Gauss Seidel Calculator

Calculator Inputs

Use 2 to 6 variables. Large screens show three columns, smaller screens two columns, and mobile shows one column.

System Size

Tolerance

Maximum Iterations

Relaxation Factor (ω)

Decimal Places

Stopping Rule

Action Panel

Tips:

Prefer a diagonally dominant matrix for stable convergence.
Use zero or close estimates as starting guesses.
Set ω = 1.00 for classic Gauss Seidel.

Coefficient Matrix A and Vector b

Initial Guesses x⁽⁰⁾

Example Data Table

This sample system is preloaded by default and converges to x₁ = 0.5, x₂ = 1.0, and x₃ = 0.5.

Equation	a₁	a₂	a₃	b	Initial Guess
Equation 1	4	1	2	4	x₁⁽⁰⁾ = 0
Equation 2	3	5	1	7	x₂⁽⁰⁾ = 0
Equation 3	1	1	3	3	x₃⁽⁰⁾ = 0

Formula Used

The Gauss Seidel method updates one variable at a time using the newest available values inside the same iteration.

x_i^(k+1) = ( b_i - Σ(a_ij x_j^(k+1), j<i) - Σ(a_ij x_j^(k), j>i) ) / a_ii

An optional relaxation factor is included for advanced control:

x_i^(k+1, relaxed) = (1 - ω) x_i^(k) + ω x̂_i^(k+1)

Residual checking verifies the current approximation against the original system:

r = b - Ax, ||r||∞ = max |r_i|, Update Error = max |x_i^(k+1) - x_i^(k)|

How to Use This Calculator

Choose the system size from 2 × 2 up to 6 × 6.
Enter the coefficient matrix A and constants vector b.
Provide starting guesses for all variables.
Set tolerance, maximum iterations, decimal places, and optional relaxation factor.
Pick a stopping rule based on update error, residual norm, or both.
Press Solve System to display results above the form.
Review the convergence chart, solution verification table, and full iteration history.
Download iteration history as CSV or generate a PDF report when needed.

Frequently Asked Questions

1) What does this calculator solve?

It solves linear systems of equations using the Gauss Seidel iterative method. You enter the matrix coefficients, constants, and starting guesses, then the page estimates the unknown values through repeated updates.

2) When does Gauss Seidel usually converge?

It commonly converges when the matrix is strictly diagonally dominant or otherwise well conditioned for iteration. Convergence can fail or slow down when diagonal terms are weak or the system is poorly arranged.

3) What is the difference between update error and residual norm?

Update error measures how much the solution changes between iterations. Residual norm measures how well the current solution satisfies the original equations. Using both provides a stronger convergence check.

4) What does the relaxation factor do?

The relaxation factor adjusts how strongly each new estimate replaces the old one. A value of 1.00 is classic Gauss Seidel. Values below or above one can sometimes improve stability or speed.

5) Why is a zero diagonal value rejected?

Each update divides by the diagonal element of the current row. A zero diagonal would cause division by zero, so the method cannot proceed until the system is rearranged or reformulated.

6) Why do results sometimes stop without convergence?

The method stops when it meets the chosen tolerance or reaches the maximum iteration limit. If convergence is not achieved, try improving the matrix arrangement, guesses, tolerance, or relaxation factor.

7) Can I use this for larger systems?

This page supports systems from 2 to 6 variables for practical manual entry. The same numerical method extends to larger systems, but bigger models usually benefit from programmatic matrix handling.

8) What is the best stopping rule to choose?

Using both update error and residual norm is usually the safest option. It checks whether the estimates have stabilized and whether the solution also matches the original equations closely.