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<B> Next:</B> <A NAME="tex2html2896" HREF="node60.html">Generalized RQ factorization</A>
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<H3><A NAME="SECTION04333100000000000000">Generalized <I>QR</I> Factorization</A></H3>
<A NAME="1829"> </A>
<P>
<A NAME="1830"> </A><A NAME="1831"> </A>
The <B>generalized</B> <B><I>QR</I></B> <B>(GQR) factorization</B> of an <I>n</I>-by-<I>m</I> matrix <I>A</I> and
an <I>n</I>-by-<I>p</I> matrix <I>B</I> is given by the pair of factorizations
<BR><IMG WIDTH=353 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath13656" SRC="img156.gif"><BR>
where <I>Q</I> and <I>Z</I> are respectively <I>n</I>-by-<I>n</I> and <I>p</I>-by-<I>p</I> orthogonal
matrices
(or unitary matrices if <I>A</I> and <I>B</I> are complex).
<I>R</I> has the form
<BR><IMG WIDTH=378 HEIGHT=63 ALIGN=BOTTOM ALT="displaymath13657" SRC="img157.gif"><BR>
or
<BR><IMG WIDTH=389 HEIGHT=43 ALIGN=BOTTOM ALT="displaymath13658" SRC="img158.gif"><BR>
where <IMG WIDTH=26 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline13492" SRC="img140.gif"> is upper triangular. <I>T</I> has the form
<BR><IMG WIDTH=380 HEIGHT=43 ALIGN=BOTTOM ALT="displaymath13659" SRC="img159.gif"><BR>
or
<BR><IMG WIDTH=369 HEIGHT=63 ALIGN=BOTTOM ALT="displaymath13660" SRC="img160.gif"><BR>
where <IMG WIDTH=23 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline13706" SRC="img161.gif"> or <IMG WIDTH=23 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline13708" SRC="img162.gif"> is upper triangular.
<P>
Note that if <I>B</I> is square and nonsingular, the GQR factorization
of <I>A</I> and <I>B</I> implicitly gives the <I>QR</I> factorization of the matrix <IMG WIDTH=45 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline13718" SRC="img163.gif">:
<BR><IMG WIDTH=324 HEIGHT=21 ALIGN=BOTTOM ALT="displaymath13661" SRC="img164.gif"><BR>
without explicitly computing the matrix inverse <IMG WIDTH=30 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline13720" SRC="img165.gif"> or the product <IMG WIDTH=45 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline13718" SRC="img163.gif">.
<P>
The routine PxGGQRF computes the GQR<A NAME="1854"> </A> factorization by<A NAME="1855"> </A><A NAME="1856"> </A><A NAME="1857"> </A><A NAME="1858"> </A>
computing first the <I>QR</I> factorization of <I>A</I> and then
the <I>RQ</I> factorization of <IMG WIDTH=36 HEIGHT=31 ALIGN=MIDDLE ALT="tex2html_wrap_inline13730" SRC="img166.gif">.
The orthogonal (or unitary) matrices <I>Q</I> and <I>Z</I>
can be formed explicitly or can be used just to multiply another given matrix
in the same way as the
orthogonal (or unitary) matrix in the <I>QR</I> factorization
(see section <A HREF="node52.html#subseccomporthog">3.3.2</A>).
<P>
The GQR factorization was introduced in [<A HREF="node189.html#hammarling86">73</A>, <A HREF="node189.html#paige90">100</A>].
The implementation of the GQR factorization here follows that in [<A HREF="node189.html#lawn31">5</A>].
Further generalizations of the GQR<A NAME="1862"> </A> factorization can be found in
[<A HREF="node189.html#demoorvandooren92">36</A>].
<P>
<BR> <HR>
<P><ADDRESS>
<I>Susan Blackford <BR>
Tue May 13 09:21:01 EDT 1997</I>
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