Table Of ContentClaude Brezinski
Pade-Type Approximation
and General Orthogonal Polynomials
ISNM 50:
International Series of Numerical Mathematics
Internationale Schriftenreihe zur N umerischen Mathematik
Serie internationale d' Analyse numerique
Vol. 50
Claude Brezinski
Pade-Type Approximation
and General
Orthogonal
Polynomials
1980
lJ
Springer Basel AG
Library of Congress Cataloging CIP-Kurztitelaufnahme
in Publication Data der Deutschen Bibliothek
Brezinski, Claude, 1941- Brezinski, Claude:
Pade-type approximation and general Pade-type approximation and general
orthogonal polynomials. orthogonal polynomials / by Claude
(International series of numerical Brezinski. - Basel, Boston, Stuttgart:
mathematics; 50 Birkhiiuser,1980.
Bibliography: p.250 (International series of numerical
Includes index. mathematics; Vol. 50)
I. Orthogonal polynomials.
2. Pade approximant.
I. Title. II. Series. All rights reserved. No part of this
QA404.4B73 515'.55 79-21387 publication may be reproduzed, stored
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© Springer Basel AG 1980
Originally published by Birkhauser Verlag Basel in 1980.
Softcover reprint of the hardcover 1s t edition 1980
ISBN 978-3-0348-6559-3 ISBN 978-3-0348-6558-6 (eBook)
DOI 10.1007/978-3-0348-6558-6
Contents
Introduction. . . . . . . . . . 7
Chapter 1: Pade-type approximants. . 9
1.1 Definition of the approximants. 9
1.2 Basic properties. . . . . 14
1.3 Convergence theorems .24
1.4 Some applications . . . . · 28
1.5 Higher order approximants . .32
Chapter 2: General orthogonal polynomials . .40
2.1 Definition.... .40
2.2 Recurrence relation . . . . .43
2.3 Algebraic properties . . . . · 50
2.3.1 Christoffel-Darboux relation. · 50
2.3.2 Reproducing kernels. . . . · 51
2.3.3 Associated polynomials. . . · 53
2.4 Properties of the zeros · 57
2.5 Interpolatory quadrature methods. · 61
2.6 Matrix formalism 67
2.6.1 Jacobi matrices . . . . . . · 67
2.6.2 Matrix interpretation . . . . .69
2.7 Orthogonal polynomials and projection methods .. 75
2.7.1 The moment method · 75
2.7.2 Lanczos method . . . . . . . . . . .79
2.7.3 Conjugate gradient method ..... · 84
2.8 Adjacent systems of orthogonal polynomials. · 91
2.9 Reciprocal orthogonal polynomials . . . 105
2.10 Positive functionals. . . . . . . . . 115
Chapter 3: Pade approximants and related matters 126
3.1 Pade approximants. . . . . . . . . 126
3.1.1 Determinantal expression and matrix formalism 126
3.1.2 The cross rule. . . . . . . . . . . . 133
3.1.3 Recursive computation of Pade approximants . 135
3.1.4 Normality 147
3.2 Continued fractions. . 152
3.3 The scalar B-algorithm 159
3.3.1 The algorithm . 159
3.3.2 Connection with orthogonal polynomials. 171
3.3.3 Connection with the moment method 176
Chapter 4: Generalizations 178
4.1 The topological B-algorithm. . . . . 178
6 Contents
4.1.1 The algorithm. . . . . . 178
4.1.2 Solution of equations. . . . 184
4.2 Double power series . . . . 190
4.2.1 Definition of the approximants 191
4.2.2 Basic properties . . . . 195
4.2.3 Higher order approximants 208
4.3 Series of functions 220
Appendix . 227
Bibliography 240
Index 249
Introduction
In the last few years Pade approximants became more and more widely used
in various fields of physics, chemistry and mathematics. They provide
rational approximations to functions which are formally defined by a power
series expansion. Pade approximants are also closely related to some
methods which are used in numerical analysis to accelerate the convergence
of sequences and iterative processes.
Many books recently appeared on this subject dealing with algebraic
properties, study of the convergence, applications and so on. Chapters on
Pade approximants can also be found in older books on continued fractions
because these two subjects have a strong connection. The scope of this book
is quite different. A Parle approximant is defined so that its power series
expansion matches the power series to be approximated as far as possible.
This property completely defines the denominator as well as the numerator
of the Pade approximant under consideration. The trouble arising with Pade
approximants is the location of the poles, that is the location of the zeros of
the denominator. One has no control on these poles and it is impossible to
force them to be in some region of the complex plane.
This was the reason for the definition and study of the so-called
Pade-type approximants. In such approximants it is possible to choose some
of the poles and then to define the denominator and the numerator so that the
expansion of the approximant matches the series to be approximated as far
as possible. On one hand it is possible to choose all the poles and, on the
other hand, it is possible to choose no pole - which is nothing but the
definition of a Pade approximant. Such an approach to the problem directly
leads to the introduction of general orthogonal polynomials into the theory
of Pade approximants. This connection, known for a long time, had not been
fully exploited. Thus the aim of this book is twofold: first to introduce
Pade-type approximants and secondly to study Pade approximants on the
basis of general orthogonal polynomials. The complete algebraic theory of
Pade approximants is unified on this basis; old and new results can be easily
obtained such as recurrence schemes for computing Pade approximants,
error formulas, matrix interpretation and so on. Properties of some con
vergence acceleration methods for sequences of numbers or for sequences of
vectors can be also derived from the theory. Pade-type approximants are
also very useful in applications since they can provide better results than
Pade approximants.
The material contained in this book leads to many research problems,
especially in the new field of Pade-type approximants. Many results appear
for the first time.
The contents are as follows. Chapter 1 deals with the definition
and general properties of Pade-type approximants. General orthogonal
8 Introduction
polynomials are studied in Chapter 2. Chapter 3 contains the theory of Pade
approximants and related matters as derived from orthogonal polynomials.
Chapter 4 is devoted to the study of some generalizations of Pade-type
approximants.
As far as possible this book has been written as to be self contained.
This book was partly written during a stay at the Tata Institute of Funda
mental Research, Bangalore, India. It is a pleasure to thank Professor K. G.
Ramanathan for his kind invitation to Bangalore and for providing me all
the facilities. I also acknowledge the T.I.F.R. for its financial support. I wish
to thank Professor J. Todd who accepted the book for publication in this
series and encouraged me for a long time. He also greatly improved my
insecure English. I am grateful to Professor A. Ruttan who checked
some parts of the manuscript. Sincere thanks are also due to many friends
and colleagues for their comments and especially to Professor R. S. Varga.
Finally I thank Mr. C. Einsele and his staff of Birkhauser Verlag, Basel, for
assistance during the editing.
Chapter 1
Pade-type approximants
1.1. Definition of the approximants
Let f be a formal power series in one variable
=
f(t) = L c/,
i=O
This is a formal equality in the sense that if the series on the right-hand side
converges for some t then f(t) is equal to its sum; if the series diverges f
represents its analytic continuation (assumed to exist). In the sequel we shall
only be dealing with formal power series and formal equalities which means
that the series developments of both sides of an equality are the same.
Our purpose is to construct a rational fraction whose denominator has
degree k and whose numerator has degree k - 1 so that its expansion in
ascending powers of t coincides with the expansion of f up to the degree
k-l.
There are several reasons for looking for such rational approximations
to series. The first is to obtain an approximation to a function which can, for
example, be used in computations. The second is that the series may
converge too slowly to be of any use and that we want to accelerate its
convergence. The third reason is that only few coefficients of the series may
be known and that a good approximation to the series is needed to obtain
properties of the function that it represents.
Let us now begin our investigation by defining a linear functional acting
on the space of real polynomials by
C(Xi) = Ci, for i = 0, 1, ....
The number Ci is called the moment of order i of the functional c.
Lemma 1.1.
f(t) = c«l-xt)-l).
Proof. Let us formally expand (1-Xt)-l in a power series, then
c«l-xt)-l) = c(l + xt + x2t2 + ...)
= c(1)+c(x)t+c(x2)f+ ...
= co+C1t+C2t2+ ... = f(t). •
Let v be an arbitrary polynomial of degree k,
v(x) = bo+ b1x + ... + bkxk,
