Page 4 - Compact Numerical Methods For Computers

CONTENTS

Preface to the Second Edition ix
Preface to the First Edition xi

1. A STARTING POINT 1
1.1. Purpose and scope 1
1.2. Machine characteristics 3
1.3. Sources of programs 9
1.4. Programming languages used and structured programming 11
1.5. Choice of algorithms 13
1.6. A method for expressing algorithms 15
1.7. General notation 17
1.8. Software engineering issues 17

2. FORMAL PROBLEMS IN LINEAR ALGEBRA 19
2.1. Introduction 19
2.2. Simultaneous linear equations 19
2.3. The linear least-squares problem 21
2.4. The inverse and generalised inverse of a matrix 24
2.5. Decompositions of a matrix 26
2.6. The matrix eigenvalue problem 28

3. THE SINGULAR-VALUE DECOMPOSITION AND ITS USE
TO SOLVE LEAST-SQUARES PROBLEMS 30
3.1. Introduction 30
3.2. A singular-value decomposition algorithm 31
3.3. Orthogonalisation by plane rotations 32
3.4. A fine point 35
3.5. An alternative implementation of the singular-value decomposi-
tion 38
3.6. Using the singular-value decomposition to solve least-squares
problems 40

4. HANDLING LARGER PROBLEMS 49
4.1. Introduction 4 9
4.2. The Givens’ reduction 49
4.3. Extension to a singular-value decomposition 54
4.4. Some labour-saving devices 54
4.5. Related calculations 63
5. SOME COMMENTS ON THE FORMATION OF THE CROSS-
T
PRODUCTS MATRIX A A 66
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