Compact numerical methods

CONTENTS......Page 4
Preface to the Second Edition......Page 7
Preface to the First Edition......Page 9
1.1. Purpose and scope......Page 11
1.2. Machine characteristics......Page 13
1.3. Sources of programs......Page 19
1.4. Programming languages used and structured programming......Page 21
1.5. Choice of algorithms......Page 23
1.6. A method for expressing algorithms......Page 25
1.8. Software engineering issues......Page 27
2.2. Simultaneous linear equations......Page 29
2.3. The linear least-squares problem......Page 31
2.4. The inverse and generalised inverse of a matrix......Page 34
2.5. Decompositions of a matrix......Page 36
2.6. The matrix eigenvalue problem......Page 38
3.1. Introduction......Page 40
3.2. A singular-value decomposition algorithm......Page 41
3.3. Orthogonalisation by plane rotations......Page 42
3.4. A fine point......Page 45
3.5. An alternative implementation of the singular-value decomposition......Page 48
3.6. Using the singular-value decomposition to solve least-squares problems......Page 50
4.2. The Givens\' reduction......Page 59
4.4. Some labour-saving devices......Page 64
4.5. Related calculations......Page 73
5. SOME COMMENTS ON THE FORMATION OF THE CROSS- PRODUCTS MATRIX ATA......Page 76
6.2. Gauss elimination......Page 82
6.3. Variations on the theme of Gauss elimination......Page 90
6.4. Complex systems of equations......Page 92
6.5. Methods for special matrices......Page 93
7.1. The Choleski decomposition......Page 94
7.2. Extension of the Choleski decomposition to non-negative definite matrices......Page 96
7.3. Some organisational details......Page 100
8.1. The Gauss-Jordan reduction......Page 104
8.2. The Gauss-Jordan algorithm for the inverse of a symmetric positive definite matrix......Page 107
9.2. The power method and inverse iteration......Page 112
9.3. Some notes on the behaviour of inverse iteration......Page 118
9.4. Eigensolutions of non-symmetric and complex matrices......Page 120
10.1. The eigensolutions of a real symmetric matrix......Page 129
10.2. Extension to matrices which are not positive definite......Page 131
10.3. The Jacobi algorithm for the eigensolutions of a real symmetric matrix......Page 136
10.4. Organisation of the Jacobi algorithm......Page 138
10.5. A brief comparison of methods for the eigenproblem of a real symmetric matrix......Page 143
11. THE GENERALISED SYMMETRIC MATRIX EIGENVALUE PROBLEM......Page 145
12.1. Formal problems in unconstrained optimisation and nonlinear equations......Page 152
12.2. Difficulties encountered in the solution of optimisation and nonlinear-equation problems......Page 156
13.2. The linear search problem......Page 158
13.3. Real roots of functions of one variable......Page 170
14.1. The Nelder-Mead simplex search for the minimum of a function of several parameters......Page 178
14.2. Possible modifications of the Nelder-Mead algorithm......Page 182
14.3. An axial search procedure......Page 188
14.4. Other direct search methods......Page 192
15.1. Descent methods for minimisation......Page 196
15.2. Variable metric algorithms......Page 197
15.3. A choice of strategies......Page 200
16.1. Conjugate gradients methods......Page 207
16.2. A particular conjugate gradients algorithm......Page 208
17.1. Introduction......Page 217
17.2. Two methods......Page 218
17.3. Hartley\'s modification......Page 220
17.4. Marquardt\'s method......Page 221
17.5. Critique and evaluation......Page 222
17.6. Related methods......Page 225
18.2. Numerical approximation of derivatives......Page 228
18.3. Constrained optimization......Page 231
18.4. A comparison of function minimisation and nonlinear least-squares methods......Page 236
19.1. Introduction......Page 244
19.2. Solution of linear equations and least-squares problems by conjugate gradients......Page 245
19.3. Inverse iteration by algorithm......Page 251
19.4. Eigensolutions by minimising the Rayleigh quotient......Page 253
1. Nine test matrices......Page 263
2. List of algorithms......Page 265
3. List of examples......Page 266
4. Files on the software diskette......Page 268
BIBLIOGRAPHY......Page 273
INDEX......Page 281