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\begin{document}
\title[CME 324 ASSIGNMENT 1]{CME 324: ITERATIVE METHODS\\SPRING 2005/06\\ASSIGNMENT 1}
\author{Gene~H.~Golub}
\thanks{This assignment is due in class on Monday, May 1.}
\date{\today, version 1.0}
\maketitle
\begin{enumerate}[\bfseries 1.]
\item Consider an $n\times n$ tridiagonal matrix of the form%
\[
T_{\alpha}=%
\begin{bmatrix}
\alpha & -1 & & & & \\
-1 & \alpha & -1 & & & \\
& -1 & \alpha & -1 & & \\
& & -1 & \alpha & -1 & \\
& & & -1 & \alpha & -1\\
& & & & -1 & \alpha
\end{bmatrix}
,
\]
where $\alpha$ is a real parameter.
\begin{enumerate}
\item Verify that the eigenvalues of $T_{\alpha}$ are given by%
\[
\lambda_{j}=\alpha-2\cos(j\theta),\qquad j=1,\dots,n,
\]
where%
\[
\theta=\frac{\pi}{n+1}%
\]
and that an eigenvector associated with each $\lambda_{j}$ is%
\[
\mathbf{q}_{j}=[\sin(j\theta),\sin(2j\theta),\dots,\sin(nj\theta)]^{\top}.
\]
Under what condition on $\alpha$ does this matrix become positive definite?
\item Now take $\alpha=2$.
\begin{enumerate}
\item Will the Jacobi iteration converge for this matrix? If so, what will its
convergence factor be?
\item Will the Gauss-Seidel iteration converge for this matrix? If so, what
will its convergence factor be?
\item For which values of $\omega$ will the SOR iteration converge?
\end{enumerate}
\end{enumerate}
\item Prove that the iteration matrix $G_{\omega}$ of SSOR can be expressed as%
\[
G_{\omega}=I-\omega(2-\omega)(D-\omega F)^{-1}D(D-\omega E)^{-1}A.
\]
\item We are interested in solving Poisson's equation on a rectangle with
$h=1/(n+1)$. We want to use a nine-point formula; i.e.\newline
\begin{center}
\includegraphics[height = 1.5in]{hw1fig1.jpg}
\end{center}
Thus,%
\[
A=%
\begin{bmatrix}
T & B & & & \\
B & \ddots & \ddots & & \\
& \ddots & \ddots & \ddots & \\
& & \ddots & \ddots & B\\
& & & B & T
\end{bmatrix}
.
\]
where the matrices $T$ and $B$ are tridiagonal.
\begin{enumerate}
\item Write down the matrices $T$ and $B$.
\item Give the eigenvalues and eigenvectors of $T$ and $B$.
\item Show that $TB=BT$.
\item Find the eigenvalues and eigenvectors of $A$. (\textit{Hint}: First,
diagonalize $T$ and $B$ and then reorder the rows and columns so that the
matrix is block diagonal.)
\item Consider the block Jacobi algorithm:%
\[
T\mathbf{x}_{j}^{(k+1)}=\mathbf{b}_{j}-B\mathbf{x}_{j-1}^{(k)}-B\mathbf{x}%
_{j+1}^{(k)}.
\]
Give the spectral radius of $M^{-1}N$.
\item Determine the optimal $\hat{\omega}$ for the SOR method.
\item Consider the differential equation%
\begin{align*}
-u_{xx}-u_{yy} & =-12x^{2}-24,\qquad0