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I am numerically simulating the Mathieu equation using ODE45 and I have to keep changing the parameters delta and epsilon for each simulation to get the required peiodic solution.

Following is the MATLAB code I am running:

function xdot=trial2(t,x) 
delta=0.1045;epsilon=0.0048685;
xdot=[x(2);(-delta-epsilon*cos(t))*x(1)-0.7*delta*abs(x(1))];

[t,x]=ode45('trial2',[0 10000000],[0;1]); 
hold on; plot(t,x(:,1),'r'); 
clear all;

It is taking me 10 minutes for each simulation if I simulate for 10000000 time and it doubles if I increase the time 5 fold. After each simulation, I analyze the result and change the parameters accordingly so that I can come closer to a periodic solution. I have to do this for maybe about 30-40 parameters maybe more. I am running this on a 64 bit desktop with 8gb ram. The program runs out of memory on a 32 bit desktop.

1) Is there somehow I can improve my computational time? I might have to go from 10000000 to 50000000, maybe even more. 2) Can you help me out in implementing the parfor loop so that I can run simultaneous simulations for more number of parameters. That might be of some help too.3) WIll it help if I run this on a faster processor than i5-2400 CPU?

Thank you.

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  • 3
    $\begingroup$ Could you say more about your ultimate objective? Have you seen this example mathworks.com/help/matlab/ref/bvp4c.html where a Mathieu equation is solved as a boundary value problem? As you've observed, solving this equation by repeatedly solving the initial value problem for different values of the parameter is not very efficient. $\endgroup$ – Bill Greene Jul 19 '14 at 13:08
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How accurate must your propagation be? You can play with the accuracy using the odeset function (to create a set of options for ode45). Since ode45 is variable step, the accuracy has a huge influence on the computational speed. The higher the required accuracy, the more steps ode45 will take. By default, the relative accuracy is set to 1e-3 and the absolute accuracy to 1e-6.

You could also consider integrating using another language (such as C or Fortran) and Mex functions or loading libraries (see loadlibrary). This will definitely result in huge speedups, since compiled languages are much faster than Matlab for this type of operations.

EDIT: added link to Mex functions (restrictions for new user allowed only 2 links per post)

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1) Unless the function you provided in your question changes, the period of the function should be constant. In that case, I suggest using the ability of ode45 to locate "events" -- such as zero crossings (see the documentation for odeset). That way you can simulate for a shorter stretch of time, and use the accurate estimations of zero-crossings to measure the period. That should reduce your computational time by reducing the number of solver steps required.

2) You can certainly use a parfor loop to explore parameters:

vfDelta = linspace(0, 1, 100);
vfEpsilon = linspace(0, 1, 100);
[mfD, mfE] = ndgrid(vfDelta, vfEpsilon);
mfParams = [mfD(:) mfE(:)];

parfor nParam = 1:size(mfParams, 1)
   % - Set up an anonymous function to pass to ode45, that contains your parameters
   fhSystem = @(t,x)trial2_params(t, x, mfParams(nParam, 1), mfParams(nParam, 2));

   % - Solve the ODE
   [t,x]=ode45(trial2, [0 1000], [0;1]);

   % - Do your analysis here...
   vfPeriod(nParam) = ...
end

function trial2_params(t, x, fDelta, fEpsilon)
   ...
end

3) A faster CPU will always help...

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  • $\begingroup$ Alternatively, you can write a small function that uses ode45 to solve the system for a given set of parameters and return the period, and then use an optimisation function like fminunc or so on to find your optimal parameter set. $\endgroup$ – Dylan Richard Muir Aug 6 '14 at 19:38

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