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+%%%%%%%%%%%%%%%%%%%%%%%%%CUT HERE%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+% This is ptexproc.tex, an example file for use with the SIAM Plain TeX
+% Proceedings Series macros. 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. Please ignore the underfull \vbox on page 3.
+
+% 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. This macro is set up for three levels of headings. Use the commands
+%% \headone, \headtwo, and \headthree. The macro will automatically
+%% number the headings.
+%%
+%% 3. Theorems, Lemmas, Definitions, etc. are to be double-numbered,
+%% indicating the section and the occurrence of that element
+%% within that section. (For example, the first theorem in the second
+%% section would be numbered 2.1.) This numbering must
+%% be done manually.
+%%
+%% 4. Proofs are handled by \prf\endprf. If you want to use an end-of-proof
+%% box, insert \qed right before the \endprf command.
+%%
+%% 5. Figures and equations must be manually single-numbered. Use \leqno
+%% for equation numbering. The macro provides the \fig for including
+%% figures. This command consists of three fields. The first field is
+%% used for inserting the appropriate space for the figure. The second
+%% field is the figure number. The third field is the caption. See the
+%% example included in this file. SIAM supports the use of psfig for
+%% including Postscript figures. All Postscript figures should be sent
+%% as separate files. A hardcopy version of all Postscript figures is
+%% also required. See note regarding this under How to Submit Your Paper.
+%%
+%% 6. Use of \title\endtitle and \lasttitle\endlastitle.
+%% This macro package provides two possible commands for handling the
+%% title of your paper. The commands \title\endtitle should be used for
+%% all lines except the last line of multiple line titles. The commands
+%% \lasttitle\endlasttitle should be used for the last line of multiple
+%% line titles. In the case of a single line title, \lasttitle\endlasttitle
+%% should be used.
+%%
+%% 7. Use of \author\endauthor and \lastauthor\endlastauthor.
+%% As in the title macro, two possible commands are provided for the
+%% author. The commands \author\endauthor should be used for the first
+%% line of authors if there are more than one line. The commands
+%% \lastauthor\endlastauthor should be used for the last line
+%% of multiple lines of authors. In the case of a single line of authors,
+%% \lastauthor\endlastauthor should be used. A maximum of four authors
+%% should be placed on any one line. The appropriate space must also be
+%% hard coded between authors on the same line. The spacing is as follows:
+%%
+%% If 2 authors; \hskip4pc between
+%% If 3 authors; \hskip3pc between
+%% If 4 authors; \hskip2pc between
+%%
+%% 8. Grant information and author affiliations.
+%% This information is included by using the \footnote command and the
+%% appropriate footnote symbol. SIAM uses footnote symbols in a
+%% particular order. Below is a list of these symbols:
+%%
+%% asterisk
+%% single-dagger
+%% double-dagger
+%% section sign
+%% paragraph
+%% parallel
+%% double asterisk
+%% double single-dagger
+%% double double-dagger
+%%
+%% For illustrative purposes, all footnote symbols have been used in the
+%% example file.
+%%
+%% A note regarding \footnote. This command seems to leave extra white
+%% space between footnotes. This is quite evident in the example file. If
+%% any user of these macros has a solution to this problem, I would
+%% appreciate hearing from you. Send your comments to: gray@siam.org.
+%%
+%% The following general rules apply for including grants and affiliations:
+%% a) If there is a single grant for the paper, then the grant
+%% information should be footnoted to the title.
+%% b) If there is more than one grant, included the grant information
+%% with each authors affiliation.
+%% c) If there are different grants for the paper but the authors share
+%% the same affiliation, footnote the grant information to the title.
+%% For example, The work of the first author was supported by xyz.
+%% The work of the second author was supported by abc. And so on.
+%%
+%%
+%% 9. Special fonts.
+%% SIAM supports the use of AMS-TeX fonts version 2.0 and later. As
+%% described in the manual for these fonts, they can be included by
+%% \input{amssym.def} and \input{amssym.tex}.
+%%
+%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\input ptexproc.sty
+
+\def\leftrh{}
+\def\rightrh{}
+
+\startchapter %Place this command at the beginning of the file immediately
+ %after the \input command.
+
+
+
+\title SIAM Proceedings Series Macros\endtitle
+\lasttitle for Use with Plain TeX\footnote*{Any information
+regarding grants should be placed here.}\endlasttitle
+
+\author J. Corey Gray\footnote{$^{\dag}$}{Production Manager, Society for
+Industrial and Applied Mathematics, Philadelphia, PA.}
+\hskip2truepc Tricia Manning\footnote{$^{\ddag}$}{Publications
+Specialist, Society for Industrial and Applied Mathematics, Philadelphia,
+PA.}
+\hskip2truepc Vickie Kearn\footnote{$^{\S}$}{Publisher, Society for Industrial
+and Applied Mathematics, Philadelphia, PA.}
+\hskip2truepc Nancy Abbott\footnote{$^{\P}$}{Design Supervisor, Society for
+Industrial and Applied Mathematics, Philadelphia, PA.}
+\endauthor
+\lastauthor Sue Ciambrano\footnote{$^{\parallel}$}{Acquisition Editor, Society for
+Industrial and Applied Mathematics, Philadelphia, PA.}
+\hskip2truepc Paul Duggan\footnote{$^{**}$}{Composition
+Specialist, Society for Industrial and Applied Mathematics, Philadelphia,
+PA.}
+\hskip2truepc Robbi Anne Albert\footnote{$^{\dagger\dagger}$}{Production,
+Assistant, Society for Industrial
+and Applied Mathematics, Philadelphia, PA.}
+\hskip2truepc Jean Anderson\footnote{$^{\ddag\ddag}$}{Composition Coordinator, Society for
+Industrial and Applied Mathematics, Philadelphia, PA.}
+\endlastauthor
+
+% Use \headone for the first level headings. The macro will automatically
+% number the headings.
+
+\centerline{\bf Abstract}
+\abstract An equivalence is shown between realizability of input/output (i/o) operators by
+rational control systems and high-order algebraic differential equations for
+i/o pairs. This generalizes, to nonlinear systems, the equivalence
+between autoregressive representations and finite dimensional linear
+realizability.\endabstract
+
+
+\headone{Problem Specification}
+In this paper, we consider the solution of the $N \times N$ linear
+system
+$$A x = b\leqno (1)$$
+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
+$$
+P A P^{t} = L D L^{t}
+\leqno (2)$$
+
+\noindent 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 \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 and \endprf to begin and end a proof.
+% The use of \qed will produce an end-of-proof box.
+
+\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.\qed\endprf
+
+% Use \dfn and \enddfn to begin and end definitions.
+
+\dfn{Definition 2.1.}We describe the two methods in \S\ 1.2. This is a
+definition in the plain tex macro.\enddfn
+
+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 $P$.
+
+% 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
+
+% Use \fig to insert space for figures .
+
+\fig{10pc}{Fig. 1}{This is the caption for figure one.}
+
+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.
+
+\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].
+
+
+% Use \headthree for third level headings.
+
+\headthree{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
+$O(n^{2} (\log n)^{2})$.
+
+% 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 Sparse 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 cholesky 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
+
+
+\bye
+
+%end of example file