
\subsection{\bf Conjugate Gradient and Method of Steepest Descent}

\begin{slide}

\begin{itemize}
\item We can complete the square
\begin{eqnarray*}
F (\alpha) & = & \frac{1}{2} \underline{r}^{(k)T} A
\underline{r}^{(k)} \left( \alpha - \alpha_k \right)^2 \\
& + & \mbox{Terms Independent of}\ \alpha
\end{eqnarray*}

\item where $\alpha_k = \underline{r}^{(k)T}\cdot
\underline{r}^{(k)} / \underline{r}^{(k)T} A
\underline{r}^{(k)}$

\item Note that if
\[
\underline{x}^{(k+1)} = \underline{x}^{(k)} + \alpha_k
\underline{r}^{(k)}
\]
\[
F \left( \underline{x}^{(k+1)} \right) \leq F \left(
\underline{x}^{(k)} \right)\ \mbox{rigorously}
\]

\item This is Poor Person's Conjugate Gradient or when applied to
minimization, the method of steepest descent.  It is called this
because we move from $\underline{x}^{(k)}$ along direction $-\nabla
F$---the direction along which $F$ varies most rapidly.
\end{itemize}

\end{slide}

