
\subsection{\bf Poor Person's Conjugate Gradient}
\subsection{\bf and Eigenvalues of Matrix}

\begin{slide}

\begin{itemize}
\item What is wrong with this version (PPCG) of conjugate gradient?
\item More generally, what is wrong with steepest descent?
\item The matrix {\bf A} is symmetric and can be diagonalized as
usual
\begin{eqnarray*}
A & = & P^{-1} \Lambda P \\
\Lambda & = & \mbox{diag.}\ \left( \lambda_1\,, \lambda_2\,, \ldots
\lambda_n\, \right) \\
& = & \left\lbrack \begin{array}{cccc}
\lambda_1 & 0 & 0 & \ldots \\
0 & \lambda_2 & 0 & \ldots \\
0 & 0 & \lambda_3 & \ldots \end{array} \right\rbrack \\
&& \lambda_1 \geq \lambda_2 \geq \ldots
\end{eqnarray*}

\item $P$ is orthogonal matrix $P^T P = I$

\item Let
\begin{eqnarray*}
\underline{\tilde{x}}^{(k)} & = & P\underline{x}^{(k)} \\
\underline{\tilde{b}} & = & P\underline{b} \\
\underline{\tilde{r}}^{(k)} & = & P\underline{r}^{(k)} \\
\end{eqnarray*}
\end{itemize}

\end{slide}

