Q.3 (c) Apply Gauss-Seidel iteration method to solve the following linear equations up to the three iterations. \(20x+y-2z=17,\ 3x+20y-z=-18,\ 2x-3y+20z=25\).
Best answer
The system is diagonally dominant. \(x=\frac1{20}(17-y+2z),\quad y=\frac1{20}(-18-3x+z),\quad z=\frac1{20}(25-2x+3y)\). Start \(y=z=0\), using latest values.
Iteration 1: \(x_1=\frac{17}{20}=0.85\); \(y_1=\frac{-18-2.55+0}{20}=-1.0275\); \(z_1=\frac{25-1.7-3.0825}{20}=1.0109\)
Iteration 2: \(x_2=\frac{17+1.0275+2.0218}{20}=1.0025\); \(y_2=\frac{-18-3.0074+1.0109}{20}=-0.9998\); \(z_2=\frac{25-2.0049-2.9995}{20}=0.9998\)
Iteration 3: \(x_3=\frac{17+0.9998+1.9996}{20}=1.0000\); \(y_3=\frac{-18-3.0000+0.9998}{20}=-1.0000\); \(z_3=\frac{25-2.0000-3.0000}{20}=1.0000\)
\(x=1,\ y=-1,\ z=1\) (check: \(20-1-2=17,\ 3-20-1=-18,\ 2+3+20=25\) ✓).
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