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<H3><A NAME="SECTION04332200000000000000"><I>LQ</I> Factorization</A></H3>
<P>
The <B><I>LQ</I></B> <B>factorization</B><A NAME="1566"> </A><A NAME="1567"> </A>
is given by
<BR><IMG WIDTH=467 HEIGHT=48 ALIGN=BOTTOM ALT="displaymath13356" SRC="img132.gif"><BR>
where <I>L</I> is <I>m</I>-by-<I>m</I> lower triangular, <I>Q</I> is <I>n</I>-by-<I>n</I>
orthogonal (or unitary), <IMG WIDTH=19 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline13270" SRC="img118.gif"> consists of the first <I>m</I> rows of <I>Q</I>,
and <IMG WIDTH=19 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline13276" SRC="img119.gif"> consists of the remaining <I>n</I>-<I>m</I> rows.
<P>
This factorization is computed by the routine PxGELQF, and again <I>Q</I> is
<A NAME="1578"> </A><A NAME="1579"> </A><A NAME="1580"> </A><A NAME="1581"> </A>
represented as a product of elementary reflectors; PxORGLQ
<A NAME="1582"> </A><A NAME="1583"> </A>
<A NAME="1584"> </A>
(or PxUNGLQ<A NAME="1585"> </A><A NAME="1586"> </A> in the complex case) can generate
all or part of <I>Q</I>, and PxORMLQ<A NAME="1587"> </A><A NAME="1588"> </A> (or PxUNMLQ<A NAME="1589"> </A><A NAME="1590"> </A>) can pre- or post-multiply a given
matrix
by <I>Q</I> or <IMG WIDTH=23 HEIGHT=31 ALIGN=MIDDLE ALT="tex2html_wrap_inline13300" SRC="img123.gif"> (<IMG WIDTH=25 HEIGHT=31 ALIGN=MIDDLE ALT="tex2html_wrap_inline13302" SRC="img124.gif"> if <I>Q</I> is complex).
<P>
The <I>LQ</I> factorization of <I>A</I> is essentially the same as the <I>QR</I> factorization
of <IMG WIDTH=23 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline12722" SRC="img63.gif"> (<IMG WIDTH=25 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline12724" SRC="img64.gif"> if <I>A</I> is complex), since
<BR><IMG WIDTH=423 HEIGHT=48 ALIGN=BOTTOM ALT="displaymath13357" SRC="img133.gif"><BR>
<P>
The <I>LQ</I> factorization may be used to find a minimum norm solution<A NAME="1597"> </A> of
an underdetermined<A NAME="1598"> </A><A NAME="1599"> </A> system of linear equations <I>A x</I> = <I>b</I>, where <I>A</I> is
<I>m</I>-by-<I>n</I> with <I>m</I> < <I>n</I> and has rank <I>m</I>. The solution is given by
<BR><IMG WIDTH=317 HEIGHT=48 ALIGN=BOTTOM ALT="displaymath13358" SRC="img134.gif"><BR>
and may be computed by calls to PxTRTRS and PxORMLQ.
<A NAME="1604"> </A><A NAME="1605"> </A><A NAME="1606"> </A><A NAME="1607"> </A>
<A NAME="1608"> </A><A NAME="1609"> </A>
<P>
<BR> <HR>
<P><ADDRESS>
<I>Susan Blackford <BR>
Tue May 13 09:21:01 EDT 1997</I>
</ADDRESS>
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