%% This is sodaptex.all. This file is to be used for creating a paper
%% in the ACM/SIAM Preprint series with Plain TeX. It consists of the following
%% two files:
%%
%% ptexpprt.tex ---- an example and documentation file
%% ptexpprt.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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%% ptexpprt.tex %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% This is ptexpprt.tex, an example file for use with the ACM/SIAM Plain TeX
% Preprint Series macros. It is designed to produce double-column output.
% Comments are placed at the beginning and throughout this file. Please
% take the time to read them 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. You must use the numbered reference style([1],[2]), listing the
%% references at the end of the chapter either by order of citation
%% or alphabetically.
%%
%% 2. 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:
%%
%% Go into the style file (ptexfrnt.sty) and search for the
%% \def\chapter#1 definition. At the end of this definition
%% there is a command \headcount=1. Change the 1 to
%% the appropriate number. This change will cause the headings
%% in your chapter to match the chapter number.
%%
%% 3. This macro is set up for two levels of headings. The macro will
%% automatically number the headings for you.
%%
%% 4. The running heads are defined in the output routine. It will be
%% necessary for you to alter the information currently included.
%% To do this, go into the style file and search for OUTPUT. Once there,
%% scroll through the file until you see the command \def\rhead. Replace
%% CHAPTER TITLE with the title (or shortened title) of your paper.
%% Replace AUTHORS NAMES with the appropriate names.
%% Neither running head may be longer than 50 characters.
%%
%% 5. 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. This numbering must
%% be done manually.
%%
%% 6. Figures and equations must be manually double-numbered, indicating
%% chapter and occurence. Use \leqno for equation numbering. See the
%% example of \caption for figure numbering.
%% Note. Although not shown, tables must also be double-numbered. The
%% command \caption can also be used for table captions.
%%
%% 7. At the first occurence of each new element there is a description
%% of how to use the coding.
%%
%%%%%%% PLEASE NOTE THE FOLLOWING POTENTIAL PROBLEMS:
%
%% 1. A bug exists that prevents a page number from printing on the first
%% page of the paper. Please ignore this problem. It will be handled
%% after you submit your paper.
%%
%% 2. The use of \topinsert and \midinsert to allow space for figures can
%% result in unusual page breaks, or unusual looking pages in general.
%% If you encounter such a situation, contact the SIAM office at the
%% address listed above for instructions.
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\input ptexpprt.sty
\voffset=.25in
\titlepage
% It will be necessary to hard code the chapter title and chapter authors.
% You must decide where to break the lines. For the authors, please follow
% the following conventions:
% 1. If 2 authors are on a line, use \hskip4pc between them. If 3 authors,
% use \hskip2pc. Do not put more than 3 authors on the same line.
% 2. Use the following notation: asterisk, dagger, double-dagger, section
% symbol, paragraph symbol, double asterisk. If more are needed, contact
% the SIAM office.
\centerline{\chapterfont Chapter 1}
\vskip2pt
\centerline{\titlefont SIAM/ACM Preprint Series Macros for
Plain TeX\footnote*{Supported by GSF grants ABC123, DEF456, and GHI 789.}}
\vskip15pt
\centerline{\authorfont J. Corey Gray\footnote\dag{Society for Industrial and
Applied Mathematics.}\hskip2pc Tricia Manning\footnote\ddag{Society for
Industrial and Applied Mathematics.}\hskip2pc Vickie Kearn\footnote\S{Society
for Industrial and Applied Mathematics.}}
\vskip2pc
\begindoublecolumns
% Use \headone for the first level headings. The macro will automatically
% number the headings.
\headone{Problem Specification}
In this paper, we consider the solution of the $N \times N$ linear
system
$$A x = b\leqno(1.1)$$
where $A$ is large, sparse, symmetric, and positive definite. We consider
the direct solution of 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.
\headone{Design Considerations}
Several good ordering algorithms (nested dissection and minimum degree)
are available for computing $P$ [1], [2].
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 $P$.
% Use \headtwo for second level headings. They will be numbered automatically.
\headtwo{Robustness}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.
\headtwo{Versatility} 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.
Our purpose here is to examine the nonnumerical complexity of the
sparse elimination algorithm given in [3].
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.
% Use \thm and \endthm for theorems. They must be numbered manually.
% Lemmas (\lem \endlem), corollaries (\cor \endcor), and
% propositions (\prop \endprop) are coded the same as theorems and must
% also be numbered manually.
\thm{Theorem 2.1.} 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.\endthm
Several good ordering algorithms (nested dissection and minimum degree)
are available for computing $P$ [1], [2].
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 $P$.
Several good ordering algorithms (nested dissection and minimum degree)
are available for computing $P$ [1], [2].
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 $P$.
% Use \prf to begin a proof.
\prf{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 each.
% Use \dfn to begin definitions.
\dfn{Definition 1.2.1.}We describe the two methods in \S\ 1.2. This is a
definition in the plain tex macro.
This is accomplished by exploiting the m-tree,
a particular spanning tree for the graph of the filled-in matrix.
Our purpose here is to examine the nonnumerical complexity of the
sparse elimination algorithm given in [3].
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.
Our purpose here is to examine the nonnumerical complexity of the
sparse elimination algorithm given in [3].
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.
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 $P$.
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].
Several good ordering algorithms (nested dissection and minimum degree)
are available for computing $P$ [1], [2].
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 $P$.
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
$O(n^{2} (\log n)^{2})$.
% Use \midinsert along with \caption to allow space for
% figures. See note above in problem section.
%\midinsert\vskip15.5pc\caption{Fig. 1.1. {\nineit This is figure 1.}}
% \endcaption\endinsert
In this paper, we consider the solution of the $N \times N$ linear
system
where $A$ is large, sparse, symmetric, and positive definite. We consider
the direct solution of by means of general sparse Gaussian
elimination. In such a procedure, we find a permutation matrix $P$, and
compute the decomposition
where $L$ is unit lower triangular and $D$ is diagonal.
\headone{Design Considerations}
Several good ordering algorithms (nested dissection and minimum degree)
are available for computing $P$ [1], [2].
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 $P$.
Several good ordering algorithms (nested dissection and minimum degree)
are available for computing $P$ [1], [2].
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 $P$.
Several good ordering algorithms (nested dissection and minimum degree)
are available for computing $P$ [1], [2].
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 [3].
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.
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 $P$.
% Use \lem and \endlem to begin and end lemmas.
\lem{Lemma 2.1.}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 $G_{z}^k$.
\endlem
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].
% Use \headtwo for second level headings. They will be numbered automatically.
\headone{Problem Solving}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.
\headtwo{Versatility} 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].
\headtwo{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. 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].
% The command \Refs sets the word Reference as a heading and allows the proper
% amount of space before the start of the references. Each reference must
% begin with \ref\\. The article or title of the reference should be in
% italic. Use the \it command within brackets. End each reference with
% \endref and allow two returns between references. Use the command
% \sameauthor (see reference 8) when the same author or group of authors
% is listed consecutively.
\Refs
\ref 1\\R.~E. Bank, {\it PLTMG users' guide, edition 5.0}, tech. report,
Department of Mathematics, University of California, San Diego, CA,
1988.\endref
\ref 2\\R.~E. Bank, T.~F. Dupont, and H.~Yserentant, {\it The hierarchical basis
multigrid method}, Numer. Math., 52 (1988), pp.~427--458.\endref
\ref 3\\R.~E. Bank and R.~K. Smith, {\it General sparse elimination requires no
permanent integer storage}, SIAM J. Sci. Stat. Comput., 8 (1987),
pp.~574--584.\endref
\ref 4\\S.~C. Eisenstat, M.~C. Gursky, M.~Schultz, and A.~Sherman, {\it
Algorithms and data structures for sparse symmetric gaussian elimination},
SIAM J. Sci. Stat. Comput., 2 (1982), pp.~225--237.\endref
\ref 5\\A.~George and J.~Liu, {\it Computer Solution of Large Positive
Definite Systems}, Prentice Hall, Englewood Cliffs, NJ, 1981.\endref
\ref 6\\K.~H. Law and S.~J. Fenves, {\it A node addition model for symbolic
factorization}, ACM TOMS, 12 (1986), pp.~37--50.\endref
\ref 7\\J.~W.~H. Liu, {\it A compact row storage scheme for factors
using elimination trees}, ACM TOMS, 12 (1986), pp.~127--148.\endref
\ref 8\\\sameauthor , {\it The role of
elimination trees in sparse factorization}, Tech. Report CS-87-12,Department
of Computer Science, York University, Ontario, Canada, 1987.\endref
\ref 9\\D.~J. Rose, {\it A graph theoretic study of the numeric solution of
sparse positive definite systems}, in Graph Theory and Computing,
Academic Press, New York, 1972.\endref
\ref 10\\D.~J. Rose, R.~E. Tarjan, and G.~S. Lueker, {\it Algorithmic aspects of
vertex elimination on graphs}, SIAM J. Comput., 5 (1976), pp.~226--283.\endref
\enddoublecolumns
\bye
%%
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CUT HERE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%% ptexpprt.sty %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% This is a file of macros and definitions for creating a chapter
% for publication in the ACM/SIAM Preprint Series using Plain TeX.
% 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 Plain TeX Preprint Series macro package, version 1.0,
September 24, 1990.***}
% Make the @ sign a letter for internal control sequences.
\catcode`\@=11
%
%
%
%%% DIMENSIONS %%%
\newdimen\pagewidth
\hsize=41pc
\pagewidth=\hsize
\newdimen\pageheight
\vsize=50pc
\pageheight=\vsize
\newdimen\ruleht
\ruleht=.5pt
\maxdepth=2.2pt
\parindent=18truept
\def\firstpar{\parindent=0pt\global\everypar{\parindent=18truept}}
\parskip=0pt plus 1pt
%%% FONTS %%%
\font\tenrm=cmr10
\font\tenbf=cmbx10
\font\tenit=cmti10
\font\tensmc=cmcsc10
\def\tenpoint{%
\def\rm{\tenrm}\def\bf{\tenbf}%
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\font\nineit=cmti9
\def\ninepoint{%
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\def\titlefont{\sixteenrm}
\def\chapterfont{\twelvebf}
\def\authorfont{\twelverm}
\def\rheadfont{\tenrm}
\def\smc{\tensmc}
%%% COUNTERS FOR HEADINGS %%%
\newcount\headcount
\headcount=1
\newcount\seccount
\seccount=1
\newcount\subseccount
\subseccount=1
\def\reset{\global\seccount=1}
\global\headcount=0
%%% HEADINGS %%%
\def\headone#1{\global\advance\headcount by 1
\vskip12truept\parindent=0pt{\tenpoint\bf\the\headcount
\hskip11truept #1.}\par\nobreak\firstpar\global\advance\headcount by 0
%\global\advance\seccount by 1
\reset\vskip2truept}
\def\headtwo#1{%\advance\seccount by -1%
\vskip12truept\parindent=0pt{\tenpoint\bf\the\headcount.%
\the\seccount\hskip11truept #1.}\enspace\ignorespaces\firstpar
\global\advance\headcount by 0\global\advance\seccount by 1}
% \global\advance\subseccount by 1}
%%% THEOREMS, PROOFS, DEFINITIONS, etc. %%%
\def\thm#1{{\smc
#1\enspace}
\begingroup\it\ignorespaces\firstpar}
\let\lem=\thm
\let\cor=\thm
\let\prop=\thm
\def\endthm{\endgroup}
\let\endlem=\endthm
\let\endcor=\endthm
\let\endprop=\endthm
\def\prf#1{{\it #1.}\rm\enspace\ignorespaces}
\let\rem=\prf
\let\case=\prf
\def\dfn#1{{\smc
#1\enspace}
\rm\ignorespaces}
%%% FIGURES AND CAPTIONS %%%
\def\caption#1\endcaption{\vskip18pt\ninerm\centerline{#1}\vskip18pt\tenrm}
\newinsert\topins \newif\ifp@ge \newif\if@mid
\def\topinsert{\@midfalse\p@gefalse\@ins}
\def\midinsert{\@midtrue\@ins}
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%%% REFERENCES %%%
\newdimen\refindent@
\newdimen\refhangindent@
\newbox\refbox@
\setbox\refbox@=\hbox{\ninepoint\rm\baselineskip=11pt [00]}% Default 2 digits
\refindent@=\wd\refbox@
\def\resetrefindent#1{%
\setbox\refbox@=\hbox{\ninepoint\rm\baselineskip=11pt [#1]}%
\refindent@=\wd\refbox@}
\def\Refs{%
\unskip\vskip1pc
\leftline{\noindent\tenpoint\bf References}%
\penalty10000
\vskip4pt
\penalty10000
\refhangindent@=\refindent@
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\hbox to 2em{\leaders\hrule\hfil}\vskip 0pt plus 300pt}}
\def\ref#1\\#2\endref{\leavevmode\hbox to \refindent@{\hfil[#1]}\enspace #2\par}
%%% OUTPUT %%%
\newinsert\margin
\dimen\margin=\maxdimen
\count\margin=0 \skip\margin=0pt
\def\footnote#1{\edef\@sf{\spacefactor\the\spacefactor}#1\@sf
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\def\rhead{\ifodd\pageno CHAPTER TITLE
\else AUTHORS NAMES\fi}
\def\makefootline{\ifnum\pageno>1\global\footline={\hfill}\fi
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\unvbox\footins\fi
\boxmaxdepth=\maxdepth}}
\advancepageno}
\def\setcornerrules{\hbox to \pagewidth{
\vrule width 1pc height\ruleht \hfil \vrule width 1pc}
\hbox to \pagewidth{\llap{\sevenrm(page \folio)\kern1pc}
\vrule height1pc width\ruleht depth0pt
\hfil \vrule width\ruleht depth0pt}}
\output{\onepageout{\unvbox255}}
\newbox\partialpage
\def\begindoublecolumns{\begingroup
\output={\global\setbox\partialpage=\vbox{\unvbox255\bigskip}}\eject
\output={\doublecolumnout} \hsize=20pc \vsize=101pc}
\def\enddoublecolumns{\output={\balancecolumns}\eject
\endgroup \pagegoal=\vsize}
\def\doublecolumnout{\splittopskip=\topskip \splitmaxdepth=\maxdepth
\dimen@=50pc \advance\dimen@ by-\ht\partialpage
\setbox0=\vsplit255 to\dimen@ \setbox2=\vsplit255 to\dimen@
\onepageout\pagesofar \unvbox255 \penalty\outputpenalty}
\def\pagesofar{\unvbox\partialpage
\wd0=\hsize \wd2=\hsize \hbox to\pagewidth{\box0\hfil\box2}}
\def\balancecolumns{\setbox0=\vbox{\unvbox255} \dimen@=\ht0
\advance\dimen@ by\topskip \advance\dimen@ by-\baselineskip
\divide\dimen@ by2 \splittopskip=\topskip
{\vbadness=10000 \loop \global\setbox3=\copy0
\global\setbox1=\vsplit3 to\dimen@
\ifdim\ht3>\dimen@ \global\advance\dimen@ by1pt \repeat}
\setbox0=\vbox to\dimen@{\unvbox1} \setbox2=\vbox to\dimen@{\unvbox 3}
\pagesofar}
% Turn off @ as being a letter.
%
\catcode`\@=13
% End of ptexpprt.sty
CUT HERE............