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So we've just figured out that we can take our numerical solution and do if we're it will transform

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and we will see dominant peaks at the three eigen frequencies that we have previously calculated by

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calculating the eigenvalues of the matrix that enters the equation of motion.

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So the idea would now be to calculate or to generate our numerical results by superimposing three sine

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functions.

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Or we could also take three cosine functions.

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And for the frequencies of these sine or cosine functions, we take, of course, the ion frequencies

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omega one, two and three.

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And I told you also already that we have three parameters for the amplitudes now.

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And additionally, we have three parameters for the phase.

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And so the idea is now that we fit these three, these six parameters in total so that we fit this model

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function to our numerical solution.

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And this is really exactly the same procedure that we have done in one of the previous sections with

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where we have first discussed the fitting procedures.

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So in fact, you can see here that I have copied even the code from these sections.

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So as I wrote and as I told you back then, there are many reasonable definitions of an error function.

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But the very common choice is this one here where you take the y values of the actual calculation,

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which would be the numerical simulation in our case and then the values of our model functions.

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And this will be then the error function and to minimize the error.

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We do gradient descent.

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We could also do other methods like Monte Carlo that we discussed in a different section.

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But here we will do gradient descent.

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So basically, we calculate the gradient of the error function with respect to these parameters.

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So it would be this one.

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And then we go along the opposite direction of the gradient several times until the error function is

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minimized.

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So I know that this is a bit difficult, but I think it's a good time for you to practice this because

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we have already done a fit together.

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And you know, now a lot about these coupled oscillators.

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And now your task is to implement the fit.

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So you can, of course, take the routines from the notebook that we have programmed back then.

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The only thing that you have to take care of that is that now the gradient is a tiny bit different.

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So until here, it's exactly the same thing as previously.

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But the values of these derivatives of the model function is, of course, different because our model

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function is now such a superposition of three harmonic functions and not on any anymore a polynomial.

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So we have to calculate the gradient of this function with respect to eight.

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And this would, of course, be for the first component cosine omega one t times a plus find one because

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the derivative is just with respect to this one here.

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So we get this term and then the derivative for respect to PHI one would be a one times minus sine omega

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one T plus phi one, and there is no in a derivative here.

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So it would just be this one.

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And then for the other, it's really the same thing.

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So yeah, pretty easy.

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Calculating the gradient analytically.

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So no, you just have to take the code from the other section and just change a few things and change

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the gradient and then run the fit and see if you can really restore the numerical results by our fitted

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function.

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So please give it a try, or at least think about it.

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And then you can, of course, watch my next video where I will show you my solution.