Wigner proved that the distribution of eigen values from a random symmetric matrix is a semi-circle when the size of the matrix is very large. We test the
semi-circle law with several small sized
100x100 matrices, instead of solving one huge matrix. Because, it is easy to solve several small matrices rather than one very large matrix. In general, ensemble computations are great fit for high throughput computing. In this tutorial, we learn how to concurrently compute the eigen values of ensemble of matrices with MATLAB utilizing the MATLAB
runtime on OSG Connect and see the validity of the famous Wigner's semicircle law.
Fig.1. The probability density as a function of eigen values for a 100x100 random symmetric matrix. The ensemble averaged probability density converges to a semi-circle (red line).
Let us utilize the
tutorial command. In the command prompt, type
$ tutorial matlab-RandomMatrix # Copies input and script files to the directory tutorial-matlab-RandomMatrix.
This will create a directory
tutorial-matlab-RandomMatrix with the following files
wigner_distribution.m # matlab script - computes eigen values of a random matrix and their histogram wigner_distribution # compiled binary of wigner_distribution.m wigner_distribution.submit # condor job description file wigner_distribution.sh # Execution script Log/ # Directory to copy the standard output, error and log files from condor jobs. average_prob.py # utility python script used by post-script.bash post-script.bash # script gathers output data after completing condor jobs
MATLAB script - Wigner's semi circle distribution
The matlab script takes two argument
fnumber. The argument
n defines the size of matrix. The argument
fnumber is used to label the output file. The matlab script generates a random symmetric matrix of size
nxn, computes the eigen values and finds the probability density of the eigen values.
function my_function(n, fnumber) %intialize variables: n is the size of matrix, dx is bin width, fnumber is used to label the output filename filenumber = num2str(fnumber); if ischar(n) n = str2num(n); end dx = 0.05; % Compute eignvalues of a nxn random matrix⋅⋅ rng('shuffle') a = randn(n); M = (a + a')/2; % construct the symetric random matrix⋅ e = eig(M); % solve for the eigen values⋅ e = e/(sqrt(2*n)); %compute histogram [m x] = hist(e, -1.1:dx:1.1); % compute histogram m = m/(n*dx); values = [x;m]; % print outputs outfilename = sprintf ( '%s%s%s', 'prob_wigner', filenumber, '.dat' ); fileID = fopen(outfilename,'w'); fprintf(fileID,'%9.3f %9.4f\n', values); fclose(fileID);
In the above script, the line
rng('shuffle') means the random number is non-repeated. The script
produces the probability density of eigen values in a file
prob_wigner$fnumber.dat where filenumber is an input integer attached with the filename.
MATLAB runtime execution
As mentioned in the lesson on basics of MATLAB compilation, we need to compile the matlab script on a machine with license. At present, OSG connect does not have license for matlab. On a machine with matlab license, invoke the compiler
mcc. We turn off all graphical options (-nodisplay), disable Java (-nojvm), and instruct MATLAB to run this program as a single-threaded application (-singleCompThread).
mcc -m -R -singleCompThread -R -nodisplay -R -nojvm wigner_distribution.m
The flag -m means
c language translation during compilation and the flag
-R is the option for runtime. The compilation would produce the files: wigner_distribution, run_wigner_distribution.sh, mccExcludedFiles.log and readme.txt files.
wigner_distribution is the compiled binary file which we run on OSG Connect as HTCondor job.
Job execution and submission files
Let us take a look at the condor job description file
Universe = vanilla # The job universe is "vanilla" Executable = wigner_distribution.sh # The job execution file which is transferred to worker machine Arguments = 100 $(Process) # "list of arguments": (1) Size of matrix. (2) process ID. transfer_input_files = wigner_distribution # list of file(s) need be transferred to the remote worker machine Output = Log/job.$(Process).out⋅ # standard output Error = Log/job.$(Process).err # standard error Log = Log/job.$(Process).log # log information about job execution requirements = Arch == "X86_64" && HAS_MODULES == True # Check if the worker machine has CVMFS queue 100 # Submit 100 jobs
The above job description instructs condor to submit 100 jobs. The executable is a wrapper
#!/bin/bash⋅ source /cvmfs/oasis.opensciencegrid.org/osg/modules/lmod/current/init/bash module load matlab/2014b chmod +x wigner_distribution ./wigner_distribution $1 $2⋅
that loads the module
matlab/2014b and executes the MATLAB compiled binary
execution requires two
arguments. The first argument is the size of the random matrix and the next argument is a numerical
label attached with the name of the output file.
We submit the job using
condor_submit command as follows
$ condor_submit wigner_distribution.submit //Submit the condor job description file "wigner_distribution.submit"
Now you have submitted an ensemble of 100 MATLAB jobs that solves the eigen values of a random matrix of size 100. The present job should be finished quickly (less than an hour). You can check the status of the submitted job by using the
condor_q command as follows
$ condor_q username # The status of the job is printed on the screen. Here, username is your login name.
Each job produces prob_wigner$(Process).dat file, where $(Process) is the process ID runs from 0 to 99. The probability distribution of eigen values computed for each random matrix is written on a output file.
After all jobs finished running, we find the average probability density and compare with the density from just one matrix to see how the averaging improves the convergence of semi-circle distribution. The
post-script.bash computes the average of probability density, generates
gnuplot plot of
comparing the average with the density distribution from one matrix and finally saves the plot in a
wigner-semi-circle.png. To get the plot from the output data, type
The script calls the python program
average_prob.py to compute the average probability density from all the output files and
gnuplot for plot.
This page was updated on Aug 24, 2019 at 19:00 from tutorials/tutorial-matlab-RandomMatrix/README.md.