boost/math/special_functions/detail/gamma_inva.hpp
// (C) Copyright John Maddock 2006. // Use, modification and distribution are subject to the // Boost Software License, Version 1.0. (See accompanying file // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) // // This is not a complete header file, it is included by gamma.hpp // after it has defined it's definitions. This inverts the incomplete // gamma functions P and Q on the first parameter "a" using a generic // root finding algorithm (TOMS Algorithm 748). // #ifndef BOOST_MATH_SP_DETAIL_GAMMA_INVA #define BOOST_MATH_SP_DETAIL_GAMMA_INVA #ifdef _MSC_VER #pragma once #endif #include <boost/math/tools/toms748_solve.hpp> #include <boost/cstdint.hpp> namespace boost{ namespace math{ namespace detail{ template <class T, class Policy> struct gamma_inva_t { gamma_inva_t(T z_, T p_, bool invert_) : z(z_), p(p_), invert(invert_) {} T operator()(T a) { return invert ? p - boost::math::gamma_q(a, z, Policy()) : boost::math::gamma_p(a, z, Policy()) - p; } private: T z, p; bool invert; }; template <class T, class Policy> T inverse_poisson_cornish_fisher(T lambda, T p, T q, const Policy& pol) { BOOST_MATH_STD_USING // mean: T m = lambda; // standard deviation: T sigma = sqrt(lambda); // skewness T sk = 1 / sigma; // kurtosis: // T k = 1/lambda; // Get the inverse of a std normal distribution: T x = boost::math::erfc_inv(p > q ? 2 * q : 2 * p, pol) * constants::root_two<T>(); // Set the sign: if(p < 0.5) x = -x; T x2 = x * x; // w is correction term due to skewness T w = x + sk * (x2 - 1) / 6; /* // Add on correction due to kurtosis. // Disabled for now, seems to make things worse? // if(lambda >= 10) w += k * x * (x2 - 3) / 24 + sk * sk * x * (2 * x2 - 5) / -36; */ w = m + sigma * w; return w > tools::min_value<T>() ? w : tools::min_value<T>(); } template <class T, class Policy> T gamma_inva_imp(const T& z, const T& p, const T& q, const Policy& pol) { BOOST_MATH_STD_USING // for ADL of std lib math functions // // Special cases first: // if(p == 0) { return tools::max_value<T>(); } if(q == 0) { return tools::min_value<T>(); } // // Function object, this is the functor whose root // we have to solve: // gamma_inva_t<T, Policy> f(z, (p < q) ? p : q, (p < q) ? false : true); // // Tolerance: full precision. // tools::eps_tolerance<T> tol(policies::digits<T, Policy>()); // // Now figure out a starting guess for what a may be, // we'll start out with a value that'll put p or q // right bang in the middle of their range, the functions // are quite sensitive so we should need too many steps // to bracket the root from there: // T guess; T factor = 8; if(z >= 1) { // // We can use the relationship between the incomplete // gamma function and the poisson distribution to // calculate an approximate inverse, for large z // this is actually pretty accurate, but it fails badly // when z is very small. Also set our step-factor according // to how accurate we think the result is likely to be: // guess = 1 + inverse_poisson_cornish_fisher(z, q, p, pol); if(z > 5) { if(z > 1000) factor = 1.01f; else if(z > 50) factor = 1.1f; else if(guess > 10) factor = 1.25f; else factor = 2; if(guess < 1.1) factor = 8; } } else if(z > 0.5) { guess = z * 1.2f; } else { guess = -0.4f / log(z); } // // Max iterations permitted: // boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>(); // // Use our generic derivative-free root finding procedure. // We could use Newton steps here, taking the PDF of the // Poisson distribution as our derivative, but that's // even worse performance-wise than the generic method :-( // std::pair<T, T> r = bracket_and_solve_root(f, guess, factor, false, tol, max_iter, pol); if(max_iter >= policies::get_max_root_iterations<Policy>()) policies::raise_evaluation_error<T>("boost::math::gamma_p_inva<%1%>(%1%, %1%)", "Unable to locate the root within a reasonable number of iterations, closest approximation so far was %1%", r.first, pol); return (r.first + r.second) / 2; } } // namespace detail template <class T1, class T2, class Policy> inline typename tools::promote_args<T1, T2>::type gamma_p_inva(T1 x, T2 p, const Policy& pol) { typedef typename tools::promote_args<T1, T2>::type result_type; typedef typename policies::evaluation<result_type, Policy>::type value_type; typedef typename policies::normalise< Policy, policies::promote_float<false>, policies::promote_double<false>, policies::discrete_quantile<>, policies::assert_undefined<> >::type forwarding_policy; if(p == 0) { return tools::max_value<result_type>(); } if(p == 1) { return tools::min_value<result_type>(); } return policies::checked_narrowing_cast<result_type, forwarding_policy>( detail::gamma_inva_imp( static_cast<value_type>(x), static_cast<value_type>(p), static_cast<value_type>(1 - static_cast<value_type>(p)), pol), "boost::math::gamma_p_inva<%1%>(%1%, %1%)"); } template <class T1, class T2, class Policy> inline typename tools::promote_args<T1, T2>::type gamma_q_inva(T1 x, T2 q, const Policy& pol) { typedef typename tools::promote_args<T1, T2>::type result_type; typedef typename policies::evaluation<result_type, Policy>::type value_type; typedef typename policies::normalise< Policy, policies::promote_float<false>, policies::promote_double<false>, policies::discrete_quantile<>, policies::assert_undefined<> >::type forwarding_policy; if(q == 1) { return tools::max_value<result_type>(); } if(q == 0) { return tools::min_value<result_type>(); } return policies::checked_narrowing_cast<result_type, forwarding_policy>( detail::gamma_inva_imp( static_cast<value_type>(x), static_cast<value_type>(1 - static_cast<value_type>(q)), static_cast<value_type>(q), pol), "boost::math::gamma_q_inva<%1%>(%1%, %1%)"); } template <class T1, class T2> inline typename tools::promote_args<T1, T2>::type gamma_p_inva(T1 x, T2 p) { return boost::math::gamma_p_inva(x, p, policies::policy<>()); } template <class T1, class T2> inline typename tools::promote_args<T1, T2>::type gamma_q_inva(T1 x, T2 q) { return boost::math::gamma_q_inva(x, q, policies::policy<>()); } } // namespace math } // namespace boost #endif // BOOST_MATH_SP_DETAIL_GAMMA_INVA