boost/math/special_functions/gamma.hpp
// Copyright John Maddock 2006-7, 2013-14. // Copyright Paul A. Bristow 2007, 2013-14. // Copyright Nikhar Agrawal 2013-14 // Copyright Christopher Kormanyos 2013-14 // 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) #ifndef BOOST_MATH_SF_GAMMA_HPP #define BOOST_MATH_SF_GAMMA_HPP #ifdef _MSC_VER #pragma once #endif #include <boost/config.hpp> #include <boost/math/tools/series.hpp> #include <boost/math/tools/fraction.hpp> #include <boost/math/tools/precision.hpp> #include <boost/math/tools/promotion.hpp> #include <boost/math/policies/error_handling.hpp> #include <boost/math/constants/constants.hpp> #include <boost/math/special_functions/math_fwd.hpp> #include <boost/math/special_functions/log1p.hpp> #include <boost/math/special_functions/trunc.hpp> #include <boost/math/special_functions/powm1.hpp> #include <boost/math/special_functions/sqrt1pm1.hpp> #include <boost/math/special_functions/lanczos.hpp> #include <boost/math/special_functions/fpclassify.hpp> #include <boost/math/special_functions/detail/igamma_large.hpp> #include <boost/math/special_functions/detail/unchecked_factorial.hpp> #include <boost/math/special_functions/detail/lgamma_small.hpp> #include <boost/math/special_functions/bernoulli.hpp> #include <boost/math/special_functions/polygamma.hpp> #include <boost/type_traits/is_convertible.hpp> #include <boost/assert.hpp> #include <boost/mpl/greater.hpp> #include <boost/mpl/equal_to.hpp> #include <boost/mpl/greater.hpp> #include <boost/config/no_tr1/cmath.hpp> #include <algorithm> #ifdef BOOST_MSVC # pragma warning(push) # pragma warning(disable: 4702) // unreachable code (return after domain_error throw). # pragma warning(disable: 4127) // conditional expression is constant. # pragma warning(disable: 4100) // unreferenced formal parameter. // Several variables made comments, // but some difficulty as whether referenced on not may depend on macro values. // So to be safe, 4100 warnings suppressed. // TODO - revisit this? #endif namespace boost{ namespace math{ namespace detail{ template <class T> inline bool is_odd(T v, const boost::true_type&) { int i = static_cast<int>(v); return i&1; } template <class T> inline bool is_odd(T v, const boost::false_type&) { // Oh dear can't cast T to int! BOOST_MATH_STD_USING T modulus = v - 2 * floor(v/2); return static_cast<bool>(modulus != 0); } template <class T> inline bool is_odd(T v) { return is_odd(v, ::boost::is_convertible<T, int>()); } template <class T> T sinpx(T z) { // Ad hoc function calculates x * sin(pi * x), // taking extra care near when x is near a whole number. BOOST_MATH_STD_USING int sign = 1; if(z < 0) { z = -z; } T fl = floor(z); T dist; if(is_odd(fl)) { fl += 1; dist = fl - z; sign = -sign; } else { dist = z - fl; } BOOST_ASSERT(fl >= 0); if(dist > 0.5) dist = 1 - dist; T result = sin(dist*boost::math::constants::pi<T>()); return sign*z*result; } // template <class T> T sinpx(T z) // // tgamma(z), with Lanczos support: // template <class T, class Policy, class Lanczos> T gamma_imp(T z, const Policy& pol, const Lanczos& l) { BOOST_MATH_STD_USING T result = 1; #ifdef BOOST_MATH_INSTRUMENT static bool b = false; if(!b) { std::cout << "tgamma_imp called with " << typeid(z).name() << " " << typeid(l).name() << std::endl; b = true; } #endif static const char* function = "boost::math::tgamma<%1%>(%1%)"; if(z <= 0) { if(floor(z) == z) return policies::raise_pole_error<T>(function, "Evaluation of tgamma at a negative integer %1%.", z, pol); if(z <= -20) { result = gamma_imp(T(-z), pol, l) * sinpx(z); BOOST_MATH_INSTRUMENT_VARIABLE(result); if((fabs(result) < 1) && (tools::max_value<T>() * fabs(result) < boost::math::constants::pi<T>())) return -boost::math::sign(result) * policies::raise_overflow_error<T>(function, "Result of tgamma is too large to represent.", pol); result = -boost::math::constants::pi<T>() / result; if(result == 0) return policies::raise_underflow_error<T>(function, "Result of tgamma is too small to represent.", pol); if((boost::math::fpclassify)(result) == (int)FP_SUBNORMAL) return policies::raise_denorm_error<T>(function, "Result of tgamma is denormalized.", result, pol); BOOST_MATH_INSTRUMENT_VARIABLE(result); return result; } // shift z to > 1: while(z < 0) { result /= z; z += 1; } } BOOST_MATH_INSTRUMENT_VARIABLE(result); if((floor(z) == z) && (z < max_factorial<T>::value)) { result *= unchecked_factorial<T>(itrunc(z, pol) - 1); BOOST_MATH_INSTRUMENT_VARIABLE(result); } else if (z < tools::root_epsilon<T>()) { if (z < 1 / tools::max_value<T>()) result = policies::raise_overflow_error<T>(function, 0, pol); result *= 1 / z - constants::euler<T>(); } else { result *= Lanczos::lanczos_sum(z); T zgh = (z + static_cast<T>(Lanczos::g()) - boost::math::constants::half<T>()); T lzgh = log(zgh); BOOST_MATH_INSTRUMENT_VARIABLE(result); BOOST_MATH_INSTRUMENT_VARIABLE(tools::log_max_value<T>()); if(z * lzgh > tools::log_max_value<T>()) { // we're going to overflow unless this is done with care: BOOST_MATH_INSTRUMENT_VARIABLE(zgh); if(lzgh * z / 2 > tools::log_max_value<T>()) return boost::math::sign(result) * policies::raise_overflow_error<T>(function, "Result of tgamma is too large to represent.", pol); T hp = pow(zgh, (z / 2) - T(0.25)); BOOST_MATH_INSTRUMENT_VARIABLE(hp); result *= hp / exp(zgh); BOOST_MATH_INSTRUMENT_VARIABLE(result); if(tools::max_value<T>() / hp < result) return boost::math::sign(result) * policies::raise_overflow_error<T>(function, "Result of tgamma is too large to represent.", pol); result *= hp; BOOST_MATH_INSTRUMENT_VARIABLE(result); } else { BOOST_MATH_INSTRUMENT_VARIABLE(zgh); BOOST_MATH_INSTRUMENT_VARIABLE(pow(zgh, z - boost::math::constants::half<T>())); BOOST_MATH_INSTRUMENT_VARIABLE(exp(zgh)); result *= pow(zgh, z - boost::math::constants::half<T>()) / exp(zgh); BOOST_MATH_INSTRUMENT_VARIABLE(result); } } return result; } // // lgamma(z) with Lanczos support: // template <class T, class Policy, class Lanczos> T lgamma_imp(T z, const Policy& pol, const Lanczos& l, int* sign = 0) { #ifdef BOOST_MATH_INSTRUMENT static bool b = false; if(!b) { std::cout << "lgamma_imp called with " << typeid(z).name() << " " << typeid(l).name() << std::endl; b = true; } #endif BOOST_MATH_STD_USING static const char* function = "boost::math::lgamma<%1%>(%1%)"; T result = 0; int sresult = 1; if(z <= -tools::root_epsilon<T>()) { // reflection formula: if(floor(z) == z) return policies::raise_pole_error<T>(function, "Evaluation of lgamma at a negative integer %1%.", z, pol); T t = sinpx(z); z = -z; if(t < 0) { t = -t; } else { sresult = -sresult; } result = log(boost::math::constants::pi<T>()) - lgamma_imp(z, pol, l) - log(t); } else if (z < tools::root_epsilon<T>()) { if (0 == z) return policies::raise_pole_error<T>(function, "Evaluation of lgamma at %1%.", z, pol); if (fabs(z) < 1 / tools::max_value<T>()) result = -log(fabs(z)); else result = log(fabs(1 / z - constants::euler<T>())); if (z < 0) sresult = -1; } else if(z < 15) { typedef typename policies::precision<T, Policy>::type precision_type; typedef typename mpl::if_< mpl::and_< mpl::less_equal<precision_type, mpl::int_<64> >, mpl::greater<precision_type, mpl::int_<0> > >, mpl::int_<64>, typename mpl::if_< mpl::and_< mpl::less_equal<precision_type, mpl::int_<113> >, mpl::greater<precision_type, mpl::int_<0> > >, mpl::int_<113>, mpl::int_<0> >::type >::type tag_type; result = lgamma_small_imp<T>(z, T(z - 1), T(z - 2), tag_type(), pol, l); } else if((z >= 3) && (z < 100) && (std::numeric_limits<T>::max_exponent >= 1024)) { // taking the log of tgamma reduces the error, no danger of overflow here: result = log(gamma_imp(z, pol, l)); } else { // regular evaluation: T zgh = static_cast<T>(z + Lanczos::g() - boost::math::constants::half<T>()); result = log(zgh) - 1; result *= z - 0.5f; // // Only add on the lanczos sum part if we're going to need it: // if(result * tools::epsilon<T>() < 20) result += log(Lanczos::lanczos_sum_expG_scaled(z)); } if(sign) *sign = sresult; return result; } // // Incomplete gamma functions follow: // template <class T> struct upper_incomplete_gamma_fract { private: T z, a; int k; public: typedef std::pair<T,T> result_type; upper_incomplete_gamma_fract(T a1, T z1) : z(z1-a1+1), a(a1), k(0) { } result_type operator()() { ++k; z += 2; return result_type(k * (a - k), z); } }; template <class T> inline T upper_gamma_fraction(T a, T z, T eps) { // Multiply result by z^a * e^-z to get the full // upper incomplete integral. Divide by tgamma(z) // to normalise. upper_incomplete_gamma_fract<T> f(a, z); return 1 / (z - a + 1 + boost::math::tools::continued_fraction_a(f, eps)); } template <class T> struct lower_incomplete_gamma_series { private: T a, z, result; public: typedef T result_type; lower_incomplete_gamma_series(T a1, T z1) : a(a1), z(z1), result(1){} T operator()() { T r = result; a += 1; result *= z/a; return r; } }; template <class T, class Policy> inline T lower_gamma_series(T a, T z, const Policy& pol, T init_value = 0) { // Multiply result by ((z^a) * (e^-z) / a) to get the full // lower incomplete integral. Then divide by tgamma(a) // to get the normalised value. lower_incomplete_gamma_series<T> s(a, z); boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>(); T factor = policies::get_epsilon<T, Policy>(); T result = boost::math::tools::sum_series(s, factor, max_iter, init_value); policies::check_series_iterations<T>("boost::math::detail::lower_gamma_series<%1%>(%1%)", max_iter, pol); return result; } // // Fully generic tgamma and lgamma use Stirling's approximation // with Bernoulli numbers. // template<class T> std::size_t highest_bernoulli_index() { const float digits10_of_type = (std::numeric_limits<T>::is_specialized ? static_cast<float>(std::numeric_limits<T>::digits10) : static_cast<float>(boost::math::tools::digits<T>() * 0.301F)); // Find the high index n for Bn to produce the desired precision in Stirling's calculation. return static_cast<std::size_t>(18.0F + (0.6F * digits10_of_type)); } template<class T> int minimum_argument_for_bernoulli_recursion() { const float digits10_of_type = (std::numeric_limits<T>::is_specialized ? static_cast<float>(std::numeric_limits<T>::digits10) : static_cast<float>(boost::math::tools::digits<T>() * 0.301F)); const float limit = std::ceil(std::pow(1.0f / std::ldexp(1.0f, 1-boost::math::tools::digits<T>()), 1.0f / 20.0f)); return (int)((std::min)(digits10_of_type * 1.7F, limit)); } template <class T, class Policy> T scaled_tgamma_no_lanczos(const T& z, const Policy& pol, bool islog = false) { BOOST_MATH_STD_USING // // Calculates tgamma(z) / (z/e)^z // Requires that our argument is large enough for Sterling's approximation to hold. // Used internally when combining gamma's of similar magnitude without logarithms. // BOOST_ASSERT(minimum_argument_for_bernoulli_recursion<T>() <= z); // Perform the Bernoulli series expansion of Stirling's approximation. const std::size_t number_of_bernoullis_b2n = policies::get_max_series_iterations<Policy>(); T one_over_x_pow_two_n_minus_one = 1 / z; const T one_over_x2 = one_over_x_pow_two_n_minus_one * one_over_x_pow_two_n_minus_one; T sum = (boost::math::bernoulli_b2n<T>(1) / 2) * one_over_x_pow_two_n_minus_one; const T target_epsilon_to_break_loop = sum * boost::math::tools::epsilon<T>(); const T half_ln_two_pi_over_z = sqrt(boost::math::constants::two_pi<T>() / z); T last_term = 2 * sum; for (std::size_t n = 2U;; ++n) { one_over_x_pow_two_n_minus_one *= one_over_x2; const std::size_t n2 = static_cast<std::size_t>(n * 2U); const T term = (boost::math::bernoulli_b2n<T>(static_cast<int>(n)) * one_over_x_pow_two_n_minus_one) / (n2 * (n2 - 1U)); if ((n >= 3U) && (abs(term) < target_epsilon_to_break_loop)) { // We have reached the desired precision in Stirling's expansion. // Adding additional terms to the sum of this divergent asymptotic // expansion will not improve the result. // Break from the loop. break; } if (n > number_of_bernoullis_b2n) return policies::raise_evaluation_error("scaled_tgamma_no_lanczos<%1%>()", "Exceeded maximum series iterations without reaching convergence, best approximation was %1%", T(exp(sum) * half_ln_two_pi_over_z), pol); sum += term; // Sanity check for divergence: T fterm = fabs(term); if(fterm > last_term) return policies::raise_evaluation_error("scaled_tgamma_no_lanczos<%1%>()", "Series became divergent without reaching convergence, best approximation was %1%", T(exp(sum) * half_ln_two_pi_over_z), pol); last_term = fterm; } // Complete Stirling's approximation. T scaled_gamma_value = islog ? T(sum + log(half_ln_two_pi_over_z)) : T(exp(sum) * half_ln_two_pi_over_z); return scaled_gamma_value; } // Forward declaration of the lgamma_imp template specialization. template <class T, class Policy> T lgamma_imp(T z, const Policy& pol, const lanczos::undefined_lanczos&, int* sign = 0); template <class T, class Policy> T gamma_imp(T z, const Policy& pol, const lanczos::undefined_lanczos&) { BOOST_MATH_STD_USING static const char* function = "boost::math::tgamma<%1%>(%1%)"; // Check if the argument of tgamma is identically zero. const bool is_at_zero = (z == 0); if((boost::math::isnan)(z) || (is_at_zero) || ((boost::math::isinf)(z) && (z < 0))) return policies::raise_domain_error<T>(function, "Evaluation of tgamma at %1%.", z, pol); const bool b_neg = (z < 0); const bool floor_of_z_is_equal_to_z = (floor(z) == z); // Special case handling of small factorials: if((!b_neg) && floor_of_z_is_equal_to_z && (z < boost::math::max_factorial<T>::value)) { return boost::math::unchecked_factorial<T>(itrunc(z) - 1); } // Make a local, unsigned copy of the input argument. T zz((!b_neg) ? z : -z); // Special case for ultra-small z: if(zz < tools::cbrt_epsilon<T>()) { const T a0(1); const T a1(boost::math::constants::euler<T>()); const T six_euler_squared((boost::math::constants::euler<T>() * boost::math::constants::euler<T>()) * 6); const T a2((six_euler_squared - boost::math::constants::pi_sqr<T>()) / 12); const T inverse_tgamma_series = z * ((a2 * z + a1) * z + a0); return 1 / inverse_tgamma_series; } // Scale the argument up for the calculation of lgamma, // and use downward recursion later for the final result. const int min_arg_for_recursion = minimum_argument_for_bernoulli_recursion<T>(); int n_recur; if(zz < min_arg_for_recursion) { n_recur = boost::math::itrunc(min_arg_for_recursion - zz) + 1; zz += n_recur; } else { n_recur = 0; } if (!n_recur) { if (zz > tools::log_max_value<T>()) return policies::raise_overflow_error<T>(function, 0, pol); if (log(zz) * zz / 2 > tools::log_max_value<T>()) return policies::raise_overflow_error<T>(function, 0, pol); } T gamma_value = scaled_tgamma_no_lanczos(zz, pol); T power_term = pow(zz, zz / 2); T exp_term = exp(-zz); gamma_value *= (power_term * exp_term); if(!n_recur && (tools::max_value<T>() / power_term < gamma_value)) return policies::raise_overflow_error<T>(function, 0, pol); gamma_value *= power_term; // Rescale the result using downward recursion if necessary. if(n_recur) { // The order of divides is important, if we keep subtracting 1 from zz // we DO NOT get back to z (cancellation error). Further if z < epsilon // we would end up dividing by zero. Also in order to prevent spurious // overflow with the first division, we must save dividing by |z| till last, // so the optimal order of divides is z+1, z+2, z+3...z+n_recur-1,z. zz = fabs(z) + 1; for(int k = 1; k < n_recur; ++k) { gamma_value /= zz; zz += 1; } gamma_value /= fabs(z); } // Return the result, accounting for possible negative arguments. if(b_neg) { // Provide special error analysis for: // * arguments in the neighborhood of a negative integer // * arguments exactly equal to a negative integer. // Check if the argument of tgamma is exactly equal to a negative integer. if(floor_of_z_is_equal_to_z) return policies::raise_pole_error<T>(function, "Evaluation of tgamma at a negative integer %1%.", z, pol); gamma_value *= sinpx(z); BOOST_MATH_INSTRUMENT_VARIABLE(gamma_value); const bool result_is_too_large_to_represent = ( (abs(gamma_value) < 1) && ((tools::max_value<T>() * abs(gamma_value)) < boost::math::constants::pi<T>())); if(result_is_too_large_to_represent) return policies::raise_overflow_error<T>(function, "Result of tgamma is too large to represent.", pol); gamma_value = -boost::math::constants::pi<T>() / gamma_value; BOOST_MATH_INSTRUMENT_VARIABLE(gamma_value); if(gamma_value == 0) return policies::raise_underflow_error<T>(function, "Result of tgamma is too small to represent.", pol); if((boost::math::fpclassify)(gamma_value) == static_cast<int>(FP_SUBNORMAL)) return policies::raise_denorm_error<T>(function, "Result of tgamma is denormalized.", gamma_value, pol); } return gamma_value; } template <class T, class Policy> inline T log_gamma_near_1(const T& z, Policy const& pol) { // // This is for the multiprecision case where there is // no lanczos support, use a taylor series at z = 1, // see https://www.wolframalpha.com/input/?i=taylor+series+lgamma(x)+at+x+%3D+1 // BOOST_MATH_STD_USING // ADL of std names BOOST_ASSERT(fabs(z) < 1); T result = -constants::euler<T>() * z; T power_term = z * z / 2; int n = 2; T term = 0; do { term = power_term * boost::math::polygamma(n - 1, T(1)); result += term; ++n; power_term *= z / n; } while (fabs(result) * tools::epsilon<T>() < fabs(term)); return result; } template <class T, class Policy> T lgamma_imp(T z, const Policy& pol, const lanczos::undefined_lanczos&, int* sign) { BOOST_MATH_STD_USING static const char* function = "boost::math::lgamma<%1%>(%1%)"; // Check if the argument of lgamma is identically zero. const bool is_at_zero = (z == 0); if(is_at_zero) return policies::raise_domain_error<T>(function, "Evaluation of lgamma at zero %1%.", z, pol); if((boost::math::isnan)(z)) return policies::raise_domain_error<T>(function, "Evaluation of lgamma at %1%.", z, pol); if((boost::math::isinf)(z)) return policies::raise_overflow_error<T>(function, 0, pol); const bool b_neg = (z < 0); const bool floor_of_z_is_equal_to_z = (floor(z) == z); // Special case handling of small factorials: if((!b_neg) && floor_of_z_is_equal_to_z && (z < boost::math::max_factorial<T>::value)) { if (sign) *sign = 1; return log(boost::math::unchecked_factorial<T>(itrunc(z) - 1)); } // Make a local, unsigned copy of the input argument. T zz((!b_neg) ? z : -z); const int min_arg_for_recursion = minimum_argument_for_bernoulli_recursion<T>(); T log_gamma_value; if (zz < min_arg_for_recursion) { // Here we simply take the logarithm of tgamma(). This is somewhat // inefficient, but simple. The rationale is that the argument here // is relatively small and overflow is not expected to be likely. if (sign) * sign = 1; if(fabs(z - 1) < 0.25) { log_gamma_value = log_gamma_near_1(T(zz - 1), pol); } else if(fabs(z - 2) < 0.25) { log_gamma_value = log_gamma_near_1(T(zz - 2), pol) + log(zz - 1); } else if (z > -tools::root_epsilon<T>()) { // Reflection formula may fail if z is very close to zero, let the series // expansion for tgamma close to zero do the work: if (sign) *sign = z < 0 ? -1 : 1; return log(abs(gamma_imp(z, pol, lanczos::undefined_lanczos()))); } else { // No issue with spurious overflow in reflection formula, // just fall through to regular code: T g = gamma_imp(zz, pol, lanczos::undefined_lanczos()); if (sign) { *sign = g < 0 ? -1 : 1; } log_gamma_value = log(abs(g)); } } else { // Perform the Bernoulli series expansion of Stirling's approximation. T sum = scaled_tgamma_no_lanczos(zz, pol, true); log_gamma_value = zz * (log(zz) - 1) + sum; } int sign_of_result = 1; if(b_neg) { // Provide special error analysis if the argument is exactly // equal to a negative integer. // Check if the argument of lgamma is exactly equal to a negative integer. if(floor_of_z_is_equal_to_z) return policies::raise_pole_error<T>(function, "Evaluation of lgamma at a negative integer %1%.", z, pol); T t = sinpx(z); if(t < 0) { t = -t; } else { sign_of_result = -sign_of_result; } log_gamma_value = - log_gamma_value + log(boost::math::constants::pi<T>()) - log(t); } if(sign != static_cast<int*>(0U)) { *sign = sign_of_result; } return log_gamma_value; } // // This helper calculates tgamma(dz+1)-1 without cancellation errors, // used by the upper incomplete gamma with z < 1: // template <class T, class Policy, class Lanczos> T tgammap1m1_imp(T dz, Policy const& pol, const Lanczos& l) { BOOST_MATH_STD_USING typedef typename policies::precision<T,Policy>::type precision_type; typedef typename mpl::if_< mpl::or_< mpl::less_equal<precision_type, mpl::int_<0> >, mpl::greater<precision_type, mpl::int_<113> > >, typename mpl::if_< mpl::and_<is_same<Lanczos, lanczos::lanczos24m113>, mpl::greater<precision_type, mpl::int_<0> > >, mpl::int_<113>, mpl::int_<0> >::type, typename mpl::if_< mpl::less_equal<precision_type, mpl::int_<64> >, mpl::int_<64>, mpl::int_<113> >::type >::type tag_type; T result; if(dz < 0) { if(dz < -0.5) { // Best method is simply to subtract 1 from tgamma: result = boost::math::tgamma(1+dz, pol) - 1; BOOST_MATH_INSTRUMENT_CODE(result); } else { // Use expm1 on lgamma: result = boost::math::expm1(-boost::math::log1p(dz, pol) + lgamma_small_imp<T>(dz+2, dz + 1, dz, tag_type(), pol, l)); BOOST_MATH_INSTRUMENT_CODE(result); } } else { if(dz < 2) { // Use expm1 on lgamma: result = boost::math::expm1(lgamma_small_imp<T>(dz+1, dz, dz-1, tag_type(), pol, l), pol); BOOST_MATH_INSTRUMENT_CODE(result); } else { // Best method is simply to subtract 1 from tgamma: result = boost::math::tgamma(1+dz, pol) - 1; BOOST_MATH_INSTRUMENT_CODE(result); } } return result; } template <class T, class Policy> inline T tgammap1m1_imp(T z, Policy const& pol, const ::boost::math::lanczos::undefined_lanczos&) { BOOST_MATH_STD_USING // ADL of std names if(fabs(z) < 0.55) { return boost::math::expm1(log_gamma_near_1(z, pol)); } return boost::math::expm1(boost::math::lgamma(1 + z, pol)); } // // Series representation for upper fraction when z is small: // template <class T> struct small_gamma2_series { typedef T result_type; small_gamma2_series(T a_, T x_) : result(-x_), x(-x_), apn(a_+1), n(1){} T operator()() { T r = result / (apn); result *= x; result /= ++n; apn += 1; return r; } private: T result, x, apn; int n; }; // // calculate power term prefix (z^a)(e^-z) used in the non-normalised // incomplete gammas: // template <class T, class Policy> T full_igamma_prefix(T a, T z, const Policy& pol) { BOOST_MATH_STD_USING T prefix; if (z > tools::max_value<T>()) return 0; T alz = a * log(z); if(z >= 1) { if((alz < tools::log_max_value<T>()) && (-z > tools::log_min_value<T>())) { prefix = pow(z, a) * exp(-z); } else if(a >= 1) { prefix = pow(z / exp(z/a), a); } else { prefix = exp(alz - z); } } else { if(alz > tools::log_min_value<T>()) { prefix = pow(z, a) * exp(-z); } else if(z/a < tools::log_max_value<T>()) { prefix = pow(z / exp(z/a), a); } else { prefix = exp(alz - z); } } // // This error handling isn't very good: it happens after the fact // rather than before it... // if((boost::math::fpclassify)(prefix) == (int)FP_INFINITE) return policies::raise_overflow_error<T>("boost::math::detail::full_igamma_prefix<%1%>(%1%, %1%)", "Result of incomplete gamma function is too large to represent.", pol); return prefix; } // // Compute (z^a)(e^-z)/tgamma(a) // most if the error occurs in this function: // template <class T, class Policy, class Lanczos> T regularised_gamma_prefix(T a, T z, const Policy& pol, const Lanczos& l) { BOOST_MATH_STD_USING if (z >= tools::max_value<T>()) return 0; T agh = a + static_cast<T>(Lanczos::g()) - T(0.5); T prefix; T d = ((z - a) - static_cast<T>(Lanczos::g()) + T(0.5)) / agh; if(a < 1) { // // We have to treat a < 1 as a special case because our Lanczos // approximations are optimised against the factorials with a > 1, // and for high precision types especially (128-bit reals for example) // very small values of a can give rather eroneous results for gamma // unless we do this: // // TODO: is this still required? Lanczos approx should be better now? // if(z <= tools::log_min_value<T>()) { // Oh dear, have to use logs, should be free of cancellation errors though: return exp(a * log(z) - z - lgamma_imp(a, pol, l)); } else { // direct calculation, no danger of overflow as gamma(a) < 1/a // for small a. return pow(z, a) * exp(-z) / gamma_imp(a, pol, l); } } else if((fabs(d*d*a) <= 100) && (a > 150)) { // special case for large a and a ~ z. prefix = a * boost::math::log1pmx(d, pol) + z * static_cast<T>(0.5 - Lanczos::g()) / agh; prefix = exp(prefix); } else { // // general case. // direct computation is most accurate, but use various fallbacks // for different parts of the problem domain: // T alz = a * log(z / agh); T amz = a - z; if(((std::min)(alz, amz) <= tools::log_min_value<T>()) || ((std::max)(alz, amz) >= tools::log_max_value<T>())) { T amza = amz / a; if(((std::min)(alz, amz)/2 > tools::log_min_value<T>()) && ((std::max)(alz, amz)/2 < tools::log_max_value<T>())) { // compute square root of the result and then square it: T sq = pow(z / agh, a / 2) * exp(amz / 2); prefix = sq * sq; } else if(((std::min)(alz, amz)/4 > tools::log_min_value<T>()) && ((std::max)(alz, amz)/4 < tools::log_max_value<T>()) && (z > a)) { // compute the 4th root of the result then square it twice: T sq = pow(z / agh, a / 4) * exp(amz / 4); prefix = sq * sq; prefix *= prefix; } else if((amza > tools::log_min_value<T>()) && (amza < tools::log_max_value<T>())) { prefix = pow((z * exp(amza)) / agh, a); } else { prefix = exp(alz + amz); } } else { prefix = pow(z / agh, a) * exp(amz); } } prefix *= sqrt(agh / boost::math::constants::e<T>()) / Lanczos::lanczos_sum_expG_scaled(a); return prefix; } // // And again, without Lanczos support: // template <class T, class Policy> T regularised_gamma_prefix(T a, T z, const Policy& pol, const lanczos::undefined_lanczos& l) { BOOST_MATH_STD_USING if((a < 1) && (z < 1)) { // No overflow possible since the power terms tend to unity as a,z -> 0 return pow(z, a) * exp(-z) / boost::math::tgamma(a, pol); } else if(a > minimum_argument_for_bernoulli_recursion<T>()) { T scaled_gamma = scaled_tgamma_no_lanczos(a, pol); T power_term = pow(z / a, a / 2); T a_minus_z = a - z; if ((0 == power_term) || (fabs(a_minus_z) > tools::log_max_value<T>())) { // The result is probably zero, but we need to be sure: return exp(a * log(z / a) + a_minus_z - log(scaled_gamma)); } return (power_term * exp(a_minus_z)) * (power_term / scaled_gamma); } else { // // Usual case is to calculate the prefix at a+shift and recurse down // to the value we want: // const int min_z = minimum_argument_for_bernoulli_recursion<T>(); long shift = 1 + ltrunc(min_z - a); T result = regularised_gamma_prefix(T(a + shift), z, pol, l); if (result != 0) { for (long i = 0; i < shift; ++i) { result /= z; result *= a + i; } return result; } else { // // We failed, most probably we have z << 1, try again, this time // we calculate z^a e^-z / tgamma(a+shift), combining power terms // as we go. And again recurse down to the result. // T scaled_gamma = scaled_tgamma_no_lanczos(T(a + shift), pol); T power_term_1 = pow(z / (a + shift), a); T power_term_2 = pow(a + shift, -shift); T power_term_3 = exp(a + shift - z); if ((0 == power_term_1) || (0 == power_term_2) || (0 == power_term_3) || (fabs(a + shift - z) > tools::log_max_value<T>())) { // We have no test case that gets here, most likely the type T // has a high precision but low exponent range: return exp(a * log(z) - z - boost::math::lgamma(a, pol)); } result = power_term_1 * power_term_2 * power_term_3 / scaled_gamma; for (long i = 0; i < shift; ++i) { result *= a + i; } return result; } } } // // Upper gamma fraction for very small a: // template <class T, class Policy> inline T tgamma_small_upper_part(T a, T x, const Policy& pol, T* pgam = 0, bool invert = false, T* pderivative = 0) { BOOST_MATH_STD_USING // ADL of std functions. // // Compute the full upper fraction (Q) when a is very small: // T result; result = boost::math::tgamma1pm1(a, pol); if(pgam) *pgam = (result + 1) / a; T p = boost::math::powm1(x, a, pol); result -= p; result /= a; detail::small_gamma2_series<T> s(a, x); boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>() - 10; p += 1; if(pderivative) *pderivative = p / (*pgam * exp(x)); T init_value = invert ? *pgam : 0; result = -p * tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter, (init_value - result) / p); policies::check_series_iterations<T>("boost::math::tgamma_small_upper_part<%1%>(%1%, %1%)", max_iter, pol); if(invert) result = -result; return result; } // // Upper gamma fraction for integer a: // template <class T, class Policy> inline T finite_gamma_q(T a, T x, Policy const& pol, T* pderivative = 0) { // // Calculates normalised Q when a is an integer: // BOOST_MATH_STD_USING T e = exp(-x); T sum = e; if(sum != 0) { T term = sum; for(unsigned n = 1; n < a; ++n) { term /= n; term *= x; sum += term; } } if(pderivative) { *pderivative = e * pow(x, a) / boost::math::unchecked_factorial<T>(itrunc(T(a - 1), pol)); } return sum; } // // Upper gamma fraction for half integer a: // template <class T, class Policy> T finite_half_gamma_q(T a, T x, T* p_derivative, const Policy& pol) { // // Calculates normalised Q when a is a half-integer: // BOOST_MATH_STD_USING T e = boost::math::erfc(sqrt(x), pol); if((e != 0) && (a > 1)) { T term = exp(-x) / sqrt(constants::pi<T>() * x); term *= x; static const T half = T(1) / 2; term /= half; T sum = term; for(unsigned n = 2; n < a; ++n) { term /= n - half; term *= x; sum += term; } e += sum; if(p_derivative) { *p_derivative = 0; } } else if(p_derivative) { // We'll be dividing by x later, so calculate derivative * x: *p_derivative = sqrt(x) * exp(-x) / constants::root_pi<T>(); } return e; } // // Asymptotic approximation for large argument, see: https://dlmf.nist.gov/8.11#E2 // template <class T> struct incomplete_tgamma_large_x_series { typedef T result_type; incomplete_tgamma_large_x_series(const T& a, const T& x) : a_poch(a - 1), z(x), term(1) {} T operator()() { T result = term; term *= a_poch / z; a_poch -= 1; return result; } T a_poch, z, term; }; template <class T, class Policy> T incomplete_tgamma_large_x(const T& a, const T& x, const Policy& pol) { BOOST_MATH_STD_USING incomplete_tgamma_large_x_series<T> s(a, x); boost::uintmax_t max_iter = boost::math::policies::get_max_series_iterations<Policy>(); T result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter); boost::math::policies::check_series_iterations<T>("boost::math::tgamma<%1%>(%1%,%1%)", max_iter, pol); return result; } // // Main incomplete gamma entry point, handles all four incomplete gamma's: // template <class T, class Policy> T gamma_incomplete_imp(T a, T x, bool normalised, bool invert, const Policy& pol, T* p_derivative) { static const char* function = "boost::math::gamma_p<%1%>(%1%, %1%)"; if(a <= 0) return policies::raise_domain_error<T>(function, "Argument a to the incomplete gamma function must be greater than zero (got a=%1%).", a, pol); if(x < 0) return policies::raise_domain_error<T>(function, "Argument x to the incomplete gamma function must be >= 0 (got x=%1%).", x, pol); BOOST_MATH_STD_USING typedef typename lanczos::lanczos<T, Policy>::type lanczos_type; T result = 0; // Just to avoid warning C4701: potentially uninitialized local variable 'result' used if(a >= max_factorial<T>::value && !normalised) { // // When we're computing the non-normalized incomplete gamma // and a is large the result is rather hard to compute unless // we use logs. There are really two options - if x is a long // way from a in value then we can reliably use methods 2 and 4 // below in logarithmic form and go straight to the result. // Otherwise we let the regularized gamma take the strain // (the result is unlikely to unerflow in the central region anyway) // and combine with lgamma in the hopes that we get a finite result. // if(invert && (a * 4 < x)) { // This is method 4 below, done in logs: result = a * log(x) - x; if(p_derivative) *p_derivative = exp(result); result += log(upper_gamma_fraction(a, x, policies::get_epsilon<T, Policy>())); } else if(!invert && (a > 4 * x)) { // This is method 2 below, done in logs: result = a * log(x) - x; if(p_derivative) *p_derivative = exp(result); T init_value = 0; result += log(detail::lower_gamma_series(a, x, pol, init_value) / a); } else { result = gamma_incomplete_imp(a, x, true, invert, pol, p_derivative); if(result == 0) { if(invert) { // Try http://functions.wolfram.com/06.06.06.0039.01 result = 1 + 1 / (12 * a) + 1 / (288 * a * a); result = log(result) - a + (a - 0.5f) * log(a) + log(boost::math::constants::root_two_pi<T>()); if(p_derivative) *p_derivative = exp(a * log(x) - x); } else { // This is method 2 below, done in logs, we're really outside the // range of this method, but since the result is almost certainly // infinite, we should probably be OK: result = a * log(x) - x; if(p_derivative) *p_derivative = exp(result); T init_value = 0; result += log(detail::lower_gamma_series(a, x, pol, init_value) / a); } } else { result = log(result) + boost::math::lgamma(a, pol); } } if(result > tools::log_max_value<T>()) return policies::raise_overflow_error<T>(function, 0, pol); return exp(result); } BOOST_ASSERT((p_derivative == 0) || (normalised == true)); bool is_int, is_half_int; bool is_small_a = (a < 30) && (a <= x + 1) && (x < tools::log_max_value<T>()); if(is_small_a) { T fa = floor(a); is_int = (fa == a); is_half_int = is_int ? false : (fabs(fa - a) == 0.5f); } else { is_int = is_half_int = false; } int eval_method; if(is_int && (x > 0.6)) { // calculate Q via finite sum: invert = !invert; eval_method = 0; } else if(is_half_int && (x > 0.2)) { // calculate Q via finite sum for half integer a: invert = !invert; eval_method = 1; } else if((x < tools::root_epsilon<T>()) && (a > 1)) { eval_method = 6; } else if ((x > 1000) && ((a < x) || (fabs(a - 50) / x < 1))) { // calculate Q via asymptotic approximation: invert = !invert; eval_method = 7; } else if(x < 0.5) { // // Changeover criterion chosen to give a changeover at Q ~ 0.33 // if(-0.4 / log(x) < a) { eval_method = 2; } else { eval_method = 3; } } else if(x < 1.1) { // // Changover here occurs when P ~ 0.75 or Q ~ 0.25: // if(x * 0.75f < a) { eval_method = 2; } else { eval_method = 3; } } else { // // Begin by testing whether we're in the "bad" zone // where the result will be near 0.5 and the usual // series and continued fractions are slow to converge: // bool use_temme = false; if(normalised && std::numeric_limits<T>::is_specialized && (a > 20)) { T sigma = fabs((x-a)/a); if((a > 200) && (policies::digits<T, Policy>() <= 113)) { // // This limit is chosen so that we use Temme's expansion // only if the result would be larger than about 10^-6. // Below that the regular series and continued fractions // converge OK, and if we use Temme's method we get increasing // errors from the dominant erfc term as it's (inexact) argument // increases in magnitude. // if(20 / a > sigma * sigma) use_temme = true; } else if(policies::digits<T, Policy>() <= 64) { // Note in this zone we can't use Temme's expansion for // types longer than an 80-bit real: // it would require too many terms in the polynomials. if(sigma < 0.4) use_temme = true; } } if(use_temme) { eval_method = 5; } else { // // Regular case where the result will not be too close to 0.5. // // Changeover here occurs at P ~ Q ~ 0.5 // Note that series computation of P is about x2 faster than continued fraction // calculation of Q, so try and use the CF only when really necessary, especially // for small x. // if(x - (1 / (3 * x)) < a) { eval_method = 2; } else { eval_method = 4; invert = !invert; } } } switch(eval_method) { case 0: { result = finite_gamma_q(a, x, pol, p_derivative); if(normalised == false) result *= boost::math::tgamma(a, pol); break; } case 1: { result = finite_half_gamma_q(a, x, p_derivative, pol); if(normalised == false) result *= boost::math::tgamma(a, pol); if(p_derivative && (*p_derivative == 0)) *p_derivative = regularised_gamma_prefix(a, x, pol, lanczos_type()); break; } case 2: { // Compute P: result = normalised ? regularised_gamma_prefix(a, x, pol, lanczos_type()) : full_igamma_prefix(a, x, pol); if(p_derivative) *p_derivative = result; if(result != 0) { // // If we're going to be inverting the result then we can // reduce the number of series evaluations by quite // a few iterations if we set an initial value for the // series sum based on what we'll end up subtracting it from // at the end. // Have to be careful though that this optimization doesn't // lead to spurious numberic overflow. Note that the // scary/expensive overflow checks below are more often // than not bypassed in practice for "sensible" input // values: // T init_value = 0; bool optimised_invert = false; if(invert) { init_value = (normalised ? 1 : boost::math::tgamma(a, pol)); if(normalised || (result >= 1) || (tools::max_value<T>() * result > init_value)) { init_value /= result; if(normalised || (a < 1) || (tools::max_value<T>() / a > init_value)) { init_value *= -a; optimised_invert = true; } else init_value = 0; } else init_value = 0; } result *= detail::lower_gamma_series(a, x, pol, init_value) / a; if(optimised_invert) { invert = false; result = -result; } } break; } case 3: { // Compute Q: invert = !invert; T g; result = tgamma_small_upper_part(a, x, pol, &g, invert, p_derivative); invert = false; if(normalised) result /= g; break; } case 4: { // Compute Q: result = normalised ? regularised_gamma_prefix(a, x, pol, lanczos_type()) : full_igamma_prefix(a, x, pol); if(p_derivative) *p_derivative = result; if(result != 0) result *= upper_gamma_fraction(a, x, policies::get_epsilon<T, Policy>()); break; } case 5: { // // Use compile time dispatch to the appropriate // Temme asymptotic expansion. This may be dead code // if T does not have numeric limits support, or has // too many digits for the most precise version of // these expansions, in that case we'll be calling // an empty function. // typedef typename policies::precision<T, Policy>::type precision_type; typedef typename mpl::if_< mpl::or_<mpl::equal_to<precision_type, mpl::int_<0> >, mpl::greater<precision_type, mpl::int_<113> > >, mpl::int_<0>, typename mpl::if_< mpl::less_equal<precision_type, mpl::int_<53> >, mpl::int_<53>, typename mpl::if_< mpl::less_equal<precision_type, mpl::int_<64> >, mpl::int_<64>, mpl::int_<113> >::type >::type >::type tag_type; result = igamma_temme_large(a, x, pol, static_cast<tag_type const*>(0)); if(x >= a) invert = !invert; if(p_derivative) *p_derivative = regularised_gamma_prefix(a, x, pol, lanczos_type()); break; } case 6: { // x is so small that P is necessarily very small too, // use http://functions.wolfram.com/GammaBetaErf/GammaRegularized/06/01/05/01/01/ result = !normalised ? pow(x, a) / (a) : pow(x, a) / boost::math::tgamma(a + 1, pol); result *= 1 - a * x / (a + 1); if (p_derivative) *p_derivative = regularised_gamma_prefix(a, x, pol, lanczos_type()); break; } case 7: { // x is large, // Compute Q: result = normalised ? regularised_gamma_prefix(a, x, pol, lanczos_type()) : full_igamma_prefix(a, x, pol); if (p_derivative) *p_derivative = result; result /= x; if (result != 0) result *= incomplete_tgamma_large_x(a, x, pol); break; } } if(normalised && (result > 1)) result = 1; if(invert) { T gam = normalised ? 1 : boost::math::tgamma(a, pol); result = gam - result; } if(p_derivative) { // // Need to convert prefix term to derivative: // if((x < 1) && (tools::max_value<T>() * x < *p_derivative)) { // overflow, just return an arbitrarily large value: *p_derivative = tools::max_value<T>() / 2; } *p_derivative /= x; } return result; } // // Ratios of two gamma functions: // template <class T, class Policy, class Lanczos> T tgamma_delta_ratio_imp_lanczos(T z, T delta, const Policy& pol, const Lanczos& l) { BOOST_MATH_STD_USING if(z < tools::epsilon<T>()) { // // We get spurious numeric overflow unless we're very careful, this // can occur either inside Lanczos::lanczos_sum(z) or in the // final combination of terms, to avoid this, split the product up // into 2 (or 3) parts: // // G(z) / G(L) = 1 / (z * G(L)) ; z < eps, L = z + delta = delta // z * G(L) = z * G(lim) * (G(L)/G(lim)) ; lim = largest factorial // if(boost::math::max_factorial<T>::value < delta) { T ratio = tgamma_delta_ratio_imp_lanczos(delta, T(boost::math::max_factorial<T>::value - delta), pol, l); ratio *= z; ratio *= boost::math::unchecked_factorial<T>(boost::math::max_factorial<T>::value - 1); return 1 / ratio; } else { return 1 / (z * boost::math::tgamma(z + delta, pol)); } } T zgh = static_cast<T>(z + Lanczos::g() - constants::half<T>()); T result; if(z + delta == z) { if(fabs(delta) < 10) result = exp((constants::half<T>() - z) * boost::math::log1p(delta / zgh, pol)); else result = 1; } else { if(fabs(delta) < 10) { result = exp((constants::half<T>() - z) * boost::math::log1p(delta / zgh, pol)); } else { result = pow(zgh / (zgh + delta), z - constants::half<T>()); } // Split the calculation up to avoid spurious overflow: result *= Lanczos::lanczos_sum(z) / Lanczos::lanczos_sum(T(z + delta)); } result *= pow(constants::e<T>() / (zgh + delta), delta); return result; } // // And again without Lanczos support this time: // template <class T, class Policy> T tgamma_delta_ratio_imp_lanczos(T z, T delta, const Policy& pol, const lanczos::undefined_lanczos& l) { BOOST_MATH_STD_USING // // We adjust z and delta so that both z and z+delta are large enough for // Sterling's approximation to hold. We can then calculate the ratio // for the adjusted values, and rescale back down to z and z+delta. // // Get the required shifts first: // long numerator_shift = 0; long denominator_shift = 0; const int min_z = minimum_argument_for_bernoulli_recursion<T>(); if (min_z > z) numerator_shift = 1 + ltrunc(min_z - z); if (min_z > z + delta) denominator_shift = 1 + ltrunc(min_z - z - delta); // // If the shifts are zero, then we can just combine scaled tgamma's // and combine the remaining terms: // if (numerator_shift == 0 && denominator_shift == 0) { T scaled_tgamma_num = scaled_tgamma_no_lanczos(z, pol); T scaled_tgamma_denom = scaled_tgamma_no_lanczos(T(z + delta), pol); T result = scaled_tgamma_num / scaled_tgamma_denom; result *= exp(z * boost::math::log1p(-delta / (z + delta), pol)) * pow((delta + z) / constants::e<T>(), -delta); return result; } // // We're going to have to rescale first, get the adjusted z and delta values, // plus the ratio for the adjusted values: // T zz = z + numerator_shift; T dd = delta - (numerator_shift - denominator_shift); T ratio = tgamma_delta_ratio_imp_lanczos(zz, dd, pol, l); // // Use gamma recurrence relations to get back to the original // z and z+delta: // for (long long i = 0; i < numerator_shift; ++i) { ratio /= (z + i); if (i < denominator_shift) ratio *= (z + delta + i); } for (long long i = numerator_shift; i < denominator_shift; ++i) { ratio *= (z + delta + i); } return ratio; } template <class T, class Policy> T tgamma_delta_ratio_imp(T z, T delta, const Policy& pol) { BOOST_MATH_STD_USING if((z <= 0) || (z + delta <= 0)) { // This isn't very sofisticated, or accurate, but it does work: return boost::math::tgamma(z, pol) / boost::math::tgamma(z + delta, pol); } if(floor(delta) == delta) { if(floor(z) == z) { // // Both z and delta are integers, see if we can just use table lookup // of the factorials to get the result: // if((z <= max_factorial<T>::value) && (z + delta <= max_factorial<T>::value)) { return unchecked_factorial<T>((unsigned)itrunc(z, pol) - 1) / unchecked_factorial<T>((unsigned)itrunc(T(z + delta), pol) - 1); } } if(fabs(delta) < 20) { // // delta is a small integer, we can use a finite product: // if(delta == 0) return 1; if(delta < 0) { z -= 1; T result = z; while(0 != (delta += 1)) { z -= 1; result *= z; } return result; } else { T result = 1 / z; while(0 != (delta -= 1)) { z += 1; result /= z; } return result; } } } typedef typename lanczos::lanczos<T, Policy>::type lanczos_type; return tgamma_delta_ratio_imp_lanczos(z, delta, pol, lanczos_type()); } template <class T, class Policy> T tgamma_ratio_imp(T x, T y, const Policy& pol) { BOOST_MATH_STD_USING if((x <= 0) || (boost::math::isinf)(x)) return policies::raise_domain_error<T>("boost::math::tgamma_ratio<%1%>(%1%, %1%)", "Gamma function ratios only implemented for positive arguments (got a=%1%).", x, pol); if((y <= 0) || (boost::math::isinf)(y)) return policies::raise_domain_error<T>("boost::math::tgamma_ratio<%1%>(%1%, %1%)", "Gamma function ratios only implemented for positive arguments (got b=%1%).", y, pol); if(x <= tools::min_value<T>()) { // Special case for denorms...Ugh. T shift = ldexp(T(1), tools::digits<T>()); return shift * tgamma_ratio_imp(T(x * shift), y, pol); } if((x < max_factorial<T>::value) && (y < max_factorial<T>::value)) { // Rather than subtracting values, lets just call the gamma functions directly: return boost::math::tgamma(x, pol) / boost::math::tgamma(y, pol); } T prefix = 1; if(x < 1) { if(y < 2 * max_factorial<T>::value) { // We need to sidestep on x as well, otherwise we'll underflow // before we get to factor in the prefix term: prefix /= x; x += 1; while(y >= max_factorial<T>::value) { y -= 1; prefix /= y; } return prefix * boost::math::tgamma(x, pol) / boost::math::tgamma(y, pol); } // // result is almost certainly going to underflow to zero, try logs just in case: // return exp(boost::math::lgamma(x, pol) - boost::math::lgamma(y, pol)); } if(y < 1) { if(x < 2 * max_factorial<T>::value) { // We need to sidestep on y as well, otherwise we'll overflow // before we get to factor in the prefix term: prefix *= y; y += 1; while(x >= max_factorial<T>::value) { x -= 1; prefix *= x; } return prefix * boost::math::tgamma(x, pol) / boost::math::tgamma(y, pol); } // // Result will almost certainly overflow, try logs just in case: // return exp(boost::math::lgamma(x, pol) - boost::math::lgamma(y, pol)); } // // Regular case, x and y both large and similar in magnitude: // return boost::math::tgamma_delta_ratio(x, y - x, pol); } template <class T, class Policy> T gamma_p_derivative_imp(T a, T x, const Policy& pol) { BOOST_MATH_STD_USING // // Usual error checks first: // if(a <= 0) return policies::raise_domain_error<T>("boost::math::gamma_p_derivative<%1%>(%1%, %1%)", "Argument a to the incomplete gamma function must be greater than zero (got a=%1%).", a, pol); if(x < 0) return policies::raise_domain_error<T>("boost::math::gamma_p_derivative<%1%>(%1%, %1%)", "Argument x to the incomplete gamma function must be >= 0 (got x=%1%).", x, pol); // // Now special cases: // if(x == 0) { return (a > 1) ? 0 : (a == 1) ? 1 : policies::raise_overflow_error<T>("boost::math::gamma_p_derivative<%1%>(%1%, %1%)", 0, pol); } // // Normal case: // typedef typename lanczos::lanczos<T, Policy>::type lanczos_type; T f1 = detail::regularised_gamma_prefix(a, x, pol, lanczos_type()); if((x < 1) && (tools::max_value<T>() * x < f1)) { // overflow: return policies::raise_overflow_error<T>("boost::math::gamma_p_derivative<%1%>(%1%, %1%)", 0, pol); } if(f1 == 0) { // Underflow in calculation, use logs instead: f1 = a * log(x) - x - lgamma(a, pol) - log(x); f1 = exp(f1); } else f1 /= x; return f1; } template <class T, class Policy> inline typename tools::promote_args<T>::type tgamma(T z, const Policy& /* pol */, const mpl::true_) { BOOST_FPU_EXCEPTION_GUARD typedef typename tools::promote_args<T>::type result_type; typedef typename policies::evaluation<result_type, Policy>::type value_type; typedef typename lanczos::lanczos<value_type, Policy>::type evaluation_type; typedef typename policies::normalise< Policy, policies::promote_float<false>, policies::promote_double<false>, policies::discrete_quantile<>, policies::assert_undefined<> >::type forwarding_policy; return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::gamma_imp(static_cast<value_type>(z), forwarding_policy(), evaluation_type()), "boost::math::tgamma<%1%>(%1%)"); } template <class T, class Policy> struct igamma_initializer { struct init { init() { typedef typename policies::precision<T, Policy>::type precision_type; typedef typename mpl::if_< mpl::or_<mpl::equal_to<precision_type, mpl::int_<0> >, mpl::greater<precision_type, mpl::int_<113> > >, mpl::int_<0>, typename mpl::if_< mpl::less_equal<precision_type, mpl::int_<53> >, mpl::int_<53>, typename mpl::if_< mpl::less_equal<precision_type, mpl::int_<64> >, mpl::int_<64>, mpl::int_<113> >::type >::type >::type tag_type; do_init(tag_type()); } template <int N> static void do_init(const mpl::int_<N>&) { // If std::numeric_limits<T>::digits is zero, we must not call // our inituialization code here as the precision presumably // varies at runtime, and will not have been set yet. Plus the // code requiring initialization isn't called when digits == 0. if(std::numeric_limits<T>::digits) { boost::math::gamma_p(static_cast<T>(400), static_cast<T>(400), Policy()); } } static void do_init(const mpl::int_<53>&){} void force_instantiate()const{} }; static const init initializer; static void force_instantiate() { initializer.force_instantiate(); } }; template <class T, class Policy> const typename igamma_initializer<T, Policy>::init igamma_initializer<T, Policy>::initializer; template <class T, class Policy> struct lgamma_initializer { struct init { init() { typedef typename policies::precision<T, Policy>::type precision_type; typedef typename mpl::if_< mpl::and_< mpl::less_equal<precision_type, mpl::int_<64> >, mpl::greater<precision_type, mpl::int_<0> > >, mpl::int_<64>, typename mpl::if_< mpl::and_< mpl::less_equal<precision_type, mpl::int_<113> >, mpl::greater<precision_type, mpl::int_<0> > >, mpl::int_<113>, mpl::int_<0> >::type >::type tag_type; do_init(tag_type()); } static void do_init(const mpl::int_<64>&) { boost::math::lgamma(static_cast<T>(2.5), Policy()); boost::math::lgamma(static_cast<T>(1.25), Policy()); boost::math::lgamma(static_cast<T>(1.75), Policy()); } static void do_init(const mpl::int_<113>&) { boost::math::lgamma(static_cast<T>(2.5), Policy()); boost::math::lgamma(static_cast<T>(1.25), Policy()); boost::math::lgamma(static_cast<T>(1.5), Policy()); boost::math::lgamma(static_cast<T>(1.75), Policy()); } static void do_init(const mpl::int_<0>&) { } void force_instantiate()const{} }; static const init initializer; static void force_instantiate() { initializer.force_instantiate(); } }; template <class T, class Policy> const typename lgamma_initializer<T, Policy>::init lgamma_initializer<T, Policy>::initializer; template <class T1, class T2, class Policy> inline typename tools::promote_args<T1, T2>::type tgamma(T1 a, T2 z, const Policy&, const mpl::false_) { BOOST_FPU_EXCEPTION_GUARD typedef typename tools::promote_args<T1, T2>::type result_type; typedef typename policies::evaluation<result_type, Policy>::type value_type; // typedef typename lanczos::lanczos<value_type, Policy>::type evaluation_type; typedef typename policies::normalise< Policy, policies::promote_float<false>, policies::promote_double<false>, policies::discrete_quantile<>, policies::assert_undefined<> >::type forwarding_policy; igamma_initializer<value_type, forwarding_policy>::force_instantiate(); return policies::checked_narrowing_cast<result_type, forwarding_policy>( detail::gamma_incomplete_imp(static_cast<value_type>(a), static_cast<value_type>(z), false, true, forwarding_policy(), static_cast<value_type*>(0)), "boost::math::tgamma<%1%>(%1%, %1%)"); } template <class T1, class T2> inline typename tools::promote_args<T1, T2>::type tgamma(T1 a, T2 z, const mpl::false_ tag) { return tgamma(a, z, policies::policy<>(), tag); } } // namespace detail template <class T> inline typename tools::promote_args<T>::type tgamma(T z) { return tgamma(z, policies::policy<>()); } template <class T, class Policy> inline typename tools::promote_args<T>::type lgamma(T z, int* sign, const Policy&) { BOOST_FPU_EXCEPTION_GUARD typedef typename tools::promote_args<T>::type result_type; typedef typename policies::evaluation<result_type, Policy>::type value_type; typedef typename lanczos::lanczos<value_type, Policy>::type evaluation_type; typedef typename policies::normalise< Policy, policies::promote_float<false>, policies::promote_double<false>, policies::discrete_quantile<>, policies::assert_undefined<> >::type forwarding_policy; detail::lgamma_initializer<value_type, forwarding_policy>::force_instantiate(); return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::lgamma_imp(static_cast<value_type>(z), forwarding_policy(), evaluation_type(), sign), "boost::math::lgamma<%1%>(%1%)"); } template <class T> inline typename tools::promote_args<T>::type lgamma(T z, int* sign) { return lgamma(z, sign, policies::policy<>()); } template <class T, class Policy> inline typename tools::promote_args<T>::type lgamma(T x, const Policy& pol) { return ::boost::math::lgamma(x, 0, pol); } template <class T> inline typename tools::promote_args<T>::type lgamma(T x) { return ::boost::math::lgamma(x, 0, policies::policy<>()); } template <class T, class Policy> inline typename tools::promote_args<T>::type tgamma1pm1(T z, const Policy& /* pol */) { BOOST_FPU_EXCEPTION_GUARD typedef typename tools::promote_args<T>::type result_type; typedef typename policies::evaluation<result_type, Policy>::type value_type; typedef typename lanczos::lanczos<value_type, Policy>::type evaluation_type; typedef typename policies::normalise< Policy, policies::promote_float<false>, policies::promote_double<false>, policies::discrete_quantile<>, policies::assert_undefined<> >::type forwarding_policy; return policies::checked_narrowing_cast<typename remove_cv<result_type>::type, forwarding_policy>(detail::tgammap1m1_imp(static_cast<value_type>(z), forwarding_policy(), evaluation_type()), "boost::math::tgamma1pm1<%!%>(%1%)"); } template <class T> inline typename tools::promote_args<T>::type tgamma1pm1(T z) { return tgamma1pm1(z, policies::policy<>()); } // // Full upper incomplete gamma: // template <class T1, class T2> inline typename tools::promote_args<T1, T2>::type tgamma(T1 a, T2 z) { // // Type T2 could be a policy object, or a value, select the // right overload based on T2: // typedef typename policies::is_policy<T2>::type maybe_policy; return detail::tgamma(a, z, maybe_policy()); } template <class T1, class T2, class Policy> inline typename tools::promote_args<T1, T2>::type tgamma(T1 a, T2 z, const Policy& pol) { return detail::tgamma(a, z, pol, mpl::false_()); } // // Full lower incomplete gamma: // template <class T1, class T2, class Policy> inline typename tools::promote_args<T1, T2>::type tgamma_lower(T1 a, T2 z, const Policy&) { BOOST_FPU_EXCEPTION_GUARD typedef typename tools::promote_args<T1, T2>::type result_type; typedef typename policies::evaluation<result_type, Policy>::type value_type; // typedef typename lanczos::lanczos<value_type, Policy>::type evaluation_type; typedef typename policies::normalise< Policy, policies::promote_float<false>, policies::promote_double<false>, policies::discrete_quantile<>, policies::assert_undefined<> >::type forwarding_policy; detail::igamma_initializer<value_type, forwarding_policy>::force_instantiate(); return policies::checked_narrowing_cast<result_type, forwarding_policy>( detail::gamma_incomplete_imp(static_cast<value_type>(a), static_cast<value_type>(z), false, false, forwarding_policy(), static_cast<value_type*>(0)), "tgamma_lower<%1%>(%1%, %1%)"); } template <class T1, class T2> inline typename tools::promote_args<T1, T2>::type tgamma_lower(T1 a, T2 z) { return tgamma_lower(a, z, policies::policy<>()); } // // Regularised upper incomplete gamma: // template <class T1, class T2, class Policy> inline typename tools::promote_args<T1, T2>::type gamma_q(T1 a, T2 z, const Policy& /* pol */) { BOOST_FPU_EXCEPTION_GUARD typedef typename tools::promote_args<T1, T2>::type result_type; typedef typename policies::evaluation<result_type, Policy>::type value_type; // typedef typename lanczos::lanczos<value_type, Policy>::type evaluation_type; typedef typename policies::normalise< Policy, policies::promote_float<false>, policies::promote_double<false>, policies::discrete_quantile<>, policies::assert_undefined<> >::type forwarding_policy; detail::igamma_initializer<value_type, forwarding_policy>::force_instantiate(); return policies::checked_narrowing_cast<result_type, forwarding_policy>( detail::gamma_incomplete_imp(static_cast<value_type>(a), static_cast<value_type>(z), true, true, forwarding_policy(), static_cast<value_type*>(0)), "gamma_q<%1%>(%1%, %1%)"); } template <class T1, class T2> inline typename tools::promote_args<T1, T2>::type gamma_q(T1 a, T2 z) { return gamma_q(a, z, policies::policy<>()); } // // Regularised lower incomplete gamma: // template <class T1, class T2, class Policy> inline typename tools::promote_args<T1, T2>::type gamma_p(T1 a, T2 z, const Policy&) { BOOST_FPU_EXCEPTION_GUARD typedef typename tools::promote_args<T1, T2>::type result_type; typedef typename policies::evaluation<result_type, Policy>::type value_type; // typedef typename lanczos::lanczos<value_type, Policy>::type evaluation_type; typedef typename policies::normalise< Policy, policies::promote_float<false>, policies::promote_double<false>, policies::discrete_quantile<>, policies::assert_undefined<> >::type forwarding_policy; detail::igamma_initializer<value_type, forwarding_policy>::force_instantiate(); return policies::checked_narrowing_cast<result_type, forwarding_policy>( detail::gamma_incomplete_imp(static_cast<value_type>(a), static_cast<value_type>(z), true, false, forwarding_policy(), static_cast<value_type*>(0)), "gamma_p<%1%>(%1%, %1%)"); } template <class T1, class T2> inline typename tools::promote_args<T1, T2>::type gamma_p(T1 a, T2 z) { return gamma_p(a, z, policies::policy<>()); } // ratios of gamma functions: template <class T1, class T2, class Policy> inline typename tools::promote_args<T1, T2>::type tgamma_delta_ratio(T1 z, T2 delta, const Policy& /* pol */) { BOOST_FPU_EXCEPTION_GUARD 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; return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::tgamma_delta_ratio_imp(static_cast<value_type>(z), static_cast<value_type>(delta), forwarding_policy()), "boost::math::tgamma_delta_ratio<%1%>(%1%, %1%)"); } template <class T1, class T2> inline typename tools::promote_args<T1, T2>::type tgamma_delta_ratio(T1 z, T2 delta) { return tgamma_delta_ratio(z, delta, policies::policy<>()); } template <class T1, class T2, class Policy> inline typename tools::promote_args<T1, T2>::type tgamma_ratio(T1 a, T2 b, const Policy&) { 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; return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::tgamma_ratio_imp(static_cast<value_type>(a), static_cast<value_type>(b), forwarding_policy()), "boost::math::tgamma_delta_ratio<%1%>(%1%, %1%)"); } template <class T1, class T2> inline typename tools::promote_args<T1, T2>::type tgamma_ratio(T1 a, T2 b) { return tgamma_ratio(a, b, policies::policy<>()); } template <class T1, class T2, class Policy> inline typename tools::promote_args<T1, T2>::type gamma_p_derivative(T1 a, T2 x, const Policy&) { BOOST_FPU_EXCEPTION_GUARD 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; return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::gamma_p_derivative_imp(static_cast<value_type>(a), static_cast<value_type>(x), forwarding_policy()), "boost::math::gamma_p_derivative<%1%>(%1%, %1%)"); } template <class T1, class T2> inline typename tools::promote_args<T1, T2>::type gamma_p_derivative(T1 a, T2 x) { return gamma_p_derivative(a, x, policies::policy<>()); } } // namespace math } // namespace boost #ifdef BOOST_MSVC # pragma warning(pop) #endif #include <boost/math/special_functions/detail/igamma_inverse.hpp> #include <boost/math/special_functions/detail/gamma_inva.hpp> #include <boost/math/special_functions/erf.hpp> #endif // BOOST_MATH_SF_GAMMA_HPP