How to use the Gauss-Seidel Method Calculator
- 1. Enter the equationsOne equation per line as coefficients then the right-hand side: 20 1 -2 17.
- 2. Set tolerance and startThe default start is all zeros. Change the tolerance if needed.
- 3. Read the solutionEach row shows the new values and the largest change.
How Gauss-Seidel works
Rewrite each equation for its diagonal unknown: x = (b₁ − a₁₂y − a₁₃z)/a₁₁, and so on. Start from a guess, usually zeros. Gauss-Seidel uses each new value straight away in the next equation, which is why it usually converges in fewer steps than the Jacobi method.
Convergence is guaranteed when the matrix is diagonally dominant: in every row, the diagonal entry is at least as large as the sum of the other entries (in absolute value). The tool checks this for you.
Worked example
Solve 20x + y − 2z = 17, 3x + 20y − z = −18, 2x − 3y + 20z = 25. The matrix is diagonally dominant.
Iteration 1 from (0, 0, 0): x = 17/20 = 0.85; y = (−18 − 3·0.85)/20 = −1.0275; z = (25 − 2·0.85 + 3·(−1.0275))/20 = 1.0109. Iteration 2: x = 1.0025, y = −0.9998, z = 0.9998. Iteration 3: (0.99997, −1.00001, 1.00000). By iteration 4 the change is below 0.0001, giving x = 1, y = −1, z = 1.
If it does not converge
Reorder the equations so that the biggest coefficient of each unknown sits on the diagonal. In the example, the equation with 20x goes first. If no order makes the matrix dominant, the method may still work, but if values grow the tool stops and warns you. For an exact check of the system, find the rank with the matrix rank calculator.
Frequently asked questions
What is the difference between Jacobi and Gauss-Seidel?
Jacobi uses only values from the previous iteration. Gauss-Seidel uses each new value as soon as it is found, so it usually needs fewer iterations.
Why must the diagonal entries be non-zero?
Each update divides by the diagonal entry. Swap equations so none of them is zero.
How accurate is the answer?
It stops when no unknown changes by more than your tolerance. The residual |Ax − b| is shown as a check.
Is this calculator free?
Yes. You enter your name and WhatsApp number once when you see your result, and other Uncodemy tools then open without the form.