BwUniCluster2.0/Software/Mathematica

Mathematica might not be available on all locations.

Mathematica
module load	math/mathematica
License	Commercial
Citing
Links	Homepage; Documentation;
Graphical Interface	Yes (See VNC)
Included in module

Mathematica is a software from Wolfram for symbolic and numerical computation with many features such as powerful visualization and application specific functions.

Loading

To check if Mathematica is available execute

$ module avail math/mathematica

If Mathematica is available you can load a specific version or you can load the default version with

$ module load math/mathematica

General Usage

Mathematica can be used interactively on the command line or with a graphical front-end. Alternatively Mathematica can run a script in batch mode which is useful when submitting batch jobs to the cluster. After loading Mathematica the different modes can be used as follows.

• Interactive with GUI (needs X11 forwarding or VNC):
  

  $ mathematica

• Interactive with command line:
  

  $ math

• Non-Interactive. Looks like interactive with command line but input is taken from script.m:
  

  $ math < script.m

• Non-Interactive. You have to explicitly specify what you want to print
  

  $ math -script script.m

  However, the output is done in InputForm which is suitable for input in other Mathematica calculations but if you want pretty output you have to change the output format to OutputForm like this
  

  SetOptions[$Output, FormatType -> OutputForm]

For an introduction to Mathematica we refer to the online documentation (Mathematica Documentation Center). Specific information on the use in a compute cluster is in the next section.

Parallel Computation

Obviously parallel computation can be useful to speed up time-consuming computations, but it should also be used for multiple computations with different input data files (e.g. for parametric studies). The reason for this is the license model from Wolfram. There are two types of licenses.

• Each time an instance of Mathematica starts, a so called MathKernel license is used up.
• Each time Mathematica spawns a subprocess, a license called SubMathKernel is used up.

Because usually there are much more SubMathKernel licenses than MathKernel licenses it is recommended to start multiple subprocesses instead of submitting multiple jobs.

Remember to request the correct amount of processors in your Batch Jobs script but note that Mathematica will not automatically use these processors. In general you have to adjust your code to benefit from more cores. To do this you first have to start a number of kernels which are then used by ParallelTable to run the computations in parallel. This basic example computes the first eight square numbers in parallel.

LaunchKernels[8]
f[i_] := i^2
DistributeDefinitions[f]
ParallelTable[f[i], {i,0,7}]
CloseKernels[]

Note that the use of DistributeDefinitions is necessary for f is a user defined function and the definition of this function must be available to all kernels.

The next example is the computation of a numerical solution for the following initial value problem

x'(t) = x(t)^2 - x(t)^3
x(0) = d

It is difficult to solve this equation with high accuracy at the point 1/d. We decrease the step size of the algorithm to see how the execution time and the relative error at the point 1/d change.

d = 0.00001

y[t_] := 1/(ProductLog[(1/d-1)*Exp[1/d-1-t]]+1)	(* analytical solution *)
relerr[t_, s_] := Abs[(y[t] - x[t]/.s)/y[t]]			(* relative error of solution s at time t *)
g[v_] := {v[[1]], v[[2]][[1]], relerr[1/d, v[[2]][[2]][[1]]]}	(* helper function *)

(* compute numerical solutions for 6 different step sizes *)
LaunchKernels[6]
tbl = ParallelTable[{step, Timing[NDSolve[{x'[t] == x[t]^2 - x[t]^3, x[0] == d}, x, {t, 0, 2/d}, MaxStepSize->step, MaxSteps->10000000]]}, {step,5,3,-0.4}]
CloseKernels[]

SetOptions[$Output, FormatType -> OutputForm]    (* better output when called with -script *)
Print[ Grid[Join[{{"Stepsize", "Time", "Error at 1/d"}}, Map[g, tbl]]] ]    (* print the result *)

Mathematica features some further functions similar to ParallelTable which are documented on the Mathematica parallel computing webpage from Wolfram.