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<BR> <P>
<H3><A NAME="SECTION04323100000000000000">Symmetric Eigenproblems </A></H3>
<A NAME="subsecdriveeigSEP"> </A>
<A NAME="1051"> </A>
<P>
The <B>symmetric eigenvalue problem (SEP)</B> is to find the <B>eigenvalues</B><A NAME="1054"> </A><A NAME="1055"> </A>,
<IMG WIDTH=8 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline12778" SRC="img69.gif">, and corresponding <B>eigenvectors</B><A NAME="1057"> </A>, <IMG WIDTH=39 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline12780" SRC="img70.gif">, such that
<BR><IMG WIDTH=391 HEIGHT=21 ALIGN=BOTTOM ALT="displaymath12772" SRC="img71.gif"><BR>
For the <B>Hermitian eigenvalue problem</B><A NAME="1061"> </A> we have
<BR><IMG WIDTH=328 HEIGHT=21 ALIGN=BOTTOM ALT="displaymath12773" SRC="img72.gif"><BR>
For both problems the eigenvalues <IMG WIDTH=8 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline12778" SRC="img69.gif"> are real.
<P>
When all eigenvalues and eigenvectors have been computed, we write
<BR><IMG WIDTH=295 HEIGHT=21 ALIGN=BOTTOM ALT="displaymath12774" SRC="img73.gif"><BR>
where <IMG WIDTH=12 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline12784" SRC="img74.gif"> is a diagonal matrix whose diagonal elements are the
eigenvalues<A NAME="1062"> </A>, and <I>Z</I> is an orthogonal (or unitary) matrix whose columns
are the eigenvectors. This is the classical <B>spectral factorization</B>
<A NAME="1064"> </A> of <I>A</I>.
<P>
Two types of driver routines<A NAME="1065"> </A> are provided for symmetric or Hermitian
eigenproblems:
<P>
<UL>
<LI> a <B>simple</B> driver (name ending -EV)<A NAME="1068"> </A><A NAME="1069"> </A>,
which computes all the eigenvalues and
(optionally) the eigenvectors of a symmetric or Hermitian matrix <I>A</I>;
<LI> an <B>expert</B> driver (name ending -EVX)<A NAME="1071"> </A><A NAME="1072"> </A>,
which can compute either all or a selected subset of the eigenvalues,
and (optionally) the corresponding eigenvectors.
<P>
</UL>
<P>
The driver routines are shown in table <A HREF="node48.html#tabdriveseig">3.4</A>. Currently
the only simple drivers provided are PSSYEV and PDSYEV.
<P>
<BR> <HR>
<P><ADDRESS>
<I>Susan Blackford <BR>
Tue May 13 09:21:01 EDT 1997</I>
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