The method for solving $ \boldsymbol{x}$ in the iterative
$\boldsymbol{x^{n+1}}=\boldsymbol{Tx^n} +\boldsymbol{ C}$
is said to converge given that $\rho(\boldsymbol{T}) <1$
Also, a quick way of inspecting convergence is checking if $\boldsymbol{A}$ diagonally dominant.
Now, without tedious checking for the spectral radius of $\boldsymbol{T}$, what will happen if the matrix $\boldsymbol{A}$ is not diagonally dominant?
Best Answer
Assuming that $Ax = b \Leftrightarrow x = Tx+C$, if the matrix is not diagonally dominant, the iteration may or may not converge, as this is just a sufficient condition. However, if it converges, it converges to the solution of $Ax = b$. So, you can just run your code with a stoping criteria based on the difference between consecutive iterations, and you'll probably be fine.