PARALLEL PACKAGE TUTORIAL
Parallel Evaluation
Sending Commands to Remote Kernels
Recall that connections to remote kernels, as opened by LaunchKernels, are represented as kernel objects. See Connection Methods for details. The commands in this section take parallel kernels as arguments and use them to carry out computations.
Low-Level Parallel Evaluation
| ParallelEvaluate[cmd,kernel] | sends cmd for evaluation to the parallel kernel kernel, then waits for the result and returns |
| ParallelEvaluate[cmd,{kernels}] | sends cmd for evaluation to the parallel kernels given, then waits for the results and returns them |
| ParallelEvaluate[cmd] | sends cmd for evaluation to all parallel kernels and returns the list of results; equivalent to ParallelEvaluate[cmd, Kernels[]] |
Sending and receiving commands to and from remote kernels.
ParallelEvaluate has the attribute HoldFirst.
You cannot use ParallelEvaluate while a concurrent computation involving ParallelSubmit or WaitAll is in progress. See "Concurrency: Managing Parallel Processes" for details.
Values of Variables
Values of variables defined on the local master kernel are usually not available to remote kernels. If a command you send for evaluation refers to a variable, it usually will not work as expected.
In these examples, a command is always sent to a single kernel.
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The following program will return
False because the symbol
will most likely not have any value at all on the remote kernel.
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A convenient way to insert variable values into unevaluated commands is to use
With, as demonstrated in the following command. The symbol
is replaced by
, then the expression
is sent to the remote kernel, where it evaluates to
True.
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If you need variable values and definitions carried over to the remote kernels, use DistributeDefinitions or shared variables.
Iterators, such as Table and Do, work in the same way with respect to the iterator variable. Therefore, a statement like the following will not do the expected thing.
The variable
will not have a value on the remote kernel.
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You can use the following command to accomplish the intended iteration on the remote kernel. This substitutes the value of
into the argument of
ParallelEvaluate.
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Pattern variables, constants, and pure function variables will work as expected on the remote kernel. Each of the following three examples will produce the expected result.
Formal parameters of pure functions are inserted into the body before the expression is sent to the parallel kernels.
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Pattern variables are also inserted on the right side of the definition.
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Constants are also inserted.
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Parallel Evaluation of Expressions
| ParallelCombine[f,h[e1,e2,...,en],comb] | evaluates  in parallel by distributing chunks  to all kernels and combining the results with comb[] |
| ParallelCombine[f,h[e1,e2,...,en]] | the default combiner comb is h, if h has attribute Flat, and Join otherwise |
Basic parallel dispatch of evaluations.
ParallelCombine[f, h[e1, e2, ..., en], comb] breaks 
into pieces 
, evaluates 
in parallel, then combines the results 
using 
. ParallelCombine has the attribute HoldFirst, so that 
is not evaluated on the master kernel before the parallelization.
The size of the pieces of the input expression is chosen to be proportional to the remote processor speed estimates for optimal load balancing.
ParallelCombine
ParallelCombine is a general and powerful command with default values for its arguments that are suitable for evaluating elements of containers, such as lists and associative functions.
Evaluating List-like Containers
If the result of applying the function f to a list is again a list, ParallelCombine[f, h[e1, e2, ..., en], comb] simply applies f to pieces of the input list and joins the partial results together.
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The result is the same as that of Prime[{1, 2, 3, 4, 5, 6, 7, 8, 9}], but the computation is done in parallel.
If the function is Identity, ParallelCombine simply evaluates the elements 
in parallel.
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If the result of applying the function f to a list is not a list, a custom combiner has to be chosen.
The function Function[li, Count[li, _?OddQ] counts the number of odd elements in a list. To find the total number of odd elements, add the partial results together.
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Evaluating Associative Operations
If the operation h in 
is associative (has attribute Flat), the identity
holds; with the default combiner being h itself, the operation is parallelized in a natural way. Here all numbers are added in parallel.
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Parallel Mapping and Iterators
The commands in this section are fundamental to parallel programming in Mathematica.
| ParallelMap[f,h[e1,e2,...] | evaluates  in parallel |
| ParallelTable[expr,{i,i0,i1,di},{j,j0,j1,dj},...]
| builds Table[expr, {i, i0, i1, di, j, j0, j1, dj, ...] in parallel; parallelization occurs along the first (outermost) iterator  |
| ParallelSum[...],ParallelProduct[...] | computes sums and products in parallel |
Parallel evaluation, mapping, and tables.
ParallelMap[f, h[e1, e2, ...]] is a parallel version of 
evaluating the individual 
in parallel rather than sequentially.
Side Effects
Unless you use shared variables, the parallel evaluations performed are completely independent and cannot influence each other. Furthermore, any side effects, such as assignments to variables, that happen as part of evaluations will be lost. The only effect of a parallel evaluation is that its result is returned at the end.
Examples
First, start several remote kernels.
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The sine function is applied to the given arguments. Each computation takes place on a remote kernel.
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This particular computation is almost certainly too trivial to benefit from parallel evaluation. The overhead required to send the expressions Sin[0], Sin[
], and so on to the remote kernels and to collect the results will be larger than the gain obtained from parallelization.
Factoring integers of the form 
takes more time, so this computation can benefit from parallelization.
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Alternatively, you can use ParallelTable. Here a list of the number of factors in 
is generated.
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Automatic Distribution of Definitions
Parallel commands such as ParallelTable will automatically distribute the values and functions needed, using effectively DistributeDefinitions.
For this parallel table, the function
and the iterator bound
will evaluate on the subkernels, so their definitions need to be distributed to make it work.
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This automatic distribution happens for any functions and variables you define interactively, within the same notebook (technically, for all symbols in the default context). Definitions from other contexts, such as functions from packages, are not distributed automatically.
Lower-level functions, such as ParallelEvaluate, do not perform any automatic distribution of values.
The symbol
will return the value of
$KernelID when evaluated.
On the remote kernels,
$KernelID is nonzero, but the symbol is not evaluated there, only later on the master kernel, where the value is zero.
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After the definition of
is distributed, it is evaluated on the remote kernels.
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Automatic Parallelization
Parallelize[cmd[list, arguments...]] recognizes if cmd is a Mathematica function that operates on a list or other long expression in a way that can be easily parallelized and performs the parallelization automatically.
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Not all uses of these commands can be parallelized. A message is generated and the evaluation is performed sequentially on the master kernel if necessary.
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