Compact Numerical Methods for Computers_ - Nash_ J. C_.pdf

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COMPACT NUMERICAL
METHODS
FOR COMPUTERS
linear algebra and
function minimisation
Second Edition
J C NASH
Adam Hilger, Bristol and New York
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Copyright © 1979, 1990 J C Nash
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system
or transmitted in any form or by any means, electronic, mechanical, photocopying or
otherwise, without the prior permission of the publisher. Multiple copying is only permitted
under the terms of the agreement between the Committee of Vice-Chancellors and Principals
and the Copyright Licensing Agency.
British Library Cataloguing in Publication Data
Nash, J. C.
Compact numerical methods for computers: linear algebra
and function minimisation - 2nd ed.
1!. Numerical analysis. Applications of microcomputer &
minicomputer systems. Algorithms
I. Title
519.4
ISBN 0-85274-318-1
ISBN 0-85274-319-X (pbk)
ISBN 0-7503-0036-1 (5¼" IBM disc)
ISBN 0-7503-0043-4 (3½" IBM disc)
Library of Congress Cataloging-in-Publication Data are available
First published, 1979
Reprinted, 1980
Second edition, 1990
Published under the Adam Hilger imprint by IOP Publishing Ltd
Techno House, Redcliffe Way, Bristol BSl 6NX, England
335 East 45th Street, New York, NY 10017-3483, USA
Filmset by Bath Typesetting Ltd, Bath, Avon
Printed in Great Britain by Page Bros (Norwich) Ltd
CONTENTS
Preface to the Second Edition
Preface to the First Edition
ix
xi
1
1
3
9
11
13
15
17
17
1. A STARTING POINT
1.1. Purpose and scope
1.2. Machine characteristics
1.3. Sources of programs
1.4. Programming languages used and structured programming
1.5. Choice of algorithms
1.6. A method for expressing algorithms
1.7. General notation
1.8. Software engineering issues
19
19
19
21
24
26
28
2. FORMAL PROBLEMS IN LINEAR ALGEBRA
2.1. Introduction
2.2. Simultaneous linear equations
2.3. The linear least-squares problem
2.4. The inverse and generalised inverse of a matrix
2.5. Decompositions of a matrix
2.6. The matrix eigenvalue problem
3. THE SINGULAR-VALUE DECOMPOSITION AND ITS USE
TO SOLVE LEAST-SQUARES PROBLEMS
3.1. Introduction
3.2. A singular-value decomposition algorithm
3.3. Orthogonalisation by plane rotations
3.4. A fine point
3.5. An alternative implementation of the singular-value decomposi-
tion
3.6. Using the singular-value decomposition to solve least-squares
problems
30
30
31
32
35
38
40
49
4. HANDLING LARGER PROBLEMS
49
4.1. Introduction
49
4.2. The Givens’ reduction
54
4.3. Extension to a singular-value decomposition
54
4.4. Some labour-saving devices
63
4.5. Related calculations
5. SOME COMMENTS ON THE FORMATION OF THE CROSS-
PRODUCTS MATRIX A T A
66
v
vi Compact numerical methods for computers
6. LINEAR EQUATIONS-A DIRECT APPROACH
6.1. Introduction
6.2. Gauss elimination
6.3. Variations on the theme of Gauss elimination
6.4. Complex systems of equations
6.5. Methods for special matrices
72
72
72
80
82
83
84
84
7. THE CHOLESKI DECOMPOSITION
7.1. The Choleski decomposition
7.2. Extension of the Choleski decomposition to non-negative defi-
nite matrices
7.3. Some organisational details
86
90
94
94
8. THE SYMMETRIC POSITIVE DEFINITE MATRIX AGAIN
8.1. The Gauss-Jordan reduction
8.2. The Gauss-Jordan algorithm for the inverse of a symmetric
positive definite matrix
97
9. THE ALGEBRAIC EIGENVALUE PROBLEM
9.1. Introduction
9.2. The power method and inverse iteration
9.3. Some notes on the behaviour of inverse iteration
9.4. Eigensolutions of non-symmetric and complex matrices
102
102
102
108
110
119
119
121
10. REAL SYMMETRIC MATRICES
10.1. The eigensolutions of a real symmetric matrix
10.2. Extension to matrices which are not positive definite
10.3. The Jacobi algorithm for the eigensolutions of a real symmetric
matrix
10.4. Organisation of the Jacobi algorithm
10.5. A brief comparison of methods for the eigenproblem of a real
symmetric matrix
126
128
133
11. THE GENERALISED SYMMETRIC MATRIX EIGENVALUE
PROBLEM
135
142
12. OPTIMISATION AND NONLINEAR EQUATIONS
12.1. Formal problems in unconstrained optimisation and nonlinear
equations
12.2. Difficulties encountered in the solution of optimisation and
nonlinear-equation problems
142
146
148
148
148
160
13. ONE-DIMENSIONAL PROBLEMS
13.1. Introduction
13.2. The linear search problem
13.3. Real roots of functions of one variable

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