
\subsection{\bf Successive Overrelaxation Iteration}
\subsection{\bf Method (SOR)}

\begin{slide}

\begin{itemize}
\item This is relaxation (acceleration) method applied to
Gauss-Seidel.

\item In Jacobi, the optimal case was $\omega=1$ (i.e., no gain in
using relaxation).

\item However, for Gauss-Seidel, we will find $\omega\sim 2$ is best
and does improve the method.

\item The spectral radius $\left( \vert\lambda\vert_{\max} \right)$ of
Jacobi is given by
\begin{displaymath}
\rho_J = 1 - \frac{\pi^2}{2N^2}
\end{displaymath}
and for conventional (unaccelerated) Gauss Seidel by
\begin{displaymath}
\rho_{GS} = 1 - \frac{\pi^2}{N^2}
\end{displaymath}

\item For choice
\begin{displaymath}
\omega_{opt} \sim  2\left( 1 - \frac{\pi}{N} \right) 
\end{displaymath}
one can show that
\begin{displaymath}
\rho_{SOR} \sim 1 - \frac{\pi}{N}
\end{displaymath} 
\end{itemize}

\end{slide}

