gamma_inc_inv for bigFloats

[lrnv]

Hi,

We have an issue there iitis/DatagenCopulaBased.jl#7 that would require the gamma_inv_inc function to work with bigfloats. Is there in the bigfloat package an implementation of such a function, or will the underlying code for Float64 be transferable to BigFloats (and really other kind of floats as well) ?

[stevengj]

I'm not aware of such a function in MPFR.

No, the Float64 implementation is not transferrable to BigFloats. In general, special-function implementations use sets of polynomial approximations that are tuned to the desired precision, and which don't give accurate results to arbitrary precision.

One could use the strategy of #278: start with the double-precision answer, and then apply a root-finding algorithm in order to polish the solution to BigFloat precision. Ideally, one would use the multidimensional Newton method, but the derivative of an incomplete Gamma function with respect to the first argument seems to require another special function that we don't have (#265). So you might have to use a derivative-free algorithm, or some hybrid algorithm since the derivative with respect to the second argument is easy.