Jacobi iteration: xᵢ^(k+1) = (bᵢ − Σⱼ≠ᵢ aᵢⱼxⱼ^(k))/aᵢᵢ. All components updated simultaneously using previous iteration. Naturally parallel. Converges for diagonally dominant matrices. Decomposition: A = D + L + U, iteration: x^(k+1) = D⁻¹(b − (L+U)x^(k)).
How to Use
Set up the system:
- Matrix A: Coefficients
- Vector b: Right-hand side
- Output: Solution + iterations
Parallelism
Each xᵢ^(k+1) depends only on x^(k) — all can be computed independently. Ideal for GPU/distributed computing. Gauss-Seidel is sequential but usually converges faster.
Convergence
Sufficient: strict diagonal dominance |aᵢᵢ| > Σⱼ≠ᵢ|aᵢⱼ|. Necessary and sufficient: spectral radius ρ(D⁻¹(L+U)) < 1.
Step-by-Step Instructions
- 1Select a system.
- 2Set tolerance.
- 3View iterations.
- 4Check convergence.
- 5Get solution.