Solve Equations 2x+y=8,x+2y=1 using Gauss Seidel method
Solution:
Total Equations are `2`
`2x+y=8`
`x+2y=1`
From the above equations
`x_(k+1)=1/2(8-y_(k))`
`y_(k+1)=1/2(1-x_(k+1))`
Initial gauss `(x,y) = (0,0)`
Solution steps are
`1^(st)` Approximation
`x_1=1/2[8-(0)]=1/2[8]=4`
`y_1=1/2[1-(4)]=1/2[-3]=-1.5`
`2^(nd)` Approximation
`x_2=1/2[8-(-1.5)]=1/2[9.5]=4.75`
`y_2=1/2[1-(4.75)]=1/2[-3.75]=-1.875`
`3^(rd)` Approximation
`x_3=1/2[8-(-1.875)]=1/2[9.875]=4.9375`
`y_3=1/2[1-(4.9375)]=1/2[-3.9375]=-1.9688`
`4^(th)` Approximation
`x_4=1/2[8-(-1.9688)]=1/2[9.9688]=4.9844`
`y_4=1/2[1-(4.9844)]=1/2[-3.9844]=-1.9922`
`5^(th)` Approximation
`x_5=1/2[8-(-1.9922)]=1/2[9.9922]=4.9961`
`y_5=1/2[1-(4.9961)]=1/2[-3.9961]=-1.998`
`6^(th)` Approximation
`x_6=1/2[8-(-1.998)]=1/2[9.998]=4.999`
`y_6=1/2[1-(4.999)]=1/2[-3.999]=-1.9995`
`7^(th)` Approximation
`x_7=1/2[8-(-1.9995)]=1/2[9.9995]=4.9998`
`y_7=1/2[1-(4.9998)]=1/2[-3.9998]=-1.9999`
Solution By Gauss Seidel Method.
`x=4.9998~=5`
`y=-1.9999~=-2`
Iterations are tabulated as below
| Iteration | x | y |
| 1 | 4 | -1.5 |
| 2 | 4.75 | -1.875 |
| 3 | 4.9375 | -1.9688 |
| 4 | 4.9844 | -1.9922 |
| 5 | 4.9961 | -1.998 |
| 6 | 4.999 | -1.9995 |
| 7 | 4.9998 | -1.9999 |
This material is intended as a summary. Use your textbook for detail explanation.
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