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<H2><A NAME="SECTION03432000000000000000"></A><A NAME="seccomponentwise"></A>
<BR>
Improved Error Bounds
</H2>
<P>
The standard error analysis just outlined has a drawback: by using the
infinity norm
<!-- MATH
$\| \delta \|_{\infty}$
-->
<IMG
WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img391.png"
ALT="$\Vert \delta \Vert _{\infty}$">
to measure the backward error,
entries of equal magnitude in <IMG
WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img366.png"
ALT="$\delta$">
contribute equally to the final
error bound
<!-- MATH
$\kappa (f,z) (\| \delta \|/\|z\|)$
-->
<IMG
WIDTH="128" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img392.png"
ALT="$\kappa (f,z) (\Vert \delta \Vert/\Vert z\Vert)$">.
This means that
if <B><I>z</I></B> is sparse or has some very tiny entries, a normwise backward
stable algorithm may make very large changes in these entries
compared to their original values. If these tiny values are known accurately
by the user, these errors may be unacceptable,
or the error bounds may be unacceptably large.
<P>
For example, consider solving a diagonal system of linear equations <B><I>Ax</I>=<I>b</I></B>.
Each component of the solution is computed accurately by
Gaussian elimination:
<!-- MATH
$x_i = b_i / a_{ii}$
-->
<B><I>x</I><SUB><I>i</I></SUB> = <I>b</I><SUB><I>i</I></SUB> / <I>a</I><SUB><I>ii</I></SUB></B>.
The usual error bound is approximately
<!-- MATH
$\epsilon \cdot \kappa_{\infty}(A) = \epsilon \cdot \max_i |a_{ii}| / \min_i |a_{ii}|$
-->
<IMG
WIDTH="265" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img393.png"
ALT="$\epsilon \cdot \kappa_{\infty}(A) = \epsilon \cdot \max_i \vert a_{ii}\vert / \min_i \vert a_{ii}\vert$">,
which can arbitrarily overestimate the true error, <IMG
WIDTH="12" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img262.png"
ALT="$\epsilon$">,
if at least one
<B><I>a</I><SUB><I>ii</I></SUB></B> is tiny and another one is large.
<P>
LAPACK addresses this inadequacy by providing some algorithms
whose backward error <IMG
WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img366.png"
ALT="$\delta$">
is a tiny relative change in
each component of <B><I>z</I></B>:
<!-- MATH
$| \delta_{i} | = O( \epsilon ) | z_{i} |$
-->
<IMG
WIDTH="108" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img394.png"
ALT="$\vert \delta_{i} \vert = O( \epsilon ) \vert z_{i} \vert$">.
This backward error retains both the sparsity structure of <B><I>z</I></B> as
well as the information in tiny entries. These algorithms are therefore
called <B>componentwise relatively backward stable</B>.
Furthermore, computed error bounds reflect this stronger form of backward
error<A NAME="tex2html2039"
HREF="footnode.html#foot13141"><SUP>4.9</SUP></A>.
<A NAME="10386"></A>
<A NAME="10387"></A>
<A NAME="10388"></A>
<A NAME="10389"></A>
<A NAME="10390"></A>
<P>
If the input data has independent uncertainty in each component,
each component must have at least a small <EM>relative</EM> uncertainty,
since each is a floating-point number.
In this case, the extra uncertainty contributed by the algorithm is not much
worse than the uncertainty in the input data, so
one could say the answer provided by a componentwise
relatively backward stable algorithm is as accurate as the data
warrants [<A
HREF="node151.html#lawn20">1</A>].
<P>
When solving <B><I>Ax</I>=<I>b</I></B> using expert driver xyySVX or computational routine xyyRFS,
for example, we almost always
compute <IMG
WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img295.png"
ALT="$\hat{x}$">
satisfying
<!-- MATH
$(A+E) \hat{x}= b+f$
-->
<IMG
WIDTH="139" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img379.png"
ALT="$(A+E)\hat{x}=b+f$">,
where
<B><I>e</I><SUB><I>ij</I></SUB></B> is a small relative change in <B><I>a</I><SUB><I>ij</I></SUB></B> and
<B><I>f</I><SUB><I>k</I></SUB></B> is a small relative change in <B><I>b</I><SUB><I>k</I></SUB></B>. In particular, if <B><I>A</I></B> is diagonal,
the corresponding error bound is always tiny, as one would
expect (see the next section).
<P>
LAPACK can achieve this accuracy <A NAME="10395"></A>
for linear equation solving,
the bidiagonal singular value decomposition, and
the symmetric tridiagonal eigenproblem,
and provides facilities for achieving this accuracy for least squares problems.
Future versions of LAPACK will also achieve this
accuracy for other linear algebra problems, as discussed below.
<P>
<P>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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