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glm/test/external/boost/math/special_functions/gamma.hpp

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// 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);
BOOST_MATH_INSTRUMENT_VARIABLE(result);
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);
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
{
result *= L::lanczos_sum(z);
BOOST_MATH_INSTRUMENT_VARIABLE(result);
BOOST_MATH_INSTRUMENT_VARIABLE(tools::log_max_value<T>());
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>());
BOOST_MATH_INSTRUMENT_VARIABLE(zgh);
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));
BOOST_MATH_INSTRUMENT_VARIABLE(hp);
result *= hp / exp(zgh);
BOOST_MATH_INSTRUMENT_VARIABLE(result);
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;
BOOST_MATH_INSTRUMENT_VARIABLE(result);
}
else
{
T zgh = (z + static_cast<T>(L::g()) - boost::math::constants::half<T>());
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 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, T(z - 1), T(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<T>("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(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;
}
//
// 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)(T(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(T(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<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;
}
//
// 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(T(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