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<H2><A NAME="SECTION04543000000000000000">Tuning the Distribution Parameters for Better Performance</A></H2>
<P>
<A NAME="subsectuning"> </A>
<A NAME="4261"> </A><A NAME="4262"> </A>
<P>
By adjusting the data distribution of the matrices,
users may be able to achieve 10-50 % greater
performance than by using
the standard data distribution suggested in section <A HREF="node106.html#distmemcomp">5.1.1</A>.
<P>
The performance attained using the standard data distribution
is usually fairly close to optimal; hence, if one is
getting poor performance, it is unlikely that
modifying the data distribution will solve the performance
problem.
<P>
An optimal data distribution
depends upon several factors including
the performance
characteristics
of the hardware,
the ScaLAPACK routine invoked,
and (to a certain
extent) the
problem size.
The algorithms
currently
implemented in
ScaLAPACK
fall
into two main
classes.
<P>
The first class
of algorithms
is distinguished
by the fact
that at each
step a block
of rows or
columns is
replicated
in all process
rows or columns.
Furthermore, the
process row or
column source of this
broadcast operation
is the one immediately
following -- or
preceding depending
on the algorithm --
the process row
or column source
of the broadcast
operation performed
at the previous
step of the algorithm.
The <I>QR</I> factorization
and the right looking
variant of the <I>LU</I>
factorization
are typical
examples of
such algorithms,
where it is thus
possible to
establish and
maintain a
communication
pipeline in
order to overlap
computation and
communication.
The direction
of the pipeline
determines the
best possible
shapes of the
process grid.
For instance,
the <I>LU</I>, <I>QR</I>, and
<I>QL</I> factorizations
perform better
for ``flat''
process grids
(<IMG WIDTH=59 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline17130" SRC="img423.gif">). These
factorizations
perform a reduction
operation for each
matrix column for
pivoting in the
<I>LU</I> factorization
and for computing
the Householder
transformation
in the <I>QR</I> and <I>QL</I>
decompositions.
Moreover, after
this reduction
has been performed,
it is important
to update the
next block of
columns as fast
as possible.
This update is done
by broadcasting
the current block
of columns using
a ring topology,
that is, feeding the
ongoing communication
pipe. Similarly,
the performance
of the <I>LQ</I> and <I>RQ</I>
factorizations
take advantage
of ``tall'' grids
(<IMG WIDTH=59 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline17142" SRC="img424.gif">) for
the same, but
transposed,
reasons.
<P>
The second group
of algorithms
is characterized
by the physical
transposition of
a block of rows
and/or columns
at each step.
Square or near
square grids
are more adequate
from a performance
point of view for
these transposition
operations. Examples
of such algorithms
implemented in
ScaLAPACK include
the right-looking
variant of the
Cholesky factorization,
the matrix inversion
algorithm, and the
reductions to bidiagonal
form (PxGEBRD),
to Hessenberg form
(PxGEHRD), and
to tridiagonal form
(PxSYTRD). It
is interesting to
note that if
square grids are
more efficient
for these matrix
reduction operations,
the corresponding
eigensolver usually
prefers flatter grids.
<P>
Table <A HREF="node130.html#tabsugg">5.17</A>
summarizes this
paragraph and
provides suggestions
for selecting the
most appropriate
shape of the
logical <IMG WIDTH=57 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline12182" SRC="img25.gif">
process grid from
a performance
point of view.
The results
presented in
this table may
need to be refined
depending on
the physical
characteristics
of the physical
interconnection
network.
<P><A NAME="4272"> </A><A NAME="tabsugg"> </A><IMG WIDTH=547 HEIGHT=197 ALIGN=BOTTOM ALT="table4271" SRC="img425.gif"><BR>
<STRONG>Table 5.17:</STRONG> Process grid suggestions for some ScaLAPACK drivers<BR>
<P>
<P>
Assume that
at most <I>P</I> nodes
are available. A
natural question
is: Could
we decide which <IMG WIDTH=94 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline17180" SRC="img426.gif"> process grid
should be used?
Similarly,
depending on
the value of
<I>P</I>, it is not
always possible
to factor <IMG WIDTH=94 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline17184" SRC="img427.gif"> to create an
appropriate
grid shape.
For example,
if the number
of nodes
available is
a prime number
and a square
grid is suitable
with respect to
performance, it
may be beneficial
to let some nodes
remain idle so
that the remaining
nodes can be arranged
in a ``squarer''
grid.
<P>
If the BLACS
implementation
or the interconnection
network features
high latency, a
one-dimensional data
distribution will improve
the performance for
small and medium
problem sizes.
The number
of messages
significantly
impacts the
performance
achieved for
small problem
sizes, whereas
the total
message volume
becomes a
dominant
factor
for medium-sized problems. The
performance
cost due to
floating-point
operations
dominates for
large problem
sizes.
One-dimensional
data distributions
reduce the
total number
of messages
exchanged on the
interconnection
network but
increase the
total volume of message traffic.
Therefore,
one-dimensional data distributions are better
for small problem sizes but are worse
for large problem sizes, especially when
one is using eight or more processors.
<P>
Determining optimal,
or near-optimal,
distribution
block sizes with
respect to performance
<A NAME="4312"> </A> for a given
platform is a
difficult task.
However, it is
empirically true
that as soon as a
good block size
or even a set of
good distribution
parameters is
found, the
performance
is not highly
sensitive to
small changes
of the values
of these parameters.
<P>
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<P><ADDRESS>
<I>Susan Blackford <BR>
Tue May 13 09:21:01 EDT 1997</I>
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