Enter the linear system
Use matrix inputs, paste equations, import CSV data, or load an example. Enter decimals or fractions such as 1/2 and -3/4.
Formula used
The Gauss-Seidel method updates one variable at a time. Each new value immediately helps calculate the next variable. This usually improves convergence over simultaneous updates.
Setting ω to one gives standard Gauss-Seidel iteration. Values below one apply under-relaxation for added stability. Values above one apply successive over-relaxation.
How to use this calculator
- Select the number of variables and equations.
- Enter every coefficient, constant, and initial guess.
- Choose tolerance, precision, iteration limit, and error type.
- Set the relaxation factor or keep its default value.
- Enable reordering or Jacobi comparison when needed.
- Press Solve system and review every result.
For dependable convergence, place strong coefficients on the diagonal. A strictly diagonally dominant matrix is especially suitable. Some valid systems still fail to converge iteratively.
Worked example data
| Equation | x1 | x2 | x3 | b |
|---|---|---|---|---|
| 1 | 10 | -1 | 2 | 6 |
| 2 | -1 | 11 | -1 | 22 |
| 3 | 2 | -1 | 10 | -10 |
Start with zero for every initial guess. Use tolerance 0.000001 and one hundred iterations. The expected solution approaches x1 = 1, x2 = 2, and x3 = -1.
Understanding convergence and residuals
Convergence means successive estimates stop changing beyond the chosen tolerance. The calculator reports absolute or relative change. It also measures the residual vector Ax minus b.
A small update error alone may be misleading. The residual confirms whether calculated values satisfy the original equations. Smaller residual norms generally indicate stronger numerical agreement.
Diagonal dominance often supports reliable convergence. A small determinant may indicate singularity or instability. Equation reordering can help without changing the solution.
Advantages and limitations
Gauss-Seidel uses little memory and supports large sparse systems. Updated values are reused immediately during each pass. Relaxation can improve speed or control instability.
However, convergence is not guaranteed for every matrix. Poor scaling and weak diagonal coefficients cause difficulties. Direct methods may suit small problematic systems better.
Frequently asked questions
What does the Gauss-Seidel method solve?
It approximates solutions for square systems of linear equations using repeated updates.
Why are initial guesses required?
Iteration needs a starting vector. Better guesses may reduce the required iterations.
What tolerance should I use?
A tolerance near 0.000001 works for many educational calculations. Stricter values need more iterations.
What is diagonal dominance?
Each diagonal coefficient is at least as large as the other absolute row coefficients combined.
What does relaxation factor mean?
It controls how strongly each newly calculated value replaces the previous estimate.
Why can the method diverge?
The matrix structure, equation order, scaling, or relaxation setting may prevent convergence.
How is Gauss-Seidel different from Jacobi?
Gauss-Seidel immediately reuses updated values. Jacobi completes each iteration using only previous values.
What is the residual norm?
It summarizes how closely the calculated solution satisfies all original equations.
Can this calculator prove a unique solution exists?
It checks useful indicators, but rigorous existence and uniqueness analysis may require additional methods.