
\subsection{\bf Conjugate Gradient Iteration for Quadratic Form}

\begin{slide}

\begin{itemize}
\item $\underline{x}^{(k)}$ is a point in an $n$ dimension space
($n$ is number of equations in $A\underline{x}=\underline{b}$)
\[
- \frac{\partial F}{\partial x} = - \nabla F = \underline{b} -
A\underline{x}^{(k)} = \underline{r}^{(k)}
\]

\item $\underline{r}^{(k)}$ is residual

\item Let $\underline{x}^{(k+1)} = \underline{x}^{(k)} + \alpha
\underline{r}^{(k)}$ ($\alpha$ a scalar, i.e., a number).

\item $F$ becomes a function of $\alpha$ for this special choice of
$\underline{x}^{(k+1)}$.  This is useful as it is much easier to
minimize functions of one variable than those that depend on $n$.

\begin{eqnarray*}
F (\alpha) & = & \alpha^2 \left\lbrack \frac{1}{2}\,
\underline{r}^{(k)T} A \underline{r}^{(k)} \right\rbrack \\
& - & \alpha \underline{r}^{(k)T} \cdot \underline{r}^{(k)} \\
& + & \mbox{Terms Independent of}\ \alpha
\end{eqnarray*}
\end{itemize}

\end{slide}

