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Solve the Following Equation by Gauss Seidal Method: 10 X 1 + X 2 + X 3 = 12 2 X 1 + 10 X 2 + X 3 − 13 2 X 1 + 2 X 2 + 10 X 3 = 14 - Applied Mathematics 1

Sum

Solve the following equation by Gauss Seidal method:

`10x_1+x_2+x_3=12`
`2x_1+10x_2+x_3-13`
`2x_1+2x_2+10x_3=14`

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Solution

By Gauss Seidal method ,

Given eqn : `10x_1+x_2+x_3=12`
                    `2x_1+10x_2+x_3-13`
                   `2x_1+2x_2+10x_3=14`

From given eqn : |10|>|1|+|1|
                            |10|>|2|+|1|
                            |10|>|2|+|2|
The given eqn are in correct order.

`therefore x_1=1/10[12-x_2-x_3]`

`therefore x_2=1/10[13-2x_1-x_3]`

`therefore x_3=1/10[14-2x_2-2x_1]`

I) For 1st iteration : take `x_2=0, x_3=0`

`x_1=1/10[12]=1.2`

`x_1=1.2,  x_3=0` gives `x_2=1.06`

`x_1=1.2,  x_2=1.06` gives `x_3=0.948`

II) For 2nd iteration : take `x_2=1.06, x_3=0.948`

`x_1=1/10[12-1.06-0.948]=0.9992`

`x_1=0.992,  x_3=0.948` gives `x_2=1.0068`

`x_1=0.992,  x_2=1.0068` gives `x_3=1.0002`

III) For 3rd iteration : `x_2=1.0068`, `x_3=1.0002`

`x_1=1/10[12-1.0068-1.0002]=0.9993`

`x_1=0.993,  x_3=1.0002` gives `x_2=1.00`

`x_1=0.993,  x_2=1.00` gives `x_3=1.00`

Result : `x_1=1.00, x_2=1.00, x_3=1.00`

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