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<H3><A NAME="SECTION03235100000000000000"></A>
<A NAME="secGSEP"></A>
<BR>
Generalized Symmetric Definite Eigenproblems (GSEP)
</H3>
<P>
<A NAME="1695"></A><A NAME="1696"></A>
Drivers are provided to compute all the eigenvalues and
(optionally) the eigenvectors of the following types of problems:
<P>
<DL COMPACT>
<DT>1.
<DD>
<!-- MATH
$A z = \lambda B z$
-->
<IMG
WIDTH="83" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img39.png"
ALT="$A z = \lambda B z$">
<P>
<DT>2.
<DD>
<!-- MATH
$A B z = \lambda z$
-->
<IMG
WIDTH="83" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img40.png"
ALT="$A B z = \lambda z$">
<P>
<DT>3.
<DD>
<!-- MATH
$B A z = \lambda z$
-->
<IMG
WIDTH="83" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$B A z = \lambda z$">
<P>
</DL>
<P>
where <I>A</I> and <I>B</I> are symmetric or Hermitian and <I>B</I> is positive definite.
For all these problems the eigenvalues<A NAME="1699"></A> <IMG
WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img23.png"
ALT="$\lambda$">
are real. The matrices <I>Z</I>
of computed eigenvectors<A NAME="1700"></A> satisfy
<!-- MATH
$Z^T A Z = \Lambda$
-->
<IMG
WIDTH="90" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img42.png"
ALT="$Z^T A Z = \Lambda$">
(problem types 1 and 3) or
<!-- MATH
$Z^{-1} A Z^{-T} = I$
-->
<I>Z</I><SUP>-1</SUP> <I>A Z</I><SUP>-<I>T</I></SUP> = <I>I</I>
(problem type 2), where <IMG
WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img28.png"
ALT="$\Lambda$">
is a diagonal matrix with the eigenvalues
on the diagonal. <I>Z</I> also satisfies
<I>Z</I><SUP><I>T</I></SUP> <I>B Z</I> = <I>I</I> (problem types 1 and 2) or
<!-- MATH
$Z^T B^{-1} Z = I$
-->
<I>Z</I><SUP><I>T</I></SUP> <I>B</I><SUP>-1</SUP> <I>Z</I> = <I>I</I> (problem type 3).
<P>
There are three types of driver routines for generalized symmetric and
Hermitian eigenproblems. Originally LAPACK had just the simple and expert
drivers described below, and the other one was added after an improved algorithm
was discovered.
<P>
<UL><LI>a <B>simple</B> driver (name ending -GV)<A NAME="1706"></A>
computes all the eigenvalues and (optionally) eigenvectors.
<P>
<LI>an <B>expert</B> driver
(name ending -GVX)<A NAME="1708"></A> computes
all or a selected subset of the eigenvalues and (optionally) eigenvectors.
If few enough eigenvalues or eigenvectors are desired, the expert driver
is faster than the simple driver.
<P>
<LI>a <B>divide-and-conquer</B> driver
(name ending -GVD)<A NAME="1710"></A> solves the
same problem as the simple driver. It is much faster than the simple
driver for large matrices, but uses more workspace. The name
divide-and-conquer<A NAME="1711"></A> refers to the underlying
algorithm (see sections <A HREF="node48.html#subseccompsep">2.4.4</A> and <A HREF="node70.html#subsecblockeig">3.4.3</A>).
<P>
</UL>
<P>
Different driver routines are provided to take advantage of special
structure or storage of the matrices <I>A</I> and <I>B</I>, as shown in
Table <A HREF="node36.html#tabdrivegeig">2.6</A>.
<P>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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