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
|
<!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>Complete Orthogonal Factorization</TITLE>
<META NAME="description" CONTENT="Complete Orthogonal Factorization">
<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="tex2html2859" HREF="node57.html"><IMG WIDTH=37 HEIGHT=24 ALIGN=BOTTOM ALT="next" SRC="http://www.netlib.org/utk/icons/next_motif.gif"></A> <A NAME="tex2html2857" HREF="node52.html"><IMG WIDTH=26 HEIGHT=24 ALIGN=BOTTOM ALT="up" SRC="http://www.netlib.org/utk/icons/up_motif.gif"></A> <A NAME="tex2html2851" HREF="node55.html"><IMG WIDTH=63 HEIGHT=24 ALIGN=BOTTOM ALT="previous" SRC="http://www.netlib.org/utk/icons/previous_motif.gif"></A> <A NAME="tex2html2861" 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="tex2html2862" 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="tex2html2860" HREF="node57.html">Other Factorizations</A>
<B>Up:</B> <A NAME="tex2html2858" HREF="node52.html">Orthogonal Factorizations and Linear </A>
<B> Previous:</B> <A NAME="tex2html2852" HREF="node55.html">QR Factorization with Column </A>
<BR> <P>
<H3><A NAME="SECTION04332400000000000000">Complete Orthogonal Factorization</A></H3>
<P>
The <I>QR</I> factorization with column pivoting does not enable us to compute
a <I>minimum norm</I> solution to a rank-deficient linear least squares problem
<A NAME="1675"> </A>
unless <IMG WIDTH=59 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline13551" SRC="img145.gif">. However,
by applying further orthogonal (or unitary) transformations<A NAME="1677"> </A>
from the right to the upper trapezoidal matrix
<IMG WIDTH=100 HEIGHT=38 ALIGN=MIDDLE ALT="tex2html_wrap_inline13553" SRC="img146.gif">,
using the routine PxTZRZF, <IMG WIDTH=26 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline13555" SRC="img147.gif"> can be eliminated:
<A NAME="1684"> </A><A NAME="1685"> </A><A NAME="1686"> </A><A NAME="1687"> </A>
<BR><IMG WIDTH=366 HEIGHT=29 ALIGN=BOTTOM ALT="displaymath13543" SRC="img148.gif"><BR>
This gives the
<B>complete orthogonal
factorization</B><A NAME="1698"> </A><A NAME="1699"> </A>
<BR><IMG WIDTH=341 HEIGHT=48 ALIGN=BOTTOM ALT="displaymath13544" SRC="img149.gif"><BR>
from which the minimum norm solution<A NAME="1705"> </A> can be obtained as
<BR><IMG WIDTH=328 HEIGHT=48 ALIGN=BOTTOM ALT="displaymath13545" SRC="img150.gif"><BR>
<P>
The matrix <I>Z</I> is not
formed explicitly but is represented as a product of elementary
reflectors,
<A NAME="1712"> </A>
<A NAME="1713"> </A>
as described in section <A HREF="node66.html#secorthog">3.4</A>.
Users need not be aware of the details of this representation,
because associated routines are provided to work with <I>Z</I>:
PxORMRZ<A NAME="1715"> </A><A NAME="1716"> </A> (or
PxUNMRZ<A NAME="1717"> </A><A NAME="1718"> </A>) can pre- or post-multiply
a given matrix by <I>Z</I> or <IMG WIDTH=22 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline13563" SRC="img151.gif">
(<IMG WIDTH=24 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline13565" SRC="img152.gif"> if complex).
<P>
<BR> <HR>
<P><ADDRESS>
<I>Susan Blackford <BR>
Tue May 13 09:21:01 EDT 1997</I>
</ADDRESS>
</BODY>
</HTML>
|
