where func can be both "_general_function" and "_weighted_general_function", is not correct.
M = σ 2 I ok unit weights
M = W^(-1) your case, W has the estimates of the error covariance
M = σ 2 W^(-1) I think this is what curve_fit uses, and what is in (my) textbooks defined as weighted least squares
Do we use definition 2 or 3 by default? both are reasonable
3 is what I expected when I looked at curve_fit 2 might be more useful for two stage estimation, but I didn't have time to check the details
Ehmm, no, curve_fit does not use def 3, as the errors would scale with W, but they don't. By the way, it does not have the correct units. Curve_fit calculates M = W \sigma^2 W^(-1) = \sigma^2 so it gives exactly the same results of case 1, irrespective the W's. This is why the errors do not scale with W. Gianfranco