
\subsection{\bf Intuitive Description of Poor Person's Conjugate Gradient 
Algorithm}

\begin{slide}

\begin{itemize}
\item So Poor Person's Conjugate Gradient is very good at large
eigenvalues, but converges slowly for small eigenvalues of $A$.

\item Note $A$ is positive definite for simple pde's $\Rightarrow
\lambda_i \geq 0$.

\item What we would like to do is to arrange first that $\tilde{r}^{(k)}
\big|_1 = 0$, then we see that equations give $\tilde{r}^{(k+1)}
\big|_2 = 0$, and then the next iteration will give $\tilde{r}^{(k+2)}
\big|_3 = 0$, and so on.

\item So qualitative goal of real conjugate gradient is to successively
minimize along the ``principal directions of matrix {\bf A}.''

\item As described in my optimization notes, this is easily understood
in case $n=2$ in terms moving in hills and valleys.
\end{itemize}

\end{slide}

