Minimizing by trisecting a function dependent on 3 parameters

[borson]

Hi and thanks to everyone for his/her attention. I have to minimize a function that depends on several parameters. The aim of minimizing that function is to actually guess these parameters, which are unknown. The thing is that the author of the pdf from which I have to make the calculations, does not specify very well how to carry it out. There are 3 parameters the function depends on (one of them nonlinear), and the author says that first off we have to figure out the nonlinear parameter, by trisecting a determined interval, and afterwards guess the spare ones by means of another function that will also have to be minimized. So here is where I am confused. How am I supossed to minimize a function by trisection paying attention to just one parameter? I mean, what do I have to do with the other ones? How should I do it?

Here is the part of the pdf in which that function is shown:

you can find the whole pdf here:http://www.roulettephysics.com/wp-content/uploads/2014/01/Roulette_Physik.pdf

that function is in the 13 page.

thank you all for your attention and I hope you can help me :)

 

[haruspex]

Hi and thanks to everyone for his/her attention. I have to minimize a function that depends on several parameters. The aim of minimizing that function is to actually guess these parameters, which are unknown. The thing is that the author of the pdf from which I have to make the calculations, does not specify very well how to carry it out. There are 3 parameters the function depends on (one of them nonlinear), and the author says that first off we have to figure out the nonlinear parameter, by trisecting a determined interval, and afterwards guess the spare ones by means of another function that will also have to be minimized. So here is where I am confused. How am I supossed to minimize a function by trisection paying attention to just one parameter? I mean, what do I have to do with the other ones? How should I do it?

Here is the part of the pdf in which that function is shown:

you can find the whole pdf here:http://www.roulettephysics.com/wp-content/uploads/2014/01/Roulette_Physik.pdf

that function is in the 13 page.

thank you all for your attention and I hope you can help me :)

Not sure, but I think the idea is to iterate as follows:

1. set the range [φ_min, φ_max] for φ as [0, 2π]
2. trisect the range, i.e. consider values φ_min, 2φ_min/3+ φ_max/3, φ_min/3+2φ_max/3, φ_max for φ
3. for each of the four values, v₁ to v₄, use "robust linear estimation" (whatever that is) to find the corresponding η and Ω²_f;
4. discard the first or last third of the range for φ which appears not to be promising; this assumes the behaviour is not too jerky
5. repeat from 2.

The discarding rule, I guess, would be
if f(v₂)<f(v₃) discard [v₃, v₄], else discard [v₁, v₂].

 

[borson]

Not sure, but I think the idea is to iterate as follows:

1. set the range [φ_min, φ_max] for φ as [0, 2π]
2. trisect the range, i.e. consider values φ_min, 2φ_min/3+ φ_max/3, φ_min/3+2φ_max/3, φ_max for φ
3. for each of the four values, v₁ to v₄, use "robust linear estimation" (whatever that is) to find the corresponding η and Ω²_f;
4. discard the first or last third of the range for φ which appears not to be promising; this assumes the behaviour is not too jerky
5. repeat from 2.

The discarding rule, I guess, would be
if f(v₂)<f(v₃) discard [v₃, v₄], else discard [v₁, v₂].

It seems a good idea.
I will try to do that, though I do not understand the last function either. I do not have any idea of what the yk and the xk stand for.
Also, considering the discarding rule, regarding what you have said, it seems that I only need the second function for discarding. So then, why is the first function for?
At the beginning I am obliged to use the f(v) of the second function (as I do not have any guess of the linear parameters to use in the first one yet), to see which set of values for the parameters yield the most minimum value, but once I do that and by means of that I obtain values for that parameters, should I keep using the second function for discardingn or the first one?
Thanks for replying :)

 

[haruspex]

I do not have any idea of what the yk and the xk stand for.

These just stand for terms in the equation at (42) which do not depend on η or Ω. So y_k=c₁e^-2aθf_k and x_k represents everything inside the square brackets.