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<H3><A NAME="SECTION03248200000000000000">
Balancing</A>
</H3>
<P>
The routine xGGBAL <A NAME="3698"></A><A NAME="3699"></A><A NAME="3700"></A><A NAME="3701"></A>
may be used to <B>balance</B> the matrix pair <B>(<I>A</I>,<I>B</I>)</B>
prior to reduction to generalized Hessenberg form.
<A NAME="3703"></A>
Balancing involves two steps, either of which is optional:
<DL COMPACT>
<DT>1.
<DD>First, xGGBAL attempts to permute <A NAME="3705"></A> <B>(<I>A</I>,<I>B</I>)</B>
by an equivalence transformation
to block upper triangular form:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
P_1(A,B)P_2 = (A^\prime,B^\prime) = \left(
\left[
\begin{array}{ccc}
A^\prime_{11} & A^\prime_{12} & A^\prime_{13} \\
0 & A^\prime_{22} & A^\prime_{23} \\
0 & 0 & A^\prime_{33} \\
\end{array}\right],\;\;
\left[
\begin{array}{ccc}
B^\prime_{11} & B^\prime_{12} & B^\prime_{13} \\
0 & B^\prime_{22} & B^\prime_{23} \\
0 & 0 & B^\prime_{33} \\
\end{array} \right]
\right)
\end{displaymath}
-->
<IMG
WIDTH="537" HEIGHT="73" BORDER="0"
SRC="img193.png"
ALT="\begin{displaymath}P_1(A,B)P_2 = (A^\prime,B^\prime) = \left(
\left[
\begin{ar...
... \\
0 & 0 & B^\prime_{33} \\
\end{array} \right]
\right)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <B><I>P</I><SUB>1</SUB></B> and <B><I>P</I><SUB>2</SUB></B> are permutation matrices and <IMG
WIDTH="32" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img194.png"
ALT="$A^\prime_{11}$">,
<IMG
WIDTH="32" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img195.png"
ALT="$A^\prime_{33}$">,
<IMG
WIDTH="32" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img196.png"
ALT="$B^\prime_{11}$">
and <IMG
WIDTH="32" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img197.png"
ALT="$B^\prime_{33}$">
are <I>upper
triangular</I>. Thus the matrix pair is already in generalized Schur
form outside the central diagonal blocks <IMG
WIDTH="32" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img198.png"
ALT="$A^\prime_{22}$">
and
<IMG
WIDTH="32" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img199.png"
ALT="$B^\prime_{22}$">
in rows and columns ILO to IHI. Subsequent operations
by xGGBAL, xGGHRD or xHGEQZ need only be applied to these rows and
columns; therefore ILO and IHI are passed as arguments to xGGHRD and
xHGEQZ. This can save a significant amount of work if ILO <B>> 1</B>
or IHI <B>< <I>n</I></B>. If no suitable permutations can be found (as is very
often the case), xGGBAL sets ILO <B>= 1</B> and IHI <B>= <I>n</I></B>, and
<IMG
WIDTH="32" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img198.png"
ALT="$A^\prime_{22}$">
is the whole of <B><I>A</I></B> and <IMG
WIDTH="32" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img199.png"
ALT="$B^\prime_{22}$">
is the whole
of <B><I>B</I></B>.
<P>
<DT>2.
<DD>Secondly, xGGBAL applies diagonal equivalence transformations
to
<!-- MATH
$(A^\prime,B^\prime)$
-->
<IMG
WIDTH="62" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img200.png"
ALT="$(A^\prime,B^\prime)$">
to attempt to make the matrix norm smaller
with respect
to the eigenvalues and tries to reduce the inaccuracy contributed by
roundoff [<A
HREF="node151.html#ward81">100</A>]:
<BR><P></P>
<DIV ALIGN="CENTER">
<IMG
WIDTH="679" HEIGHT="104" BORDER="0"
SRC="img201.png"
ALT="\begin{eqnarray*}
(A^{\prime\prime},B^{\prime\prime}) & = &
D_1(A^\prime,B^\pr...
...& 0 & 0 \\
0 & D'_2 & 0 \\
0 & 0 & I \\ \end{array} \right]
\end{eqnarray*}">
</DIV><P></P>
<BR CLEAR="ALL">
This can improve the accuracy of later processing in some cases; see
subsection <A HREF="node103.html#GENP32">4.11.1.2</A>.
<P>
</DL>
<P>
If the matrix pair <B>(<I>A</I>,<I>B</I>)</B> was balanced by xGGBAL, then eigenvectors
computed by subsequent operations are eigenvectors of the balanced
matrix pair
<!-- MATH
$(A^{\prime\prime},B^{\prime\prime})$
-->
<IMG
WIDTH="69" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img202.png"
ALT="$(A^{\prime\prime},B^{\prime\prime})$">.
xGGBAK
<A NAME="3766"></A><A NAME="3767"></A><A NAME="3768"></A><A NAME="3769"></A>
must then
be called to transform them back to eigenvectors of the original matrix
pair <B>(<I>A</I>,<I>B</I>)</B>.
Note that these transformations can improve speed and accuracy of
later processing in some cases; however, the diagonal transformation
step can occasionally make the norm of the pencil
<!-- MATH
$(A^\prime,B^\prime)$
-->
<IMG
WIDTH="62" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img200.png"
ALT="$(A^\prime,B^\prime)$">
larger
and hence degrade the accuracy.
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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