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<H3><A NAME="SECTION03481100000000000000"></A><A NAME="secnepsummary"></A>
<BR>
Overview
</H3>

<P>
In this subsection, we will summarize all the available error bounds.
Later subsections will provide further details. The reader may also
refer to [<A
 HREF="node151.html#baidemmelmckenney">12</A>,<A
 HREF="node151.html#stewartsun90">95</A>].

<P>
Bounds for individual eigenvalues and eigenvectors are provided by
driver xGEEVX (subsection&nbsp;<A HREF="node29.html#subsecdriveeig">2.3.4</A>) or computational
routine xTRSNA (subsection&nbsp;<A HREF="node49.html#subseccompnep">2.4.5</A>).
<A NAME="11403"></A><A NAME="11404"></A><A NAME="11405"></A><A NAME="11406"></A>
<A NAME="11407"></A><A NAME="11408"></A><A NAME="11409"></A><A NAME="11410"></A>
Bounds for
clusters<A NAME="11411"></A> of eigenvalues
and their associated invariant subspace are
provided by driver xGEESX (subsection&nbsp;<A HREF="node29.html#subsecdriveeig">2.3.4</A>) or
<A NAME="11413"></A><A NAME="11414"></A><A NAME="11415"></A><A NAME="11416"></A>
<A NAME="11417"></A><A NAME="11418"></A><A NAME="11419"></A><A NAME="11420"></A>
computational routine xTRSEN (subsection&nbsp;<A HREF="node49.html#subseccompnep">2.4.5</A>).
<A NAME="11422"></A>

<P>
We let 
<!-- MATH
 $\hat{\lambda}_i$
 -->
<IMG
 WIDTH="20" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img530.png"
 ALT="$\hat{\lambda}_i$">
be the <B><I>i</I><SUP><I>th</I></SUP></B> computed eigenvalue and
<IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img523.png"
 ALT="$\lambda_i$">
an <B><I>i</I><SUP><I>th</I></SUP></B> true eigenvalue.<A NAME="tex2html2253"
 HREF="footnode.html#foot13242"><SUP>4.1</SUP></A> 

Let <IMG
 WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img564.png"
 ALT="$\hat{v}_i$">
be the
corresponding computed right eigenvector, and <B><I>v</I><SUB><I>i</I></SUB></B> a true right
eigenvector (so 
<!-- MATH
 $Av_i = \lambda_i v_i$
 -->
<IMG
 WIDTH="84" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img563.png"
 ALT="$A v_i = \lambda_i v_i$">).
If <IMG
 WIDTH="15" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img553.png"
 ALT="$\cal I$">
is a subset of the
integers from 1 to <B><I>n</I></B>, we let 
<!-- MATH
 $\lambda_{\cal I}$
 -->
<IMG
 WIDTH="24" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img582.png"
 ALT="$\lambda_{\cal I}$">
denote the average of
the selected eigenvalues:

<!-- MATH
 $\lambda_{\cal I} = (\sum_{i \in \cal I} \lambda_i)/(\sum_{i \in \cal I} 1)$
 -->
<IMG
 WIDTH="192" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img583.png"
 ALT="$\lambda_{\cal I} = (\sum_{i \in \cal I} \lambda_i)/(\sum_{i \in \cal I} 1)$">,
and similarly for 
<!-- MATH
 $\hat{\lambda}_{\cal I}$
 -->
<IMG
 WIDTH="24" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img584.png"
 ALT="$\hat{\lambda}_{\cal I}$">.
We also let 
<!-- MATH
 ${\cal S}_{\cal I}$
 -->
<IMG
 WIDTH="24" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img585.png"
 ALT="${\cal S}_{\cal I}$">
denote the subspace spanned by 
<!-- MATH
 $\{ v_i \, , \, i \in \cal I \}$
 -->
<IMG
 WIDTH="87" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img586.png"
 ALT="$\{ v_i \, , \, i \in \cal I \}$">;
it is
called a right invariant subspace because if <B><I>v</I></B> is any vector in 
<!-- MATH
 ${\cal S}_{\cal I}$
 -->
<IMG
 WIDTH="24" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img585.png"
 ALT="${\cal S}_{\cal I}$">
then
<B><I>Av</I></B> is also in 
<!-- MATH
 ${\cal S}_{\cal I}$
 -->
<IMG
 WIDTH="24" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img585.png"
 ALT="${\cal S}_{\cal I}$">.

<!-- MATH
 ${\hat{\cal S}}_{\cal I}$
 -->
<IMG
 WIDTH="24" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img587.png"
 ALT="${\hat{\cal S}}_{\cal I}$">
is the corresponding computed subspace.

<P>
The algorithms for the nonsymmetric eigenproblem are normwise backward stable:
<A NAME="11447"></A>
<A NAME="11448"></A>
they compute the exact eigenvalues, eigenvectors and invariant subspaces
of slightly perturbed matrices <B><I>A</I>+<I>E</I></B>, where 
<!-- MATH
 $\|E\| \leq p(n) \epsilon \|A\|$
 -->
<IMG
 WIDTH="129" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img588.png"
 ALT="$\Vert E\Vert \leq p(n) \epsilon \Vert A\Vert$">.
Some of the bounds are stated in terms of <B>|E|<SUB>2</SUB></B> and others in
terms of <B>|E|<SUB><I>F</I></SUB></B>; one may use 
<!-- MATH
 $p(n) \epsilon \|A\|_1$
 -->
<IMG
 WIDTH="82" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img589.png"
 ALT="$p(n) \epsilon \Vert A\Vert _1$">
to approximate
either quantity.
The code fragment in the previous subsection approximates
<B>|E|</B> by 
<!-- MATH
 $\epsilon \cdot {\tt ABNRM}$
 -->
<IMG
 WIDTH="70" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img590.png"
 ALT="$\epsilon \cdot {\tt ABNRM}$">,
where 
<!-- MATH
 ${\tt ABNRM} = \|A\|_1$
 -->
<IMG
 WIDTH="112" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img591.png"
 ALT="${\tt ABNRM} = \Vert A\Vert _1$">
is returned by xGEEVX.

<P>
xGEEVX (or xTRSNA) returns two quantities for each

<!-- MATH
 $\hat{\lambda_i}$
 -->
<IMG
 WIDTH="20" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img530.png"
 ALT="$\hat{\lambda}_i$">,
<IMG
 WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img564.png"
 ALT="$\hat{v}_i$">
pair: <B><I>s</I><SUB><I>i</I></SUB></B> and <IMG
 WIDTH="34" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img592.png"
 ALT="${\rm sep}_i$">.
xGEESX (or xTRSEN) returns two quantities for a selected subset
<IMG
 WIDTH="15" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img553.png"
 ALT="$\cal I$">
of eigenvalues: <IMG
 WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img593.png"
 ALT="$s_{\cal I}$">
and 
<!-- MATH
 ${\rm sep}_{\cal I}$
 -->
<IMG
 WIDTH="38" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img594.png"
 ALT="${\rm sep}_{\cal I}$">.
<B><I>s</I><SUB><I>i</I></SUB></B> (or <IMG
 WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img593.png"
 ALT="$s_{\cal I}$">)
is a reciprocal condition number for the
computed eigenvalue 
<!-- MATH
 $\hat{\lambda}_i$
 -->
<IMG
 WIDTH="20" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img530.png"
 ALT="$\hat{\lambda}_i$">
(or 
<!-- MATH
 $\hat{\lambda}_{\cal I}$
 -->
<IMG
 WIDTH="24" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img584.png"
 ALT="$\hat{\lambda}_{\cal I}$">),
and is referred to as <TT>RCONDE</TT> by xGEEVX (or xGEESX).
<A NAME="11460"></A>
<A NAME="11461"></A>
<IMG
 WIDTH="34" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img592.png"
 ALT="${\rm sep}_i$">
(or 
<!-- MATH
 ${\rm sep}_{\cal I}$
 -->
<IMG
 WIDTH="38" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img594.png"
 ALT="${\rm sep}_{\cal I}$">)
is a reciprocal condition number for
the right eigenvector <IMG
 WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img564.png"
 ALT="$\hat{v}_i$">
(or 
<!-- MATH
 ${\cal S}_{\cal I}$
 -->
<IMG
 WIDTH="24" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img585.png"
 ALT="${\cal S}_{\cal I}$">), and
is referred to as <TT>RCONDV</TT> by xGEEVX (or xGEESX).
<A NAME="11467"></A>
The approximate error bounds for eigenvalues, averages of eigenvalues,
eigenvectors, and invariant subspaces
provided in Table&nbsp;<A HREF="node93.html#tabasympnepbounds">4.5</A> are
true for sufficiently small <B>|E|</B>, which is why they are called asymptotic.

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

<A NAME="tabasympnepbounds"></A>
<DIV ALIGN="CENTER">
<A NAME="11470"></A>
<TABLE CELLPADDING=3 BORDER="1">
<CAPTION><STRONG>Table 4.5:</STRONG>
Asymptotic error bounds for the nonsymmetric eigenproblem</CAPTION>
<TR><TD ALIGN="LEFT">Simple eigenvalue</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $|\hat{\lambda}_i - \lambda_i | \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}}\|E\|_2 / s_i$
 -->
<IMG
 WIDTH="151" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img595.png"
 ALT="$\vert\hat{\lambda}_i - \lambda_i \vert \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle &lt;}$}}\Vert E\Vert _2 / s_i$"></TD>
</TR>
<TR><TD ALIGN="LEFT">Eigenvalue cluster</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $|\hat{\lambda}_{\cal I} - \lambda_{\cal I} | \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}}\|E\|_2 / s_{\cal I}$
 -->
<IMG
 WIDTH="162" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img596.png"
 ALT="$\vert\hat{\lambda}_{\cal I} - \lambda_{\cal I} \vert \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle &lt;}$}}\Vert E\Vert _2 / s_{\cal I}$"></TD>
</TR>
<TR><TD ALIGN="LEFT">Eigenvector</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\theta ( \hat{v}_i , v_i ) \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}}\|E\|_F / {\rm sep}_i$
 -->
<IMG
 WIDTH="166" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img597.png"
 ALT="$\theta ( \hat{v}_i , v_i ) \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle &lt;}$}}\Vert E\Vert _F / {\rm sep}_i$"></TD>
</TR>
<TR><TD ALIGN="LEFT">Invariant subspace</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\theta ( {\hat{\cal S}}_{\cal I}, {\cal S}_{\cal I}) \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}}\|E\|_F / {\rm sep}_{\cal I}$
 -->
<IMG
 WIDTH="182" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img598.png"
 ALT="$\theta ( {\hat{\cal S}}_{\cal I}, {\cal S}_{\cal I}) \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle &lt;}$}}\Vert E\Vert _F / {\rm sep}_{\cal I}$"></TD>
</TR>
</TABLE>
</DIV>
</DIV>
<BR>
<A NAME="11485"></A>
<A NAME="11486"></A>
<A NAME="11487"></A>
<A NAME="11488"></A>

<P>
If the problem is ill-conditioned, the asymptotic bounds may only hold
for extremely small <B>|E|</B>. Therefore, in Table&nbsp;<A HREF="node93.html#tabglobalnepbounds">4.6</A>
we also provide global bounds
which are guaranteed to hold for all 
<!-- MATH
 $\|E\|_F < s \cdot {\rm sep}/ 4$
 -->
<IMG
 WIDTH="133" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img599.png"
 ALT="$\Vert E\Vert _F &lt; s \cdot {\rm sep}/ 4$">.

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

<A NAME="tabglobalnepbounds"></A>
<DIV ALIGN="CENTER">
<A NAME="11491"></A>
<TABLE CELLPADDING=3 BORDER="1">
<CAPTION><STRONG>Table:</STRONG>
Global error bounds for the nonsymmetric eigenproblem
assuming 
<!-- MATH
 $\|E\|_F < s_i \cdot {\rm sep}_i / 4$
 -->
<IMG
 WIDTH="144" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img600.png"
 ALT="$\Vert E\Vert _F &lt; s_i \cdot {\rm sep}_i / 4$"></CAPTION>
<TR><TD ALIGN="LEFT">Eigenvalue cluster</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $|\hat{\lambda}_{\cal I} - \lambda_{\cal I} | \leq 2 \|E\|_2 / s_{\cal I}$
 -->
<IMG
 WIDTH="171" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img601.png"
 ALT="$\vert\hat{\lambda}_{\cal I} - \lambda_{\cal I} \vert \leq 2 \Vert E\Vert _2 / s_{\cal I}$"></TD>
</TR>
<TR><TD ALIGN="LEFT">Eigenvector</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\theta ( \hat{v}_i , v_i ) \leq \arctan (2 \|E\|_F/({\rm sep}_i - 4 \|E\|_F/s_i))$
 -->
<IMG
 WIDTH="345" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img602.png"
 ALT="$\theta ( \hat{v}_i , v_i ) \leq \arctan (2 \Vert E\Vert _F/({\rm sep}_i - 4 \Vert E\Vert _F/s_i))$"></TD>
</TR>
<TR><TD ALIGN="LEFT">Invariant subspace</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $\theta ( {\hat{\cal S}}_{\cal I}, {\cal S}_{\cal I}) \leq \arctan (2 \|E\|_F/({\rm sep}_{\cal I} - 4 \|E\|_F/s_{\cal I}))$
 -->
<IMG
 WIDTH="364" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img603.png"
 ALT="$\theta ( {\hat{\cal S}}_{\cal I}, {\cal S}_{\cal I}) \leq \arctan (2 \Vert E\Vert _F/({\rm sep}_{\cal I} - 4 \Vert E\Vert _F/s_{\cal I}))$"></TD>
</TR>
</TABLE>
</DIV>
</DIV>
<BR>

<P>
We also have the following bound, which is true for all <B><I>E</I></B>:
all the <IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img523.png"
 ALT="$\lambda_i$">
lie in the union of <B><I>n</I></B> disks,
where the <B><I>i</I></B>-th disk is centered at 
<!-- MATH
 $\hat{\lambda}_i$
 -->
<IMG
 WIDTH="20" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img530.png"
 ALT="$\hat{\lambda}_i$">
and has
radius 
<!-- MATH
 $n \|E\|_2 / s_i$
 -->
<B><I>n</I> |E|<SUB>2</SUB> / <I>s</I><SUB><I>i</I></SUB></B>. If <B><I>k</I></B> of these disks overlap,
so that any two points inside the <B><I>k</I></B> disks can be connected
by a continuous curve lying entirely inside the <B><I>k</I></B> disks,
and if no larger set of <B><I>k</I>+1</B> disks has this property,
then exactly <B><I>k</I></B> of the <IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img523.png"
 ALT="$\lambda_i$">
lie inside the
union of these <B><I>k</I></B> disks. Figure&nbsp;<A HREF="node93.html#figureBauerFike">4.1</A> illustrates
this for a 10-by-10 matrix, with 4 such overlapping unions
of disks, two containing 1 eigenvalue each, one containing 2
eigenvalues, and one containing 6 eigenvalues.

<P>

<P></P>
<DIV ALIGN="CENTER"><A NAME="figureBauerFike"></A><A NAME="11509"></A>
<TABLE>
<CAPTION><STRONG>Figure 4.1:</STRONG>
Bounding eigenvalues inside overlapping disks</CAPTION>
<TR><TD></TD></TR>
</TABLE>
</DIV><P></P>

<P>
Finally, the quantities <B><I>s</I></B> and <IMG
 WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img2.png"
 ALT="${\rm sep}$">
tell use how we can best
(block) diagonalize a matrix <B><I>A</I></B> by a similarity,

<!-- MATH
 $V^{-1}AV = {\mbox {\rm diag}}(A_{11} , \ldots , A_{bb})$
 -->
<IMG
 WIDTH="224" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img605.png"
 ALT="$V^{-1}AV = {\mbox {\rm diag}}(A_{11} , \ldots , A_{bb})$">,
where each diagonal block
<B><I>A</I><SUB><I>ii</I></SUB></B> has a selected subset of the eigenvalues of <B><I>A</I></B>. Such a decomposition
may be useful in computing functions of matrices, for example.
The goal is to choose a <B><I>V</I></B> with a nearly minimum condition number
<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img606.png"
 ALT="$\kappa_2 (V)$">
<A NAME="11516"></A>
which performs this decomposition, since this generally minimizes the error
in the decomposition.
This may be done as follows. Let <B><I>A</I><SUB><I>ii</I></SUB></B> be
<B><I>n</I><SUB><I>i</I></SUB></B>-by-<B><I>n</I><SUB><I>i</I></SUB></B>. Then columns 
<!-- MATH
 $1+\sum_{j=1}^{i-1} n_j$
 -->
<IMG
 WIDTH="96" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img607.png"
 ALT="$1+\sum_{j=1}^{i-1} n_j$">
through

<!-- MATH
 $\sum_{j=1}^{i} n_j$
 -->
<IMG
 WIDTH="66" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img608.png"
 ALT="$\sum_{j=1}^{i} n_j$">
of <B><I>V</I></B> span the invariant
subspace<A NAME="11522"></A> of <B><I>A</I></B> corresponding
to the eigenvalues of <B><I>A</I><SUB><I>ii</I></SUB></B>; these columns should be chosen to be any
orthonormal basis of this space (as computed by xGEESX, for example).
Let 
<!-- MATH
 $s_{{\cal I}_i}$
 -->
<IMG
 WIDTH="26" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img609.png"
 ALT="$s_{{\cal I}_i}$">
be the value corresponding to the
cluster of
eigenvalues of <B><I>A</I><SUB><I>ii</I></SUB></B>, as computed by xGEESX or xTRSEN. Then

<!-- MATH
 $\kappa_2 (V) \leq b/ \min_i (s_{{\cal I}_i})$
 -->
<IMG
 WIDTH="161" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img610.png"
 ALT="$\kappa_2 (V) \leq b/ \min_i (s_{{\cal I}_i})$">,
and no other choice of <B><I>V</I></B> can make
its condition number smaller than 
<!-- MATH
 $1/ \min_i (s_{{\cal I}_i})$
 -->
<IMG
 WIDTH="94" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img611.png"
 ALT="$1/ \min_i (s_{{\cal I}_i})$">
[<A
 HREF="node151.html#demmel83">26</A>].
Thus choosing orthonormal
subblocks of <B><I>V</I></B> gets <IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img606.png"
 ALT="$\kappa_2 (V)$">
to within a factor <B><I>b</I></B> of its minimum
value.

<P>
In the case of a real symmetric (or complex Hermitian) matrix,
<B><I>s</I>=1</B> and <IMG
 WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img2.png"
 ALT="${\rm sep}$">
is the absolute gap, as defined in subsection&nbsp;<A HREF="node89.html#secsym">4.7</A>.
The bounds in Table&nbsp;<A HREF="node93.html#tabasympnepbounds">4.5</A> then reduce to the
bounds in subsection&nbsp;<A HREF="node89.html#secsym">4.7</A>.

<P>
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<B> Next:</B> <A NAME="tex2html5498"
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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