
\subsection{\bf Eigenvalues of Gauss Seidel Iteration Matrix}

\begin{slide}

\begin{itemize}
\item It was relatively easy to find the eigenvalues and
eigenfunctions of Jacobi's method for a simple geometry.

\item Gauss-Seidel it is not so easy as matrix $M^{-1}N$ is not
symmetric.  There is a nice discussion in

\begin{description}
\item[] ``Numerical Computation of Internal and External Flows,''
C.~Hirsch, Wiley, 1988.
\end{description}

\item Eigenvalues $\lambda_{GS}$ of Gauss-Seidel Iteration Matrix are
related to those of Jacobi Iteration $\lambda_J$ by $\lambda_{GS} =
\lambda_{J}^2$

\item We showed that

\begin{displaymath}
1 - \epsilon \lea \lambda_J \lea -1 + \epsilon \hspace{2em} \epsilon
\sim \frac{\pi^2}{2N^2}
\end{displaymath}

\item Therefore,

\begin{displaymath}
\left\vert \lambda_{GS}\right\vert_{\rm max} = \left( \left\vert \lambda_J 
\right\vert_{\max}\right) ^2 \sim 1 - 2\epsilon 
\end{displaymath}

\end{itemize}

\end{slide}
