Use GaussSeidel iterative technique to find approximate solu

Use Gauss-Seidel iterative technique to find approximate solutions to -x_1 + 11x_2 - x_3 + 3x_4 = 25 10x_1 - x_2 + 2x_3 = 6 3x_2 - x_3 + 8x_4 = 15 2x_1 - x_2 + 10x_3 - x_4 = -11 start with initial guesses of x = (0, 0, 0, 0) and iterate until % epsilon_a

Solution

To guarantee convergence first we need to set the eqution 1st which has highest coefficient of x1 and equation 2nd which has highest coefficient of x2 and so on. Below will be the set of equatios:

10x1 - x2 - 2x3 = 6

-x1 + 11x2 - x3 +3x4 = 25

2x1 - x2 + 10x3 -x4 = -11

0x1 + 3x2 - x3 +8x4 = 15

Given that initial guesses are x = (0,0,0,0)

Step : 1

x1 = 6/10- (-1 x 0)/10- (2 x 0)/10- (0 x 0)/10 = 0.6
x2 = 25/11- (-1 x 0)/11- (-1 x 0)/11- (3 x 0)/11 = 2.273
x3 = -11/10- (2 x 0)/10- (-1 x 0)/10- (-1 x 0)/10 = -1.1
x4 = 15/8- (0 x 0)/8- (3 x 0)/8- (-1 x 0)/8 = 1.875

Step : 2

x1 = 6/10- (-1 x 2.273)/10- (2 x -1.1)/10- (0 x 1.875)/10 = 1.047
x2 = 25/11- (-1 x 0.6)/11- (-1 x -1.1)/11- (3 x 1.875)/11 = 1.716
x3 = -11/10- (2 x 0.6)/10- (-1 x 2.273)/10- (-1 x 1.875)/10 = -0.805
x4 = 15/8- (0 x 0.6)/8- (3 x 2.273)/8- (-1 x -1.1)/8 = 0.885

Step : 3

x1 = 6/10- (-1 x 1.716)/10- (2 x -0.805)/10- (0 x 0.885)/10 = 0.933
x2 = 25/11- (-1 x 1.047)/11- (-1 x -0.805)/11- (3 x 0.885)/11 = 2.053
x3 = -11/10- (2 x 1.047)/10- (-1 x 1.716)/10- (-1 x 0.885)/10 = -1.049
x4 = 15/8- (0 x 1.047)/8- (3 x 1.716)/8- (-1 x -0.805)/8 = 1.131

Step : 4

x1 = 6/10- (-1 x 2.053)/10- (2 x -1.049)/10- (0 x 1.131)/10 = 1.015
x2 = 25/11- (-1 x 0.933)/11- (-1 x -1.049)/11- (3 x 1.131)/11 = 1.954
x3 = -11/10- (2 x 0.933)/10- (-1 x 2.053)/10- (-1 x 1.131)/10 = -0.968
x4 = 15/8- (0 x 0.933)/8- (3 x 2.053)/8- (-1 x -1.049)/8 = 0.974

Step : 5

x1 = 6/10- (-1 x 1.954)/10- (2 x -0.968)/10- (0 x 0.974)/10 = 0.989
x2 = 25/11- (-1 x 1.015)/11- (-1 x -0.968)/11- (3 x 0.974)/11 = 2.011
x3 = -11/10- (2 x 1.015)/10- (-1 x 1.954)/10- (-1 x 0.974)/10 = -1.01
x4 = 15/8- (0 x 1.015)/8- (3 x 1.954)/8- (-1 x -0.968)/8 = 1.021

Step : 6

x1 = 6/10- (-1 x 2.011)/10- (2 x -1.01)/10- (0 x 1.021)/10 = 1.003
x2 = 25/11- (-1 x 0.989)/11- (-1 x -1.01)/11- (3 x 1.021)/11 = 1.992
x3 = -11/10- (2 x 0.989)/10- (-1 x 2.011)/10- (-1 x 1.021)/10 = -0.995
x4 = 15/8- (0 x 0.989)/8- (3 x 2.011)/8- (-1 x -1.01)/8 = 0.995

Step : 7

x1 = 6/10- (-1 x 1.992)/10- (2 x -0.995)/10- (0 x 0.995)/10 = 0.998
x2 = 25/11- (-1 x 1.003)/11- (-1 x -0.995)/11- (3 x 0.995)/11 = 2.002
x3 = -11/10- (2 x 1.003)/10- (-1 x 1.992)/10- (-1 x 0.995)/10 = -1.002
x4 = 15/8- (0 x 1.003)/8- (3 x 1.992)/8- (-1 x -0.995)/8 = 1.004

Step : 8

x1 = 6/10- (-1 x 2.002)/10- (2 x -1.002)/10- (0 x 1.004)/10 = 1.001
x2 = 25/11- (-1 x 0.998)/11- (-1 x -1.002)/11- (3 x 1.004)/11 = 1.999
x3 = -11/10- (2 x 0.998)/10- (-1 x 2.002)/10- (-1 x 1.004)/10 = -0.999
x4 = 15/8- (0 x 0.998)/8- (3 x 2.002)/8- (-1 x -1.002)/8 = 0.999

Step : 9

x1 = 6/10- (-1 x 1.999)/10- (2 x -0.999)/10- (0 x 0.999)/10 = 1
x2 = 25/11- (-1 x 1.001)/11- (-1 x -0.999)/11- (3 x 0.999)/11 = 2
x3 = -11/10- (2 x 1.001)/10- (-1 x 1.999)/10- (-1 x 0.999)/10 = -1
x4 = 15/8- (0 x 1.001)/8- (3 x 1.999)/8- (-1 x -0.999)/8 = 1.001

Step : 10

x1 = 6/10- (-1 x 2)/10- (2 x -1)/10- (0 x 1.001)/10 = 1
x2 = 25/11- (-1 x 1)/11- (-1 x -1)/11- (3 x 1.001)/11 = 2
x3 = -11/10- (2 x 1)/10- (-1 x 2)/10- (-1 x 1.001)/10 = -1
x4 = 15/8- (0 x 1)/8- (3 x 2)/8- (-1 x -1)/8 = 1