Question

solve the equation using guass seidel method 10x+2y+z=9,2x+20y-2z=-44,-2x+3y+10z=22​

Answer 1

Given:

10x+2y+z=9

2x+20y-2z=-44  

-2x+3y+10z=22​

To find:

Solve the equation using Gauss Seidel method.

Solution:

Total Equations are 3

10x+2y+z=9

2x+20y-2z=-44  

-2x+3y+10z=22

From the above equations

$x_{k+1} = \frac{1}{10} (9 - 2y_k - z_k)\\\\y_{k+1} = \frac{1}{20} (-44 - 2x_{k+1} - 2z_k)\\\\z_{k+1} = \frac{1}{10} (22 + 2x_{k+1} - 3y_{k+1})$

Initial gauss (x,y,z)=(0,0,0)

Solution steps are

1st Approximation

$x_1 = \frac{1}{10} [9 - 2(0) - 0] = \frac{1}{10} [9] = 0.9\\\\y_1 = \frac{1}{20} [-44 - 2(0.9) - 2(0)] = \frac{1}{20} [-45.8] = -2.29\\\\z_1 = \frac{1}{10} [22 - 2(0.9) - 3(-2.29)] = \frac{1}{10} [30.67] = 3.067$

2nd Approximation

$x_2 = \frac{1}{10}[9 - 2(-2.29) - (3.067)] = 1.0513\\\\y_2 = \frac{1}{20}[-44 - 2(1.0513) + 2(3.076)] = -1.9984\\\\z_2 = \frac{1}{10}[22 + 2(1.0513) - 3(-1.9984)] = 3.0098$

3rd Approximation

$x_3 = \frac{1}{10} [9 - 2(-1.9984) - (3.0098)] = 0.9987\\\\y_3 = \frac{1}{20}[-44 - 2(0.9987) + 2(3.0098)] = -1.9989\\\\z_3 = \frac{1}{10}[22 + 2(0.9987) - 3(-1.9989)] = 2.9994$

4th Approximation

$x_4 = \frac{1}{10}[9 - 2(-1.9989) - (2.9994)] = 0.9998\\\\y_4 = \frac{1}{20}[-44 - 2(0.9998) + 2(2.9994)] = -2\\\\z_4 = \frac{1}{10}[22 + 2(0.9998) - 3(-2)] = 3$

5th Approximation

$x_5 = \frac{1}{10} [9 - 2(-2) - (3)] = 1\\\\y_5 = \frac{1}{20}[-44 - 2(1) + 2(3)] = -2\\\\z_5 = \frac{1}{10}[22 + 2(1) - 3(-2)] = 3$

Solution by Gauss Seidel Method.

x = 1

y = -2

z = 3