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Derivative of the incomplete Gamma function and the sample of a Gamma random variable.

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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.
This commit is contained in:
Michael Figurnov
2018-06-06 18:49:26 +01:00
parent e206f8d4a4
commit 4bd158fa37
12 changed files with 791 additions and 207 deletions
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15
unsupported/Eigen/src/SpecialFunctions/SpecialFunctionsPacketMath.h
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@@ -42,6 +42,21 @@ Packet perfc(const Packet& a) { using numext::erfc; return erfc(a); }
template<typename Packet> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
Packet pigamma(const Packet& a, const Packet& x) { using numext::igamma; return igamma(a, x); }
/** \internal \returns the derivative of the incomplete gamma function
* igamma_der_a(\a a, \a x) */
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet pigamma_der_a(const Packet& a, const Packet& x) {
using numext::igamma_der_a; return igamma_der_a(a, x);
}
/** \internal \returns compute the derivative of the sample
* of Gamma(alpha, 1) random variable with respect to the parameter a
* gamma_sample_der_alpha(\a alpha, \a sample) */
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet pgamma_sample_der_alpha(const Packet& alpha, const Packet& sample) {
using numext::gamma_sample_der_alpha; return gamma_sample_der_alpha(alpha, sample);
}
/** \internal \returns the complementary incomplete gamma function igammac(\a a, \a x) */
template<typename Packet> EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE
Packet pigammac(const Packet& a, const Packet& x) { using numext::igammac; return igammac(a, x); }