PARALLEL PACKAGE TUTORIAL
Parallel kernels do not have access to the values of variables defined in the master kernel, nor do they have access to locally defined functions.
Mathematica contains a command DistributeDefinitions that makes it easy to transport local variables and definitions to all parallel kernels. The main advantage of this method is that the application package does not need to be installed on the remote kernels. All definitions are sent through the existing connection to the remote kernels.
DistributeDefinitions has the attribute HoldAll to prevent the evaluation of the symbols.
DistributeDefinitions exports the following kinds of definitions: OwnValues, DownValues, , UpValues, , .
DistributeDefinitions sets the attributes of the remote symbols equal to the locally defined attributes, except for attributes such as Protected and Locked.
Any old definitions existing on the remote side are cleared before the new definitions are made.
Start several parallel kernels.
Define a variable with a value and a function with attributes.
The remote kernels do not know the symbol
, as demonstrated with the next evaluation.
Here is a subtle point. The following remote evaluation seems to work, even though the symbols are not defined on the remote side.
The reason is that the remote kernels return the unevaluated expression , because the function and variable are not defined on the remote kernel. The master kernel evaluates the returned results further, but it does so sequentially.
You can easily produce an example where the difference between remote and local evaluation becomes apparent.
On the local kernel, the symbol
, and the Head
On the remote kernels,
stays a symbol, and its head is Symbol
You can export the local definitions to the remote kernels.
Now the remote evaluation gives the same result as the local one.
Automatic Distribution of Definitions
Higher-level parallel commands, such as Parallelize, ParallelTable, ParallelSum, ... will automatically distribute definitions of symbols occurring in their arguments.
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.
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.
from a context other than the default context is not distributed automatically.
As a result, the symbol is returned unevaluated from the remote kernels, and is evaluated only after the parallel computation is done, where $KernelID
DistributeDefinitions["Context`"] exports all definitions for all symbols in the given context. Thus, you can use the following to make all your interactively entered definitions known to the remote kernels.
Exporting the context of a package you have loaded may not have the same effect on the remote kernels as loading the package on each remote kernel. The reason is that loading a package may perform certain initializations and it may also define auxiliary functions in other contexts (such as a private context). Also, a package may load additional auxiliary packages that establish their own contexts.
DistributeDefinitions["Context`"] is useful for exporting contexts for definitions that you have explicitly set up to be used on remote kernels. There is a separate command ParallelNeeds for remote loading of packages.
clears any definitions for symbols in the context on remote kernels.
Loading Packages on Remote Kernels
ParallelNeeds["Context`"] is essentially equivalent to ParallelEvaluate[Needs["Context`"]], but it is remembered, and any newly launched remote kernels will be initialized as well.
Exporting the context of a package you have loaded may not have the same effect on the remote kernels as loading the package on each remote kernel with ParallelNeeds. The reason is that loading a package may perform certain initializations, and it may also define auxiliary functions in other contexts (such as a private context). Also, a package may load additional auxiliary packages that establish their own contexts.
The next two commands load the Mathematica
on the master kernel and all remote kernels.
Note that Mathematica packages available to the master kernel may not be available on remote kernels from older versions of Mathematica.
Example: Eigenvalues of Matrices
gives the desired precision for the computation of the eigenvalues of a random
generates a random
matrix with numeric entries.
measures the time it takes to find the eigenvalues.
It is enough to distribute the definition of the main function
. Any values it depends on will be distributed automatically.
A Sample Run
Here you measure the time it takes to find the eigenvalues of 5×5 to 25×25 matrices. Because the computations may happen on remote computers that differ in their processor speeds, the results do not necessarily form an increasing sequence.
Alternatively, you can perform the same computation on each parallel processor to measure their relative speed. Here you find the speed of calculation of the eigenvalues of a 20×20 matrix on each of the parallel processors.