File: node90.html

package info (click to toggle)
lapack 3.0.20000531a-28
links: PTS
area: main
in suites: sarge
size: 61,920 kB
ctags: 46,200
sloc: fortran: 584,835; perl: 8,226; makefile: 2,331; awk: 71; sh: 45
file content (582 lines) | stat: -rw-r--r-- 17,058 bytes
parent folder | download | duplicates (4)
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
<!--Converted with LaTeX2HTML 98.2 beta6 (August 14th, 1998)
original version by:  Nikos Drakos, CBLU, University of Leeds
* revised and updated by:  Marcus Hennecke, Ross Moore, Herb Swan
* with significant contributions from:
  Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
<HTML>
<HEAD>
<TITLE>Further Details: Error Bounds for the Symmetric Eigenproblem </TITLE>
<META NAME="description" CONTENT="Further Details: Error Bounds for the Symmetric Eigenproblem ">
<META NAME="keywords" CONTENT="lug_l2h">
<META NAME="resource-type" CONTENT="document">
<META NAME="distribution" CONTENT="global">
<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
<LINK REL="STYLESHEET" HREF="lug_l2h.css">
<LINK REL="previous" HREF="node89.html">
<LINK REL="up" HREF="node89.html">
<LINK REL="next" HREF="node91.html">
</HEAD>
<BODY >
<!--Navigation Panel-->
<A NAME="tex2html5450"
 HREF="node91.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
 SRC="next_motif.png"></A> 
<A NAME="tex2html5444"
 HREF="node89.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
 SRC="up_motif.png"></A> 
<A NAME="tex2html5440"
 HREF="node89.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
 SRC="previous_motif.png"></A> 
<A NAME="tex2html5446"
 HREF="node1.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
 SRC="contents_motif.png"></A> 
<A NAME="tex2html5448"
 HREF="node152.html">
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
 SRC="index_motif.png"></A> 
<BR>
<B> Next:</B> <A NAME="tex2html5451"
 HREF="node91.html">Error Bounds for the</A>
<B> Up:</B> <A NAME="tex2html5445"
 HREF="node89.html">Error Bounds for the</A>
<B> Previous:</B> <A NAME="tex2html5441"
 HREF="node89.html">Error Bounds for the</A>
 &nbsp <B>  <A NAME="tex2html5447"
 HREF="node1.html">Contents</A></B> 
 &nbsp <B>  <A NAME="tex2html5449"
 HREF="node152.html">Index</A></B> 
<BR>
<BR>
<!--End of Navigation Panel-->

<H2><A NAME="SECTION03471000000000000000"></A><A NAME="secDetailsym"></A>
<BR>
Further Details:  Error Bounds for the Symmetric Eigenproblem 
</H2>

<P>
The usual error analysis of the
symmetric<A NAME="11258"></A>
eigenproblem (using any LAPACK
routine in subsection&nbsp;<A HREF="node29.html#subsecdriveeig">2.3.4</A>
or any EISPACK routine) is as follows [<A
 HREF="node151.html#parlett">85</A>]:

<P>
<BLOCKQUOTE>
The computed eigendecomposition 
<!-- MATH
 $\hat{Z} \hat{\Lambda} \hat{Z}^T$
 -->
<IMG
 WIDTH="53" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="img541.png"
 ALT="$\hat{Z} \hat{\Lambda} \hat{Z}^T$">
is nearly
the exact
eigendecomposition of <B><I>A</I>+<I>E</I></B>, i.e.,

<!-- MATH
 $A+E = (\hat{Z}+ \delta \hat{Z}) \hat{\Lambda} (\hat{Z}+ \delta \hat{Z})^T$
 -->
<IMG
 WIDTH="238" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img542.png"
 ALT="$A+E = (\hat{Z}+ \delta \hat{Z}) \hat{\Lambda} (\hat{Z}+ \delta \hat{Z})^T$">
is a true eigendecomposition so that 
<!-- MATH
 $\hat{Z}+ \delta \hat{Z}$
 -->
<IMG
 WIDTH="60" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img543.png"
 ALT="$\hat{Z}+ \delta \hat{Z}$">
is
orthogonal,
where 
<!-- MATH
 $\|E\|_2 / \|A\|_2 \leq p(n) \epsilon$
 -->
<IMG
 WIDTH="153" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img544.png"
 ALT="$\Vert E\Vert _2 / \Vert A\Vert _2 \leq p(n) \epsilon$">
and

<!-- MATH
 $\| \delta \hat{Z} \|_2 \leq p(n) \epsilon$
 -->
<IMG
 WIDTH="114" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img545.png"
 ALT="$\Vert \delta \hat{Z} \Vert _2 \leq p(n) \epsilon$">.
Here <B><I>p</I>(<I>n</I>)</B> is a modestly growing
function of <B><I>n</I></B>. We take <B><I>p</I>(<I>n</I>)=1</B> in the above code fragment.
Each computed eigenvalue 
<!-- MATH
 $\hat{\lambda}_i$
 -->
<IMG
 WIDTH="20" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img530.png"
 ALT="$\hat{\lambda}_i$">
differs from a
true <IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img523.png"
 ALT="$\lambda_i$">
by at most
</BLOCKQUOTE>
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
| \hat{\lambda}_i - \lambda_i |  \leq p(n) \cdot \epsilon \cdot \|A\|_2
= {\tt EERRBD} \; .
\end{displaymath}
 -->


<IMG
 WIDTH="275" HEIGHT="31" BORDER="0"
 SRC="img546.png"
 ALT="\begin{displaymath}
\vert \hat{\lambda}_i - \lambda_i \vert \leq p(n) \cdot \epsilon \cdot \Vert A\Vert _2
= {\tt EERRBD} \; .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><BLOCKQUOTE>
Thus large eigenvalues (those near 
<!-- MATH
 $\max_i | \lambda_i | = \|A\|_2$
 -->
<IMG
 WIDTH="132" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img547.png"
 ALT="$\max_i \vert \lambda_i \vert = \Vert A\Vert _2$">)
are computed to high relative accuracy <A NAME="11274"></A> and small ones may not be.
<A NAME="11275"></A>
</BLOCKQUOTE>
<P>
<BLOCKQUOTE>There are two questions to ask about the computed eigenvectors:
``Are they orthogonal?'' and ``How much do they differ from the
true eigenvectors?''
The answer to the first question is that except for eigenvectors
computed by computational routine xSTEIN (which is called by drivers
with names ending in -EVX when only a subset of the eigenvalues and
eigenvectors are requested),
the computed eigenvectors
are always nearly orthogonal to working precision, independent of
how much they differ from the true eigenvectors. In other words
</BLOCKQUOTE>
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
|\hat{z}_i^T \hat{z}_j | = O( \epsilon )
\end{displaymath}
 -->


<IMG
 WIDTH="99" HEIGHT="32" BORDER="0"
 SRC="img548.png"
 ALT="\begin{displaymath}
\vert\hat{z}_i^T \hat{z}_j \vert = O( \epsilon )
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><BLOCKQUOTE>
for <IMG
 WIDTH="42" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img549.png"
 ALT="$i \neq j$">.
xSTEIN almost
always returns orthogonal eigenvectors, but can occasionally fail when
eigenvalues are tightly clustered.
<A NAME="11278"></A><A NAME="11279"></A><A NAME="11280"></A><A NAME="11281"></A>
</BLOCKQUOTE>
<P>
<BLOCKQUOTE>Here is the answer to the second question about eigenvectors.
The angular difference between the computed unit eigenvector
<IMG
 WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img527.png"
 ALT="$\hat{z}_i$">
and a true unit eigenvector <B><I>z</I><SUB><I>i</I></SUB></B> satisfies the approximate bound
<A NAME="11282"></A>
</BLOCKQUOTE>
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\theta ( \hat{z}_i , z_i ) \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}}\frac{p(n) \epsilon \|A\|_2}{{\rm gap}_i}
= {\tt ZERRBD}(i)
\end{displaymath}
 -->


<IMG
 WIDTH="257" HEIGHT="48" BORDER="0"
 SRC="img550.png"
 ALT="\begin{displaymath}
\theta ( \hat{z}_i , z_i ) \mathrel{\raisebox{-.75ex}{$\math...
...{p(n) \epsilon \Vert A\Vert _2}{{\rm gap}_i}
= {\tt ZERRBD}(i)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><BLOCKQUOTE>
if <IMG
 WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img455.png"
 ALT="$p(n) \epsilon$">
is small enough.
Here 
<!-- MATH
 ${\rm gap}_i = \min_{j \neq i} | \lambda_i - \lambda_j |$
 -->
<IMG
 WIDTH="178" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img551.png"
 ALT="${\rm gap}_i = \min_{j \neq i} \vert \lambda_i - \lambda_j \vert$">
is the
<B>absolute gap</B><A NAME="11288"></A><A NAME="11289"></A>
between <IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img523.png"
 ALT="$\lambda_i$">
and the nearest other eigenvalue.  Thus, if <IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img523.png"
 ALT="$\lambda_i$">
is close to other eigenvalues, its corresponding eigenvector <B><I>z</I><SUB><I>i</I></SUB></B>
may be inaccurate.
The gaps may be easily computed from the array of computed eigenvalues
using subroutine <TT>SDISNA</TT><A NAME="11291"></A><A NAME="11292"></A>.
The gaps computed by <TT>SDISNA</TT> are ensured not to be so small as
to cause overflow when used as divisors.
<A NAME="11294"></A>
<A NAME="11295"></A>
</BLOCKQUOTE>
<P>
<BLOCKQUOTE>Let 
<!-- MATH
 ${\hat{\cal S}}$
 -->
<IMG
 WIDTH="16" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="img320.png"
 ALT="$\hat{\cal S}$">
be the invariant subspace spanned by a collection of eigenvectors

<!-- MATH
 $\{\hat{z}_i \, , \, i \in {\cal I}\}$
 -->
<IMG
 WIDTH="87" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img552.png"
 ALT="$\{\hat{z}_i \, , \, i \in {\cal I}\}$">,
where <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>. Let <IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img318.png"
 ALT="$\cal S$">
be the corresponding true subspace. Then
</BLOCKQUOTE>
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\theta ( {\hat{\cal S}}, {\cal S}) \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}}\frac{p(n) \epsilon \|A\|_2}
{{\rm gap}_{\cal I}}
\end{displaymath}
 -->


<IMG
 WIDTH="157" HEIGHT="48" BORDER="0"
 SRC="img554.png"
 ALT="\begin{displaymath}
\theta ( {\hat{\cal S}}, {\cal S}) \mathrel{\raisebox{-.75ex...
...&lt;}$}}\frac{p(n) \epsilon \Vert A\Vert _2}
{{\rm gap}_{\cal I}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><BLOCKQUOTE>
<A NAME="11300"></A>
where
</BLOCKQUOTE>
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
{\rm gap}_{\cal I} = \min_{i \in {\cal I} \atop j \not\in {\cal I}}
| \lambda_i - \lambda_j |
\end{displaymath}
 -->


<IMG
 WIDTH="153" HEIGHT="50" BORDER="0"
 SRC="img555.png"
 ALT="\begin{displaymath}
{\rm gap}_{\cal I} = \min_{i \in {\cal I} \atop j \not\in {\cal I}}
\vert \lambda_i - \lambda_j \vert
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><BLOCKQUOTE>
is the absolute gap between the eigenvalues in <IMG
 WIDTH="15" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img553.png"
 ALT="$\cal I$">
and the nearest
other eigenvalue. Thus, a cluster<A NAME="11304"></A> of
close eigenvalues which is
far away from any other eigenvalue may have a well determined
invariant subspace 
<!-- MATH
 ${\hat{\cal S}}$
 -->
<IMG
 WIDTH="16" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="img320.png"
 ALT="$\hat{\cal S}$">
even if its individual eigenvectors are
ill-conditioned<A NAME="tex2html2204"
 HREF="footnode.html#foot13239"><SUP>4.3</SUP></A>.

</BLOCKQUOTE>

<P>
In the special case of a real symmetric tridiagonal matrix <B><I>T</I></B>, the eigenvalues
and eigenvectors can be computed much more accurately. xSYEV (and the other
symmetric eigenproblem drivers) computes the eigenvalues and eigenvectors of
a dense symmetric matrix by first reducing it to tridiagonal form<A NAME="11307"></A> <B><I>T</I></B>, and then
finding the eigenvalues and eigenvectors of <B><I>T</I></B>.
Reduction of a dense matrix to tridiagonal form<A NAME="11308"></A> <B><I>T</I></B> can introduce
additional errors, so the following bounds for the tridiagonal case do not
apply to the dense case.

<P>
<BLOCKQUOTE>
The eigenvalues of <B><I>T</I></B> may be computed with small componentwise relative
backward error
<A NAME="11310"></A>
<A NAME="11311"></A>
(<IMG
 WIDTH="39" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img250.png"
 ALT="$O(\epsilon)$">)
by using subroutine xSTEBZ (subsection
<A NAME="11312"></A><A NAME="11313"></A>
<A HREF="node48.html#subseccompsep">2.4.4</A>)
<A NAME="11315"></A><A NAME="11316"></A>
or driver xSTEVX (subsection <A HREF="node29.html#subsecdriveeig">2.3.4</A>). If <B><I>T</I></B> is also positive definite,
they may also be computed at least as accurately by
xPTEQR<A NAME="11318"></A><A NAME="11319"></A><A NAME="11320"></A><A NAME="11321"></A>
or
xSTEGR<A NAME="11322"></A><A NAME="11323"></A><A NAME="11324"></A><A NAME="11325"></A>
(subsection <A HREF="node48.html#subseccompsep">2.4.4</A>).
To compute error bounds
for the computed eigenvalues 
<!-- MATH
 $\hat{\lambda}_i$
 -->
<IMG
 WIDTH="20" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img530.png"
 ALT="$\hat{\lambda}_i$">
we must make some assumptions
about <B><I>T</I></B>. The bounds discussed here are from [<A
 HREF="node151.html#barlowdemmel">14</A>].
Suppose <B><I>T</I></B> is positive definite, and
write <B><I>T</I>=<I>DHD</I></B> where 
<!-- MATH
 $D = {\mbox {\rm diag}}( t_{11}^{1/2} , \ldots , t_{nn}^{1/2})$
 -->
<IMG
 WIDTH="182" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="img556.png"
 ALT="$D = {\mbox {\rm diag}}( t_{11}^{1/2} , \ldots , t_{nn}^{1/2})$">
and <B><I>h</I><SUB><I>ii</I></SUB>= 1</B>.
Then the computed eigenvalues

<!-- MATH
 $\hat{\lambda}_i$
 -->
<IMG
 WIDTH="20" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img530.png"
 ALT="$\hat{\lambda}_i$">
can differ from true eigenvalues <IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img523.png"
 ALT="$\lambda_i$">
by
</BLOCKQUOTE>
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
| \hat{\lambda}_i - \lambda_i | \leq p(n) \cdot \epsilon \cdot \kappa_2(H)
\cdot \lambda_i
\end{displaymath}
 -->


<IMG
 WIDTH="224" HEIGHT="31" BORDER="0"
 SRC="img557.png"
 ALT="\begin{displaymath}
\vert \hat{\lambda}_i - \lambda_i \vert \leq p(n) \cdot \epsilon \cdot \kappa_2(H)
\cdot \lambda_i
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><BLOCKQUOTE>
where <B><I>p</I>(<I>n</I>)</B> is a modestly growing function of <B><I>n</I></B>.
Thus if <IMG
 WIDTH="52" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img558.png"
 ALT="$\kappa_2 (H)$">
is moderate, each eigenvalue will be computed
to high relative accuracy, <A NAME="11333"></A> no matter how tiny it is.
The eigenvectors <IMG
 WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img527.png"
 ALT="$\hat{z}_i$">
computed by xPTEQR
can differ from true eigenvectors <B><I>z</I><SUB><I>i</I></SUB></B> by
at most about
</BLOCKQUOTE>
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\theta ( \hat{z}_i , z_i ) \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}}\frac{p(n) \cdot \epsilon \cdot \kappa_2 (H)}
{{\rm relgap}_i}
\end{displaymath}
 -->


<IMG
 WIDTH="194" HEIGHT="48" BORDER="0"
 SRC="img559.png"
 ALT="\begin{displaymath}
\theta ( \hat{z}_i , z_i ) \mathrel{\raisebox{-.75ex}{$\math...
...\frac{p(n) \cdot \epsilon \cdot \kappa_2 (H)}
{{\rm relgap}_i}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><BLOCKQUOTE>
if <IMG
 WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img455.png"
 ALT="$p(n) \epsilon$">
is small enough, where

<!-- MATH
 ${\rm relgap}_i = \min_{j \neq i} | \lambda_i - \lambda_j |/ ( \lambda_i + \lambda_j )$
 -->
<IMG
 WIDTH="274" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img560.png"
 ALT="${\rm relgap}_i = \min_{j \neq i} \vert \lambda_i - \lambda_j \vert/ ( \lambda_i + \lambda_j )$">
is the <B>relative gap</B> between <IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img523.png"
 ALT="$\lambda_i$">
and the nearest other eigenvalue.
<A NAME="11338"></A><A NAME="11339"></A>
Since the relative gap may be much larger than the absolute gap, this
error bound may be much smaller than the previous one.
Independent of the difference between the computed and true eigenvectors,
the computed eigenvectors are orthogonal to nearly full precision,
i.e. 
<!-- MATH
 $| \hat{z}_i^T \hat{z}_j | = O( \epsilon )$
 -->
<IMG
 WIDTH="106" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img561.png"
 ALT="$\vert \hat{z}_i^T \hat{z}_j \vert = O( \epsilon )$">
for <IMG
 WIDTH="42" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img549.png"
 ALT="$i \neq j$">.
</BLOCKQUOTE>
<P>
<BLOCKQUOTE><IMG
 WIDTH="52" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img558.png"
 ALT="$\kappa_2 (H)$">
could be computed by applying
xPTCON (subsection <A HREF="node38.html#subseccomplineq">2.4.1</A>) to <B><I>H</I></B>.
<A NAME="11343"></A><A NAME="11344"></A><A NAME="11345"></A><A NAME="11346"></A>
The relative gaps are easily computed from the
array of computed eigenvalues.
</BLOCKQUOTE>
<P>

<P>
<HR>
<!--Navigation Panel-->
<A NAME="tex2html5450"
 HREF="node91.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
 SRC="next_motif.png"></A> 
<A NAME="tex2html5444"
 HREF="node89.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
 SRC="up_motif.png"></A> 
<A NAME="tex2html5440"
 HREF="node89.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
 SRC="previous_motif.png"></A> 
<A NAME="tex2html5446"
 HREF="node1.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
 SRC="contents_motif.png"></A> 
<A NAME="tex2html5448"
 HREF="node152.html">
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
 SRC="index_motif.png"></A> 
<BR>
<B> Next:</B> <A NAME="tex2html5451"
 HREF="node91.html">Error Bounds for the</A>
<B> Up:</B> <A NAME="tex2html5445"
 HREF="node89.html">Error Bounds for the</A>
<B> Previous:</B> <A NAME="tex2html5441"
 HREF="node89.html">Error Bounds for the</A>
 &nbsp <B>  <A NAME="tex2html5447"
 HREF="node1.html">Contents</A></B> 
 &nbsp <B>  <A NAME="tex2html5449"
 HREF="node152.html">Index</A></B> 
<!--End of Navigation Panel-->
<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
</ADDRESS>
</BODY>
</HTML>