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diff --git a/Master/texmf-dist/doc/latex/siam/lexample.tex b/Master/texmf-dist/doc/latex/siam/lexample.tex new file mode 100644 index 00000000000..a8502d3788a --- /dev/null +++ b/Master/texmf-dist/doc/latex/siam/lexample.tex @@ -0,0 +1,658 @@ +%% +%% 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. + + +\begin{thebibliography}{10} +\bibitem{bs} {\sc R.~A. Brualdi and B.~L. Shader}, +{\em On sign-nonsingular matrices and the conversion of the +permanent into the determinant}, in Applied Geometry and +Discrete Mathematics, The Victor Klee Festschrift, P. +Gritzmann and B. Sturmfels, eds., American Mathematical +Society, Providence, RI, 1991, pp. 117--134. + +\bibitem{djd} {\sc J. Drew, C.~R. Johnson, and P. van den Driessche}, +{\em Strong forms of nonsingularity}, Linear Algebra Appl., +162 (1992), to appear. + +\bibitem{g} {\sc P.~M. Gibson}, {\em Conversion of the permanent into the +determinant}, Proc. Amer. Math. Soc., 27 (1971), +pp.~471--476. + +\bibitem{klm} +{\sc V.~Klee, R.~Ladner, and R.~Manber}, {\it +Signsolvability revisited}, Linear Algebra Appl., 59 +(1984), pp.~131--157. + +\bibitem{m} +{\sc K. 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