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diff --git a/Master/texmf-dist/doc/latex/siam/lexample.tex b/Master/texmf-dist/doc/latex/siam/lexample.tex deleted file mode 100644 index a8502d3788a..00000000000 --- a/Master/texmf-dist/doc/latex/siam/lexample.tex +++ /dev/null @@ -1,658 +0,0 @@ -%% -%% 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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