Jacobi Method

Updated 01 Oct 2026

•
What it is
- •
  iteratively solves a linear system by updating every component using only values from the previous iteration
•
Formula
- •
  $x_i^{(k+1)}=[b_i-\sum_{j\ne i}a_{ij}x_j^{(k)}]/a_{ii}$
•
Convergence
- •
  with $A=D+L+U$ , the iteration matrix is $T_J=-D^{-1}(L+U)$ .
- •
  convergence from every starting vector requires $\rho(T_J)<1$ .
- •
  strict row diagonal dominance is a sufficient condition, not a necessary one.
•
Distinguish from
- •
  Gauss-Seidel Method uses newly computed components immediately
- •
  symmetric positive definiteness alone does not guarantee Jacobi convergence

Linked references 4