for Which values $w$ the SOR method is convergent?

Question

hope you can help me, I've got stuck on this.

Consider the following matrix. $$A =\begin{bmatrix} 1 & p\\ -p & 1 \end{bmatrix}$$

for which $w$ values the SOR method is convergent?

thanks!, Cheers.

Answer 1

Express $A$ as $A = D - L - U$, where $D$ is diagonal, $L$ is strictly lower triangular, and $U$ is strictly upper triangular. Then the SOR iteration matrix is

$$ H_\omega = (D-\omega L)^{-1}((1-\omega)D + \omega U) $$

The SOR method converges if the spectral radius $\rho(H_\omega) < 1$. Thus, you need to find all $\omega$ such that $\rho(H_\omega) < 1$.