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\title[CME 324 ASSIGNMENT 2]{CME 324: ITERATIVE METHODS\\SPRING 2005/06\\ASSIGNMENT 2}
\author{Gene~H.~Golub}
\thanks{This assignment is due in class on Wednesday, May 24.}
\date{\today, version 2.2}
\maketitle
\begin{enumerate}[\bfseries 1.] \setlength{\itemsep}{2ex}
\item Consider the matrix $A$ given in Problem \textbf{3} of Homework
\textbf{1}, and the equation given in part (g) of that problem. We want to
conduct a number of experiments with the CG method and make comparisons.
\begin{enumerate}
\item Solve the linear system with the two variants described in class, using
the preconditioner $M=I$. Compute the residual vector $\mathbf{r}%
_{k}=\mathbf{b}_{k}-A\mathbf{x}_{k}$ in one set of experiments, and then
repeat the experiments using the recursion for the residual vector. Graph the
behavior of $\lVert\mathbf{x}-\mathbf{x}_{k}\rVert_{2}$, $\lVert
\mathbf{x}-\mathbf{x}_{k}\rVert_{A}$, $\lVert\mathbf{r}_{k}\rVert_{2}$.
Described the termination rule for determining your approximate solution.
Which method seems to perform best in terms of computational efficiency and accuracy.
\item Repeat part (a) using the preconditioner $M=\operatorname*{blockdiag}%
(A)$. Compare the convergence properties with those given by the bound.
\end{enumerate}
\item Let $\sigma>0$. Consider the differential equation%
\begin{align*}
& -u^{\prime\prime}+\sigma u^{\prime}=f,\qquad\qquad0