Page 126 - Applied Numerical Methods Using MATLAB

PROBLEMS 115
2.9 Gauss–Seidel Iterative Method with Relaxation Technique
(a) Try the relaxation technique (introduced in Section 2.5.3) with several
values of the relaxation factor ω = 0.2, 0.4,..., 1.8 for the following
problems. Find the best one among these values of the relaxation factor
for each problem, together with the number of iterations required for
−6
satisfying the termination criterion ||x k+1 − x k ||/||x k || < 10 .

5 −4 x 1 1
(i) A 1 x = = = b 1 (P2.9.1)
−9 10 x 2 1

2 −1 x 1 1
(ii) A 2 x = = = b 2 (P2.9.2)
−1 4 x 2 3
(iii) The nonlinear equations (E2.5.1) given in Example 2.5.
(b) Which of the two matrices A 1 and A 2 has stronger diagonal dominancy
in the above equations? For which equation does Gauss–Seidel iteration
converge faster, Eq. (P2.9.1) or Eq. (P2.9.2)? What would you conjecture
about the relationship between the convergence speed of Gauss–Seidel
iteration for a set of linear equations and the diagonal dominancy of the
coefficient matrix A?
(c) Is the relaxation technique always helpful for improving the convergence
speed of the Gauss–Seidel iterative method regardless of the value of
the relaxation factor ω?