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<H3><A NAME="SECTION03242400000000000000">
Complete Orthogonal Factorization</A>
</H3>
<P>
The <B><I>QR</I></B> factorization with column pivoting does not enable us to compute
a <I>minimum norm</I> solution to a rank-deficient linear least squares problem,
<A NAME="2737"></A>
unless <B><I>R</I><SUB>12</SUB> = 0</B>. However,
by applying further orthogonal (or unitary) transformations<A NAME="2739"></A>
from the right to the upper trapezoidal matrix
<!-- MATH
$\left( \begin{array}{cc}R_{11} & R_{12}\end{array} \right)$
-->
<IMG
WIDTH="110" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
SRC="img118.png"
ALT="$\left( \begin{array}{cc}R_{11} & R_{12}\end{array} \right)$">,
using the routine xTZRQF (or xTZRZF), <B><I>R</I><SUB>12</SUB></B> can be eliminated:
<A NAME="2746"></A><A NAME="2747"></A><A NAME="2748"></A><A NAME="2749"></A>
<A NAME="2750"></A><A NAME="2751"></A><A NAME="2752"></A><A NAME="2753"></A>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\left( \begin{array}{cc}R_{11} & R_{12}\end{array}\right) Z =
\left( \begin{array}{cc}T_{11} & 0\end{array}\right) .
\end{displaymath}
-->
<IMG
WIDTH="234" HEIGHT="37" BORDER="0"
SRC="img119.png"
ALT="\begin{displaymath}
\left( \begin{array}{cc}R_{11} & R_{12}\end{array}\right) Z =
\left( \begin{array}{cc}T_{11} & 0\end{array}\right) .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
This gives the
<B>complete orthogonal
factorization</B><A NAME="2764"></A>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A P = Q \left(
\begin{array}{cc}T_{11} & 0 \\0 & 0\end{array}
\right) Z^{T}
\end{displaymath}
-->
<IMG
WIDTH="184" HEIGHT="54" BORDER="0"
SRC="img120.png"
ALT="\begin{displaymath}
A P = Q \left(
\begin{array}{cc}T_{11} & 0 \\ 0 & 0\end{array}\right) Z^{T}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
from which the minimum norm solution<A NAME="2770"></A> can be obtained as
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
x = P Z \left( \begin{array}{c} T_{11}^{-1} \hat{c}_1 \\0\end{array}\right) .
\end{displaymath}
-->
<IMG
WIDTH="156" HEIGHT="54" BORDER="0"
SRC="img121.png"
ALT="\begin{displaymath}
x = P Z \left( \begin{array}{c} T_{11}^{-1} \hat{c}_1 \\ 0\end{array}\right) .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<P>
The matrix <B><I>Z</I></B> is not
formed explicitly, but is represented as a product of elementary
reflectors,
<A NAME="2777"></A>
<A NAME="2778"></A>
as described in section <A HREF="node128.html#secorthog">5.4</A>.
Users need not be aware of the details of this representation,
because associated routines are provided to work with <B><I>Z</I></B>:
xORMRZ<A NAME="2780"></A><A NAME="2781"></A> (or
xUNMRZ<A NAME="2782"></A><A NAME="2783"></A>) can pre- or post-multiply
a given matrix by <B><I>Z</I></B> or <B><I>Z</I><SUP><I>T</I></SUP></B>
(<B><I>Z</I><SUP><I>H</I></SUP></B> if complex).
<P>
The subroutine xTZRZF is a faster and blocked version of xTZRQF.
xTZRQF has been retained for compatibility
with Release 2.0 of LAPACK, but we omit references to this routine
in the remainder of this users' guide.
<P>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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