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<H3><A NAME="SECTION04333200000000000000">Generalized <I>RQ</I> factorization</A></H3>
<P>
<A NAME="1865"> </A><A NAME="1866"> </A><A NAME="1867"> </A>
The <B>generalized</B> <B><I>RQ</I></B> <B>(GRQ) factorization</B> of an <I>m</I>-by-<I>n</I> matrix <I>A</I> and
a <I>p</I>-by-<I>n</I> matrix <I>B</I> is given by the pair of factorizations
<BR><IMG WIDTH=354 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath13747" SRC="img167.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=391 HEIGHT=43 ALIGN=BOTTOM ALT="displaymath13748" SRC="img168.gif"><BR>
or
<BR><IMG WIDTH=378 HEIGHT=63 ALIGN=BOTTOM ALT="displaymath13749" SRC="img169.gif"><BR>
where <IMG WIDTH=26 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline13555" SRC="img147.gif"> or <IMG WIDTH=26 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline13795" SRC="img170.gif"> is upper triangular. <I>T</I> has the form
<BR><IMG WIDTH=369 HEIGHT=63 ALIGN=BOTTOM ALT="displaymath13750" SRC="img171.gif"><BR>
or
<BR><IMG WIDTH=382 HEIGHT=43 ALIGN=BOTTOM ALT="displaymath13751" SRC="img172.gif"><BR>
where <IMG WIDTH=23 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline13799" SRC="img173.gif"> is upper triangular.
<P>
Note that if <I>B</I> is square and nonsingular, the GRQ factorization of
<I>A</I> and <I>B</I> implicitly gives the <I>RQ</I> factorization of the matrix <IMG WIDTH=44 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline13809" SRC="img174.gif">:
<BR><IMG WIDTH=324 HEIGHT=21 ALIGN=BOTTOM ALT="displaymath13752" SRC="img175.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=44 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline13809" SRC="img174.gif">.
<P>
The routine PxGGRQF computes the GRQ factorization<A NAME="1890"> </A><A NAME="1891"> </A><A NAME="1892"> </A><A NAME="1893"> </A><A NAME="1894"> </A><A NAME="1895"> </A>
by computing first the <I>RQ</I> factorization of <I>A</I> and then
the <I>QR</I> factorization of <IMG WIDTH=38 HEIGHT=31 ALIGN=MIDDLE ALT="tex2html_wrap_inline13821" SRC="img176.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>RQ</I> factorization
(see section <A HREF="node52.html#subseccomporthog">3.3.2</A>).
<P>
<BR> <HR>
<P><ADDRESS>
<I>Susan Blackford <BR>
Tue May 13 09:21:01 EDT 1997</I>
</ADDRESS>
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