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<BLOCKQUOTE><CODE>
Start(5)
</CODE></BLOCKQUOTE>

MATLAB is then started on five additional processors taken from
a predetermined list.
Or perhaps the second author is a sitting at a machine connected
to Cornell's Computer Science Department network.  He
types

<BLOCKQUOTE><CODE>
Start(['gemini'; 'orion'; 'rigel'; 'castor'; 'pollux'])
</CODE></BLOCKQUOTE>

Now MATLAB is started on the five processors with the names
indicated.  (Some names could be repeated, in which case multiple
MATLAB processes would be started on a single processor.)
In either case, when all the processes are started the message is returned,

<BLOCKQUOTE><CODE>
6 MultiMATLAB processes running.
</CODE></BLOCKQUOTE>

This total number of processors can subsequently
be accessed by the MultiMATLAB command <CODE>Nproc</CODE>.
<P>

The standard MultiMATLAB command for executing commands on
one or more processors is <CODE>Eval</CODE>.
If the user now types

<BLOCKQUOTE><CODE>
Eval( 'sqrt(2)' )</CODE></BLOCKQUOTE>

then the MATLAB command <CODE>sqrt(2)</CODE> is executed
on all six processors.
The result is six repetitions of <CODE>1.4142</CODE>,
which is not very interesting.
On the other hand the command

<BLOCKQUOTE><CODE>
Eval( 'ID' )</CODE></BLOCKQUOTE>

calls the MultiMATLAB command <CODE>ID</CODE>
on each of the processors running.  This command
returns the number of the current process, an integer
from 0 to <CODE>Nproc</CODE>-1.  Running it on all
nodes might give the result

<BLOCKQUOTE><CODE>
ans = 0<BR>
ans = 1<BR>
ans = 5<BR>
ans = 2<BR>
ans = 3<BR>
ans = 4</CODE></BLOCKQUOTE>

The ordering of these numbers is arbitrary,
since the processors are not synchronized and output
is sent to the master process as soon as it is ready.
(It is a good idea to
type <CODE>Eval('format compact')</CODE> at the beginning
to keep the
output from the various processes as condensed as possible.)
The command

<BLOCKQUOTE><CODE>
Eval( 'ID^ID' )</CODE></BLOCKQUOTE>

might produce

<BLOCKQUOTE><CODE>
ans = 1<BR>
ans = 1<BR>
ans = 256<BR>
ans = 3125<BR>
ans = 27<BR>
ans = 4</CODE></BLOCKQUOTE>
<P>

In the above examples, in keeping with our orientation toward
SPMD programming, each command passed to <CODE>Eval</CODE> was
executed on all MATLAB processes.  Alternatively, one can
select a subset of the processes by passing two arguments to
the <CODE>Eval</CODE> command, the first being
a vector of process IDs.  Thus

<BLOCKQUOTE><CODE>
Eval( [4 5] , 'cond(hilb(ID))' )</CODE></BLOCKQUOTE>

might return
<BLOCKQUOTE><CODE>
ans = 1.5514e+04<BR>
ans = 4.7661e+05
</CODE>,</BLOCKQUOTE>

the condition numbers of the Hilbert matrices of dimensions
4 and 5, and

<BLOCKQUOTE><CODE>
Eval( 0:2:4 , 'quad(''exp'',ID,ID+1)' )
</CODE></BLOCKQUOTE>
<P>

might return
<BLOCKQUOTE><CODE>
ans = 1.7183<BR>
ans = 93.8151<BR>
ans = 12.6965
</CODE></BLOCKQUOTE>

the integrals of <VAR>e</VAR><SUP><VAR>x</VAR></SUP> from <VAR>n</VAR> to
<VAR>n</VAR>+1 for <VAR>n</VAR> = 0, 2, and 4.
Note how the double quote is used to obtain a string
within a string.  This calling of the MATLAB command
<CODE>quad</CODE> gives a hint of the high-level power
available that is so characteristic of MATLAB.  In this
case, adaptive numerical quadrature has been carried out
to compute the desired integral.  MATLAB users are
accustomed to treating problems like integration, zerofinding,
minimization, and computation of eigenvalues as routine
matters to be handled silently by appropriate single-word commands.
<P>

None of these examples were costly enough for the use of
multiple processors to serve much purpose, but it is
easy to devise such examples.  Suppose we want to
find the spectral radii (maximum of the absolute values of the
eigenvalues) of six
matrices of dimension 400.  The command

<BLOCKQUOTE><CODE>
Eval( 'max(abs(eig(randn(400))))' )</CODE></BLOCKQUOTE>
<P>

does not do the trick; we get six copies of the number
<CODE>20.8508</CODE>, since the random number generators deliver
identical results on all processors.  Preceding the eigenvalue
computation by

<BLOCKQUOTE><CODE>
Eval( 'randn(''seed'',ID)' )</CODE>,</BLOCKQUOTE>
<P>

however, leads to the result desired:

<BLOCKQUOTE><CODE>
ans = 20.9729<BR>
ans = 20.8508<BR>
ans = 21.0364<BR>
ans = 21.0312<BR>
ans = 21.6540<BR>
ans = 20.4072</CODE></BLOCKQUOTE>

(The spectral radius of an <VAR>n</VAR> by <VAR>n</VAR>
random matrix is approximately the square root of <VAR>n</VAR>,
for large <VAR>n</VAR>.)  In a typical experiment
this example might run in 23 seconds on six thin
nodes of our SP2; the elapsed time would be six times greater
if one used a <CODE>for</CODE> loop on a single machine.
Of course, Monte Carlo experiments like this one are
always the easiest examples of embarrassingly parallel computations.
<P>

For simplicity, the examples above call <CODE>Eval</CODE>
with an explicit MATLAB command as an argument string.
For most applications, however, a user will want to execute a program
(an m-file) rather than a single line of text.  A command such as

<BLOCKQUOTE><CODE>
Eval( 'filename' )</CODE></BLOCKQUOTE>

achieves this effect.

<H3>2.2. Put, Get</H3>

So far, we have not communicated between processes except
to send screen output to the master process.  Of course, a nontrivial
MultiMATLAB system depends on such communication.
<P>

One form of communication we have implemented is puts and gets,
executable solely by the master MATLAB process.  For example,
the command

<BLOCKQUOTE><CODE>
Put(1:4,'A')</CODE>,</BLOCKQUOTE>

sends the matrix <CODE>A</CODE> from the master process 0
to processes 1 through 4; an
optional argument permits the name of <CODE>A</CODE> to
be changed at the destination.  The command

<BLOCKQUOTE><CODE>
Get(3,'B')</CODE>,</BLOCKQUOTE>

gets the matrix <CODE>B</CODE> back from process 3 to the master.
<P>

<H3>2.3. Send, Recv, Probe, Barrier</H3>

More general point-to-point communication is accomplished by send and receive
commands, which can be executed on any of the MATLAB processes.
For example, the sequence

<BLOCKQUOTE><CODE>
x = [pi pi^2];<BR>
Send(3,x)<BR>
Eval(3, 'Recv' )<BR>
</CODE></BLOCKQUOTE>

passes a message containing a 2-vector from the master
process to process 3, leading to the output

<BLOCKQUOTE><CODE>
3.1416   9.8696
</CODE></BLOCKQUOTE>

An optional argument can be added in <CODE>Recv</CODE>
to specify the source.  Another optional argument may be added
in both <CODE>Send</CODE> and <CODE>Recv</CODE> to
specify a message tag so as to ensure that
sends and receives are properly matched and to aid in error
checking.
The command

<BLOCKQUOTE><CODE>
Probe</CODE>,</BLOCKQUOTE>

run on any process, again with optional source process
number and message tag, returns 1 (true) if a message