// Copyright John Maddock 2006-7. | |
// Copyright Paul A. Bristow 2007. | |
// 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> | |
#ifdef BOOST_MSVC | |
# pragma warning(push) | |
# pragma warning(disable: 4127 4701) | |
// // For lexical_cast, until fixed in 1.35? | |
// // conditional expression is constant & | |
// // Potentially uninitialized local variable 'name' used | |
#endif | |
#include <boost/lexical_cast.hpp> | |
#ifdef BOOST_MSVC | |
# pragma warning(pop) | |
#endif | |
#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/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_MATH_INSTRUMENT | |
#include <iostream> | |
#include <iomanip> | |
#include <typeinfo> | |
#endif | |
#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; | |
} | |
else | |
{ | |
sign = -sign; | |
} | |
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 L> | |
T gamma_imp(T z, const Policy& pol, const L& 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); | |
if((fabs(result) < 1) && (tools::max_value<T>() * fabs(result) < boost::math::constants::pi<T>())) | |
return 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); | |
return result; | |
} | |
// shift z to > 1: | |
while(z < 0) | |
{ | |
result /= z; | |
z += 1; | |
} | |
} | |
if((floor(z) == z) && (z < max_factorial<T>::value)) | |
{ | |
result *= unchecked_factorial<T>(itrunc(z, pol) - 1); | |
} | |
else | |
{ | |
result *= L::lanczos_sum(z); | |
if(z * log(z) > tools::log_max_value<T>()) | |
{ | |
// we're going to overflow unless this is done with care: | |
T zgh = (z + static_cast<T>(L::g()) - boost::math::constants::half<T>()); | |
if(log(zgh) * z / 2 > tools::log_max_value<T>()) | |
return policies::raise_overflow_error<T>(function, "Result of tgamma is too large to represent.", pol); | |
T hp = pow(zgh, (z / 2) - T(0.25)); | |
result *= hp / exp(zgh); | |
if(tools::max_value<T>() / hp < result) | |
return policies::raise_overflow_error<T>(function, "Result of tgamma is too large to represent.", pol); | |
result *= hp; | |
} | |
else | |
{ | |
T zgh = (z + static_cast<T>(L::g()) - boost::math::constants::half<T>()); | |
result *= pow(zgh, z - boost::math::constants::half<T>()) / exp(zgh); | |
} | |
} | |
return result; | |
} | |
// | |
// lgamma(z) with Lanczos support: | |
// | |
template <class T, class Policy, class L> | |
T lgamma_imp(T z, const Policy& pol, const L& 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 <= 0) | |
{ | |
// 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 < 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, z - 1, z - 2, tag_type(), pol, l); | |
} | |
else if((z >= 3) && (z < 100)) | |
{ | |
// 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 + L::g() - boost::math::constants::half<T>()); | |
result = log(zgh) - 1; | |
result *= z - 0.5f; | |
result += log(L::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("boost::math::detail::lower_gamma_series<%1%>(%1%)", max_iter, pol); | |
return result; | |
} | |
// | |
// Fully generic tgamma and lgamma use the incomplete partial | |
// sums added together: | |
// | |
template <class T, class Policy> | |
T gamma_imp(T z, const Policy& pol, const lanczos::undefined_lanczos& l) | |
{ | |
static const char* function = "boost::math::tgamma<%1%>(%1%)"; | |
BOOST_MATH_STD_USING | |
if((z <= 0) && (floor(z) == z)) | |
return policies::raise_pole_error<T>(function, "Evaluation of tgamma at a negative integer %1%.", z, pol); | |
if(z <= -20) | |
{ | |
T result = gamma_imp(-z, pol, l) * sinpx(z); | |
if((fabs(result) < 1) && (tools::max_value<T>() * fabs(result) < boost::math::constants::pi<T>())) | |
return 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); | |
return result; | |
} | |
// | |
// The upper gamma fraction is *very* slow for z < 6, actually it's very | |
// slow to converge everywhere but recursing until z > 6 gets rid of the | |
// worst of it's behaviour. | |
// | |
T prefix = 1; | |
while(z < 6) | |
{ | |
prefix /= z; | |
z += 1; | |
} | |
BOOST_MATH_INSTRUMENT_CODE(prefix); | |
if((floor(z) == z) && (z < max_factorial<T>::value)) | |
{ | |
prefix *= unchecked_factorial<T>(itrunc(z, pol) - 1); | |
} | |
else | |
{ | |
prefix = prefix * pow(z / boost::math::constants::e<T>(), z); | |
BOOST_MATH_INSTRUMENT_CODE(prefix); | |
T sum = detail::lower_gamma_series(z, z, pol) / z; | |
BOOST_MATH_INSTRUMENT_CODE(sum); | |
sum += detail::upper_gamma_fraction(z, z, ::boost::math::policies::get_epsilon<T, Policy>()); | |
BOOST_MATH_INSTRUMENT_CODE(sum); | |
if(fabs(tools::max_value<T>() / prefix) < fabs(sum)) | |
return policies::raise_overflow_error<T>(function, "Result of tgamma is too large to represent.", pol); | |
BOOST_MATH_INSTRUMENT_CODE((sum * prefix)); | |
return sum * prefix; | |
} | |
return prefix; | |
} | |
template <class T, class Policy> | |
T lgamma_imp(T z, const Policy& pol, const lanczos::undefined_lanczos& l, int*sign) | |
{ | |
BOOST_MATH_STD_USING | |
static const char* function = "boost::math::lgamma<%1%>(%1%)"; | |
T result = 0; | |
int sresult = 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); | |
T t = detail::sinpx(z); | |
z = -z; | |
if(t < 0) | |
{ | |
t = -t; | |
} | |
else | |
{ | |
sresult = -sresult; | |
} | |
result = log(boost::math::constants::pi<T>()) - lgamma_imp(z, pol, l, 0) - log(t); | |
} | |
else if((z != 1) && (z != 2)) | |
{ | |
T limit = (std::max)(z+1, T(10)); | |
T prefix = z * log(limit) - limit; | |
T sum = detail::lower_gamma_series(z, limit, pol) / z; | |
sum += detail::upper_gamma_fraction(z, limit, ::boost::math::policies::get_epsilon<T, Policy>()); | |
result = log(sum) + prefix; | |
} | |
if(sign) | |
*sign = sresult; | |
return result; | |
} | |
// | |
// 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 L> | |
T tgammap1m1_imp(T dz, Policy const& pol, const L& 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_< | |
is_same<L, lanczos::lanczos24m113>, | |
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 dz, Policy const& pol, | |
const ::boost::math::lanczos::undefined_lanczos& l) | |
{ | |
BOOST_MATH_STD_USING // ADL of std names | |
// | |
// There should be a better solution than this, but the | |
// algebra isn't easy for the general case.... | |
// Start by subracting 1 from tgamma: | |
// | |
T result = gamma_imp(1 + dz, pol, l) - 1; | |
BOOST_MATH_INSTRUMENT_CODE(result); | |
// | |
// Test the level of cancellation error observed: we loose one bit | |
// for each power of 2 the result is less than 1. If we would get | |
// more bits from our most precise lgamma rational approximation, | |
// then use that instead: | |
// | |
BOOST_MATH_INSTRUMENT_CODE((dz > -0.5)); | |
BOOST_MATH_INSTRUMENT_CODE((dz < 2)); | |
BOOST_MATH_INSTRUMENT_CODE((ldexp(1.0, boost::math::policies::digits<T, Policy>()) * fabs(result) < 1e34)); | |
if((dz > -0.5) && (dz < 2) && (ldexp(1.0, boost::math::policies::digits<T, Policy>()) * fabs(result) < 1e34)) | |
{ | |
result = tgammap1m1_imp(dz, pol, boost::math::lanczos::lanczos24m113()); | |
BOOST_MATH_INSTRUMENT_CODE(result); | |
} | |
return result; | |
} | |
// | |
// 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; | |
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) | |
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 L> | |
T regularised_gamma_prefix(T a, T z, const Policy& pol, const L& l) | |
{ | |
BOOST_MATH_STD_USING | |
T agh = a + static_cast<T>(L::g()) - T(0.5); | |
T prefix; | |
T d = ((z - a) - static_cast<T>(L::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 - L::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>()) / L::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&) | |
{ | |
BOOST_MATH_STD_USING | |
T limit = (std::max)(T(10), a); | |
T sum = detail::lower_gamma_series(a, limit, pol) / a; | |
sum += detail::upper_gamma_fraction(a, limit, ::boost::math::policies::get_epsilon<T, Policy>()); | |
if(a < 10) | |
{ | |
// special case for small a: | |
T prefix = pow(z / 10, a); | |
prefix *= exp(10-z); | |
if(0 == prefix) | |
{ | |
prefix = pow((z * exp((10-z)/a)) / 10, a); | |
} | |
prefix /= sum; | |
return prefix; | |
} | |
T zoa = z / a; | |
T amz = a - z; | |
T alzoa = a * log(zoa); | |
T prefix; | |
if(((std::min)(alzoa, amz) <= tools::log_min_value<T>()) || ((std::max)(alzoa, amz) >= tools::log_max_value<T>())) | |
{ | |
T amza = amz / a; | |
if((amza <= tools::log_min_value<T>()) || (amza >= tools::log_max_value<T>())) | |
{ | |
prefix = exp(alzoa + amz); | |
} | |
else | |
{ | |
prefix = pow(zoa * exp(amza), a); | |
} | |
} | |
else | |
{ | |
prefix = pow(zoa, a) * exp(amz); | |
} | |
prefix /= sum; | |
return prefix; | |
} | |
// | |
// 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("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; | |
} | |
// | |
// 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) | |
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) | |
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 | |
BOOST_ASSERT((p_derivative == 0) || (normalised == true)); | |
bool is_int, is_half_int; | |
bool is_small_a = (a < 30) && (a <= x + 1); | |
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 < 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) | |
{ | |
T init_value = 0; | |
if(invert) | |
{ | |
init_value = -a * (normalised ? 1 : boost::math::tgamma(a, pol)) / result; | |
} | |
result *= detail::lower_gamma_series(a, x, pol, init_value) / a; | |
if(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; | |
} | |
} | |
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 L> | |
T tgamma_delta_ratio_imp_lanczos(T z, T delta, const Policy& pol, const L&) | |
{ | |
BOOST_MATH_STD_USING | |
T zgh = z + L::g() - constants::half<T>(); | |
T result; | |
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>()); | |
} | |
result *= pow(constants::e<T>() / (zgh + delta), delta); | |
result *= L::lanczos_sum(z) / L::lanczos_sum(z + 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&) | |
{ | |
BOOST_MATH_STD_USING | |
// | |
// The upper gamma fraction is *very* slow for z < 6, actually it's very | |
// slow to converge everywhere but recursing until z > 6 gets rid of the | |
// worst of it's behaviour. | |
// | |
T prefix = 1; | |
T zd = z + delta; | |
while((zd < 6) && (z < 6)) | |
{ | |
prefix /= z; | |
prefix *= zd; | |
z += 1; | |
zd += 1; | |
} | |
if(delta < 10) | |
{ | |
prefix *= exp(-z * boost::math::log1p(delta / z, pol)); | |
} | |
else | |
{ | |
prefix *= pow(z / zd, z); | |
} | |
prefix *= pow(constants::e<T>() / zd, delta); | |
T sum = detail::lower_gamma_series(z, z, pol) / z; | |
sum += detail::upper_gamma_fraction(z, z, ::boost::math::policies::get_epsilon<T, Policy>()); | |
T sumd = detail::lower_gamma_series(zd, zd, pol) / zd; | |
sumd += detail::upper_gamma_fraction(zd, zd, ::boost::math::policies::get_epsilon<T, Policy>()); | |
sum /= sumd; | |
if(fabs(tools::max_value<T>() / prefix) < fabs(sum)) | |
return policies::raise_overflow_error<T>("boost::math::tgamma_delta_ratio<%1%>(%1%, %1%)", "Result of tgamma is too large to represent.", pol); | |
return sum * prefix; | |
} | |
template <class T, class Policy> | |
T tgamma_delta_ratio_imp(T z, T delta, const Policy& pol) | |
{ | |
BOOST_MATH_STD_USING | |
if(z <= 0) | |
policies::raise_domain_error<T>("boost::math::tgamma_delta_ratio<%1%>(%1%, %1%)", "Gamma function ratios only implemented for positive arguments (got a=%1%).", z, pol); | |
if(z+delta <= 0) | |
policies::raise_domain_error<T>("boost::math::tgamma_delta_ratio<%1%>(%1%, %1%)", "Gamma function ratios only implemented for positive arguments (got b=%1%).", 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 gamma_p_derivative_imp(T a, T x, const Policy& pol) | |
{ | |
// | |
// Usual error checks first: | |
// | |
if(a <= 0) | |
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) | |
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); | |
} | |
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 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; | |
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; | |
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; | |
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; | |
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; | |
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_delta_ratio_imp(static_cast<value_type>(a), static_cast<value_type>(static_cast<value_type>(b) - static_cast<value_type>(a)), 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 | |