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<H2><A NAME="SECTION04337000000000000000">Generalized Symmetric Definite Eigenproblems</A></H2>
<P>
<A NAME="2118"> </A><A NAME="2119"> </A>
<P>
This section is concerned with the solution of the generalized eigenvalue
problems <IMG WIDTH=77 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline12875" SRC="img83.gif">, <IMG WIDTH=77 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline12877" SRC="img84.gif">, and <IMG WIDTH=77 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline12879" SRC="img85.gif">, where
<I>A</I> and <I>B</I> are real symmetric or complex Hermitian and <I>B</I> is positive definite.
Each of these problems can be reduced to a standard symmetric
eigenvalue problem, using a Cholesky factorization of <I>B</I> as either
<IMG WIDTH=71 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline14317" SRC="img219.gif"> or <IMG WIDTH=74 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline14319" SRC="img220.gif"> (<IMG WIDTH=36 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline14321" SRC="img221.gif"> or <IMG WIDTH=38 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline14323" SRC="img222.gif"> in the Hermitian case).
<P>
With <IMG WIDTH=71 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline14317" SRC="img219.gif">, we have
<BR><IMG WIDTH=428 HEIGHT=21 ALIGN=BOTTOM ALT="displaymath14301" SRC="img223.gif"><BR>
Hence the eigenvalues of <IMG WIDTH=77 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline12875" SRC="img83.gif"> are those of <IMG WIDTH=64 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline14329" SRC="img224.gif">,
where <I>C</I> is the symmetric matrix <IMG WIDTH=111 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline14333" SRC="img225.gif"> and <IMG WIDTH=63 HEIGHT=31 ALIGN=MIDDLE ALT="tex2html_wrap_inline14335" SRC="img226.gif">.
In the complex case <I>C</I> is Hermitian with <IMG WIDTH=113 HEIGHT=15 ALIGN=BOTTOM ALT="tex2html_wrap_inline14339" SRC="img227.gif"> and <IMG WIDTH=65 HEIGHT=31 ALIGN=MIDDLE ALT="tex2html_wrap_inline14341" SRC="img228.gif">.
<P>
Table <A HREF="node65.html#tabgst">3.11</A> summarizes how each of the three types of problem
may be reduced to standard form<A NAME="2127"> </A><A NAME="2128"> </A>
<IMG WIDTH=64 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline14329" SRC="img224.gif">, and how the eigenvectors <I>z</I>
of the original problem may be recovered from the eigenvectors <I>y</I> of the
reduced problem. The table applies to real problems; for complex problems,
transposed matrices must be replaced by conjugate transposes.
<P>
<P><A NAME="2130"> </A><A NAME="tabgst"> </A><IMG WIDTH=507 HEIGHT=176 ALIGN=BOTTOM ALT="table2129" SRC="img229.gif"><BR>
<STRONG>Table 3.11:</STRONG> Reduction of generalized symmetric definite eigenproblems to standard
problems<BR>
<P>
<P>
Given <I>A</I> and a Cholesky factorization of <I>B</I>,
the routines PxyyGST overwrite <I>A</I>
<A NAME="2161"> </A><A NAME="2162"> </A><A NAME="2163"> </A><A NAME="2164"> </A>
with the matrix <I>C</I> of the corresponding standard problem
<IMG WIDTH=64 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline14329" SRC="img224.gif"> (see table <A HREF="node65.html#tabcompgeig">3.12</A>).
This may then be solved by using the routines described in
subsection <A HREF="node61.html#subseccompsep">3.3.4</A>.
No special routines are needed
to recover the eigenvectors <I>z</I> of the generalized problem from
the eigenvectors <I>y</I> of the standard problem, because these
computations are simple applications of Level 2 or Level 3 BLAS.
<P>
<P><A NAME="2168"> </A><A NAME="tabcompgeig"> </A><IMG WIDTH=697 HEIGHT=67 ALIGN=BOTTOM ALT="table2167" SRC="img230.gif"><BR>
<STRONG>Table 3.12:</STRONG> Computational routines for the generalized symmetric definite eigenproblem<BR>
<P><BR> <HR>
<P><ADDRESS>
<I>Susan Blackford <BR>
Tue May 13 09:21:01 EDT 1997</I>
</ADDRESS>
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