summaryrefslogtreecommitdiff
path: root/texmf-dist/doc/latex/siam/ptexproc.tex
blob: b25d65601db616ee5a0d029c99b9b7eec8eeaffd (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
%%%%%%%%%%%%%%%%%%%%%%%%%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