Enter System Details
This solver handles 2 to 6 equations, supports relaxation, and logs every iteration for review.
Example Data Table
Use this sample system to test convergence and compare your manual work.
| Equation | Linear Expression | Constant |
|---|---|---|
| 1 | 10x₁ - x₂ + 2x₃ | 6 |
| 2 | -x₁ + 11x₂ - x₃ | 25 |
| 3 | 2x₁ - x₂ + 10x₃ | -11 |
| Approximate solution: x₁ = 1.043269, x₂ = 2.269231, x₃ = -1.081731 | ||
Formula Used
The Gauss-Seidel method updates each variable with the latest available values from the same iteration. For the i-th equation:
If relaxation is enabled, the adjusted update becomes:
The solver also tracks:
- Maximum change: max |x_i^(k+1) - x_i^(k)|
- Residual norm: ||b - Ax||∞
- Maximum relative error: max |(x_i^(k+1) - x_i^(k)) / x_i^(k+1)| × 100
How to Use This Calculator
- Select the number of equations in your linear system.
- Enter all matrix coefficients, constant values, and initial guesses.
- Choose a tolerance, maximum iterations, precision, and stopping rule.
- Set ω = 1 for standard Gauss-Seidel or another value for relaxed updates.
- Press Solve System to view the solution summary and full iteration history.
- Use the export buttons to save iteration details as CSV or PDF.
FAQs
1. What does this calculator solve?
It solves square systems of linear equations using the Gauss-Seidel iterative technique. You can enter coefficients, constants, initial guesses, and convergence controls for detailed numerical analysis.
2. When does Gauss-Seidel usually converge?
It often converges when the matrix is strictly diagonally dominant or symmetric positive definite. Some other systems also converge, but those conditions are common practical indicators.
3. What is the role of the initial guess?
The initial guess starts the iteration. A better guess can reduce the number of iterations, although the final solution should stay the same if the method converges properly.
4. What does the relaxation factor mean?
The relaxation factor adjusts each update. A value of 1 gives the standard method. Values below 1 under-relax, while values above 1 may speed convergence or cause instability.
5. Why do I see a non-dominant matrix warning?
That warning means convergence is less predictable. The solver still tries the iteration, but the values may oscillate, diverge, or converge slowly depending on the matrix structure.
6. What is the residual norm?
The residual norm measures how closely the current approximation satisfies Ax = b. Smaller residuals usually indicate a more accurate solution estimate.
7. Can I use decimals and negative values?
Yes. The calculator accepts decimal coefficients, constants, and guesses, including negative values. That makes it suitable for many classroom and numerical-methods problems.
8. Why might the method fail?
Failure can happen with zero diagonal entries, poor matrix properties, unsuitable relaxation, or extremely tight settings. In such cases, try reordering equations or changing parameters.