src/boost/math/special_functions/digamma.hpp

//  (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/tools/rational.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 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_<0>*)
{
   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;
}
//
// 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_<0>*)
{
   //
   // 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[] = {    
      1,
      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[] = {    
      1,
      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),
      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[] = {    
      0.25479851023250261e0,
      -0.44981331915268368e0,
      -0.43916936919946835e0,
      -0.61041765350579073e-1
   };
   static const T Q[] = {    
      0.1e1,
      0.15890202430554952e1,
      0.65341249856146947e0,
      0.63851690523355715e-1
   };
   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 < 0)
   {
      // 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 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:
      //
      if(x < 1)
      {
         result = -1/x;
         x += 1;
      }
      result += digamma_imp_1_2(x, 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()
      {
         boost::math::digamma(T(1.5), Policy());
         boost::math::digamma(T(500), Policy());
      }
      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& pol)
{
   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_<64> >
      >,
      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>,
            mpl::int_<64>
         >::type
      >::type
   >::type tag_type;

   // Force initialization of constants:
   detail::digamma_initializer<result_type, 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), pol), "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
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