- Asked 2/13
- 2068 Bhadra · 8 marks
Iterative method: 4 equations
Short answer
The Gauss-Seidel method is used after reordering equations for better diagonal dominance. Starting from zero, after about 19 iterations the solution converges to x≈5.000, y≈6.000, z≈-10.000, w≈8.000.
Solve the following set of linear equations using a suitable iterative method. 2x + y + z - 2w = -10 4x + 2z + w = 8 3x + 2y + 2z = 7 x + 3y + 2z - w = -5
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
As given, the system is not diagonally dominant (for example, the coefficient of y in the second equation is 0 and the x coefficient in the first equation is only 2). Reorder the equations so that the largest available coefficient is on the diagonal:
4x + 2z + w = 8 (given eq. 2)
3x + 2y + 2z = 7 (given eq. 3)
x + 3y + 2z − w = −5 (given eq. 4)
2x + y + z − 2w = −10 (given eq. 1)
The diagonal entries are 4, 2, 2, −2. Only the first row is strictly dominant, but for this order the Gauss-Seidel iteration matrix has spectral radius 0.59 < 1 (checked), so the iteration converges.
Step 2: Iteration formulas
x = (8 − 2z − w) / 4
y = (7 − 3x − 2z) / 2
z = (−5 − x − 3y + w) / 2
w = (10 + 2x + y + z) / 2
Start with x(0) = 0, y(0) = 0, z(0) = 0, w(0) = 0.
Step 3: Iterations
| Iter | x | y | z | w |
|---|---|---|---|---|
| 0 | 0.000 | 0.000 | 0.000 | 0.000 |
| 1 | 2.000 | 0.500 | -4.250 | 5.125 |
| 2 | 2.844 | 3.484 | -6.586 | 6.293 |
| 3 | 3.720 | 4.506 | -7.973 | 6.986 |
| 4 | 4.240 | 5.113 | -8.796 | 7.398 |
| 5 | 4.549 | 5.473 | -9.285 | 7.643 |
| 6 | 4.732 | 5.687 | -9.576 | 7.788 |
| 7 | 4.841 | 5.814 | -9.748 | 7.874 |
| 8 | 4.906 | 5.890 | -9.850 | 7.925 |
| 9 | 4.944 | 5.935 | -9.911 | 7.956 |
| 10 | 4.967 | 5.961 | -9.947 | 7.974 |
| 11 | 4.980 | 5.977 | -9.969 | 7.984 |
| 12 | 4.988 | 5.986 | -9.981 | 7.991 |
| 13 | 4.993 | 5.992 | -9.989 | 7.994 |
| 14 | 4.996 | 5.995 | -9.993 | 7.997 |
| 15 | 4.998 | 5.997 | -9.996 | 7.998 |
| 16 | 4.999 | 5.998 | -9.998 | 7.999 |
| 17 | 4.999 | 5.999 | -9.999 | 7.999 |
| 18 | 4.999 | 5.999 | -9.999 | 8.000 |
| 19 | 5.000 | 6.000 | -10.000 | 8.000 |
Between the last two iterations every unknown changes by less than 0.0005, so we stop. (Check: x = 5, y = 6, z = −10, w = 8 satisfy all four equations exactly.)
Answer: x ≈ 5.000, y ≈ 6.000, z ≈ -10.000, w ≈ 8.000
✓ Answer checked by Claude Opus 5.5 (high effort)
Related questions
- LU factorization: x-3y+10z=3 · Numerical Method, chapter 3 · asked 2×
- Gauss elimination: x+2y+3z=6 · Numerical Method, chapter 3 · asked 3×
- Gauss-Seidel: 20x+y-2z=17 · Numerical Method, chapter 3
- Gauss elimination/Jordan with pivoting · Numerical Method, chapter 3 · asked 2×
- Least squares significance, normal equations · Numerical Method, chapter 4
- BVP: y'' + 3y' = y + x² · Numerical Method, chapter 6
- RK-4: y'' - 3y' + y = sin x · Numerical Method, chapter 6
- Solve ODE by Laplace transform y''−3y'+2y=4x+e^(3x) · Mathematics III, chapter 3 · asked 14×
- Solve y''+3y'+2y=e^(2x) sin x · Mathematics I, chapter 4 · asked 2×
- Centre, axes of ellipse 2x²+3y²−4x−12y+13=0 · Mathematics I, chapter 3 · asked 7×
- Convergence of 2x+3x²/8+4x³/27+... · Mathematics, chapter 6 · asked 3×