%%%%		-*- Mode: LaTeX -*-
%%%% Author:    J. Robert Buchanan
%%%% Address:   Department of Mathematics
%%%%		Millersville University
%%%%            P.O. Box 1002
%%%%            Millersville, PA 17551-0302
%%%% Phone:     717-871-7305
%%%% FAX:	717-871-7948
%%%%
%%%% Email:     Robert.Buchanan@millersville.edu
%%%%
%%%% Date:      04/06/2025
%%%%
%%%% Purpose:   Convergence and Stability
%%%%

\documentclass[10pt]{beamer}

\mode<presentation>
{
  %\usetheme{Warsaw}
  \usetheme{default}
  %\usetheme{CambridgeUS}
  % or ...

  %\setbeamercovered{transparent}
  % or whatever (possibly just delete it)
}


\usepackage[english]{babel}
% or whatever

\usepackage[latin1]{inputenc}
% or whatever

\usepackage{times}
\usepackage[T1]{fontenc}
% Or whatever. Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.


\title{Convergence and Stability}
\subtitle{\textit{Partial Differential Equations}} % (optional)


\author % (optional, use only with lots of authors)
{J~Robert~Buchanan}
% - Use the \inst{?} command only if the authors have different
%   affiliation.

\institute[Millersville University of Pennsylvania]
% (optional, but mostly needed)
{
  Department of Mathematics
}
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.

\date{ % (optional)
Fall
%Spring
%Summer
\the\year}

\subject{Talks}
% This is only inserted into the PDF information catalog. Can be left
% out. 



% If you have a file called "university-logo-filename.xxx", where xxx
% is a graphic format that can be processed by latex or pdflatex,
% resp., then you can add a logo as follows:

% \pgfdeclareimage[height=0.5cm]{university-logo}{university-logo-filename}
% \logo{\pgfuseimage{university-logo}}



% Delete this, if you do not want the table of contents to pop up at
% the beginning of each subsection:
\AtBeginSubsection[]
{
  \begin{frame}<beamer>
    \frametitle{Outline}
    \tableofcontents[currentsection,currentsubsection]
  \end{frame}
}


% If you wish to uncover everything in a step-wise fashion, uncomment
% the following command: 

%\beamerdefaultoverlayspecification{<+->}


\begin{document}

\begin{frame}
  \titlepage
\end{frame}

%\begin{frame}
%  \frametitle{Outline}
%  \tableofcontents
  % You might wish to add the option [pausesections]
%\end{frame}


% Since this a solution template for a generic talk, very little can
% be said about how it should be structured. However, the talk length
% of between 15min and 45min and the theme suggest that you stick to
% the following rules:  

% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.

%\section{Introduction}

\begin{frame}\frametitle{Objectives}

In this lesson we will:
\begin{itemize}
\item develop formulas for and properties of eigenvalues and
  eigenvectors of tridiagonal matrices, and
\item explore the stability properties of iterative methods for
  solving the heat, wave, and Poisson's equations.
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{Eigensystems of Tridiagonal Matrices}

\begin{lemma}
%\label{lma:findiff:tridiagonaleigs}
Suppose $A$ is an $(n-1) \times (n-1)$ tridiagonal matrix
of the form
\[
A = \left[ \begin{array}{ccccccc}
a & b & 0 & \cdots & 0 & 0 & 0	\\
c & a & b & \cdots & 0 & 0 & 0	\\
\vdots & \vdots & \vdots & {} & \vdots & \vdots & \vdots	\\
0 & 0 & 0 & \cdots & c & a & b	\\
0 & 0 & 0 & \cdots & 0 & c & a
\end{array} \right]
\]
with real or complex entries.
If $b \, c \ne 0$ then the eigenvalues of $A$ are
\[
%\label{eq:findiff:eigenvalues}
\lambda_{j}
	= a + 2 b \sqrt{\frac{c}{b}} \cos \frac{j \pi}{n}	,
\]
with corresponding eigenvectors
\[
%\label{eq:findiff:eigenvector}
\mathbf{v}_{j}
	= \left( \left( \frac{c}{b} \right)^{1/2} \sin \frac{j \pi}{n},
%\left( \frac{c}{b} \right) \sin \frac{2 j \pi}{n}	,
\ldots	,
\left( \frac{c}{b} \right)^{(n-1)/2} \sin \frac{(n-1) j \pi}{n}
\right)^{T}	,
\]
for $j = 1, 2, \ldots, n-1$.
\end{lemma}

\end{frame}
%%

\begin{frame}\frametitle{Eigensystems of Block-Structured Matrices}
\fontsize{9pt}{10.8}\selectfont

\begin{theorem}
%\label{thm:findiff:blockmatrix}
Let matrix $A$ be an $N M \times N M$ matrix written in
block form as
\[
A
	= \left[ \begin{array}{cccc}
A_{1, 1} & A_{1, 2} & \cdots & A_{1, M}	\\
A_{2, 1} & A_{2, 2} & \cdots & A_{2, M}	\\
\vdots & \vdots & {} & \vdots	\\
A_{M, 1} & A_{M, 2} & \cdots & A_{M, M}
\end{array} \right]	.
\]
Suppose each block $A_{i, j}$ is an $N \times N$ matrix and all
the matrices $A_{i, j}$ have a set of $N$ linearly independent
eigenvectors $\{ \mathbf{v}_{1}, \mathbf{v}_{2}, \ldots,
\mathbf{v}_{N} \}$ in common.
Then the eigenvalues of matrix $A$ are the eigenvalues of the matrices
\[
\Lambda_{k}
	= \left[ \begin{array}{cccc}
\lambda_{1,1}^{(k)} & \lambda_{1,2}^{(k)} & \cdots & \lambda_{1,M}^{(k)}	\\
\lambda_{2,1}^{(k)} & \lambda_{2,2}^{(k)} & \cdots & \lambda_{2,M}^{(k)}	\\
\vdots & \vdots & {} & \vdots	\\
\lambda_{M,1}^{(k)} & \lambda_{M,2}^{(k)} & \cdots & \lambda_{M,M}^{(k)}
\end{array} \right]
\mbox{ for $k = 1, 2, \ldots, N$,}
\]
where $\lambda_{i, j}^{(k)}$ is the eigenvalue of $A_{i, j}$
corresponding to the common eigenvector $\mathbf{v}_{k}$.
\end{theorem} %\cite[p.~107]{Smith:1985::}

\end{frame}
%%

\begin{frame}\frametitle{Stability}

\begin{itemize}
\item An algorithm (such as a finite difference
scheme) is \textbf{stable} if small changes in the input data result
in proportionately small changes in the output data.
\item If an algorithm is not stable, then it is labeled \textbf{unstable}.
\item If $e_{0}$ is the error present in the data and $e_{n}$ is the
  error after $n$ subsequent calculations, then the growth rate of the error is
\textbf{linear} if $e_{n} \propto n \,
e_{0}$ and the growth rate of
the error is \textbf{exponential}
if $e_{n} \propto \gamma^{n} \, e_{0}$ for some $\gamma > 1$.
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{Sources of Error}

There are three primary types of error: truncation error, measurement
error, and rounding error.
\begin{itemize}
\item Truncation error is due to the use of a finite number of terms taken
from a Taylor series to develop the approximations to various
derivatives and derivative operators used in the partial differential equations.
\item Measurement error comes from approximations used to set
the initial and boundary conditions of an initial boundary value
problem.
\item Round-off errors result from the machine arithmetic
used by computing devices.
\end{itemize}
Even if measurement errors are eliminated, truncation error and rounding
error will still be present.

\end{frame}
%%

\begin{frame}\frametitle{Heat Equation}

Recall the explicit scheme for approximating the solution to the
heat/diffusion equation.
\begin{align*}
\mathbf{u}^{(j+1)} 
	&= A(r) \, \mathbf{u}^{(j)}	\\
%\label{eq:findiff:explicitrmatrix}
\left[ \begin{array}{c}
u_{1}^{j+1} \\ u_{2}^{j+1} \\ u_{3}^{j+1} \\ \vdots %\\ u_{N-3}^{j+1}
         \\ u_{N-2}^{j+1} \\ u_{N-1}^{j+1}
\end{array} \right]
	&= \left[ \begin{array}{cccccc}
1-2r & r & 0 & \cdots & 0 & 0	\\
r & 1-2r & r & \cdots & 0 & 0	\\
0 & r & 1-2r & \cdots & 0 & 0	\\
\vdots & \vdots & \vdots & {} & \vdots & \vdots	\\
%0 & 0 & 0 & \cdots & 1-2r & r & 0	\\
0 & 0 & 0 & \cdots & 1-2r & r	\\
0 & 0 & 0 & \cdots & r & 1-2r
\end{array} \right]
\left[ \begin{array}{c}
u_{1}^{j} \\ u_{2}^{j} \\u_{3}^{j} \\ \vdots %\\ u_{N-3}^{j}
	\\ u_{N-2}^{j} \\ u_{N-1}^{j}
\end{array} \right]	.
\end{align*}
Matrix $A(r)$
is real, tridiagonal, and symmetric, thus the eigenvalues
are all distinct and the eigenvectors form a basis for the vector
space $\mathbb{R}^{N-1}$.

\end{frame}
%%

\begin{frame}\frametitle{Heat Equation}

Suppose after the completion of the calculation of $\mathbf{u}^{(j)}$,
the vector can be expressed as $\mathbf{u}^{(j)} =
\hat{\mathbf{u}}^{(j)} + \mathbf{e}^{(0)}$ where
$\hat{\mathbf{u}}^{(j)}$ is the exact solution of the finite
difference equations.
The error in the calculation of $\mathbf{u}^{(j)}$ is therefore
$\mathbf{e}^{(0)}$.

\vspace*{0.10in}

After $n$ additional time steps forward
\begin{align*}
\mathbf{u}^{(j+n)}
	&= (A(r))^{n} \mathbf{u}^{(j)}	\\
\hat{\mathbf{u}}^{(j+n)} + \mathbf{e}^{(n)}
	&= (A(r))^{n} (\hat{\mathbf{u}}^{(j)} + \mathbf{e}^{(0)})	\\
\mathbf{e}^{(n)}
	&= (A(r))^{n} \mathbf{e}^{(0)}	.
\end{align*}
Since the eigenvectors of $A(r)$ form a basis for $\mathbb{R}^{N-1}$
then there exist constants $c_{1}$, $c_{2}$, \ldots, $c_{N-1}$ such
that $\mathbf{e}^{(0)} = c_{1} \mathbf{v}_{1} + c_{2} \mathbf{v}_{2} +
\cdots + c_{N-1} \mathbf{v}_{N-1}$.
\[
\mathbf{e}^{(n)}
	= (A(r))^{n} \sum_{k=1}^{N-1} c_{k} \mathbf{v}_{k}
	= \sum_{k=1}^{N-1} c_{k} \lambda_{k}^{n} \mathbf{v}_{k}	.
\]

\end{frame}
%%

\begin{frame}\frametitle{Heat Equation}

The explicit finite difference scheme for the heat equation is stable if $|
\lambda_{k} | \le 1$ for $k = 1, 2, \ldots, N-1$.
\[
\lambda_{k}
	= 1 - 2r + 2r \cos \frac{k \pi}{N}
	= 1 - 4 r \sin^{2} \frac{k \pi}{2N}
\]
and $| \lambda_{k} | \le 1$ if $r \le 1/2$.
Thus the explicit finite difference scheme given in
is stable if $r \le 1/2$ and
unstable for $r > 1/2$.

\end{frame}
%%

\begin{frame}\frametitle{Crank-Nicolson Scheme}

The implicit Crank-Nicolson scheme
expressed is
\[
\mathbf{u}^{(j+1)}
	= (A(r))^{-1} A(-r) \, \mathbf{u}^{(j)}	.
\]
By the Lemma the eigenvalues of $A(r)$ are
\[
\lambda_{k}
	= 2 (1+r) - 2 r \cos \frac{k \pi}{N}
	= 2 + 4 r \sin^{2} \frac{k \pi}{2 N}
\mbox{ for $k = 1, 2, \ldots, N-1$,}
\]
while the eigenvalues of $A(-r)$ are
\[
\mu_{k}
	= 2 (1-r) + 2 r \cos \frac{k \pi}{N}
	= 2 - 4 r \sin^{2} \frac{k \pi}{2 N}
\mbox{ for $k = 1, 2, \ldots, N-1$}	.
\]
Matrices $A(r)$ and $A(-r)$ share the same set of eigenvectors
\[
\mathbf{v}_{k}
	= \left( \sin \frac{k \pi}{N}	,
\ldots	,
\sin \frac{(N-1) k \pi}{N} \right)^{T}	.
\]

\end{frame}
%%

\begin{frame}\frametitle{Crank-Nicolson Scheme}

\begin{itemize}
\item If $\lambda$ is an eigenvalue of an invertible
square matrix $A$ with corresponding eigenvector $\mathbf{v}$ then
$1/\lambda$ is an eigenvalue of $A^{-1}$ with the same
eigenvector $\mathbf{v}$.
\item If $\mathbf{v}$ is an eigenvector of matrix
$A$ corresponding to eigenvalue $\lambda$ and also an eigenvector of
matrix $B$ corresponding to eigenvalue $\mu$ then $\mathbf{v}$ is an
eigenvector of matrix $A \, B$ corresponding to eigenvalue $\lambda \,
\mu$.
\item The Crank-Nicolson finite difference scheme
is stable for all $r > 0$ since for $k =
1, 2, \ldots, N-1$ the eigenvalues of $(A(r))^{-1} A(-r)$ all have magnitudes
bounded by 1,
\[
\left| \frac{\mu_{k}}{\lambda_{k}} \right|
	= \left| \frac{2 - 4 r \sin^{2} \frac{k \pi}{2 N}}{2 + 4 r \sin^{2} \frac{k
    \pi}{2 N}} \right|
	\le 1	.
\]
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{Wave Equation}

Recall the $O(k^{2}) + O(h^{2})$ explicit scheme for approximating
solutions to the wave equation.
\[
%\label{eq:findiff:explicitwave}
u_{i}^{j+1}
	= r^{2} u_{i+1}^{j} + 2 (1 - r^{2}) u_{i}^{j} + r^{2} u_{i-1}^{j} - u_{i}^{j-1}	.
\]
Suppose the solution $u_{i}^{j}$ for $i = 1, 2, \ldots, N-1$ and $j =
1, 2, \ldots$ is a product solution of the form $u_{i}^{j} = X_{i} T_{j}$.
Substitute the product solution into the finite difference scheme
divide both sides by $X_{i} T_{j}$ produce
\begin{align*}
X_{i} T_{j+1}
	&= r^{2} X_{i+1} T_{j} + 2 (1-r^{2}) X_{i} T_{j} + r^{2} X_{i-1} T_{j} -
              X_{i} T_{j-1}	\\
\frac{T_{j+1}}{T_{j}} + \frac{T_{j-1}}{T_{j}}
	&= r^{2} \frac{X_{i+1}}{X_{i}} +
2 (1-r^{2}) +
r^{2} \frac{X_{i-1}}{X_{i}}	.
\end{align*}

\end{frame}
%%

\begin{frame}\frametitle{Wave Equation}

\[
\frac{T_{j+1}}{T_{j}} + \frac{T_{j-1}}{T_{j}}
	= r^{2} \frac{X_{i+1}}{X_{i}} + 2 (1-r^{2}) + r^{2} \frac{X_{i-1}}{X_{i}}
\]
The left-hand side of the equation
above depends only on the index $j$ while the right-hand side depends
on the index $i$ and therefore the left- and right-hand sides must be
constant with respect to $i$ and $j$.
The constant will be denoted as $\lambda$.

\vspace*{0.10in}

Thus two difference equations are implied:
\begin{align*}
%\label{eq:findiff:wavesepspace}
r^{2} X_{i-1} + [2(1-r^{2}) - \lambda] X_{i} + r^{2} X_{i+1}
	&= 0
\mbox{ for $i = 1, \ldots, N-1$}	\\
%\label{eq:findiff:waveseptime}
T_{j-1} - \lambda T_{j} + T_{j+1}
	&= 0
\mbox{ for $j = 1, 2, \ldots$.}
\end{align*}

\end{frame}
%%

\begin{frame}\frametitle{Wave Equation}

\[
r^{2} X_{i-1} + [ 2(1-r^{2}) - \lambda ] X_{i} + r^{2} X_{i+1}
	= 0
\]
This is a discrete boundary value problem
with boundary conditions $X_{0} = X_{N} = 0$ (assuming homogeneous
Dirichlet boundary conditions).
Hence the values of $\lambda$ are:
\[
\lambda_{k}
	= 2 (1 - r^{2}) + 2 r^{2} \cos \frac{k \pi}{N}
	= 2 - 4 r^{2} \sin^{2} \frac{k \pi}{2 N}
\]
and $X_{i} = \sin (i k \pi/N)$ for $i = 0, 1, \ldots, N$.

\end{frame}
%%

\begin{frame}\frametitle{Wave Equation}

\[
T_{j-1} - \lambda T_{j} + T_{j+1}
	= 0
\]
This is an initial value problem.
Assume the initial conditions $T_{0}$ and $T_{1}$ are known, then
\[
T_{j-1} - \lambda_{k} T_{j} + T_{j+1}
	= 0
\]
for $j = 1, 2, \ldots$.
If $T_{j} = s^{j}$ then
\[
0
	= s^{j-1} - \lambda_{k} s^{j} + s^{j+1}
\iff
s^{2} - \lambda_{k} s + 1
	= 0	.
\]
Solving this quadratic equation implies $s$ takes on the value
\[
s_{1}
	= \frac{1}{2} \left( \lambda_{k} - \sqrt{\lambda_{k}^{2} - 4}
\right)
\mbox{ or }
s_{2}
	= \frac{1}{2} \left( \lambda_{k} + \sqrt{\lambda_{k}^{2} - 4}
\right)	.
\]

\end{frame}
%%

\begin{frame}\frametitle{Wave Equation}
\fontsize{9pt}{10.8}\selectfont

By the Principle of Superposition the general solution is
$T_{j} = \alpha \, s_{1}^{j} + \beta \, s_{2}^{j}$ where $\alpha$ and
$\beta$ are arbitrary constants.
Let $\alpha_{k}$ and $\beta_{k}$ be the solutions to the simultaneous
equations
\begin{align*}
T_{0}
	&= \alpha_{k} + \beta_{k}	\\
T_{1}
	&= \alpha_{k} s_{1} + \beta_{k} s_{2}	.
\end{align*}
The product solution $u_{i}^{j}$ can be written as
\[
u_{i}^{j}
	= \sum_{k=1}^{N-1} \left(
\alpha_{k} s_{1}^{j} + \beta_{k} s_{2}^{j}
\right) \sin \frac{i k \pi}{N}	.
\]
The finite difference scheme is
stable whenever $u_{i}^{j}$ remains bounded for all $j$.
This condition is met if and only if $|s_{1}| \le 1$ and $|s_{2}| \le
1$.
If $\lambda_{k}^{2} \le 4$ then $s_{1}$ and $s_{2}$ are complex
conjugates and
\[
|s_{i}|^{2}
	= s_{1} s_{2}
	= \frac{1}{4} \left( \lambda_{k}^{2} - (\lambda_{k}^{2} - 4) \right)
	= 1	.
\]
If $\lambda_{k}^{2} > 4$ then $\lambda_{k} < -2$ in which case
$|s_{1}| > 1$.
A necessary and sufficient condition for the finite difference scheme
to be stable is that $-2 \le
\lambda_{k} \le 2$.
This inequality is equivalent to having $r = k/h \le 1$.

\end{frame}
%%

\begin{frame}\frametitle{Poisson's Equation}

\begin{itemize}
\item In matrix/vector form the iterative methods can be
expressed as the iterative equation $\mathbf{u}^{(k+1)} = T \,
\mathbf{u}^{(k)} + \mathbf{b}$.
\item The vector $\mathbf{b}$ encompasses the boundary conditions for
a Laplace or Poisson problem and, in the case of a Poisson problem, the
nonhomogeneous function on the right-hand side of the equation.
\item The matrix $T$ will be one of
$T_{J}$, $T_{GS}$, or $T_{SOR}$.
\item Let $\hat{\mathbf{u}}$ be the exact solution of the linear system,
then $\hat{\mathbf{u}} = T \, \hat{\mathbf{u}} + \mathbf{b}$.
\item Define the error vector of the $k$th iterate as $\mathbf{e}^{(k)} =
\mathbf{u}^{(k)} - \hat{\mathbf{u}}$, then
\begin{align*}
\hat{\mathbf{u}} + \mathbf{e}^{(k+1)}
	&= T (\hat{\mathbf{u}} + \mathbf{e}^{(k)}) + \mathbf{b}	\\
%\label{eq:findiff:itererror}
\mathbf{e}^{(k+1)}
	&= T \mathbf{e}^{(k)}
\end{align*}
\item Let the expression $\rho(T)$ denotes the spectral radius (eigenvalue of largest magnitude) of matrix $T$.
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{Jacobi and Gauss-Seidel Convergence}

\begin{theorem}
%\label{thm:findiff:iterconv}
If $\hat{\mathbf{u}} = T \, \hat{\mathbf{u}} + \mathbf{b}$ then the
iterates $\mathbf{u}^{(k)} = T \, \mathbf{u}^{(k-1)} +
\mathbf{b}$ for $k = 1, 2, \ldots$ converge to $\hat{\mathbf{u}}$ for
any choice of $\mathbf{u}^{(0)}$ if and only if $\rho(T) < 1$.
\end{theorem}

\vspace*{0.10in}

\textbf{Remark}: one method for establishing that all the eigenvalues of a matrix
lie within the unit circle in the complex plane is to demonstrate that
$\| T \|_{2} < 1$.

\end{frame}
%%

\begin{frame}\frametitle{Jacobi Convergence}

\begin{theorem}
%\label{thm:findiff:AubJacobi}
If $A$ is an $n \times n$ strictly diagonally dominant or irreducibly
diagonally dominant matrix and
$\mathbf{b}$ is an $n \times 1$ vector, then the Jacobi iterative
method converges to the solution of $A \, \mathbf{u} =
\mathbf{b}$ for any choice of initial estimate $\mathbf{u}^{(0)}$.
\end{theorem}

\end{frame}
%%

\begin{frame}\frametitle{Gauss-Seidel Convergence}

\begin{theorem}
%\label{thm:findiff:AubGS}
If $A$ is an $n \times n$ strictly diagonally dominant or irreducibly
diagonally dominant matrix and
$\mathbf{b}$ is an $n \times 1$ vector, then the Gauss-Seidel iterative
method converges to the solution of $A \, \mathbf{u} =
\mathbf{b}$ for any choice of initial estimate $\mathbf{u}^{(0)}$.
\end{theorem}


\end{frame}
%%

\begin{frame}\frametitle{Successive Over-Relaxation Convergence}

\begin{theorem}
%\label{thm:findiff:boundomega}
If $A \, \mathbf{u} = \mathbf{b}$ is a linear system of equations with
$a_{ii} \ne 0$ for $i = 1, 2, \ldots, n$, the successive
over-relaxation method converges to the solution $\mathbf{u}$ for an
arbitrary initial estimate $\mathbf{u}^{(0)}$ only if $0 < \omega < 2$.
\end{theorem}

\vspace*{0.10in}

\uncover<2->{%
If matrix $A$ is positive definite, the
condition in the previous theorem is also sufficient.
\begin{theorem}[Ostrowski-Reich]
%\label{thm:findiff:SORposdef}
Let $A \, \mathbf{u} = \mathbf{b}$ be a linear system of equations
where $A$ is a positive definite matrix.
The successive over-relaxation method converges to the solution
$\mathbf{u}$ for an arbitrary initial estimate $\mathbf{u}^{(0)}$ if
and only if $0 < \omega < 2$.
\end{theorem}
}

\end{frame}
%%

\begin{frame}\frametitle{Remarks}

\begin{itemize}
\item The matrices of the finite difference approximation methods for
Laplace's or Poisson's equation are all positive definite and
therefore the successive over-relaxation iterative method will converge.
\item The optimal choice for the successive over-relaxation parameter
$\omega$ is the one which minimizes the magnitude of the largest
magnitude eigenvalue of $T_{SOR}$.
The optimal value of $\omega$ is
\[
\omega^{*}
	= \frac{2}{1 + \sqrt{1 - |\lambda|^{2}}}
\]
where $\lambda$ is the eigenvalue of $A^{-1} (L + U)$ with the largest
magnitude.
\end{itemize}

\end{frame}
%%

%\begin{frame}\frametitle{Homework}

%\begin{itemize}
%\item Read Section $x.y$
%\item Exercises: 
%\end{itemize}

%\end{frame}
%%

%%%
\end{document}

%%%%
%%%% EOF
%%%%
