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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<H2><A NAME="SECTION03247000000000000000"></A>
<A NAME="3530"></A>
<BR>
Generalized Symmetric Definite Eigenproblems
</H2>
<P>
This section is concerned with the solution of the generalized eigenvalue
problems <IMG
WIDTH="83" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img39.png"
ALT="$A z = \lambda B z$">,
<IMG
WIDTH="83" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img40.png"
ALT="$A B z = \lambda z$">,
and <IMG
WIDTH="83" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$B A z = \lambda z$">,
where
<B><I>A</I></B> and <B><I>B</I></B> are real symmetric or complex Hermitian and <B><I>B</I></B> is positive definite.
Each of these problems can be reduced to a standard symmetric
eigenvalue problem, using a Cholesky factorization of <B><I>B</I></B> as either
<B><I>B</I>=<I>LL</I><SUP><I>T</I></SUP></B> or <B><I>B</I>=<I>U</I><SUP><I>T</I></SUP><I>U</I></B> (<B><I>LL</I><SUP><I>H</I></SUP></B> or <B><I>U</I><SUP><I>H</I></SUP><I>U</I></B> in the Hermitian case).
In the case
<!-- MATH
$Ax = \lambda Bz$
-->
<IMG
WIDTH="84" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img171.png"
ALT="$Ax = \lambda Bz$">,
if <B><I>A</I></B> and <B><I>B</I></B> are banded then this may also
be exploited to get a faster algorithm.
<P>
With <B><I>B</I> = <I>LL</I><SUP><I>T</I></SUP></B>, we have
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
Az=\lambda Bz \quad \Rightarrow \quad (L ^{-1}AL ^{-T})(L^Tz)= \lambda(L^Tz).
\end{displaymath}
-->
<IMG
WIDTH="357" HEIGHT="31" BORDER="0"
SRC="img172.png"
ALT="\begin{displaymath}
Az=\lambda Bz \quad \Rightarrow \quad (L ^{-1}AL ^{-T})(L^Tz)= \lambda(L^Tz).
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Hence the eigenvalues of <IMG
WIDTH="83" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img39.png"
ALT="$A z = \lambda B z$">
are those of <IMG
WIDTH="70" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img173.png"
ALT="$Cy=\lambda y$">,
where <B><I>C</I></B> is the symmetric matrix
<!-- MATH
$C = L^{-1} A L^{-T}$
-->
<B><I>C</I> = <I>L</I><SUP>-1</SUP> <I>A L</I><SUP>-<I>T</I></SUP></B> and <B><I>y</I> = <I>L</I><SUP><I>T</I></SUP> <I>z</I></B>.
In the complex case <B><I>C</I></B> is Hermitian with
<!-- MATH
$C = L^{-1} A L^{-H}$
-->
<B><I>C</I> = <I>L</I><SUP>-1</SUP> <I>A L</I><SUP>-<I>H</I></SUP></B> and <B><I>y</I> =
<I>L</I><SUP><I>H</I></SUP> <I>z</I></B>.
<P>
Table <A HREF="node54.html#tabgst">2.13</A> summarizes how each of the three types of problem
may be reduced to standard form <IMG
WIDTH="70" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img173.png"
ALT="$Cy=\lambda y$">,
and how the eigenvectors <B><I>z</I></B>
of the original problem may be recovered from the eigenvectors <B><I>y</I></B> of the
reduced problem. The table applies to real problems; for complex problems,
transposed matrices must be replaced by conjugate-transposes.
<P>
<BR>
<DIV ALIGN="CENTER">
<A NAME="tabgst"></A>
<DIV ALIGN="CENTER">
<A NAME="3539"></A>
<TABLE CELLPADDING=3 BORDER="1">
<CAPTION><STRONG>Table 2.13:</STRONG>
Reduction of generalized symmetric definite eigenproblems to standard
problems</CAPTION>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="CENTER">Type of</TD>
<TD ALIGN="CENTER" COLSPAN=1>Factorization</TD>
<TD ALIGN="CENTER" COLSPAN=1>Reduction</TD>
<TD ALIGN="CENTER" COLSPAN=1>Recovery of</TD>
</TR>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="CENTER">problem</TD>
<TD ALIGN="CENTER" COLSPAN=1>of <B><I>B</I></B></TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="CENTER" COLSPAN=1>eigenvectors</TD>
</TR>
<TR><TD ALIGN="LEFT">1.</TD>
<TD ALIGN="CENTER">
<!-- MATH
$Az = \lambda Bz$
-->
<IMG
WIDTH="83" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img39.png"
ALT="$A z = \lambda B z$"></TD>
<TD ALIGN="LEFT"><B><I>B</I> = <I>LL</I><SUP><I>T</I></SUP></B></TD>
<TD ALIGN="LEFT">
<!-- MATH
$C = L^{-1} A L^{-T}$
-->
<B><I>C</I> = <I>L</I><SUP>-1</SUP> <I>A L</I><SUP>-<I>T</I></SUP></B></TD>
<TD ALIGN="LEFT"><B><I>z</I> = <I>L</I><SUP>-<I>T</I></SUP> <I>y</I></B></TD>
</TR>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="CENTER"> </TD>
<TD ALIGN="LEFT"><B><I>B</I> = <I>U</I><SUP><I>T</I></SUP><I>U</I></B></TD>
<TD ALIGN="LEFT">
<!-- MATH
$C = U^{-T} A U^{-1}$
-->
<B><I>C</I> = <I>U</I><SUP>-<I>T</I></SUP> <I>A U</I><SUP>-1</SUP></B></TD>
<TD ALIGN="LEFT"><B><I>z</I> = <I>U</I><SUP>-1</SUP> <I>y</I></B></TD>
</TR>
<TR><TD ALIGN="LEFT">2.</TD>
<TD ALIGN="CENTER">
<!-- MATH
$ABz = \lambda z$
-->
<IMG
WIDTH="83" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img40.png"
ALT="$A B z = \lambda z$"></TD>
<TD ALIGN="LEFT"><B><I>B</I> = <I>LL</I><SUP><I>T</I></SUP></B></TD>
<TD ALIGN="LEFT"><B><I>C</I> = <I>L</I><SUP><I>T</I></SUP> <I>A L</I></B></TD>
<TD ALIGN="LEFT"><B><I>z</I> = <I>L</I><SUP>-<I>T</I></SUP> <I>y</I></B></TD>
</TR>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="CENTER"> </TD>
<TD ALIGN="LEFT"><B><I>B</I> = <I>U</I><SUP><I>T</I></SUP><I>U</I></B></TD>
<TD ALIGN="LEFT"><B><I>C</I> = <I>U A U</I><SUP><I>T</I></SUP></B></TD>
<TD ALIGN="LEFT"><B><I>z</I> = <I>U</I><SUP>-1</SUP> <I>y</I></B></TD>
</TR>
<TR><TD ALIGN="LEFT">3.</TD>
<TD ALIGN="CENTER">
<!-- MATH
$BAz = \lambda z$
-->
<IMG
WIDTH="83" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$B A z = \lambda z$"></TD>
<TD ALIGN="LEFT"><B><I>B</I> = <I>LL</I><SUP><I>T</I></SUP></B></TD>
<TD ALIGN="LEFT"><B><I>C</I> = <I>L</I><SUP><I>T</I></SUP> <I>A L</I></B></TD>
<TD ALIGN="LEFT"><B><I>z</I> = <I>L y</I></B></TD>
</TR>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="CENTER"> </TD>
<TD ALIGN="LEFT"><B><I>B</I> = <I>U</I><SUP><I>T</I></SUP><I>U</I></B></TD>
<TD ALIGN="LEFT"><B><I>C</I> = <I>U A U</I><SUP><I>T</I></SUP></B></TD>
<TD ALIGN="LEFT"><B><I>z</I> = <I>U</I><SUP><I>T</I></SUP> <I>y</I></B></TD>
</TR>
</TABLE>
</DIV>
</DIV>
<BR>
<P>
Given <B><I>A</I></B> and a Cholesky factorization of <B><I>B</I></B>,
the routines xyyGST overwrite <B><I>A</I></B>
with the matrix <B><I>C</I></B> of the corresponding standard problem
<!-- MATH
$Cy = \lambda y$
-->
<IMG
WIDTH="70" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img173.png"
ALT="$Cy=\lambda y$">
(see Table <A HREF="node54.html#tabcompgeig">2.14</A>).
This may then be solved using the routines described in
subsection <A HREF="node48.html#subseccompsep">2.4.4</A>.
No special routines are needed
to recover the eigenvectors <B><I>z</I></B> of the generalized problem from
the eigenvectors <B><I>y</I></B> of the standard problem, because these
computations are simple applications of Level 2 or Level 3 BLAS.
<P>
If the problem is
<!-- MATH
$A z = \lambda B z$
-->
<IMG
WIDTH="83" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img39.png"
ALT="$A z = \lambda B z$">
and the matrices <B><I>A</I></B> and <B><I>B</I></B> are banded,
the matrix <B><I>C</I></B> as defined above is, in general, full.
We can reduce the problem to a banded standard problem by modifying the
definition of <B><I>C</I></B> thus:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
C = X^T A X, \quad \mbox{ where } X = U^{-1} Q \quad \mbox{ or } \quad L^{-T} Q,
\end{displaymath}
-->
<IMG
WIDTH="369" HEIGHT="30" BORDER="0"
SRC="img174.png"
ALT="\begin{displaymath}
C = X^T A X, \quad \mbox{ where } X = U^{-1} Q \quad \mbox{ or } \quad L^{-T} Q,
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <B><I>Q</I></B> is an orthogonal matrix chosen to ensure that <B><I>C</I></B> has bandwidth
no greater than that of <B><I>A</I></B>.
<B><I>Q</I></B> is determined as a product of Givens rotations.
<A NAME="3576"></A>
This is known as Crawford's algorithm<A NAME="3577"></A>
<A NAME="3578"></A>
(see Crawford [<A
HREF="node151.html#crawford">19</A>]).
If <B><I>X</I></B> is required, it must be formed explicitly by the reduction routine.
<P>
A further refinement is possible when <B><I>A</I></B> and <B><I>B</I></B> are banded, which halves
the amount of work required to form <B><I>C</I></B> (see Wilkinson [<A
HREF="node151.html#wilkinsona">104</A>]).
Instead of the standard Cholesky factorization of <B><I>B</I></B> as <B><I>U</I><SUP><I>T</I></SUP> <I>U</I></B> or <B><I>L L</I><SUP><I>T</I></SUP></B>,
we use a ``split Cholesky'' factorization <B><I>B</I> = <I>S</I><SUP><I>T</I></SUP> <I>S</I></B>
<A NAME="3581"></A>
<A NAME="3582"></A>
(<B><I>S</I><SUP><I>H</I></SUP> <I>S</I></B> if <B><I>B</I></B> is complex), where:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
S = \left( \begin{array}{cc}
U_{11} & \\
M_{21} & L_{22} \\
\end{array} \right)
\end{displaymath}
-->
<IMG
WIDTH="144" HEIGHT="54" BORDER="0"
SRC="img175.png"
ALT="\begin{displaymath}
S = \left( \begin{array}{cc}
U_{11} & \\
M_{21} & L_{22} \\
\end{array} \right)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
with <B><I>U</I><SUB>11</SUB></B> upper triangular and <B><I>L</I><SUB>22</SUB></B> lower triangular of
order approximately <B><I>n</I>/2</B>;
<B><I>S</I></B> has the same bandwidth as <B><I>B</I></B>. After <B><I>B</I></B> has been factorized in this way
by the routine
xPBSTF<A NAME="3591"></A><A NAME="3592"></A><A NAME="3593"></A><A NAME="3594"></A>,
the reduction of the banded generalized
problem
<!-- MATH
$A z = \lambda B z$
-->
<IMG
WIDTH="83" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img39.png"
ALT="$A z = \lambda B z$">
to a banded standard problem
<!-- MATH
$C y = \lambda y$
-->
<IMG
WIDTH="70" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img173.png"
ALT="$Cy=\lambda y$">
is performed by the routine xSBGST<A NAME="3595"></A><A NAME="3596"></A>
(or xHBGST<A NAME="3597"></A><A NAME="3598"></A> for complex matrices).
This routine implements a vectorizable form of the algorithm, suggested by
Kaufman [<A
HREF="node151.html#vbandr">77</A>].
<P>
<BR>
<DIV ALIGN="CENTER">
<A NAME="tabcompgeig"></A>
<DIV ALIGN="CENTER">
<A NAME="3601"></A>
<TABLE CELLPADDING=3 BORDER="1">
<CAPTION><STRONG>Table 2.14:</STRONG>
Computational routines for the generalized symmetric definite eigenproblem</CAPTION>
<TR><TD ALIGN="LEFT">Type of matrix</TD>
<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">and storage scheme</TD>
<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">symmetric/Hermitian</TD>
<TD ALIGN="LEFT">reduction</TD>
<TD ALIGN="LEFT">SSYGST<A NAME="3613"></A></TD>
<TD ALIGN="LEFT">CHEGST<A NAME="3614"></A></TD>
<TD ALIGN="LEFT">DSYGST<A NAME="3615"></A></TD>
<TD ALIGN="LEFT">ZHEGST<A NAME="3616"></A></TD>
</TR>
<TR><TD ALIGN="LEFT">symmetric/Hermitian</TD>
<TD ALIGN="LEFT">reduction</TD>
<TD ALIGN="LEFT">SSPGST<A NAME="3617"></A></TD>
<TD ALIGN="LEFT">CHPGST<A NAME="3618"></A></TD>
<TD ALIGN="LEFT">DSPGST<A NAME="3619"></A></TD>
<TD ALIGN="LEFT">ZHPGST<A NAME="3620"></A></TD>
</TR>
<TR><TD ALIGN="LEFT">(packed storage)</TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
</TR>
<TR><TD ALIGN="LEFT">symmetric/Hermitian</TD>
<TD ALIGN="LEFT">split Cholesky</TD>
<TD ALIGN="LEFT">SPBSTF<A NAME="3621"></A></TD>
<TD ALIGN="LEFT">CPBSTF<A NAME="3622"></A></TD>
<TD ALIGN="LEFT">DPBSTF<A NAME="3623"></A></TD>
<TD ALIGN="LEFT">ZPBSTF<A NAME="3624"></A></TD>
</TR>
<TR><TD ALIGN="LEFT">banded</TD>
<TD ALIGN="LEFT">factorization</TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
</TR>
<TR><TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT">reduction</TD>
<TD ALIGN="LEFT">SSBGST<A NAME="3625"></A></TD>
<TD ALIGN="LEFT">DSBGST<A NAME="3626"></A></TD>
<TD ALIGN="LEFT">CHBGST<A NAME="3627"></A></TD>
<TD ALIGN="LEFT">ZHBGST<A NAME="3628"></A></TD>
</TR>
</TABLE>
</DIV>
</DIV>
<BR>
<P>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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