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-%% This is ltexpprt.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:
-%%
-%% ltexpprt.tex ---- an example and documentation file
-%% ltexpprt.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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%% ltexpprt.tex %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%
-% This is ltexpprt.tex, an example file for use with the SIAM LaTeX
-% 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. Unless otherwise stated by your editor, do your chapter as if it
-%% is Chapter 1.
-%% If you know which number your chapter is, you must do the following:
-%%
-%% Use the \setcounter command to set the counters for chapter,
-%% section, and page number to the appropriate number. The counter
-%% for chapter is incremental, so it should be set one less than
-%% the actual chapter number. The section counter is not
-%% incremental. Set the page counter to 1. The following example
-%% is set up as if it were chapter 3. Please note the placement of
-%% the three \setcounter commands and follow it exactly.
-%%
-%% 4. This macro is set up for two levels of headings (\section and
-%% \subsection). The macro will automatically number the headings for you.
-%%
-%% 5. The running heads are defined by the \markboth command. The left running
-%% head (the first field of the \markboth command) should be defined with
-%% the authors names. The right running head (the second field of the
-%% \markboth command) should be defined with the title (or shortened title)
-%% of your chapter. Neither running head may be more than 40 characters.
-%%
-%% 6. Theorems, Lemmas, Definitions, etc. are to be triple numbered,
-%% indicating the chapter, section, and the occurence of that element
-%% within that section. (For example, the first theorem in the second
-%% section of chapter three would be numbered 3.2.1. The macro will
-%% automatically do the numbering for you.
-%%
-%% 7. Figures, equations, and tables must be double-numbered indicating
-%% chapter and occurence. Use existing LaTeX tags for these elements.
-%% Numbering will be done automatically.
-%%
-%%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-
-
-\documentstyle[twoside,leqno,twocolumn,ltexpprt]{article}
-
-\begin{document}
-
-
-%\setcounter{chapter}{2} % If you are doing your chapter as chapter one,
-%\setcounter{section}{3} % comment these two lines out.
-
-\title{\large\bf Chapter 1 \\
-\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{AUTHORS NAMES}{CHAPTER TITLE}
-
-%\pagenumbering{arabic}
-%\setcounter{page}{1}%Leave this line commented out.
-
-\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, AcademicΠPress, 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 ltexpprt.tex
-%
-%
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CUT HERE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%
-%
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ltexpprt.sty %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%
-% This is ltexpprt.sty, a file of macros and definitions for creating a
-% chapter for publication in the ACM/SIAM Preprint series using LaTeX.
-% 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 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 ltexpprt.sty
-%
-%%%%%%%%%%%%%%%%%%%%%%%%% End of ltexpprt.all %%%%%%%%%%%%%%%%%%%%%%%
- \ No newline at end of file