grad_reg_inc_gamma()

template<typename T1 , typename T2 >

return_type_t< T1, T2 > stan::math::grad_reg_inc_gamma	(	T1	a,
		T2	z,
		T1	g,
		T1	dig,
		double	precision = 1e-6,
		int	max_steps = 1e5
	)

Gradient of the regularized incomplete gamma functions igamma(a, z)

For small z, the gradient is computed via the series expansion; for large z, the series is numerically inaccurate due to cancellation and the asymptotic expansion is used.

Template Parameters

T1	type of the shape parameter
T2	type of the location parameter

Parameters

a	shape parameter, a > 0
z	location z >= 0
g	stan::math::tgamma(a) (precomputed value)
dig	boost::math::digamma(a) (precomputed value)
precision	required precision; applies to series expansion only
max_steps	number of steps to take.

Exceptions

throws

std::domain_error if not converged after max_steps or increment overflows to inf.

For the asymptotic expansion, the gradient is given by:

\[ \begin{array}{rcl} \Gamma(a, z) & = & z^{a-1}e^{-z} \sum_{k=0}^N \frac{(a-1)_k}{z^k} \qquad , z \gg a\\ Q(a, z) & = & \frac{z^{a-1}e^{-z}}{\Gamma(a)} \sum_{k=0}^N \frac{(a-1)_k}{z^k}\\ (a)_k & = & (a)_{k-1}(a-k)\\ \frac{d}{da} (a)_k & = & (a)_{k-1} + (a-k)\frac{d}{da} (a)_{k-1}\\ \frac{d}{da}Q(a, z) & = & (log(z) - \psi(a)) Q(a, z)\\ && + \frac{z^{a-1}e^{-z}}{\Gamma(a)} \sum_{k=0}^N \left(\frac{d}{da} (a-1)_k\right) \frac{1}{z^k} \end{array} \]

Definition at line 52 of file grad_reg_inc_gamma.hpp.