Gauss-Seidel Calculator
Solve a diagonally-dominant linear system iteratively using the Gauss-Seidel method.
Inputs
x₁
1.04326923
x₂
2.26923077
x₃
-1.08173077
Iterations
8
Max residual
8.7408 × 10⁻¹⁰
Status
Converged
Step by step
Values used
a₁₁ = 10; a₁₂ = -1; a₁₃ = 2; b₁ = 6; a₂₁ = -1; a₂₂ = 11; a₂₃ = -1; b₂ = 25; a₃₁ = 2; a₃₂ = -1; a₃₃ = 10; b₃ = -11; Tolerance = 0.0000; Max iterations = 100
Gauss-Seidel iteration
xᵢ^(k+1) = (bᵢ - Σⱼ<ᵢ aᵢⱼ·xⱼ^(k+1) - Σⱼ>ᵢ aᵢⱼ·xⱼ^(k)) / aᵢᵢ
x₁
= 1.04326923
x₂
= 2.26923077
x₃
= -1.08173077
Iterations
= 8
Max residual
= 0.0000
Status
= Converged
How it works
Gauss-Seidel improves on Jacobi by using updated values immediately: when computing xᵢ^(k+1), it uses xⱼ^(k+1) for j < i (already computed this iteration) and xⱼ^(k) for j > i. This typically converges faster than Jacobi. Guaranteed to converge for diagonally dominant or symmetric positive definite matrices.
Formula
Gauss-Seidel iteration
xᵢ^(k+1) = (bᵢ - Σⱼ<ᵢ aᵢⱼ·xⱼ^(k+1) - Σⱼ>ᵢ aᵢⱼ·xⱼ^(k)) / aᵢᵢ
- xⱼ^(k+1)
- Already-updated values (this iteration)
- xⱼ^(k)
- Previous iteration values
Frequently Asked Questions
Why is Gauss-Seidel faster than Jacobi?
By using the most recent values immediately, Gauss-Seidel propagates corrections faster through the system, typically converging in fewer iterations.
Can I use any initial guess?
Yes, but a good initial guess (close to the solution) reduces iteration count. The zero vector is a common default.