PROBLEMS 115 2.9 Gauss-Seidel Iterative Method with Relaxation Technique (a) Try the relaxation technique (introduced in

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Problems 115 2 9 Gauss Seidel Iterative Method With Relaxation Technique A Try The Relaxation Technique Introduced In 1 (73.73 KiB) Viewed 30 times

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 @= 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 satisfying the termination criterion ||Xk+1 X/X || < 10-6. X1 (i) A₁x = 5-4 -9 10 =b₁ (P2.9.1) X2 [x]-8- 4[*]=[]= 2 (ii) A₂x = : b₂ (P2.9.2) (iii) The nonlinear equations (E2.5.1) given in Example 2.5. (b) Which of the two matrices A₁ and A₂ 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 w?