
\subsection{\bf Convergence of SOR Compared to Jacobi and Gauss Seidel}

\begin{slide}

\begin{itemize}
\item Remember number of iterations is proportional to
\begin{eqnarray*}
\frac{1}{1-\rho} & = & \frac{2N^2}{\pi^2} \hspace{1em} \mbox{Jacobi} \\
& = & \frac{N^2}{\pi^2} \hspace{1.5em} \mbox{Gauss\ Seidel} \\
& = & \frac{N}{\pi} \hspace{2em} \mbox{SOR} \\
\end{eqnarray*}

\begin{description}
\item[---] i.e., number of iterations decreases in SOR by a factor
$N/\pi$, which is large!
\end{description}

\item Unfortunately, the above analysis is only valid in simple cases

\hspace{2em}$\bullet$ simple geometry

\hspace{2em}$\bullet$ some orderings in Gauss-Seidel
\end{itemize}

\end{slide}


