Q.6 (c) Find the three sets of iterative solutions of the following equations by Gauss Jacobi method: \(5x-y+z=10;\ 2x+4y=12;\ x+y+5z=-1\).
Best answer
The system is diagonally dominant (\(5>1+1,\ 4>2,\ 5>1+1\)).
\(x=\frac15(10+y-z),\quad y=\frac14(12-2x),\quad z=\frac15(-1-x-y)\)
Start with \(x_0=y_0=z_0=0\); in Jacobi's method every new value uses only the previous iteration's values.
| Iteration | x | y | z |
|---|---|---|---|
| 1 | \(\frac{10}{5}=2\) | \(\frac{12}{4}=3\) | \(\frac{-1}{5}=-0.2\) |
| 2 | \(\frac{10+3+0.2}{5}=2.64\) | \(\frac{12-4}{4}=2\) | \(\frac{-1-2-3}{5}=-1.2\) |
| 3 | \(\frac{10+2+1.2}{5}=2.64\) | \(\frac{12-5.28}{4}=1.68\) | \(\frac{-1-2.64-2}{5}=-1.128\) |
Three sets: (2, 3, −0.2), (2.64, 2, −1.2), (2.64, 1.68, −1.128). (They converge to the exact solution \(x=\frac{23}{9}\approx2.556,\ y=\frac{31}{18}\approx1.722,\ z=-\frac{19}{18}\approx-1.056\).)
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