . . . . . . . . . . . 2.1. Iterative methods based on Pad4 approximation . 2.2. Iterative methods based on rational interpolation 2.3. Iterative methods using continued fractions . . . 2.4. The qd-algorithm . . . . . . . . . . . . . . . 2.5. The generalized qd-algorithm . . . . . . . . .

. . . . . 220 . . . . . . 220 . . . .

. . . . . 227 . . . . . 233 . . . . 233 . . . . . 236

53. Initial value problems . . . . . . . . . . . . 3.1. The use of Pad4 approximants . . . . . 3.2. The use of rational interpolants . . . . 3.3. Predictor-corrector methods . . . . . 3.4. Numerical results . . . . . . . . . . . 3.5. Systems of first order ordinary differential

. . . . .

. 238 . 241 . 243

. . . . .

. . . . .

. . . . .

. . . . .

. . . . . . . . . . . . . . . . . . . . equations . . . .

238

244

. 247

.

54 Numerical integration . . . . . . . . . . . . . . . . . . . . . 250 4.1. Methods using Pad4 approximants . . . . . . . . . . . . . 251 4.2. Methods using rational interpolants . . . . . . . . . . . . 252 4.3. Methods without the evaluation of derivatives . . . . . . . . 253 4.4. Numerical results for singular integrands . . . . . . . . . . 254

55 . Partial differential equations

. . . . . . . . . . . . . . . . . . 257

§6.Integral equations . . . . . . . . . . . . . . . . . . . . . . . 260 0.1. Kernels of finite rank . . . . . . . . . . . . . . . . . . 260 . . . . . . . . . . . . . . 262 6.2. Completely continuous kernels

Problems

. . . . . . . . . . . . . . . . . . . . . . . . . . .

265

Remarks

. . . . . . . . . . . . . . . . . . . . . . . . . . .

267

. . . . . . . . . . . . . . . . . . . . . . . . . .

268

References

196

“It i s my hope that by demonstratdng the e m e with which the various transformations may be effected, their field of application might be widened, and deeper insight thereby obtained into the problems for whoae aoh6tion the tran8formatdons have been uaed.

P . WYNN (1956).

- “On a

device for computing the e m ( & )

tran~f~rmation~’

Z V . l . l . The univariate €-algorithm

197

The approximations introduced in the previous chapters will now be used to develop techniques for the solution of various mathematical problems: convergence acceleration, numerical integration, the solution of one or more simultaneous nonlinear equations, the solution of initial value problems, boundary value problems, partial differential equations, integral equations, etc. Since these techniques are based on nonlinear approximations they shall be nonlinear thernselves. We shall discuss advantages and disadvantages in each of the sections separately and illustrate their use by means of numerical examples.

§I. Convergence acceleration. 1.1. The univariate r-algorithm. Consider a sequence { a ; } ; , ~ of real or complex numbers with lim ai = A

i-

00

Since we are interested in the limiting value A of the sequence we shall try to construct a sequence (b;),.,N that converges faster to A , or

We shall describe here some nonlinear techniques that can be used for the construction of { b;},N. Consider the univariate power series

i-t

with Vai = a; - a;-1. Then clearly for the partial sums

we have

Fk(1)= ak

k = 0, 1,2, . . .

If we approximate f(z) by r ; , ; ( z ) , the Pad4 approximant of order (i,i) for f , then we can put b; = r;,;(l)

i = 0 , 1 , 2 , . ..

198

I V . l . l . The univariate €-algorithm

For the computation of b; the c-algorithm can be used:

Then

The convergence of the sequence {biIiEm depends very much on the given sequence { a , } ; , N . Of course the convergence properties of { b ; } ; , ~are th e same as those of the diagonal Pad6 approximants evaluated a t z = 1 and for this we refer to section 4 of chapter 11. In some special cases it is possible to prove acceleration of the convergence of ( a ; } i E N . A sequence {a;}iEN is called totally monotone if

Ak u ; > O

d,k=0,1,2

,...

where A k a , = Ak-' ai+i - Ak-' ai and Ao a; = a;. In other words, {ai}iEN is totally monotone if

a0 2 a1 2 a2 2 ... 2 0 Aao 5 Aal 5 h a 2 5 . .. 5 0 A2ao 2 A 2 a l 2 A2a2 2 . . . 2 0 and so on.

IV.1.1. The uniuariate 6-algorithm

199

Note t ha t every totally monotone sequence actually converges. The sequences (Xi}i,-~ for 0 0 can be interesting because the asymptotic error constant C* may be smaller than when n = 0 [43]. The iterative procedures (4.3) and (4.4) can also be derived as inverse methods (see problem (5)). Let us apply Newton’s and Halley’s method t o the solution of

+ +

The root Z* = 0. We use xo = 0.09 as initial point. The next iteration steps can be found in table 4.13. Computations were performed in double precision accuracy (56-digit binary arithmetic).

222

IV.8.1. Iterative methods baaed on P a d 4 approzimation

Table 4.13.

i

Newton

Halley

Zi

2;

0

O.QOOOOOO0D-01

1

0.80997588D-01

2

0.6559965QD-01

3

0.43022008D-01

4

~.18500311D-01

5

0.34212128D-02

6

0.11703452D-03

7

0.13697026D -06

8

0.18760851D-12

9

0.35197080D-24

10

0.00000000D+00

0.90000000D -0 1 -0.40647645D-02 0.35821646D-06 -0.24514986D-18 0.00000000D+00

It is obvious that a method based on the use of (m,n) Pad4 approximants for f with n > 0 gives better results here: the function f has a singularity at z = 0.1. Observe that in the Newton-iteration zg is a good initial point in the sense that from there on quadratic convergence is guaranteed: 1zi+1

- 2.1 = 1zi+l I N c ' J z ~- z * I 2 for i 2

o

with C’ = 10. For the Halley iteration we clearly have cubic convergence from X I on. The formulas (4.3.) and (4.4.)can also be generalized for the solution of a system of nonlinear equations

which we shall write as

IV.d.1. Iterative methods based on Pad6 approzimation

223

Newton's method can then be expressed as [45]

where F’(xr),. . . ,xr)) is the Jacobiau matrix of first partial derivatives evaluated at (zy),. .. ,zt))with

Let us now introduce the abbreviations Fi = F ( z f ) ,. . . ,xk(i))

F , ! = F ( z (4 , ,... ,zk (4) To generalize Halley's method we first rewrite (4.4.) as

Then for the solution of a system of equations it becomes [I41

224

IV.2.1. Iterative method8 based o n P a d 6 appton'mdion

where the division and the square are performed componentwise and F " ( x 1 , ... ,x k ) is the hypermatrix of second partial derivatives given by

~~

a 2f k a2f k ... a2f k axkaxs ... axlaxk

dxlax2

.

.

I

az:

which we have t o multiply twice with the vector --F,!-'Fi. This multiplication is performed as follows. The hypermatrix F " ( x 1 , . . . , Z k ) is a row of k matrices, each le x k. If we use the usual matrix-vector nlultiplication for each element in the row we obtain a2fi

c

a2fk

c

k

asl ax;Yi ... i= 1 axlax;

Yi

...

k

i= 1

jin) vk

In [14] is proved that the iterative procedure (4.6.) actually results from the use of multivariate Pad6 approximants of order ( 1 , l ) for the inverse operator of ~ ( x l. ., . , x k ) at ( x v ) , . . . , x g ) ).

IV.8.1. Iterative method8 baaed on Pod6 approzimation

225

To illustrate the use of the formulas (4.5.) and (4.6.) we shall now solve the nonlinear system

{ (-

f 1 (‘J

Y) = -=+I/ = e-=-y

f&,y)

- 0.1

=0

- 0.1 = 0

which has a simple root at 12.0.1)) = (2.302585092994O46.. 0.

.

As initial point we take ( d 0 ) , g ( ’ ) ) = (4.3,2.0). In table 4.14.one finds the consecutive iteration steps of Newton’s and Halley’s method. Again Halley’s method behaves much better than the polynomial method of Newton. Here the inverse operator G of the system of equations F has a singularity near to the origin and this singularity causes trouble if we get close to it. For

we can write -0.5(ln(0.1 0.5(ln(0.1

+ u) + ln(O.l + v)) + u) - ln(O.l + v))

With (do), y(O)) = (4.3,2.0) the value do)= f2(z(O),y(O)) is close t o -0.1 which is close t o the singularity of G. For the computation of th e Pad6 approximants involved in all these methods the €-algorithm can be used. Another iterative procedure for the solution of a system of nonlinear equations based on the e-algorithm but without the evaluation of derivatives can be found in [5]. Since it does not result from approximating the multivariate nonlinear problem by a multivariate rational function, we do not discuss it here.

a

J9 0

Ne ton

0.43000000D+01

H Iev

,A4 0.20000000D+01

N N

0.43000000D+01

0.20000000D+01 0.5705728dD-kOO

1

-0.22427305D+02

-0.24720886D+02

0.287Q8400D+01

2

-0.21927303D+02

-0.24228888D-k.02

0.22495475D+Ol

-0.52625816D-01

3

-0.21427303D+02

-0.23720888D-kO2

0.23018229D+Ol

-0.44Q01947D-02

4

-0.20827303D+02

-0.23228888D+02

0.23025841D+01

-0.57154737D-05

5

-0.20427303D+02

-0.2272Q888D-kO2

0.23025851D-l-01

-0.Q7680305D--11

6

-0.1 QQ27303DS-02

-0.22228888D-kO2

0.23025851D+Ol

-0.17438130D--16

7

-0.18427303D+02

-0.21720888D-bO2

8

-0.18Q27303D+02

-0.2122Q888D-tO2

Q

0.18427303D+02

-0.20728888D)+02

10

-0.17827303D)+02

-0.20229888D-kO2

11

-0.17427303D+02

-0.19720888D-kO2

12

-0.16927303D+02

-0.10228888D+02

13

-0.16427303D+02

-0.18728888D+02

14

-0.15Q27303D+02

-0.18228888D-I-02

15

-0.15427303D+02

-0.17729888D+02

16

-0.14927303D+02

-0.17229888D+02

17

-0.14427303D+02

-0.16729888D+02

18

-0.13027303D+02

-0.1622Q888D-tO2

18

-0.13427303D+O2

-O.l572Q888D+02

20

-0,12!327303D+02

-0.15228888D-tO2

rn

y(4

3 e

m

J

?:

;r a

R e

Iv.8.B. Iterative methods based on rational interpolation

227

3.2.Iterative methods baa e d o n rut i o n a1 int erpol at i o n. Let

with pi and 9, respectively of degree m and n, be such that in an approximation z i for the root z * of f

fr)(zi-j)

with m that

= f(')(Zi-j)

I!

= 0 , . . .,8j - 1

(4.7.)

-+ n + 1 = C'tz08 ~ Then . the next iteration step z;+] is computed such Pi(zi4-1)

=0

For the calculation of zi+l we now use information in more than one previous point. Hence such methods are called multipoint. Their order of convergence can be calculated as follows. Theorem 4.5.

If

(Z;};~Nconverges to a simple root z* of f and f(n+n+l)(z) continuous in a neighbourhood of z* with

with n

> 0 is

where f ( k ) ( z *= ) 0 if k < 0, then the order of the iterative method based on the use of r,(z) satisfying (4.7.) is the unique positive root of the polynomial .i+1

- 8ozj- B1zj-l

The proof can be found in 1521.

- .. .- 8j = 0

228

IV.2.2.Iterative methods baaed on rational interpolation

If we restrict ourselves to the case 8[

=8

l = 0 , . , ., j

then it is interesting t o note that t h e unique positive root of

!=O

increases with j but is bounded above by 8 + 1 [53 pp. 46-52]. As a conclusion we may say t h a t the use of large j is not recommendable. We give some examples. Take m = 1, n = 1, 8 = I and j = 2. Then z;+1 is given by

The order of this method is 1.84, which is already very close to 8 + 1 = 2. Take m = 1, n = 1, 80 = 2, 81 = 1 and j = 1 . Then x;+1 is given by

The order of this procedure is 2.41. The ease m = I , n = 0,8 = 1 and j = 1 reduces to the secant method with order I .82. Let us again calculate the root of

with initial points close t o the singularity in x = 0.1. The successive iteration steps computed in double precision (56-digit binary arithmetic) are shown in table 4.15.

IV.2.2. Iterative method8 based on rational interpolation

229

Table 4.15.

(4.8.)

(4.0.)

secant method

Xi

zi

Xi

d

0.80000000D-01

0

1 2 3

0.90OOOO0OD-01

0.80000000D-01

0.80000000D-01

0.85000000D-01

O.QO0OOOOOD-01

O.QOOO00OOD-01

-0.15847802D-03 0.15131452D-06 -0.51538368D-13

4

0.20630843D-24

5

0.00000000D+00

- 0.17324153D-03

0.71095917D-01

0 . 4 6 6 2 1186D-10

0.64701421D-01

-0.6139231 1 D-25

0.46636684D-01

0.00000000D+00

0.30206020D-01 0.14080364D-01

6

0.425123451)-02

7

0.59843882D-03

8

0.28 4 30 08 1 D -0 4

0

0.15223576D-06

10

0.38727362D--10

11

0.58956894D-16

12

0.22832450D-25

By means of the multivariate Newton-Pad4 approximants introduced in section 8.2. of chapter 111 the previous formulas can be generalized for the solution of systems of nonlinear equations. We use the same notations as in chapter 111 and as in the previous section. For each of the multivariate functions f,(q,.. .,zk) with j = 1 , . ..,k we choose

D = N = ((0,. . . , O ) , ( l , O , . . .,O ) , ( 0 , 1 , 0 , .. . , O ) , . .., (0,...,o, 1)) H={(2,0,

..., 0 ) , ( 0 , 2 , 0,...10),..., (0,..., 0,2)) 2 INk

c Nk

IV. 2.2. Iterative methods baaed on rational interpolation

230

Here the interpolationset N U H expresses interpolation conditions in the points

zp)). . ., ($',

..

.)

Remark that this set of interpolation points is constructed from only three successive iteration points. The numerator of

with possible coalescence of points, is then given by

NO ,...,O ( 21, . . ,zk) c02,00,. ..(00

coo, ...,00,02

where

Nl,O

,...,0 (21> .

1

* 3

c12,00,...,00

0

zk

. . . NO ,...,0,1(21, . . ... 0 ...

*

coo, ...,00,12

>

zk)

IV.B.8. Iterative methods based o n rational interpolation

231

The values e,,tl,...,,Ltk are multivariate divided differences with possible coalescence of points. Remark that this formula is only valid if the set H provides a sys( i + l ), . . ., zk (i+l)) tern of linearly independent equations. The next iterationstep (21 is then constructed such that

p i k ( z y l ) ., .

.J

Z k( i + l ) ) = 0

For k = 1 and without coalescence of points this procedure coincides with the iterative method (4.8.). With k = 2 and without coalescence of points we obtain a bivariate generalixation of (4.8.). Let us use this technique t o solve the sgslem

{

e--l+Y e-Z-9

= 0.1 = 0.1

with initial points (3.2, -0.95), (3.4, -1.15) and (3.3, -1.00). The numerical results computed in double precision (56-digit binary arithmetic) are displayed in table 4.16. The simple root is (2.302585092994046.. . ,O.). In this way we can also derive a discretized Newton method in which the partial derivatives of the Jacobian matrix are approximated by difference quotients

N = H = ( ( 0 , ...,Q ) , ( l , O , ..., 0 ),,..J(Ol...lO,l)}

D

= ( ( 0 , . . .,n))

If we call this matrix of difference quotients AFi, then the next iterate is computed by meana of

we take the same system of equations and the same but fewer initial pointa as above. The consecutive iteration steps computed in double preciaion (56-digit binary arithmetic) can now be found in table 4.17.

As an example

232

IV.2.2. Iterative method8 baaed on rational interpolation Table 4.16.

i

~

y")

~-

0.320000OOD+Ol

-0.Q50000OOD-l-00

0.34000000D-k01

-0.11500000Df01

0

0.33000000D-k01

-0.10000000D~01

1

0.25249070D-kOl

-0.22072875D-kOO

2

0.22618832D-tOl

3

0.231278OQD)+Ol

-0.101644QOD-01 -0.51269373D-03

0.41 9 7 l Q 4 4 D - 0 1

4

0.23030978D-kOl

5

0.23025801D-kOl

0.40675854D-05

6

0.28025851Df01

-0.2568682QD-08

7

0.23025851 D+Ol

-0.1277881 6D-13

8

0.23025851Df01

-0.1 1350Q32D-16

Table 4.17.

i 0 1

~~

0.34000000D-k01

-0.11500000D-k01

0.33000000D-b0 1

-O.lO0OOOOOD+Ol

-0.29618530D-kOO

0.21743833Df01 0.20884Q33D-kOl

2

0.32743183D-l-01

3

0 . 2 2 1 1 4 2 1 1 Di-0 1

-0.84011352D+01

4

0.3651533RDt-0 1

-0.72149651D-kOl

5

-0.17Q00083D-kO4

0.208541 11 D i - 0 4

divergence

IV.2.3. Iteratave method3 wing continued fractions

233

The rational method is again giving better results. Now the initial points are

such t h at u = f l ( z , y ) is close t o -0.1 which is precisely a singularity of the inverse operator for the considered system of nonlinear equations. For a more stable variant of thc diacretized Newton method we refer to [24]. 2.3. Iterative method8 wing continued fractions.

If ri(z) is t h e rational interpolant of order (m, 1) for $(z), satisfying 1 ri(zw)) = --(&-L))

L = 0,.. . , m + 1

f

then ri(z) can be written in the form rj(z) = do

+ d l ( z - z(i-m-')

+ ldm(z

z(i-m-

-

) + . . . + dm-l(Z - z(i-m-1) 1)

). . .(. - z(i-2J)l

t

+I

2

-

1.. .(z- z ( i - 3 ) )

&-q

&+I

The coefficients d , ( j = 0 , 1 , . . ., m ) are divided differences while dm+l is an inverse difference. The root of ,!;(z) can be considered as an approximation for the root z* of f . SO z(i+l)

-2

(i-1)

- dm+l

This method can be compared with methods based on the use of rational interpolants of order (1, rn) for f(z).

2.4. The qd-algorithm. First we s t at e an important analytical property of the qd-scheme. To this end we introduce the following nokition. For the function f(s) given by its Taylor series development f(.) = co c 1 2 c 2 22

+

we define the H ankel-det erminant s

+

+

234

IV.Z.4. The gd-algorithm

We now call the series f(z)ultimately k-normal if for every R with 0 5 n 5 k there exists an integer M,, such t h a t for m 2 M,,the determinant Ifm,,,is nonzero.

Theorem 4.8. Let the meromorphic function f(z) be given by its Taylor series development in the disk B(0,r) = {z E C 121 < r ) and let the poles wi of f in B(0, r ) be numbered such that

1

0

< Iw] L

Iwzl

I ... < r

each pole occuring as many times as indicated by its order. If the series f(z) is ultimately k-normal for some integer k > 0, then the qd-scheme associated with this series has the following properties: a) for each n such that 0 < n 5 k and such t h a t Iwn-~l < (tun( < Iw,,+l(, where wg = 0 and, if f has exactly k poles, wk+l = 00, we have Iim m-+m

b) for each n such t h a t 0

be asymptotic to the sequence { ZO, zl,.. . , z,, 20, z1, . . . ,z i , 2 0 , z1, . . . ,z j , . . .} in the sense that

lim

k-oo

sk(j+l)+i =

d = 0 , . . . ,j

z;

Let the function f(s) be meromorphic in

..

~ ( 2 0 , ., Z j , r )

= {z E C

I

- zo)(z - 21).

. .( 5 - zj)l 5 r)

and analytic in the sequence of points { s ~ } ~and ~ Nlet the poles w i of B(z0,. . . , z j , r ) be numbered such that for wi

we have

=

I(Wi

- 20)

0 < w1 Iwp

1 . .

(Wi

- 2j)l

s . .. < r

f in

IV.2.5. The generalized qd-algorithm

237

where the poles are counted with their multiplicities. If the Newton series is ultimately k-normal for some integer k > 0, then the generalized qd-scheme associated with this series has the following properties : a) for each n such that 0 < n 5 k and such that w,-~ < w, < w,+1 where wo = 0 and, if f has exactly k poles, wk+l = r , we have

b) for each ra such that 0 < n 5 k and such that w , < w,+1, we have

For the proof we refer to [9].

238

IV.3.1. The we of Pad4 approzirnants

53. Initial value problems. Consider the following first order ordinary differential equation: dv

- =j(z,y)

dz

for

2

E [a,bj

(4.10.)

with y(a) = yo. When we solve (4.10.) numerically, we do not look for an explicit formula giving y ( z ) as a function of z but we content ourselves with the knowledge of y(zi) a t several points zi in [a, 61. If we subdivide the interval [a,b ] , k

U Izi-1 ,z i ]

[a, 61 =

i= 1

where zi =a

+ ik

i = 0,. . ., k

with

b-a h=--

k

for k

>0

then we can calculate approximations yi+l for y ( z i + ~ )by constructing local approximations for the solution y ( z ) of (4.10.) a t the point 2 ; . We restrict ourselves now t o methods based on the use of nonlinear approximations.

3.1. The use of Pad6 approximantcr. Let us try the following technique. If 8 i ( z ) is the Pad6 approximant of a certain order for y(z) a t z i then we can put Yi+l

=ai(Zi+l)

which is an approximation for y ( z i + l ) . For the calculation of ai(z) we would need the Taylor series expansion of y(z) at z;, in other words

Since the exact value of y(z;) is not known itself, but only approximately by y i , this Taylor series development is not known and hence this technique cannot be applied. However, we can proceed as follows. Consider the power series

IV.3.1. The u8e of Pad6 approzirnants

239

Let r i ( z ) be the Pad6 approximant of order ( m , n )for this power series. If we put z = zi+l, in other words z - x i = h , we obtain 8' = 0,..., k - 1

yi+r = r i ( z i + I )

Hence we can write ui+l

= Y i + hg(zir Yip h)

i = 0,. . .,k - 1

(4.11.)

where g is determined by r i . Such a technique uses only the value of zi and y i t o determine yi+1. Consequently such methods are called one-step methods. Moreover (4.11.) is an explicit method for t he calculation of v i + l . It is called a method of order p if the Taylor series expansion for g(z,y,h) satisfies Y ( z i + l ) - u ( z i ) - hg(ziJ Y ( z i ) ,h) = 0(hp+')

Clearly (4.11.) is a method of order (m + n) if r i ( z ) is a normal Pad6 approximant. The convergence of (4.11.) follows if g(z, y , h) satisfies the conditions of the following classical theorem [30 p. 711.

Theorem 4.8. Let the function g(zyy , h) be continuous and let there exist a constant L such that

g('J YJ

= f(’,

Y)

is a necessary and sufficient condition for the convergence of the method (4.11.), meaning that for fixed z E [ a ,b ] .

2 40

IV.3.1. The we of Podi approzarnants

From the fact that t i is a Pad4 approximant it follows th a t the relation g(z,y, 0) = f(z,y) is always satisfied (see problem (8)). The case n = 0 results in the classical Taylor series method for the problem (4.10.), If we take rn = n = 1 we get (4.12.)

If y(z) is a rational function itself, then using (4.11.) we get the exact solution Yi+l = Y(z;+I)

at least theoretically, if the degrees of numerator and denominator of r i ( z ) are chosen in a n appropriate way. Techniques based on the use of Pad4 approximants can be interesting if we consider stiff differential equations, i.e. if has a large negative real part [ZO]. An example of such a problem is the equation dY = xy dz

(4.13.)

-

with Re(1) large and negative. Since the exact solution of (4.13.) is

we have lim y ( z ) = lim

z-00

2-00

2% =o

and we want our approximations y; to behave in the same way. Dahlquist [15] defined a method to be A-stable if it yields a numerical solution of (4.13.) with Re(1) < 0 which tends to zero as i -+ 00 for any fixed positive h. He also proved t hat there are no A-stable explicit linear one-step methods. Take for instance the method of Euler (rn = 1, n = 0): Y i + l = ~i

+ hf ( z i j y i )

Y i + l = (1

+ hX)Yi

We get

= (1

+ hX)'+1 yo

IV.3.2. The

t~t: of

rational anterpolanta

241

Clearly lim yi+l = yo Iim (I i-

i-+W

00

+ /A)'+' = o

only if

)I

+ hX( < 1

So for large negative X the steplength h has to be intolerably small before acceptable accuracy is obtained. lo practice h is so small t h a t round-off errors and computation time become critical. The problem is t o develop methods t h a t do not restrict the stepsize for slability reasons. If (4.11.) results from the use of t h e Pad6 approximant of order (rn,m), (m, rn 1) or (m,m + 2) then one gets an A-stable method [16]. This can be seen as follows. If f ( z , y) = Xy then r,(z) is the Pad6 approximant for the power series

+

Hence Y i + l = rm,rt(hA) ~i

with h = ~ i + - lz;, where rm+(x) is the Pad6 approximant of order (rn,n)for e 2 . A-stability now follows from the following theorem.

Theorem 4.9. If m = n or m = n - 1 or m = n - 2 then the Pad6 approximant of order (m,n) for ez satisfies

For the proof we refer to [3, 181. 3.2. The uae of rational interpolants.

It is clear t h a t if the interpolation conditions are spread over several points, then the computation of yi+l will need several xi-^ and yi-t(t = 0, 1, . . .). Such methods are called multistep methods. Let r i ( z ) be the rational Hermite iuterpolant of order (rn,n) satisfying

242

IV.8.2. The me of rational interpolants

where (j

+ I)(R + 1) = rn + n + 1

Here

Then an approximation for y(zi+l) can be computed by putting

This is a nonlinear explicit multistep method. A two-step formula is obtained for instance by putting j = 1, m = 2, n = 1, and 8 = 1 and using theorem 3.17.:

where f, = j(x;,y;) and fi-1 = f(z;-1, yi-1). We can also derive implicit methods which require an approximation the caIculation of y;+1 itself, by demanding

v;+l

for

where (j

+ 2)(8 + 1) - 1 = m + n +

For m = 1 = n, j = 0 and

8

1

= 1 we get the formula

(4.15 .)

IV.3.3. Predictor-corrector method8

243

For more information concerning such techniques we refer t o [36]. Remark th a t multistep methods are never selfstarting. Both explicit and implicit ( j + 1)- step methods are of the form Yi+l =

j

C

acyi-c

+ h g ( z i + l , . . ., z i - j l

yi+lJ..

-J

~ i - j lh)

L=O

and they have order p if the Taylor series expansion of g satisfies i

y(zi+t) -

C

aty(zi-t)

- hg(zi+l

J

.

‘1

zi-j] y ( z i + l ) j , ..,y(zi-j), h ) = 0(hp+l)

L= 0

Hence (4.14.) is a third-order method if the starting values are third-order and (4.15.) is second-order. When applied t o a stiff differential equation one should keep in mind that linear multistep methods are not A-stable if their order is greater than two. The following result is helpful if af&’yl is real and negative. We know th a t we can write for problem (4.13.) y(z) = v(2i-j)

e(~-~i-j)X

with Re(X) large and negative. €leiice it is interesting to take a closer look a t rational Hermitc interpolants for exp(z) in some real and negative interpolation points and also in 0. Theorem 4.10.

5 m and 8qi 5 n is such th a t ri( t ) (0) = 1 = exp(*)(O)for 0 5 t 5 rn + n - t2 with e 5 m

If ri(z) = pi(z)/qi(z) with api a)

b) ri(tt) = exp(&) for & then lri(z)l

< 1 if

m

< 0, 1 5 k’ 5 j

5 n and

with &

# &, 1 5 !# k 5 j

z is real and negative.

For the proof we refer to [32]. 3.3. Predict0 r-correct or methode. For a solution of the initial value problem (4.10.) we have y’(z) = f ( z , y(z)) for every z in [a, b]

244

IV.S.4. Numeracal results

If we integrate this equation on the interval [zi-i, z i + t ] with j , 8 2 0 we get PZitl

Now f can be replaced by an interpolating function, through the points j(zi-1 ,yj-l)), . . ., which is easily integrated. ( z 8 ,f(z;, yi)), If t! = 1 we get p r e d i c t o r - m e t h o d s because they are explicit. If 8 = 0 we get corrector-methods because the value of y; is needed for the computation of y;. These implicit formulas can be used t o update an estimate of yfz;)iteratively. W h e n f is replaced by an interpolating polynomial we get the well-known methods of Adarns-Bashforth ( j = 0 and !. = 1) and Adams-Moulton ( 3 = 1 and 6 = 0). When f is replaced by an intc3rpolating rational function we can get nonliriear formulas of predictor or corrector type. Sincc the rational interpolant must be integrated, it is not recommendable to choose rational functions with a denominator of high degree.

3.4. Nu me rical r es ult 8 . Let u s compare two Taylor series methods ( m= 2 and m = 3) with the explicit method (4.12.) for t h e solution of the equation y‘ = I

+ y2

for z

2o

Y(0) = 1

The theoretical solution is g = tg(z -t 7r/4). We take the steplength h = 0.05. As can be seen in table 4.19. the second order rational method gives even better results t h a n t h e third order Taylor series method, a fact which can be explained by the singularity of the solution y(z) at z = 7 . To illustrate A-stability we will compare Euler’s method ( m= 1,n = 0) with the formulas (4.12.))(4.14.) and (4.15.) for the equation y’ = -25y

for z E [0,I]

Y(0) = 1

The solution is known to be y = exp(-25z). For the results in table 4.20. we chose the steplength h = 0.1. As expected Eiiler’s solution blows up while formulas based on the use of a “diagonal” entry of the Pad6 table or the rational Hermite interpolation table for t h e exponential decay quite rapidly. Similar results would have been obtained if “superdiagonal” entries of the Pad6 table were used. Remark that formula

Table 4.19. exact solution

+7)

Taylor series

Pad6 approximant

Taylor series

m=3

i

zi

1

0.05

0.1 10536D+01

0.1 105OOD+Ol

0.110526D+01

0.110533D+01

2

0.10

0.122305D+01

O.l22210D+Ol

0.122284D-kOl

0.122299D)+Ol

3

0.15

0.135600D-kOl

0.135449D+Ol

0.135573D+01

0.135598D)+01

4

0.20

0.150850D+01

0.1 5 0 5 8 2 D t 0 1

0.150795D+01

0.150831D+Ol

5

0.25

0.1 6858oD+oi

0.168150D-kOl

O.l68500D+Ol

0.188547D+01

6

0.30

0.188577D$-01

O.l88886D+Ol

0.188462D-kOl

0.189522D-t-01

7

0.35

0.214875D-kOl

0.213884D-kOl

0.214811D+Ol

0.214883D-kOl

8

0.40

0.246496D-kOl

0.244751D-kOl

0.24626lD+Ol

0.246335D-kOl

9

0.45

0.286888D+01

0.28398OD+Ol

0.286543D-I-01

0.286584D-I-01

10

0.50

0.340822D+Ol

0.335737D-t-01

0.340298D-t-01

0.340248D-kO 1

11

0.55

0.4 16936D4-01

0.407388D-kOl

0.416097D+Ol

0.415703D+01

12

0.60

0.533186D-t-01

0.513307D+Ol

0.531720D+Ol

0.5301 31D-t-01

13

0.65

0.7 3 4 0 4 4 D t 0 1

0.685144D-t-01

0.731087D+01

0.724568D-kOl

14

0.70

0.1 16814D+02

0.1 00697D-kO2

0.1160 1QDi-02

0.112431D-kO2

15

0.75

0.282383D-kO2

0 .I 77676D-kO2

0.277486D-i-02

0.232131D-kO2

tg(zi

m=2

m=l=n

*CD e

5 e

Table 4.20.

exact solution exp(-25si)

i 1

0.1

0.820850D-01

2

0.2

0.6737851)-02

3

0.3

0.553084D-03

4

0.4

0.453QQQD-04

5

0.5

0.372665D-05

6

0.6

0.305902D-06

7

0.7

0.251 100D-07

8

0.8

0.2061 15D-08

Q

0.Q

0.16QlQOD-OB

10

1.0

0.13887QD--10

m=l=n explicit one-step

Euler -0.150OOOD+01 0.225000D+01 -0.337500D+01 0.506250D-l-01 -0.758375D+01 0.113Q06D+02 -0.17085QD)+02 0.25628OD-l-02 -0.384434D)+02 0.576650D4-02

1

-0.111111D+OO 0.123457D-01 -0.137174D-02 0.152416D-03 -0.189351 D-04 0.188168D-05 -0.200075D-06 0.2323061)-07 -0.258117D-08 0.286797D-00

rn = 2,n = I

explicit multisteD

rn=l=n

implicit one-step

0.820850D-01

0.123047D+00

0.840728D-01

0.151 407D-01

-0.542641D-01

0.186302D-02

-0.4Q474OD-kOO

0.22023QD-03

-0.252173D+00

0.282073D-04

0.314055D-kOO

0.347083D-05

0.102175D+O1

0.427077D-06

0.16Ql37D-kOO

0.525507D-07

-0.lQl5Q6D-I-00

0.646622D-08

-0.6Q8115D-kOO

0.7Q5651D-OQ

IV.3.5. Systems of first order ordinary differential equatiow

247

(4.14.) based on a "subdiagonal" rational approximation is surely not producing an A-stable method. To obtain the results of table 4.20. by means of (4.14.) a second starting value yl was necessary. We took yl = exp(-2.5) = y(z1). For (4.15.) the expression f(zi,y;) = -25y; was substituted and y;+l was solved

from the quadratic equation. All these schemes can be coupled to mesh refinement and the use of extrapolation methods. If an asymptotic error expansion of y(zi) in powers of h exists, then the convergence of the sequence of approximations for y(z;), witb z i fixed, obtained by letting the stepsize decrease, ran be accelerated by the use of techniques described in section 1 [23]. 3.5. Systems of first order ordinary differential equations.

Nonlinear techniques can also be used to solve a system of first order ordinary differential equations

dzj = f j ( X , 2 1 , 2 2 , . . . , Z t ) dX

where the values z,,! and the functions approaches are possible. If we introduce vectors

fj

j = 1, ..., k are given for j = I , . . .,k. Several

then one method is t o approximate the solution componentwise using similar techniques as in the preceding sections. So for instance (4.12.) becomes

Y ( z ; + ~N) Y;:+l where

and

= Y;

Y;) ] + h [*F(zi,2F2(Zi, Y;)- hF'(xil yi)

(4.16.)

248

IV.3.5. Systems of first order ordinary differential equation8

and the addition and multiplication of vectors is performed componentwise. In other words (4.16.) is equivalent with

For more information on such techniques we refer to [56,35, 381. Another approach is not based on componentwise approximation of the solution vector Y ( s )but is more vectorial in nature. Examples of such methods and a discussion of their properties is given in [57, 7 , 27, 131. The nonlinear techniques introduced here c a n also be used to solve higher order ordinary differential equations and boundary value problems because these can be rewritten as systems of first order ordinary differential equations. Again same of the nonlinear techniques prove to be especially useful if we are dealing with stiff problems. A system of differential equations

dY -= F ( z , Y ) dx Y ( a ) = Yo with

is called stiff if the matrix

IV.3.5. System8 of first order ordinary differential equations

249

has eigenvalues with small and large negative real part.

{

Consider for example

+

= 9 9 8 ( ~x )~ 199822 (z)

Zl(0j = 1

- - - -99921(2) - 19992z(x)

22(0)

=0

The solution is

so that again

lim z I ( x ) = 0 = lim

z 4 m

DO’%

z2(x)

where both z1 and a2 contain fast and slow decaying components. For a discussion of stiff problems we refer to [20 pp.209-2221.

IV.4. Numerical integration

250

§4. Numerical integration.

s,

b

Consider I = j ( z ) d z . Many methods to calculate approximate values for I are based on replacing j by a function which can easily be integrated. The classical Newton-Cotes formulas are obtained in this way: j is replaced by an interpolating polynomial and hence I is approximated by a linear combination of function values. In some cases the values of the derivatives of j ( x ) are also taken into consideration and then linear combinations of the values of f ( z ) and its derivatives a t certain points are formed t o approximate the value I of the integral. This is for instance the case if polynomial Hermite interpolation is used. In many cases the linear methods for approximating I give good results. There are however situations, for example if f has singularities, for which linear methods are unsatisfactory. So one could t r y t o replace f by a rational function r and consider

[

r(x)dz

as an approximation for I . But rational functions are not that easy to integrate unless the poles of r are known and the partial fraction decomposition can be formed. Hence we use another technique. Let us put

Then

1 = y(b) If f is Riemann integrable on [a,b ] , then y is continuous on [a, b ] . If f is continuous on [a, 61, then y is differentiable on [a,b] with y’(z) = j ( z ) and y(a) = 0

So I can be considered as the solution of a n initial value problem and hence the techniques from the previous section can be used. We group them in different categories.

I V . 4 . l . Methoda w i n g Pad6 approximantd

251

4.1. Methods using Pad6 approzirnants. Let us partition the interval [a, 61 with steplength h = ( b - a ) / k and write Z; = a

+ ih

a = 0, ...,k

and (4.17.)

s,”’

f(t)dt. where y ; approximates y ( z i ) = If r; is the Pad6 approximant of order ( m ,n ) for t l , ; ( h )then we can put

and consider y k as an approximation for I . In this way Yi+l

which means

= yi

+ hg(zi, h)

i = 0, ...,k - 1

(4.18.)

l;’’

f ( t ) d t N hg(z;,h)

If m = 1 = ra we can easily read from (4.12.) t h a t (4.18.) results in (4.19.) Formulas like (4.18.) use derivatives of f(z) and are nonlinear if n > 0. From the previous section we know that (4.18.) is exact, in other words that yk = I , if y ( z ) is a rational function with numerator of degree m and denominator of degree ra. For n = 0 formula (4.18.) is exact if y ( z ) is a polynomial of degree m, i.e. f ( z ) is a polynomial of degree m - 1. The obtained integration rule is then said t o be of order m - 1. The convergence of formula (4.18.) is described in the following theorem which is only a reformulation of theorem 4.8.

252

IV.4.2. Method8 wing rational interpolanta

Theorem 4.11. Let y; be defined by (4.18.). Then lim

h-0

yi(h) = y(z) for fixed z E [a, b]

====i(h)

if and only if g(z,O) = f(z). Instead of (4.17.) one can also write t l , i ( h )= T/i

+ lata,i(h)

with

and compute Pad6 approximants r , of a certain order for tz,;(h).Let us take

m = 1 = n . If we define

Yi+i

= yi

+ hr;(h)

then we get

4.2. Method8 using rational interpolanta.

If we again proceed as in the section on initial value problems we can construct nonlinear methods using information in more than one point. Since these methods are not self starting but need more than one starting value their use is somewhat limited and rather unpractical. An example of such a procedure is the following. Let ti(.) be the rational Hermite interpolant of order ( 2 , l ) satisfying

and let

IV.4.3. Methods without the evaluation of derivatives

253

Then we know from formula (4.14.) t h a t

with f; = f(zi) and f i - 1 = f ( z i - I ) . Often rational interpolants are preferred t o Pad6 approximants for the solution of numerical problems because the use of derivatives of f is avoided. As mentioned, a drawback here is the necessity of more starting values. Another way t o eliminate the use of derivatives, now without the need of more starting values, is the following.

4.3. Methods without the evaluation of derivatives. One can replace derivatives of j ( z ) in formula (4.18.) by linear combinations of function values of f ( z ) without disturbing the order of the integration rule. To illustrate this procedure we consider the case m = 1 = n. Then 1591

We will compute constants a,p and 7 such t h a t for

we have gfZ,

h) - t ( s ,h) = O(hrn+%) = O(h2)

For ~ ; + i= ~i

+ h t ( z ; ,h)

this would imply y(si+l) - Yi+l = ~(hm+m+l)= O ( P )

Condition (4.20.) is satisfied when a+B=2 B7 = -1

(4.20.)

254

IV.4.4. Numerical resulta for singular integranda

In other words, for 7 # 0, 27 a=---

p = -1

+1 7

-.

7

so

For 7 = 1 we get the integration rule

In this way we approximate (4.21.)

4.4. Numerical results for singular integranda We will now especially be interested in integrands regular in [a,b] but with a singularity close t o the interval of integration and on the other hand in integrands singular in Q or b . The problem of integrating a function with several singularities within [a, 61 can always be reduced t o the computation of a sum of integrals with endpoint singularities. If f ( z ) is singular in 6 then the value of the integral is defined by

and is assumed t o exist. We shall compare formula (4.19.) with Simpson’s rule ( m = 2, n = 0 ) and with a (2k)-point Gaussian quadrature rule that isolates the singularity in the weight function. If f(t) can be written as w ( t ) h ( t ) where w ( t ) contains the endpoint singularity of f(t) and h(t) is regular then the approximation

I

wlh(tl) + . . .

+ 402kh(t2k)

does not involve function evaluations in singular points. We use a (2t)-point formula because on [zi,2;+1] for i = 0 , . . ., k - 1 both (4.19.) and Simpson’s rule

IV.4.4. Numerical reaulta for aingular integranda

255

need two function evaluations. Since / is singular in b = zk, we take f ( z k ) = 0 in Simpson's rule which means that the singularity is ignored. In (4.19.) the singularity of f is no problem since f and f’ are only evaluated in 2 0 , . . ., xk--1 [58]. Our first numerical example is

d t = 3.04964677.. with a singularity in

t = In 3 = I .09861228. . . and the second example is

+

2(1_ _ ~ t)sint _ _ _ _ _cost d t = 2

We shall also compare the different integration rules for the calculation of e'dt = 1.71828182. . .

which has a smooth integrand. Because of the second example the weight function w ( t ) in the Gaussian quadrature rule was taken t o be W ( t )=

1

J1-t

.___

All the computations were performed in double precision arithmetic and the double precision values for the weights w i (i = 1 , . . .,2k) and the nodes ti (d = 1 , . . .,2k) were taken from [I pp. 916-9191. Remark t ha t the nonlinear techniques behave better than the linear techniques in case of singular integrands. However, for smooth integrands such as in table 4.23., the classical linear methods give better results th a n the nonlinear techniques. Also, if the singularity can be isolated in the weight function such as in table 4.22., Gaussian quadrature rules are very accurate. In general, little accuracy is gained by using nonlinear techniques if other methods are available for the type of integrand considered [51].

256

IV.4.4. Numerical reeults f o r singular integranda

As in the previous section all these schemes can be coupled to mesh refinement and extrapolation.

Table 4.21. 1

I=J

et

o (3 - et)2

k = 16

dt = 3.04964677.. .

Gaussian

Simpson's

formula

quadrature

rule

(4.19 .)

3.1734660 3.0773081 3.0564777

3.28067 65 3.0841993 3.0531052

3.1006213 3.0573798 3.0510553

Table 4.22.

I = J 2(1 - t ) sin t -tc o s t dt = 2 0 G - j

k=4 k=8 k = 16

Gaussian

Simpson's

formula

quadrature

rule

(4.19.)

2.0000000 2.0000000 2.0000000

1.7504074 1 A267581 1.8786406

1.9033557 1.8797832 1.go48831

Table 4.23. 1

I = 1etdt = 1.71828182.. . 0

k=4 k=8 k = 18

Gaussian

Simpson's

formula

quadrature

rule

(4.18.)

1.7205746 1.7204038 1.7188196

1.7182842 1.7182820 1.7182818

1.7284991 1.7206677 1.7188592

IV.5. Partial diflerential equatiom

257

§5. Partial differential equations. Nonlinear techniques are not frequently used for the solution of partial differential equations. We will describe here a method based on the use of Pad6 approximants t o solve the heat conduction equation which is a linear problem. For other illustrations we refer to [54, 48, 17, 22, 281. All these techniques first discretize the problem so that thr: original partial differential equation is replaced by a system of equations which is nonlinear if the partial differential equation is. Another type of techniques which we do not consider here are methods which do not discretize the original problem but solve it iteratively by means of a procedure in which subsequent iteration steps are differentiable functions [13]. Linear techniques of this type arc recommended for linear problems and nonlinear techniques can be used for nonlinear partial differential equations. Let us now concentrate on the heat conduction equation. Suppose we want t o find a solution u ( z , t )of the linear problem

a+, t ) - a%+, t ) ~

at

.~

82

~

a 0 and i = 1, . . . , k. In general we c a n write

(4.23.)

where A is a real symmetric positive definite k x k matrix and depends on the chosen approximation for the operator d 2 / d s 2 . If we introduce the notations

then the exact solution of (4.23.) is U ( t ) = eWtAV where e-tA is defined by

Using the discretization in the time variable t we can also write (4.24.)

I V.5. Partial d ifler e nt ial equ at i o ns

259

If rm,.,(t)= p ( t ) / q ( t ) is the Pad6 approximant of order (n,n) for e-t then (4.24.) can be approximated by

U ( t + A t ) = [q(At.A)]-' [P(At.A)] U ( t )

(4.25 .)

Varga proved t ha t for n 2 m this is an unconditionally stable method [ 5 5 ] , meaning t ha t initial rounding errors remain within reasonable bounds as the computation proceeds independent of the stepsize used. If rn = 1 and n = 0 then (4.25.) means

U(t + At) = ( I

- AtA)U(t)

or equivalently if A = (&.!)kXk

ui(ti+l) = u i ( t j ) - ~t

k

C aitut(tj)

.!=I

which is the well-known explicit method to solve (4.22.). The solution a t level t = ti+l is determined from the solution at t = t , , For nt = 1 = n we obtain from (4.25.)

or equivalently

which is the method of Crank-Nicholson [50]. The operator a2/ax2is replaced by the mean of a n approximation for the partial second derivative a t level t = and the same approximation at t = t i .

IV.6.1. Kernel8 of finite rank

260 56. Integral equations.

As in the previous section we shall discuss linear equations for which the use of nonlinear methods is recommendable. Those interested in nonlinear integral equations are referred to [lo, 121 where methods are indicated for their solution. If the integral equation is rewritten as a differential equation then techniques developed for the solution of initial value problems can also be used. We restrict ourselves t o the discussion of an inhomogeneous Fredholm integral equation of the second kind (the unknown function f appears once outside the integral sign and once behind it):

f ( 4-

J'." K ( z ,y)f(v)dy

= g(z)

xE

1% bl

(4.28.)

Here the kernel K ( z , y ) and the inhomogeneous right hand side g(z) are given real-valued continuous functions. Fredholm equations reduce to Volterra integral equations if the kernel K ( z , y ) vanishes for g > z which produces a variable integration limit.

6.1.K e r n e l 8 of finite rank. Formally the solution of (4.26.) can be written as a series. P u t

and

If we define

and

then (4.26.) reduces to

IV.6.1. Kernel8 of finite rank

261

The series (4.27.) which is a power series in X, is called the Neumann series of the equation (4.26.). Convergence of the Neurnann series for certain values of X depends on the properties of the kernel K ( z ,y). If K ( z ,y ) is bounded by IWz,y)l

<M

for (z,Y) E

b,b]

X [a, b ]

then clearly the series (4.27.) converges uniformly t o f(s)in [a, b] if 1

1x1 < ---___ M ( b - a) If Pad6 approximants in the variable X are constructed for (4.27.) then they may have a larger convergence region than the series itself. Especially interesting is the case t ha t the kernel is degenerate, in other words k

~ ( zY), =

C Xi(Z)Yi(y)

i= 1

with ( X i } and {Y;} each linearly independent sets of functions. Such a kernel is also said to be of finite rank k. Let us try to determine f(z) in this case. If we put

then (4.26.) can be written as (4.28.) Multiply (4.28.) by Yj and integrate to get

or equivalently

IV.6.2. Completely continuoua kernela

262

(4.29 .)

which is the determinant of the coefficient matrix of system (4.29.), then D(X) is a polynomial in X of degree at most k. In case D(X) # 0 the solution of (4.29.) is given by

with

Dij

the minor of the ( i , j ) t h element in D(X). SO (4.28.) can be written as

which is a rational function in X of degree a t most k both in numerator and denominator. Since the series development of (4.30.) coincides with the Neumann series (4.27.) we know that the solution f(z) is equal to the Pad6 approximant of order (k,k) for the Neumann series. So in this case the Pad6 approximant is the exact sum of the series because the sum is a rational function [2 p. 1761. 6.8. Completely continuoua kernels.

The equation (4.26.) can be rewritten

EUI

(I-XK)f=g where the linear operators I and K are defined by

(4.31.)

263

IV.6.2. Completely continuoua kernela

We shall consider square-integrable functions f with

Suppose now t ha t { f;}iEN is a bounded converging sequence of functions. Then the operator K is said t o be completely continuous if for all bounded { f ; ) , ~ the sequence { K fi}iENcontains s subsequence converging t o some function h ( s ) with

- h(2)l2ds -+ 0 as i -+

llKfi - hll = \i[lKfi(Z)

00

A basic property of completely continuous transformations is th a t they can be uniformly approximated by transformations of finite rank. Thus there is a n infinite sequence { K ; } ; Lof~ kernels of finite rank i such th a t

where lim

i-+w

t;

=0

When a completely continuous kernel K is replaced by Ki then the solution f; of

( I - Mi)!; = g is given by the Pad4 approximant ri,i of order (i,;), as explained in the previous section. In case K is completely contiuuous the exact solution f of (4.31.)is a meromorphic function of X [31 p. 311 arid

lim f, = f

i-

00

2 64

Iv.6.2. Completely continuous kernels

in any compact set of the X-plane except at the finite number of X-poles of f [8].Since

we consequently have

lim

ri,i

1-m

=f

in any compact set of the X-plane except at the finite number of X-poles of f.Thus the solution f is the limit of a sequence of diagonal Pad6 approximants. This result is not very useful since each approximant in the sequence is derived from a different Neumann series, with kernel K ; and not with kernel K . However a similar result exists for the sequence r i , i derived from the Neumann series (4.32.) i= 1

where the operator K' is defined by

Theorem 4.12. If ri,i is the Pad4 approximant of order (d, i) t o the Neumann series (4.32.) of the integral equation (4.28.) with completely continuous kernel K then the solution f is given by lim

i-oc

ri,i

=f

in any compact set of the X-plane except a t the finite number of poles of f and at limit points of poles of ri,i as i -+ 00. More information on this subject can be found in [2 pp. 178-1821.

I V . Problems

265

Problems. Show t ha t with

EF’= an for n 2 0,

which coincides with Aitken’s A2-process to accelerate the convergence of a sequence. Give a n algorithm similar to the one in section 1.4. for the calculation of t,,i, but now based on the use of inverse differences instead of reciprocal differences, Show that if the algorithm of Bulirsch and Stoer is used with the interpolation points zi (lim,+m s; = 0) and the p-algorithm is used with the interpolation points z: = 1,s; (1imi4- 2;. = oo),then t i ; = B~( 0 ) . Compare the amount of additions and multiplications performed in the c-algorithm and the qd-algorithm when used for convergence acceleration. Derive the formulas (4.3.) and (4.4.) using inverse interpolation instead of direct interpolation. Derive the formulas (4.8.) and (4.9.) based on the use of rational interpolants. in section 2.3. for successive values of Organize the computation of d,+l d such t ha t a mininum number of operations is involved. Prove that one-step explicit methods for the solution of initial value problems based on the use of Pad6 approximants are convergent if g(z, g, h) given by (4.11.) is continuous and satisfies

IV. Problem8

266

(9)

Check the formulas (4.12.) and (4.14.).

(10)

Write down formula (4.16.) for the solution of

- 99821 (2)+ 199822( Z) - - - -9992+) (11)

- 199922(.)

ZI(0) = 1 ZZ(0) = 0

Construct a nonlinear numerical integration rule based on the use of Pad4 approximants of order ( 2 , l ) . Afterwards eliminate the use of derivatives as explained in section 4.3.

I V . Remnrka

267

Remarks. Nonlinear methods can also be used for the solution of other numerical problems. We refer for instance to [40] where the solution of linear systems of equations is treated, t o [19] for analytic continuation, to [34] for numerical differentiation and to (421 for the inversion of Laplace transforms.

An important link with the theory of numerical linear algebra is through QR-factorization. Rutishauser [47] proved th a t the determination of the eigenvalues of a square malrix A can be reduced to the determination of the poles of a rational functiom .f built from th e given matrix. In this way decomposition techniques for A to compute its eigenvalues are related t o the qd-algorithm when used to compute poles of meromorphic functions. Univariate and multivariate continued fractions and rational functions are also often used t o approximate given functions. For univariate examples we refer t o (33, 421. The bivariate Beta function is a popular multivariate example because it has numerous singularities in a quite regular pattern. For numerical results we refer to [89, 26, 121.

As a conclusion we can say t h a t every linear method has its nonlinear analogue. In case linear methods are inaccurate or divergent, it is recommendable t o use a similar nonlinear technique. The price we have to pay for the ability of the nonlinear method to cope with the singularities is the programming difficulty to avoid division by small numbers.

2 68 References.

1

1) Abramowite M. and Steguo Handbook o Mathematical functions. Dover publications, New York, 1968.

[

21 Baker G. and Gamrnel J. The Pad6 approximant in theoretical physics. Academic Press, New York, 1970.

[

31 33aker G. and Graves-Morris P. Pad4 approximants: basic theory. Encyclopedia of Mathematics and its applications: vol 13. AddisonWesley, Reading, 1981.

\

41 Brezinski C. Algorithnies d’accklkration de la convergence. Editions Tecbnip, Paris, 1978.

[

51 Rrezinski C. Application de 1’6-algorithme B la r6solution des syst6mes non linhaires. C. R. Acad. Sci. Paris Sbr. A 271, 1970, 1174-1177.

[

61 Bulirsch R . and Stoer J. Fehlerabschatzungen und Extrapolation mit rationalen Funktionen bei Verfahren vom Richardson-Typus. Numer. Math. 6, 1964, 413-427.

[

On the stability of rational Runge 71 Calvo M. and Mar Quemada M. Kutta methods. J. Comput. Appl. Math. 8, 1982, 289-292.

[

81 Chisholm J. Solution of linear integral equations using Pad6 Approximants. J. Math. Phys. 4, 1963, 1506-1510.

[

91 Claessens G. Convergence of multipoint Pad6 approximants. Report 77-26, University of Antwerp, 1977.

[ lo] Clarysse 7 .

Rational predictor-corrector methods for nonlinear Volterra integral equations of the second kind. In [60], 278-294.

[ 111 Cuyt A.

Accelerating the convergence of a table with multiple entry. Numer. Math. 41, 1983, 281-286.

[ 121 Cuyt A.

Pad6 approximants for operators: theory and applications. Lecture Notes in Mathematics 1065, Springer Verlag, Berlin, 1984.

[ 131 Cuyt A.

Pad6 approximants in operator theory for the solution of nonlinear differential and integral equations. Comput. Math. Appl. 6, 1982, 445-466.

Iv. Reference8

269

Abstract Pad6 approximants for 141 Cuyt A. and Van der Cruyssen 1’. the solution of a system of nonlinear equations. Comput. Math. Appl. 9, 1983, 617-624.

[ 151 Dahiquist G.

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[ 161 E61e B.

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[ 171 Fairweather G.

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[ 181 Frame J .

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[ 191 Gammei J.

Continuation of functions beyond natural boundaries. Rocky Mountain J. Math. 4, 1974, 203-206.

[ 201 Gear C .

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[ 211 Genz A .

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[ 221 Gerber P. and Miranker W .

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[ 231 Gragg W.

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[ 241 Gragg W. and Stewart G .

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[ 251 Graves-Morris P .

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[ 261 Graves-Morris P. , Hughes Jones R . and Makiuson G . The calculation of some rational approximants in two variables. J. Inst. Math. Appl. 13, 1974, 311-320.

[ 271 Hairer E.

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Pad6 approximants, fractional step methods 281 Hall C. and Porsching T. and Navier-Stokes discretizations. SLAM J . Numer. Anal. 17, 1980, 840851.

[ 291 Henrici P.

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[ 341 Kopal Z.

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[ 351 Lam bert J.

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[ 371 Langer R.

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[ 381 Lee D.and Preiser S.

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I 391 Levin D.

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1

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BIT 3, 1963,

273

SUBJECT INDEX A Aitken A2-process, 265 A-stable method, 240 asymptotic error constant, 221

B backward algorithm, 3 1 block-structure, 68 multivariate -, 109 Bulirsch-Stoer algorithm, 209

c continued fraction, 3 associated -, 18 branched -, 41, 169 contraction of -, 7 convergence of -, 20 convergent of -, 3 corresponding -, 17,76 equivalent -, 15 evaluation of -, 31 even part of --, 9, 80, 84 - expansion, 25 extension of--, 56 generalized -, 56 interpolating -, 138, 169 odd part. of -, 9,80,84 reduced -, 7 value of -, 3 convergence - acceleration, 197 - of continued fraction, 20 - in measure, 98 - of Pad6 approximants, 96 - of rational interpolants, 167 corrector method. 244

274

Subject in dez

covariance homografic --, 118, 119 reciprocal -, 118,119

D denominator partial -, 3 nth --, 3 determinant formulas, 85, 162, 181 direct method, 220 divided difference, 145, 163, 233 multivariate -, 175

E c- algorithm

univariate -, 92, 197 multivariate -, 114 €-table, 94,198 equivalence transformation, 6 Euler-Minding series, 5,46 explicit method 239 extrapolation, 209 - point, 209

F

forward algorithm, 32 Fredholm integral equation, 260

G Gauss quadrature, 254 generalized - polynomial, 190 - rational interpolant, 190 - continued fraction, 56

Subject in dez

H Hankel determinant, 233 Halley’s method, 221, 223

I

implicit method, 242 inverse difference, 139,233 bivariate -, 169 inverse method, 220 iteration one-point -, 221 multipoint -, 227

K kernel degenerate -, 261 - of finite rank, 261 completely continuous -,

L

lemniscate, 167

M modified convergent, 35 modifying factor, 35 multipoint method, 227 multistep method, 241

N

Neumann series, 261 Newton’s method, 220, 223 discretized -, 231

263

275

2 76

Subject indez

Newton-Coates formulas, 250 Newton-Pad6 approximation problem, 168 Newton-Pad4 approximant, 159 Newton series, 157 multivariate -, 236 numerator partial -, 3 nth 3 -.-)

0 one-point method, 221 one-step method, 221, 239 order - of iteration method, 221, 227 - of one-step method, 239 - of multistep met,hod, 243

P Pad6 aproximant, 65 convergence of -, 96 homographic covariance of -, 118 multivariate -, 109 normal -, 72,114, 119 reciprocal covariance of -, 118 Pad6 operator, 100 continuity of --, 100 Padk-type approximation, 120 predictor-corrector method, 244

Q qd- algorithm , 79 extended 81 generalized -, 148, 236 multivariate -, 116 progressive form of -, 236 -$

Subject indez

qd-table, 80 extended

-, 82

R rational interpolant, 130 generalized -, 190 multivariate -, 169 normal -, 135 table of -, 132 rational interpolation problem, 129 rational Hermite interpolant, 157 table of -, 159 rational Hermite interpolation problem, 156 reciprocal difference, 143 recursive algorithm, 90, 153, 209 recurrence relations, 4 p-algorithm, 213 S

sequence totally monotone -, 198 totally oscillating -, 199 series Stieltjes -, 29, 99 ultimately k-normal -, 234, 236 Stieltjes series, 29, 99 Stieltjes transform, 29 stiff differential equation, 240 Stoer’s method, 153

T

Thiele continued fraction expansion, 164 Thiele interpolating continued fraction, 19 branched -, 141. 170

277

Subject indez

2 78

U unconditionally stable method, 259 ultimately k-normal series, 234, 236

V Viscovat ov method of -, 16 multivariate algorithm of

W weights, 255

-,

49