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<BR> <P>
<H2><A NAME="SECTION04642000000000000000">Improved Error Bounds</A></H2>
<A NAME="seccomponentwise"> </A>
<P>
The standard error analysis just outlined has a drawback: by using the
infinity norm <IMG WIDTH=36 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline18154" SRC="img559.gif"> to measure the backward error,
entries of equal magnitude in <IMG WIDTH=7 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline18009" SRC="img532.gif"> contribute equally to the final
error bound <IMG WIDTH=120 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline18158" SRC="img560.gif">. This means that
if <I>z</I> is sparse or has some tiny entries, a normwise backward
stable algorithm may make large changes in these entries
compared wit 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 <I>Ax</I>=<I>b</I>.
Each component of the solution is computed accurately by
Gaussian elimination: <IMG WIDTH=78 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline18164" SRC="img561.gif">.
The usual error bound is approximately
<IMG WIDTH=258 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline18166" SRC="img562.gif">,
which can arbitrarily overestimate the true error, <IMG WIDTH=6 HEIGHT=8 ALIGN=BOTTOM ALT="tex2html_wrap_inline17202" SRC="img429.gif">, if at least one
<IMG WIDTH=17 HEIGHT=17 ALIGN=MIDDLE ALT="tex2html_wrap_inline18170" SRC="img563.gif"> is tiny and another one is large.
<P>
LAPACK addresses this inadequacy by providing some algorithms
whose backward error <IMG WIDTH=7 HEIGHT=13 ALIGN=BOTTOM ALT="tex2html_wrap_inline18009" SRC="img532.gif"> is a tiny relative change in
each component of <I>z</I>: <IMG WIDTH=99 HEIGHT=27 ALIGN=MIDDLE ALT="tex2html_wrap_inline18176" SRC="img564.gif">.
This backward error retains both the sparsity structure of <I>z</I> 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="tex2html1247" HREF="footnode.html#8056"><IMG ALIGN=BOTTOM ALT="gif" SRC="http://www.netlib.org/utk/icons/foot_motif.gif"></A>
<A NAME="5075"> </A>
<A NAME="5076"> </A>
<A NAME="5077"> </A>
<A NAME="5078"> </A>
<A NAME="5079"> </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="node189.html#lawn20">4</A>].
<P>
When solving <I>Ax</I>=<I>b</I> using expert driver PxyySVX or computational routine
PxyyRFS,
for example, we almost always
compute <IMG WIDTH=9 HEIGHT=12 ALIGN=BOTTOM ALT="tex2html_wrap_inline17482" SRC="img449.gif"> satisfying <IMG WIDTH=132 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline18075" SRC="img546.gif">, where
<IMG WIDTH=16 HEIGHT=17 ALIGN=MIDDLE ALT="tex2html_wrap_inline18186" SRC="img565.gif"> is a small relative change in <IMG WIDTH=18 HEIGHT=17 ALIGN=MIDDLE ALT="tex2html_wrap_inline18188" SRC="img566.gif"> and
<IMG WIDTH=14 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline18190" SRC="img567.gif"> is a small relative change in <IMG WIDTH=13 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline18192" SRC="img568.gif">. In particular, if <I>A</I> is diagonal,
the corresponding error bound is always tiny, as one would
expect (see the next section).
<P>
ScaLAPACK can achieve this accuracy <A NAME="5084"> </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.
<P>
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<P><ADDRESS>
<I>Susan Blackford <BR>
Tue May 13 09:21:01 EDT 1997</I>
</ADDRESS>
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