-
Notifications
You must be signed in to change notification settings - Fork 3
/
givens-parallel.html
185 lines (172 loc) · 9.56 KB
/
givens-parallel.html
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
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
<!--Converted with LaTeX2HTML 99.2beta6 (1.42)
original version by: Nikos Drakos, CBLU, University of Leeds
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
* with significant contributions from:
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
<html><head>
<title>6.1.3 Givens rotation on a parallel computer</title>
<meta name="description" content="6.1.3 Givens rotation on a parallel computer">
<meta name="keywords" content="parvor">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta http-equiv="Content-Type" content="text/html; charset=windows-1252">
<meta name="Generator" content="LaTeX2HTML v99.2beta6">
<meta http-equiv="Content-Style-Type" content="text/css">
<link rel="STYLESHEET" href="givens-parallel_files/parvor.css">
<link rel="previous" href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node117.html">
<link rel="up" href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node115.html">
<link rel="next" href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node119.html">
</head>
<body>
<!--Navigation Panel-->
<a name="tex2html1967" href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node119.html">
<img alt="next" src="givens-parallel_files/next.html" width="37" height="24" border="0" align="BOTTOM"></a>
<a name="tex2html1963" href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node115.html">
<img alt="up" src="givens-parallel_files/up.html" width="26" height="24" border="0" align="BOTTOM"></a>
<a name="tex2html1959" href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node117.html">
<img alt="previous" src="givens-parallel_files/prev.html" width="63" height="24" border="0" align="BOTTOM"></a>
<a name="tex2html1965" href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node1.html">
<img alt="contents" src="givens-parallel_files/contents.html" width="65" height="24" border="0" align="BOTTOM"></a>
<br>
<b> Next:</b> <a name="tex2html1968" href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node119.html">6.2 LU factorization</a>
<b> Up:</b> <a name="tex2html1964" href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node115.html">6.1 Elimination by rotation</a>
<b> Previous:</b> <a name="tex2html1960" href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node117.html">6.1.2 Givens rotation on</a>
<b> <a name="tex2html1966" href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node1.html">Contents</a></b>
<br>
<br>
<!--End of Navigation Panel-->
<h2><a name="SECTION00813000000000000000"></a>
<a name="kap6.1.3"></a>
<br>
6.1.3 Givens rotation on a parallel computer
</h2>
Denote by <img src="givens-parallel_files/img16.gif" alt="$ g(k,s)$" width="54" height="35" border="0" align="MIDDLE"> the Givens rotation of rows <img src="givens-parallel_files/img17.gif" alt="$ k-1$" width="45" height="32" border="0" align="MIDDLE"> and <img src="givens-parallel_files/img18.gif" alt="$ k$" width="13" height="16" border="0" align="BOTTOM"> in column <img src="givens-parallel_files/img19.gif" alt="$ s$" width="11" height="16" border="0" align="BOTTOM"> .
<br>
<img src="givens-parallel_files/img733.gif" alt="\begin{algorithmus}
% latex2html id marker 26285
\caption {$g(k,s)$\ - Givens ro...
... \quad\qquad\quad a_{k,j} \,:=\, c a_{k,j} + h
\end{eqnarray*}\end{algorithmus}" width="424" height="203" align="MIDDLE">
<br>
The following data dependencies occur :
<p></p>
<div align="CENTER"><a name="fig6.1"></a><a name="26517"></a>
<table>
<caption align="BOTTOM"><strong>Figure 6.1:</strong>
Static data dependencies in the Givens rotation</caption>
<tbody><tr><td><img src="givens-parallel_files/img734.gif" alt="\begin{figure}\unitlength0.09\textwidth
\mbox{}\hfill
\begin{picture}(8,5)(0,0....
...e n$}}
\put(7.4,2.8){\line(4,-1){1}}
%
\end{picture}\hfill\mbox{}
\end{figure}" width="310" height="165" align="MIDDLE"></td></tr>
</tbody></table>
</div><p></p>
<br>
We transform the static data dependency in Fig. <a href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node118.html#fig6.1">6.1</a> by means
of markers <img src="givens-parallel_files/img190.gif" alt="$ \ast$" width="12" height="16" border="0" align="BOTTOM"> into a <i>dynamic data flow chart</i>
with an arbitrary number of processors.
<br>
<br>
Let <img src="givens-parallel_files/img735.gif" alt="$ m(k,s)$" width="61" height="35" border="0" align="MIDDLE"> the number of markers for process <img src="givens-parallel_files/img16.gif" alt="$ g(k,s)$" width="54" height="35" border="0" align="MIDDLE">.
<br>
<br><!-- MATH
$m(k,s) = 2$
-->
<img src="givens-parallel_files/img736.gif" alt="$ m(k,s) = 2$" width="96" height="35" border="0" align="MIDDLE"> <!-- MATH
$\Longrightarrow$
-->
<img src="givens-parallel_files/img34.gif" alt="$ \Longrightarrow$" width="33" height="16" border="0" align="BOTTOM"> process <img src="givens-parallel_files/img16.gif" alt="$ g(k,s)$" width="54" height="35" border="0" align="MIDDLE"> starts.
<br>
<br>
Initial values :
<table cellpadding="3">
<tbody><tr><td align="LEFT"><!-- MATH
$m(n,1) := 2$
-->
<img src="givens-parallel_files/img737.gif" alt="$ m(n,1) := 2$" width="103" height="35" border="0" align="MIDDLE"></td>
<td align="LEFT"> </td><td align="LEFT"> </td>
</tr>
<tr><td align="LEFT"><!-- MATH
$m(k,1) := 1$
-->
<img src="givens-parallel_files/img738.gif" alt="$ m(k,1) := 1$" width="101" height="35" border="0" align="MIDDLE"></td>
<td align="LEFT"> </td><td align="LEFT"><img src="givens-parallel_files/img739.gif" alt="$ k<n$" width="50" height="32" border="0" align="MIDDLE"></td>
</tr>
<tr><td align="LEFT"><!-- MATH
$m(k,s) := 0$
-->
<img src="givens-parallel_files/img740.gif" alt="$ m(k,s) := 0$" width="101" height="35" border="0" align="MIDDLE"></td>
<td align="LEFT"> </td><td align="LEFT"><img src="givens-parallel_files/img741.gif" alt="$ s>1$" width="45" height="30" border="0" align="MIDDLE"></td>
</tr>
</tbody></table>
<br>
<br>
On a <i>shared memory</i> computer, we achieve the flow chart in Fig. <a href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node118.html#fig6.2">6.2</a>
<p></p>
<div align="CENTER"><a name="fig6.2"></a><a name="26558"></a>
<table>
<caption align="BOTTOM"><strong>Figure 6.2:</strong>
Flow chart on a shared memory computer : Givens rotation</caption>
<tbody><tr><td><img src="givens-parallel_files/img734.gif" alt="\begin{figure}\unitlength0.09\textwidth
\mbox{}\hfill
\begin{picture}(8,5)(0,0....
...e n$}}
\put(7.4,2.8){\line(4,-1){1}}
%
\end{picture}\hfill\mbox{}
\end{figure}" width="310" height="165" align="MIDDLE"></td></tr>
</tbody></table>
</div><p></p>
<br>
The markers <img src="givens-parallel_files/img742.gif" alt="$ m(k-1,s)$" width="93" height="35" border="0" align="MIDDLE"> and <!-- MATH
$m(k+1,s+1)$
-->
<img src="givens-parallel_files/img743.gif" alt="$ m(k+1,s+1)$" width="125" height="35" border="0" align="MIDDLE"> are incremented after
process <img src="givens-parallel_files/img16.gif" alt="$ g(k,s)$" width="54" height="35" border="0" align="MIDDLE"> has finished.
<br>
<br>
If the processors have only access on a part of the <i>distributed memory</i>
then we have to investigate the data dependencies between the processors.
Here, a rowise distribution of blocks of matrix <img src="givens-parallel_files/img303.gif" alt="$ A$" width="17" height="17" border="0" align="BOTTOM"> is preferred, see Fig. <a href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node116.html#form6.3">6.3</a>.
<p></p>
<div align="CENTER"><a name="fig6.3"></a><a name="26604"></a>
<table>
<caption align="BOTTOM"><strong>Figure 6.3:</strong>
Flow chart on a distributed memory computer: Givens rotation</caption>
<tbody><tr><td><img src="givens-parallel_files/img744.gif" alt="\begin{figure}\unitlength0.09\textwidth
\mbox{}\hfill
\begin{picture}(8,5)(0,0....
...0){3}} \put(5.5,0.5){\line(0,-1){0.5}}
%
\end{picture}\hfill\mbox{}
\end{figure}" width="315" height="196" align="MIDDLE"></td></tr>
</tbody></table>
</div><p></p>
<br><i>Remark :</i> A similar distribution of rows can also be uses in the
Gauß elimination.
<br>
<hr>
<!--Navigation Panel-->
<a name="tex2html1967" href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node119.html">
<img alt="next" src="givens-parallel_files/next.html" width="37" height="24" border="0" align="BOTTOM"></a>
<a name="tex2html1963" href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node115.html">
<img alt="up" src="givens-parallel_files/up.html" width="26" height="24" border="0" align="BOTTOM"></a>
<a name="tex2html1959" href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node117.html">
<img alt="previous" src="givens-parallel_files/prev.html" width="63" height="24" border="0" align="BOTTOM"></a>
<a name="tex2html1965" href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node1.html">
<img alt="contents" src="givens-parallel_files/contents.html" width="65" height="24" border="0" align="BOTTOM"></a>
<br>
<b> Next:</b> <a name="tex2html1968" href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node119.html">6.2 LU factorization</a>
<b> Up:</b> <a name="tex2html1964" href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node115.html">6.1 Elimination by rotation</a>
<b> Previous:</b> <a name="tex2html1960" href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node117.html">6.1.2 Givens rotation on</a>
<b> <a name="tex2html1966" href="http://www.numa.uni-linz.ac.at/Staff/haase/parvor_e/node1.html">Contents</a></b>
<!--End of Navigation Panel-->
<address>
Gundolf Haase
2000-03-20
</address>
</body></html>