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<H3><A NAME="SECTION03245200000000000000">
Balancing</A>
</H3>
<P>
The routine xGEBAL<A NAME="3246"></A><A NAME="3247"></A><A NAME="3248"></A><A NAME="3249"></A>
may be used to <B>balance</B> the
matrix <B><I>A</I></B> prior to reduction to Hessenberg form<A NAME="3251"></A>. Balancing involves two
steps, either of which is optional:
<UL><LI>first, xGEBAL<A NAME="3253"></A><A NAME="3254"></A><A NAME="3255"></A><A NAME="3256"></A>
attempts to permute <B><I>A</I></B> by a similarity transformation<A NAME="3257"></A> to
block upper triangular form:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
P A P^T = A'
= \left( \begin{array}{ccc} A'_{11} & A'_{12} & A'_{13} \\
0 & A'_{22} & A'_{23} \\
0 & 0 & A'_{33} \end{array} \right)
\end{displaymath}
-->
<IMG
WIDTH="269" HEIGHT="73" BORDER="0"
SRC="img152.png"
ALT="\begin{displaymath}
P A P^T = A'
= \left( \begin{array}{ccc} A'_{11} & A'_{12} ...
...0 & A'_{22} & A'_{23} \\
0 & 0 & A'_{33} \end{array} \right)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <B><I>P</I></B> is a permutation matrix and <B><I>A</I>'<SUB>11</SUB></B> and <B><I>A</I>'<SUB>33</SUB></B> are
<EM>upper triangular</EM>.
Thus the matrix is already in Schur form<A NAME="3270"></A> outside the
central diagonal block <B><I>A</I>'<SUB>22</SUB></B> in rows and columns ILO to IHI.
<A NAME="3272"></A>
Subsequent operations by xGEBAL, xGEHRD or xHSEQR need only be applied to
these rows and columns; therefore ILO and IHI are passed as arguments to
xGEHRD<A NAME="3273"></A><A NAME="3274"></A><A NAME="3275"></A><A NAME="3276"></A> and
xHSEQR<A NAME="3277"></A><A NAME="3278"></A><A NAME="3279"></A><A NAME="3280"></A>. This can save a significant amount of
work if ILO <B>></B> 1 or IHI <B>< <I>n</I></B>.
If no suitable permutation can be found (as is very often the case),
xGEBAL sets ILO = 1 and IHI = <B><I>n</I></B>, and <B><I>A</I>'<SUB>22</SUB></B> is
the whole of <B><I>A</I></B>.
<P>
<LI>secondly, xGEBAL applies a diagonal similarity transformation to
<B><I>A</I>'</B>
to make the rows and columns of <B><I>A</I>'<SUB>22</SUB></B>
as close in norm in possible:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A'' = D A' D^{-1} = \left( \begin{array}{ccc} I & 0 & 0 \\
0 & D_{22} & 0 \\
0 & 0 & I \end{array} \right)
\left( \begin{array}{ccc} A'_{11} & A'_{12} & A'_{13} \\
0 & A'_{22} & A'_{23} \\
0 & 0 & A'_{33} \end{array} \right)
\left( \begin{array}{ccc} I & 0 & 0 \\
0 & D_{22}^{-1} & 0 \\
0 & 0 & I \end{array} \right)
\end{displaymath}
-->
<IMG
WIDTH="542" HEIGHT="73" BORDER="0"
SRC="img153.png"
ALT="\begin{displaymath}
A'' = D A' D^{-1} = \left( \begin{array}{ccc} I & 0 & 0 \\
...
... 0 \\
0 & D_{22}^{-1} & 0 \\
0 & 0 & I \end{array} \right)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
This can improve the
accuracy of later processing in some cases; see subsection <A HREF="node94.html#secbalance">4.8.1.2</A>.
<P>
</UL>
<P>
If <B><I>A</I></B> was balanced by xGEBAL, then eigenvectors computed by subsequent
operations are eigenvectors of the balanced matrix <B><I>A</I>''</B>;
xGEBAK<A NAME="3304"></A><A NAME="3305"></A><A NAME="3306"></A><A NAME="3307"></A> must then be called
to transform them back to eigenvectors of the original matrix <B><I>A</I></B>.
<P>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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