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<H3><A NAME="SECTION03234200000000000000">
Nonsymmetric Eigenproblems (NEP)</A>
</H3>
<P>
The <B>nonsymmetric eigenvalue problem</B> is to find the <B>eigenvalues</B><A NAME="1535"></A><A NAME="1536"></A>,
<IMG
WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img23.png"
ALT="$\lambda$">,
and corresponding <B>eigenvectors</B><A NAME="1538"></A>, <IMG
WIDTH="45" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img29.png"
ALT="$v \ne 0$">,
such that
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
Av = \lambda v.
\end{displaymath}
-->
<IMG
WIDTH="68" HEIGHT="27" BORDER="0"
SRC="img30.png"
ALT="\begin{displaymath}
Av = \lambda v.
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
A real matrix <I>A</I> may have complex eigenvalues, occurring as complex conjugate
pairs. More precisely, the vector <I>v</I> is called a <B>right
eigenvector</B><A NAME="1540"></A> of <I>A</I>, and a vector <IMG
WIDTH="46" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img31.png"
ALT="$u\neq 0$">
satisfying
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
u^HA = \lambda u^H
\end{displaymath}
-->
<IMG
WIDTH="91" HEIGHT="27" BORDER="0"
SRC="img32.png"
ALT="\begin{displaymath}
u^HA = \lambda u^H
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
is called a <B>left eigenvector</B><A NAME="1542"></A> of <I>A</I>.
<P>
This problem can be solved
via the <B>Schur factorization</B><A NAME="1544"></A> of <I>A</I>,
defined in the real case as
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A = ZTZ^T,
\end{displaymath}
-->
<I>A</I> = <I>ZTZ</I><SUP><I>T</I></SUP>,
</DIV>
<BR CLEAR="ALL">
<P></P>
where <I>Z</I> is an orthogonal matrix and <I>T</I> is an upper quasi-triangular matrix
with 1-by-1 and 2-by-2 diagonal blocks, the 2-by-2 blocks
corresponding to complex conjugate pairs of eigenvalues of <I>A</I>. In the complex
case the Schur factorization is
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A = ZTZ^H,
\end{displaymath}
-->
<I>A</I> = <I>ZTZ</I><SUP><I>H</I></SUP>,
</DIV>
<BR CLEAR="ALL">
<P></P>
where <I>Z</I> is unitary and <I>T</I> is a complex upper triangular matrix.
<P>
The columns of <I>Z</I> are called the <B>Schur vectors</B><A NAME="1546"></A>.
For each <I>k</I>
<!-- MATH
$(1 \leq k \leq n)$
-->
<IMG
WIDTH="93" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img33.png"
ALT="$(1 \leq k \leq n)$">,
the first <I>k</I> columns of <I>Z</I> form an orthonormal
<A NAME="1547"></A>
basis for the <B>invariant subspace</B> corresponding to the
first <I>k</I> eigenvalues on the diagonal of <I>T</I>. Because this
basis is orthonormal, it is preferable in many
applications to compute Schur vectors rather than
eigenvectors. It is possible to order the Schur
factorization so that any desired set of <I>k</I> eigenvalues
occupy the <I>k</I> leading positions on the diagonal of <I>T</I>.
<P>
Two pairs of drivers<A NAME="1549"></A> are provided, one pair focusing on the Schur
factorization, and the other pair on the eigenvalues and eigenvectors
as shown in Table <A HREF="node32.html#tabdriveseig">2.5</A>:
<P>
<UL><LI>xGEES<A NAME="1552"></A><A NAME="1553"></A><A NAME="1554"></A><A NAME="1555"></A>: a simple driver that computes all or part of the Schur
factorization of <I>A</I>, with optional ordering of the eigenvalues;
<A NAME="1556"></A>
<P>
<LI>xGEESX<A NAME="1557"></A><A NAME="1558"></A><A NAME="1559"></A><A NAME="1560"></A>: an expert driver that can additionally compute condition
numbers for the average of a selected subset of the eigenvalues, and for
the corresponding right invariant subspace;
<P>
<LI>xGEEV<A NAME="1561"></A><A NAME="1562"></A><A NAME="1563"></A><A NAME="1564"></A>: a simple driver that computes all the eigenvalues of <I>A</I>,
and (optionally) the right or left eigenvectors (or both);
<P>
<LI>xGEEVX<A NAME="1565"></A><A NAME="1566"></A><A NAME="1567"></A><A NAME="1568"></A>: an expert driver that can additionally balance the
matrix to improve the conditioning of the eigenvalues and eigenvectors,
and compute condition numbers for the eigenvalues or right eigenvectors
(or both).
<P>
</UL>
<P>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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