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<H3><A NAME="SECTION034111200000000000000"></A><A NAME="GENP32"></A>
<BR>
Balancing and Conditioning
</H3>
<P>
As in the standard nonsymmetric eigenvalue problem (section <A HREF="node94.html#secbalance">4.8.1.2</A>),
two preprocessing steps<A NAME="12491"></A>
may be performed on the input matrix pair <B>(<I>A</I>, <I>B</I>)</B>.
The first one is a <B>permutation</B>, reordering the rows and columns
to attempt to make <B><I>A</I></B> and <B><I>B</I></B> block upper triangular, and
therefore to reduce the order of the eigenvalue problems to be solved:
we let
<!-- MATH
$(A',B') = P_1 (A, B) P_2$
-->
<B>(<I>A</I>',<I>B</I>') = <I>P</I><SUB>1</SUB> (<I>A</I>, <I>B</I>) <I>P</I><SUB>2</SUB></B>, where <B><I>P</I><SUB>1</SUB></B> and <B><I>P</I><SUB>2</SUB></B> are
permutation matrices.
The second one is a <B>scaling</B><A NAME="12494"></A>
by two-sided diagonal transformation <B><I>D</I><SUB>1</SUB></B>
and <B><I>D</I><SUB>2</SUB></B> to make the elements of
<!-- MATH
$A''= D_1 A' D_2$
-->
<B><I>A</I>''= <I>D</I><SUB>1</SUB> <I>A</I>' <I>D</I><SUB>2</SUB></B> and
<!-- MATH
$B'' = D_1 B' D_2$
-->
<B><I>B</I>'' = <I>D</I><SUB>1</SUB> <I>B</I>' <I>D</I><SUB>2</SUB></B>
have magnitudes as close to unity as possible, so as to reduce the effect
of the roundoff error made by the later algorithm [<A
HREF="node151.html#ward81">100</A>].
We refer to these two operations as <EM>balancing</EM>.
<P>
Balancing is performed by driver xGGEVX, which calls computational
routine xGGBAL. The user may choose to optionally
permute, scale, do both or do either;
<A NAME="12497"></A><A NAME="12498"></A><A NAME="12499"></A><A NAME="12500"></A>
this is specified by the input parameter
<TT>BALANC</TT><A NAME="12502"></A> when xGGEVX is called.
Permuting has no effect on the condition numbers<A NAME="12503"></A>
or their interpretation as described in the previous subsections. Scaling does,
however, change their interpretation, as we now describe.
<P>
The output parameters of xGGEVX -
<TT>ILO</TT>(integer), <TT>IHI</TT>(integer),
<TT>LSCALE</TT>(real array of length N),
<TT>RSCALE</TT>(real array of length N),
<TT>ABNRM</TT>(real) and
<TT>BBNRM</TT>(real) -
<A NAME="12510"></A>
<A NAME="12511"></A>
<A NAME="12512"></A>
<A NAME="12513"></A>
<A NAME="12514"></A>
describe the result of balancing the matrix pair <B>(<I>A</I>, <I>B</I>)</B> to
<B>(<I>A</I>'',<I>B</I>'')</B>, where N is the dimension of <B>(<I>A</I>,<I>B</I>)</B>.
The matrix pair <B>(<I>A</I>'',<I>B</I>'')</B> has block upper triangular
structure, with at most three blocks: from 1 to <TT>ILO</TT>-1,
from <TT>ILO</TT> to <TT>IHI</TT>, and from <TT>IHI</TT>+1 to N (see section
<A HREF="node55.html#sec_gnep_comp">2.4.8</A>). The first and last blocks are upper
triangular, and so already in generalized Schur form. These blocks are not
scaled; only the block from <TT>ILO</TT> to <TT>IHI</TT> is scaled.
Details of the left permutations (<B><I>P</I><SUB>1</SUB></B>) and scaling (<B><I>D</I><SUB>1</SUB></B>)
and the right permutations (<B><I>P</I><SUB>2</SUB></B>) and scaling (<B><I>D</I><SUB>2</SUB></B>)
are described in <TT>LSCALE</TT> and <TT>RSCALE</TT>, respectively.
(See the specification of xGGEVX or xGGBAL for more information).
The one-norms of <B><I>A</I>''</B> and <B><I>B</I>''</B> are returned in
<TT>ABNRM</TT> and <TT>BBNRM</TT>, respectively.
<P>
The condition numbers
<A NAME="12526"></A>
described in earlier subsections are computed
for the balanced matrix pair <B>(<I>A</I>'',<I>B</I>'')</B> in xGGEVX, and so
some interpretation is needed to apply them to the eigenvalues
and eigenvectors of the original matrix pair <B>(<I>A</I>, <I>B</I>)</B>.
To use the bounds for eigenvalues in Tables <A HREF="node102.html#asymp">4.7</A> and <A HREF="node102.html#global">4.8</A>,
we must replace <IMG
WIDTH="82" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img773.png"
ALT="$\Vert(E, F)\Vert _F$">
by
<!-- MATH
$O(\epsilon)\|(A'',B'')\|_F =
O(\epsilon)\sqrt{ {\tt ABNRM}^2 + {\tt BBNRM}^2 }$
-->
<IMG
WIDTH="332" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
SRC="img814.png"
ALT="$O(\epsilon)\Vert(A'',B'')\Vert _F =
O(\epsilon)\sqrt{ {\tt ABNRM}^2 + {\tt BBNRM}^2 }$">.
To use the bounds for eigenvectors, we also need to take
into account that bounds on rotation of the right and left eigenvectors
are for the right and left eigenvectors <B><I>x</I>''</B> and <B><I>y</I>''</B> of
<B><I>A</I>''</B> and <B><I>B</I>''</B>, respectively, which are related to
the right and left eigenvectors <B><I>x</I></B> and <B><I>y</I></B> by
<!-- MATH
$x'' = D^{-1}_2 P^T_2 x$
-->
<B><I>x</I>'' = <I>D</I><SUP>-1</SUP><SUB>2</SUB> <I>P</I><SUP><I>T</I></SUP><SUB>2</SUB> <I>x</I></B> and
<!-- MATH
$y'' = D^{-1}_1 P_1 y$
-->
<B><I>y</I>'' = <I>D</I><SUP>-1</SUP><SUB>1</SUB> <I>P</I><SUB>1</SUB> <I>y</I></B>,
or
<!-- MATH
$x = P_2 D_2 x''$
-->
<B><I>x</I> = <I>P</I><SUB>2</SUB> <I>D</I><SUB>2</SUB> <I>x</I>''</B> and
<!-- MATH
$y = P^T_1 D_1 x''$
-->
<B><I>y</I> = <I>P</I><SUP><I>T</I></SUP><SUB>1</SUB> <I>D</I><SUB>1</SUB> <I>x</I>''</B> respectively.
Let <IMG
WIDTH="21" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img613.png"
ALT="$\theta''$">
be the bound on the rotation of <B><I>x</I>''</B>
from Table <A HREF="node102.html#asymp">4.7</A> and Table <A HREF="node102.html#global">4.8</A> and
let <IMG
WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img328.png"
ALT="$\theta$">
be the desired bound on the rotation of <B><I>x</I></B>. Let
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\kappa(D_2)=\frac{ \max_{ {\tt ILO} \leq i \leq {\tt IHI} } {\tt RSCALE}(i) }
{ \min_{ {\tt ILO} \leq i \leq {\tt IHI} } {\tt RSCALE}(i) }
\end{displaymath}
-->
<IMG
WIDTH="247" HEIGHT="48" BORDER="0"
SRC="img815.png"
ALT="\begin{displaymath}
\kappa(D_2)=\frac{ \max_{ {\tt ILO} \leq i \leq {\tt IHI} } ...
...
{ \min_{ {\tt ILO} \leq i \leq {\tt IHI} } {\tt RSCALE}(i) }
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
be the condition number of the right scaling <B><I>D</I><SUB>2</SUB></B> with respect
to matrix inversion. Then
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\sin \theta \leq \kappa(D_2) \sin\theta''.
\end{displaymath}
-->
<IMG
WIDTH="150" HEIGHT="31" BORDER="0"
SRC="img816.png"
ALT="\begin{displaymath}
\sin \theta \leq \kappa(D_2) \sin\theta''.
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Similarly, for the bound of the angles <IMG
WIDTH="15" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img817.png"
ALT="$\phi$">
and <IMG
WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img818.png"
ALT="$\phi''$">
of the
left eigenvectors <B><I>y</I>''</B> and <B><I>y</I></B>, we have
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\sin \phi \leq \kappa(D_1) \sin \phi'',
\end{displaymath}
-->
<IMG
WIDTH="153" HEIGHT="31" BORDER="0"
SRC="img819.png"
ALT="\begin{displaymath}
\sin \phi \leq \kappa(D_1) \sin \phi'',
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <IMG
WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img820.png"
ALT="$\kappa(D_1)$">
is the condition number of the left
scaling <B><I>D</I><SUB>1</SUB></B> with respect to inversion,
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\kappa(D_1)=\frac{ \max_{ {\tt ILO} \leq i \leq {\tt IHI} } {\tt LSCALE}(i) }
{ \min_{ {\tt ILO} \leq i \leq {\tt IHI} } {\tt LSCALE}(i) }.
\end{displaymath}
-->
<IMG
WIDTH="254" HEIGHT="48" BORDER="0"
SRC="img821.png"
ALT="\begin{displaymath}
\kappa(D_1)=\frac{ \max_{ {\tt ILO} \leq i \leq {\tt IHI} } ...
... { \min_{ {\tt ILO} \leq i \leq {\tt IHI} } {\tt LSCALE}(i) }.
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<A NAME="12547"></A>
<A NAME="12548"></A>
<P>
The numerical example in section <A HREF="node100.html#sec_GNEPErrorBounds">4.11</A>
does no scaling, just permutation.
<P>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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