Question

solve 2x + y + 6z = 9; 8x + 3y + 2z = 13; x + 5y + z = 17 by using gauss Seidel iteration method​

Answer 1

Step-by-step explanation:

Given:

$2x+y+6z=9\\8x+3y+2z=13\\x+5y+z=17$

To find: Solution of system of equations using Gauss Seidel iteration method

Solution:

In this method;First write the given equations in terms of x,y and z respectively.Then approximate the values of x,y and z by iteration.

The whole process is shown below:

$2x+y+6z=9\\x=\frac{1}{2}(9-y-6z)...eq1\\\\8x+3y+2z=13\\y=\frac{1}{3}(13-8x-2z)..eq2\\\\x+5y+z=17\\z=17-x-5y... eq3$

Now,Start the process of approximation

Put

$x_0=y_0=z_0=0$

put

$y=y_0\\z=z_0$

in first equation

$x_1=\frac{1}{2}(9-y_0-6z_0)\\\\x_1=\frac{1}{2}(9)\\\\\bold{x_1=4.500}$

$y_1=\frac{1}{3}(13-8x_1-2z_0)\\\\y_1=\frac{1}{3}(13-8(4.500)-2(0))\\\\\bold{y_1=-7.6666}$

$z_1=17-x_1-5y_1\\\\\bold{z_1=50.8333}$

For second iteration we have the values of x1,y1,z1

apply second iteration

$x_2=\frac{1}{2}(9-y_1-6z_1)\\\\\bold{x_2=-144.1666}$

$y_2=\frac{1}{3}(13-8x_2-2z_1)\\\\\bold{y_2=354.8887}$

$z_2=17-x_2-5y_2\\\\\bold{z_2=-1613.2769}$

For third iteration

$x_3=\frac{1}{2}(9-y_2-6z_2)\\\\\bold{x_3=4666.8863}$

$y_3=\frac{1}{3}(13-8x_3-2z_2)\\\\\bold{y_3=-11365.1788}$

$z_3=17-x_3-5y_3\\\\\bold{z_3=52176. 0077}$

On observing the values of x,y and z in each iteration,one can analyze that the values are very different(Deviation is very large)

Thus,

These linear equations can not be solve by Gauss Seidel iteration method.

Remark*: These equation can be solve by Gauss elimination Method.

Hope it helps you.

To learn more on brainly:

What is the limitation of Gauss-Seidel method?

a) It cannot be used for the matrices with non-zero diagonal elem

b) ...

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