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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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42
unsupported/Eigen/src/SpecialFunctions/SpecialFunctionsArrayAPI.h
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@@ -33,6 +33,48 @@ igamma(const Eigen::ArrayBase<Derived>& a, const Eigen::ArrayBase<ExponentDerive
);
}
/** \cpp11 \returns an expression of the coefficient-wise igamma_der_a(\a a, \a x) to the given arrays.
*
* This function computes the coefficient-wise derivative of the incomplete
* gamma function with respect to the parameter a.
*
* \note This function supports only float and double scalar types in c++11
* mode. To support other scalar types,
* or float/double in non c++11 mode, the user has to provide implementations
* of igamma_der_a(T,T) for any scalar
* type T to be supported.
*
* \sa Eigen::igamma(), Eigen::lgamma()
*/
template <typename Derived, typename ExponentDerived>
inline const Eigen::CwiseBinaryOp<Eigen::internal::scalar_igamma_der_a_op<typename Derived::Scalar>, const Derived, const ExponentDerived>
igamma_der_a(const Eigen::ArrayBase<Derived>& a, const Eigen::ArrayBase<ExponentDerived>& x) {
return Eigen::CwiseBinaryOp<Eigen::internal::scalar_igamma_der_a_op<typename Derived::Scalar>, const Derived, const ExponentDerived>(
a.derived(),
x.derived());
}
/** \cpp11 \returns an expression of the coefficient-wise gamma_sample_der_alpha(\a alpha, \a sample) to the given arrays.
*
* This function computes the coefficient-wise derivative of the sample
* of a Gamma(alpha, 1) random variable with respect to the parameter alpha.
*
* \note This function supports only float and double scalar types in c++11
* mode. To support other scalar types,
* or float/double in non c++11 mode, the user has to provide implementations
* of gamma_sample_der_alpha(T,T) for any scalar
* type T to be supported.
*
* \sa Eigen::igamma(), Eigen::lgamma()
*/
template <typename AlphaDerived, typename SampleDerived>
inline const Eigen::CwiseBinaryOp<Eigen::internal::scalar_gamma_sample_der_alpha_op<typename AlphaDerived::Scalar>, const AlphaDerived, const SampleDerived>
gamma_sample_der_alpha(const Eigen::ArrayBase<AlphaDerived>& alpha, const Eigen::ArrayBase<SampleDerived>& sample) {
return Eigen::CwiseBinaryOp<Eigen::internal::scalar_gamma_sample_der_alpha_op<typename AlphaDerived::Scalar>, const AlphaDerived, const SampleDerived>(
alpha.derived(),
sample.derived());
}
/** \cpp11 \returns an expression of the coefficient-wise igammac(\a a, \a x) to the given arrays.
*
* This function computes the coefficient-wise complementary incomplete gamma function.