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
|
<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML 2.0//EN">
<!--Converted with LaTeX2HTML 96.1-h (September 30, 1996) by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds -->
<HTML>
<HEAD>
<TITLE>Singular Value Decomposition</TITLE>
<META NAME="description" CONTENT="Singular Value Decomposition">
<META NAME="keywords" CONTENT="slug">
<META NAME="resource-type" CONTENT="document">
<META NAME="distribution" CONTENT="global">
<LINK REL=STYLESHEET HREF="slug.css">
</HEAD>
<BODY LANG="EN" >
<A NAME="tex2html2745" HREF="node49.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="http://www.netlib.org/utk/icons/next_motif.gif"></A> <A NAME="tex2html2743" HREF="node46.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="http://www.netlib.org/utk/icons/up_motif.gif"></A> <A NAME="tex2html2739" HREF="node47.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="http://www.netlib.org/utk/icons/previous_motif.gif"></A> <A NAME="tex2html2747" HREF="node1.html"><IMG WIDTH=65 HEIGHT=24 ALIGN=BOTTOM ALT="contents" SRC="http://www.netlib.org/utk/icons/contents_motif.gif"></A> <A NAME="tex2html2748" HREF="node190.html"><IMG WIDTH=43 HEIGHT=24 ALIGN=BOTTOM ALT="index" SRC="http://www.netlib.org/utk/icons/index_motif.gif"></A> <BR>
<B> Next:</B> <A NAME="tex2html2746" HREF="node49.html">Generalized Symmetric Definite Eigenproblems </A>
<B>Up:</B> <A NAME="tex2html2744" HREF="node46.html">Standard Eigenvalue and Singular </A>
<B> Previous:</B> <A NAME="tex2html2740" HREF="node47.html">Symmetric Eigenproblems </A>
<BR> <P>
<H3><A NAME="SECTION04323200000000000000">Singular Value Decomposition</A></H3>
<P>
The <B>singular value decomposition (SVD)</B> of an <I>m</I>-by-<I>n</I> matrix <I>A</I> is given by
<A NAME="1077"> </A><A NAME="1078"> </A>
<BR><IMG WIDTH=443 HEIGHT=21 ALIGN=BOTTOM ALT="displaymath12804" SRC="img75.gif"><BR>
where <I>U</I> and <I>V</I> are orthogonal (unitary)
and <IMG WIDTH=11 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline12820" SRC="img76.gif"> is an <I>m</I>-by-<I>n</I> diagonal matrix with real
diagonal elements, <IMG WIDTH=13 HEIGHT=17 ALIGN=MIDDLE ALT="tex2html_wrap_inline12826" SRC="img77.gif">, such that
<BR><IMG WIDTH=353 HEIGHT=18 ALIGN=BOTTOM ALT="displaymath12805" SRC="img78.gif"><BR>
The <IMG WIDTH=13 HEIGHT=17 ALIGN=MIDDLE ALT="tex2html_wrap_inline12826" SRC="img77.gif"> are the <B>singular values</B> of <I>A</I> and the
first min(<I>m</I>,<I>n</I>) columns of <I>U</I> and <I>V</I>
are the <B>left</B> and <B>right singular vectors</B> of <I>A</I>.
<A NAME="1084"> </A><A NAME="1085"> </A>
<P>
The singular values and singular vectors satisfy
<BR><IMG WIDTH=448 HEIGHT=21 ALIGN=BOTTOM ALT="displaymath12806" SRC="img79.gif"><BR>
where <IMG WIDTH=14 HEIGHT=17 ALIGN=MIDDLE ALT="tex2html_wrap_inline12840" SRC="img80.gif"> and <IMG WIDTH=12 HEIGHT=17 ALIGN=MIDDLE ALT="tex2html_wrap_inline12842" SRC="img81.gif"> are the <I>i</I>th columns of <I>U</I> and <I>V</I>, respectively.
<P>
A single driver<A NAME="1088"> </A> routine, PxGESVD<A NAME="1089"> </A><A NAME="1090"> </A>, computes the ``economy size'' or
``thin'' singular value decomposition of a general nonsymmetric matrix
(see table <A HREF="node48.html#tabdriveseig">3.4</A>). Thus, if <I>A</I> is <I>m</I>-by-<I>n</I> with
<I>m</I>><I>n</I>, then only the first <I>n</I> columns of <I>U</I> are computed and <IMG WIDTH=11 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline12820" SRC="img76.gif"> is an
<I>n</I>-by-<I>n</I> matrix. For a detailed discussion of the ``thin'' singular
value decomposition, refer to [<A HREF="node189.html#GVL2">71</A>, p. 72,].
<P>
Currently, only PSGESVD and PDGESVD are provided.
<P>
<P><A NAME="1094"> </A><A NAME="tabdriveseig"> </A><IMG WIDTH=703 HEIGHT=111 ALIGN=BOTTOM ALT="table1093" SRC="img82.gif"><BR>
<STRONG>Table 3.4:</STRONG> Driver routines for standard eigenvalue and singular value problems<BR>
<P><BR> <HR>
<P><ADDRESS>
<I>Susan Blackford <BR>
Tue May 13 09:21:01 EDT 1997</I>
</ADDRESS>
</BODY>
</HTML>
|
