Page 6 - Compact Numerical Methods For Computers

Contents vii
14. DIRECT SEARCH METHODS 168
14.1. The Nelder-Mead simplex search for the minimum of a
function of several parameters 168
14.2. Possible modifications of the Nelder-Mead algorithm 172
14.3. An axial search procedure 178
14.4. Other direct search methods 182
15. DESCENT TO A MINIMUM I: VARIABLE METRIC
ALGORITHMS 186
15.1. Descent methods for minimisation 186
15.2. Variable metric algorithms 187
15.3. A choice of strategies 190
16. DESCENT TO A MINIMUM II: CONJUGATE GRADIENTS 197
16.1. Conjugate gradients methods 197
16.2. A particular conjugate gradients algorithm 198
17. MINIMISING A NONLINEAR SUM OF SQUARES 207
17.1. Introduction 207
17.2. Two methods 208
17.3. Hartley’s modification 210
17.4. Marquardt’s method 211
17.5. Critique and evaluation 212
17.6. Related methods 215
18. LEFT-OVERS 218
18.1. Introduction 218
18.2. Numerical approximation of derivatives 218
18.3. Constrained optimisation 221
18.4. A comparison of function minimisation and nonlinear least-
squares methods 226

19. THE CONJUGATE GRADIENTS METHOD APPLIED TO
PROBLEMS IN LINEAR ALGEBRA 234
19.1. Introduction 234
19.2. Solution of linear equations and least-squares problems by
conjugate gradients 235
19.3. Inverse iteration by algorithm 24 241
19.4. Eigensolutions by minimising the Rayleigh quotient 243
APPENDICES 253
1. Nine test matrices 253
2. List of algorithms 255
3. List of examples 256
4. Files on the software diskette 258

BIBLIOGRAPHY 263

INDEX 271