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<P>

SPMD programs can be built upon
<CODE>Send</CODE> and <CODE>Recv</CODE> commands.  Typically
the program contains <CODE>if</CODE> and <CODE>else</CODE> commands
that specify different actions for different processes.
For example, suppose the m-file <CODE>cycle.m</CODE> consists of
the following program:

<BLOCKQUOTE><CODE><PRE>
if ID==0                % first process: send
   a = 1
   Send(ID+1,a)
elseif ID == Nproc-1    % last process: receive and double
   a = 2*Recv
else                    % middle processes: receive, double, and send
   a = 2*Recv
   Send(ID+1,a)
end;
</PRE></CODE></BLOCKQUOTE>

Process 0 creates the variable <CODE>a</CODE> with value 1
and sends it to process 1.  Process 1 receives the message,
doubles the value of <CODE>a</CODE>, and sends it along to process
2; and so on.  If there are six processors the command
<CODE>Eval( 'cycle' )</CODE> produces the output

<BLOCKQUOTE><CODE>
a = 1<BR>
a = 2<BR>
a = 4<BR>
a = 8<BR>
a = 16<BR>
a = 32<BR>
</CODE></BLOCKQUOTE>

The processes run asynchronously, but since each
<CODE>Send</CODE> command is only executed after the corresponding
<CODE>Recv</CODE> has completed, the proper sequence of computations
and final value 32 are guaranteed so long as all of the
nodes are functioning.
<P>

Alternatively, a MultiMATLAB command is
available for explicit synchronization.  The command

<BLOCKQUOTE><CODE>
Barrier
</CODE></BLOCKQUOTE>

returns only when called on each process.

<H3>2.4. Bcast, Min, Max, Sum</H3>

Although <CODE>Send</CODE> takes a vector of processor
IDs as its destination list, the underlying idea is
that of point-to-point communication.  For more efficient
communication between multiple processes, as well as greater
convenience for the programmer, MultiMATLAB also has various commands
for collective communication.  These commands must be
evaluated simultaneously on all processes.
<P>

The <CODE>Bcast</CODE> command is used to broadcast a matrix from
one process to all other processes, using a tree-structured algorithm.
For example,

<BLOCKQUOTE><CODE>
Eval( 'Bcast(1,ID)' )
</CODE></BLOCKQUOTE>

returns the number 1 on all processes.  <CODE>Bcast</CODE> is much more
efficient than a corresponding <CODE>Send</CODE> and <CODE>Recv</CODE>.
<P>

The same kind of a tree algorithm is used for various computations
that reduce data from many processes to one.  For example, the commands 
<CODE>Min</CODE>, <CODE>Max</CODE>, and <CODE>Sum</CODE>
compute vectors obtained by reducing data over the copies of a vector or
matrix located on all processors.   Thus the command

<BLOCKQUOTE><CODE>
Eval( 'Sum(1,[1 ID Nproc])' )
</CODE></BLOCKQUOTE>

executed on six processes will return the vector
<BLOCKQUOTE><CODE>
[6 15 36]
</CODE></BLOCKQUOTE>
to process 1.
If the first argument is omitted, the result is returned (broadcast) to
all processes.
<P>

<H3>2.5. Higher-level MultiMATLAB Commands</H3>

The MultiMATLAB commands described so far represent
communication primitives as they are used in the
message-passing paradigm of programming.  One of the aims of this
project, however, is to provide also an interface at a higher level
by building on these routines, hiding
as much of the message passing as possible.
<P>

We can do this
by taking a data-parallel approach in a simplistic fashion.  We have
developed a number of routines such as <CODE>Distribute</CODE> and
<CODE>Collect</CODE> that
allow a user to distribute a matrix or to collect a set of matrices
into one large matrix.  These functions operate using a mask that
indicates which processors hold which portions of the matrix.  This
allows us also to develop routines such as <CODE>Shift</CODE>
and <CODE>Copy</CODE> that are useful in data-parallel computing,
keeping the communication to a more abstract level.
<P>

Additional geometry routines such as <CODE>Grid</CODE> and <CODE>Coord</CODE>
have also been constructed that allow the user to create
a grid of processors in 1,2 or 3 dimensions. These
provide a powerful tool for more sophisticated parallel coding.  An optional
argument on the communication routines allows communication within a
given set of nodes, for example along a column or row of the grid.
We do not give further details, as these facilities are under development.

<P><BR><P>
<H2>3. Multiprocessor Graphics</H2>

One of the great strengths of MATLAB is graphics.  A primary
goal of the MultiMATLAB project has been to ensure that this
strength carries over to multiprocessor computations.
<P>

In many applications, the user will find it most convenient to
compute on multiple processors but produce plots on the master 
process, after sending data as necessary.  Equally often, however,
it may be desirable to produce plots in a distributed fashion that
are then sent to the user's screen.  This can be particularly useful
when one wishes to monitor the progress of computations on several
processors graphically.
<P>

We have found the following simple method of doing this to be very useful.
As mentioned above, many calculations
with a geometric flavor divide easily into, say, four or eight
subdomains assigned to a corresponding set of processors.  We
set up a MATLAB figure window in each process and arrange them in
a grid on the screen.  This is easily done using standard MATLAB
handle graphics commands, and we expect shortly to develop
MultiMATLAB commands for this purpose that are integrated with
the grid operations mentioned earlier.
<P>

The figure below shows an example of
this kind of computing;
in this case we have a 4 by 1 grid
of windows.  In this particular example, what has been
computed are the pseudospectra of a 64 by 64 matrix known as
the "Grcar matrix" [<!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><!WA7><A HREF=#ref_trefethen>17</A>]. 
This is an easy application for MultiMATLAB
since the computation requires a very large number of floating point
operations (1024
singular value decompositions of dimension 64 by 64) but minimal communication
(just the global minimum and maximum
of the data with <CODE>Min</CODE> and <CODE>Max</CODE>,
so that all panels can be on the same scale). 

<CENTER>
<PRE>
<!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><!WA8><img src="http://www.cs.cornell.edu/Info/People/lnt/pseudospectra.gif"></PRE>
</CENTER>

Another kind of application that might benefit from this kind of graphics would be
as follows.  Suppose we wish to solve the wave equation by an explicit
finite difference scheme and watch waves bounce around in the computational
domain.   It is a straightforward matter to divide the computation into
a grid of processors as in the figure above, communicating just one row
or column of boundary data to adjacent processors at each step.  Waves
can then be seen to propagate from one window to another.  This kind
of visualization can be very convenient for interactive experimentation, and
higher-quality plots can be produced at selected time steps as necessary
by sending data to a single processor.
<P>

Our second computed example illustrates the use of multiple
figure windows for monitoring a process of numerical optimization.
MATLAB contains powerful programs
for minimization of functions of several variables; one of the original such
programs is <CODE>fminu</CODE>.  Unfortunately, such programs generally
find local minima, not global ones.  If one requires the global minimum
it is customary to run the search multiple times from distinct initial points,
which in many cases might as well be taken to be random.  With sufficiently
many trials leading to a single smallest minimum found over and over again,
one acquires confidence that the global minimum has been found, but the
cost of this confidence may be considerable computing time.
<P>

Such a problem is easily parallelizable, and the next figure
shows a case
in which it has been distributed to four processors.  A function
<VAR>f(x,y)</VAR> of two variables has been constructed that has many local
minima but just one global minimum, the value 0 taken at the origin.
On each of four processors, the optimization is carried out from twenty
random initial points, and the result is displayed in the corresponding figure
window as a straight line from the initial guess to the converged value.
The background curves are contours of the objective function 
<VAR>f(x,y)</VAR>.  Note that in three of the windows, the smallest
value obtained is <VAR>f(x,y)</VAR>=0.1935, whereas the fourth window
has found the global minimum <VAR>f(x,y)</VAR>=0. 

<CENTER>
<PRE>
<!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><!WA9><img src="http://www.cs.cornell.edu/Info/People/lnt/opt4.gif"></PRE>
</CENTER>

In these examples we have set up a grid of windows, one to each processor.
As an alternative it might be desirable sometimes to have multiple MATLAB processes
all draw to one common window.
This arrangement is possible within XWindows, for example.  However,
it is not possible within MultiMATLAB at present, because a figure's window ID
is a read-only property in the current version of MATLAB, which cannot be
set or reset by the user.


<P><BR><P>
<H2>4. Implementation of MultiMATLAB</H2>

MultiMATLAB is built upon 
<!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><!WA10><A HREF=http://www.mcs.anl.gov/mpi/index.html>MPI</A>
(Message Passing Interface), a
highly functional and portable message passing standard