Theory and Applications of Numerical Analysis, 2 ed, Elsevier (1996).part1.rar

Theory and Applications of Numerical Analysis, 2 ed, Elsevier (1996)



Theory and Applications of
Numerical Analysis
Second edition, by G. M. M. Phillips, Peter J. Taylor

  ISBN: 0125535600
· Pub. Date: September 1996
· Publisher: Elsevier Science & Technology Books

PREFACE
Although no text of reasonable length can cover all possible topics, it
seemed to us that the first edition of this text would be improved by
including material on two further items. Thus the main change in the second
edition is the addition of two entirely new chapters, Chapter 6 (Splines and
other approximations) and Chapter 11 (Matrix eigenvalues and eigenvectors).
In the preface to the first edition we stated that the material was
equivalent to about sixty lectures. This new edition has adequate material for
two separate semester courses.
When this book first appeared in 1973, computers were much less
'friendly' than they are now (1995) and, in general, it was only specialists
who had access to them. The only calculating aids available to the majority
of our readers were mathematical tables and desk calculators; the latter were
mainly mechanical machines, which are now museum pieces. In contrast,
the readership of our second edition will be able to appreciate the power and
elegance of the algorithms which we discuss by implementing them, and
experimenting with them, on the computer. (Sounding like our parents of
old, we can say to our students 'We never had it so easy: you are lucky!')
To encourage the active pursuit of the algorithms in the text, we have
included computing exercises amongst the problems at the end of each
chapter.
Despite the changes made in the second edition, the flavour of this text
remains the same, reflecting the authors' tastes: in short, we like both
theorems and algorithms and we remain in awe of the masters of our craft
who discovered or created them. The theorems show how the mathematics
hangs together and point the way to the algorithms.
We are deeply grateful to our readership and to our publishers for keeping
this text in print for such a long time. We also owe much to the many
colleagues from many countries with whom we have shared the sheer fun of
studying mathematics in our research collaborations, and we mention their
X Preface
names here as a sign of our respect and gratitude: G. E. Bell, L. Brutman,
B. L. Chalmers, E. W. Cheney, F. Deutsch, D. K. Dimitrov, D. Elliott, Feng
Shun-xi, D. M. E. Foster, H. T. Freitag, R. E. Grundy, A. S. B. Holland,
Z. F. Koqak, S. L. Lee, W. Light, J. H. McCabe, A R. Mitchell, D. F. Paget,
A. Sri Ranga, B. N. Sahney, S. P. Singh, E. L. Wachspress, M. A. Wolfe,
D. Yahaya.
G. M. PHILLIPS
Mathematical Institute
University of St Andrews
Scotland
December 1995
P. J. TAYLOR
formerly of the
Department of Mathematics
University of Strathclyde
Scotland

FROM THE PREFACE TO THE FIRST EDITION
We have written this book as an introductory text on numerical analysis for
undergraduate mathematicians, computer scientists, engineers and other
scientists. The material is equivalent to about sixty lectures, to be taken after
a first year calculus course, although some calculus is included in the early
chapters both to refresh the reader's memory and to provide a foundation on
which we may build. We do not assume that the reader has a knowledge of
matrix algebra and so have included a brief introduction to matrices. It
would, however, help the understanding of the reader if he has taken a basic
course in matrix algebra or is taking one concurrently with any course based
on this book.
We have tried to give a logical, self-contained development of our
subject ab initio, with equal emphasis on practical methods and mathematical
theory. Thus we have stated algorithms precisely and have usually given
proofs of theorems, since each of these aspects of the subject illuminates
the other. We believe that numerical analysis can be invaluable in the
teaching of mathematics. Numerical analysis is well motivated and uses
many important mathematical concepts. Where possible we have used the
theme of approximation to give a unified treatment throughout the text.
Thus the different types of approximation introduced are used in the
chapters on the solution of non-linear algebraic equations, numerical
integration and differential equations.
It is not easy to be a good numerical analyst since, as in other branches of
mathematics, it takes considerable skill and experience to be able to leap
nimbly to and fro from the general and abstract to the particular and
practical. A large number of worked examples has been included to help the
reader develop these skills. The problems, which are given at the end of
each chapter, are to be regarded as an extension of the text as well as a test
of the reader's understanding.
Some of our colleagues have most kindly read all or part of the
manuscript and offered wise and valuable advice. These include Dr J. D.
xii From the preface to the first edition
Lambert, Prof. P. Lancaster, Dr J. H. McCabe and Prof. A. R. Mitchell. Our
thanks go to them and, of course, any errors or omissions which remain are
entirely our responsibility We are also much indebted to our students and to
our colleagues for the stimulus given by their encouragement.
G. M. PHILLIPS
Department of Applied Mathematics
University of St Andrews
Scotland
P. J. TAYLOR
Department of Computing Science
University of Stirling
Scotland
December 1972

Table of Contents
Preface
From the preface to the first edition
1 Introduction 1
2 Basic analysis 11
3 Taylor's polynomial and series 39
4 The interpolating polynomial 52
5 'Best' approximation 86
6 Splines and other approximations 131
7 Numerical integration and differentiation 160
8 Solution of algebraic equations of one variable 196
9 Linear equations 221
10 Matrix norms and applications 265
11 Matrix eigenvalues and eigenvectors 299
12 Systems of non-linear equations 323
13 Ordinary differential equations 335
14
Boundary value and other methods for ordinary
differential equations
396
App Computer arithmetic 418
Solutions to selected problems 424
References and further reading 440
Index 443

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