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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<TITLE>Error Bounds for the Nonsymmetric Eigenproblem </TITLE>
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<H1><A NAME="SECTION03480000000000000000"></A><A NAME="secnonsym"></A>
<BR>
Error Bounds for the Nonsymmetric Eigenproblem
</H1>
<P>
The nonsymmetric eigenvalue
problem<A NAME="11350"></A> is more
complicated than the
symmetric eigenvalue problem. In this subsection,
we state the simplest bounds and leave the more complicated ones to
subsequent subsections.
<P>
Let <B><I>A</I></B> be an <B><I>n</I></B>-by-<B><I>n</I></B> nonsymmetric matrix, with eigenvalues
<!-- MATH
$\lambda_1, \ldots , \lambda_n$
-->
<IMG
WIDTH="80" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img562.png"
ALT="$\lambda_1, \ldots , \lambda_n$">.
Let <B><I>v</I><SUB><I>i</I></SUB></B> be a right eigenvector
corresponding to <IMG
WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img523.png"
ALT="$\lambda_i$">:
<!-- MATH
$A v_i = \lambda_i v_i$
-->
<IMG
WIDTH="84" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img563.png"
ALT="$A v_i = \lambda_i v_i$">.
Let
<!-- MATH
$\hat{\lambda}_i$
-->
<IMG
WIDTH="20" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img530.png"
ALT="$\hat{\lambda}_i$">
and <IMG
WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img564.png"
ALT="$\hat{v}_i$">
be the corresponding
computed eigenvalues and eigenvectors, computed by expert driver routine
xGEEVX (see subsection <A HREF="node29.html#subsecdriveeig">2.3.4</A>).
<A NAME="11354"></A><A NAME="11355"></A><A NAME="11356"></A><A NAME="11357"></A>
<P>
The approximate error bounds<A NAME="footfnm 0"><SUP>4.10</SUP></A>for the computed eigenvalues are
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
| \hat{\lambda}_i - \lambda_i | \leq {\tt EERRBD}(i) \; \; .
\end{displaymath}
-->
<IMG
WIDTH="173" HEIGHT="31" BORDER="0"
SRC="img565.png"
ALT="\begin{displaymath}
\vert \hat{\lambda}_i - \lambda_i \vert \leq {\tt EERRBD}(i) \; \; .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
The approximate error
bounds<A NAME="11361"></A><A NAME="11362"></A>
for the computed eigenvectors <IMG
WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img564.png"
ALT="$\hat{v}_i$">,
which bound the acute angles between the computed eigenvectors and true
eigenvectors <B><I>v</I><SUB><I>i</I></SUB></B>, are
<A NAME="11364"></A>
<A NAME="11365"></A>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\theta ( \hat{v}_i , v_i ) \leq {\tt VERRBD}(i) \; .
\end{displaymath}
-->
<IMG
WIDTH="164" HEIGHT="31" BORDER="0"
SRC="img566.png"
ALT="\begin{displaymath}
\theta ( \hat{v}_i , v_i ) \leq {\tt VERRBD}(i) \; .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
These bounds can be computed by the following code fragment:
<P>
<PRE>
EPSMCH = SLAMCH( 'E' )
* Compute the eigenvalues and eigenvectors of A
* WR contains the real parts of the eigenvalues
* WI contains the real parts of the eigenvalues
* VL contains the left eigenvectors
* VR contains the right eigenvectors
CALL SGEEVX( 'P', 'V', 'V', 'B', N, A, LDA, WR, WI,
$ VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM,
$ RCONDE, RCONDV, WORK, LWORK, IWORK, INFO )
IF( INFO.GT.0 ) THEN
PRINT *,'SGEEVX did not converge'
ELSE IF ( N.GT.0 ) THEN
DO 10 I = 1, N
EERRBD(I) = EPSMCH*ABNRM/RCONDE(I)
VERRBD(I) = EPSMCH*ABNRM/RCONDV(I)
10 CONTINUE
ENDIF
</PRE>
<P>
For example<A NAME="footfnm 0"><SUP>4.11</SUP></A>, if
<!-- MATH
${\tt SLAMCH('E')} = 2^{-24} = 5.961 \cdot 10^{-8}$
-->
<IMG
WIDTH="259" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
SRC="img397.png"
ALT="${\tt SLAMCH('E')} = 2^{-24} = 5.961 \cdot 10^{-8}$">
and
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A = \left( \begin{array}{ccc} 58 & 9 & 2 \\186 & 383 & 96 \\-912 & -1551 & -388 \end{array} \right)
\end{displaymath}
-->
<IMG
WIDTH="235" HEIGHT="73" BORDER="0"
SRC="img567.png"
ALT="\begin{displaymath}
A = \left( \begin{array}{ccc} 58 & 9 & 2 \\ 186 & 383 & 96 \\ -912 & -1551 & -388 \end{array} \right)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
then true eigenvalues, approximate eigenvalues, approximate error bounds,
and true errors are
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="CENTER"><B><I>i</I></B></TD>
<TD ALIGN="CENTER"><IMG
WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img523.png"
ALT="$\lambda_i$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
$\hat{\lambda}_i$
-->
<IMG
WIDTH="20" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img530.png"
ALT="$\hat{\lambda}_i$"></TD>
<TD ALIGN="CENTER"><TT> EERRBD</TT><B>(<I>i</I>)</B></TD>
<TD ALIGN="CENTER">true
<!-- MATH
$| \hat{\lambda}_i - \lambda_i |$
-->
<IMG
WIDTH="67" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img531.png"
ALT="$\vert \hat{\lambda}_i - \lambda_i \vert$"></TD>
<TD ALIGN="CENTER"><TT> VERRBD</TT><B>(<I>i</I>)</B></TD>
<TD ALIGN="CENTER">true
<!-- MATH
$\theta ( \hat{v}_i , v_i )$
-->
<IMG
WIDTH="62" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img568.png"
ALT="$\theta ( \hat{v}_i , v_i )$"></TD>
</TR>
<TR><TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER"><B>50</B></TD>
<TD ALIGN="CENTER"><B>50.00</B></TD>
<TD ALIGN="CENTER">
<!-- MATH
$8.5 \cdot 10^{-4}$
-->
<IMG
WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img569.png"
ALT="$8.5 \cdot 10^{-4}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
$2.7 \cdot 10^{-4}$
-->
<IMG
WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img570.png"
ALT="$2.7 \cdot 10^{-4}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
$2.6 \cdot 10^{-5}$
-->
<IMG
WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img571.png"
ALT="$2.6 \cdot 10^{-5}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
$8.5 \cdot 10^{-6}$
-->
<IMG
WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img572.png"
ALT="$8.5 \cdot 10^{-6}$"></TD>
</TR>
<TR><TD ALIGN="CENTER">2</TD>
<TD ALIGN="CENTER"><B>2</B></TD>
<TD ALIGN="CENTER"><B>1.899</B></TD>
<TD ALIGN="CENTER">
<!-- MATH
$3.1 \cdot 10^{-1}$
-->
<IMG
WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img573.png"
ALT="$3.1 \cdot 10^{-1}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
$1.0 \cdot 10^{-1}$
-->
<IMG
WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img574.png"
ALT="$1.0 \cdot 10^{-1}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
$1.9 \cdot 10^{-4}$
-->
<IMG
WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img575.png"
ALT="$1.9 \cdot 10^{-4}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
$7.7 \cdot 10^{-5}$
-->
<IMG
WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img576.png"
ALT="$7.7 \cdot 10^{-5}$"></TD>
</TR>
<TR><TD ALIGN="CENTER">3</TD>
<TD ALIGN="CENTER"><B>1</B></TD>
<TD ALIGN="CENTER"><B>1.101</B></TD>
<TD ALIGN="CENTER">
<!-- MATH
$3.1 \cdot 10^{-1}$
-->
<IMG
WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img573.png"
ALT="$3.1 \cdot 10^{-1}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
$1.0 \cdot 10^{-1}$
-->
<IMG
WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img574.png"
ALT="$1.0 \cdot 10^{-1}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
$1.8 \cdot 10^{-4}$
-->
<IMG
WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img577.png"
ALT="$1.8 \cdot 10^{-4}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
$7.5 \cdot 10^{-5}$
-->
<IMG
WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img578.png"
ALT="$7.5 \cdot 10^{-5}$"></TD>
</TR>
</TABLE>
</DIV>
<P>
<BR><HR>
<!--Table of Child-Links-->
<A NAME="CHILD_LINKS"></A>
<UL>
<LI><A NAME="tex2html5466"
HREF="node92.html">Further Details: Error Bounds for the Nonsymmetric Eigenproblem</A>
<UL>
<LI><A NAME="tex2html5467"
HREF="node93.html">Overview</A>
<LI><A NAME="tex2html5468"
HREF="node94.html">Balancing and Conditioning</A>
<LI><A NAME="tex2html5469"
HREF="node95.html">Computing <B><I>s</I></B> and <IMG
WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img2.png"
ALT="${\rm sep}$"></A>
</UL></UL>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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