
\subsection{\bf Improvement of Poor Person's}
\subsection{\bf Conjugate Gradient with}
\subsection{\bf Orthonormal Iteration}

\begin{slide}

\begin{itemize}
\item In practice
$ \left| \lambda_1 = \lambda_{\max} \right| \gg \left| \lambda_n =
\lambda_{\min} \right| $ and in simple cases
$ \frac{\lambda_n}{\lambda_1} \sim h^2 $ where $h$ is grid spacing

\item We must modify PPCG to keep search directions properly
orthogonal so that eigenvectors with large eigenvalues do not
dominate

\item We set 
\[ \underline{x}^{(k)} = \underline{x}^{(k-1)} + \alpha_k
\underline{p}^{(k)} \]
where we choose
\[ \underline{p}^{(k)T} A \underline{p}^{(k-1)} = 0 \]
\end{itemize}

\end{slide}

