Question

QUESTION 2 i 0.2 -1.425 2 Consider the linear system 13 0.52 + 22 0.25.13 = Ii 0.522 + 23 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)* 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

consider the linear system -X3= 0.2 re 0.5X1 + X2-0.25X3 X, +0.5X +Xz 7.495 -20 ras from 6 we get efficient matrix AX=B - A = 0 0.5 l -| -0.25 - -0.5 1 Here al = 1; Ala = 0 ; 013 =-1; Call = 0.5; Agal Agg = -0.25 ; A 3 = 1 ; Az. 2-05; O2z=1 11 la

laga l>lag H 19:31 and A as not strictly diagenel dominant matrix b. Plere lail= laialt laial 1033|< lazıl+lazal 80 from o X= [(0.2) - (0 X 2+x3] Xa - [-|-103 -x + 6 - 35x50 Xa = (3 - X + 0.5 x1 By grauss piedel Method ) , CK+1). [0.2-X te cett) = (-1.42-* CH41) + (0.35) X, Cku). + 0.5 XC71) tock) + ticky (left) 2- Xe

.. first X()_ 0.2 ; Xa ) -1.625 1 second iteration + (a) - (6) . ; x = 0 ; x = 0 literation [o.9-0+0] = 0.2 [-1425-0.4+0-2560) - -7.625 - (2-(0-9)+(0-5) 61.685)]- 0-9875 ġ to 0.9875 PO. 2-0 6-41.625)+ (0.9875)]- 11875 x ) = (-1425-01-1875)+(6:35) (6 5625) -3-3452 (- (1.1875) +(0.5) (-2,36563) -0.3703) 1-1875 3 x ) = -2,3656; Xce) - = -0.37031 for SoR method X{kt) = (10) x(k) + (W) [0.2 - xylk + x3 (8] (1-W) X (K) +(w) [-1.425 – x(+12460.25) xkey to Pear) - (-2) ik + (as) [ 2 - X CHAD + 0.5 x H)] YpsoX,9.05 kg ? first iteration : (1) Xi (-0.70) +(0.7)(0.2-0+0]= 0.14 (1-0.7) (0)+(0.7) 6.485-0-14+0) = -0.9975 (1-0-7)(0) + (0.7) (2-0.94 +(0.5)(-0-9975) 0.952875 = 0.14 tal- -0.9995 ; talent P: 0.952875 Answe ģ W=0.7 x, C+1) X.CO 9