| // (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> |
| |
| 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[] = { |
| 0.083333333333333333333333333333333333333333333333333L, |
| -0.0083333333333333333333333333333333333333333333333333L, |
| 0.003968253968253968253968253968253968253968253968254L, |
| -0.0041666666666666666666666666666666666666666666666667L, |
| 0.0075757575757575757575757575757575757575757575757576L, |
| -0.021092796092796092796092796092796092796092796092796L, |
| 0.083333333333333333333333333333333333333333333333333L, |
| -0.44325980392156862745098039215686274509803921568627L, |
| 3.0539543302701197438039543302701197438039543302701L, |
| -26.456212121212121212121212121212121212121212121212L, |
| 281.4601449275362318840579710144927536231884057971L, |
| -3607.510546398046398046398046398046398046398046398L, |
| 54827.583333333333333333333333333333333333333333333L, |
| -974936.82385057471264367816091954022988505747126437L, |
| 20052695.796688078946143462272494530559046688078946L, |
| -472384867.72162990196078431372549019607843137254902L, |
| 12635724795.916666666666666666666666666666666666667L |
| }; |
| 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[] = { |
| 0.083333333333333333333333333333333333333333333333333L, |
| -0.0083333333333333333333333333333333333333333333333333L, |
| 0.003968253968253968253968253968253968253968253968254L, |
| -0.0041666666666666666666666666666666666666666666666667L, |
| 0.0075757575757575757575757575757575757575757575757576L, |
| -0.021092796092796092796092796092796092796092796092796L, |
| 0.083333333333333333333333333333333333333333333333333L, |
| -0.44325980392156862745098039215686274509803921568627L, |
| 3.0539543302701197438039543302701197438039543302701L, |
| -26.456212121212121212121212121212121212121212121212L, |
| 281.4601449275362318840579710144927536231884057971L, |
| }; |
| 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[] = { |
| 0.083333333333333333333333333333333333333333333333333L, |
| -0.0083333333333333333333333333333333333333333333333333L, |
| 0.003968253968253968253968253968253968253968253968254L, |
| -0.0041666666666666666666666666666666666666666666666667L, |
| 0.0075757575757575757575757575757575757575757575757576L, |
| -0.021092796092796092796092796092796092796092796092796L, |
| 0.083333333333333333333333333333333333333333333333333L, |
| -0.44325980392156862745098039215686274509803921568627L |
| }; |
| 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[] = { |
| 0.083333333333333333333333333333333333333333333333333L, |
| -0.0083333333333333333333333333333333333333333333333333L, |
| 0.003968253968253968253968253968253968253968253968254L |
| }; |
| 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 = 1569415565.0 / 1073741824uL; |
| static const T root2 = (381566830.0 / 1073741824uL) / 1073741824uL; |
| static const T root3 = ((111616537.0 / 1073741824uL) / 1073741824uL) / 1073741824uL; |
| static const T root4 = (((503992070.0 / 1073741824uL) / 1073741824uL) / 1073741824uL) / 1073741824uL; |
| static const T root5 = 0.52112228569249997894452490385577338504019838794544e-36L; |
| |
| static const T P[] = { |
| 0.25479851061131551526977464225335883769L, |
| -0.18684290534374944114622235683619897417L, |
| -0.80360876047931768958995775910991929922L, |
| -0.67227342794829064330498117008564270136L, |
| -0.26569010991230617151285010695543858005L, |
| -0.05775672694575986971640757748003553385L, |
| -0.0071432147823164975485922555833274240665L, |
| -0.00048740753910766168912364555706064993274L, |
| -0.16454996865214115723416538844975174761e-4L, |
| -0.20327832297631728077731148515093164955e-6L |
| }; |
| static const T Q[] = { |
| 1, |
| 2.6210924610812025425088411043163287646L, |
| 2.6850757078559596612621337395886392594L, |
| 1.4320913706209965531250495490639289418L, |
| 0.4410872083455009362557012239501953402L, |
| 0.081385727399251729505165509278152487225L, |
| 0.0089478633066857163432104815183858149496L, |
| 0.00055861622855066424871506755481997374154L, |
| 0.1760168552357342401304462967950178554e-4L, |
| 0.20585454493572473724556649516040874384e-6L, |
| -0.90745971844439990284514121823069162795e-11L, |
| 0.48857673606545846774761343500033283272e-13L, |
| }; |
| T g = x - root1; |
| g -= root2; |
| g -= root3; |
| g -= root4; |
| g -= root5; |
| T r = tools::evaluate_polynomial(P, x-1) / tools::evaluate_polynomial(Q, 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 = 1569415565.0 / 1073741824uL; |
| static const T root2 = (381566830.0 / 1073741824uL) / 1073741824uL; |
| static const T root3 = 0.9016312093258695918615325266959189453125e-19L; |
| |
| static const T P[] = { |
| 0.254798510611315515235L, |
| -0.314628554532916496608L, |
| -0.665836341559876230295L, |
| -0.314767657147375752913L, |
| -0.0541156266153505273939L, |
| -0.00289268368333918761452L |
| }; |
| static const T Q[] = { |
| 1, |
| 2.1195759927055347547L, |
| 1.54350554664961128724L, |
| 0.486986018231042975162L, |
| 0.0660481487173569812846L, |
| 0.00298999662592323990972L, |
| -0.165079794012604905639e-5L, |
| 0.317940243105952177571e-7L |
| }; |
| T g = x - root1; |
| g -= root2; |
| g -= root3; |
| T r = tools::evaluate_polynomial(P, x-1) / tools::evaluate_polynomial(Q, 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 = 1569415565.0 / 1073741824uL; |
| static const T root2 = (381566830.0 / 1073741824uL) / 1073741824uL; |
| static const T root3 = 0.9016312093258695918615325266959189453125e-19L; |
| |
| static const T P[] = { |
| 0.25479851061131551L, |
| -0.32555031186804491L, |
| -0.65031853770896507L, |
| -0.28919126444774784L, |
| -0.045251321448739056L, |
| -0.0020713321167745952L |
| }; |
| static const T Q[] = { |
| 1L, |
| 2.0767117023730469L, |
| 1.4606242909763515L, |
| 0.43593529692665969L, |
| 0.054151797245674225L, |
| 0.0021284987017821144L, |
| -0.55789841321675513e-6L |
| }; |
| T g = x - root1; |
| g -= root2; |
| g -= root3; |
| T r = tools::evaluate_polynomial(P, x-1) / tools::evaluate_polynomial(Q, 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, x-1) / tools::evaluate_polynomial(Q, 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; |
| } |
| |
| } // 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; |
| |
| 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 |
| |