I want to check the convergence of Jacobi (from now on J) and of Gauss-Seidel (from now on GS) methods applied to a linear system $Ax=b$, where $A$ is square and non-singular.
I was wondering if the convergence of one of the two methods shed light on the convergence of the other. I mean:
When can I say that if J converges then GS converges? When can I say that if GS converges then J converges?
I know that if $A$ is (for example, but this is true with other very similar hypothesis) strictly diagonally dominant then the spectral radius of J and GS matrices is $<1$ and so both J and GS converge. Also I know that for example if $A$ is Hermitian with real positive diagonal elements then $A$ is positive defined iff GS converges.