Gauss-Seidel: for each equation i, solve for xᵢ using latest values. xᵢ^(k+1) = (bᵢ − Σⱼ≠ᵢ aᵢⱼxⱼ)/aᵢᵢ. Uses updated values within same iteration (unlike Jacobi). Converges for diagonally dominant or symmetric positive definite matrices.
How to Use
Set up the system:
- Matrix A: Coefficient matrix
- Vector b: Right-hand side
- x₀: Initial guess (zeros)
vs Jacobi
Gauss-Seidel uses new values immediately: x₂ uses updated x₁ from same iteration. Typically converges ~2× faster. Uses less memory (no second array). But is sequential (harder to parallelize).
Convergence
Guaranteed for: diagonally dominant matrices (|aᵢᵢ| > Σⱼ≠ᵢ|aᵢⱼ|), SPD matrices. Spectral radius ρ(T_GS) < 1. Usually ρ(T_GS) ≤ ρ(T_J)² for SPD.
Step-by-Step Instructions
- 1Select or enter system.
- 2Set tolerance.
- 3View iterations.
- 4Check convergence.
- 5Get solution.