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Gauss-Seidel: 20x+y-2z=17

Short answer

The system is diagonally dominant, so Gauss-Seidel converges quickly from zero initial guess. After 3 iterations the solution is x = 1.0000, y = -1.0000, z = 1.0000.

Use Gauss Seidel method to solve the following equations: 20x + y - 2z = 17 3x + 20y - z = -18 2x - 3y + 20z = 25

Gauss-Seidel method: solve each equation for its diagonal unknown and use the newest values immediately in the same iteration. It converges when the system is diagonally dominant.

Step 1: Check dominance and rearrange

Diagonal coefficients: 20 > 1 + 2, 20 > 3 + 1, 20 > 2 + 3. The system is diagonally dominant, so the method converges.

Step 2: Iteration formulas

x(k+1) = (17 − y(k) + 2z(k)) / 20
y(k+1) = (−18 − 3x(k+1) + z(k)) / 20
z(k+1) = (25 − 2x(k+1) + 3y(k+1)) / 20

Start with x(0) = 0, y(0) = 0, z(0) = 0.

Step 3: Iterations

Iterxyz
00.00000.00000.0000
10.8500-1.02751.0109
21.0025-0.99980.9998
31.0000-1.00001.0000
41.0000-1.00001.0000
51.0000-1.00001.0000

Values of two successive iterations agree to 4 decimal places, so we stop.

Answer: x ≈ 1.0000, y ≈ -1.0000, z ≈ 1.0000

✓ Answer checked by Claude Opus 5.5 (high effort)

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