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<H1><A NAME="SECTION03350000000000000000">
LAPACK Benchmark</A>
</H1>

<P>
This section contains performance numbers for selected LAPACK driver
routines.
These routines provide complete solutions for the most common problems of
numerical
linear algebra, and are the routines users are most likely to call:

<P>

<UL><LI>Solve an <B><I>n</I></B>-by-<B><I>n</I></B> system of linear equations with 1 right hand side
using DGESV.
<A NAME="8029"></A>
side

<LI>Find only the eigenvalues of an <B><I>n</I></B>-by-<B><I>n</I></B> nonsymmetric matrix using
DGEEV.
<A NAME="8030"></A>

<LI>Find the eigenvalues and right eigenvectors of an <B><I>n</I></B>-by-<B><I>n</I></B>
nonsymmetric matrix using
<BR>
DGEEV.
DSYEVD.
matrix using DSYEVD.

<LI>Find only the singular values of an <B><I>n</I></B>-by-<B><I>n</I></B> matrix using DGESDD.
<A NAME="8031"></A>

<LI>Find the singular values and right and left singular vectors of an
<B><I>n</I></B>-by-<B><I>n</I></B> matrix using 
<BR>
DGESVD.
<A NAME="8032"></A>

<LI>Find the singular values and right and left singular vectors of an
<B><I>n</I></B>-by-<B><I>n</I></B> matrix using 
<BR>
DGESDD.

</UL>

<P>
We only present data on DGESDD for singular values only, and not DGESVD,
because
both use the same algorithm. We include both DGESVD and DGESDD for computing
all the singular values and singular vectors to illustrate the speedup of
the new algorithm DGESDD over its predecessor DGESVD:
For 1000-by-1000 matrices DGESDD is between 6 and 7 times faster than
DGESVD on most machines.

<P>
The above drivers are timed on a variety of computers. In addition, we
present
data on fewer machines to compare the performance of
the five different routines for solving linear least squares problems,
and several different
routines for the symmetric eigenvalue problem. Again, the purpose is to
illustrate the performance improvements in LAPACK 3.0.

<P>
Data is provided for PCs, shared memory parallel
computers, and high performance workstations. All timings were obtained by
using the machine-specific optimized BLAS available on each machine.  For
machines running the Linux operating system, the ATLAS[<A
 HREF="node151.html#atlas_sc98">102</A>] BLAS
were used.
In all cases the data consisted
of 64-bit floating point numbers (double precision).
For each machine and each driver, a small problem
(<B><I>N</I>=100</B> with <B><I>LDA</I>=101</B>) and a large problem (<B><I>N</I>=1000</B> with <B><I>LDA</I>=1001</B>) were
run.  Block sizes NB = 1, 16, 32 and 64 were
tried, with data only for the fastest run reported in the tables below.
For DGEEV, <B><I>ILO</I>=1</B> and
<B><I>IHI</I>=<I>N</I></B>.  The test matrices were generated with randomly distributed
entries.
All run times are reported in seconds, and block size is denoted by
<B><I>nb</I></B>.  The value of <B><I>nb</I></B> was chosen to make <B><I>N</I>=1000</B> optimal.  It is not
necessarily the best choice for <B><I>N</I>=100</B>.  See Section&nbsp;<A HREF="node131.html#secilaenv">6.2</A>
for details.

<P>
The performance data is reported using three or four statistics. First, the
run-time in seconds is given. The second statistic measures
how well our performance compares to the speed of the BLAS, specifically
DGEMM.  This ``equivalent matrix multiplies'' statistic is calculated as
<DIV ALIGN="CENTER">

<!-- MATH
 $\frac{\rm run-time(LAPACK~Driver~on~an~{\it n}-by-{\it n}~matrix)}{\rm run-time(DGEMM~used~to~multiply~two~{\it n}-by-{\it n}~matrices)}$
 -->
<IMG
 WIDTH="378" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="img252.png"
 ALT="$\frac{\rm run-time(LAPACK~Driver~on~an~{\it n}-by-{\it n}~matrix)}{\rm run-time(DGEMM~used~to~multiply~two~{\it n}-by-{\it n}~matrices)}$">
</DIV>
and labeled as 
<!-- MATH
 $\frac{\rm Time}{\rm T(MM)}$
 -->
<IMG
 WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img253.png"
 ALT="$\frac{\rm Time}{\rm T(MM)}$">
in the tables.
The performance information for the BLAS routines 
<BR>
DGEMV (TRANS='N')
and DGEMM (TRANSA='N', TRANSB='N') is provided in Table&nbsp;<A HREF="node71.html#emmtable">3.12</A>,
along with the clock speed for each machine in
Tables&nbsp;<A HREF="node67.html#tab:node1">3.2</A>, <A HREF="node67.html#tab:node2">3.3</A>, <A HREF="node67.html#tab:node3">3.4</A>, <A HREF="node67.html#tab:node4">3.5</A>,
and <A HREF="node67.html#tab:node5">3.6</A>.
The third statistic is the true megaflop rating. For
the eigenvalue and singular value drivers, a fourth ``synthetic megaflop''
statistic is also presented. We provide this statistic because the number
of floating point operations needed to find eigenvalues and singular values
depends on the input data, unlike linear equation solving or linear least
squares solving with DGELS, and on the algorithm.
The synthetic megaflop rating is defined
to be a ``standard'' number of flops required to solve the problem, divided
by the run-time in microseconds.  This ``standard'' number of flops is taken
to be the average for a standard algorithm over a variety of problems, as
given in Table&nbsp;<A HREF="node71.html#standardflopcount">3.13</A>. In all cases we ignore terms of
<B><I>O</I>(<I>N</I><SUP>2</SUP>)</B>.
The flop count in this table for the nonsymmetric eigenvalue problem
assumes the unblocked algorithm is used, and that two QR sweeps are needed
to deflate each eigenvalue [<A
 HREF="node151.html#GVL2">55</A>]. The flop count for the symmetric
eigenproblem
and SVD assumes that all the work done on the tridiagonal and bidiagonal
matrices
is <B><I>O</I>(<I>N</I><SUP>2</SUP>)</B>, as is the case with xSTEGR and will be the case with the SVD
in the future.
<A NAME="8052"></A><A NAME="8053"></A><A NAME="8054"></A><A NAME="8055"></A>

<P>
We also include several figures comparing the speed of
several routines for the symmetric eigenvalue problem
and
several least squares drivers
to highlight the
performance improvements in LAPACK 3.0.

<P>
First consider Figure&nbsp;<A HREF="node71.html#fig:STEVcomparison">3.1</A>,
which compares the performance of three routines,
DSTEQR, DSTEDC and DSTEGR, for computing all the
eigenvalues and eigenvectors of
a symmetric tridiagonal matrix.
The times are shown on a Compaq AlphaServer DS-20 for
matrix dimensions from 100 to 1000.
The symmetric tridiagonal matrix was
obtained by taking a random dense symmetric matrix and reducing it to
tridiagonal
form (the performance can vary depending on the distribution of the
eigenvalues
of the matrix, but the data shown here is typical).
DSTEQR (used in driver DSYEV) was the only algorithm available in LAPACK
1.0,
DSTEDC (used in driver DSYEVD) was introduced in LAPACK 2.0, and
DSTEGR (used in driver DSYEVR) was introduced in LAPACK 3.0.
As can be seen, for large matrices DSTEGR is about 14 times faster than
DSTEDC and
nearly 50 times faster than DSTEQR.
<A NAME="8057"></A><A NAME="8058"></A><A NAME="8059"></A>
<A NAME="8060"></A><A NAME="8061"></A><A NAME="8062"></A>

<P>
Next consider Figure&nbsp;<A HREF="node71.html#fig:SYEVcomparison">3.2</A>,
which compares the performance of four driver routines,
DSYEV, DSYEVX, DSYEVD and DSYEVR, for computing all
the eigenvalues and eigenvectors of a dense symmetric matrix.
The times are shown on an IBM Power 3 for
matrix dimensions from 100 to 2000.
The symmetric matrix was chosen randomly.
The cost of these drivers is essentially the cost of phases 1 and
3 (reduction to tridiagonal form and backtransformation) plus
the cost of phase 2 (the symmetric tridiagonal eigenproblem) discussed
in the last paragraph. Since the cost of phases 1 and 3 is large,
performance differences in phase 2 are no longer as visible.
We note that if we had chosen a test matrix with a large cluster
of nearby eigenvalues, then the cost of DSYEVX would have been
much larger, without significantly affecting the timings
of the other drivers.  DSYEVR is the driver of choice.
<A NAME="8064"></A><A NAME="8065"></A><A NAME="8066"></A><A NAME="8067"></A>

<P>
Finally consider Figure&nbsp;<A HREF="node71.html#fig:GELScomparison">3.3</A>,
which compares the performance of five drivers for the
linear least squares problem,
DGELS, DGELSY, DGELSX, DGELSD and DGELSS, which are shown in
order of decreasing speed.
DGELS is the fastest. DGELSY and DGELSX use QR with pivoting, and
so handle rank-deficient problems more reliably than DGELS
but can be more expensive.
DGELSD and DGELSS use the SVD, and so are the most reliable
(and expensive) ways to solve rank deficient least squares problems.
DGELS, DGELSX and DGELSS were in LAPACK 1.0, and DGELSY and
DGELSD were introduced in LAPACK 3.0.
The times are shown on a Compaq AlphaServer DS-20 for
squares matrices with dimensions from 100 to 1000, and for one
right-hand-side.
The matrices were chosen at random (which means they are full rank).
First consider DGELSY, which
is meant to replace DGELSX. We can see that the speed of DGELSY is
nearly indistinguishable from the fastest routine DGELS,
whereas DGELSX is over 2.5 times slower for large matrices.
Next consider DGELSD, which is meant to replace DGELSS.
It is 3 to 5 times slower than the fastest routine, DGELS,
whereas its predecessor DGELSS was 7 to 34 times slower.
Thus both DGELSD and DGELSY are significantly faster than
their predecessors.
<A NAME="8069"></A><A NAME="8070"></A><A NAME="8071"></A><A NAME="8072"></A><A NAME="8073"></A>

<P>

<P></P>
<DIV ALIGN="CENTER"><A NAME="fig:STEVcomparison"></A><A NAME="8077"></A>
<TABLE>
<CAPTION><STRONG>Figure 3.1:</STRONG>
Timings of routines for computing all eigenvalues and eigenvectors
of a symmetric tridiagonal matrix.
The upper graph shows times in seconds on a Compaq AlphaServer DS-20.
The lower graph shows times relative to the fastest routine DSTEGR, which
appears as a horizontal line at 1.</CAPTION>
<TR><TD><IMG
 WIDTH="511" HEIGHT="827" BORDER="0"
 SRC="img254.png"
 ALT="\begin{figure}
\centerline{\psfig{file=SEPtbw.eps,width=4.5in}}\centerline{\psfig{file=SEPrbw.eps,width=4.5in}}\end{figure}"></TD></TR>
</TABLE>
</DIV><P></P>

<P>

<P></P>
<DIV ALIGN="CENTER"><A NAME="fig:SYEVcomparison"></A><A NAME="8083"></A>
<TABLE>
<CAPTION><STRONG>Figure 3.2:</STRONG>
Timings of driver routines for computing all eigenvalues and
eigenvectors
of a dense symmetric matrix.
The upper graph shows times in seconds on an IBM Power3.
The lower graph shows times relative to the fastest routine DSYEVR, which
appears as a horizontal line at 1.</CAPTION>
<TR><TD><IMG
 WIDTH="512" HEIGHT="843" BORDER="0"
 SRC="img255.png"
 ALT="\begin{figure}
\centerline{\psfig{file=SEPDtbw.eps,width=4.5in}}\centerline{\psfig{file=SEPDrbw.eps,width=4.5in}}\end{figure}"></TD></TR>
</TABLE>
</DIV><P></P>

<P>

<P></P>
<DIV ALIGN="CENTER"><A NAME="fig:GELScomparison"></A><A NAME="8089"></A>
<TABLE>
<CAPTION><STRONG>Figure 3.3:</STRONG>
Timings of driver routines for the least squares problem.
The upper graph shows times in seconds on a Compaq AlphaServer DS-20.
The lower graph shows times relative to the fastest routine DGELS, which
appears as a horizontal line at 1.</CAPTION>
<TR><TD><IMG
 WIDTH="511" HEIGHT="827" BORDER="0"
 SRC="img256.png"
 ALT="\begin{figure}
\centerline{\psfig{file=LLStbw.eps,width=4.5in}}\centerline{\psfig{file=LLSrbw.eps,width=4.5in}}\end{figure}"></TD></TR>
</TABLE>
</DIV><P></P>

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

<A NAME="emmtable"></A>
<DIV ALIGN="CENTER">
<A NAME="8093"></A>
<TABLE CELLPADDING=3 BORDER="1">
<CAPTION><STRONG>Table 3.12:</STRONG>
Execution time and Megaflop rates for DGEMV and DGEMM</CAPTION>
<TR><TD></TD>
<TD ALIGN="CENTER" COLSPAN=4>DGEMV</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=4>DGEMM</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">  
</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=8>Values of <B><I>n</I>=<I>m</I>=<I>k</I></B></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">  
</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=2><B>100</B></FONT></TD>
<TD ALIGN="CENTER" COLSPAN=2><B>1000</B></FONT></TD>
<TD ALIGN="CENTER" COLSPAN=2><B>100</B></FONT></TD>
<TD ALIGN="CENTER" COLSPAN=2><B>1000</B></FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
</FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> </FONT><FONT SIZE="-1"> Time</FONT><FONT SIZE="-1"> </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> </FONT><FONT SIZE="-1"> Mflops</FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> </FONT><FONT SIZE="-1"> Time</FONT><FONT SIZE="-1"> </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> </FONT><FONT SIZE="-1"> Mflops</FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> </FONT><FONT SIZE="-1"> Time</FONT><FONT SIZE="-1"> </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> </FONT><FONT SIZE="-1"> Mflops</FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> </FONT><FONT SIZE="-1"> Time</FONT><FONT SIZE="-1"> </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> </FONT><FONT SIZE="-1"> Mflops</FONT><FONT SIZE="-1"> </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
Dec Alpha Miata    </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> .0151 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 66 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 27.778 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 36 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> .0018 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 543  </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 1.712 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 584
</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
Compaq AlphaServer DS-20 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> .0027 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 376</FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1">  8.929 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 112</FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> .0019 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 522  </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 2.000 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1">
500 </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
IBM Power 3          </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> .0032 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 304 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 2.857 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 350</FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> .0018 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 567 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 1.385 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 722
</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
IBM PowerPC          </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> .0435 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 23 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 40.000 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 25 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> .0063 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 160 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 4.717 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 212
</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
Intel Pentium II     </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> .0075 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 134 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 16.969 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 59 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> .0031 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 320 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 3.003 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 333
</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
Intel Pentium III    </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> .0071 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 141 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 14.925 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 67 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> .0030 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 333  </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 2.500 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1">
400 </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
SGI O2K (1 proc)     </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> .0046 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 216 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 4.762  </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 210 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1">.0018 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 563  </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 1.801 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1">
555 </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
SGI O2K (4 proc)     </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 5.000 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 0.2  </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 2.375 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 421 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> .0250 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 40 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 0.517 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1">
1936 </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
Sun Ultra 2 (1 proc) </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> .0081 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 124 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 17.544  </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 57 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> .0033 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 302  </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 3.484 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1">
287 </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
Sun Enterprise 450 (1 proc) </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> .0037 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 267 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 11.628  </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 86 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> .0021 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 474 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1">
1.898 </FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> 527 </FONT></TD>
</TR>
</TABLE></DIV>
</DIV>
<BR>

<P>
megaflop
SSYEV/DSYEV.

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

<A NAME="standardflopcount"></A>
<DIV ALIGN="CENTER">
<A NAME="8134"></A>
<TABLE CELLPADDING=3 BORDER="1">
<CAPTION><STRONG>Table 3.13:</STRONG>
``Standard'' floating point operation counts for LAPACK drivers
for <B><I>n</I></B>-by-<B><I>n</I></B> matrices</CAPTION>
<TR><TD ALIGN="LEFT">Driver</TD>
<TD ALIGN="LEFT">Options</TD>
<TD ALIGN="CENTER" COLSPAN=1>Operation</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=1>Count</TD>
</TR>
<TR><TD ALIGN="LEFT">xGESV</TD>
<TD ALIGN="LEFT">1 right hand side</TD>
<TD ALIGN="RIGHT"><IMG
 WIDTH="63" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img257.png"
 ALT="$ .67
\cdot N^3$"></TD>
</TR>
<TR><TD ALIGN="LEFT">xGEEV</TD>
<TD ALIGN="LEFT">eigenvalues only</TD>
<TD ALIGN="RIGHT">
<!-- MATH
 $10.00
\cdot N^3$
 -->
<IMG
 WIDTH="80" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img258.png"
 ALT="$10.00
\cdot N^3$"></TD>
</TR>
<TR><TD ALIGN="LEFT">xGEEV</TD>
<TD ALIGN="LEFT">eigenvalues and right eigenvectors</TD>
<TD ALIGN="RIGHT">
<!-- MATH
 $26.33
\cdot N^3$
 -->
<IMG
 WIDTH="80" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img259.png"
 ALT="$26.33
\cdot N^3$"></TD>
</TR>
<TR><TD ALIGN="LEFT">xGES{VD,DD}</TD>
<TD ALIGN="LEFT">singular values only</TD>
<TD ALIGN="RIGHT">
<!-- MATH
 $2.67
\cdot N^3$
 -->
<IMG
 WIDTH="72" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img260.png"
 ALT="$2.67
\cdot N^3$"></TD>
</TR>
<TR><TD ALIGN="LEFT">xGES{VD,DD}</TD>
<TD ALIGN="LEFT">singular values and left and right singular vectors</TD>
<TD ALIGN="RIGHT">
<!-- MATH
 $6.67
\cdot N^3$
 -->
<IMG
 WIDTH="72" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img261.png"
 ALT="$6.67
\cdot N^3$"></TD>
</TR>
</TABLE>
</DIV>
</DIV>
<BR>

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

<A NAME="tabludriver"></A>
<DIV ALIGN="CENTER">
<A NAME="8149"></A>
<TABLE CELLPADDING=3 BORDER="1">
<CAPTION><STRONG>Table 3.14:</STRONG>
Performance of DGESV for <B><I>n</I></B>-by-<B><I>n</I></B> matrices</CAPTION>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER"><FONT SIZE="-1"> No. of</FONT></TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=6>Values of <B><I>n</I></B></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER"><FONT SIZE="-1"> proc.</FONT></TD>
<TD ALIGN="CENTER"><B><I>nb</I></B></TD>
<TD ALIGN="CENTER" COLSPAN=3>100</TD>
<TD ALIGN="CENTER" COLSPAN=3>1000</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Time</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> 
<!-- MATH
 $\frac{\rm Time}{\rm T(MM)}$
 -->
<IMG
 WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img253.png"
 ALT="$\frac{\rm Time}{\rm T(MM)}$"></FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> Mflops</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Time</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> 
<!-- MATH
 $\frac{\rm Time}{\rm T(MM)}$
 -->
<IMG
 WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img253.png"
 ALT="$\frac{\rm Time}{\rm T(MM)}$"></FONT></TD>
<TD ALIGN="RIGHT"><FONT SIZE="-1"> Mflops</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT">Dec Alpha Miata</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">28</TD>
<TD ALIGN="RIGHT">.004</TD>
<TD ALIGN="RIGHT">2.2</TD>
<TD ALIGN="RIGHT">164</TD>
<TD ALIGN="RIGHT">1.903</TD>
<TD ALIGN="RIGHT">1.11</TD>
<TD ALIGN="RIGHT">351</TD>
</TR>
<TR><TD ALIGN="LEFT">Compaq AlphaServer DS-20</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">28</TD>
<TD ALIGN="RIGHT">.002</TD>
<TD ALIGN="RIGHT">1.05</TD>
<TD ALIGN="RIGHT">349</TD>
<TD ALIGN="RIGHT">1.510</TD>
<TD ALIGN="RIGHT">0.76</TD>
<TD ALIGN="RIGHT">443</TD>
</TR>
<TR><TD ALIGN="LEFT">IBM Power 3</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">32</TD>
<TD ALIGN="RIGHT">.003</TD>
<TD ALIGN="RIGHT">1.67</TD>
<TD ALIGN="RIGHT">245</TD>
<TD ALIGN="RIGHT">1.210</TD>
<TD ALIGN="RIGHT">0.87</TD>
<TD ALIGN="RIGHT">552</TD>
</TR>
<TR><TD ALIGN="LEFT">Intel Pentium II</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">40</TD>
<TD ALIGN="RIGHT">.006</TD>
<TD ALIGN="RIGHT">1.94</TD>
<TD ALIGN="RIGHT">123</TD>
<TD ALIGN="RIGHT">2.730</TD>
<TD ALIGN="RIGHT">0.91</TD>
<TD ALIGN="RIGHT">245</TD>
</TR>
<TR><TD ALIGN="LEFT">Intel Pentium III</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">40</TD>
<TD ALIGN="RIGHT">.005</TD>
<TD ALIGN="RIGHT">1.67</TD>
<TD ALIGN="RIGHT">136</TD>
<TD ALIGN="RIGHT">2.270</TD>
<TD ALIGN="RIGHT">0.91</TD>
<TD ALIGN="RIGHT">294</TD>
</TR>
<TR><TD ALIGN="LEFT">SGI Origin 2000</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.003</TD>
<TD ALIGN="RIGHT">1.67</TD>
<TD ALIGN="RIGHT">227</TD>
<TD ALIGN="RIGHT">1.454</TD>
<TD ALIGN="RIGHT">0.81</TD>
<TD ALIGN="RIGHT">460</TD>
</TR>
<TR><TD ALIGN="LEFT">SGI Origin 2000</TD>
<TD ALIGN="CENTER">4</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.004</TD>
<TD ALIGN="RIGHT">0.16</TD>
<TD ALIGN="RIGHT">178</TD>
<TD ALIGN="RIGHT">1.204</TD>
<TD ALIGN="RIGHT">2.33</TD>
<TD ALIGN="RIGHT">555</TD>
</TR>
<TR><TD ALIGN="LEFT">Sun Ultra 2</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.008</TD>
<TD ALIGN="RIGHT">2.42</TD>
<TD ALIGN="RIGHT">81</TD>
<TD ALIGN="RIGHT">5.460</TD>
<TD ALIGN="RIGHT">1.57</TD>
<TD ALIGN="RIGHT">122</TD>
</TR>
<TR><TD ALIGN="LEFT">Sun Enterprise 450</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.006</TD>
<TD ALIGN="RIGHT">2.86</TD>
<TD ALIGN="RIGHT">114</TD>
<TD ALIGN="RIGHT">3.698</TD>
<TD ALIGN="RIGHT">1.95</TD>
<TD ALIGN="RIGHT">181</TD>
</TR>
</TABLE>
</DIV>
</DIV>
<BR>

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

<A NAME="tabnepdriver1"></A>
<DIV ALIGN="CENTER">
<A NAME="8187"></A>
<TABLE CELLPADDING=3 BORDER="1">
<CAPTION><STRONG>Table 3.15:</STRONG>
Performance of DGEEV, eigenvalues only</CAPTION>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER"><FONT SIZE="-1"> No. of</FONT></TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=8>Values of <B><I>n</I></B></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER"><FONT SIZE="-1"> proc.</FONT></TD>
<TD ALIGN="CENTER"><B><I>nb</I></B></TD>
<TD ALIGN="CENTER" COLSPAN=4>100</TD>
<TD ALIGN="CENTER" COLSPAN=4>1000</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="RIGHT">&nbsp;</TD>
<TD ALIGN="RIGHT">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> True</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Synth</FONT></TD>
<TD ALIGN="RIGHT">&nbsp;</TD>
<TD ALIGN="RIGHT">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> True</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Synth</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Time</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> 
<!-- MATH
 $\frac{\rm Time}{\rm T(MM)}$
 -->
<IMG
 WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img253.png"
 ALT="$\frac{\rm Time}{\rm T(MM)}$"></FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Mflops</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Mflops</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Time</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> 
<!-- MATH
 $\frac{\rm Time}{\rm T(MM)}$
 -->
<IMG
 WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img253.png"
 ALT="$\frac{\rm Time}{\rm T(MM)}$"></FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Mflops</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Mflops</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT">Dec Alpha Miata</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">28</TD>
<TD ALIGN="RIGHT">.157</TD>
<TD ALIGN="RIGHT">87.22</TD>
<TD ALIGN="RIGHT">70</TD>
<TD ALIGN="RIGHT">64</TD>
<TD ALIGN="RIGHT">116.480</TD>
<TD ALIGN="RIGHT">68.04</TD>
<TD ALIGN="RIGHT">81</TD>
<TD ALIGN="RIGHT">86</TD>
</TR>
<TR><TD ALIGN="LEFT">Compaq AS DS-20</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">28</TD>
<TD ALIGN="RIGHT">.044</TD>
<TD ALIGN="RIGHT">23.16</TD>
<TD ALIGN="RIGHT">423</TD>
<TD ALIGN="RIGHT">228</TD>
<TD ALIGN="RIGHT">52.932</TD>
<TD ALIGN="RIGHT">26.47</TD>
<TD ALIGN="RIGHT">177</TD>
<TD ALIGN="RIGHT">189</TD>
</TR>
<TR><TD ALIGN="LEFT">IBM Power 3</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">32</TD>
<TD ALIGN="RIGHT">.060</TD>
<TD ALIGN="RIGHT">33.33</TD>
<TD ALIGN="RIGHT">183</TD>
<TD ALIGN="RIGHT">167</TD>
<TD ALIGN="RIGHT">91.210</TD>
<TD ALIGN="RIGHT">65.86</TD>
<TD ALIGN="RIGHT">103</TD>
<TD ALIGN="RIGHT">110</TD>
</TR>
<TR><TD ALIGN="LEFT">Intel Pentium II</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">40</TD>
<TD ALIGN="RIGHT">.100</TD>
<TD ALIGN="RIGHT">32.26</TD>
<TD ALIGN="RIGHT">110</TD>
<TD ALIGN="RIGHT">100</TD>
<TD ALIGN="RIGHT">107.940</TD>
<TD ALIGN="RIGHT">35.94</TD>
<TD ALIGN="RIGHT">87</TD>
<TD ALIGN="RIGHT">93</TD>
</TR>
<TR><TD ALIGN="LEFT">Intel Pentium III</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">40</TD>
<TD ALIGN="RIGHT">.080</TD>
<TD ALIGN="RIGHT">26.67</TD>
<TD ALIGN="RIGHT">137</TD>
<TD ALIGN="RIGHT">133</TD>
<TD ALIGN="RIGHT">91.230</TD>
<TD ALIGN="RIGHT">36.49</TD>
<TD ALIGN="RIGHT">103</TD>
<TD ALIGN="RIGHT">110</TD>
</TR>
<TR><TD ALIGN="LEFT">SGI Origin 2000</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.074</TD>
<TD ALIGN="RIGHT">41.11</TD>
<TD ALIGN="RIGHT">148</TD>
<TD ALIGN="RIGHT">135</TD>
<TD ALIGN="RIGHT">54.852</TD>
<TD ALIGN="RIGHT">30.46</TD>
<TD ALIGN="RIGHT">172</TD>
<TD ALIGN="RIGHT">182</TD>
</TR>
<TR><TD ALIGN="LEFT">SGI Origin 2000</TD>
<TD ALIGN="CENTER">4</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.093</TD>
<TD ALIGN="RIGHT">3.72</TD>
<TD ALIGN="RIGHT">117</TD>
<TD ALIGN="RIGHT">107</TD>
<TD ALIGN="RIGHT">42.627</TD>
<TD ALIGN="RIGHT">82.45</TD>
<TD ALIGN="RIGHT">222</TD>
<TD ALIGN="RIGHT">235</TD>
</TR>
<TR><TD ALIGN="LEFT">Sun Ultra 2</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.258</TD>
<TD ALIGN="RIGHT">78.18</TD>
<TD ALIGN="RIGHT">43</TD>
<TD ALIGN="RIGHT">38</TD>
<TD ALIGN="RIGHT">246.151</TD>
<TD ALIGN="RIGHT">70.65</TD>
<TD ALIGN="RIGHT">38</TD>
<TD ALIGN="RIGHT">41</TD>
</TR>
<TR><TD ALIGN="LEFT">Sun Enterprise 450</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.178</TD>
<TD ALIGN="RIGHT">84.76</TD>
<TD ALIGN="RIGHT">62</TD>
<TD ALIGN="RIGHT">56</TD>
<TD ALIGN="RIGHT">163.141</TD>
<TD ALIGN="RIGHT">85.95</TD>
<TD ALIGN="RIGHT">57</TD>
<TD ALIGN="RIGHT">61</TD>
</TR>
</TABLE>
</DIV>
</DIV>
<BR>

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

<A NAME="tabnepdriver2"></A>
<DIV ALIGN="CENTER">
<A NAME="8247"></A>
<TABLE CELLPADDING=3 BORDER="1">
<CAPTION><STRONG>Table 3.16:</STRONG>
Performance of DGEEV, eigenvalues and right eigenvectors</CAPTION>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER"><FONT SIZE="-1"> No. of</FONT></TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=8>Values of <B><I>n</I></B></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER"><FONT SIZE="-1"> proc.</FONT></TD>
<TD ALIGN="CENTER"><B><I>nb</I></B></TD>
<TD ALIGN="CENTER" COLSPAN=4>100</TD>
<TD ALIGN="CENTER" COLSPAN=4>1000</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="RIGHT">&nbsp;</TD>
<TD ALIGN="RIGHT">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> True</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Synth</FONT></TD>
<TD ALIGN="RIGHT">&nbsp;</TD>
<TD ALIGN="RIGHT">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> True</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Synth</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Time</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> 
<!-- MATH
 $\frac{\rm Time}{\rm T(MM)}$
 -->
<IMG
 WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img253.png"
 ALT="$\frac{\rm Time}{\rm T(MM)}$"></FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Mflops</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Mflops</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Time</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> 
<!-- MATH
 $\frac{\rm Time}{\rm T(MM)}$
 -->
<IMG
 WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img253.png"
 ALT="$\frac{\rm Time}{\rm T(MM)}$"></FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Mflops</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Mflops</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT">Dec Alpha Miata</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">28</TD>
<TD ALIGN="RIGHT">.308</TD>
<TD ALIGN="RIGHT">171.11</TD>
<TD ALIGN="RIGHT">86</TD>
<TD ALIGN="RIGHT">86</TD>
<TD ALIGN="RIGHT">325.650</TD>
<TD ALIGN="RIGHT">190.22</TD>
<TD ALIGN="RIGHT">73</TD>
<TD ALIGN="RIGHT">81</TD>
</TR>
<TR><TD ALIGN="LEFT">Compaq AS DS-20</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">28</TD>
<TD ALIGN="RIGHT">.092</TD>
<TD ALIGN="RIGHT">48.42</TD>
<TD ALIGN="RIGHT">290</TD>
<TD ALIGN="RIGHT">287</TD>
<TD ALIGN="RIGHT">159.409</TD>
<TD ALIGN="RIGHT">79.70</TD>
<TD ALIGN="RIGHT">149</TD>
<TD ALIGN="RIGHT">165</TD>
</TR>
<TR><TD ALIGN="LEFT">IBM Power 3</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">32</TD>
<TD ALIGN="RIGHT">.130</TD>
<TD ALIGN="RIGHT">72.22</TD>
<TD ALIGN="RIGHT">204</TD>
<TD ALIGN="RIGHT">203</TD>
<TD ALIGN="RIGHT">230.650</TD>
<TD ALIGN="RIGHT">166.53</TD>
<TD ALIGN="RIGHT">103</TD>
<TD ALIGN="RIGHT">114</TD>
</TR>
<TR><TD ALIGN="LEFT">Intel Pentium II</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">40</TD>
<TD ALIGN="RIGHT">.200</TD>
<TD ALIGN="RIGHT">64.52</TD>
<TD ALIGN="RIGHT">133</TD>
<TD ALIGN="RIGHT">132</TD>
<TD ALIGN="RIGHT">284.020</TD>
<TD ALIGN="RIGHT">94.58</TD>
<TD ALIGN="RIGHT">84</TD>
<TD ALIGN="RIGHT">93</TD>
</TR>
<TR><TD ALIGN="LEFT">Intel Pentium III</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">40</TD>
<TD ALIGN="RIGHT">.170</TD>
<TD ALIGN="RIGHT">56.67</TD>
<TD ALIGN="RIGHT">156</TD>
<TD ALIGN="RIGHT">155</TD>
<TD ALIGN="RIGHT">239.070</TD>
<TD ALIGN="RIGHT">95.63</TD>
<TD ALIGN="RIGHT">100</TD>
<TD ALIGN="RIGHT">110</TD>
</TR>
<TR><TD ALIGN="LEFT">SGI Origin 2000</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.117</TD>
<TD ALIGN="RIGHT">65.00</TD>
<TD ALIGN="RIGHT">228</TD>
<TD ALIGN="RIGHT">226</TD>
<TD ALIGN="RIGHT">197.455</TD>
<TD ALIGN="RIGHT">109.64</TD>
<TD ALIGN="RIGHT">121</TD>
<TD ALIGN="RIGHT">133</TD>
</TR>
<TR><TD ALIGN="LEFT">SGI Origin 2000</TD>
<TD ALIGN="CENTER">4</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.159</TD>
<TD ALIGN="RIGHT">6.36</TD>
<TD ALIGN="RIGHT">167</TD>
<TD ALIGN="RIGHT">166</TD>
<TD ALIGN="RIGHT">146.975</TD>
<TD ALIGN="RIGHT">284.28</TD>
<TD ALIGN="RIGHT">164</TD>
<TD ALIGN="RIGHT">179</TD>
</TR>
<TR><TD ALIGN="LEFT">Sun Ultra 2</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.460</TD>
<TD ALIGN="RIGHT">139.39</TD>
<TD ALIGN="RIGHT">58</TD>
<TD ALIGN="RIGHT">58</TD>
<TD ALIGN="RIGHT">601.732</TD>
<TD ALIGN="RIGHT">172.71</TD>
<TD ALIGN="RIGHT">39</TD>
<TD ALIGN="RIGHT">44</TD>
</TR>
<TR><TD ALIGN="LEFT">Sun Enterprise 450</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.311</TD>
<TD ALIGN="RIGHT">148.10</TD>
<TD ALIGN="RIGHT">85</TD>
<TD ALIGN="RIGHT">85</TD>
<TD ALIGN="RIGHT">418.011</TD>
<TD ALIGN="RIGHT">220.24</TD>
<TD ALIGN="RIGHT">57</TD>
<TD ALIGN="RIGHT">63</TD>
</TR>
</TABLE>
</DIV>
</DIV>
<BR>

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

<A NAME="tabsvddriver3"></A>
<DIV ALIGN="CENTER">
<A NAME="8367"></A>
<TABLE CELLPADDING=3 BORDER="1">
<CAPTION><STRONG>Table 3.17:</STRONG>
Performance of DGESDD, singular values only</CAPTION>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER"><FONT SIZE="-1"> No. of</FONT></TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=8>Values of <B><I>n</I></B></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER"><FONT SIZE="-1"> proc.</FONT></TD>
<TD ALIGN="CENTER"><B><I>nb</I></B></TD>
<TD ALIGN="CENTER" COLSPAN=4>100</TD>
<TD ALIGN="CENTER" COLSPAN=4>1000</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="RIGHT">&nbsp;</TD>
<TD ALIGN="RIGHT">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> True</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Synth</FONT></TD>
<TD ALIGN="RIGHT">&nbsp;</TD>
<TD ALIGN="RIGHT">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> True</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Synth</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Time</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> 
<!-- MATH
 $\frac{\rm Time}{\rm T(MM)}$
 -->
<IMG
 WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img253.png"
 ALT="$\frac{\rm Time}{\rm T(MM)}$"></FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Mflops</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Mflops</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Time</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> 
<!-- MATH
 $\frac{\rm Time}{\rm T(MM)}$
 -->
<IMG
 WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img253.png"
 ALT="$\frac{\rm Time}{\rm T(MM)}$"></FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Mflops</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Mflops</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT">Dec Alpha Miata</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">28</TD>
<TD ALIGN="RIGHT">.043</TD>
<TD ALIGN="RIGHT">23.89</TD>
<TD ALIGN="RIGHT">61</TD>
<TD ALIGN="RIGHT">61</TD>
<TD ALIGN="RIGHT">36.581</TD>
<TD ALIGN="RIGHT">21.37</TD>
<TD ALIGN="RIGHT">73</TD>
<TD ALIGN="RIGHT">73</TD>
</TR>
<TR><TD ALIGN="LEFT">Compaq AS DS-20</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">28</TD>
<TD ALIGN="RIGHT">.011</TD>
<TD ALIGN="RIGHT">5.79</TD>
<TD ALIGN="RIGHT">236</TD>
<TD ALIGN="RIGHT">236</TD>
<TD ALIGN="RIGHT">11.789</TD>
<TD ALIGN="RIGHT">5.89</TD>
<TD ALIGN="RIGHT">226</TD>
<TD ALIGN="RIGHT">226</TD>
</TR>
<TR><TD ALIGN="LEFT">IBM Power 3</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">32</TD>
<TD ALIGN="RIGHT">.020</TD>
<TD ALIGN="RIGHT">11.11</TD>
<TD ALIGN="RIGHT">133</TD>
<TD ALIGN="RIGHT">133</TD>
<TD ALIGN="RIGHT">8.090</TD>
<TD ALIGN="RIGHT">5.84</TD>
<TD ALIGN="RIGHT">330</TD>
<TD ALIGN="RIGHT">330</TD>
</TR>
<TR><TD ALIGN="LEFT">Intel Pentium II</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">40</TD>
<TD ALIGN="RIGHT">.040</TD>
<TD ALIGN="RIGHT">12.90</TD>
<TD ALIGN="RIGHT">67</TD>
<TD ALIGN="RIGHT">67</TD>
<TD ALIGN="RIGHT">29.120</TD>
<TD ALIGN="RIGHT">9.70</TD>
<TD ALIGN="RIGHT">92</TD>
<TD ALIGN="RIGHT">92</TD>
</TR>
<TR><TD ALIGN="LEFT">Intel Pentium III</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">40</TD>
<TD ALIGN="RIGHT">.030</TD>
<TD ALIGN="RIGHT">10.00</TD>
<TD ALIGN="RIGHT">89</TD>
<TD ALIGN="RIGHT">89</TD>
<TD ALIGN="RIGHT">25.830</TD>
<TD ALIGN="RIGHT">10.33</TD>
<TD ALIGN="RIGHT">103</TD>
<TD ALIGN="RIGHT">103</TD>
</TR>
<TR><TD ALIGN="LEFT">SGI Origin 2000</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.024</TD>
<TD ALIGN="RIGHT">13.33</TD>
<TD ALIGN="RIGHT">113</TD>
<TD ALIGN="RIGHT">113</TD>
<TD ALIGN="RIGHT">12.407</TD>
<TD ALIGN="RIGHT">6.89</TD>
<TD ALIGN="RIGHT">215</TD>
<TD ALIGN="RIGHT">215</TD>
</TR>
<TR><TD ALIGN="LEFT">SGI Origin 2000</TD>
<TD ALIGN="CENTER">4</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.058</TD>
<TD ALIGN="RIGHT">2.32</TD>
<TD ALIGN="RIGHT">46</TD>
<TD ALIGN="RIGHT">46</TD>
<TD ALIGN="RIGHT">4.926</TD>
<TD ALIGN="RIGHT">9.53</TD>
<TD ALIGN="RIGHT">541</TD>
<TD ALIGN="RIGHT">541</TD>
</TR>
<TR><TD ALIGN="LEFT">Sun Ultra 2</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.088</TD>
<TD ALIGN="RIGHT">26.67</TD>
<TD ALIGN="RIGHT">30</TD>
<TD ALIGN="RIGHT">30</TD>
<TD ALIGN="RIGHT">60.478</TD>
<TD ALIGN="RIGHT">17.36</TD>
<TD ALIGN="RIGHT">44</TD>
<TD ALIGN="RIGHT">44</TD>
</TR>
<TR><TD ALIGN="LEFT">Sun Enterprise 450</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.060</TD>
<TD ALIGN="RIGHT">28.57</TD>
<TD ALIGN="RIGHT">92</TD>
<TD ALIGN="RIGHT">45</TD>
<TD ALIGN="RIGHT">47.813</TD>
<TD ALIGN="RIGHT">25.19</TD>
<TD ALIGN="RIGHT">56</TD>
<TD ALIGN="RIGHT">56</TD>
</TR>
</TABLE>
</DIV>
</DIV>
<BR>

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

<A NAME="tabsvddriver2"></A>
<DIV ALIGN="CENTER">
<A NAME="8427"></A>
<TABLE CELLPADDING=3 BORDER="1">
<CAPTION><STRONG>Table 3.18:</STRONG>
Performance of DGESVD, singular values and left and right singular
vectors</CAPTION>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER"><FONT SIZE="-1"> No. of</FONT></TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=8>Values of <B><I>n</I></B></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER"><FONT SIZE="-1"> proc.</FONT></TD>
<TD ALIGN="CENTER"><B><I>nb</I></B></TD>
<TD ALIGN="CENTER" COLSPAN=4>100</TD>
<TD ALIGN="CENTER" COLSPAN=4>1000</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="RIGHT">&nbsp;</TD>
<TD ALIGN="RIGHT">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> True</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Synth</FONT></TD>
<TD ALIGN="RIGHT">&nbsp;</TD>
<TD ALIGN="RIGHT">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> True</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Synth</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Time</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> 
<!-- MATH
 $\frac{\rm Time}{\rm T(MM)}$
 -->
<IMG
 WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img253.png"
 ALT="$\frac{\rm Time}{\rm T(MM)}$"></FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Mflops</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Mflops</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Time</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> 
<!-- MATH
 $\frac{\rm Time}{\rm T(MM)}$
 -->
<IMG
 WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img253.png"
 ALT="$\frac{\rm Time}{\rm T(MM)}$"></FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Mflops</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Mflops</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT">Dec Alpha Miata</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">28</TD>
<TD ALIGN="RIGHT">.222</TD>
<TD ALIGN="RIGHT">123.33</TD>
<TD ALIGN="RIGHT">77</TD>
<TD ALIGN="RIGHT">30</TD>
<TD ALIGN="RIGHT">320.985</TD>
<TD ALIGN="RIGHT">187.49</TD>
<TD ALIGN="RIGHT">48</TD>
<TD ALIGN="RIGHT">21</TD>
</TR>
<TR><TD ALIGN="LEFT">Compaq AS DS-20</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">28</TD>
<TD ALIGN="RIGHT">.053</TD>
<TD ALIGN="RIGHT">27.89</TD>
<TD ALIGN="RIGHT">326</TD>
<TD ALIGN="RIGHT">126</TD>
<TD ALIGN="RIGHT">142.843</TD>
<TD ALIGN="RIGHT">71.42</TD>
<TD ALIGN="RIGHT">107</TD>
<TD ALIGN="RIGHT">47</TD>
</TR>
<TR><TD ALIGN="LEFT">IBM Power 3</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">32</TD>
<TD ALIGN="RIGHT">.070</TD>
<TD ALIGN="RIGHT">38.89</TD>
<TD ALIGN="RIGHT">245</TD>
<TD ALIGN="RIGHT">95</TD>
<TD ALIGN="RIGHT">251.940</TD>
<TD ALIGN="RIGHT">181.91</TD>
<TD ALIGN="RIGHT">61</TD>
<TD ALIGN="RIGHT">26</TD>
</TR>
<TR><TD ALIGN="LEFT">Intel Pentium II</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">40</TD>
<TD ALIGN="RIGHT">.150</TD>
<TD ALIGN="RIGHT">48.39</TD>
<TD ALIGN="RIGHT">114</TD>
<TD ALIGN="RIGHT">44</TD>
<TD ALIGN="RIGHT">282.550</TD>
<TD ALIGN="RIGHT">94.09</TD>
<TD ALIGN="RIGHT">54</TD>
<TD ALIGN="RIGHT">24</TD>
</TR>
<TR><TD ALIGN="LEFT">Intel Pentium III</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">40</TD>
<TD ALIGN="RIGHT">.120</TD>
<TD ALIGN="RIGHT">40.00</TD>
<TD ALIGN="RIGHT">142</TD>
<TD ALIGN="RIGHT">56</TD>
<TD ALIGN="RIGHT">244.690</TD>
<TD ALIGN="RIGHT">97.88</TD>
<TD ALIGN="RIGHT">62</TD>
<TD ALIGN="RIGHT">27</TD>
</TR>
<TR><TD ALIGN="LEFT">SGI Origin 2000</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.074</TD>
<TD ALIGN="RIGHT">41.11</TD>
<TD ALIGN="RIGHT">232</TD>
<TD ALIGN="RIGHT">90</TD>
<TD ALIGN="RIGHT">176.134</TD>
<TD ALIGN="RIGHT">97.80</TD>
<TD ALIGN="RIGHT">87</TD>
<TD ALIGN="RIGHT">38</TD>
</TR>
<TR><TD ALIGN="LEFT">SGI Origin 2000</TD>
<TD ALIGN="CENTER">4</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.145</TD>
<TD ALIGN="RIGHT">5.80</TD>
<TD ALIGN="RIGHT">118</TD>
<TD ALIGN="RIGHT">46</TD>
<TD ALIGN="RIGHT">198.656</TD>
<TD ALIGN="RIGHT">384.25</TD>
<TD ALIGN="RIGHT">77</TD>
<TD ALIGN="RIGHT">34</TD>
</TR>
<TR><TD ALIGN="LEFT">Sun Ultra 2</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.277</TD>
<TD ALIGN="RIGHT">83.94</TD>
<TD ALIGN="RIGHT">62</TD>
<TD ALIGN="RIGHT">24</TD>
<TD ALIGN="RIGHT">570.290</TD>
<TD ALIGN="RIGHT">163.69</TD>
<TD ALIGN="RIGHT">27</TD>
<TD ALIGN="RIGHT">12</TD>
</TR>
<TR><TD ALIGN="LEFT">Sun Enterprise 450</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.181</TD>
<TD ALIGN="RIGHT">86.19</TD>
<TD ALIGN="RIGHT">95</TD>
<TD ALIGN="RIGHT">37</TD>
<TD ALIGN="RIGHT">402.456</TD>
<TD ALIGN="RIGHT">212.04</TD>
<TD ALIGN="RIGHT">38</TD>
<TD ALIGN="RIGHT">17</TD>
</TR>
</TABLE>
</DIV>
</DIV>
<BR>

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

<A NAME="tabsvddriver4"></A>
<DIV ALIGN="CENTER">
<A NAME="8487"></A>
<TABLE CELLPADDING=3 BORDER="1">
<CAPTION><STRONG>Table 3.19:</STRONG>
Performance of DGESDD, singular values and left and right singular
vectors</CAPTION>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER"><FONT SIZE="-1"> No. of</FONT></TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=8>Values of <B><I>n</I></B></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER"><FONT SIZE="-1"> proc.</FONT></TD>
<TD ALIGN="CENTER"><B><I>nb</I></B></TD>
<TD ALIGN="CENTER" COLSPAN=4>100</TD>
<TD ALIGN="CENTER" COLSPAN=4>1000</TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="RIGHT">&nbsp;</TD>
<TD ALIGN="RIGHT">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> True</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Synth</FONT></TD>
<TD ALIGN="RIGHT">&nbsp;</TD>
<TD ALIGN="RIGHT">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> True</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Synth</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER">&nbsp;</TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Time</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> 
<!-- MATH
 $\frac{\rm Time}{\rm T(MM)}$
 -->
<IMG
 WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img253.png"
 ALT="$\frac{\rm Time}{\rm T(MM)}$"></FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Mflops</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Mflops</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Time</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> 
<!-- MATH
 $\frac{\rm Time}{\rm T(MM)}$
 -->
<IMG
 WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="img253.png"
 ALT="$\frac{\rm Time}{\rm T(MM)}$"></FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Mflops</FONT></TD>
<TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> Mflops</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT">Dec Alpha Miata</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">28</TD>
<TD ALIGN="RIGHT">.055</TD>
<TD ALIGN="RIGHT">30.56</TD>
<TD ALIGN="RIGHT">123</TD>
<TD ALIGN="RIGHT">121</TD>
<TD ALIGN="RIGHT">47.206</TD>
<TD ALIGN="RIGHT">27.57</TD>
<TD ALIGN="RIGHT">141</TD>
<TD ALIGN="RIGHT">141</TD>
</TR>
<TR><TD ALIGN="LEFT">Compaq AS DS-20</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">28</TD>
<TD ALIGN="RIGHT">.021</TD>
<TD ALIGN="RIGHT">11.05</TD>
<TD ALIGN="RIGHT">310</TD>
<TD ALIGN="RIGHT">318</TD>
<TD ALIGN="RIGHT">20.658</TD>
<TD ALIGN="RIGHT">10.33</TD>
<TD ALIGN="RIGHT">323</TD>
<TD ALIGN="RIGHT">323</TD>
</TR>
<TR><TD ALIGN="LEFT">IBM Power 3</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">32</TD>
<TD ALIGN="RIGHT">.025</TD>
<TD ALIGN="RIGHT">13.89</TD>
<TD ALIGN="RIGHT">268</TD>
<TD ALIGN="RIGHT">267</TD>
<TD ALIGN="RIGHT">15.230</TD>
<TD ALIGN="RIGHT">11.00</TD>
<TD ALIGN="RIGHT">438</TD>
<TD ALIGN="RIGHT">438</TD>
</TR>
<TR><TD ALIGN="LEFT">Intel Pentium II</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">40</TD>
<TD ALIGN="RIGHT">.060</TD>
<TD ALIGN="RIGHT">19.35</TD>
<TD ALIGN="RIGHT">112</TD>
<TD ALIGN="RIGHT">111</TD>
<TD ALIGN="RIGHT">44.270</TD>
<TD ALIGN="RIGHT">14.74</TD>
<TD ALIGN="RIGHT">151</TD>
<TD ALIGN="RIGHT">151</TD>
</TR>
<TR><TD ALIGN="LEFT">Intel Pentium III</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">40</TD>
<TD ALIGN="RIGHT">.050</TD>
<TD ALIGN="RIGHT">16.67</TD>
<TD ALIGN="RIGHT">134</TD>
<TD ALIGN="RIGHT">133</TD>
<TD ALIGN="RIGHT">38.930</TD>
<TD ALIGN="RIGHT">15.57</TD>
<TD ALIGN="RIGHT">171</TD>
<TD ALIGN="RIGHT">171</TD>
</TR>
<TR><TD ALIGN="LEFT">SGI Origin 2000</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.035</TD>
<TD ALIGN="RIGHT">19.44</TD>
<TD ALIGN="RIGHT">189</TD>
<TD ALIGN="RIGHT">191</TD>
<TD ALIGN="RIGHT">24.985</TD>
<TD ALIGN="RIGHT">13.87</TD>
<TD ALIGN="RIGHT">267</TD>
<TD ALIGN="RIGHT">267</TD>
</TR>
<TR><TD ALIGN="LEFT">SGI Origin 2000</TD>
<TD ALIGN="CENTER">4</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.091</TD>
<TD ALIGN="RIGHT">3.64</TD>
<TD ALIGN="RIGHT">73</TD>
<TD ALIGN="RIGHT">73</TD>
<TD ALIGN="RIGHT">8.779</TD>
<TD ALIGN="RIGHT">16.89</TD>
<TD ALIGN="RIGHT">759</TD>
<TD ALIGN="RIGHT">760</TD>
</TR>
<TR><TD ALIGN="LEFT">Sun Ultra 2</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.149</TD>
<TD ALIGN="RIGHT">45.15</TD>
<TD ALIGN="RIGHT">45</TD>
<TD ALIGN="RIGHT">45</TD>
<TD ALIGN="RIGHT">93.417</TD>
<TD ALIGN="RIGHT">26.81</TD>
<TD ALIGN="RIGHT">72</TD>
<TD ALIGN="RIGHT">71</TD>
</TR>
<TR><TD ALIGN="LEFT">Sun Enterprise 450</TD>
<TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">64</TD>
<TD ALIGN="RIGHT">.102</TD>
<TD ALIGN="RIGHT">48.57</TD>
<TD ALIGN="RIGHT">66</TD>
<TD ALIGN="RIGHT">65</TD>
<TD ALIGN="RIGHT">70.597</TD>
<TD ALIGN="RIGHT">37.20</TD>
<TD ALIGN="RIGHT">94</TD>
<TD ALIGN="RIGHT">94</TD>
</TR>
</TABLE>
</DIV>
</DIV>
<BR>

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