Solve Equations 2x+y+z=5,3x+5y+2z=15,2x+y+4z=8 using Gauss Seidel method
Solution:
Total Equations are `3`
`2x+y+z=5`
`3x+5y+2z=15`
`2x+y+4z=8`
From the above equations
`x_(k+1)=1/2(5-y_(k)-z_(k))`
`y_(k+1)=1/5(15-3x_(k+1)-2z_(k))`
`z_(k+1)=1/4(8-2x_(k+1)-y_(k+1))`
Initial gauss `(x,y,z) = (0,0,0)`
Solution steps are
`1^(st)` Approximation
`x_1=1/2[5-(0)-(0)]=1/2[5]=2.5`
`y_1=1/5[15-3(2.5)-2(0)]=1/5[7.5]=1.5`
`z_1=1/4[8-2(2.5)-(1.5)]=1/4[1.5]=0.375`
`2^(nd)` Approximation
`x_2=1/2[5-(1.5)-(0.375)]=1/2[3.125]=1.5625`
`y_2=1/5[15-3(1.5625)-2(0.375)]=1/5[9.5625]=1.9125`
`z_2=1/4[8-2(1.5625)-(1.9125)]=1/4[2.9625]=0.7406`
`3^(rd)` Approximation
`x_3=1/2[5-(1.9125)-(0.7406)]=1/2[2.3469]=1.1734`
`y_3=1/5[15-3(1.1734)-2(0.7406)]=1/5[9.9984]=1.9997`
`z_3=1/4[8-2(1.1734)-(1.9997)]=1/4[3.6534]=0.9134`
`4^(th)` Approximation
`x_4=1/2[5-(1.9997)-(0.9134)]=1/2[2.087]=1.0435`
`y_4=1/5[15-3(1.0435)-2(0.9134)]=1/5[10.0429]=2.0086`
`z_4=1/4[8-2(1.0435)-(2.0086)]=1/4[3.9045]=0.9761`
`5^(th)` Approximation
`x_5=1/2[5-(2.0086)-(0.9761)]=1/2[2.0153]=1.0077`
`y_5=1/5[15-3(1.0077)-2(0.9761)]=1/5[10.0248]=2.005`
`z_5=1/4[8-2(1.0077)-(2.005)]=1/4[3.9797]=0.9949`
`6^(th)` Approximation
`x_6=1/2[5-(2.005)-(0.9949)]=1/2[2.0001]=1.0001`
`y_6=1/5[15-3(1.0001)-2(0.9949)]=1/5[10.01]=2.002`
`z_6=1/4[8-2(1.0001)-(2.002)]=1/4[3.9979]=0.9995`
`7^(th)` Approximation
`x_7=1/2[5-(2.002)-(0.9995)]=1/2[1.9985]=0.9993`
`y_7=1/5[15-3(0.9993)-2(0.9995)]=1/5[10.0033]=2.0007`
`z_7=1/4[8-2(0.9993)-(2.0007)]=1/4[4.0008]=1.0002`
`8^(th)` Approximation
`x_8=1/2[5-(2.0007)-(1.0002)]=1/2[1.9991]=0.9996`
`y_8=1/5[15-3(0.9996)-2(1.0002)]=1/5[10.0009]=2.0002`
`z_8=1/4[8-2(0.9996)-(2.0002)]=1/4[4.0007]=1.0002`
Solution By Gauss Seidel Method.
`x=0.9996~=1`
`y=2.0002~=2`
`z=1.0002~=1`
Iterations are tabulated as below
| Iteration | x | y | z |
| 1 | 2.5 | 1.5 | 0.375 |
| 2 | 1.5625 | 1.9125 | 0.7406 |
| 3 | 1.1734 | 1.9997 | 0.9134 |
| 4 | 1.0435 | 2.0086 | 0.9761 |
| 5 | 1.0077 | 2.005 | 0.9949 |
| 6 | 1.0001 | 2.002 | 0.9995 |
| 7 | 0.9993 | 2.0007 | 1.0002 |
| 8 | 0.9996 | 2.0002 | 1.0002 |
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
