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-%%
-%% This is file `lexample.tex',
-%% Sample file for siam macros for use with LaTeX 2e
-%%
-%% By Paul Duggan for the Society for Industrial and Applied
-%% Mathematics.
-%%
-%% October 1, 1995
-%%
-%% Version 1.0
-%%
-%% You are not allowed to change this file.
-%%
-%% You are allowed to distribute this file under the condition that
-%% it is distributed together with all of the files in the siam macro
-%% distribution. These are:
-%%
-%% siamltex.cls (main LaTeX macro file for SIAM)
-%% siamltex.sty (includes siamltex.cls for compatibility mode)
-%% siam10.clo (size option for 10pt papers)
-%% subeqn.clo (allows equation numbners with lettered subelements)
-%% siam.bst (bibliographic style file for BibTeX)
-%% docultex.tex (documentation file)
-%% lexample.tex (this file)
-%%
-%% If you receive only some of these files from someone, complain!
-%%
-%% You are NOT ALLOWED to distribute this file alone. You are NOT
-%% ALLOWED to take money for the distribution or use of either this
-%% file or a changed version, except for a nominal charge for copying
-%% etc.
-%% \CharacterTable
-%% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
-%% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
-%% Digits \0\1\2\3\4\5\6\7\8\9
-%% Exclamation \! Double quote \" Hash (number) \#
-%% Dollar \$ Percent \% Ampersand \&
-%% Acute accent \' Left paren \( Right paren \)
-%% Asterisk \* Plus \+ Comma \,
-%% Minus \- Point \. Solidus \/
-%% Colon \: Semicolon \; Less than \<
-%% Equals \= Greater than \> Question mark \?
-%% Commercial at \@ Left bracket \[ Backslash \\
-%% Right bracket \] Circumflex \^ Underscore \_
-%% Grave accent \` Left brace \{ Vertical bar \|
-%% Right brace \} Tilde \~}
-
-
-\documentclass[final]{siamltex}
-
-% definitions used by included articles, reproduced here for
-% educational benefit, and to minimize alterations needed to be made
-% in developing this sample file.
-
-\newcommand{\pe}{\psi}
-\def\d{\delta}
-\def\ds{\displaystyle}
-\def\e{{\epsilon}}
-\def\eb{\bar{\eta}}
-\def\enorm#1{\|#1\|_2}
-\def\Fp{F^\prime}
-\def\fishpack{{FISHPACK}}
-\def\fortran{{FORTRAN}}
-\def\gmres{{GMRES}}
-\def\gmresm{{\rm GMRES($m$)}}
-\def\Kc{{\cal K}}
-\def\norm#1{\|#1\|}
-\def\wb{{\bar w}}
-\def\zb{{\bar z}}
-
-% some definitions of bold math italics to make typing easier.
-% They are used in the corollary.
-
-\def\bfE{\mbox{\boldmath$E$}}
-\def\bfG{\mbox{\boldmath$G$}}
-
-\title{Sample file for SIAM \LaTeX\ macro package\thanks{This
- work was supported by the Society for Industrial and
- Applied Mathematics, Philadelphia, Pennsylvania.}}
-
-% The thanks line in the title should be filled in if there is
-% any support acknowledgement for the overall work to be included
-% This \thanks is also used for the received by date info, but
-% authors are not expected to provide this.
-
-\author{Paul Duggan\thanks{Composition Department, Society
- for Industrial and Applied Mathematics, 3600 Univeristy
- City Science Center, Philadelphia, Pennsylvania,
- 19104-2688 ({\tt duggan@siam.org}).}
- \and Various A.~U. Thors\thanks{Various Affiliations,
- supported by various foundation grants.}}
-
-\begin{document}
-
-\maketitle
-
-\begin{abstract}
-An example of SIAM \LaTeX\ macros is presented. Various
-aspects of composing manuscripts for SIAM's journal series
-are illustrated with actual examples from accepted
-manuscripts. SIAM's stylistic standards are adhered to
-throughout, and illustrated.
-\end{abstract}
-
-\begin{keywords}
-sign-nonsingular matrix, LU-factorization, indicator
-polynomial
-\end{keywords}
-
-\begin{AMS}
-15A15, 15A09, 15A23
-\end{AMS}
-
-\pagestyle{myheadings}
-\thispagestyle{plain}
-\markboth{P. DUGGAN AND V. A. U. THORS}{SIAM MACRO EXAMPLES}
-
-
-\section{Introduction and examples}
-This paper presents a sample file for the use of SIAM's
-\LaTeX\ macro package. It illustrates the features of the
-macro package, using actual examples culled from various
-papers published in SIAM's journals. It is to be expected
-that this sample will provide examples of how to use the
-macros to generate standard elements of journal papers,
-e.g., theorems, definitions, or figures. This paper also
-serves as an example of SIAM's stylistic preferences for
-the formatting of such elements as bibliographic references,
-displayed equations, and equation arrays, among others.
-Some special circumstances are not dealt with in this
-sample file; for such information one should see the
-included documentation file.
-
-{\em Note:} This paper is not to be read in any form for content.
-The conglomeration of equations, lemmas, and other text elements were
-put together solely for typographic illustrative purposes and don't
-make any sense as lemmas, equations, etc.
-
-\subsection{Sample text}
-Let $S=[s_{ij}]$ ($1\leq i,j\leq n$) be a $(0,1,-1)$-matrix
-of order $n$. Then $S$ is a {\em sign-nonsingular matrix}
-(SNS-matrix) provided that each real matrix with the same
-sign pattern as $S$ is nonsingular. There has been
-considerable recent interest in constructing and
-characterizing SNS-matrices \cite{bs}, \cite{klm}. There
-has also been interest in strong forms of
-sign-nonsingularity \cite{djd}. In this paper we give a new
-generalization of SNS-matrices and investigate some of
-their basic properties.
-
-Let $S=[s_{ij}]$ be a $(0,1,-1)$-matrix of order $n$ and
-let $C=[c_{ij}]$ be a real matrix of order $n$. The pair
-$(S,C)$ is called a {\em matrix pair of order} $n$.
-Throughout, $X=[x_{ij}]$ denotes a matrix of order $n$
-whose entries are algebraically independent indeterminates
-over the real field. Let $S\circ X$ denote the Hadamard
-product (entrywise product) of $S$ and $X$. We say that the
-pair $(S,C)$ is a {\em sign-nonsingular matrix pair of
-order} $n$, abbreviated SNS-{\em matrix pair of order} $n$,
-provided that the matrix \[A=S\circ X+C\] is nonsingular
-for all positive real values of the $x_{ij}$. If $C=O$
-then the pair $(S,O)$ is a SNS-matrix pair if and only if
-$S$ is a SNS-matrix. If $S=O$ then the pair $(O,C)$ is a
-SNS-matrix pair if and only if $C$ is nonsingular. Thus
-SNS-matrix pairs include both nonsingular matrices and
-sign-nonsingular matrices as special cases.
-
-The pairs $(S,C)$ with
-\[S=\left[\begin{array}{cc}1&0\\0&0\end{array}\right],\qquad
-C=\left[\begin{array}{cc}1&1\\1&1\end{array}\right]\] and
-\[S=\left[\begin{array}{ccc}1&1&0\\1&1&0\\0&0&0\end{array}\right],\qquad
-C=\left[\begin{array}{ccc}0&0&1\\0&2&0\\
-3&0&0\end{array}\right]\] are examples of SNS-matrix pairs.
-
-\subsection{A remuneration list}
-In this paper we consider the evaluation of integrals of the
-following forms:
-\begin{equation}
-\int_a^b \left( \sum_i E_i B_{i,k,x}(t) \right)
- \left( \sum_j F_j B_{j,l,y}(t) \right) dt,\label{problem}
-\end{equation}
-\begin{equation}
-\int_a^b f(t) \left( \sum_i E_i B_{i,k,x}(t) \right) dt,\label{problem2}
-\end{equation}
-where $B_{i,k,x}$ is the $i$th B-spline of order $k$ defined over the
-knots $x_i, x_{i+1}, \ldots, x_{i+k}$.
-We will consider B-splines normalized so that their integral is one.
-The splines may be of different orders and
-defined on different knot sequences $x$ and $y$.
-Often the limits of integration will be the entire real line, $-\infty$
-to $+\infty$. Note that (\ref{problem}) is a special case of (\ref{problem2})
-where $f(t)$ is a spline.
-
-
-There are five different methods for calculating (\ref{problem})
-that will be considered:
-\begin{remunerate}
-\item Use Gauss quadrature on each interval.
-\item Convert the integral to a linear combination of
- integrals of products of B-splines and provide a recurrence for
- integrating the product of a pair of B-splines.
-\item Convert the sums of B-splines to piecewise
- B\'{e}zier format and integrate segment
- by segment using the properties of the Bernstein polynomials.
-\item Express the product of a pair of B-splines as a linear combination
- of B-splines.
- Use this to reformulate the integrand as a linear combination
- of B-splines, and integrate term by term.
-\item Integrate by parts.
-\end{remunerate}
-Of these five, only methods 1 and 5 are suitable for calculating
-(\ref{problem2}). The first four methods will be touched on and the
-last will be discussed at length.
-
-
-\subsection{Some displayed equations and \{{\tt eqnarray}\}s}
- By introducing the product topology on $R^{m \times m} \times
-R^{n \times n}$ with the induced inner product
-\begin{equation}
-\langle (A_{1},B_{1}), (A_{2},B_{2})\rangle := \langle A_{1},A_{2}\rangle
-+ \langle B_{1},B_{2}\rangle,\label{eq2.10}
-\end{equation}
-we calculate the Fr\'{e}chet derivative of $F$ as follows:
-\begin{eqnarray}
- F'(U,V)(H,K) &=& \langle R(U,V),H\Sigma V^{T} + U\Sigma K^{T} -
-P(H\Sigma V^{T} + U\Sigma K^{T})\rangle \nonumber \\
- &=& \langle R(U,V),H\Sigma V^{T} + U\Sigma K^{T}\rangle \label{eq2.11} \\
-&=& \langle R(U,V)V\Sigma^{T},H\rangle + \langle \Sigma^{T}U^{T}R(U,V),K^{T}\rangle. \nonumber
-\end{eqnarray}
-In the middle line of (\ref{eq2.11}) we have used the fact that the range of
-$R$ is always perpendicular to the range of $P$. The gradient $\nabla F$ of
-$F$, therefore, may be interpreted as the
-pair of matrices:
-\begin{equation}
- \nabla F(U,V) = (R(U,V)V\Sigma^{T},R(U,V)^{T}U\Sigma ) \in
-R^{m \times m} \times R^{n \times n}. \label{eq2.12}
-\end{equation}
-Because of the product topology, we know
-\begin{equation}
- {\cal T}_{(U,V)}({\cal O} (m) \times {\cal O} (n)) =
-{\cal T}_{U}{\cal O} (m) \times {\cal T}_{V}{\cal O} (n), \label{eq2.13}
-\end{equation}
-where ${\cal T}_{(U,V)}({\cal O} (m) \times {\cal O} (n))$ stands for the
-tangent space to the manifold ${\cal O} (m) \times {\cal O} (n)$ at $(U,V)
-\in {\cal O} (m) \times {\cal O} (n)$ and so on. The projection of
-$\nabla F(U,V)$ onto ${\cal T}_{(U,V)}({\cal O} (m) \times {\cal O} (n))$,
-therefore, is the product of the projection of the first component of
-$\nabla F(U,V)$ onto ${\cal T}_{U}{\cal O} (m)$ and the projection of the
-second component of $\nabla F(U,V)$ onto ${\cal T}_{V}{\cal O} (n)$.
-In particular, we claim that the
-projection $ g(U,V)$ of the gradient $\nabla F(U,V)$ onto
-${\cal T}_{(U,V)}({\cal O} (m) \times {\cal O} (n))$ is given by the pair of
-matrices:
-\begin{eqnarray}
-g(U,V) = && \left( \frac{R(U,V)V\Sigma^{T}U^{T}-U\Sigma V^{T}R(U,V)^{T}}{2}U,
-\right. \nonumber \\[-1.5ex]
-\label{eq2.14}\\[-1.5ex]
-&&\quad \left. \frac{R(U,V)^{T}U\Sigma V^{T}-V
- \Sigma^{T}U^{T}R(U,V)}{2}V \right).\nonumber
-\end{eqnarray}
-Thus, the vector field
-\begin{equation}
-\frac{d(U,V)}{dt} = -g(U,V) \label{eq2.15}
-\end{equation}
-defines a steepest descent flow on the manifold ${\cal O} (m) \times
-{\cal O} (n)$ for the objective function $F(U,V)$.
-
-
-\section{Main results}
-
-Let $(S,C)$ be a matrix pair of order $n$. The determinant
-\[\det (S\circ X+C)\]
-is a polynomial in the indeterminates of $X$ of degree at
-most $n$ over the real field. We call this polynomial the
-{\em indicator polynomial} of the matrix pair $(S,C)$
-because of the following proposition.
-
-\begin{theorem}
-\label{th:prop}
-The matrix pair $(S,C)$ is a {\rm SNS}-matrix pair if and
-only if all the nonzero coefficients in its indicator
-polynomial have the same sign and there is at least one
-nonzero coefficient.
-\end{theorem}
-
-\begin{proof}
-Assume that $(S,C)$ is a SNS-matrix pair. Clearly the
-indicator polynomial has a nonzero coefficient. Consider a
-monomial
-\begin{equation}
-\label{eq:mono}
-b_{i_{1},\ldots,i_{k};j_{1},\ldots,j_{k}}x_{i_{1}j_{1}}\cdots
-x_{i_{k}j_{k}}
-\end{equation}
-occurring in the indicator polynomial with a nonzero
-coefficient. By taking the $x_{ij}$ that occur in
-(\ref{eq:mono}) large and all others small, we see that any
-monomial that occurs in the indicator polynomial with a
-nonzero coefficient can be made to dominate all others.
-Hence all the nonzero coefficients have the same sign. The
-converse is im-\linebreak mediate. \qquad\end{proof}
-
-
-For SNS-matrix pairs $(S,C)$ with $C=O$ the indicator
-polynomial is a homogeneous polynomial of degree $n$. In
-this case Theorem \ref{th:prop} is a standard fact about
-SNS-matrices.
-
-\begin{lemma}[{\rm Stability}]
-\label{stability}
-Given $T>0$, suppose that $\| \epsilon (t) \|_{1,2} \leq h^{q-2}$
-for $0 \leq t \leq T$ and $q \geq 6$.
-Then there exists a positive number $B$ that depends on
-$T$ and the exact solution $\pe$ only such that for all $0 \leq t \leq T$,
-\begin{equation}
-\label{Gron}
-\frac {d}{dt} \| \epsilon (t) \| _{1,2} \leq B
- ( h^{q-3/2} + \| \epsilon (t) \|_{1,2})\;.
-\end{equation}
-The function $B(T)$ can be chosen to be nondecreasing in time.
-\end{lemma}
-
-
-\begin{theorem}
-\label{th:gibson}
-The maximum number of nonzero entries in a {\rm SNS}-matrix
-$S$ of order $n$ equals \[\frac{n^{2}+3n-2}{2}\] with
-equality if and only if there exist permutation matrices
-such that $P|S|Q=T_{n}$ where
-\begin{equation}
-\label{eq:gibson}
-T_{n}=\left[\begin{array}{cccccc} 1&1&\cdots&1&1&1\\
-1&1&\cdots&1&1&1\\ 0&1&\cdots&1&1&1\\
-\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\
-0&0&\cdots&1&1&1\\ 0&0&\cdots&0&1&1\end{array}\right].
-\end{equation}
-\end{theorem}
-
-We note for later use that each submatrix of $T_{n}$ of
-order $n-1$ has all 1s on its main diagonal.
-
-We now obtain a bound on the number of nonzero entries of
-$S$ in a SNS-matrix pair $(S,C)$ in terms of the degree of
-the indicator polynomial. We denote the strictly upper
-triangular (0,1)-matrix of order $m$ with all 1s above the
-main diagonal by $U_{m}$. The all 1s matrix of size $m$ by
-$p$ is denoted by $J_{m,p}$.
-
-
-\begin{proposition}[{\rm Convolution theorem}]
-\label{pro:2.1} Let
-\begin{eqnarray*}
-a\ast u(t) = \int_0^t a(t- \tau) u(\tau) d\tau, \hspace{.2in} t \in
-(0, \infty).
-\end{eqnarray*}
-Then
-\begin{eqnarray*}
-\widehat{a\ast u}(s) = \widehat{a}(s)\widehat{u}(s).
-\end{eqnarray*}
-\end{proposition}
-
-\begin{lemma}
-\label{lem:3.1}
-For $s_0 >0$, if
-$$
-\int_0^{\infty} e^{-2s_0 t}v^{(1)}(t) v(t) dt \; \leq 0 \;,
-$$
-then
-\begin{eqnarray*}
-\int_0^{\infty} e^{-2s_0 t} v^2(t) dt \; \leq \; \frac{1}{2s_0} v^2(0).
-\end{eqnarray*}
-\end{lemma}
-
-{\em Proof}. Applying integration by parts, we obtain
-\begin{eqnarray*}
-\int_0^{\infty} e^{-2s_0 t} [v^2(t)-v^2(0)] dt
-&=&\lim_{t\rightarrow \infty}\left (
--\frac{1}{2s_0}e^{-2s_0 t}v^2(t) \right ) +\frac{1}{s_0}
-\int_0^{\infty} e^{-2s_0 t}v^{(1)}(t)v(t)dt\\
-&\leq& \frac{1}{s_0} \int_0^{\infty} e^{-2s_0 t} v^{(1)}(t)v(t) dt \;\;
-\leq \;\; 0.
-\end{eqnarray*}
-Thus
-$$
-\int_0^{\infty} e^{-2s_0 t} v^2(t) dt \;\;\leq v^2(0) \int_0^{\infty}
-\;\;e^{-2s_0 t} dt\;\;=\;\;\frac{1}{2s_0} v^2(0).\eqno\endproof
-$$
-
-\begin{corollary}\label{c4.1}
-Let $ \bfE $ satisfy $(5)$--$(6)$ and
-suppose $ \bfE^h $ satisfies $(7)$ and $(8)$
-with a general $ \bfG $. Let $ \bfG= \nabla \times {\bf \Phi} + \nabla p,$
-$p \in H_0^1 (\Omega) $. Suppose that $\nabla p$ and $ \nabla \times
-{\bf \Phi} $ satisfy all the assumptions of Theorems $4.1$ and
-$4.2$, respectively. In addition suppose all the regularity
-assumptions of Theorems $4.1$--$4.2$ are satisfied. Then
-for $ 0 \le t \le T $ and $ 0 < \epsilon \le \epsilon_0 $ there exists a
-constant $ C = C(\epsilon, T) $ such that
-$$
-\Vert (\bfE - \bfE^h)(t) \Vert_0 \le C h^{k+1- \epsilon},
-$$
-where $ C $ also depends on the constants given in Theorems
-$4.1$ and $4.2$.
-\end{corollary}
-
-
-\begin{definition}
-Let $S$ be an isolated invariant set with isolating neighborhood $N$.
-An {\em index pair} for $S$ is a pair of compact sets $(N_{1},N_{0})$
-with $N_{0} \subset N_{1} \subset N$ such that:
-\begin{romannum}
-\item $cl(N_{1} \backslash N_{0})$
-is an isolating neighborhood for $S$.
-\item $N_{i}$ is positively invariant relative to $N$ for $i=0,1$,
-i.e., given
-$x \in N_{i}$ and $x \cdot [0,t] \subset N$, then $x \cdot [0,t] \subset
-N_{i}$.
-\item $N_{0}$ is an exit set for $N_{1}$, i.e. if $x \in N_{1}$,
-$x \cdot [0, \infty ) \not\subset N_{1}$, then there is a $T \geq 0$ such
-that $x \cdot [0,T] \subset N_{1}$ and $x \cdot T \in N_{0}$.
-\end{romannum}
-\end{definition}
-
-\subsection{Numerical experiments} We conducted numerical experiments
-in computing inexact Newton steps for discretizations of a
-{\em modified Bratu problem}, given by
-\begin{eqnarray}
-{\ds \Delta w + c e^w + d{ {\partial w}\over{\partial x} } }
-&=&{\ds f \quad {\rm in}\ D, }\nonumber\\[-1.5ex]
-\label{bratu} \\[-1.5ex]
-{\ds w }&=&{\ds 0 \quad {\rm on}\ \partial D , } \nonumber
-\end{eqnarray}
-where $c$ and $d$ are constants. The actual Bratu problem has $d=0$ and
-$f \equiv0$. It provides a simplified model of nonlinear diffusion
-phenomena, e.g., in combustion and semiconductors, and has been
-considered by Glowinski, Keller, and Rheinhardt \cite{GloKR85},
-as well as by a number of other investigators; see \cite{GloKR85}
-and the references therein. See also problem 3 by Glowinski and Keller
-and problem 7 by Mittelmann in the collection of nonlinear model
-problems assembled by Mor\'e \cite{More}. The modified problem
-(\ref{bratu}) has been used as a test problem for inexact Newton
-methods by Brown and Saad \cite{Brown-Saad1}.
-
-In our experiments, we took $D = [0,1]\times[0,1]$, $f \equiv0$,
-$c=d=10$, and discretized (\ref{bratu}) using the usual second-order
-centered differences over a $100\times100$ mesh of equally
-spaced points in $D$. In \gmres($m$), we took $m=10$ and used fast
-Poisson right preconditioning as in the experiments in \S2. The computing
-environment was as described in \S2. All computing was done
-in double precision.
-
-
-
-\begin{figure}[ht]
-\vspace{2.5in}
-\caption{{\rm Log}$_{10}$ of the residual norm versus the number of
-{\rm GMRES$(m)$} iterations for the finite difference methods.}
-\label{diff}
-\end{figure}
-
-
-
-In the first set of experiments, we allowed each method to
-run for $40$ {\gmresm} iterations, starting with zero as the initial
-approximate solution, after which the limit of residual norm
-reduction had been reached. The results are shown in Fig.~\ref{diff}.
-In Fig.~\ref{diff}, the top curve was produced by method FD1.
-The second curve from the top is actually a superposition of
-the curves produced by methods EHA2 and FD2; the two curves are
-visually indistinguishable. Similarly, the third curve from
-the top is a superposition of the curves produced by methods EHA4
-and FD4, and the fourth curve from the top, which lies barely above
-the bottom curve, is a superposition of the curves produced by
-methods EHA6 and FD6. The bottom curve was produced by method A.
-
-In the second set of experiments, our purpose was to assess the
-relative amount of computational work required by the methods
-which use higher-order differencing to reach comparable levels
-of residual norm reduction. We compared pairs of methods EHA2
-and FD2, EHA4 and FD4, and EHA6 and FD6 by observing in each of
-20 trials the number of {\gmresm} iterations, number of $F$-evaluations,
-and run time required by each method to reduce the residual norm
-by a factor of $\e$, where for each pair of methods $\e$ was chosen
-to be somewhat greater than the limiting ratio of final to
-initial residual norms obtainable by the methods. In these trials,
-the initial approximate solutions were obtained by generating random
-components as in the similar experiments in \S2. We note that for every
-method, the numbers of {\gmresm} iterations and $F$-evaluations required
-before termination did not vary at all over the 20 trials. The {\gmresm}
-iteration counts, numbers of $F$-evaluations, and means and standard
-deviations of the run times are given in Table \ref{diffstats}.
-
-\begin{table}
-\caption{Statistics over $20$ trials of {\rm GMRES$(m)$} iteration numbers,
-$F$-evaluations, and run times required to reduce the residual norm by
-a factor of $\e$. For each method, the number of {\rm GMRES$(m)$} iterations
-and $F$-evaluations was the same in every trial.}
-
-\begin{center} \footnotesize
-\begin{tabular}{|c|c|c|c|c|c|} \hline
-&& Number of & Number of & Mean Run Time & Standard \\
-Method & $\e$ & Iterations & $F$-Evaluations& (Seconds) & Deviation \\ \hline
-\lower.3ex\hbox{EHA2} & \lower.3ex\hbox{$10^{-10}$} & \lower.3ex\hbox{26} &
-\lower.3ex\hbox{32} & \lower.3ex\hbox{47.12} & \lower.3ex\hbox{.1048} \\
-FD2 & $10^{-10}$ & 26 & 58 & 53.79 & .1829 \\ \hline
-\lower.3ex\hbox{EHA4} & \lower.3ex\hbox{$10^{-12}$} & \lower.3ex\hbox{30} &
-\lower.3ex\hbox{42} & \lower.3ex\hbox{56.76} & \lower.3ex\hbox{.1855} \\
-FD4 & $10^{-12}$ & 30 & 132 & 81.35 & .3730 \\ \hline
-\lower.3ex\hbox{EHA6} & \lower.3ex\hbox{$10^{-12}$} & \lower.3ex\hbox{30} &
-\lower.3ex\hbox{48} & \lower.3ex\hbox{58.56} & \lower.3ex\hbox{.1952} \\
-FD6 & $10^{-12}$ & 30 & 198 & 100.6 & .3278 \\ \hline
-\end{tabular}
-\end{center}
-\label{diffstats}
-\end{table}
-
-In our first set of experiments, we took $c=d=10$ and used right
-preconditioning with a fast Poisson solver from {\fishpack}
-\cite{Swarztrauber-Sweet}, which is very effective for these
-fairly small values of $c$ and $d$. We first started each method
-with zero as the initial approximate solution and allowed it
-to run for 40 {\gmresm} iterations, after which the limit of residual
-norm reduction had been reached. Figure \ref{pdep} shows plots
-of the logarithm of the Euclidean norm of the residual versus
-the number of {\gmresm} iterations for the three methods. We note
-that in Fig.~\ref{pdep} and in all other figures below, the plotted
-residual norms were not the values maintained by {\gmresm}, but rather
-were computed as accurately as possible ``from scratch.'' That is,
-at each {\gmresm} iteration, the current approximate solution was
-formed and its product with the coefficient matrix was subtracted
-from the right-hand side, all in double precision.
-It was important to compute the residual norms in this way because
-the values maintained by {\gmresm} become increasingly untrustworthy
-as the limits of residual norm reduction are neared; see \cite{Walker88}.
-It is seen in Fig.~\ref{pdep} that Algorithm EHA achieved
-the same ultimate level of residual norm reduction as the FDP
-method and required only a few more {\gmresm} iterations to do
-so.
-
-\begin{figure}[t]
-\vspace{3in}
-\caption{{\rm Log}$_{10}$ of the residual norm versus the number of
-{\rm GMRES}$(m)$ iterations for $c=d=10$ with fast Poisson
-preconditioning. Solid curve: Algorithm {\rm EHA}; dotted
-curve: {\rm FDP} method; dashed curve: {\rm FSP} method.}
-\label{pdep}
-\end{figure}
-
-
-In our second set of experiments, we took $c=d=100$ and carried out
-trials analogous to those in the first set above. No preconditioning
-was used in these experiments, both because we wanted to compare
-the methods without preconditioning and because the fast
-Poisson preconditioning used in the first set of experiments is
-not cost effective for these large values of $c$ and $d$. We first
-allowed each method to run for 600 {\gmresm} iterations,
-starting with zero as the initial approximate solution, after which
-the limit of residual norm reduction had been reached.
-
-
-\section*{Acknowledgments}
-The author thanks the anonymous authors whose work largely
-constitutes this sample file. He also thanks the INFO-TeX mailing
-list for the valuable indirect assistance he received.
-
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