Google Git
Sign in
nest-open-source / nest-cam / 4320010 / boost / 62d1066a6d52d5ac4e10c106f0eab14c820be0ce / . / boost / boost / math / special_functions / digamma.hpp
blob: 718eaf9278fc31a1d0650a717f6f3dd63518ce93 [file] [log] [blame]
// (C) Copyright John Maddock 2006.
// Use, modification and distribution are subject to the
// Boost Software License, Version 1.0. (See accompanying file
// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
#ifndef BOOST_MATH_SF_DIGAMMA_HPP
#define BOOST_MATH_SF_DIGAMMA_HPP
#ifdef _MSC_VER
#pragma once
#endif
#include <boost/math/special_functions/math_fwd.hpp>
#include <boost/math/tools/rational.hpp>
#include <boost/math/tools/series.hpp>
#include <boost/math/tools/promotion.hpp>
#include <boost/math/policies/error_handling.hpp>
#include <boost/math/constants/constants.hpp>
#include <boost/mpl/comparison.hpp>
#include <boost/math/tools/big_constant.hpp>
namespace boost{
namespace math{
namespace detail{
//
// Begin by defining the smallest value for which it is safe to
// use the asymptotic expansion for digamma:
//
inline unsigned digamma_large_lim(const mpl::int_<0>*)
{ return 20; }
inline unsigned digamma_large_lim(const mpl::int_<113>*)
{ return 20; }
inline unsigned digamma_large_lim(const void*)
{ return 10; }
//
// Implementations of the asymptotic expansion come next,
// the coefficients of the series have been evaluated
// in advance at high precision, and the series truncated
// at the first term that's too small to effect the result.
// Note that the series becomes divergent after a while
// so truncation is very important.
//
// This first one gives 34-digit precision for x >= 20:
//
template <class T>
inline T digamma_imp_large(T x, const mpl::int_<113>*)
{
BOOST_MATH_STD_USING // ADL of std functions.
static const T P[] = {
BOOST_MATH_BIG_CONSTANT(T, 113, 0.083333333333333333333333333333333333333333333333333),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.0083333333333333333333333333333333333333333333333333),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.003968253968253968253968253968253968253968253968254),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.0041666666666666666666666666666666666666666666666667),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0075757575757575757575757575757575757575757575757576),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.021092796092796092796092796092796092796092796092796),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.083333333333333333333333333333333333333333333333333),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.44325980392156862745098039215686274509803921568627),
BOOST_MATH_BIG_CONSTANT(T, 113, 3.0539543302701197438039543302701197438039543302701),
BOOST_MATH_BIG_CONSTANT(T, 113, -26.456212121212121212121212121212121212121212121212),
BOOST_MATH_BIG_CONSTANT(T, 113, 281.4601449275362318840579710144927536231884057971),
BOOST_MATH_BIG_CONSTANT(T, 113, -3607.510546398046398046398046398046398046398046398),
BOOST_MATH_BIG_CONSTANT(T, 113, 54827.583333333333333333333333333333333333333333333),
BOOST_MATH_BIG_CONSTANT(T, 113, -974936.82385057471264367816091954022988505747126437),
BOOST_MATH_BIG_CONSTANT(T, 113, 20052695.796688078946143462272494530559046688078946),
BOOST_MATH_BIG_CONSTANT(T, 113, -472384867.72162990196078431372549019607843137254902),
BOOST_MATH_BIG_CONSTANT(T, 113, 12635724795.916666666666666666666666666666666666667)
};
x -= 1;
T result = log(x);
result += 1 / (2 * x);
T z = 1 / (x*x);
result -= z * tools::evaluate_polynomial(P, z);
return result;
}
//
// 19-digit precision for x >= 10:
//
template <class T>
inline T digamma_imp_large(T x, const mpl::int_<64>*)
{
BOOST_MATH_STD_USING // ADL of std functions.
static const T P[] = {
BOOST_MATH_BIG_CONSTANT(T, 64, 0.083333333333333333333333333333333333333333333333333),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.0083333333333333333333333333333333333333333333333333),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.003968253968253968253968253968253968253968253968254),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.0041666666666666666666666666666666666666666666666667),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0075757575757575757575757575757575757575757575757576),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.021092796092796092796092796092796092796092796092796),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.083333333333333333333333333333333333333333333333333),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.44325980392156862745098039215686274509803921568627),
BOOST_MATH_BIG_CONSTANT(T, 64, 3.0539543302701197438039543302701197438039543302701),
BOOST_MATH_BIG_CONSTANT(T, 64, -26.456212121212121212121212121212121212121212121212),
BOOST_MATH_BIG_CONSTANT(T, 64, 281.4601449275362318840579710144927536231884057971),
};
x -= 1;
T result = log(x);
result += 1 / (2 * x);
T z = 1 / (x*x);
result -= z * tools::evaluate_polynomial(P, z);
return result;
}
//
// 17-digit precision for x >= 10:
//
template <class T>
inline T digamma_imp_large(T x, const mpl::int_<53>*)
{
BOOST_MATH_STD_USING // ADL of std functions.
static const T P[] = {
BOOST_MATH_BIG_CONSTANT(T, 53, 0.083333333333333333333333333333333333333333333333333),
BOOST_MATH_BIG_CONSTANT(T, 53, -0.0083333333333333333333333333333333333333333333333333),
BOOST_MATH_BIG_CONSTANT(T, 53, 0.003968253968253968253968253968253968253968253968254),
BOOST_MATH_BIG_CONSTANT(T, 53, -0.0041666666666666666666666666666666666666666666666667),
BOOST_MATH_BIG_CONSTANT(T, 53, 0.0075757575757575757575757575757575757575757575757576),
BOOST_MATH_BIG_CONSTANT(T, 53, -0.021092796092796092796092796092796092796092796092796),
BOOST_MATH_BIG_CONSTANT(T, 53, 0.083333333333333333333333333333333333333333333333333),
BOOST_MATH_BIG_CONSTANT(T, 53, -0.44325980392156862745098039215686274509803921568627)
};
x -= 1;
T result = log(x);
result += 1 / (2 * x);
T z = 1 / (x*x);
result -= z * tools::evaluate_polynomial(P, z);
return result;
}
//
// 9-digit precision for x >= 10:
//
template <class T>
inline T digamma_imp_large(T x, const mpl::int_<24>*)
{
BOOST_MATH_STD_USING // ADL of std functions.
static const T P[] = {
BOOST_MATH_BIG_CONSTANT(T, 24, 0.083333333333333333333333333333333333333333333333333),
BOOST_MATH_BIG_CONSTANT(T, 24, -0.0083333333333333333333333333333333333333333333333333),
BOOST_MATH_BIG_CONSTANT(T, 24, 0.003968253968253968253968253968253968253968253968254)
};
x -= 1;
T result = log(x);
result += 1 / (2 * x);
T z = 1 / (x*x);
result -= z * tools::evaluate_polynomial(P, z);
return result;
}
//
// Fully generic asymptotic expansion in terms of Bernoulli numbers, see:
// http://functions.wolfram.com/06.14.06.0012.01
//
template <class T>
struct digamma_series_func
{
private:
int k;
T xx;
T term;
public:
digamma_series_func(T x) : k(1), xx(x * x), term(1 / (x * x)) {}
T operator()()
{
T result = term * boost::math::bernoulli_b2n<T>(k) / (2 * k);
term /= xx;
++k;
return result;
}
typedef T result_type;
};
template <class T, class Policy>
inline T digamma_imp_large(T x, const Policy& pol, const mpl::int_<0>*)
{
BOOST_MATH_STD_USING
digamma_series_func<T> s(x);
T result = log(x) - 1 / (2 * x);
boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter, -result);
result = -result;
policies::check_series_iterations<T>("boost::math::digamma<%1%>(%1%)", max_iter, pol);
return result;
}
//
// Now follow rational approximations over the range [1,2].
//
// 35-digit precision:
//
template <class T>
T digamma_imp_1_2(T x, const mpl::int_<113>*)
{
//
// Now the approximation, we use the form:
//
// digamma(x) = (x - root) * (Y + R(x-1))
//
// Where root is the location of the positive root of digamma,
// Y is a constant, and R is optimised for low absolute error
// compared to Y.
//
// Max error found at 128-bit long double precision: 5.541e-35
// Maximum Deviation Found (approximation error): 1.965e-35
//
static const float Y = 0.99558162689208984375F;
static const T root1 = T(1569415565) / 1073741824uL;
static const T root2 = (T(381566830) / 1073741824uL) / 1073741824uL;
static const T root3 = ((T(111616537) / 1073741824uL) / 1073741824uL) / 1073741824uL;
static const T root4 = (((T(503992070) / 1073741824uL) / 1073741824uL) / 1073741824uL) / 1073741824uL;
static const T root5 = BOOST_MATH_BIG_CONSTANT(T, 113, 0.52112228569249997894452490385577338504019838794544e-36);
static const T P[] = {
BOOST_MATH_BIG_CONSTANT(T, 113, 0.25479851061131551526977464225335883769),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.18684290534374944114622235683619897417),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.80360876047931768958995775910991929922),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.67227342794829064330498117008564270136),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.26569010991230617151285010695543858005),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.05775672694575986971640757748003553385),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.0071432147823164975485922555833274240665),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.00048740753910766168912364555706064993274),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.16454996865214115723416538844975174761e-4),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.20327832297631728077731148515093164955e-6)
};
static const T Q[] = {
BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
BOOST_MATH_BIG_CONSTANT(T, 113, 2.6210924610812025425088411043163287646),
BOOST_MATH_BIG_CONSTANT(T, 113, 2.6850757078559596612621337395886392594),
BOOST_MATH_BIG_CONSTANT(T, 113, 1.4320913706209965531250495490639289418),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.4410872083455009362557012239501953402),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.081385727399251729505165509278152487225),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0089478633066857163432104815183858149496),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00055861622855066424871506755481997374154),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.1760168552357342401304462967950178554e-4),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.20585454493572473724556649516040874384e-6),
BOOST_MATH_BIG_CONSTANT(T, 113, -0.90745971844439990284514121823069162795e-11),
BOOST_MATH_BIG_CONSTANT(T, 113, 0.48857673606545846774761343500033283272e-13),
};
T g = x - root1;
g -= root2;
g -= root3;
g -= root4;
g -= root5;
T r = tools::evaluate_polynomial(P, T(x-1)) / tools::evaluate_polynomial(Q, T(x-1));
T result = g * Y + g * r;
return result;
}
//
// 19-digit precision:
//
template <class T>
T digamma_imp_1_2(T x, const mpl::int_<64>*)
{
//
// Now the approximation, we use the form:
//
// digamma(x) = (x - root) * (Y + R(x-1))
//
// Where root is the location of the positive root of digamma,
// Y is a constant, and R is optimised for low absolute error
// compared to Y.
//
// Max error found at 80-bit long double precision: 5.016e-20
// Maximum Deviation Found (approximation error): 3.575e-20
//
static const float Y = 0.99558162689208984375F;
static const T root1 = T(1569415565) / 1073741824uL;
static const T root2 = (T(381566830) / 1073741824uL) / 1073741824uL;
static const T root3 = BOOST_MATH_BIG_CONSTANT(T, 64, 0.9016312093258695918615325266959189453125e-19);
static const T P[] = {
BOOST_MATH_BIG_CONSTANT(T, 64, 0.254798510611315515235),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.314628554532916496608),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.665836341559876230295),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.314767657147375752913),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.0541156266153505273939),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.00289268368333918761452)
};
static const T Q[] = {
BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
BOOST_MATH_BIG_CONSTANT(T, 64, 2.1195759927055347547),
BOOST_MATH_BIG_CONSTANT(T, 64, 1.54350554664961128724),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.486986018231042975162),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0660481487173569812846),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.00298999662592323990972),
BOOST_MATH_BIG_CONSTANT(T, 64, -0.165079794012604905639e-5),
BOOST_MATH_BIG_CONSTANT(T, 64, 0.317940243105952177571e-7)
};
T g = x - root1;
g -= root2;
g -= root3;
T r = tools::evaluate_polynomial(P, T(x-1)) / tools::evaluate_polynomial(Q, T(x-1));
T result = g * Y + g * r;
return result;
}
//
// 18-digit precision:
//
template <class T>
T digamma_imp_1_2(T x, const mpl::int_<53>*)
{
//
// Now the approximation, we use the form:
//
// digamma(x) = (x - root) * (Y + R(x-1))
//
// Where root is the location of the positive root of digamma,
// Y is a constant, and R is optimised for low absolute error
// compared to Y.
//
// Maximum Deviation Found: 1.466e-18
// At double precision, max error found: 2.452e-17
//
static const float Y = 0.99558162689208984F;
static const T root1 = T(1569415565) / 1073741824uL;
static const T root2 = (T(381566830) / 1073741824uL) / 1073741824uL;
static const T root3 = BOOST_MATH_BIG_CONSTANT(T, 53, 0.9016312093258695918615325266959189453125e-19);
static const T P[] = {
BOOST_MATH_BIG_CONSTANT(T, 53, 0.25479851061131551),
BOOST_MATH_BIG_CONSTANT(T, 53, -0.32555031186804491),
BOOST_MATH_BIG_CONSTANT(T, 53, -0.65031853770896507),
BOOST_MATH_BIG_CONSTANT(T, 53, -0.28919126444774784),
BOOST_MATH_BIG_CONSTANT(T, 53, -0.045251321448739056),
BOOST_MATH_BIG_CONSTANT(T, 53, -0.0020713321167745952)
};
static const T Q[] = {
BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
BOOST_MATH_BIG_CONSTANT(T, 53, 2.0767117023730469),
BOOST_MATH_BIG_CONSTANT(T, 53, 1.4606242909763515),
BOOST_MATH_BIG_CONSTANT(T, 53, 0.43593529692665969),
BOOST_MATH_BIG_CONSTANT(T, 53, 0.054151797245674225),
BOOST_MATH_BIG_CONSTANT(T, 53, 0.0021284987017821144),
BOOST_MATH_BIG_CONSTANT(T, 53, -0.55789841321675513e-6)
};
T g = x - root1;
g -= root2;
g -= root3;
T r = tools::evaluate_polynomial(P, T(x-1)) / tools::evaluate_polynomial(Q, T(x-1));
T result = g * Y + g * r;
return result;
}
//
// 9-digit precision:
//
template <class T>
inline T digamma_imp_1_2(T x, const mpl::int_<24>*)
{
//
// Now the approximation, we use the form:
//
// digamma(x) = (x - root) * (Y + R(x-1))
//
// Where root is the location of the positive root of digamma,
// Y is a constant, and R is optimised for low absolute error
// compared to Y.
//
// Maximum Deviation Found: 3.388e-010
// At float precision, max error found: 2.008725e-008
//
static const float Y = 0.99558162689208984f;
static const T root = 1532632.0f / 1048576;
static const T root_minor = static_cast<T>(0.3700660185912626595423257213284682051735604e-6L);
static const T P[] = {
0.25479851023250261e0f,
-0.44981331915268368e0f,
-0.43916936919946835e0f,
-0.61041765350579073e-1f
};
static const T Q[] = {
0.1e1,
0.15890202430554952e1f,
0.65341249856146947e0f,
0.63851690523355715e-1f
};
T g = x - root;
g -= root_minor;
T r = tools::evaluate_polynomial(P, T(x-1)) / tools::evaluate_polynomial(Q, T(x-1));
T result = g * Y + g * r;
return result;
}
template <class T, class Tag, class Policy>
T digamma_imp(T x, const Tag* t, const Policy& pol)
{
//
// This handles reflection of negative arguments, and all our
// error handling, then forwards to the T-specific approximation.
//
BOOST_MATH_STD_USING // ADL of std functions.
T result = 0;
//
// Check for negative arguments and use reflection:
//
if(x <= -1)
{
// Reflect:
x = 1 - x;
// Argument reduction for tan:
T remainder = x - floor(x);
// Shift to negative if > 0.5:
if(remainder > 0.5)
{
remainder -= 1;
}
//
// check for evaluation at a negative pole:
//
if(remainder == 0)
{
return policies::raise_pole_error<T>("boost::math::digamma<%1%>(%1%)", 0, (1-x), pol);
}
result = constants::pi<T>() / tan(constants::pi<T>() * remainder);
}
if(x == 0)
return policies::raise_pole_error<T>("boost::math::digamma<%1%>(%1%)", 0, x, pol);
//
// If we're above the lower-limit for the
// asymptotic expansion then use it:
//
if(x >= digamma_large_lim(t))
{
result += digamma_imp_large(x, t);
}
else
{
//
// If x > 2 reduce to the interval [1,2]:
//
while(x > 2)
{
x -= 1;
result += 1/x;
}
//
// If x < 1 use recurrance to shift to > 1:
//
while(x < 1)
{
result -= 1/x;
x += 1;
}
result += digamma_imp_1_2(x, t);
}
return result;
}
template <class T, class Policy>
T digamma_imp(T x, const mpl::int_<0>* t, const Policy& pol)
{
//
// This handles reflection of negative arguments, and all our
// error handling, then forwards to the T-specific approximation.
//
BOOST_MATH_STD_USING // ADL of std functions.
T result = 0;
//
// Check for negative arguments and use reflection:
//
if(x <= -1)
{
// Reflect:
x = 1 - x;
// Argument reduction for tan:
T remainder = x - floor(x);
// Shift to negative if > 0.5:
if(remainder > 0.5)
{
remainder -= 1;
}
//
// check for evaluation at a negative pole:
//
if(remainder == 0)
{
return policies::raise_pole_error<T>("boost::math::digamma<%1%>(%1%)", 0, (1 - x), pol);
}
result = constants::pi<T>() / tan(constants::pi<T>() * remainder);
}
if(x == 0)
return policies::raise_pole_error<T>("boost::math::digamma<%1%>(%1%)", 0, x, pol);
//
// If we're above the lower-limit for the
// asymptotic expansion then use it, the
// limit is a linear interpolation with
// limit = 10 at 50 bit precision and
// limit = 250 at 1000 bit precision.
//
T lim = 10 + (tools::digits<T>() - 50) * 240 / 950;
T two_x = ldexp(x, 1);
if(x >= lim)
{
result += digamma_imp_large(x, pol, t);
}
else if(floor(x) == x)
{
//
// Special case for integer arguments, see
// http://functions.wolfram.com/06.14.03.0001.01
//
result = -constants::euler<T, Policy>();
T val = 1;
while(val < x)
{
result += 1 / val;
val += 1;
}
}
else if(floor(two_x) == two_x)
{
//
// Special case for half integer arguments, see:
// http://functions.wolfram.com/06.14.03.0007.01
//
result = -2 * constants::ln_two<T, Policy>() - constants::euler<T, Policy>();
int n = itrunc(x);
if(n)
{
for(int k = 1; k < n; ++k)
result += 1 / T(k);
for(int k = n; k <= 2 * n - 1; ++k)
result += 2 / T(k);
}
}
else
{
//
// Rescale so we can use the asymptotic expansion:
//
while(x < lim)
{
result -= 1 / x;
x += 1;
}
result += digamma_imp_large(x, pol, t);
}
return result;
}
//
// Initializer: ensure all our constants are initialized prior to the first call of main:
//
template <class T, class Policy>
struct digamma_initializer
{
struct init
{
init()
{
typedef typename policies::precision<T, Policy>::type precision_type;
do_init(mpl::bool_<precision_type::value && (precision_type::value <= 113)>());
}
void do_init(const mpl::true_&)
{
boost::math::digamma(T(1.5), Policy());
boost::math::digamma(T(500), Policy());
}
void do_init(const mpl::false_&){}
void force_instantiate()const{}
};
static const init initializer;
static void force_instantiate()
{
initializer.force_instantiate();
}
};
template <class T, class Policy>
const typename digamma_initializer<T, Policy>::init digamma_initializer<T, Policy>::initializer;
} // namespace detail
template <class T, class Policy>
inline typename tools::promote_args<T>::type
digamma(T x, const Policy&)
{
typedef typename tools::promote_args<T>::type result_type;
typedef typename policies::evaluation<result_type, Policy>::type value_type;
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_<114> >
>,
mpl::int_<0>,
typename mpl::if_<
mpl::less<precision_type, mpl::int_<25> >,
mpl::int_<24>,
typename mpl::if_<
mpl::less<precision_type, mpl::int_<54> >,
mpl::int_<53>,
typename mpl::if_<
mpl::less<precision_type, mpl::int_<65> >,
mpl::int_<64>,
mpl::int_<113>
>::type
>::type
>::type
>::type tag_type;
typedef typename policies::normalise<
Policy,
policies::promote_float<false>,
policies::promote_double<false>,
policies::discrete_quantile<>,
policies::assert_undefined<> >::type forwarding_policy;
// Force initialization of constants:
detail::digamma_initializer<value_type, forwarding_policy>::force_instantiate();
return policies::checked_narrowing_cast<result_type, Policy>(detail::digamma_imp(
static_cast<value_type>(x),
static_cast<const tag_type*>(0), forwarding_policy()), "boost::math::digamma<%1%>(%1%)");
}
template <class T>
inline typename tools::promote_args<T>::type
digamma(T x)
{
return digamma(x, policies::policy<>());
}
} // namespace math
} // namespace boost
#endif