Moffat1D and Moffat2D derivatives

The derivatives in Moffat1D are wrong:

https://github.com/astropy/astropy/blob/2b8fe1305b2689d8b2715f98985e36f2c7d5c0d9/astropy/modeling/functional_models.py#L2223

They should be replaced by:

    @staticmethod
    def fit_deriv(x, amplitude, x_0, gamma, alpha):
        """One dimensional Moffat model derivative with respect to parameters"""

        fac = (1 + (x - x_0) ** 2 / gamma ** 2)
        d_A = fac ** (-alpha)
        d_x_0 = (2 * amplitude * alpha * (x - x_0) * d_A / (fac * gamma ** 2))
        d_gamma = (2 * amplitude * alpha * (x - x_0) ** 2 * d_A /
                   (fac * gamma ** 3))
        d_alpha = -amplitude * d_A * np.log(fac)
        return [d_A, d_x_0, d_gamma, d_alpha]

In Moffat2D:

https://github.com/astropy/astropy/blob/2b8fe1305b2689d8b2715f98985e36f2c7d5c0d9/astropy/modeling/functional_models.py#L2301

only derivative wrt gamma is wrong. It should be

        d_gamma = 2 * amplitude * alpha * d_A * rr_gg / ((1 + rr_gg) * gamma ** 3)

[adonath]

I remember that for the Beta1D and Beta2D models I computed the derivatives and reference values in a notebook using using Sympy (http://nbviewer.jupyter.org/gist/adonath/6088712). So I’m pretty sure it used to be correct at that time. Maybe when the rename to Moffat1D and Moffat2D happened, the parametrization changed as well and the derivatives where not adapted? From a quick look in the code it seems there are no unit tests for the model derivatives.

@ycopin I didn’t check whether your proposed formulae are correct, please check against Sympy or some other algebra library and post the result as a reference. In case you still find the current formulae are incorrect, would you mind preparing a minimal pull request with the correct ones? Ideally one should add unit tests for the model derivatives, but this is a somewhat larger project…

[ycopin]

I’m sorry I introduced a bug in Moffat2D derivatives… 😕 As mentioned earlier, verifying the formal derivatives using e.g. Wolfram is not necessarily sufficient, but one should check the actual implementation, thanks @saimn for doing so (even if the tested implementation might not be the most efficient). I still wonder why this implementation error was not caught by a test comparing formal and numerical derivatives…