Page 5 - Compact Numerical Methods For Computers

vi Compact numerical methods for computers
6. LINEAR EQUATIONS-A DIRECT APPROACH 72
6.1. Introduction 72
6.2. Gauss elimination 72
6.3. Variations on the theme of Gauss elimination 80
6.4. Complex systems of equations 82
6.5. Methods for special matrices 83

7. THE CHOLESKI DECOMPOSITION 84
7.1. The Choleski decomposition 84
7.2. Extension of the Choleski decomposition to non-negative defi-
nite matrices 86
7.3. Some organisational details 90

8. THE SYMMETRIC POSITIVE DEFINITE MATRIX AGAIN 94
8.1. The Gauss-Jordan reduction 94
8.2. The Gauss-Jordan algorithm for the inverse of a symmetric
positive definite matrix 97

9. THE ALGEBRAIC EIGENVALUE PROBLEM 102
9.1. Introduction 102
9.2. The power method and inverse iteration 102
9.3. Some notes on the behaviour of inverse iteration 108
9.4. Eigensolutions of non-symmetric and complex matrices 110

10. REAL SYMMETRIC MATRICES 119
10.1. The eigensolutions of a real symmetric matrix 119
10.2. Extension to matrices which are not positive definite 121
10.3. The Jacobi algorithm for the eigensolutions of a real symmetric
matrix 126
10.4. Organisation of the Jacobi algorithm 128
10.5. A brief comparison of methods for the eigenproblem of a real
symmetric matrix 133

11. THE GENERALISED SYMMETRIC MATRIX EIGENVALUE
PROBLEM 135

12. OPTIMISATION AND NONLINEAR EQUATIONS 142
12.1. Formal problems in unconstrained optimisation and nonlinear
equations 142
12.2. Difficulties encountered in the solution of optimisation and
nonlinear-equation problems 146
13. ONE-DIMENSIONAL PROBLEMS 148
13.1. Introduction 148
13.2. The linear search problem 148
13.3. Real roots of functions of one variable 160