Solve Equations x+y+z=7,x+2y+2z=13,x+3y+z=13 using Gauss Seidel method
Solution:
Total Equations are `3`
`x+y+z=7`
`x+2y+2z=13`
`x+3y+z=13`
The coefficient matrix of the given system is not diagonally dominant.
Hence, we re-arrange the equations as follows, such that the elements in the coefficient matrix are diagonally dominant.
`x+y+z=7`
`x+3y+z=13`
`x+2y+2z=13`
From the above equations
`x_(k+1)=1/1(7-y_(k)-z_(k))`
`y_(k+1)=1/3(13-x_(k+1)-z_(k))`
`z_(k+1)=1/2(13-x_(k+1)-2y_(k+1))`
Initial gauss `(x,y,z) = (0,0,0)`
Solution steps are
`1^(st)` Approximation
`x_1=1/1[7-(0)-(0)]=1/1[7]=7`
`y_1=1/3[13-(7)-(0)]=1/3[6]=2`
`z_1=1/2[13-(7)-2(2)]=1/2[2]=1`
`2^(nd)` Approximation
`x_2=1/1[7-(2)-(1)]=1/1[4]=4`
`y_2=1/3[13-(4)-(1)]=1/3[8]=2.6667`
`z_2=1/2[13-(4)-2(2.6667)]=1/2[3.6667]=1.8333`
`3^(rd)` Approximation
`x_3=1/1[7-(2.6667)-(1.8333)]=1/1[2.5]=2.5`
`y_3=1/3[13-(2.5)-(1.8333)]=1/3[8.6667]=2.8889`
`z_3=1/2[13-(2.5)-2(2.8889)]=1/2[4.7222]=2.3611`
`4^(th)` Approximation
`x_4=1/1[7-(2.8889)-(2.3611)]=1/1[1.75]=1.75`
`y_4=1/3[13-(1.75)-(2.3611)]=1/3[8.8889]=2.963`
`z_4=1/2[13-(1.75)-2(2.963)]=1/2[5.3241]=2.662`
`5^(th)` Approximation
`x_5=1/1[7-(2.963)-(2.662)]=1/1[1.375]=1.375`
`y_5=1/3[13-(1.375)-(2.662)]=1/3[8.963]=2.9877`
`z_5=1/2[13-(1.375)-2(2.9877)]=1/2[5.6497]=2.8248`
`6^(th)` Approximation
`x_6=1/1[7-(2.9877)-(2.8248)]=1/1[1.1875]=1.1875`
`y_6=1/3[13-(1.1875)-(2.8248)]=1/3[8.9877]=2.9959`
`z_6=1/2[13-(1.1875)-2(2.9959)]=1/2[5.8207]=2.9104`
`7^(th)` Approximation
`x_7=1/1[7-(2.9959)-(2.9104)]=1/1[1.0938]=1.0938`
`y_7=1/3[13-(1.0938)-(2.9104)]=1/3[8.9959]=2.9986`
`z_7=1/2[13-(1.0938)-2(2.9986)]=1/2[5.909]=2.9545`
`8^(th)` Approximation
`x_8=1/1[7-(2.9986)-(2.9545)]=1/1[1.0469]=1.0469`
`y_8=1/3[13-(1.0469)-(2.9545)]=1/3[8.9986]=2.9995`
`z_8=1/2[13-(1.0469)-2(2.9995)]=1/2[5.954]=2.977`
`9^(th)` Approximation
`x_9=1/1[7-(2.9995)-(2.977)]=1/1[1.0234]=1.0234`
`y_9=1/3[13-(1.0234)-(2.977)]=1/3[8.9995]=2.9998`
`z_9=1/2[13-(1.0234)-2(2.9998)]=1/2[5.9769]=2.9884`
`10^(th)` Approximation
`x_10=1/1[7-(2.9998)-(2.9884)]=1/1[1.0117]=1.0117`
`y_10=1/3[13-(1.0117)-(2.9884)]=1/3[8.9998]=2.9999`
`z_10=1/2[13-(1.0117)-2(2.9999)]=1/2[5.9884]=2.9942`
`11^(th)` Approximation
`x_11=1/1[7-(2.9999)-(2.9942)]=1/1[1.0059]=1.0059`
`y_11=1/3[13-(1.0059)-(2.9942)]=1/3[8.9999]=3`
`z_11=1/2[13-(1.0059)-2(3)]=1/2[5.9942]=2.9971`
`12^(th)` Approximation
`x_12=1/1[7-(3)-(2.9971)]=1/1[1.0029]=1.0029`
`y_12=1/3[13-(1.0029)-(2.9971)]=1/3[9]=3`
`z_12=1/2[13-(1.0029)-2(3)]=1/2[5.9971]=2.9985`
`13^(th)` Approximation
`x_13=1/1[7-(3)-(2.9985)]=1/1[1.0015]=1.0015`
`y_13=1/3[13-(1.0015)-(2.9985)]=1/3[9]=3`
`z_13=1/2[13-(1.0015)-2(3)]=1/2[5.9985]=2.9993`
`14^(th)` Approximation
`x_14=1/1[7-(3)-(2.9993)]=1/1[1.0007]=1.0007`
`y_14=1/3[13-(1.0007)-(2.9993)]=1/3[9]=3`
`z_14=1/2[13-(1.0007)-2(3)]=1/2[5.9993]=2.9996`
Solution By Gauss Seidel Method.
`x=1.0007~=1`
`y=3~=3`
`z=2.9996~=3`
Iterations are tabulated as below
| Iteration | x | y | z |
| 1 | 7 | 2 | 1 |
| 2 | 4 | 2.6667 | 1.8333 |
| 3 | 2.5 | 2.8889 | 2.3611 |
| 4 | 1.75 | 2.963 | 2.662 |
| 5 | 1.375 | 2.9877 | 2.8248 |
| 6 | 1.1875 | 2.9959 | 2.9104 |
| 7 | 1.0938 | 2.9986 | 2.9545 |
| 8 | 1.0469 | 2.9995 | 2.977 |
| 9 | 1.0234 | 2.9998 | 2.9884 |
| 10 | 1.0117 | 2.9999 | 2.9942 |
| 11 | 1.0059 | 3 | 2.9971 |
| 12 | 1.0029 | 3 | 2.9985 |
| 13 | 1.0015 | 3 | 2.9993 |
| 14 | 1.0007 | 3 | 2.9996 |
This material is intended as a summary. Use your textbook for detail explanation.
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