Methods of Numerical Approximation. Lectures Delivered at a Summer School Held at Oxford University, September 1965

203 Pages·1966·14.39 MB·German

by  D.C. ed. HANDSCOMB

METHODS OF
NUMERICAL
APPROXIMATION
Lectures delivered at a Summer School
held at Oxford University
September, 1965
Edited by
D. C. HANDSCOMB
Oxford University Computing Laboratory
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Copyright © 1966
Pergamon Press Ltd.
First edition 1966
Library of Congress Catalog Card No. 66-23045
PRINTED IN GREAT BRITAIN BY BELL AND BAIN LTD., GLASGOW
2955/66
Lecturers taking part:
J. D. P. Donnelly
L. Fox
Oxford University Computing Laboratory
D. C. Handscomb
D. F. Mayers
A. R. Curtis Ί Atomic Energy Research Establishment,
M. J. D. Powell J Harwell
EDITOR'S PREFACE
FOR several years now it has been the practice of Oxford University Com­
puting Laboratory, with the co-operation of external lecturers and of the
Delegacy for Extra-Mural Studies, to arrange short Summer Schools in
Oxford on various topics in or around the field of computation. This book is
based on the Summer School held in September 1965, which was attended by
some seventy people drawn mainly from university and other computing
installations throughout Great Britain.
The Editor would like to take this opportunity to record his gratitude to
Professor Fox for the invitation to organize this course and for his much-
needed encouragement and advice, to all his fellow authors for the long time
spent in preparation and the short time spent in redrafting for publication, to
the staff of the Delegacy for the smoothness of all the administrative and
domestic arrangements, to Miss O. Moon for accurately and uncomplainingly
typing the manuscript, and to the staff of the Pergamon Press for seeing it
through to publication. Finally he must thank all the participants in the
Summer School, and particularly those whose criticisms or other remarks
have helped him in the making of this book.
D. C. HANDSCOMB
IX
CHAPTER 1
INTRODUCTION
D. C. HANDSCOMB
THE subject of this book is the approximation of functions of one or more
variables by means of more elementary functions, regarded as a tool in
numerical computation. This is not to say that we present the reader with
rules of thumb so that he can programme for his computer without further
thought, for none of us would claim to be able to predict the best method to
apply to any particular problem. On the other hand we have been somewhat
selective in discarding material of purely theoretical interest while retaining
everything that seems practically relevant.
Some may have thought, on glancing at the table of contents, that we should
have been even more selective; why, for instance, pay so much attention to
questions of uniqueness or rates of convergence? The answer to this question
lies partly in the fact that most present-day computing is performed by
automatic machinery. Previously, when the numerical analyst had to do his
own calculation with desk machine, pencil and paper, he could leave many
decisions to his own instinctive feel of the direction the computation was
taking. The automatic computer, on the other hand, has no instincts, and
needs to be told under precisely what conditions to take any course of action.
It is therefore advisable for the programmer to find out at least what even­
tualities he need or need not allow for. Quite apart from such practical
reasons, however, we believe that it is fatal to productive research for the
specialist to be completely ignorant of the foundations or the applications of
his speciality, as so many specialists tend to be.
The foundations of the modern theory of approximation were laid in the
middle of the nineteenth century by P. L. Chebyshev (whose name has
appeared in various other spellings) in the course of some very practical
investigations. His work began by considering approximation by poly­
nomials alone. Other systems of approximating functions have since been
taken into consideration; in this book we mention the systems of trigono­
metric sums, rational functions, continued fractions, and spline functions, and
one or two others. These systems may be divided into linear (polynomials,
trigonometric sums, and splines) and non-linear. The theory and techniques
connected with the linear systems are, naturally, the more highly developed;
3
4 D. C. HANDSCOMB
nevertheless a look at the bibliography will show that all departments of the
subject are becoming more and more active.
One reason for the sudden surge of interest in approximating functions in
recent years is again the rise of the automatic digital computer. Previously
the most convenient way of representing any function was by means of a table
of values, this being the form that enabled the human computer most easily
to see what was happening, and to avoid, detect, or correct errors. Tables of
values are far less suitable for an automatic computer, though. For one
thing, they fill up a great deal of storage with information most of which (such
as the first digits of a slowly-varying function) is strictly redundant, when
the computer still has to be told how to find its place in this table and how to
interpolate; on the other hand the computer can manipulate quantities with
few or many significant figures with equal ease, and so has no reason to
prefer tabular values.
Another point that could be made here is that the automatic computer is
particularly well suited to long iterative processes, such as are called for in
searching for "best" approximations. Such approximations have thus for the
first time acquired more than academic interest.
We have in this book concentrated most of our attention on the approxi­
mation of "mathematical" functions, meaning functions which can in
principle be determined as exactly as we please (defined, perhaps, by a
precisely-formulated differential equation or a contour integral). We have
said less, possibly less than we should, about the approximation of experi­
mental data. The latter problem differs from the former in important
respects. First of all one is not often able to choose the points at which
observations are to be made—they may well have been made at some time in
the past. Secondly, and most important, there is a definite limit to their
accuracy—usually two or three significant figures. And thirdly the approxi­
mating function may be required to conform to a preconceived theory, either
by taking some special form or by being nearly-linear, smooth, monotone, or
otherwise "well behaved". In such circumstances there is one injunction
above all to bear in mind: do not try to approximate the data more closely
than their experimental error warrants; too close a fit usually means that one
is using too complicated a function, and probably getting spurious humps
and inflexions.
The plan of the book is simple. After a chapter of fundamental definitions
the second section deals with linear processes of approximation—much of
this material is covered in standard texts. The third section deals with
rational functions and continued fractions, linking the latter with the QD
and ε algorithms although the main scope of these algorithms has little to do
with approximation. The final section deals with miscellaneous topics,
including some of the most recent developments.
INTRODUCTION 5
When dealing with theoretical matters we have concentrated on results and
methods of proof rather than the proofs themselves; when the reader can
easily reconstruct a proof or find it in a convenient text we have often
abbreviated or omitted it. The same can be said on the other side; when
explaining a practical method we have tried not to obscure the principle with
minute detail.
We hope that we have succeeded in presenting a comprehensive and up-to-
date survey of the approximation methods available, with some indication of
the virtues and vices associated with each, and in stimulating some readers to
explore this subject more deeply.
CHAPTER 2
SOME ABSTRACT CONCEPTS AND
DEFINITIONS
D. C. HANDSCOMB
1. INTRODUCTION
It is possible to discuss the subject of approximation without going beyond
the bounds of "traditional" analysis, and indeed many branches of the
subject do not lend themselves to any other treatment. The reader should
find, however, that some familiarity with a few of the modern relevant
concepts is a great spur to constructive thinking, besides being essential in
order to understand much of the current literature.
A fundamental concept is that of a function as a thing in itself. One can
then regard functions in some respects as if they were points of a geometric
vector space (often actually spoken of as "function-space") and make use of
geometrical reasoning, speaking for instance of "convex sets" of functions or
of the "distance" between functions.
The usefulness of this quasi-geometric approach, which is that of functional
analysis, lies mainly in establishing general theorems from which many
diverse results may be deduced as special cases. A price that we pay for this
convenience is that we are discouraged from following certain lines of thought
that cannot be made to fit in with the general scheme (cf. Hammer, 1964)*.
The important instance of this is the study of non-linear, and particularly
rational, approximations, which are discussed later in this book.
Even when dealing with linear approximation, however, we have often
found it easier to present the material in traditional form. Nevertheless, we
feel that the reader who ignores the abstract side does so to his loss, if only
because he may not appreciate the underlying pattern to all that is done.
This account will be brief. A more comprehensive account will be found
in Buck (1959), or in any of the books on approximation theory such as
Achieser (1947), Natanson (1949), Timan (1960), Davis (1963), or Rice (1964).
* Full references are given on pp. 199-214.
7
B
8 D. C. HANDSCOMB
2. FUNCTION SPACES
Let F be a set of functions defined over a domain A\ so that/(x) is defined
(real or complex) and one-valued for every xeX and every feF.
Notice that we do not restrict the nature of X; it could for instance be a
finite set of points, an interval [a, b] of the real line, or the whole of 3-dimen-
sional complex Euclidean space. We do, however, restrict the values of
f(x) to be real or (sometimes) complex.
The functions of F may be treated as if they were vectors (of possibly
infinitely many components), by defining vector addition and multiplication
by a scalar in the natural manner:
f+g=h if h(x)=f(x) + g(x) for all xeX (1)
Xf=h if h(x) = λ/(χ) for all xeX (2)
where/, g, and h are members of F and λ is a scalar. The analogy with
ordinary vectors may be made clear by considering the special case where X
is the set of integers {1, 2, ... , n}, when any function/on X may be described
by the vector of its values {/(l),/(2),... ,/(«)}. The rules (1) and (2) are then
simply the rules for vector addition and multiplication in n dimensions.
The set F is a linear vector space, or simply a vector space, over the field of
real (or complex) numbers if it is closed under addition and multiplication;
in other words, if
f+geF for all fgeF (3)
XfeF for all feF and for all real (complex) λ. (4)
The set of all functions on X is ajvector space, but in practice one normally
considers some closed subspace of this. Some important subspaces are
(a) the space of bounded functions, denoted by B{X),
(b) (if X is an interval, for example) the space of continuous functions,
denoted by C(X),
(c) the space of all functions of the form Xlf1+X2f2 + --+KL· where
fufi^ ·" >fn are given; this is the space spanned byfl,f2, ... ,/,.
Here we may introduce the idea of convexity. A set S of functions is
convex if, for all real λ in the interval [0, 1],
Xf+(l-X)geS for all fgeS (5)
or, in geometrical terms, if the line-segment joining any two points of S lies
wholly in S. The set of functions of the form kxfx +... +X„fn9 with
A^O, ...,^Ο,Σλ,- = 1,
is a convex set and is called the convex hull off, ...,/,.
SOME ABSTRACT CONCEPTS AND DEFINITIONS 9
3. METRICS, NORMS AND SEMINORMS
Any discussion of approximation requires that we have a way of measuring
the discrepancy between two functions. If quasi-geometric arguments are to
work it is reasonable to demand that this measure should have some of the
properties of geometrical distance, or in formal terms that it should be a
metric.
A *functional d(f9 g) is a metric on a set F(not necessarily a vector space)
if it satisfies the following requirements:
d(f, g) is defined, real, and non-negative, for all/, geF (6)
d(f9g)=0 if and only if j = g (7)
d(f,g) = d(g,f) (8)
d(f9h) ^ d(f,g) + d(g,h). (9)
The most general form of metric, as here defined, is not rich enough in its
properties, however, to yield enough useful results, and we do better to
confine our attention to a more restricted class of metric that is defined on
linear vector spaces alone.
A functional n(f) is a norm on a linear vector space F if it satisfies the
following:
n(f) is defined, real, and non-negative, for all feF (10)
n(f) = 0 if and only if /= 0 (that is, if/(x) = 0 for all xeX) (11)
n(kf) = \k\n(f) for every scalar k (12)
n(f+g)£n(f) + n(g). (13)
The notation n(f)=\\f\\, with or without distinguishing subscripts, is
generally employed for a norm.
It follows from (10) to (13) that any norm gives rise to a metric
d(f,g) = \\f-g\\ (14)
and almost all approximation theory is constructed with reference to a metric
of this form.
If n(f) satisfies (10), (12), (13) but not (11) except to the extent that
«(0) = 0, so that there are non-zero functions with n(f) = 0, then n is a
seminorm.
If a norm satisfies the stronger condition that n(f+g) <n(f)+n(g), unless
g = 0 or / = kg for some k, then it is a strict norm.
* A functional is a function whose arguments are functions; we use the term here merely
to avoid confusion. We examine linear functional in more detail in Chapter 20.