
\subsection{\bf Introduction to Poor Person's}
\subsection{\bf Conjugate Gradient}

\begin{slide}

\setcounter{equation}{0}

\begin{itemize}
\item Finite Element Method leads to minimization of a Quadratic
Form
\begin{equation}
F(\underline{x}) = \frac{1}{2} \underline{x}^T A\underline{x} -
\underline{x}^T b
\label{pp1.eqn}
\end{equation}
with customary solution
\begin{equation}
A\underline{x} = \underline{b}
\label{pp2.eqn}
\end{equation}
derived by differentiating (1) with respect to $\underline{x}$ or by
completing the square
\[
F(\underline{x}) = \frac{1}{2} \left( \nabla - A^{-1} \underline{b}
\right)^T \underline{A} \left( \underline{x} - A^{-1} \underline{b}
\right) - \frac{1}{2} \underline{b}^T A^{-1} \underline{b}
\]

\item We will derive an iterative approach (PPCG) to (\ref{pp2.eqn})
by an intuitive minimization method applied to (\ref{pp1.eqn}).  This
is not the true conjugate method, but has the basic idea.  As usual,
iterations are labelled by $(k)$ and we start with
$\underline{x}^{(k)}$
\end{itemize}

\end{slide}

