
\subsection{\bf Diagonalization of Conjugate Gradient Equations}

\begin{slide}

\begin{itemize}
\item We can take all the PPCG equations and multiply them by the
transformation matrix $P$ and convert them into ``$\, \widetilde{}\, $''
equations.

\item Scale factor $\alpha$
\[ \alpha^{(k)} = \frac{\underline{\tilde{r}}^{(k-1)T} \cdot
\underline{\tilde{r}}^{(k-1)}}{\underline{\tilde{r}}^{(k-1)T}
\Lambda \underline{\tilde{r}}^{(k-1)}} \]

\item Iteration
\[ \underline{\tilde{x}}^{(k)} = \underline{\tilde{x}}^{(k-1)} 
+ \alpha_k \underline{\tilde{r}}^{(k-1)} \]

\item Residual
\begin{eqnarray*}
\underline{\tilde{r}}^{(k)} & = & \underline{\tilde{b}} - \Lambda\,
\underline{\tilde{x}}^{(k)} \\
& = & \underline{\tilde{b}} - \Lambda\, \underline{\tilde{x}}^{(k-1)}
- \alpha_k \Lambda \underline{\tilde{r}}^{(k-1)} \\
& = & \left( I - \alpha_k \Lambda \right) \tilde{r}^{(k-1)}
\end{eqnarray*}

\item Residual components in diagonalized coordinate system
\[ \tilde{r}^{(k)} \big|_i = \left( 1 - \alpha_k \lambda_i
\right) \tilde{r}^{(k-1)} \big|_i \]
where $i=1\ldots n$ labels components
\end{itemize}

\end{slide}

