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<H3><A NAME="SECTION04332500000000000000">Other Factorizations</A></H3>
<P>
The <B><I>QL</I></B> and <B><I>RQ</I></B> <B>factorizations</B>
<A NAME="1723"> </A><A NAME="1724"> </A><A NAME="1725"> </A><A NAME="1726"> </A> are given by
<BR><IMG WIDTH=351 HEIGHT=48 ALIGN=BOTTOM ALT="displaymath13582" SRC="img153.gif"><BR>
and
<BR><IMG WIDTH=360 HEIGHT=29 ALIGN=BOTTOM ALT="displaymath13583" SRC="img154.gif"><BR>
These factorizations are computed by PxGEQLF and PxGERQF, respectively; they
are<A NAME="1735"> </A><A NAME="1736"> </A><A NAME="1737"> </A><A NAME="1738"> </A><A NAME="1739"> </A><A NAME="1740"> </A><A NAME="1741"> </A><A NAME="1742"> </A>
less commonly used than either the <I>QR</I> or <I>LQ</I> factorizations
described above, but have applications in, for example, the
computation of generalized <I>QR</I> factorizations [<A HREF="node189.html#lawn31">5</A>].
<A NAME="1744"> </A><A NAME="1745"> </A><A NAME="1746"> </A>
<P>
All the factorization routines discussed here (except PxTZRZF)
allow
arbitrary <I>m</I> and <I>n</I>, so that in some cases the matrices <I>R</I> or <I>L</I> are
trapezoidal rather than triangular.
A routine that performs pivoting is provided only for the <I>QR</I> factorization.
<P>
<P><A NAME="1748"> </A><A NAME="tabcompof"> </A><IMG WIDTH=755 HEIGHT=373 ALIGN=BOTTOM ALT="table1747" SRC="img155.gif"><BR>
<STRONG>Table 3.7:</STRONG> Computational routines for orthogonal factorizations<BR>
<P><BR> <HR>
<P><ADDRESS>
<I>Susan Blackford <BR>
Tue May 13 09:21:01 EDT 1997</I>
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