Question

QUESTION 2 Consider the linear system Ti 0.521 + 21 2 0.5x2 + 13 0.25.13 23 0.2 -1.425 2 whose solution is (0.9,-0.8, 0.7). (a) Determine whether the coefficient matrix is strictly diagonally dominant. (5) (b) Approximate the solution of the system by performing two iterations of the Gauss-Seidel algorithm, using (10) x(0) - (0,0,0)' as the initial guess. (c) Approximate the solution of the system using one iteration of the SOR scheme, with w = 0.7 and (5) x(0) -(0,0,0)' [20]

Answer 1

solution y consider the above given that. xi-x₂ +0.2 0.501782 -0.25*3=-14 25 2,-0.5% +83=2 written in coefficient matrix au a 2 -1 agi 12 O.S. -0.25 azh - azi a33 - -0.5 la, le 19,21 +19,31 => IZO+1-12/ lagalzlagel+lagal =) 1 z 10.5/+10.25) 12 0.75 laza) 2 193, 1+ A32) - 111 (-05)+ 111 11171115) condition is not satisfied lazak 1931+ 19321 - co-efficient matrix is not strictly diagonal dominant)

(3,1,2,) = (0:2, 7.525, 1.0345) 12 2 2nd Approximelion value is! = 1.2375 =o.2 +1.0375 Ya = -1.425-015 (12375) + 0.2511.037) -1.7844 Z2 = &-(1.2375) + 0.5(-1.78dede) --0.1297 (x21 9122) = (1.2375, 1.78444, -0.1297) c) by a using SoR method : Given . W=0.7 XK+ 1 = (1-w) Sk + Wt 10.2 - 094+24) (1-w) x x + W (0.2+ 2x) (1-W/ Yk twt (21.425-0.5 8k+ 1 +0.252 = (1-2) Yk+W(-1.425.0.500 y ktl K+1 +0152

initial value (aal kl Zk+ = (1-w) Zx t W 1 (2-XX+1 +0.594+1) (1-w) Zk T W (2-xx+1 Yk+i) tos Wro.7 subshite 24, = (1-0.7)0 +0.7 (0,2-0+0) 0.710:2) 0.14 Y,= (1-0.7)0 + 0.761.485)- 0.510.1648+0) = -1.0465 2, -1-0.7) + 07 (2-0.44*0.56-1.666 0.935725 (x, c9, 2) = (older -1.0465, 0.935725 *
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