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<H3><A NAME="SECTION034111100000000000000"></A><A NAME="GNEP311"></A>
<BR>
Overview
</H3>
In this subsection, we summarize all the available error bounds.
Later sections will provide further details.
The error bounds presented here apply to regular matrix pairs only.

<P>
Bounds for individual eigenvalues and eigenvectors are provided by
the driver
xGGEVX<A NAME="12138"></A><A NAME="12139"></A><A NAME="12140"></A><A NAME="12141"></A>
(subsection <A HREF="node35.html#sec_gnep_driver">2.3.5.2</A>)
or the computational routine xTGSNA
<A NAME="12143"></A><A NAME="12144"></A><A NAME="12145"></A><A NAME="12146"></A>
(subsection <A HREF="node55.html#sec_gnep_comp">2.4.8</A>).
Bounds for cluster of eigenvalues
<A NAME="12148"></A>
and their associated pair of deflating
subspaces are provided by the driver
xGGESX<A NAME="12149"></A><A NAME="12150"></A><A NAME="12151"></A><A NAME="12152"></A>
(subsection <A HREF="node35.html#sec_gnep_driver">2.3.5.2</A>) or the
computational routine xTGSEN
<A NAME="12154"></A><A NAME="12155"></A><A NAME="12156"></A><A NAME="12157"></A>
(subsection <A HREF="node55.html#sec_gnep_comp">2.4.8</A>).
<A NAME="12159"></A>

<P>
We let 
<!-- MATH
 $({\hat{\alpha}}_i, {\hat{\beta}}_i)$
 -->
<IMG
 WIDTH="58" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img727.png"
 ALT="$({\hat{\alpha}}_i, {\hat{\beta}}_i)$">
be the <B><I>i</I><SUP><I>th</I></SUP></B> computed
eigenvalue pair and 
<!-- MATH
 $({\alpha}_i, {\beta}_i)$
 -->
<IMG
 WIDTH="58" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img725.png"
 ALT="$({\alpha}_i, {\beta}_i)$">
the <B><I>i</I><SUP><I>th</I></SUP></B> exact eigenvalue
pair.<A NAME="tex2html2442"
 HREF="footnode.html#foot13385"><SUP>4.2</SUP></A>Let <IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img356.png"
 ALT="$\hat{x}_i$">
and <IMG
 WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img760.png"
 ALT="$\hat{y}_i$">
be the corresponding computed right
and left eigenvectors, and <B><I>x</I><SUB><I>i</I></SUB></B> and <B><I>y</I><SUB><I>i</I></SUB></B> the exact right and left
eigenvectors (so that 
<!-- MATH
 ${\beta}_i A x_i = {\alpha}_i B x_i$
 -->
<IMG
 WIDTH="117" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img726.png"
 ALT="${\beta}_i A x_i = {\alpha}_i B x_i$">
and

<!-- MATH
 ${\beta}_i y_i^H A  = {\alpha}_i y_i^H B$
 -->
<IMG
 WIDTH="131" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img761.png"
 ALT="${\beta}_i y_i^H A = {\alpha}_i y_i^H B$">).
As in the standard nonsymmetric eigenvalue problem, we also want to
bound the error in the average of a cluster of eigenvalues, corresponding
to a subset <IMG
 WIDTH="15" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img553.png"
 ALT="$\cal I$">
of the integers from 1 to <B><I>n</I></B>.
However, since there are both finite and infinite eigenvalues,
we need a proper definition for the average of the eigenvalues

<!-- MATH
 ${\lambda}_i = {\alpha}_i/{\beta}_i$
 -->
<IMG
 WIDTH="84" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img192.png"
 ALT="$\lambda_i = \alpha_i / \beta_i$">
for 
<!-- MATH
 $i \in {\cal I}$
 -->
<IMG
 WIDTH="43" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img762.png"
 ALT="$i \in {\cal I}$">.
Here we let 
<!-- MATH
 $( {\alpha}_{\cal I}, {\beta}_{\cal I} )$
 -->
<IMG
 WIDTH="65" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img763.png"
 ALT="$( {\alpha}_{\cal I}, {\beta}_{\cal I} )$">
denote the average of the selected eigenvalues<A NAME="tex2html2443"
 HREF="footnode.html#foot13386"><SUP>4.3</SUP></A>:

<!-- MATH
 $( {\alpha}_{\cal I}, {\beta}_{\cal I} ) =
  (\sum_{i \in {\cal I}} {\alpha}_i, \sum_{i \in {\cal I}} {\beta}_i )/
  (\sum_{i \in {\cal I}} 1)$
 -->
<IMG
 WIDTH="300" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img766.png"
 ALT="$( {\alpha}_{\cal I}, {\beta}_{\cal I} ) =
(\sum_{i \in {\cal I}} {\alpha}_i, \sum_{i \in {\cal I}} {\beta}_i )/
(\sum_{i \in {\cal I}} 1)$">,
and similarly for

<!-- MATH
 $( \hat{{\alpha}}_{\cal I}, \hat{{\beta}}_{\cal I} )$
 -->
<IMG
 WIDTH="65" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img767.png"
 ALT="$( \hat{{\alpha}}_{\cal I}, \hat{{\beta}}_{\cal I} )$">.
We also let 
<!-- MATH
 ${\cal L}_{\cal I}$
 -->
<IMG
 WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img768.png"
 ALT="${\cal L}_{\cal I}$">
and 
<!-- MATH
 ${\cal R}_{\cal I}$
 -->
<IMG
 WIDTH="28" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img769.png"
 ALT="${\cal R}_{\cal I}$">
denote the
exact pair of left
and right deflating subspaces associated with the cluster of selected
eigenvalues.
Similarly, 
<!-- MATH
 $\widehat{{\cal L}}_{\cal I}$
 -->
<IMG
 WIDTH="26" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img770.png"
 ALT="$\widehat{{\cal L}}_{\cal I}$">
and

<!-- MATH
 $\widehat{{\cal R}}_{\cal I}$
 -->
<IMG
 WIDTH="28" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img771.png"
 ALT="$\widehat{{\cal R}}_{\cal I}$">
are the
corresponding computed pair of left and right deflating subspaces.

<P>
The algorithms for the generalized nonsymmetric eigenproblem are normwise
backward stable;
<A NAME="12230"></A>
<A NAME="12231"></A>
the computed eigenvalues, eigenvectors and deflating
subspaces are the exact ones of slightly perturbed matrices <B><I>A</I> + <I>E</I></B> and <B><I>B</I> +<I>F</I></B>,
where 
<!-- MATH
 $\|(E, F)\|_F \leq p(n) \epsilon \|(A, B)\|_F$
 -->
<IMG
 WIDTH="222" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img772.png"
 ALT="$\Vert(E, F)\Vert _F \leq p(n) \epsilon \Vert(A, B)\Vert _F$">.
The code fragment in the previous subsection approximates
<IMG
 WIDTH="82" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img773.png"
 ALT="$\Vert(E, F)\Vert _F$">
by 
<!-- MATH
 $\epsilon \cdot {\tt ABNORM}$
 -->
<IMG
 WIDTH="79" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img774.png"
 ALT="$\epsilon \cdot {\tt ABNORM}$">,
where

<!-- MATH
 ${\tt ABNORM} = \sqrt{ {\tt ABNRM}^2 + {\tt BBNRM}^2 }$
 -->
<IMG
 WIDTH="225" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="img775.png"
 ALT="${\tt ABNORM} = \sqrt{ {\tt ABNRM}^2 + {\tt BBNRM}^2 }$">,
and the values <TT>ABNRM</TT> and <TT>BBNRM</TT> returned by xGGEVX
are the 1-norm of the matrices <B><I>A</I></B> and <B><I>B</I></B>, respectively.

<P>
xGGEVX (or xTGSNA) returns reciprocal condition numbers
for each eigenvalue pair

<!-- MATH
 $({\hat{\alpha}}_i, {\hat{\beta}}_i)$
 -->
<IMG
 WIDTH="58" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img727.png"
 ALT="$({\hat{\alpha}}_i, {\hat{\beta}}_i)$">
and corresponding
left and right eigenvectors <IMG
 WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img760.png"
 ALT="$\hat{y}_i$">
and <IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img356.png"
 ALT="$\hat{x}_i$">:
<B><I>s</I><SUB><I>i</I></SUB></B> and 
<!-- MATH
 ${\rm Dif}_l(i)$
 -->
<IMG
 WIDTH="54" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img776.png"
 ALT="${\rm Dif}_l(i)$">.
<B><I>s</I><SUB><I>i</I></SUB></B> is a reciprocal condition
number for the computed eigenpair 
<!-- MATH
 $({\hat{\alpha}}_i, {\hat{\beta}}_i)$
 -->
<IMG
 WIDTH="58" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img727.png"
 ALT="$({\hat{\alpha}}_i, {\hat{\beta}}_i)$">,
and is referred to as <TT>RCONDE(i)</TT> by xGGEVX.
<A NAME="12249"></A>
<A NAME="12250"></A>

<!-- MATH
 ${\rm Dif}_l(i)$
 -->
<IMG
 WIDTH="54" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img776.png"
 ALT="${\rm Dif}_l(i)$">
is a reciprocal condition number for the left and right
eigenvectors <IMG
 WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img760.png"
 ALT="$\hat{y}_i$">
and <IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img356.png"
 ALT="$\hat{x}_i$">,
and is referred to as
<TT>RCONDV(i)</TT> by xGGEVX (see subsection <A HREF="node104.html#GNEP33">4.11.1.3</A> for definitions).
Similarly, xGGESX (or xTGSEN) returns condition numbers for
eigenvalue clusters and deflating subspaces corresponding to
a subset <IMG
 WIDTH="15" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img553.png"
 ALT="$\cal I$">
of the eigenvalues.
<A NAME="12256"></A>
These are <IMG
 WIDTH="19" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img4.png"
 ALT="$l_{\cal I}$">
and <IMG
 WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img5.png"
 ALT="$r_{\cal I}$">,
the reciprocal values of
the left and right projection norms <B><I>p</I></B> and <B><I>q</I></B>, and
estimates of the separation between two matrix pairs
defined by 
<!-- MATH
 ${\rm Dif}_u({\cal I})$
 -->
<IMG
 WIDTH="62" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img777.png"
 ALT="${\rm Dif}_u({\cal I})$">
and 
<!-- MATH
 ${\rm Dif}_l({\cal I})$
 -->
<IMG
 WIDTH="59" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img778.png"
 ALT="${\rm Dif}_l({\cal I})$">
(see subsection <A HREF="node104.html#GNEP33">4.11.1.3</A> for definitions).
xGGESX reports <IMG
 WIDTH="19" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img4.png"
 ALT="$l_{\cal I}$">
and <IMG
 WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img5.png"
 ALT="$r_{\cal I}$">
in <TT>RCONDE(1)</TT>
and <TT>RCONDE(2)</TT> (<TT>PL</TT> and <TT>PR</TT> in xTGSEN)), respectively,
and estimates of 
<!-- MATH
 ${\rm Dif}_u({\cal I})$
 -->
<IMG
 WIDTH="62" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img777.png"
 ALT="${\rm Dif}_u({\cal I})$">
and 
<!-- MATH
 ${\rm Dif}_l({\cal I})$
 -->
<IMG
 WIDTH="59" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img778.png"
 ALT="${\rm Dif}_l({\cal I})$">
in <TT>RCONDV(1)</TT>
and <TT>RCONDV(2)</TT> (<TT>DIF(1)</TT> and <TT>DIF(2)</TT> in xTGSEN), respectively.

<P>
As for the nonsymmetric eigenvalue problem we provide both asymptotic
and global error bounds.  The asymptotic approximate error bounds for
eigenvalues, averages of eigenvalues, eigenvectors, and deflating
subspaces provided in Table <A HREF="node102.html#asymp">4.7</A> are true only for
sufficiently small <IMG
 WIDTH="82" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img773.png"
 ALT="$\Vert(E, F)\Vert _F$">.

<P>
<BR>
<DIV ALIGN="CENTER">

<A NAME="asymp"></A>
<P>
<DIV ALIGN="CENTER">
<A NAME="12281"></A>
<TABLE CELLPADDING=3 BORDER="1">
<CAPTION><STRONG>Table 4.7:</STRONG>
Asymptotic error bounds for the generalized nonsymmetric eigenvalue
problem</CAPTION>
<TR><TD ALIGN="LEFT">Simple eigenvalue:</TD>
<TD ALIGN="CENTER">
<!-- MATH
 ${\cal X}(({\hat{\alpha}}_i,{\hat{\beta}}_i), ({\alpha}_i,{\beta}_i))
    \def \theguybelow{\sim}
\def \verticalposition{\lower 2.5pt}
\def \spacingwithinsymbol{\baselineskip0pt.2pt}
\mathrel{\mathpalette\verticalposition\vbox{\spacingwithinsymbol
\everycr={}\tabskip0pt
\halign{$;\hfil##\hfil$\crcr S\crcr
\theguybelow\crcr}}PMlt;}\|(E,F)\|_F/ s_i$
 -->
<IMG
 WIDTH="320" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img779.png"
 ALT="% latex2html id marker 16841
${\cal X}(({\hat{\alpha}}_i,{\hat{\beta}}_i), ({\al...
...n{$;\hfil ..."></TD>
</TR>
<TR><TD ALIGN="LEFT">Eigenvalue cluster:</TD>
<TD ALIGN="CENTER">
<!-- MATH
 ${\cal X}( ({\hat{\alpha}}_{\cal I},{\hat{\beta}}_{\cal I}),
             ({\alpha}_{\cal I},{\beta}_{\cal I}) )
    \def \theguybelow{\sim}
\def \verticalposition{\lower 2.5pt}
\def \spacingwithinsymbol{\baselineskip0pt.2pt}
\mathrel{\mathpalette\verticalposition\vbox{\spacingwithinsymbol
\everycr={}\tabskip0pt
\halign{$;\hfil##\hfil$\crcr S\crcr
\theguybelow\crcr}}PMlt;}\|(E,F)\|_F/ l_{\cal I}$
 -->
<IMG
 WIDTH="336" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img780.png"
 ALT="% latex2html id marker 16845
${\cal X}( ({\hat{\alpha}}_{\cal I},{\hat{\beta}}_{...
...fil ..."></TD>
</TR>
<TR><TD ALIGN="LEFT">Eigenvector pair:</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;&nbsp; Left</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\theta_{\max} (\hat{{\em y}}_i , {\em y}_i) \def \theguybelow{\sim}
\def \verticalposition{\lower 2.5pt}
\def \spacingwithinsymbol{\baselineskip0pt.2pt}
\mathrel{\mathpalette\verticalposition\vbox{\spacingwithinsymbol
\everycr={}\tabskip0pt
\halign{$;\hfil##\hfil$\crcr S\crcr
\theguybelow\crcr}}PMlt;}
{\|(E,F)\|_F}/{{\rm Dif}_l(i)}$
 -->
<IMG
 WIDTH="297" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img781.png"
 ALT="% latex2html id marker 16849
$\theta_{\max} (\hat{{\em y}}_i , {\em y}_i) \def \...
...il$\crcr S\crcr
\theguybelow\crcr}}PMlt;}
{\Vert(E,F)\Vert _F}/{{\rm Dif}_l(i)}$"></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;&nbsp; Right</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\theta_{\max} (\hat{{\em x}}_i , {\em x}_i) \def \theguybelow{\sim}
\def \verticalposition{\lower 2.5pt}
\def \spacingwithinsymbol{\baselineskip0pt.2pt}
\mathrel{\mathpalette\verticalposition\vbox{\spacingwithinsymbol
\everycr={}\tabskip0pt
\halign{$;\hfil##\hfil$\crcr S\crcr
\theguybelow\crcr}}PMlt;}
{\|(E,F)\|_F}/{{\rm Dif}_l(i)}$
 -->
<IMG
 WIDTH="296" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img782.png"
 ALT="% latex2html id marker 16853
$\theta_{\max} (\hat{{\em x}}_i , {\em x}_i) \def \...
...il$\crcr S\crcr
\theguybelow\crcr}}PMlt;}
{\Vert(E,F)\Vert _F}/{{\rm Dif}_l(i)}$"></TD>
</TR>
<TR><TD ALIGN="LEFT">Deflating subspace pair:</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;&nbsp; Left</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\theta_{\max} ({\widehat{{\cal L}}}_{\cal I} , {\cal L}_{\cal I})
     \def \theguybelow{\sim}
\def \verticalposition{\lower 2.5pt}
\def \spacingwithinsymbol{\baselineskip0pt.2pt}
\mathrel{\mathpalette\verticalposition\vbox{\spacingwithinsymbol
\everycr={}\tabskip0pt
\halign{$;\hfil##\hfil$\crcr S\crcr
\theguybelow\crcr}}PMlt;}{\|(E,F)\|_F}/{{\rm Dif}_l({\cal I})}$
 -->
<IMG
 WIDTH="316" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img783.png"
 ALT="% latex2html id marker 16857
$\theta_{\max} ({\widehat{{\cal L}}}_{\cal I} , {\c...
...cr S\crcr
\theguybelow\crcr}}PMlt;}{\Vert(E,F)\Vert _F}/{{\rm Dif}_l({\cal I})}$"></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;&nbsp; Right</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\theta_{\max} ({\widehat{{\cal R}}}_{\cal I} , {\cal R}_{\cal I})
     \def \theguybelow{\sim}
\def \verticalposition{\lower 2.5pt}
\def \spacingwithinsymbol{\baselineskip0pt.2pt}
\mathrel{\mathpalette\verticalposition\vbox{\spacingwithinsymbol
\everycr={}\tabskip0pt
\halign{$;\hfil##\hfil$\crcr S\crcr
\theguybelow\crcr}}PMlt;}{\|(E,F)\|_F}/{{\rm Dif}_l({\cal I})}$
 -->
<IMG
 WIDTH="321" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img784.png"
 ALT="% latex2html id marker 16861
$\theta_{\max} ({\widehat{{\cal R}}}_{\cal I} , {\c...
...cr S\crcr
\theguybelow\crcr}}PMlt;}{\Vert(E,F)\Vert _F}/{{\rm Dif}_l({\cal I})}$"></TD>
</TR>
</TABLE>
</DIV>
</DIV>
<BR>
<A NAME="12332"></A>
<A NAME="12333"></A>
<A NAME="12334"></A>
<A NAME="12335"></A>

<P>
If the problem is ill-conditioned, the asymptotic bounds
may only hold for extremely small values of <IMG
 WIDTH="82" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img773.png"
 ALT="$\Vert(E, F)\Vert _F$">.
Therefore, we also
provide similar global error bounds, which are valid for
all perturbations that satisfy an upper bound on <IMG
 WIDTH="82" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img773.png"
 ALT="$\Vert(E, F)\Vert _F$">.
The global error bounds in Table <A HREF="node102.html#global">4.8</A> are guaranteed to hold for all

<!-- MATH
 $\|(E,F)\|_F < {\Delta}$
 -->
<IMG
 WIDTH="120" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img785.png"
 ALT="$\Vert(E,F)\Vert _F &lt; {\Delta}$">,
where

<UL><LI>
<!-- MATH
 ${\Delta} \equiv \frac{1}{4} \min (l_i,r_i)
{\min({\rm Dif}_u(i), {\rm Dif}_l(i))}$
 -->
<IMG
 WIDTH="283" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img786.png"
 ALT="${\Delta} \equiv \frac{1}{4} \min (l_i,r_i)
{\min({\rm Dif}_u(i), {\rm Dif}_l(i))}$">
for an individual eigenvector pair, and

<LI>
<!-- MATH
 ${\Delta} \equiv \frac{1}{4} \min (l_{\cal I},r_{\cal I})
{\min({\rm Dif}_u({\cal I}), {\rm Dif}_l({\cal I}))}$
 -->
<IMG
 WIDTH="300" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img787.png"
 ALT="${\Delta} \equiv \frac{1}{4} \min (l_{\cal I},r_{\cal I})
{\min({\rm Dif}_u({\cal I}), {\rm Dif}_l({\cal I}))}$">
for a cluster of
eigenvalues or a deflating subspace pair.

</UL>

<P>
We let 
<!-- MATH
 $\delta \equiv \|(E,F)\|_F / {\Delta}$
 -->
<IMG
 WIDTH="137" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img788.png"
 ALT="$\delta \equiv \Vert(E,F)\Vert _F / {\Delta}$">
in Table <A HREF="node102.html#global">4.8</A>.
If <IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img366.png"
 ALT="$\delta$">
is small, then the computed pair of left and right deflating
subspaces (or computed left and right eigenvectors) are small perturbations of
the exact pair of deflating subspaces (or the true left and right eigenvectors).
The error bounds conform with the corresponding bounds for the nonsymmetric
eigenproblem (see subsection&nbsp;<A HREF="node93.html#secnepsummary">4.8.1.1</A> ).

<P>
<BR>
<DIV ALIGN="CENTER">

<A NAME="global"></A>
<DIV ALIGN="CENTER">
<A NAME="13304"></A>
<TABLE CELLPADDING=3 BORDER="1">
<CAPTION><STRONG>Table:</STRONG>
 Global error bounds for the generalized nonsymmetric
eigenvalue problem assuming 
<!-- MATH
 $\delta \equiv \|(E,F)\|_F/{\Delta} < 1$
 -->
<IMG
 WIDTH="169" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img789.png"
 ALT="$\delta \equiv \Vert(E,F)\Vert _F/{\Delta} &lt; 1$">.</CAPTION>
<TR><TD ALIGN="LEFT">Eigenvalue cluster:</TD>
<TD ALIGN="CENTER">
<!-- MATH
 ${\cal X}( ({\hat{\alpha}}_{\cal I},{\hat{\beta}}_{\cal I}),
             ({\alpha}_{\cal I},{\beta}_{\cal I}) )
    \leq 2 \|(E,F)\|_F/ l_{\cal I}$
 -->
<IMG
 WIDTH="294" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img790.png"
 ALT="${\cal X}( ({\hat{\alpha}}_{\cal I},{\hat{\beta}}_{\cal I}),
({\alpha}_{\cal I},{\beta}_{\cal I}) )
\leq 2 \Vert(E,F)\Vert _F/ l_{\cal I}$"></TD>
</TR>
<TR><TD ALIGN="LEFT">Eigenvector pair:</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;&nbsp; Left</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\theta_{\max} (\hat{y}_i , y_i) \leq
\arctan \left(
{\delta \cdot l_i}/{(1 - \delta \sqrt{1 - l_i^2})}
\right)$
 -->
<IMG
 WIDTH="337" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="img791.png"
 ALT="$\theta_{\max} (\hat{y}_i , y_i) \leq
\arctan \left(
{\delta \cdot l_i}/{(1 - \delta \sqrt{1 - l_i^2})}
\right)$"></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;&nbsp; Right</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\theta_{\max} (\hat{x}_i , x_i) \leq
\arctan \left(
{\delta \cdot r_i}/{(1 - \delta \sqrt{1 - r_i^2})}
\right)$
 -->
<IMG
 WIDTH="345" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="img792.png"
 ALT="$\theta_{\max} (\hat{x}_i , x_i) \leq
\arctan \left(
{\delta \cdot r_i}/{(1 - \delta \sqrt{1 - r_i^2})}
\right)$"></TD>
</TR>
<TR><TD ALIGN="LEFT">Deflating subspace pair:</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;&nbsp; Left</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\theta_{\max} ({\widehat{{\cal L}}}_{\cal I},{\cal L}_{\cal I})\leq
\arctan \left(
{\delta \cdot l_{\cal I}}/{(1 - \delta \sqrt{1 - l_{\cal I}^2})}
\right)$
 -->
<IMG
 WIDTH="352" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
 SRC="img793.png"
 ALT="$\theta_{\max} ({\widehat{{\cal L}}}_{\cal I},{\cal L}_{\cal I})\leq
\arctan \left(
{\delta \cdot l_{\cal I}}/{(1 - \delta \sqrt{1 - l_{\cal I}^2})}
\right)$"></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;&nbsp; Right</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\theta_{\max}({\widehat{{\cal R}}}_{\cal I},{\cal R}_{\cal I})\leq
\arctan \left(
{\delta \cdot r_{\cal I}}/{(1 - \delta \sqrt{1 - r_{\cal I}^2 })}
\right)$
 -->
<IMG
 WIDTH="363" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
 SRC="img794.png"
 ALT="$\theta_{\max}({\widehat{{\cal R}}}_{\cal I},{\cal R}_{\cal I})\leq
\arctan \left(
{\delta \cdot r_{\cal I}}/{(1 - \delta \sqrt{1 - r_{\cal I}^2 })}
\right)$"></TD>
</TR>
</TABLE>
</DIV>
</DIV>
<BR>

<P>
For ill-conditioned problems the restriction <IMG
 WIDTH="19" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img795.png"
 ALT="$\Delta$">
on <IMG
 WIDTH="82" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img773.png"
 ALT="$\Vert(E, F)\Vert _F$">
may also be small.
Indeed, a small value of <IMG
 WIDTH="19" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img795.png"
 ALT="$\Delta$">
shows that the cluster of
eigenvalues (in the (1,1)-blocks of <B>(<I>A</I>, <I>B</I>)</B>) is ill-conditioned in
the sense that small perturbations of <B>(<I>A</I>, <I>B</I>)</B> may imply that one eigenvalue in
the cluster moves and coalesces with another eigenvalue (outside the cluster).
Accordingly, this also means that the associated (left and right)
deflating subspaces are sensitive to small perturbations,
since the size of the
perturbed subspaces may change for small perturbations of <B>(<I>A</I>, <I>B</I>)</B>.
See also the discussion of singular problems in section&nbsp;<A HREF="node105.html#sec_singular">4.11.1.4</A>.

<P>
As for the nonsymmetric eigenvalue problem we have global error bounds for
eigenvalues which are true for all <B><I>E</I></B> and <B><I>F</I></B>.
Let <B>(<I>A</I>, <I>B</I>)</B> be a diagonalizable matrix pair. We let the columns of
<IMG
 WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="img796.png"
 ALT="$\widehat{Y}$">
and <IMG
 WIDTH="20" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="img797.png"
 ALT="$\widehat{X}$">
be the computed left and right
eigenvectors associated with
the computed generalized eigenvalue pairs

<!-- MATH
 $({\hat{\alpha}}_i, {\hat{\beta}}_i), i = 1, \ldots, n$
 -->
<IMG
 WIDTH="152" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img798.png"
 ALT="$({\hat{\alpha}}_i, {\hat{\beta}}_i), i = 1, \ldots, n$">.
Moreover, we assume that <IMG
 WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="img796.png"
 ALT="$\widehat{Y}$">
and <IMG
 WIDTH="20" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="img797.png"
 ALT="$\widehat{X}$">
are normalized such that 
<!-- MATH
 $|\hat{\alpha}_i|^2 + |\hat{\beta}_i|^2 = 1$
 -->
<IMG
 WIDTH="124" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img799.png"
 ALT="$\vert\hat{\alpha}_i\vert^2 + \vert\hat{\beta}_i\vert^2 = 1$">
and 
<!-- MATH
 $\|\hat{y}_i\|_2 = 1$
 -->
<IMG
 WIDTH="75" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img800.png"
 ALT="$\Vert\hat{y}_i\Vert _2 = 1$">,
i.e., we
overwrite <IMG
 WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img760.png"
 ALT="$\hat{y}_i$">
with  
<!-- MATH
 $\hat{y_i}/ \|\hat{y_i}\|_2$
 -->
<IMG
 WIDTH="66" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img801.png"
 ALT="$\hat{y_i}/ \Vert\hat{y_i}\Vert _2$">,

<!-- MATH
 $({\hat{\alpha}}_i,
{\hat{\beta}}_i)$
 -->
<IMG
 WIDTH="58" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img727.png"
 ALT="$({\hat{\alpha}}_i, {\hat{\beta}}_i)$">
with  
<!-- MATH
 $({\hat{\alpha}}_i, {\hat{\beta}}_i) /
(|\hat{\alpha}_i|^2 + |\hat{\beta}_i|^2)^{1/2}$
 -->
<IMG
 WIDTH="188" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img802.png"
 ALT="$({\hat{\alpha}}_i, {\hat{\beta}}_i) /
(\vert\hat{\alpha}_i\vert^2 + \vert\hat{\beta}_i\vert^2)^{1/2}$">
and <IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img356.png"
 ALT="$\hat{x}_i$">
with

<!-- MATH
 $\hat{x}_i \|\hat{y}_i\|_2 / (|\hat{\alpha}_i|^2 + |\hat{\beta}_i|^2)^{1/2}$
 -->
<IMG
 WIDTH="190" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img803.png"
 ALT="$\hat{x}_i \Vert\hat{y}_i\Vert _2 / (\vert\hat{\alpha}_i\vert^2 + \vert\hat{\beta}_i\vert^2)^{1/2}$">.
Then all eigenvalues 
<!-- MATH
 $({\alpha}_i, {\beta}_i)$
 -->
<IMG
 WIDTH="58" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img725.png"
 ALT="$({\alpha}_i, {\beta}_i)$">
of <B>(<I>A</I>, <I>B</I>)</B> with

<!-- MATH
 $|{\alpha}_i|^2 + |{\beta}_i|^2 = 1$
 -->
<IMG
 WIDTH="124" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img804.png"
 ALT="$\vert{\alpha}_i\vert^2 + \vert{\beta}_i\vert^2 = 1$">
lie in the union of <B><I>n</I></B> regions
(``spheres'')
<BR>
<DIV ALIGN="RIGHT">


<!-- MATH
 \begin{equation}
\left\{  (\alpha,\beta), |\alpha|^2 + |\beta|^2 = 1:
{\cal X}((\alpha,\beta), ({\hat{\alpha}}_i, {\hat{\beta}}_i))
\leq n \|(E,F)\|_2/ s_i \right\}.
\end{equation}
 -->

<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq411.29a"></A><IMG
 WIDTH="465" HEIGHT="37" BORDER="0"
 SRC="img805.png"
 ALT="\begin{displaymath}
\left\{ (\alpha,\beta), \vert\alpha\vert^2 + \vert\beta\vert...
..._i, {\hat{\beta}}_i))
\leq n \Vert(E,F)\Vert _2/ s_i \right\}.
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(4.10)</TD></TR>
</TABLE>
<BR CLEAR="ALL"></DIV><P></P>
If <B><I>k</I></B> of the regions overlap, so that any two points inside the <B><I>k</I></B>
``spheres''
can be connected by a continuous curve lying entirely inside the <B><I>k</I></B> regions,
and if no larger set of <B><I>k</I> + 1</B> regions has this property,
then exactly <B><I>k</I></B> of the
eigenvalues  
<!-- MATH
 $({\alpha}_i, {\beta}_i)$
 -->
<IMG
 WIDTH="58" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img725.png"
 ALT="$({\alpha}_i, {\beta}_i)$">
lie inside the union of these
``spheres''.
In other words, the global error bound with respect to an individual eigenvalue

<!-- MATH
 $({\alpha}_i,{\beta}_i)$
 -->
<IMG
 WIDTH="58" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img725.png"
 ALT="$({\alpha}_i, {\beta}_i)$">
is only useful if it defines a region
that does not intersect with regions corresponding to other eigenvalues.
If two or more regions intersect,
then we can only say that a (true) eigenvalue of <B>(<I>A</I>, <I>B</I>)</B> lies in
the union of the overlapping regions. If <IMG
 WIDTH="78" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img806.png"
 ALT="$\Vert(E,F)\Vert _2$">
is so large
that <B>(<I>A</I>+<I>E</I>,<I>B</I>+<I>F</I>)</B> could be singular, which means that the eigenvalues
are not well-determined by the data, then the error bound from
(<A HREF="node102.html#eq411.29a">4.10</A>) will be so large as to not limit the eigenvalues at all;
see section&nbsp;<A HREF="node105.html#sec_singular">4.11.1.4</A> for details.

<P>
<B>Notation Conversion</B> For easy of reference,
the following table summarizes the notation used in mathematical
expression of the error bounds in tables <A HREF="node102.html#asymp">4.7</A> and <A HREF="node102.html#global">4.8</A>
and in the corresponding driver and computational routines.

<P>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="LEFT">Mathematical</TD>
<TD ALIGN="CENTER" COLSPAN=2>Driver Routines</TD>
<TD ALIGN="CENTER" COLSPAN=2>Computational Routines</TD>
</TR>
<TR><TD ALIGN="LEFT">notation</TD>
<TD ALIGN="CENTER">name</TD>
<TD ALIGN="LEFT">parameter</TD>
<TD ALIGN="CENTER">name</TD>
<TD ALIGN="LEFT">parameter</TD>
</TR>
<TR><TD ALIGN="LEFT"><B><I>s</I><SUB><I>i</I></SUB></B></TD>
<TD ALIGN="CENTER">xGGEVX</TD>
<TD ALIGN="LEFT"><TT> RCONDE(i)</TT></TD>
<TD ALIGN="CENTER">xTGSNA</TD>
<TD ALIGN="LEFT"><TT> S(i)</TT>&nbsp;&nbsp;&nbsp;</TD>
</TR>
<TR><TD ALIGN="LEFT">
<!-- MATH
 $\mbox{Dif}_l(i)$
 -->
<IMG
 WIDTH="53" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img807.png"
 ALT="$\mbox{Dif}_l(i)$"></TD>
<TD ALIGN="CENTER">xGGEVX</TD>
<TD ALIGN="LEFT"><TT> RCONDV(i)</TT></TD>
<TD ALIGN="CENTER">xTGSNA</TD>
<TD ALIGN="LEFT"><TT> DIF(i)</TT></TD>
</TR>
<TR><TD ALIGN="LEFT"><IMG
 WIDTH="19" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img4.png"
 ALT="$l_{\cal I}$"></TD>
<TD ALIGN="CENTER">xGGESX</TD>
<TD ALIGN="LEFT"><TT> RCONDE(1)</TT></TD>
<TD ALIGN="CENTER">xTGSEN</TD>
<TD ALIGN="LEFT"><TT> PL</TT></TD>
</TR>
<TR><TD ALIGN="LEFT"><IMG
 WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img5.png"
 ALT="$r_{\cal I}$"></TD>
<TD ALIGN="CENTER">xGGESX</TD>
<TD ALIGN="LEFT"><TT> RCONDE(2)</TT></TD>
<TD ALIGN="CENTER">xTGSEN</TD>
<TD ALIGN="LEFT"><TT> PR</TT></TD>
</TR>
<TR><TD ALIGN="LEFT">
<!-- MATH
 $\mbox{Dif}_u({\cal I})$
 -->
<IMG
 WIDTH="61" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img808.png"
 ALT="$\mbox{Dif}_u({\cal I})$"></TD>
<TD ALIGN="CENTER">xGGESX</TD>
<TD ALIGN="LEFT"><TT> RCONDV(1)</TT></TD>
<TD ALIGN="CENTER">xTGSEN</TD>
<TD ALIGN="LEFT"><TT> DIF(1)</TT></TD>
</TR>
<TR><TD ALIGN="LEFT">
<!-- MATH
 $\mbox{Dif}_l({\cal I})$
 -->
<IMG
 WIDTH="57" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img809.png"
 ALT="$\mbox{Dif}_l({\cal I})$"></TD>
<TD ALIGN="CENTER">xGGESX</TD>
<TD ALIGN="LEFT"><TT> RCONDV(2)</TT></TD>
<TD ALIGN="CENTER">xTGSEN</TD>
<TD ALIGN="LEFT"><TT> DIF(2)</TT></TD>
</TR>
</TABLE>
</DIV>

<P>
The quantities <B><I>l</I><SUB><I>i</I></SUB></B>, <B><I>r</I><SUB><I>i</I></SUB></B>, 
<!-- MATH
 $\mbox{Dif}_u(i)$
 -->
<IMG
 WIDTH="56" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img810.png"
 ALT="$\mbox{Dif}_u(i)$">
and 
<!-- MATH
 $\mbox{Dif}_l(i)$
 -->
<IMG
 WIDTH="53" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img807.png"
 ALT="$\mbox{Dif}_l(i)$">
used in
Table <A HREF="node102.html#global">4.8</A> (for the global error bounds of
the <IMG
 WIDTH="28" HEIGHT="23" ALIGN="BOTTOM" BORDER="0"
 SRC="img811.png"
 ALT="$i^{\mbox{th}}$">
computed eigenvalue pair 
<!-- MATH
 $(\hat{\alpha},\hat{\beta})$
 -->
<IMG
 WIDTH="48" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img812.png"
 ALT="$(\hat{\alpha},\hat{\beta})$">
and the left and right eigenvectors <IMG
 WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img760.png"
 ALT="$\hat{y}_i$">
and <IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img356.png"
 ALT="$\hat{x}_i$">)
can be obtained by calling xTGSEN with 
<!-- MATH
 ${\cal I} = \{ i \}$
 -->
<IMG
 WIDTH="62" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img813.png"
 ALT="${\cal I} = \{ i \}$">.

<P>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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