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<H2><A NAME="SECTION03249000000000000000"></A><A NAME="sectionGSVDcomputational"></A>
<BR>
Generalized (or Quotient) Singular Value Decomposition
</H2>

<P>
<A NAME="3858"></A><A NAME="3859"></A>
<A NAME="3860"></A>
<A NAME="3861"></A>
The <B>generalized (or quotient) singular value decomposition</B>
of an <B><I>m</I></B>-by-<B><I>n</I></B> matrix <B><I>A</I></B> and a <B><I>p</I></B>-by-<B><I>n</I></B> matrix <B><I>B</I></B> is described
in section&nbsp;<A HREF="node33.html#subsecdrivegeig">2.3.5</A>.
The routines described in this section, are used
to compute the decomposition. The computation proceeds in the following
two stages:
<DL COMPACT>
<DT>1.
<DD>xGGSVP<A NAME="3865"></A><A NAME="3866"></A><A NAME="3867"></A><A NAME="3868"></A> is used to reduce the matrices <B><I>A</I></B> and <B><I>B</I></B> to triangular form:
<BR><P></P>
<DIV ALIGN="CENTER">
<IMG
 WIDTH="373" HEIGHT="177" BORDER="0"
 SRC="img205.png"
 ALT="\begin{eqnarray*}
U^T_1 A Q_1 &amp; = &amp; \bordermatrix{ &amp; n-k-l &amp; k &amp; l \cr
\hfill k...
...-l &amp; k &amp; l \cr
\hfill l &amp; 0 &amp; 0 &amp; B_{13} \cr
p-l &amp; 0 &amp; 0 &amp; 0 }
\end{eqnarray*}">
</DIV><P></P>
<BR CLEAR="ALL">
where <B><I>A</I><SUB>12</SUB></B> and <B><I>B</I><SUB>13</SUB></B> are nonsingular upper triangular, and
<B><I>A</I><SUB>23</SUB></B> is upper triangular.
If <B><I>m</I>-<I>k</I>-<I>l</I> &lt; 0</B>, the bottom zero block of <B><I>U</I><SUB>1</SUB><SUP><I>T</I></SUP> <I>A Q</I><SUB>1</SUB></B> does not appear,
and <B><I>A</I><SUB>23</SUB></B> is upper trapezoidal.
<B><I>U</I><SUB>1</SUB></B>, <B><I>V</I><SUB>1</SUB></B> and <B><I>Q</I><SUB>1</SUB></B> are
orthogonal matrices (or unitary matrices if <B><I>A</I></B> and <B><I>B</I></B> are complex).
<B><I>l</I></B> is the rank of <B><I>B</I></B>, and
<B><I>k</I>+<I>l</I></B> is the rank of 
<!-- 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) $">.
<P>
<DT>2.
<DD>The generalized singular value decomposition of two <B><I>l</I></B>-by-<B><I>l</I></B>
upper triangular matrices <B><I>A</I><SUB>23</SUB></B> and <B><I>B</I><SUB>13</SUB></B> is computed using
xTGSJA<A NAME="tex2html1653"
 HREF="footnode.html#foot3940"><SUP>2.2</SUP></A>:
<A NAME="3883"></A><A NAME="3884"></A><A NAME="3885"></A><A NAME="3886"></A>
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
A_{23} =  U_2 C R Q^T_2  \quad  \mbox{and} \quad
 B_{13} =  V_2 S R Q^T_2  \; \; .
\end{displaymath}
 -->


<IMG
 WIDTH="316" HEIGHT="31" BORDER="0"
 SRC="img206.png"
 ALT="\begin{displaymath}
A_{23} = U_2 C R Q^T_2 \quad \mbox{and} \quad
B_{13} = V_2 S R Q^T_2 \; \; .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Here <B><I>U</I><SUB>2</SUB></B>, <B><I>V</I><SUB>2</SUB></B> and <B><I>Q</I><SUB>2</SUB></B> are orthogonal (or unitary) matrices,
<B><I>C</I></B> and <B><I>S</I></B> are both real
nonnegative diagonal matrices satisfying <B><I>C</I><SUP>2</SUP> + <I>S</I><SUP>2</SUP> = <I>I</I></B>, <B><I>S</I></B> is nonsingular,
and <B><I>R</I></B> is upper triangular and nonsingular.
</DL>

<P>
<BR>
<DIV ALIGN="CENTER">

<A NAME="tabcompGSVD"></A>
<DIV ALIGN="CENTER">
<A NAME="3892"></A>
<TABLE CELLPADDING=3 BORDER="1">
<CAPTION><STRONG>Table 2.16:</STRONG>
Computational routines for the generalized singular value decomposition</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">&nbsp;</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">triangular reduction of <B><I>A</I></B> and <B><I>B</I></B></TD>
<TD ALIGN="LEFT">SGGSVP<A NAME="3904"></A></TD>
<TD ALIGN="LEFT">CGGSVP<A NAME="3905"></A></TD>
<TD ALIGN="LEFT">DGGSVP<A NAME="3906"></A></TD>
<TD ALIGN="LEFT">ZGGSVP<A NAME="3907"></A></TD>
</TR>
<TR><TD ALIGN="LEFT">GSVD of a pair of triangular matrices</TD>
<TD ALIGN="LEFT">STGSJA<A NAME="3908"></A></TD>
<TD ALIGN="LEFT">CTGSJA<A NAME="3909"></A></TD>
<TD ALIGN="LEFT">DTGSJA<A NAME="3910"></A></TD>
<TD ALIGN="LEFT">ZTGSJA<A NAME="3911"></A></TD>
</TR>
</TABLE>
</DIV>
</DIV>
<BR>

<P>
The reduction to triangular form, performed by
xGGSVP, uses QR decomposition with column pivoting
<A NAME="3915"></A>
for numerical rank determination.  See [<A
 HREF="node151.html#baizha93">8</A>] for details.
<A NAME="3917"></A>

<P>
The generalized singular value decomposition of two
triangular matrices, performed by xTGSJA, is done
using a Jacobi-like method as described in [<A
 HREF="node151.html#paige86a">83</A>,<A
 HREF="node151.html#baidemmel92b">10</A>].

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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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