diff options
Diffstat (limited to 'Master/texmf-dist/source/latex/siam/soda209.all')
-rw-r--r-- | Master/texmf-dist/source/latex/siam/soda209.all | 682 |
1 files changed, 0 insertions, 682 deletions
diff --git a/Master/texmf-dist/source/latex/siam/soda209.all b/Master/texmf-dist/source/latex/siam/soda209.all deleted file mode 100644 index 862535ccfd3..00000000000 --- a/Master/texmf-dist/source/latex/siam/soda209.all +++ /dev/null @@ -1,682 +0,0 @@ -%% This is soda209.all. This file is to be used for creating a paper -%% in the ACM/SIAM Preprint series with LaTeX. It consists of the following -%% two files: -%% -%% ltexprt.tex ---- an example and documentation file -%% ltexprt.sty ---- the macro file -%% -%% To use, cut this file apart at the appropriate places. You can run the -%% example file with the macros to get sample output. -%% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CUT HERE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% -% -%%%%%%%%%%%%%%%%%%%%%%%%%% ltexprt.tex %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% -% This is ltexprt.tex, an example file for use with the SIAM LaTeX (version 2.09) -% Preprint Series macros. It is designed to provide double-column output. -% Please take the time to read the following comments, as they document -% how to use these macros. This file can be composed and printed out for -% use as sample output. - -% Any comments or questions regarding these macros should be directed to: -% -% Corey Gray -% SIAM -% 3600 University City Science Center -% Philadelphia, PA 19104-2688 -% USA -% Telephone: (215) 382-9800 -% Fax: (215) 386-7999 -% e-mail: gray@siam.org - - -% This file is to be used as an example for style only. It should not be read -% for content. - -%%%%%%%%%%%%%%% PLEASE NOTE THE FOLLOWING STYLE RESTRICTIONS %%%%%%%%%%%%%%% - -%% 1. There are no new tags. Existing LaTeX tags have been formatted to match -%% the Preprint series style. -%% -%% 2. You must use \cite in the text to mark your reference citations and -%% \bibitem in the listing of references at the end of your chapter. See -%% the examples in the following file. If you are using BibTeX, please -%% supply the bst file with the manuscript file. -%% -%% -%% 3. This macro is set up for two levels of headings (\section and -%% \subsection). The macro will automatically number the headings for you. -%% -%% 4. No running heads are to be used in this volume. -%% -%% 5. Theorems, Lemmas, Definitions, etc. are to be double numbered, -%% indicating the section and the occurence of that element -%% within that section. (For example, the first theorem in the second -%% section would be numbered 2.1. The macro will -%% automatically do the numbering for you. -%% -%% 6. Figures, equations, and tables must be single-numbered. -%% Use existing LaTeX tags for these elements. -%% Numbering will be done automatically. -%% -%% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - - -\documentstyle[twoside,leqno,twocolumn,ltexprt]{article} - -\begin{document} - - -\title{\Large SIAM/ACM Preprint Series Macros for -Use With LaTeX\thanks{Supported by GSF grants ABC123, DEF456, and GHI789.}} -\author{Corey Gray\thanks{Society for Industrial and Applied Mathematics.} \\ -\and -Tricia Manning\thanks{Society for Industrial and Applied Mathematics.}} -\date{} - -\maketitle - -\pagestyle{myheadings} -\markboth{}{} - -%\pagenumbering{arabic} - - -\begin{abstract} \small\baselineskip=9pt This is the text of my abstract. It is a brief -description of my -paper, outlining the purposes and goals I am trying to address.\end{abstract} - -\section{Problem Specification.}In this paper, we consider the solution of the $N \times -N$ linear -system -\begin{equation} \label{e1.1} -A x = b -\end{equation} -where $A$ is large, sparse, symmetric, and positive definite. We consider -the direct solution of (\ref{e1.1}) by means of general sparse Gaussian -elimination. In such a procedure, we find a permutation matrix $P$, and -compute the decomposition -\[ -P A P^{t} = L D L^{t} -\] -where $L$ is unit lower triangular and $D$ is diagonal. - - -\section{Design Considerations.}Several good ordering algorithms (nested dissection and -minimum degree) -are available for computing $P$ \cite{GEORGELIU}, \cite{ROSE72}. -Since our interest here does not -focus directly on the ordering, we assume for convenience that $P=I$, -or that $A$ has been preordered to reflect an appropriate choice of $P$. - -Our purpose here is to examine the nonnumerical complexity of the -sparse elimination algorithm given in \cite{BANKSMITH}. -As was shown there, a general sparse elimination scheme based on the -bordering algorithm requires less storage for pointers and -row/column indices than more traditional implementations of general -sparse elimination. This is accomplished by exploiting the m-tree, -a particular spanning tree for the graph of the filled-in matrix. - -\begin{theorem} The method was extended to three -dimensions. For the standard multigrid -coarsening -(in which, for a given grid, the next coarser grid has $1/8$ -as many points), anisotropic problems require plane -relaxation to -obtain a good smoothing factor.\end{theorem} - -Our purpose here is to examine the nonnumerical complexity of the -sparse elimination algorithm given in \cite{BANKSMITH}. -As was shown there, a general sparse elimination scheme based on the -bordering algorithm requires less storage for pointers and -row/column indices than more traditional implementations of general -sparse elimination. This is accomplished by exploiting the m-tree, -a particular spanning tree for the graph of the filled-in matrix. -Several good ordering algorithms (nested dissection and minimum degree) -are available for computing $P$ \cite{GEORGELIU}, \cite{ROSE72}. -Since our interest here does not -focus directly on the ordering, we assume for convenience that $P=I$, -or that $A$ has been preordered to reflect an appropriate choice of $P$. - -\begin{proof} In this paper we consider two methods. The first method -is -basically the method considered with two differences: -first, we perform plane relaxation by a two-dimensional -multigrid method, and second, we use a slightly different -choice of -interpolation operator, which improves performance -for nearly singular problems. In the second method coarsening -is done by successively coarsening in each of the three -independent variables and then ignoring the intermediate -grids; this artifice simplifies coding considerably. -\end{proof} - -Our purpose here is to examine the nonnumerical complexity of the -sparse elimination algorithm given in \cite{BANKSMITH}. -As was shown there, a general sparse elimination scheme based on the -bordering algorithm requires less storage for pointers and -row/column indices than more traditional implementations of general -sparse elimination. This is accomplished by exploiting the m-tree, -a particular spanning tree for the graph of the filled-in matrix. - -\begin{Definition}{\rm We describe the two methods in \S 1.2. In \S\ 1.3. we -discuss -some remaining details.} -\end{Definition} - -Our purpose here is to examine the nonnumerical complexity of the -sparse elimination algorithm given in \cite{BANKSMITH}. -As was shown there, a general sparse elimination scheme based on the -bordering algorithm requires less storage for pointers and -row/column indices than more traditional implementations of general -sparse elimination. This is accomplished by exploiting the m-tree, -a particular spanning tree for the graph of the filled-in matrix. -Several good ordering algorithms (nested dissection and minimum degree) -are available for computing $P$ \cite{GEORGELIU}, \cite{ROSE72}. -Since our interest here does not -focus directly on the ordering, we assume for convenience that $P=I$, -or that $A$ has been preordered to reflect an appropriate choice of $P$. - -Our purpose here is to examine the nonnumerical complexity of the -sparse elimination algorithm given in \cite{BANKSMITH}. -As was shown there, a general sparse elimination scheme based on the -bordering algorithm requires less storage for pointers and -row/column indices than more traditional implementations of general -sparse elimination. - -\begin{lemma} We discuss first the choice for $I_{k-1}^k$ -which is a generalization. We assume that $G^{k-1}$ is -obtained -from $G^k$ -by standard coarsening; that is, if $G^k$ is a tensor product -grid $G_{x}^k \times G_{y}^k \times G_{z}^k$, -$G^{k-1}=G_{x}^{k-1} \times G_{y}^{k-1} \times G_{z}^{k-1}$, -where $G_{x}^{k-1}$ is obtained by deleting every other grid -point of $G_x^k$ and similarly for $G_{y}^k$ and $G_{z}^k$. -\end{lemma} - -To our knowledge, the m-tree previously has not been applied in this -fashion to the numerical factorization, but it has been used, -directly or indirectly, in several optimal order algorithms for -computing the fill-in during the symbolic factorization phase -[4] - [10], [5], [6]. In \S 1.3., we analyze the complexity of the old and new -approaches to the intersection problem for the special case of -an $n \times n$ grid ordered by nested dissection. The special -structure of this problem allows us to make exact estimates of -the complexity. To our knowledge, the m-tree previously has not been applied in this -fashion to the numerical factorization, but it has been used, -directly or indirectly, in several optimal order algorithms for -computing the fill-in during the symbolic factorization phase -[4] - [10], [5], [6]. - -In \S 1.2, we review the bordering algorithm, and introduce -the sorting and intersection problems that arise in the -sparse formulation of the algorithm. -In \S 1.3., we analyze the complexity of the old and new -approaches to the intersection problem for the special case of -an $n \times n$ grid ordered by nested dissection. The special -structure of this problem allows us to make exact estimates of -the complexity. To our knowledge, the m-tree previously has not been applied in this -fashion to the numerical factorization, but it has been used, -directly or indirectly, in several optimal order algorithms for -computing the fill-in during the symbolic factorization phase -[4] - [10], [5], [6]. - - -For the old approach, we show that the -complexity of the intersection problem is $O(n^{3})$, the same -as the complexity of the numerical computations. For the -new approach, the complexity of the second part is reduced to -$O(n^{2} (\log n)^{2})$. - -To our knowledge, the m-tree previously has not been applied in this -fashion to the numerical factorization, but it has been used, -directly or indirectly, in several optimal order algorithms for -computing the fill-in during the symbolic factorization phase -[4] - [10], [5], [6]. In \S 1.3., we analyze the complexity of the old and new -approaches to the intersection problem for the special case of -an $n \times n$ grid ordered by nested dissection. The special -structure of this problem allows us to make exact estimates of -the complexity. To our knowledge, the m-tree previously has not been applied in this -fashion to the numerical factorization, but it has been used, -directly or indirectly, in several optimal order algorithms for -computing the fill-in during the symbolic factorization phase -[4] - [10], [5], [6]. -This is accomplished by exploiting the m-tree, -a particular spanning tree for the graph of the filled-in matrix. -To our knowledge, the m-tree previously has not been applied in this -fashion to the numerical factorization, but it has been used, -directly or indirectly, in several optimal order algorithms for -computing the fill-in during the symbolic factorization phase -\cite{EISENSTAT} - \cite{LIU2}, \cite{ROSE76}, \cite{SCHREIBER}. - -\subsection{Robustness.}\ We do not -attempt to present an overview -here, but rather attempt to focus on those results that -are relevant to our particular algorithm. -This section assumes prior knowledge of the role of graph theory -in sparse Gaussian elimination; surveys of this role are -available in \cite{ROSE72} and \cite{GEORGELIU}. More general -discussions of elimination trees are given in -\cite{LAW} - \cite{LIU2}, \cite{SCHREIBER}. -Thus, at the $k$th stage, the bordering algorithm consists of -solving the lower triangular system -\begin{equation} \label{1.2} - L_{k-1}v = c -\end{equation} -and setting -\begin{eqnarray} -\ell &=& D^{-1}_{k-1}v , \\ -\delta &=& \alpha - \ell^{t} v . -\end{eqnarray} - -\begin{figure} -\vspace{14pc} -\caption{This is a figure 1.1.} -\end{figure} - -\section{Robustness.} We do not -attempt to present an overview -here, but rather attempt to focus on those results that -are relevant to our particular algorithm. - -\subsection{Versatility.}\ The special -structure of this problem allows us to make exact estimates of -the complexity. For the old approach, we show that the -complexity of the intersection problem is $O(n^{3})$, the same -as the complexity of the numerical computations -\cite{GEORGELIU}, \cite{ROSEWHITTEN}. For the -new approach, the complexity of the second part is reduced to -$O(n^{2} (\log n)^{2})$. - -To our knowledge, the m-tree previously has not been applied in this -fashion to the numerical factorization, but it has been used, -directly or indirectly, in several optimal order algorithms for -computing the fill-in during the symbolic factorization phase -[4] - [10], [5], [6]. In \S 1.3., we analyze the complexity of the old and new -approaches to the intersection problem for the special case of -an $n \times n$ grid ordered by nested dissection. The special -structure of this problem allows us to make exact estimates of -the complexity. To our knowledge, the m-tree previously has not been applied in this -fashion to the numerical factorization, but it has been used, -directly or indirectly, in several optimal order algorithms for -computing the fill-in during the symbolic factorization phase -[4] - [10], [5], [6]. - -In \S 1.2, we review the bordering algorithm, and introduce -the sorting and intersection problems that arise in the -sparse formulation of the algorithm. -In \S 1.3., we analyze the complexity of the old and new -approaches to the intersection problem for the special case of -an $n \times n$ grid ordered by nested dissection. The special -structure of this problem allows us to make exact estimates of -the complexity. To our knowledge, the m-tree previously has not been applied in this -fashion to the numerical factorization, but it has been used, -directly or indirectly, in several optimal order algorithms for -computing the fill-in during the symbolic factorization phase -[4] - [10], [5], [6]. - - -For the old approach, we show that the -complexity of the intersection problem is $O(n^{3})$, the same -as the complexity of the numerical computations. For the -new approach, the complexity of the second part is reduced to -$O(n^{2} (\log n)^{2})$. - -To our knowledge, the m-tree previously has not been applied in this -fashion to the numerical factorization, but it has been used, -directly or indirectly, in several optimal order algorithms for -computing the fill-in during the symbolic factorization phase -[4] - [10], [5], [6]. In \S 1.3., we analyze the complexity of the old and new -approaches to the intersection problem for the special case of -an $n \times n$ grid ordered by nested dissection. The special -structure of this problem allows us to make exact estimates of -the complexity. To our knowledge, the m-tree previously has not been applied in this -fashion to the numerical factorization, but it has been used, -directly or indirectly, in several optimal order algorithms for -computing the fill-in during the symbolic factorization phase -[4] - [10], [5], [6]. -This is accomplished by exploiting the m-tree, -a particular spanning tree for the graph of the filled-in matrix. -To our knowledge, the m-tree previously has not been applied in this -fashion to the numerical factorization, but it has been used, -directly or indirectly, in several optimal order algorithms for -computing the fill-in during the symbolic factorization phase -\cite{EISENSTAT} - \cite{LIU2}, \cite{ROSE76}, \cite{SCHREIBER}. - -\begin{thebibliography}{99} - -%\bibitem{GUIDE} -%R.~E. Bank, {\em PLTMG users' guide, edition 5.0}, tech. report, -% Department of Mathematics, University of California, San Diego, CA, 1988. - -%\bibitem{HBMG} -%R.~E. Bank, T.~F. Dupont, and H.~Yserentant, {\em The hierarchical basis -% multigrid method}, Numer. Math., 52 (1988), pp.~427--458. - -\bibitem{BANKSMITH} -R.~E. Bank and R.~K. Smith, {\em General sparse elimination requires no - permanent integer storage}, SIAM J. Sci. Stat. Comput., 8 (1987), - pp.~574--584. - -\bibitem{EISENSTAT} -S.~C. Eisenstat, M.~C. Gursky, M.~Schultz, and A.~Sherman, {\em - Algorithms and data structures for sparse symmetric gaussian elimination}, - SIAM J. Sci. Stat. Comput., 2 (1982), pp.~225--237. - -\bibitem{GEORGELIU} -A.~George and J.~Liu, {\em Computer Solution of Large Sparse Positive - Definite Systems}, Prentice Hall, Englewood Cliffs, NJ, 1981. - -\bibitem{LAW} -K.~H. Law and S.~J. Fenves, {\em A node addition model for symbolic - factorization}, ACM TOMS, 12 (1986), pp.~37--50. - -\bibitem{LIU} -J.~W.~H. Liu, {\em A compact row storage scheme for cholesky factors - using elimination trees}, ACM TOMS, 12 (1986), pp.~127--148. - -\bibitem{LIU2} -\sameauthor , {\em The role of - elimination trees in sparse factorization}, Tech. Report CS-87-12,Department - of Computer Science, York University, Ontario, Canada, 1987. - -\bibitem{ROSE72} -D.~J. Rose, {\em A graph theoretic study of the numeric solution of - sparse positive definite systems}, in Graph Theory and Computing, Academic Press, New -York, 1972. - -\bibitem{ROSE76} -D.~J. Rose, R.~E. Tarjan, and G.~S. Lueker, {\em Algorithmic aspects of - vertex elimination on graphs}, SIAM J. Comput., 5 (1976), pp.~226--283. - -\bibitem{ROSEWHITTEN} -D.~J. Rose and G.~F. Whitten, {\em A recursive analysis of disection - strategies}, in Sparse Matrix Computations, Academic Press, New York, 1976. - -\bibitem{SCHREIBER} -R.~Schrieber, {\em A new implementation of sparse gaussian elimination}, - ACM TOMS, 8 (1982), pp.~256--276. - -\end{thebibliography} -\end{document} - -% End of ltexprt.tex -% -% -% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CUT HERE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% -% -% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ltexprt.sty %%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% -% This is ltexprt.sty, a file of macros and definitions for creating a -% chapter for publication in the ACM/SIAM Preprint series using LaTeX (version 2.09). -% It is designed to produce double-column output. -% This file may be freely distributed but may not be altered in any way. -% Any comments or questions regarding these macros should be directed to: - -% Corey Gray -% SIAM -% 3600 University City Science Center -% Philadelphia, PA 19104-2688 -% USA -% Telephone: (215) 382-9800 -% Fax: (215) 386-7999 -% e-mail: gray@siam.org - - -% Report the version. -\message{*** ACM/SIAM LaTeX 2.09 Preprint Series macro package, version 1.0, -September 24,1990 ***} - - -\pretolerance=800 -\tolerance=10000 -\sloppy - -\voffset=-.5in -\hoffset=-.5in -\vsize=55pc -\hsize=41pc -\baselineskip=14pt -\footskip=18pt -\topmargin 24pt -\headheight 12pt -\headsep 17pt -\textheight 52.5pc \advance\textheight by \topskip -\textwidth 41pc -\parskip 0pt -\parindent 18pt - -\font\tensmc=cmcsc10 -\def\smc{\tensmc} - -%% footnotes to be set 8/10 -\def\footnotesize{\@setsize\footnotesize{10pt}\viiipt\@viiipt - % \indent - \abovedisplayskip \z@ - \belowdisplayskip\z@ - \abovedisplayshortskip\abovedisplayskip - \belowdisplayshortskip\belowdisplayshortskip - \def\@listi{\leftmargin\leftmargini \topsep 3pt plus 1pt minus 1pt - \parsep 2pt plus 1pt minus 1pt - \itemsep \parsep}} - -\let\referencesize\footnotesize - -\footnotesep 0pt - -\skip\footins 12pt plus 12pt - -\def\footnoterule{\kern3\p@ \hrule width 3em} % the \hrule is .4pt high - -\def\ps@plain{\let\@mkboth\@gobbletwo - \def\@oddfoot{{\hfil\small\thepage\hfil}}% - \def\@oddhead{} - \def\@evenhead{}\def\@evenfoot{}} - - - - - -\def\ps@headings{\let\@mkboth\markboth - \def\@oddfoot{}\def\@evenfoot{}% - \def\@evenhead{{\rm\thepage}\hfil{\small\leftmark}}% - \def\@oddhead{{\noindent\small\rightmark}\hfil{\rm\thepage}}% - - - -\def\ps@myheadings{\let\@mkboth\@gobbletwo - \def\@oddfoot{}\def\@evenfoot{}% - \def\@oddhead{\rlap{\normalsize\rm\rightmark}\hfil{small\thepage}}% - \def\@evenhead%{\hfil{\small\@chapapp}\ - {\small\thepage}\hfil\llap{\normalsize\rm\leftmark}}% - \def\chaptermark##1{}% - \def\sectionmark##1{}\def\subsectionmark##1{}} - - -\def\theequation{\arabic{section}.\arabic{equation}} - - -\def\section{\@startsection{section}{1}{0pt}{-12pt}{3pt}{\hyphenpenalty=\@M -\exhyphenpenalty=\@M\normalsize\bf}} -\def\subsection{\@startsection{subsection}{2}{0pt}{-12pt}{0pt}{\normalsize\bf} -} -\def\subsubsection{\@startsection - {subsubsection}{3}{0pt}{-12pt}{0pt}{\normalsize\bf}} -\def\paragraph{\@startsection - {paragraph}{4}{\parindent}{0pt}{0pt}{\normalsize\bf}} -\def\subparagraph{\@startsection - {subparagraph}{4}{\parindent}{0pt}{0pt}{\normalsize\bf}} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% % -% THEOREMS, PROOFS, ALGORITHMS % -% % -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -%%% defined proof environment by theorem model (took out counter) - -\def\newproof#1{\@nprf{#1}} - -\def\@nprf#1#2{\@xnprf{#1}{#2}} - -\def\@xnprf#1#2{\expandafter\@ifdefinable\csname #1\endcsname -\global\@namedef{#1}{\@prf{#1}{#2}}\global\@namedef{end#1}{\@endproof}} - -\def\@prf#1#2{\@xprf{#1}{#2}} - -\def\@xprf#1#2{\@beginproof{#2}{\csname the#1\endcsname}\ignorespaces} - - - -%%% defined algorithm environment by theorem model - -\def\newalgorithm#1{\@ifnextchar[{\@oalg{#1}}{\@nalg{#1}}} - -\def\@nalg#1#2{% -\@ifnextchar[{\@xnalg{#1}{#2}}{\@ynalg{#1}{#2}}} - -\def\@xnalg#1#2[#3]{\expandafter\@ifdefinable\csname #1\endcsname -{\@definecounter{#1}\@addtoreset{#1}{#3}% -\expandafter\xdef\csname the#1\endcsname{\expandafter\noexpand - \csname the#3\endcsname \@thmcountersep \@thmcounter{#1}}% -\global\@namedef{#1}{\@alg{#1}{#2}}\global\@namedef{end#1}{\@endalgorithm}}} - -\def\@ynalg#1#2{\expandafter\@ifdefinable\csname #1\endcsname -{\@definecounter{#1}% -\expandafter\xdef\csname the#1\endcsname{\@thmcounter{#1}}% -\global\@namedef{#1}{\@alg{#1}{#2}}\global\@namedef{end#1}{\@endalgorithm}}} - -\def\@oalg#1[#2]#3{\expandafter\@ifdefinable\csname #1\endcsname - {\global\@namedef{the#1}{\@nameuse{the#2}}% -\global\@namedef{#1}{\@alg{#2}{#3}}% -\global\@namedef{end#1}{\@endalgorithm}}} - -\def\@alg#1#2{\refstepcounter - {#1}\@ifnextchar[{\@yalg{#1}{#2}}{\@xalg{#1}{#2}}} - -\def\@xalg#1#2{\@beginalgorithm{#2}{\csname the#1\endcsname}\ignorespaces} -\def\@yalg#1#2[#3]{\@opargbeginalgorithm{#2}{\csname - the#1\endcsname}{#3}\ignorespaces} - - - - -\def\@beginproof#1{\rm \trivlist \item[\hskip \labelsep{\it #1.\/}]} -\def\@endproof{\outerparskip 0pt\endtrivlist} - -\def\@begintheorem#1#2{\it \trivlist \item[\hskip \labelsep{\sc #1\ #2.}]} -\def\@opargbegintheorem#1#2#3{\it \trivlist - \item[\hskip \labelsep{\sc #1\ #2.\ (#3)}]} -\def\@endtheorem{\outerparskip 0pt\endtrivlist} - -%\def\@begindefinition#1#2{\rm \trivlist \item[\hskip \labelsep{\sc #1\ #2.}]} -%\def\@opargbegindefinition#1#2#3{\rm \trivlist -% \item[\hskip \labelsep{\sc #1\ #2.\ (#3)}]} -%\def\@enddefinition{\outerparskip 0pt\endtrivlist} - - -\def\@beginalgorithm#1#2{\rm \trivlist \item[\hskip \labelsep{\sc #1\ #2.}]} -\def\@opargbeginalgorithm#1#2#3{\rm \trivlist - \item[\hskip \labelsep{\sc #1\ #2.\ (#3)}]} -\def\@endalgorithm{\outerparskip 6pt\endtrivlist} - - -\newskip\outerparskip - -\def\trivlist{\parsep\outerparskip - \@trivlist \labelwidth\z@ \leftmargin\z@ - \itemindent\parindent \def\makelabel##1{##1}} - -\def\@trivlist{\topsep=0pt\@topsepadd\topsep - \if@noskipsec \leavevmode \fi - \ifvmode \advance\@topsepadd\partopsep \else \unskip\par\fi - \if@inlabel \@noparitemtrue \@noparlisttrue - \else \@noparlistfalse \@topsep\@topsepadd \fi - \advance\@topsep \parskip - \leftskip\z@\rightskip\@rightskip \parfillskip\@flushglue - \@setpar{\if@newlist\else{\@@par}\fi}% - \global\@newlisttrue \@outerparskip\parskip} - - -\def\endtrivlist{\if@newlist\@noitemerr\fi - \if@inlabel\indent\fi - \ifhmode\unskip \par\fi - \if@noparlist \else - \ifdim\lastskip >\z@ \@tempskipa\lastskip \vskip -\lastskip - \advance\@tempskipa\parskip \advance\@tempskipa -\@outerparskip - \vskip\@tempskipa - \fi\@endparenv\fi - \vskip\outerparskip} - - - - \newproof{@proof}{Proof} - \newenvironment{proof}{\begin{@proof}}{\end{@proof}} - - \newtheorem{@theorem}{Theorem}[section] - \newenvironment{theorem}{\begin{@theorem}}{\end{@theorem}} - - \newalgorithm{@algorithm}{Algorithm}[section] - \newenvironment{algorithm}{\begin{@algorithm}}{\end{@algorithm}} - - - -\newtheorem{lemma}{Lemma}[section] -\newtheorem{fact}{Fact}[section] -\newtheorem{corollary}{Corollary}[section] -\newtheorem{axiom}{Axiom}[section] -\newtheorem{cond}{Condition}[section] -\newtheorem{property}{Property}[section] -\newtheorem{proposition}{Proposition}[section] - -\newtheorem{Conjecture}{Conjecture}[section] -%\newtheorem{Corollary}[Theorem]{Corollary} -\newtheorem{Definition}{Definition}[section] -\newtheorem{Lemma}{Lemma}[section] -\newtheorem{Remark}{Remark}[section] - -\newproof{Example}{Example} -\newproof{Method}{Method} -\newproof{Exercise}{Exercise} - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%% %% -%% BIBLIOGRAPHY %% -%% %% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - -\def\thebibliography#1{% -%\cleardoublepage -\parindent 0em -\vspace{6pt} -\begin{flushleft}\normalsize\bf References\end{flushleft} -\addvspace{3pt}\nopagebreak\list - %% default is no labels, for those not using \cite or BibTeX -% {[\arabic{enumi}]} {\settowidth\labelwidth{[#1]} -{[\arabic{enumi}]}{\settowidth\labelwidth{mm} -\leftmargin\labelwidth - \advance\leftmargin\labelsep - \usecounter{enumi}\@bibsetup} -\def\newblock{\hskip .11em plus .33em minus -.07em} - \sloppy\clubpenalty4000\widowpenalty4000 - \sfcode`\.=1000\relax} - -%% setup 8/10 type -\def\@bibsetup{\itemindent=0pt \itemsep=0pt \parsep=0pt -\small} - -\def\sameauthor{\leavevmode\vrule height 2pt depth -1.6pt width 23pt} - -% -%% End of ltexprt.sty -% -%%%%%%%%%%%%%%%%%%%%%%%%% End of soda209.all %%%%%%%%%%%%%%%%%%%%%%% |