Page 269 - Compact Numerical Methods For Computers

Appendix 3

LIST OF EXAMPLES

Example 2.1. Mass-spectrograph calibration 20
Example 2.2. Ordinary differential equations: a two-point boundary
value problem 20
Example 2.3. Least squares 23
Example 2.4. Surveying-data fitting 24
Example 2.5. Illustration of the matrix eigenvalue problem 28
Example 3.1. The generalised inverse of a rectangular matrix via the
singular-value decomposition 44
Example 3.2. Illustration of the use of algorithm 2 45
Example 4.1. The operation of Givens’ reduction 52
Example 4.2. The use of algorithm 4 62
Example 6.1. The use of linear equations and linear least-squares prob-
lems 77
Example 7.1. The Choleski decomposition of the Moler matrix 91
Example 7.2. Solving least-squares problems via the normal equations 92
Example 8.1. The behaviour of the Bauer-Reinsch Gauss-Jordan in-
version 100
Example 9.1. Inverse iteration 108
Example 9.2. Eigensolutions of a complex matrix 117
Example 10.1. Principal axes of a cube 125
Example 10.2. Application of the Jacobi algorithm in celestial mechanics 131
Example 11.1. The generalised symmetric eigenproblem: the anhar-
monic oscillator 138
Example 12.1. Function minimisation-optimal operation of a public
lottery 144
Example 12.2. Nonlinear least squares 144
Example 12.3. An illustration of a system of simultaneous nonlinear
equations 144
Example 12.4. Root-finding 145
Example 12.5. Minimum of a function of one variable 146
Example 13.1. Grid and linear search 156
Example 13.2. A test of root-finding algorithms 164
Example 13.3. Actuarial calculations 165
Example 14.1. Using the Nelder-Mead simplex procedure (algorithm 19) 180
Example 15.1. Illustration of the variable metric algorithm 21 196
Example 16.1. Conjugate gradients minimisation 204
Example 17.1. Marquardt’s minimisation of a nonlinear sum of squares 216
Example 18.1. Optimal operation of a public lottery 228
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