
\subsection{\bf Preconditioning in Conjugate Gradient}

\begin{slide}

\begin{itemize}
\item This is a general technique, where one finds $M$ such that
$M\underline{x} = \underline{y}$ can be easily solved and $M^{-1} A$
is numerically faster (easier or more stable) to solve than $A$ 

i.e., $M$ is an approximation to $A$ whose inverse can be found
easily---more precisely, for which linear equations $M\underline{x} =
\underline{y}$ can be solved as one never inverts matrices in this type
of problem.

\item The simplest preconditioner is $M = $ Diagonal part of $A$.

\item The best solution methods use sophisticated preconditioners
such as those given by multigrid.
\end{itemize}

\end{slide}

