Calculator
Example Data Table
Use this sample 3 × 3 system to test the solver quickly.
| Equation | Coefficient Row | Right Side | Initial Guess |
|---|---|---|---|
| 1 | [10, -1, 2] | 6 | 0 |
| 2 | [-1, 11, -1] | 25 | 0 |
| 3 | [2, -1, 10] | -11 | 0 |
Formula Used
The Gauss Seidel method updates each variable immediately after it is computed. That makes it faster than the Jacobi method for many well-formed systems.
If relaxation is used, the adjusted update becomes:
The solver also tracks:
- Absolute error: |xᵢ(new) − xᵢ(old)|
- Relative error: |xᵢ(new) − xᵢ(old)| / |xᵢ(new)|
- Residual: r = b − Ax
How to Use This Calculator
- Select the matrix size from 2 to 8 variables.
- Choose built-in example mode or custom input mode.
- Enter the coefficient matrix A values.
- Enter vector b values for the right-hand side.
- Enter the initial guess values for all variables.
- Set tolerance, maximum iterations, and optional relaxation factor.
- Click Solve System to compute the approximate solution.
- Review the result box, iteration table, and convergence chart.
- Export the iteration data as CSV or PDF if needed.
Frequently Asked Questions
1. What does this solver calculate?
It solves linear systems of equations using the Gauss Seidel iterative method. You enter matrix coefficients, constants, and starting guesses, then the page computes approximate variable values across repeated updates.
2. When does Gauss Seidel converge well?
It usually performs well when the matrix is diagonally dominant or symmetric positive definite. Poorly conditioned systems may converge slowly, oscillate, or fail to settle within the chosen iteration limit.
3. Why are initial guesses required?
Initial guesses provide the starting point for the iteration cycle. Better starting values can reduce iteration count, although many stable systems still converge from zero guesses.
4. What does tolerance control?
Tolerance sets the stopping threshold for successive solution changes. Smaller tolerance means tighter accuracy, but it often increases the number of iterations required before the method stops.
5. What is the relaxation factor?
The relaxation factor adjusts each fresh Gauss Seidel update. A value of 1 gives standard behavior, values below 1 damp changes, and some values above 1 may accelerate convergence.
6. What does the residual show?
The residual measures how closely the current solution satisfies the original equations. Smaller residuals mean the computed vector better matches the system Ax = b.
7. Why might the solver stop without convergence?
The iteration may hit the maximum limit first, or the matrix structure may be unsuitable for stable convergence. Checking diagonal dominance and scaling the system often helps.
8. Can I export the results?
Yes. The page includes CSV and PDF export buttons for the iteration table. These are useful when you want to save convergence history or include it in reports.