Question

) Solve the following system of linear equations using Gauss-Seidel iterative method: 5 x + y = 13 4x - y = 2 n​

Answer 1

Answer:

the eigen values of the given matrix are4,12-√65and12+√65

the largest eigen value is 12+√65.

2) we construct the iterative formulae fromthe given system of non homogeneous linear equations as x =(1/20){ 17-y+2z} , y = (1/20){-18-3x+z} , z =(1/20){25-2x+3y}

to initialize, we put x=0,y=0,z=0 from this weget

x0=17/20 , y0 = -18/20 and z0 =25/20----------------(1)

to get the 1st iterations , we use y0 =-18/20 , z0=25/20 in x.

then we get x1 = (1/20){17+(18/20)+2(25/20)} =1.02

similarly using z0 = 25/20 and x0 =17/20 weget y1 = (1/20){-18-3(17/20)+(25/20)}=-0.965

using x0 = 17/20 and y0 = -18/20 we get z1 =(1/20){25-2(17/20)+3(-18/20)} =1.03

for 2nd iteration, we usex1,y1,z1

x2=(1/20){17-(-0.965)+2(1.03)}=1.00125

y2=(1/20){-18-3(1.02)+1.03}=-1.0025

z2=(1/20){25-2(1.02)+3(-0.965)}=1.04625

3rditeration :

x3= (1/20){17-(-1.0025)+2(1.04625)}=0.9524

y3=(1/20){-18-3(1.0025)+1.04625}=-0.9982625

z3=(1/20){25-2(1.00125)+3(-1.0025)}=0.9995