Jacobi Method Calculator
Solve a diagonally-dominant linear system iteratively using the Jacobi method.
Inputs
x₁
1.04326923
x₂
2.26923077
x₃
-1.08173077
Iterations
15
Max residual
8.7240 × 10⁻⁹
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
Jacobi iteration
xᵢ^(k+1) = (bᵢ - Σⱼ≠ᵢ aᵢⱼ·xⱼ^(k)) / aᵢᵢ
x₁
= 1.04326923
x₂
= 2.26923077
x₃
= -1.08173077
Iterations
= 15
Max residual
= 0.0000
Status
= Converged
How it works
The Jacobi method solves Ax = b iteratively by isolating each variable: xᵢ^(k+1) = (bᵢ − Σⱼ≠ᵢ aᵢⱼxⱼ^(k)) / aᵢᵢ. All updates use values from the previous iteration (unlike Gauss-Seidel). Convergence is guaranteed for diagonally dominant systems. The method reports iteration count and achieved residual.
Formula
Jacobi iteration
xᵢ^(k+1) = (bᵢ - Σⱼ≠ᵢ aᵢⱼ·xⱼ^(k)) / aᵢᵢ
- xᵢ^(k+1)
- New value of variable i
- xⱼ^(k)
- Previous iteration values
- aᵢⱼ
- Matrix coefficients
Frequently Asked Questions
When does Jacobi converge?
Convergence is guaranteed when the matrix is strictly diagonally dominant: |aᵢᵢ| > Σⱼ≠ᵢ |aᵢⱼ| for all rows.
How does Jacobi compare to Gauss-Seidel?
Gauss-Seidel uses updated values immediately (faster convergence) but is sequential. Jacobi can be parallelized since it uses only previous-iteration values.