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| Hennon Map for a=1.4 and b=0.3. |
Multi-core processors are considered as a de facto standard. In scientific computing it is common to face a problem of generating data based on a single functionality with many different parameters. This can be thought as "single instruction multiple parameter sets" type of computation. It is quite natural to utilise all cores available in your host machine. While many people uses MATLAB for rapid prototyping; Here I show how to generate many
Hennon Maps with different initial conditions (ICs) using MATLAB and Python. We will use Python to drive the computations and "single instruction" is being a function in MATLAB.
Let's first shortly remember the definition of the Hennon Map
$ x_{n+1} = y_{n} + 1 - \alpha x_{n}^2 $
$ y_{n+1} = \beta x_{n} $
It is known that for parameter ranges $\alpha \in [1.16, 1.41]$ and $\beta \in [0.2, 0.3]$ map generates
chaotic behaviour i.e. sensitive dependence on initial conditions.
Here is a MATLAB function that generates a png file for a given parameters, initial condition values, file name and the upper bound for the iteration.
function hennonMap(a, b, xn, yn, upper, filename)
% HENNONMAP generate hennon map at given parameters and initial conditions
%
% Mehmet Suzen
% msuzen on gmail
% (c) 2013
% GPLv3
%
% a \in [1.16 1.41]
% b \in [0.2, 0.3]
%
% Example of running this in the command line
% > matlab -nodesktop -nosplash -logfile my.log -r "hennonMap(1.4, 0.3, -1.0, -0.4, 1e4, 'hennonA1.4B0.3.png'); exit;"
%
Xval = zeros(upper, 1);
Yval = zeros(upper, 1);
for i=1:upper
Xval(i) = xn;
Yval(i) = yn;
x = yn + 1 - a*xn^2;
y = b * xn;
xn = x;
yn = y;
end
h = figure
plot(Xval, Yval,'o')
title(['Henon Map at ', sprintf(' a=%0.2f', a), sprintf(' b=%0.2f', b)])
xlabel('x')
ylabel('y')
print(h, '-dpng', filename)
end
Running this function from a command line, as described in the function help, would generate the figure shown above. Now imagine that we need to generate many plots with different ICs. It is easy to open many shells and run from command line many times. However it is a manual task and we developers do not like that at all! Python would help us in this case, specifically its
multiprocessing module, to spawn the process from a single script. The concept here is closely related to threading, while multiprocessing module mimics threading API.
For example, let's say we have 16 cores and we would like to generate Henon maps for 16 different initial conditions. Here is a Python code running the above Hennon Map function by invoking 16 different MATLAB instances with each using different ICs:
#
# Mehmet Suzen
# msuzen on gmail
# (c) 2013
# GPLv3
#
# Run Hennon Map on multi-process (spawning processes)
#
from multiprocessing import Pool
import commands
import numpy
import itertools
def f(argF):
a = '%0.1f' % argF[0]
b = '%0.1f' % argF[1]
filename = "hennonA" + a + "B" + b + ".png"
commandLine = "matlab -nodesktop -nosplash -logfile my.log -r \"hennonMap(1.4, 0.3," + a + "," + b + ", 1e4, '" + filename + "'); exit;\""
print commandLine
return commands.getstatusoutput(commandLine)
if __name__ == '__main__':
pool = Pool(processes=12) # start 12 worker processes
xns = list(numpy.linspace(-1.0, 0.5, 4))
yns = list(numpy.linspace(-0.4, 1.1, 4))
print pool.map(f, list(itertools.product(xns, yns)))
It would be interesting to do the same thing only using R. Maybe in the next post, I'll show that too.
