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diff --git a/Master/texmf-dist/doc/amstex/siam/amsamp.tex b/Master/texmf-dist/doc/amstex/siam/amsamp.tex deleted file mode 100644 index 44048937dd7..00000000000 --- a/Master/texmf-dist/doc/amstex/siam/amsamp.tex +++ /dev/null @@ -1,365 +0,0 @@ -% This is the sample paper for the AmSTeX SIAM style file, (amstex)siam.sty -% for use with AmSTeX version 2.1 or later and amsppt.sty, version 2.1a. -% RCS information: $Revision: 1.1 $, $Date: 93/01/25 15:33:19 $. -\input amstex -\documentstyle{amstexs1} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% Macro definitions for running heads and first page % -\accepted\SIMAF % -\firstpageno{10} % -\lastpageno{12} % -\issuevolume{1} % -\issuenumber{2} % -\issuemonth{February} % -\placenumber{002} % place of paper in this issue % -\issueyear{1988} % -\shortauthor{Bradley J. Lucier and Douglas N. Arnold} % -\shorttitle{A Sample Paper} % -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% Macros specific to this paper % -\define\loner{{L^1(\Bbb R)}} % -\define\linfr{{L^\infty(\Bbb R)}} % -\define\bvr{{\roman{BV}(\Bbb R)}} % -\define\TV{{\roman {TV}}} % -\define\sdot{\,\cdot\,} % -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\topmatter -\title -A SAMPLE PAPER, WITH A RATHER LONG TITLE, TO ILLUSTRATE THE -\AmSTeX\ SIAM STYLE\footnote[\boldkey*]{Unlikely to appear.} -\endtitle -\author -BRADLEY J. LUCIER\footnote[\dag]{Department of Mathematics, Purdue University, -West Lafayette, Indiana 47907. Present address, somewhere on the beach -(lucier\@math.purdue.edu). -The work of the first author was not supported by the -Wolf Foundation.}\ and DOUGLAS N. ARNOLD\footnote[\ddag]{Department -of Mathematics, Pennsylvania State University, -University Park, Pennsylvania 16802.} -\endauthor -\abstract -This sample paper illustrates many of the amstex -macros as used with the \AmSTeX\ SIAM style file amstexsiam (version 2.0a). -The \AmSTeX\ SIAM style file, which -inputs and builds upon the amsppt style (version 2.1a or later) -of Michael Spivak, gives authors easy -access to most of the typographical constructions used in SIAM journals. -It does not address the issues of the table of contents -or tables, which must be set using more primitive \TeX\ macros. -\endabstract -\keywords -porous medium, interface curves -\endkeywords -\subjclass -65N60 -\endsubjclass -\endtopmatter -\document -\subhead 1. Introduction\endsubhead -We are concerned with numerical approximations to the so-called -porous-medium equation \cite{6}, -$$ -\alignedat2 - &u_t=\phi(u)_{xx},&&\qquad x\in\Bbb R,\quad t>0,\quad\phi(u)=u^m,\quad m>1, -\\ - &u(x,0)=u_0(x),&&\qquad x\in\Bbb R. -\endalignedat -\tag 1.1 -$$ -We assume that the initial data $u_0(x)$ has bounded support, that -$0\leq u_0\leq M$, and that $\phi(u_0)_x\in\bvr$. -It is well known that a unique solution $u(x,t)$ of (1.1) exists, -and that $u$ satisfies -$$ - 0\leq u\leq M\text{ and }\TV\phi(u(\,\cdot\,,t))_x\leq\TV\phi(u_0)_x. -\tag 1.2 -$$ -If the data has slightly more regularity, then this too is satisfied -by the solution. Specifically, if $m$ is no greater than two and -$u_0$ is Lipschitz continuous, then $u(\,\cdot\,,t)$ is also Lipschitz; -if $m$ is greater than two and $(u_0^{m-1})_x\in\linfr$, then -$(u(\,\cdot\,,t)^{m-1})_x\in\linfr$ -(see [3]). (This will follow from results presented here, also.) -We also use the fact that the solution $u$ is H\"older continuous in $t$. - -\subhead 2. $\linfr$ error bounds\endsubhead -After a simple definition, we state a theorem -that expresses the error of approximations $u^h$ in -terms of the weak truncation error $E$. -\definition{Definition 2.1}\rm A {\it definition} -is the same as a theorem set in roman -type. In version 2 of the \AmSTeX\ style file for the SIAM journals, -definitions are set with their own command. -\enddefinition -\proclaim{Theorem 2.1} -Let $\{u^h\}$ be a family of approximate solutions satisfying -the following conditions for $0\leq t\leq T${\rm:} -\roster -\item For all $x\in\Bbb R$ and positive $t$, $0\leq u^h(x,t)\leq M${\rm;} -\item Both $u$ and $u^h$ are H\"older--$\alpha$ in $x$ -for some $\alpha\in(0,1\wedge 1/(m-1))${\rm;} $u^h$ is right -continuous in $t${\rm;} -and $u^h$ is H\"older continuous in $t$ on -strips $\Bbb R\times(t^n,t^{n+1})$, with the set $\{t^n\}$ having no -limit points\/{\rm;} and -\item There exists a positive function $\omega(h,\epsilon)$ such that\/{\rm:} -whenever $\{w^\epsilon\}_{0<\epsilon\leq\epsilon_0}$ is a family of functions -in $\bold X$ for which -{\roster -\item"(a)" there is a sequence of positive numbers $\epsilon$ tending -to zero, such that for these values of -$\epsilon$, $\|w^\epsilon\|_\infty\leq 1/\epsilon$, -\item"(b)" for all positive -$\epsilon$, $\|w_x^\epsilon(\sdot,t)\|_\loner\leq 1/\epsilon^2$, and -\item"(c)" for all $\epsilon>0$, -$$ -\sup\Sb -x\in\Bbb R\\0\leq t_1,t_2\leq T\endSb -\dfrac{|w^\epsilon(x,t_2)-w^\epsilon(x,t_1)|}{|t_2-t_1|^p}\leq 1/\epsilon^2, -$$ -where $p$ is some number not exceeding $1$, -\endroster}% -then\footnote{This is an obvious ploy, but we need a footnote.} - $|E (u^h,w^\epsilon,T)|\leq\omega(h,\epsilon).$ -\item -This is the fourth item in the outer roster. -\endroster -Then, there is a constant $C=C(m,M,T)$ such that -$$\multline -\|u-u^h\|_{\infty,\Bbb R\times[0,T]}\leq C\biggl[ -\sup \biggl |\int_\Bbb R(u_0(x)-u^h(x,0)) w(x,0) \,dx\biggr|\\ -+\omega(h,\epsilon)+\epsilon^\alpha\biggr],\endmultline -\tag 2.1 -$$ -where the supremum is taken over all $w\in\bold X$. -\endproclaim - -\demo{Proof} -We assume first that $Q$ is decreasing and consider the following cases: -\case{Case\/ {\rm1:} -$b'\geq 1/2$} We have $P(1/8)\geq\delta>0$ where $\delta$ -depends only on $d$, for otherwise by (3.7) applied to $P$ and $p=\infty$, -$P$ could not attain the value $1$ at $x=1$. Similarly, for -$m=(a'+b')/2$, $Q(m)\geq\delta'>0$ for some $\delta'$ depending only on $d$ -since otherwise $Q$ cannot attain the value $1$ at $x=a'$. Hence, for -$\delta''=\min(\delta,\delta')$, -$|A(y)|\geq|m-1/8|\geq b'/4\geq\frac18\max(b',1)$ for -$y\in[0,\delta'']$. On the other hand, -$|A(y)|\leq \max(b',1)$ for all $y\in[0,1]$, so (4.2) follows for -all $1\leq p\leq\infty$. -\endcase -\case{Case\/ {\rm2:} -$b'\leq 1/2$} We have $P(3/4)\leq\delta<1$ with $\delta$ -depending only on $d$ for otherwise (3.7) applied to $1-P$ and $p=\infty$ -would show that $P$ could not attain the value $0$ at $x=0$. It follows -that $|A(y)|\geq 3/4-b'\geq 1/4$, $y\in[\delta,1]$, while $|A(y)|\leq 1$ -for all $y\in[0,1]$. Hence (4.2) follows for -all $1\leq p\leq\infty$. -\endcase -We consider now when $Q$ is increasing. We can assume that $Q$ is not -a translate of $P$, i.e\., we do not have $P(x)=Q(x+\delta)$ for some $\delta$, -for then (4.2) follows trivially. In what follows, $C$ and $\delta$ -depend on $d$, and $C$ may depend on $p$. We consider the following cases: -\case{Case\/ {\rm3:} $a'\geq 1/4$ and $b'\leq 100$} -From (3.7) for $P$ -and $p=\infty$, it follows that $P(1/8)\geq\delta$ since otherwise $P$ cannot -attain the value $1$ at $x=1$. Hence $|A(y)|\geq a'-1/8\geq1/8$ on -$[0,\delta]$. On the other hand $|A(y)|\leq b'$ for all $y\in[0,1]$ and hence -(4.2) follows for all $1\leq p\leq\infty$. -\endcase -Let $z$ be in $\bold X$. Because $E(u,\sdot,\sdot)\equiv0$, -Equation (1.5) implies that -$$ -\int_\Bbb R\Delta uz|^T_0dx=\int_0^T\int_\Bbb R -\Delta u(z_t+\phi[u,u^h]z_{xx})\,dx\,dt- -E(u^h,z,t), -\tag 2.2 -$$ -where $\Delta u=u-u^h$ and -$$ -\phi[u,u^h]=\dfrac{\phi(u)-\phi(u^h)}{u-u^h}. -$$ -Extend $\phi[u,u^h](\cdot,t)=\phi[u,u^h](\cdot,0)$ for negative $t$, and -$\phi[u,u^h](\cdot,t)=\phi[u,u^h](\cdot,T)$ -for $t>T$. -Fix a point $x_0$ and a number $\epsilon>0$. Let $j_\epsilon$ -be a smooth function of $x$ with integral $1$ and support in -$[-\epsilon,\epsilon]$, -and let $J_\delta$ be a smooth function of -$x$ and $t$ with integral $1$ and support in -$[-\delta,\delta]\times[-\delta,\delta]$; $\delta$ and $\epsilon$ are -positive numbers to be specified later. -We choose $z=z^{\epsilon\delta}$ to satisfy -$$ -\aligned - &z_t+(\delta+J_\delta*\phi[u,u^h])z_{xx}=0,\qquad x\in\Bbb R,\;0 -\leq t\leq T,\\ - &z(x,T)=j_\epsilon(x-x_0). -\endaligned -\tag 2.3 -$$ -The conclusion of the theorem now follows from (2.1) and the fact that -$$ -|j_\epsilon*\Delta u(x_0,t)-\Delta u(x_0,t)|\leq C\epsilon^\alpha, -$$ -which follows from Assumption 2. -\qquad\qed -\enddemo -\example{Example\/ {\rm 1}} This is an example of an example. -\endexample -\remark{Remark\/ {\rm 1}} Examples are set the same as definitions in -some styles, -and the same as proofs in others. What convention does this style follow? -\endremark -Sometimes you want to include a figure, as in Fig.~1. -\topinsert -\def\Bif{{\bf if\/ }}\def\Bwhile{{\bf while\/ }}\def\Belse{{\bf else\/ }} -\settabs\+\qquad&\qquad&\qquad&\qquad&\cr -\+\smc Tree Partition Algorithm \{\cr -\+&Let stack size denote the number of nodes in the\cr -\+&&subtrees stored temporarily on the local stack\cr -\+&pop I from global stack\cr -\+&set stack size := 0\cr -\+&\Bwhile (stack size $\leq$ max size and stack size + -I$\rightarrow$tree size $>$ 3 (max size)) \{\cr -\+&&process I as an interior node\cr -\+&&let min tree be the smaller of the subtrees of the two children of I\cr -\+&&let max tree be the larger of the subtrees of the two children of I\cr -\+&&\Bif (min tree$\rightarrow$tree size + stack size $>$ 3 (max size)) \{\cr -\+&&&push min tree onto the global stack\cr -\+&&\} \Belse \{\cr -\+&&&push min tree onto the local stack\cr -\+&&&set stack size := stack size + min tree$\rightarrow$tree size\cr -\+&&\}\cr -\+&&set I := max tree\cr -\+&\}\cr -\+&\Bif (I$\rightarrow$tree size + stack size $>$ 3 (max size)) \{\cr -\+&&push I onto the global stack\cr -\+&\} \Belse \{\cr -\+&&push I onto the local stack\cr -\+&\}\cr -\+&Process all subtrees on the local stack\cr -\+\}\cr -\botcaption{Fig.~1} Tree partition algorithm Tree partition algorithm -Tree partition algorithm Tree partition algorithm Tree partition algorithm -Tree partition algorithm Tree partition algorithm.\endcaption -\endinsert - -We finish with a table of all SIAM journals. -\midinsert -\topcaption{Table 1}{SIAM journal acronyms and titles}\endcaption -\settabs\+\indent&Acronym\indent&Title&\cr -\hbox to \hsize{\hrulefill} -\+&Acronym&Title&\cr -\hbox to \hsize{\hrulefill} -\+&SINUM&SIAM Journal on Numerical Analysis&\cr -\+&SIREV&SIAM Review&\cr -\+&SIMA&SIAM Journal on Mathematical Analysis&\cr -\+&SIMAX&SIAM Journal on Matrix Analysis and Applications&\cr -\+&SICOMP&SIAM Journal on Computing&\cr -\+&SISC&SIAM Journal on Scientific Computing&\cr -\+&SIOPT&SIAM Journal on Optimization&\cr -\+&SIAP&SIAM Journal on Applied Mathematics&\cr -\+&SICON&SIAM Journal on Control and Optimization&\cr -\+&SIDMA&SIAM Journal on Discrete Mathematics&\cr -\+&TVP&Theory of Probability and Its Applications&\cr -\hbox to \hsize{\hrulefill} -\endinsert - -\Refs -\ref - \no 1 - \by L. A. Caffarelli and A. Friedman - \paper Regularity of the free boundary of a gas flow in an - $n$-dimensional porous medium - \jour Indiana Math. J. - \vol 29 - \yr 1980 - \pages 361--391 -\endref -\ref\no 2 - \by R. DeVore and B. Lucier - \paper High order regularity for solutions of the inviscid Burgers equation - \inbook Nonlinear Hyperbolic Problems -\procinfo Proceedings of an Advanced Research Workshop, Bordeaux, -France, June 1988 - \bookinfo Lecture Notes in Mathematics - \vol 1402 - \eds C. Carasso, P. Charrier, B. Hanouzet, and J.-L. Joly - \yr 1989 - \publ Springer-Verlag - \publaddr New York - \pages 147--154 -\endref -\ref \no 3 - \bysame - \paper Wavelets - \jour Acta Numerica - \yr 1992 - \ed A. Iserles - \publ Cambridge University Press - \publaddr New York - \pages 1--56 -\endref -\ref \no 4 - \by R. A. DeVore and V. A. Popov - \paper Interpolation spaces and non-linear approximation - \inbook Function Spaces and Applications - \bookinfo Lecture Notes in Mathematics - \procinfo Proceedings of the US--Swedish Seminar held in Lund, -Sweden, June 15--21, 1986 - \vol 1302 - \eds M. Cwikel, J. Peetre, Y. Sagher, and H. Wallin - \publ Springer-Verlag - \publaddr New York - \yr 1988 - \pages 191--205 - \endref -\ref \no 5 - \by R. A. DeVore and X. M. Yu - \paper Nonlinear $n$-widths in Besov spaces - \inbook Approximation Theory VI: Vol. 1 - \eds C. K. Chui, L. L. Schumaker, and J. D. Ward - \publ Academic Press - \publaddr New York - \yr 1989 - \pages 203--206 - \lang In Russian - \endref -\ref - \no 6 - \by K. Hollig and M. Pilant - \paper Regularity of the free boundary for the porous medium equation - \paperinfo MRC Tech. Rep. 2742 -\endref -\ref - \no 7 - \by J. Jerome - \book Approximation of Nonlinear Evolution Systems - \publ Academic Press - \publaddr New York - \yr 1983 -\endref -\ref - \no 8 - \manyby R. J. LeVeque - \paper Convergence of a large time step generalization of Godunov's method - for conservation laws - \jour Comm. Pure Appl. Math. - \vol 37 - \yr 1984 - \pages 463--478 -\endref -\ref\no 9 - \by O. Rioul and M. Vetterli - \paper Wavelets and signal processing - \jour IEEE Signal Processing Magazine - \vol 8 - \issue 4 - \yr 1991 - \toappear -\endref -\endRefs -\enddocument - diff --git a/Master/texmf-dist/doc/latex/siam/READ.ME b/Master/texmf-dist/doc/latex/siam/READ.ME deleted file mode 100644 index d362d373370..00000000000 --- a/Master/texmf-dist/doc/latex/siam/READ.ME +++ /dev/null @@ -1,38 +0,0 @@ -READ.ME 12/11/92 - -The following files constitute the full distribution -of SIAM's LaTeX macro package. The files are: - -read.me -siamltex.sty -docultex.tex -siam10.sty -siam11.sty -numinsec.sty -subeqn.sty -siam.bst -lexample.tex - -docultex.tex is the documentation file and should be refered to -for information about using the macros. - -Lines 339, 340, 348, 349, and 351 of siamltex.sty need to be -commented out to work with the NFSS. - -The SIAM macro's home is NETLIB@research.att.com, accessible through email. - -Both the LaTeX and plain TeX macros can be accessed through NETLIB: -Send an e-mail message to netlib@research.att.com, the body of the -message should be: - - send siamltex.all from typesetting - -for the LaTeX version, and - - send siamptex.all from typesetting - -for the plain TeX version. - -Questions, problems, bugs, and bug fixes should be -forwarded to SIAM TeX Support, 3600 University City Science Center, -Philadelphia PA, 19104. E-mail to duggan@siam.org. diff --git a/Master/texmf-dist/doc/latex/siam/docultex.tex b/Master/texmf-dist/doc/latex/siam/docultex.tex deleted file mode 100644 index 811a058cc91..00000000000 --- a/Master/texmf-dist/doc/latex/siam/docultex.tex +++ /dev/null @@ -1,592 +0,0 @@ -%% -%% This is file `docultex.tex', -%% Documentation for siam macros for use with LaTeX 2e -%% -%% By Paul Duggan for the Society for Industrial and Applied -%% Mathematics. -%% -%% December 1, 1995 -%% -%% Version 1.0a -%% -%% 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 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 (this file) -%% lexample.tex (example file for latex macro) -%% -%% 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} - -\title{USING SIAM'S \LaTeX\ MACROS\thanks{This work was -supported by the Society for Industrial and Applied -Mathematics}} - -\author{Paul Duggan\thanks{Society for Industrial and -Applied Mathematics, Philadelphia, Pennsylvania. -(duggan@siam. org). Questions, comments, or corrections -to this document may be directed to that email address.}} - - -\begin{document} -\maketitle - -\begin{abstract} -Documentation is given for use of the SIAM \LaTeX\ macros. These -macros are now compatible with \LaTeX$2_{\varepsilon}$. -Instructions and suggestions for compliance with SIAM style -standards are also included. Familiarity with standard \LaTeX\ commands -is assumed. -\end{abstract} - -\begin{keywords}\end{keywords} - -\begin{AMS}\end{AMS} - - -\pagestyle{myheadings} -\thispagestyle{plain} -\markboth{PAUL DUGGAN}{USING SIAM'S \LaTeX\ MACROS} - -\section{Introduction} - -This file is documentation for the SIAM \LaTeX\ macros, and -provides instruction for submission of your files. - -To accommodate authors who electronically typeset their manuscripts, -SIAM supports the use of \LaTeX. To ensure quality typesetting according -to SIAM style standards, SIAM provides a \LaTeX\ macro style file. -Using \LaTeX\ to format a manuscript should simplify the editorial process -and lessen the author's proofreading burden. However, -it is still necessary to proofread the galley proofs with care. - -Electronic files should not be submitted until the paper has been -accepted, and then not until requested to do so by someone in the SIAM -office. Once an article is slated for an issue, -someone from the SIAM office will contact the author about any or all -of the following: editorial and stylistic queries, -supplying the source files (and any supplementary macros) -for the properly formatted article, and handling figures. - -When submitting electronic files (electronic submissions) -(to {\tt tex@siam.org}) include the journal, issue, and author's -name in the subject line of the message. -Authors are responsible for ensuring that the paper generated -from the source files exactly matches the paper that -was accepted for publication by the review editor. If it does not, -information on how it differs should be indicated in the transmission -of the file. When submitting a file, please be sure to include any -additional macros (other than those provided by SIAM) that will be -needed to run the paper. - -SIAM uses MS-DOS-based computers for \LaTeX\ processing. Therefore -all filenames should be restricted to eight characters or less, -plus a three character extension. - -Once the files are corrected here at SIAM, we will mail the revised -proofs to be read against the original edited hardcopy -manuscript. We are not -set up to shuttle back and forth varying electronic versions of each -paper, so we must rely on hard copy of the galleys. The author's proofreading -is an important but easily overlooked step. Even if SIAM were not -to introduce a single editorial change into your manuscript, there -would still be a need to check, because electronic transmission -can introduce errors. - -The distribution contains the following items: {\tt -siamltex.cls}, the main macro package based on {\tt -article.cls}; {\tt siam10.clo}, for the ten-point size option;\linebreak -{\tt subeqn.clo}, a style option for equation numbering (see \S3 for -an explanation); and {\tt siam.bst}, - the style file for use with {\sc Bib}\TeX. Also included are this -file {\tt docultex.tex} and a sample file {\tt lexample.tex}. -The sample file represents a standard application of -the macros. The rest of this paper will highlight -some keys to effective macro use, as well as point out options and -special cases, and describe SIAM style standards to which -authors should conform. - - -\section{Headings} -The top matter of a journal paper falls into a standard -format. It begins of course with the \verb|\documentclass| command - -\begin{verbatim} - -\documentclass{siamltex} - -\end{verbatim} - -Other class options can be included -in the bracketed argument of the command, separated by commas. - - -The title and author parts are formatted using the -\verb|\title| and \verb|\author| commands as described in Lamport -\cite{Lamport}. The \verb|\date| -command is not used. \verb|\maketitle| produces the actual -output of the commands. - -The addresses and support acknowledgments are put into the -\verb|\author| commands via \verb|\thanks|. If support is -overall for the authors, the support acknowledgment should -be put in a \verb|\thanks| command in the \verb|\title|. -Specific support should go following the addresses of the -individual authors in the same \verb|\thanks| command. - -Sometimes authors have support or addresses in common which -necessitates having multiple \verb|\thanks| commands for -each author. Unfortunately \LaTeX\ does not normally allow this, -so a special procedure must be used. An example of this procedure -follows. Grant information can also be run into both authors' -footnotes. - -\begin{verbatim} - -\title{TITLE OF PAPER} - -\author{A.~U. Thorone\footnotemark[2]\ \footnotemark[5] -\and A.~U. Thortwo\footnotemark[3]\ \footnotemark[5] -\and A.~U. Thorthree\footnotemark[4]} - -\begin{document} -\maketitle - -\renewcommand{\thefootnote}{\fnsymbol{footnote}} - -\footnotetext[2]{Address of A.~U. Thorone} -\footnotetext[3]{Address of A.~U. Thortwo} -\footnotetext[4]{Address of A.~U. Thorthree} -\footnotetext[5]{Support in common for the first and second -authors.} - -\renewcommand{\thefootnote}{\arabic{footnote}} - -\end{verbatim} - -Notice that the footnote marks begin with {\tt [2]} -because the first mark (the asterisk) will be used in the -title for date-received information by SIAM, even if not -already used for support data. This is just one example; -other situations follow a similar pattern. - -Following the author and title is the abstract, key words -listing, and AMS subject classification number(s), -designated using the \verb|{abstract}|, \verb|{keywords}|, -and \verb|{AMS}| environments. If -there is only one AMS number, the commands -\verb|\begin{AM}| and \verb|\end{AM}| are used -instead of \verb|{AMS}|. This causes the heading to be -in the singular. Authors are responsible for providing AMS numbers. -They can be found in the Annual Index of Math Reviews, or -through {\tt e-Math} ({\tt telnet e-math.ams.com}; login -and password are both {\tt e-math}). - -Left and right running heads should be provided in the -following way. - -\begin{verbatim} - -\pagestyle{myheadings} -\thispagestyle{plain} -\markboth{A.~U. THORONE AND A.~U. THORTWO}{SHORTER PAPER -TITLE} - -\end{verbatim} - -\section{Equations and mathematics} - -One advantage of \LaTeX\ is that it can automatically number -equations and refer to these equation numbers in text. While plain \TeX's -method of equation numbering (explicit numbering using -\verb|\leqno|) works in the SIAM macro, it is not preferred -except in certain cases. SIAM style guidelines call for -aligned equations in many circumstances, and \LaTeX's -\verb|{eqnarray}| environment is not compatible with -\verb|\leqno| and \LaTeX\ is not compatible with the plain -\TeX\ command \verb|\eqalign| and \verb|\leqalignno|. Since -SIAM may have to alter or realign certain groups of -equations, it is necessary to use the \LaTeX\ system of -automatic numbering. - -Sometimes it is desirable to designate subequations of a larger -equation number. The subequations are designated with -(roman font) letters appended after the number. SIAM has -supplemented its macros with the {\tt subeqn.clo} option which -defines the environment \verb|{subequations}|. - -\begin{verbatim} - -\begin{subequations}\label{EKx} -\begin{equation} - y_k = B y_{k-1} + f, \qquad k=1,2,3,\ldots -\end{equation} -for any initial vector $ y_0$. Then -\begin{equation} - y_k\rightarrow u \mbox{\quad iff\quad} \rho( B)<1. -\end{equation} -\end{subequations} - -\end{verbatim} - -All equations within the \verb|{subequations}| environment -will keep the same overall number, but the letter -designation will increase. - -Clear equation formatting using \TeX\ can be challenging. Aside from -the regular \TeX\ documentation, authors will find Nicholas -J. Higham's book {\em Handbook of Writing for the Mathematical -Sciences\/} \cite{Higham} useful for guidelines and tips on -formatting with \TeX. The book covers many other topics related -to article writing as well. - -Authors commonly make mistakes by using - \verb|<|, \verb|>|, \verb|\mid|, and -\verb|\parallel| as delimiters, instead of -\verb|\langle|, \verb|\rangle|, \verb:|:, -and \verb:\|:. The incorrect symbols have particular -meanings distinct from the correct ones and should not be confused. - -\begin{table}[htbp] -\caption{Illustration of incorrect delimiter use.} -\begin{center}\footnotesize -\renewcommand{\arraystretch}{1.3} -\begin{tabular}{|ll|ll|}\hline -\multicolumn{2}{|c|}{{\bf Wrong}} & \multicolumn{2}{c|}{{\bf Right}}\\ \hline -\verb|<x, y>| & $<x, y>$ & \verb|\langle x, y\rangle| & $\langle x, y\rangle$\\ -\verb|5 < \mid A \mid| & $5 < \mid A \mid$ & \verb:5 < |A|: & $5 < |A|$\\ -\verb|6x = \parallel x|&&&\\ -\verb| - 1\parallel_{i}| & $6x = \parallel x - 1\parallel_{i}$ & - \verb:6x = \|x - 1\|_{i}: & $6x = \| x - 1\|_{i}$\\ \hline -\end{tabular} -\end{center} -\end{table} - -Another common author error is to put large (and even medium sized) -matrices in-line with the text, rather than displaying them. This -creates unattractive line spacing problems, and should be assiduously -avoided. Text-sized matrices (like $({a \atop b} {b \atop c})$) might -be used but anything much more complex than the example cited will -not be easy to read and should be displayed. - -More information on the formatting of equations and aligned -equations is found in Lamport \cite{Lamport}. Authors bear -primary responsibility for formatting their equations within -margins and in an aesthetically pleasing and informative manner. - -The SIAM macros include additional roman math words, or ``log-like" -functions, to those provided in standard \TeX. The following -commands are added: \verb:\const:, \verb:\diag:, \verb:\grad:, -\verb:\Range:, \verb:\rank:, and \verb:\supp:. -These commands produce the same word as the command name -in math mode, in upright type. - -\section{Special fonts} - -SIAM supports the use of the AMS-\TeX\ fonts (version 2.0 -and later). The package \verb|amsfonts| can be included with -the command\linebreak \verb|\usepackage{amsfonts}|. This package -is part of the AMS-\LaTeX distribution, available -from the AMS or from the Comprehensive TeX Archive -Network (anonymous ftp to ftp.shsu.edu). The blackboard bold font in this -font package can be used for designating number sets. -This is preferable to other methods of combining letters -(such as I and R for the real numbers) to produce pseudo-bold -letters but this is tolerable as well. Typographically speaking, -number sets may simply be designated using regular bold letters; -the blackboard bold typeface was designed to fulfil a desire -to simulate the limitations of a chalk board in printed type. - - -\subsection{Punctuation} -All standard punctuation and all numerals should be set in roman type -(upright) even within italic text. The only exceptions are periods and -commas. They may be set to match the surrounding text. - -References to sections should use the symbol \S, generated by -\verb|\S|. (If the reference begins a sentence, the term ``Section'' -should be spelled out in full.) Authors should not redefine -\verb|\S|, say, to be a calligraphic S, because \verb|\S| -must be reserved for use as the section symbol. - -Authors sometimes confuse the use of various types of dashes. -Hyphens (\verb|-|, -) are used for some compound words (many -such words should have no hyphen but must be run together, -like ``nonzero,'' or split apart, like ``well defined''). -Minus signs (\verb|$-$|, $-$) -should be used in math to represent subtraction or negative numbers. -En dashes (\verb|--|, --) are used for ranges (like 3--5, -June--August), or for joined names (like Runge--Kutta). Em dashes -(\verb|---|, ---) are used to set off a clause---such as this -one---from the rest of the sentence. - - -\subsection{Text formatting} - -SIAM style preferences do not make regular use of the \verb|{enumerate}| -and \verb|{itemize}| environments. Instead, -{\tt siamltex.cls} includes definitions of two alternate list -environments, \verb|{remunerate}| and \verb|{romannum}|. -Unlike the standard itemized lists, these environments do -not indent the secondary lines of text. The labels, whether -defaults or the optional user-defined, are always aligned -flush right. - -The \verb|{remunerate}| environment consecutively numbers -each item with an arabic numeral followed by a period. This -number is always upright, even in slanted -environments. (For those wondering at the unusual -naming of this environment, it comes from Seroul and Levy's -\cite{SerLev} definition of a similar macro for plain \TeX: -%{\tt \char"5C meti} -\verb|\meti| which is -% {\tt \char"5C item} -\protect\verb|\item| spelled backwards. Thus -%{\tt \{remunerate\}}, -\verb|{remunerate}| -a portion of -%{\tt \{enumerate\}} -\verb|{enumerate}| -spelled backwards.) - -The \verb|{romannum}| environment consecutively numbers -each item with a lower-case roman numeral enclosed in -parentheses. This number will always be upright within -slanted environments (as in theorems). - - -\section{Theorems and Lemmas} -Theorems, lemmas, corollaries, definitions, and propositions are covered -in the SIAM macros by the theorem-environments -\verb|{theorem}|, \verb|{lemma}|, \verb|{corollary}|, -\verb|{definition}| and \verb|{proposition}|. These are all -numbered in the same sequence and produce labels in small -caps with an italic body. Other environments may be specified by the -\verb|\newtheorem| command. SIAM's style is for Remarks and Examples -to appear with italic labels and an upright roman body. - -\begin{verbatim} - -\begin{theorem} -Sample theorem included for illustration. -Numbers and parentheses, like equation $(3.2)$, should be set -in roman type. Note that words (as opposed to ``log-like'' -functions) in displayed equations, such as -$$ x^2 = Y^2 \sin z^2 \mbox{ for all } x $$ -will appear in italic type in a theorem, though normally -they should appear in roman.\end{theorem} - -\end{verbatim} - -This sample produces Theorem 4.1 below. - -\begin{theorem} -Sample theorem included for illustration. -Numbers and parentheses, like equation $(3.2)$, should be set -in roman type. Note that words (as opposed to ``log-like'' -functions) in displayed equations, such as -$$ x^2 = Y^2 \sin z^2 \mbox{ for all } x $$ -will appear in italic type in a theorem, though normally -they should appear in roman. -\end{theorem} - - -Proofs are handled with the \verb|\begin{proof}| -\verb|\end{proof}| environment. A ``QED'' box \endproof\ is created -automatically by \verb|\end{proof}|, but this should be -preceded with a \verb|\qquad|. - -Named proofs, if used, must be done independently by the -authors. SIAM style specifies that proofs which end with -displayed equations should have the QED box two ems (\verb|\qquad|) -from the end of the equation on line with it horizontally. -Below is an example of how this can be done: - -\begin{verbatim} - -{\em Proof}. Proof of the previous theorem - . - . - . -thus, -$$ -a^2 + b^2 = c^2 \qquad\endproof -$$ - -\end{verbatim} - - -\section{Figures and tables} -Figures and tables sometimes require special consideration. -Tables in SIAM style are need to be set in eight point size -by using the \verb|\footnotesize| command inside the -\verb|\begin{table}| environment. Also, they should be designed -so that they do not extend beyond the text margins. - -SIAM style requires that no figures or tables appear in the -references section of the paper. \LaTeX\ is notorious for -making figure placement difficult, so it is important to -pay particular attention to figure placement near the -references in the text. All figures and tables should -be referred to in the text. - -SIAM supports the use of {\tt epsfig} for including {\sc PostScript} -figures. All {\sc Post\-Script} figures should be sent in separate -files. See the {\tt epsfig} documentation (available via -anonymous ftp from CTAN: ftp.shsu.edu) for more details on the use -of this style option. It is a good idea to submit high-quality -hardcopy of all {\sc Post\-Script} figures just in case there -is difficulty in the reproduction of the figure. Figures produced -by other non-\TeX\ methods should be included as high-quality -hardcopy when the manuscript is submitted. - -{\sc PostScript} figures that are sent should be generated with -sufficient line thickness. Some past figures authors have sent -had their line widths become very faint when SIAM set the papers -using a high-quality 1200dpi printer. - -Hardcopy for non-{\sc PostScript} figures should be included in -the submission of the hardcopy of the manuscript. Space -should be left in the \verb|{figure}| command for the -hardcopy to be inserted in production. - -\section{Bibliography and Bib\TeX} - -If using {\sc Bib}\TeX, authors need not submit the {\tt .bib} file for -their papers. Merely submit the completed {\tt .bbl} file, having used -{\tt siam.bst} as their bibliographic style file. {\tt siam.bst} -only works with Bib\TeX\ version 99i and later. The use of -Bib\TeX\ and the preparation of a {\tt .bib} file is -described in greater detail in \cite{Lamport}. - -If not using Bib\TeX, SIAM bibliographic references follow -the format of the following examples: - -\begin{verbatim} - -\bibitem{Ri} {\sc W. Riter}, -{\em Title of a paper appearing in a book}, in The Book -Title, E.~D. One, E.~D. Two, and A.~N. Othereditor, eds., -Publisher, Location, 1992, pp.~000--000. - -\bibitem{AuTh1} {\sc A.~U. Thorone}, {\em Title of paper -with lower case letters}, SIAM J. Abbrev. Correctly, 2 -(1992), pp.~000--000. - -\bibitem{A1A2} {\sc A.~U. Thorone and A.~U. Thortwo}, {\em -Title of paper appearing in book}, in Book Title: With All -Initial Caps, Publisher, Location, 1992. - -\bibitem{A1A22} \sameauthor, % generates the 3 em rule -{\em Title of Book{\rm :} Note Initial Caps and {\rm ROMAN -TYPE} for Punctuation and Acronyms}, Publisher, -Location, pp.~000--000, 1992. - -\bibitem{AuTh3} {\sc A.~U. Thorthree}, {\em Title of paper -that's not published yet}, SIAM. J. Abbrev. Correctly, to appear. - -\end{verbatim} - -Other types of references fall into the same general -pattern. See the sample file or any SIAM journal for other -examples. Authors must correctly format their bibliography to -be considered as having used the macros correctly. An incorrectly -formatted bibliography is not only time-consuming for SIAM to -process but it is possible that errors may be introduced into -it by keyboarders/copy editors. - -As an alternative to the above style of reference, an alphanumeric -code may be used in place of the number (e.g., [AUTh90]). The same -commands are used, but \verb|\bibitem| takes an optional argument -containing the desired alphanumeric code. - -Another alternative is no number, simply the authors' names and -the year of publication following in parentheses. The rest of the -format is identical. The macros do not support this alternative -directly, but modifications to the macro definition are possible -if this reference style is preferred. - - -\section{Conclusion} Many other style suggestions and tips -could be given to help authors but are beyond the scope of this -document. Simple mistakes can be avoided by increasing your familiarity -with how \LaTeX\ functions. The books referred to throughout this document -are also useful to the author who wants clear, beautiful typography -with minimal mistakes. - -\Appendix -\section{The use of appendices} -The \verb|\appendix| command may be used before the final sections -of a paper to designate them as appendices. Once \verb|\appendix| -is called, all subsequent sections will appear as - -\appendix -\section{Title of appendix} Each one will be sequentially lettered -instead of numbered. Theorem-like environments, subsections, -and equations will also have the section number changed to a letter. - -If there is only {\em one} appendix, however, the \verb|\Appendix| -(with a capital letter) should be used instead. This produces only -the word {\bf Appendix} in the section title, and does not add a letter. -Equation numbers, theorem numbers and subsections of the appendix -will have the letter ``A'' designating the section number. - -If you don't want to title your appendix, and just call it -{\bf Appendix A.} for example, use \verb|\appendix\section*{}| -and don't include anything in the title field. This works -opposite to the way \verb|\section*| usually works, by including the -section number, but not using a title. - - -Appendices should appear before the bibliography section, not after, -and any acknowledgments should be placed after the appendices and before -the bibliography. - -\begin{thebibliography}{1} -\bibitem{GoMiSa} {\sc M. Goossens, F. Mittelbach, and A. Samarin}, -{\em The} \LaTeX\ {\em Companion}, Addison-Wesley, Reading, MA, 1994. - -\bibitem{Higham} {\sc N.~J. Higham}, {\em Handbook of Writing for -the Mathematical Sciences}, Society for Industrial and Applied -Mathematics, Philadelphia, PA, 1993. - -\bibitem{Lamport} {\sc L. Lamport}, \LaTeX: {\em A Document -Preparation System}, Addison-Wesley, Reading, MA, 1986. - -\bibitem{SerLev} {\sc R. Seroul and S. Levy}, {\em A -Beginner's Book of} \TeX, Springer-Verlag, Berlin, New -York, 1991. -\end{thebibliography} - - -\end{document} -%% end of file `docultex.tex'
\ No newline at end of file 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. 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