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% This is proc2e.tex, an example file for use with the SIAM LaTeX2e
% Proceedings Series macros.
% Please take the time to read the following comments, as they describe
% how to use these macros. This file can be composed and printed out for
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%
% Corey Gray
% SIAM
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%% included for use with BiBTeX.
%%
%% 3. This macro is set up for three levels of headings (\section,
%% \subsection, and \subsubsection). The macro will automatically number
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%% 4. Theorems, Lemmas, Definitions, etc. are to be double-numbered,
%% indicating the section and the occurrence of that element
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%% section would be numbered 2.1. The macro will
%% automatically do the numbering for you.
%%
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%% If you wish to use an end of proof box, use \qed preceding \end{proof}.
%% The example uses one. It is not required.
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\documentclass[twoside,leqno,11pt]{article}
\usepackage{proc2e}
\begin{document}
\cleardoublepage
\pagestyle{headings}
\title{SIAM Proceedings Series Macros
for Use With LaTeX\thanks{Any information regarding grants should be placed
here.}}
\author{J. Corey Gray\thanks{Production Manager, Society for Industrial and Applied
Mathematics, Philadelphia, PA.}
\and
Tricia Manning\thanks{Publications Specialist, Society for Industrial and Applied
Mathematics, Philadelphia, PA.}
\and
Vickie Kearn\thanks{Publisher, Society for Industrial and Applied Mathematics,
Philadelphia, PA.}\\
\and
Nancy Abbott\thanks{Design Supervisor, Society for Industrial and Applied
Mathematics, Philadelphia, PA}
\and
Sue Ciambrano\thanks{Acquisitions Editor, Society for Industrial and Applied
Mathematics, Philadelphia, PA}
\and
Paul Duggan\thanks{Composition Specialist, Society for Industrial and Applied
Mathematics, Philadelphia, PA}
\and
Robbi Anne Albert\thanks{Production Assistant, Society for Industrial and Applied
Mathematics, Philadelphia, PA}
\and
Jean Anderson\thanks{Composition Coordinator, Society for Industrial and Applied
Mathematics, Philadelphia, PA}
}
\date{}
\maketitle
\pagenumbering{arabic}
\begin{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. \end{abstract}
\section{Problem Specification}In this paper, we consider the solution of the $N \times
N$ linear
system
\begin{equation} \label{e1.1}
\cos \sin 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}
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.\qed
\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}
\begin{figure}
\vspace*{24pc}
\caption{This is figure 1.}
\end{figure}
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}
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}
\subsubsection{Versatility.} We do not
attempt to present an overview
here, but rather attempt to focus on those results that
are relevant to our particular algorithm.
\section{Conclusions} 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
$O(n^{2} (\log n)^{2})$.
\begin{thebibliography}{99}
\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.~Schreiber, {\em A new implementation of sparse gaussian elimination},
ACM TOMS, 8 (1982), pp.~256--276.
\end{thebibliography}
\end{document}