In addition to igamma(a, x), this code implements:
* igamma_der_a(a, x) = d igamma(a, x) / da -- derivative of igamma with respect to the parameter
* gamma_sample_der_alpha(alpha, sample) -- reparameterization derivative of a Gamma(alpha, 1) random variable sample with respect to the alpha parameter
The derivatives are computed by forward mode differentiation of the igamma(a, x) code. Although gamma_sample_der_alpha can be implemented via igamma_der_a, a separate function is more accurate and efficient due to analytical cancellation of some terms. All three functions are implemented by a method parameterized with "mode" that always computes the derivatives, but does not return them unless required by the mode. The compiler is expected to (and, based on benchmarks, does) skip the unnecessary computations depending on the mode.
The functions are conventionally called i0e and i1e. The exponentially scaled version is more numerically stable. The standard Bessel functions can be obtained as i0(x) = exp(|x|) i0e(x)
The code is ported from Cephes and tested against SciPy.
TernaryFunctors and their executors allow operations on 3-tuples of inputs.
API fully implemented for Arrays and Tensors based on binary functors.
Ported the cephes betainc function (regularized incomplete beta
integral) to Eigen, with support for CPU and GPU, floats, doubles, and
half types.
Added unit tests in array.cpp and cxx11_tensor_cuda.cu
Collapsed revision
* Merged helper methods for betainc across floats and doubles.
* Added TensorGlobalFunctions with betainc(). Removed betainc() from TensorBase.
* Clean up CwiseTernaryOp checks, change igamma_helper to cephes_helper.
* betainc: merge incbcf and incbd into incbeta_cfe. and more cleanup.
* Update TernaryOp and SpecialFunctions (betainc) based on review comments.
