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<H3><A NAME="SECTION03242200000000000000">
<B><I>LQ</I></B> Factorization</A>
</H3>
<P>
The <B><I>LQ</I></B> <B>factorization</B><A NAME="2629"></A>
is given by
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A = \left( \begin{array}{cc} L & 0 \end{array}\right) Q
= \left( \begin{array}{cc} L & 0 \end{array}\right)
\left( \begin{array}{c} Q_1 \\Q_2 \end{array} \right)
= L Q_1, \quad \mbox{if $ m \le n$,}
\end{displaymath}
-->
<IMG
WIDTH="435" HEIGHT="54" BORDER="0"
SRC="img108.png"
ALT="\begin{displaymath}
A = \left( \begin{array}{cc} L & 0 \end{array}\right) Q
= \...
... Q_2 \end{array} \right)
= L Q_1, \quad \mbox{if $ m \le n$,}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <B><I>L</I></B> is <B><I>m</I></B>-by-<B><I>m</I></B> lower triangular, <B><I>Q</I></B> is <B><I>n</I></B>-by-<B><I>n</I></B>
orthogonal (or unitary), <B><I>Q</I><SUB>1</SUB></B> consists of the first <B><I>m</I></B> rows of <B><I>Q</I></B>,
and <B><I>Q</I><SUB>2</SUB></B> the remaining <B><I>n</I>-<I>m</I></B> rows.
<P>
This factorization is computed by the routine xGELQF, and again <B><I>Q</I></B> is
<A NAME="2640"></A><A NAME="2641"></A><A NAME="2642"></A><A NAME="2643"></A>
represented as a product of elementary reflectors; xORGLQ
<A NAME="2644"></A><A NAME="2645"></A>
<A NAME="2646"></A>
(or xUNGLQ<A NAME="2647"></A><A NAME="2648"></A> in the complex case) can generate
all or part of <B><I>Q</I></B>, and xORMLQ<A NAME="2649"></A><A NAME="2650"></A> (or xUNMLQ
<A NAME="2651"></A><A NAME="2652"></A>) can pre- or post-multiply a given
matrix
by <B><I>Q</I></B> or <B><I>Q</I><SUP><I>T</I></SUP></B> (<B><I>Q</I><SUP><I>H</I></SUP></B> if <B><I>Q</I></B> is complex).
<P>
The <B><I>LQ</I></B> factorization of <B><I>A</I></B> is essentially the same as the <B><I>QR</I></B> factorization
of <B><I>A</I><SUP><I>T</I></SUP></B> (<B><I>A</I><SUP><I>H</I></SUP></B> if <B><I>A</I></B> is complex), since
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A = \left( \begin{array}{cc} L & 0 \end{array}\right) Q
\quad \Longleftrightarrow
\quad
A^T = Q^T \left( \begin{array}{c} L^T \\0\end{array}\right) .
\end{displaymath}
-->
<IMG
WIDTH="346" HEIGHT="54" BORDER="0"
SRC="img109.png"
ALT="\begin{displaymath}
A = \left( \begin{array}{cc} L & 0 \end{array}\right) Q
\qua...
...A^T = Q^T \left( \begin{array}{c} L^T \\ 0\end{array}\right) .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<P>
The <B><I>LQ</I></B> factorization may be used to find a minimum norm solution<A NAME="2659"></A> of
an underdetermined<A NAME="2660"></A> system of linear equations <B><I>A x</I> = <I>b</I></B> where <B><I>A</I></B> is
<B><I>m</I></B>-by-<B><I>n</I></B> with <B><I>m</I> < <I>n</I></B> and has rank <B><I>m</I></B>. The solution is given by
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
x = Q^T \left( \begin{array}{c} L^{-1} b \\0 \end{array} \right)
\end{displaymath}
-->
<IMG
WIDTH="135" HEIGHT="54" BORDER="0"
SRC="img110.png"
ALT="\begin{displaymath}
x = Q^T \left( \begin{array}{c} L^{-1} b \\ 0 \end{array} \right)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
and may be computed by calls to xTRTRS and xORMLQ.
<A NAME="2665"></A><A NAME="2666"></A><A NAME="2667"></A><A NAME="2668"></A>
<A NAME="2669"></A><A NAME="2670"></A>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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