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<H1><A NAME="SECTION034130000000000000000"></A><A NAME="secfastblas"></A>
<BR>
Error Bounds for Fast Level 3 BLAS
</H1>

<P>
The Level 3 BLAS specifications [<A
 HREF="node151.html#blas3">40</A>] specify the input, output
and calling sequence for each routine, but allow freedom of
implementation, subject to the requirement that the routines be
numerically stable<A NAME="13071"></A>.
Level 3 BLAS implementations can therefore be
built using matrix multiplication algorithms that achieve a more
favorable operation count (for suitable dimensions) than the standard
multiplication technique, provided that these ``fast''  algorithms are
numerically stable.  The simplest fast matrix multiplication
technique is Strassen's
method<A NAME="13072"></A><A NAME="13073"></A>, which can
multiply two <B><I>n</I></B>-by-<B><I>n</I></B>
matrices in fewer than 
<!-- MATH
 $4.7 n^{\log_2 7}$
 -->
<IMG
 WIDTH="71" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img908.png"
 ALT="$4.7 n^{\log_2 7}$">
operations, where

<!-- MATH
 $\log_2 7 \approx 2.807$
 -->
<IMG
 WIDTH="109" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img909.png"
 ALT="$\log_2 7 \approx 2.807$">.

<P>
The effect on the results in this chapter of using a fast Level&nbsp;3 BLAS
implementation can be explained as follows. In general, reasonably
implemented fast Level&nbsp;3 BLAS preserve all the bounds presented here
(except those at the end of subsection <A HREF="node98.html#secgendef">4.10</A>), but the constant
<B><I>p</I>(<I>n</I>)</B> may increase somewhat. Also, the iterative refinement
routine<A NAME="13076"></A>
xyyRFS may take more steps to converge.

<P>
This is what we mean by reasonably implemented fast Level&nbsp;3 BLAS.
Here, <B><I>c</I><SUB><I>i</I></SUB></B> denotes a constant depending on the specified matrix dimensions.

<P>
(1) If <B><I>A</I></B> is <B><I>m</I></B>-by-<B><I>n</I></B>, <B><I>B</I></B> is <B><I>n</I></B>-by-<B><I>p</I></B> and <IMG
 WIDTH="18" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="img910.png"
 ALT="$\widehat C$">
is the computed
approximation to <B><I>C</I>=<I>AB</I></B>, then
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\| \widehat C- AB \|_{\infty} \le c_1(m,n,p) \epsilon \|A\|_{\infty}\|B\|_{\infty} + O(\epsilon ^2).
\end{displaymath}
 -->


<IMG
 WIDTH="359" HEIGHT="31" BORDER="0"
 SRC="img911.png"
 ALT="\begin{displaymath}
\Vert \widehat C- AB \Vert _{\infty} \le c_1(m,n,p) \epsilon \Vert A\Vert _{\infty}\Vert B\Vert _{\infty} + O(\epsilon ^2).
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>

<P>
(2)
The computed solution <IMG
 WIDTH="20" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="img797.png"
 ALT="$\widehat{X}$">
to the triangular systems <B><I>TX</I>=<I>B</I></B>,
where <B><I>T</I></B> is <B><I>m</I></B>-by-<B><I>m</I></B> and <B><I>B</I></B> is <B><I>m</I></B>-by-<B><I>p</I></B>, satisfies
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\| T \widehat X- B \|_{\infty} \le c_2(m,p) \epsilon \|T\|_{\infty} \|\widehat X\|_{\infty}
        + O(\epsilon ^2).
\end{displaymath}
 -->


<IMG
 WIDTH="344" HEIGHT="31" BORDER="0"
 SRC="img912.png"
 ALT="\begin{displaymath}
\Vert T \widehat X- B \Vert _{\infty} \le c_2(m,p) \epsilon...
...rt _{\infty} \Vert\widehat X\Vert _{\infty}
+ O(\epsilon ^2).
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
For conventional Level&nbsp;3 BLAS implementations these conditions
hold with 
<!-- MATH
 $c_1(m,n,p) = n^2$
 -->
<B><I>c</I><SUB>1</SUB>(<I>m</I>,<I>n</I>,<I>p</I>) = <I>n</I><SUP>2</SUP></B> and 
<!-- MATH
 $c_2(m,p)= m(m+1)$
 -->
<B><I>c</I><SUB>2</SUB>(<I>m</I>,<I>p</I>)= <I>m</I>(<I>m</I>+1)</B>.
Strassen's method<A NAME="13083"></A> satisfies these
bounds for slightly larger <B><I>c</I><SUB>1</SUB></B> and <B><I>c</I><SUB>2</SUB></B>.

<P>
For further details, and references to fast multiplication techniques,
see&nbsp;[<A
 HREF="node151.html#Demmel-Higham-Wnote22">27</A>].

<P>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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