Convergence
The standard convergence condition (for any iterative method) is when the spectral radius of the iteration matrix is less than 1:
The method is guaranteed to converge if the matrix A is strictly or irreducibly diagonally dominant. Strict row diagonal dominance means that for each row, the absolute value of the diagonal term is greater than the sum of absolute values of other terms:
The Jacobi method sometimes converges even if these conditions are not satisfied.
Read more about this topic: Jacobi Method