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Initial Commit
2018-02-08 21:28:33 -05:00
// 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/zeta.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;
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>
T 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));
return T(digits10_of_type * 1.7F);
}
// 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(is_at_zero)
return policies::raise_domain_error<T>(function, "Evaluation of tgamma at zero %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 T 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;
}
const T log_gamma_value = lgamma_imp(zz, pol, lanczos::undefined_lanczos());
if(log_gamma_value > tools::log_max_value<T>())
return policies::raise_overflow_error<T>(function, 0, pol);
T gamma_value = exp(log_gamma_value);
// 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...
//
BOOST_MATH_STD_USING // ADL of std names
BOOST_ASSERT(fabs(z) < 1);
T result = -constants::euler<T>() * z;
T power_term = z * z;
T term;
unsigned j = 0;
do
{
term = boost::math::zeta<T>(j + 2, pol) * power_term / (j + 2);
if(j & 1)
result -= term;
else
result += term;
power_term *= z;
++j;
} 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);
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 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 T 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(fabs(z - 1) < 0.25)
{
return log_gamma_near_1(T(zz - 1), pol);
}
else if(fabs(z - 2) < 0.25)
{
return 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:
log_gamma_value = log(abs(gamma_imp(z, pol, lanczos::undefined_lanczos())));
if (sign)
{
*sign = z < 0 ? -1 : 1;
}
return log_gamma_value;
}
else
{
// No issue with spurious overflow in reflection formula,
// just fall through to regular code:
log_gamma_value = log(abs(gamma_imp(zz, pol, lanczos::undefined_lanczos())));
}
}
else
{
// Perform the Bernoulli series expansion of Stirling's approximation.
const std::size_t number_of_bernoullis_b2n = highest_bernoulli_index<T>();
T one_over_x_pow_two_n_minus_one = 1 / zz;
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>()) * T(1.0E-10F);
for(std::size_t n = 2U; n < number_of_bernoullis_b2n; ++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 >= 8U) && (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;
}
sum += term;
}
// Complete Stirling's approximation.
const T half_ln_two_pi = log(boost::math::constants::two_pi<T>()) / 2;
log_gamma_value = ((((zz - boost::math::constants::half<T>()) * log(zz)) - zz) + half_ln_two_pi) + 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;
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
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&)
{
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)
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 < 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(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&)
{
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) || (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
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