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<B> Next:</B> <A NAME="tex2html2756" HREF="node50.html">Computational Routines</A>
<B>Up:</B> <A NAME="tex2html2754" HREF="node43.html">Driver Routines</A>
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<H2><A NAME="SECTION04324000000000000000">Generalized Symmetric Definite Eigenproblems (GSEP)</A></H2>
<P>
<A NAME="secGSEP"> </A><A NAME="1119"> </A>
<P>
<A NAME="1120"> </A><A NAME="1121"> </A>
An expert driver<A NAME="1122"> </A> is provided to compute all the eigenvalues and
(optionally) the eigenvectors of the following types of problems:
<P>
<OL>
<LI> <IMG WIDTH=77 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline12875" SRC="img83.gif">
<LI> <IMG WIDTH=77 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline12877" SRC="img84.gif">
<LI> <IMG WIDTH=77 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline12879" SRC="img85.gif"></OL>
<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="1125"> </A> <IMG WIDTH=8 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline12778" SRC="img69.gif"> are
real.
When <I>A</I> and <I>B</I> are symmetric, the matrices <I>Z</I>
of computed eigenvectors<A NAME="1126"> </A> satisfy
<IMG WIDTH=84 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline12895" SRC="img86.gif"> (problem types 1 and 3) or <IMG WIDTH=110 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline12897" SRC="img87.gif">
(problem type 2), where <IMG WIDTH=12 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline12784" SRC="img74.gif"> is a diagonal matrix with the eigenvalues
on the diagonal. <I>Z</I> also satisfies
<IMG WIDTH=82 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline12903" SRC="img88.gif"> (problem types 1 and 2) or <IMG WIDTH=100 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline12905" SRC="img89.gif"> (problem type 3).
When <I>A</I> and <I>B</I> are Hermitian, the matrices <I>Z</I>
of computed eigenvectors<A NAME="1130"> </A> satisfy
<IMG WIDTH=86 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline12913" SRC="img90.gif"> (problem types 1 and 3) or <IMG WIDTH=112 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline12915" SRC="img91.gif">
(problem type 2), where <IMG WIDTH=12 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline12784" SRC="img74.gif"> is a diagonal matrix with the eigenvalues
on the diagonal. <I>Z</I> also satisfies
<IMG WIDTH=84 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline12921" SRC="img92.gif"> (problem types 1 and 2) or <IMG WIDTH=102 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline12923" SRC="img93.gif"> (problem type 3).
<P>
The routine is listed in table <A HREF="node49.html#tabdrivegeig">3.5</A>.
<P>
<P><A NAME="1136"> </A><A NAME="tabdrivegeig"> </A><IMG WIDTH=631 HEIGHT=67 ALIGN=BOTTOM ALT="table1135" SRC="img94.gif"><BR>
<STRONG>Table 3.5:</STRONG> Driver routine for the generalized symmetric definite eigenvalue problems<BR>
<P><BR> <HR>
<P><ADDRESS>
<I>Susan Blackford <BR>
Tue May 13 09:21:01 EDT 1997</I>
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