Hello everybody.
I`m trying to fit Coffin-Menson curve to data generated by Coffin-Menson equation using least squares method.
For that I use an example code from Newton.sm (18kb) downloaded 31 time(s). wth some modifications Least_squares_method.sm (48kb) downloaded 25 time(s)., but code tells me,
that something is going wrong and I can`t find out the reason.
I will be appriciated, if someone is able to explain it.
Thank you in advance.

Hi Romanoff. Your setup it's basically correct, and I guess that it theoretically must to work, with the only addition of Clear(i) in the body of the target function. But that's only in theory. Problem here is that the Jacobian require the evaluation for a symbolic derivatives for a target function with 104 elements for each sum by four rows with four variables. Even maybe SMath could get some result for that, you also must to invert that Jacobian, which I guess that it can't be done for some reasonable time.

In the attached you found a pure numerical Jacobian and it's inversion. In green a couple of notes. Theoretically, it's works. But again, just theory. In the practice I don't understand the construction of your target function. In the second attached, a simplification for the setup. Your start point seems to be the actual solution.

Hope that this notes helps.

Least_squares_method.sm (53kb) downloaded 25 time(s).
Least_squares_method revised.sm (36kb) downloaded 36 time(s).

Best regards.
Alvaro.

Just an exercise, it suggest a model fit rather polyfit.

Romanoffff.sm (66kb) downloaded 16 time(s).

Thank you very much for your answers, any way, I need to think a little about ways you`ve proposed. It looks, like my problem is partly solved, but stability of the convergence is the issue. When I slightly change parameters of the model, even 1000 iterations is not enough. And another one thing, when you look at b derivative, evaluated by Smath it looks fine, but c derivatve despite of the same function structure looks absolutely different.
To make a script, which works, I`ve chose the simplest way, I tried to fit function to points with known function used to get them.

So, I`ve tried to adopt Smath file Jean Giraud showed, but have some issues.I`m truing to fit function f(x,Ni)el(x,1)*(Ni)^el(x,2)+el(x,3)*(Ni)^el(x,4)) to experimental points: Ni:mat(729,1000,3000,10000,30000,100000,6,1) ε_res:mat(1.9996,1.6798,1.0557,0.7854,0.6741,0.6046,6,1), where Ni- number of full cycles till rupture, ε_res,- strain range. The issue is, that in your examples derivatives are functions of one variable, but in case of my one it is function of 2 variables, for example, derivative by el(x,2) is el(x,1)*(Ni)^el(x,2)*ln(Ni). Could you please give me an advise, how to fit this function, Pure Newton`s method doesn`t give me convergence.
Thank you in advance.

If you purchase strain gauge, they come with the corresponding factor
and if applicable the reading equipment will compensate for temperature.
What I understand: you are experimenting some strain gauge material
simulating some real application. In that case, from the experimental
data as collected, you want a model function and the best representative
fit to the data. Then, only the data are needed.
If you just have a graph from the lab test, that's good enough.
Please don't be shy, Smath Community is worth a Gold Bridge
As you don't look familiar enough with fitting data, you may
obscure yourself by some examples that you will gather here/there.
Cheers ... Jean is waiting to hear more from you.

My last involvement with strain gauge was at Pratt-Whitney [1986]
designing test bench. Just purchased from Omega, plug in the
Data Logging input card.

No problem to fit either one in the elastic region !