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<H2><A NAME="SECTION03233000000000000000"></A><A NAME="subsecdrivegllsq"></A>
<BR>
Generalized Linear Least Squares (LSE and GLM) Problems
</H2>
<P>
Driver routines are provided for two types of generalized linear least squares
problems.<A NAME="1442"></A>
<A NAME="1443"></A>
<A NAME="1444"></A>
<P>
The first is
<BR>
<DIV ALIGN="RIGHT">
<!-- MATH
\begin{equation}
\min _{x} \|c - Ax\|_2 \;\;\; \mbox{subject to} \;\;\; B x = d
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eqnLSE"></A><IMG
WIDTH="270" HEIGHT="38" BORDER="0"
SRC="img17.png"
ALT="\begin{displaymath}
\min _{x} \Vert c - Ax\Vert _2 \;\;\; \mbox{subject to} \;\;\; B x = d
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(2.2)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
where <I>A</I> is an <I>m</I>-by-<I>n</I> matrix and <I>B</I> is a <I>p</I>-by-<I>n</I> matrix,
<I>c</I> is a given <I>m</I>-vector, and <I>d</I> is a given <I>p</I>-vector,
with
<!-- MATH
$p \leq n \leq m+p$
-->
<IMG
WIDTH="116" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img18.png"
ALT="$p \leq n \leq m+p$">.
This is
called a <B>linear equality-constrained least squares problem (LSE)</B>.
The routine xGGLSE<A NAME="1451"></A><A NAME="1452"></A><A NAME="1453"></A>
solves this problem using the generalized <I>RQ</I>
(GRQ) factorization,<A NAME="1454"></A><A NAME="1455"></A> on the
assumptions that <I>B</I> has full row rank <I>p</I> and
the matrix
<!-- MATH
$\left( \begin{array}{c}
A \\
B
\end{array} \right)$
-->
<IMG
WIDTH="60" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
SRC="img19.png"
ALT="$ \left( \begin{array}{c}
A \\
B
\end{array} \right) $">
has full column rank <I>n</I>.
Under these assumptions, the problem LSE<A NAME="1459"></A> has a unique solution.
<P>
The second generalized linear least squares problem is
<BR>
<DIV ALIGN="RIGHT">
<!-- MATH
\begin{equation}
\min _{x} \|y\|_2 \;\;\; \mbox{subject to} \;\;\; d = A x + B y
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="glm3"></A><A NAME="eqnGLM"></A><IMG
WIDTH="271" HEIGHT="38" BORDER="0"
SRC="img20.png"
ALT="\begin{displaymath}
\min _{x} \Vert y\Vert _2 \;\;\; \mbox{subject to} \;\;\; d = A x + B y
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(2.3)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
where <I>A</I> is an <I>n</I>-by-<I>m</I> matrix, <I>B</I> is an <I>n</I>-by-<I>p</I> matrix,
and <I>d</I> is a given <I>n</I>-vector,
with
<!-- MATH
$m \leq n \leq m+p$
-->
<IMG
WIDTH="122" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img21.png"
ALT="$m \leq n \leq m+p$">.
This is sometimes called a <B>general</B> (Gauss-Markov) <B>linear model problem (GLM)</B>.
<A NAME="1468"></A>
<A NAME="1469"></A><A NAME="1470"></A>
When <I>B</I> = <I>I</I>, the problem reduces to an ordinary linear least squares problem.
When <I>B</I> is square and nonsingular, the GLM problem is equivalent to the
<B>weighted linear least squares problem</B>:<A NAME="1472"></A>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\min_x \|B^{-1}(d-Ax) \|_2
\end{displaymath}
-->
<IMG
WIDTH="154" HEIGHT="38" BORDER="0"
SRC="img22.png"
ALT="\begin{displaymath}\min_x \Vert B^{-1}(d-Ax) \Vert _2 \end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
The routine xGGGLM<A NAME="1474"></A><A NAME="1475"></A><A NAME="1476"></A><A NAME="1477"></A>
solves this problem using the generalized <I>QR</I> (GQR)
factorization,<A NAME="1478"></A><A NAME="1479"></A> on the
assumptions that <I>A</I> has full column rank <I>m</I>, and the
matrix ( <I>A</I>, <I>B</I> ) has full row rank <I>n</I>. Under these assumptions, the
problem is always consistent, and there are unique solutions <I>x</I> and <I>y</I>.
The driver routines for generalized linear least squares problems are listed
in Table <A HREF="node28.html#tabdrivegllsq">2.4</A>.
<P>
<BR>
<DIV ALIGN="CENTER">
<A NAME="tabdrivegllsq"></A>
<DIV ALIGN="CENTER">
<A NAME="1482"></A>
<TABLE CELLPADDING=3 BORDER="1">
<CAPTION><STRONG>Table 2.4:</STRONG>
Driver routines for generalized linear least squares problems</CAPTION>
<TR><TD ALIGN="LEFT">Operation</TD>
<TD ALIGN="CENTER" COLSPAN=2>Single precision</TD>
<TD ALIGN="CENTER" COLSPAN=2>Double precision</TD>
</TR>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT">real</TD>
<TD ALIGN="LEFT">complex</TD>
<TD ALIGN="LEFT">real</TD>
<TD ALIGN="LEFT">complex</TD>
</TR>
<TR><TD ALIGN="LEFT">solve LSE problem using GRQ</TD>
<TD ALIGN="LEFT">SGGLSE<A NAME="1494"></A></TD>
<TD ALIGN="LEFT">CGGLSE<A NAME="1495"></A></TD>
<TD ALIGN="LEFT">DGGLSE<A NAME="1496"></A></TD>
<TD ALIGN="LEFT">ZGGLSE<A NAME="1497"></A></TD>
</TR>
<TR><TD ALIGN="LEFT">solve GLM problem using GQR</TD>
<TD ALIGN="LEFT">SGGGLM<A NAME="1498"></A></TD>
<TD ALIGN="LEFT">CGGGLM<A NAME="1499"></A></TD>
<TD ALIGN="LEFT">DGGGLM<A NAME="1500"></A></TD>
<TD ALIGN="LEFT">ZGGGLM<A NAME="1501"></A></TD>
</TR>
</TABLE>
</DIV>
</DIV>
<BR>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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