Question

{ 6.6 (3). Consider the linear system 2 +4.02 = 0 4.0 +02 = 0 The true solution is 11 = -1/15, 12 = 4/15. Apply the Jacobi and Gauss-Seidel methods with (0) = [0,0to the system and find out which methods diverge more rapidly. Next, interchange the two equations to write the system as 4.0 +x2 = 0 2 +4.62 = 0 and apply both methods with 7(0) = [0,0]?. Iterate until ||2 – 2(k)|| < 10-5. Which method converge faster?

Answer 1

6) linear System 174 atz=4 922 - 1 The homogeneous is not diagonally dominant an=1 lai171922) 921=4, 1922l>, lail 13, 4 So even though we apply Jacoba of Gauss Seidel melhod the solution does not Convergent it is divergent ft the Jacobi as well as Causs Seidel methods The R, R₂ The linear System becomes ace=4 9,2 =4 10) interchange Now uf (9:10] a21 = 1 a22=4 4 7. Ir laut > Pial lazzl > laril = 4711 This System slem is dominant, we get diagonally milial a convergent &duction by choosing approximation as [o,o] (0-yk) 2 K+ 90=0, Yo=0 - 6 y=0 24 = 4 Co-o] Y2 = /[o-o)=0 21=0, Solution by Gaun Seidel melhód Illy for gacobi 10-96) "K+1= I (0-ak) method Hvala 4= 0, y = 0 Both methods have a trivial Solution The unique solution is hew solution