In one line
Gauss–Seidel is Jacobi with one change: each newly computed value is used immediately in the following equations of the same iteration, which usually makes it converge about twice as fast.
Explained simply
Back to the three friends sharing a bill. In Jacobi, all three update at the same moment using old guesses. In Seidel, the first friend updates, the second friend hears that new guess and uses it straight away, and the third uses both new guesses. Fresher information means fewer rounds to agree.
Technically, \(x^{(k+1)}\) is computed from \(y^{(k)},z^{(k)}\); then \(y^{(k+1)}\) from \(x^{(k+1)},z^{(k)}\); then \(z^{(k+1)}\) from \(x^{(k+1)},y^{(k+1)}\). For diagonally dominant systems both methods converge, and Seidel generally needs fewer iterations.
How it works
- Check (or arrange) diagonal dominance.
- Make \(x\), \(y\), \(z\) the subject of equations 1, 2, 3.
- Start with \(y=z=0\) (no starting \(x\) needed).
- Compute \(x\), then \(y\) using this new \(x\), then \(z\) using the new \(x,y\).
- Repeat until values agree to the required accuracy.
Key formula
\(x^{(k+1)}=\dfrac{d_1-b_1y^{(k)}-c_1z^{(k)}}{a_1},\ y^{(k+1)}=\dfrac{d_2-a_2x^{(k+1)}-c_2z^{(k)}}{b_2},\ z^{(k+1)}=\dfrac{d_3-a_3x^{(k+1)}-b_3y^{(k+1)}}{c_3}\)
Quick example
Same system: \(10x+y+z=12\), \(x+10y+z=12\), \(x+y+10z=12\).
| Iteration | \(x\) | \(y\) | \(z\) |
|---|---|---|---|
| 1 | 1.2 | \(\frac{12-1.2-0}{10}=1.08\) | \(\frac{12-1.2-1.08}{10}=0.972\) |
| 2 | 0.9948 | 1.00332 | 1.000188 |
| 3 | 0.999649 | 1.000016 | 1.000033 |
After three iterations Seidel is within 0.0004 of the exact answer (1, 1, 1), while Jacobi after three iterations was still at 1.008.
Common mistakes
- Forgetting to use the latest \(x\) when computing \(y\), which turns it back into Jacobi.
- Not checking diagonal dominance; the method can diverge.
- Rounding intermediate values too much, so the final digits are wrong.
How MU asks it
Frequent 5/6-mark question (and SCILAB tutorial): "Solve by Gauss–Seidel method, correct to three decimal places: \(27x+6y-z=85\), \(6x+15y+2z=72\), \(x+y+54z=110\)". Sometimes "Compare with the Jacobi method". Show the rearranged equations, clearly indicate which updated values you use, tabulate, and state the solution.