I found that the relaxation methods for solving linear system as an iterative sequence are derived from the Gauss-Seidel method and not from the Jacobi method. I understand that the Gauss-Seidel method might look better behaved and in many cases have a smaller spectral radius than that of the Jacobi method. However, once we introduce a relaxation parameter $\omega$, things might change. Or, if they do not improve can anyone show why doing a Jacobi-based relaxation would be a bad idea? That is \begin{equation} \bf{x}_{k+1} = (\rm 1 - \omega) \bf{x}_k + \omega \, Jacobi\_Iteration( \bf{x}_k). \end{equation} does not converge faster in any situation that \begin{equation} \bf{x}_{k+1} = (\rm 1 - \omega) \bf{x}_k + \omega \, Gauss\_Seidel\_Iteration( \bf{x}_k) ? \end{equation}

Thanks.