boost/math/special_functions/detail/bessel_k0.hpp
// Copyright (c) 2006 Xiaogang Zhang // Copyright (c) 2017 John Maddock // 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_BESSEL_K0_HPP #define BOOST_MATH_BESSEL_K0_HPP #ifdef _MSC_VER #pragma once #pragma warning(push) #pragma warning(disable:4702) // Unreachable code (release mode only warning) #endif #include <boost/math/tools/rational.hpp> #include <boost/math/tools/big_constant.hpp> #include <boost/math/policies/error_handling.hpp> #include <boost/assert.hpp> // Modified Bessel function of the second kind of order zero // minimax rational approximations on intervals, see // Russon and Blair, Chalk River Report AECL-3461, 1969, // as revised by Pavel Holoborodko in "Rational Approximations // for the Modified Bessel Function of the Second Kind - K0(x) // for Computations with Double Precision", see // http://www.advanpix.com/2015/11/25/rational-approximations-for-the-modified-bessel-function-of-the-second-kind-k0-for-computations-with-double-precision/ // // The actual coefficients used are our own derivation (by JM) // since we extend to both greater and lesser precision than the // references above. We can also improve performance WRT to // Holoborodko without loss of precision. namespace boost { namespace math { namespace detail{ template <typename T> T bessel_k0(const T& x); template <class T, class tag> struct bessel_k0_initializer { struct init { init() { do_init(tag()); } static void do_init(const mpl::int_<113>&) { bessel_k0(T(0.5)); bessel_k0(T(1.5)); } static void do_init(const mpl::int_<64>&) { bessel_k0(T(0.5)); bessel_k0(T(1.5)); } template <class U> static void do_init(const U&){} void force_instantiate()const{} }; static const init initializer; static void force_instantiate() { initializer.force_instantiate(); } }; template <class T, class tag> const typename bessel_k0_initializer<T, tag>::init bessel_k0_initializer<T, tag>::initializer; template <typename T, int N> T bessel_k0_imp(const T& x, const mpl::int_<N>&) { BOOST_ASSERT(0); return 0; } template <typename T> T bessel_k0_imp(const T& x, const mpl::int_<24>&) { BOOST_MATH_STD_USING if(x <= 1) { // Maximum Deviation Found : 2.358e-09 // Expected Error Term : -2.358e-09 // Maximum Relative Change in Control Points : 9.552e-02 // Max Error found at float precision = Poly : 4.448220e-08 static const T Y = 1.137250900268554688f; static const T P[] = { -1.372508979104259711e-01f, 2.622545986273687617e-01f, 5.047103728247919836e-03f }; static const T Q[] = { 1.000000000000000000e+00f, -8.928694018000029415e-02f, 2.985980684180969241e-03f }; T a = x * x / 4; a = (tools::evaluate_rational(P, Q, a) + Y) * a + 1; // Maximum Deviation Found: 1.346e-09 // Expected Error Term : -1.343e-09 // Maximum Relative Change in Control Points : 2.405e-02 // Max Error found at float precision = Poly : 1.354814e-07 static const T P2[] = { 1.159315158e-01f, 2.789828686e-01f, 2.524902861e-02f, 8.457241514e-04f, 1.530051997e-05f }; return tools::evaluate_polynomial(P2, T(x * x)) - log(x) * a; } else { // Maximum Deviation Found: 1.587e-08 // Expected Error Term : 1.531e-08 // Maximum Relative Change in Control Points : 9.064e-02 // Max Error found at float precision = Poly : 5.065020e-08 static const T P[] = { 2.533141220e-01, 5.221502603e-01, 6.380180669e-02, -5.934976547e-02 }; static const T Q[] = { 1.000000000e+00, 2.679722431e+00, 1.561635813e+00, 1.573660661e-01 }; if(x < tools::log_max_value<T>()) return ((tools::evaluate_rational(P, Q, T(1 / x)) + 1) * exp(-x) / sqrt(x)); else { T ex = exp(-x / 2); return ((tools::evaluate_rational(P, Q, T(1 / x)) + 1) * ex / sqrt(x)) * ex; } } } template <typename T> T bessel_k0_imp(const T& x, const mpl::int_<53>&) { BOOST_MATH_STD_USING if(x <= 1) { // Maximum Deviation Found: 6.077e-17 // Expected Error Term : -6.077e-17 // Maximum Relative Change in Control Points : 7.797e-02 // Max Error found at double precision = Poly : 1.003156e-16 static const T Y = 1.137250900268554688; static const T P[] = { -1.372509002685546267e-01, 2.574916117833312855e-01, 1.395474602146869316e-02, 5.445476986653926759e-04, 7.125159422136622118e-06 }; static const T Q[] = { 1.000000000000000000e+00, -5.458333438017788530e-02, 1.291052816975251298e-03, -1.367653946978586591e-05 }; T a = x * x / 4; a = (tools::evaluate_polynomial(P, a) / tools::evaluate_polynomial(Q, a) + Y) * a + 1; // Maximum Deviation Found: 3.429e-18 // Expected Error Term : 3.392e-18 // Maximum Relative Change in Control Points : 2.041e-02 // Max Error found at double precision = Poly : 2.513112e-16 static const T P2[] = { 1.159315156584124484e-01, 2.789828789146031732e-01, 2.524892993216121934e-02, 8.460350907213637784e-04, 1.491471924309617534e-05, 1.627106892422088488e-07, 1.208266102392756055e-09, 6.611686391749704310e-12 }; return tools::evaluate_polynomial(P2, T(x * x)) - log(x) * a; } else { // Maximum Deviation Found: 4.316e-17 // Expected Error Term : 9.570e-18 // Maximum Relative Change in Control Points : 2.757e-01 // Max Error found at double precision = Poly : 1.001560e-16 static const T Y = 1; static const T P[] = { 2.533141373155002416e-01, 3.628342133984595192e+00, 1.868441889406606057e+01, 4.306243981063412784e+01, 4.424116209627428189e+01, 1.562095339356220468e+01, -1.810138978229410898e+00, -1.414237994269995877e+00, -9.369168119754924625e-02 }; static const T Q[] = { 1.000000000000000000e+00, 1.494194694879908328e+01, 8.265296455388554217e+01, 2.162779506621866970e+02, 2.845145155184222157e+02, 1.851714491916334995e+02, 5.486540717439723515e+01, 6.118075837628957015e+00, 1.586261269326235053e-01 }; if(x < tools::log_max_value<T>()) return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * exp(-x) / sqrt(x)); else { T ex = exp(-x / 2); return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * ex / sqrt(x)) * ex; } } } template <typename T> T bessel_k0_imp(const T& x, const mpl::int_<64>&) { BOOST_MATH_STD_USING if(x <= 1) { // Maximum Deviation Found: 2.180e-22 // Expected Error Term : 2.180e-22 // Maximum Relative Change in Control Points : 2.943e-01 // Max Error found at float80 precision = Poly : 3.923207e-20 static const T Y = 1.137250900268554687500e+00; static const T P[] = { BOOST_MATH_BIG_CONSTANT(T, 64, -1.372509002685546875002e-01), BOOST_MATH_BIG_CONSTANT(T, 64, 2.566481981037407600436e-01), BOOST_MATH_BIG_CONSTANT(T, 64, 1.551881122448948854873e-02), BOOST_MATH_BIG_CONSTANT(T, 64, 6.646112454323276529650e-04), BOOST_MATH_BIG_CONSTANT(T, 64, 1.213747930378196492543e-05), BOOST_MATH_BIG_CONSTANT(T, 64, 9.423709328020389560844e-08) }; static const T Q[] = { BOOST_MATH_BIG_CONSTANT(T, 64, 1.000000000000000000000e+00), BOOST_MATH_BIG_CONSTANT(T, 64, -4.843828412587773008342e-02), BOOST_MATH_BIG_CONSTANT(T, 64, 1.088484822515098936140e-03), BOOST_MATH_BIG_CONSTANT(T, 64, -1.374724008530702784829e-05), BOOST_MATH_BIG_CONSTANT(T, 64, 8.452665455952581680339e-08) }; T a = x * x / 4; a = (tools::evaluate_polynomial(P, a) / tools::evaluate_polynomial(Q, a) + Y) * a + 1; // Maximum Deviation Found: 2.440e-21 // Expected Error Term : -2.434e-21 // Maximum Relative Change in Control Points : 2.459e-02 // Max Error found at float80 precision = Poly : 1.482487e-19 static const T P2[] = { BOOST_MATH_BIG_CONSTANT(T, 64, 1.159315156584124488110e-01), BOOST_MATH_BIG_CONSTANT(T, 64, 2.764832791416047889734e-01), BOOST_MATH_BIG_CONSTANT(T, 64, 1.926062887220923354112e-02), BOOST_MATH_BIG_CONSTANT(T, 64, 3.660777862036966089410e-04), BOOST_MATH_BIG_CONSTANT(T, 64, 2.094942446930673386849e-06) }; static const T Q2[] = { BOOST_MATH_BIG_CONSTANT(T, 64, 1.000000000000000000000e+00), BOOST_MATH_BIG_CONSTANT(T, 64, -2.156100313881251616320e-02), BOOST_MATH_BIG_CONSTANT(T, 64, 2.315993873344905957033e-04), BOOST_MATH_BIG_CONSTANT(T, 64, -1.529444499350703363451e-06), BOOST_MATH_BIG_CONSTANT(T, 64, 5.524988589917857531177e-09) }; return tools::evaluate_rational(P2, Q2, T(x * x)) - log(x) * a; } else { // Maximum Deviation Found: 4.291e-20 // Expected Error Term : 2.236e-21 // Maximum Relative Change in Control Points : 3.021e-01 //Max Error found at float80 precision = Poly : 8.727378e-20 static const T Y = 1; static const T P[] = { BOOST_MATH_BIG_CONSTANT(T, 64, 2.533141373155002512056e-01), BOOST_MATH_BIG_CONSTANT(T, 64, 5.417942070721928652715e+00), BOOST_MATH_BIG_CONSTANT(T, 64, 4.477464607463971754433e+01), BOOST_MATH_BIG_CONSTANT(T, 64, 1.838745728725943889876e+02), BOOST_MATH_BIG_CONSTANT(T, 64, 4.009736314927811202517e+02), BOOST_MATH_BIG_CONSTANT(T, 64, 4.557411293123609803452e+02), BOOST_MATH_BIG_CONSTANT(T, 64, 2.360222564015361268955e+02), BOOST_MATH_BIG_CONSTANT(T, 64, 2.385435333168505701022e+01), BOOST_MATH_BIG_CONSTANT(T, 64, -1.750195760942181592050e+01), BOOST_MATH_BIG_CONSTANT(T, 64, -4.059789241612946683713e+00), BOOST_MATH_BIG_CONSTANT(T, 64, -1.612783121537333908889e-01) }; static const T Q[] = { BOOST_MATH_BIG_CONSTANT(T, 64, 1.000000000000000000000e+00), BOOST_MATH_BIG_CONSTANT(T, 64, 2.200669254769325861404e+01), BOOST_MATH_BIG_CONSTANT(T, 64, 1.900177593527144126549e+02), BOOST_MATH_BIG_CONSTANT(T, 64, 8.361003989965786932682e+02), BOOST_MATH_BIG_CONSTANT(T, 64, 2.041319870804843395893e+03), BOOST_MATH_BIG_CONSTANT(T, 64, 2.828491555113790345068e+03), BOOST_MATH_BIG_CONSTANT(T, 64, 2.190342229261529076624e+03), BOOST_MATH_BIG_CONSTANT(T, 64, 9.003330795963812219852e+02), BOOST_MATH_BIG_CONSTANT(T, 64, 1.773371397243777891569e+02), BOOST_MATH_BIG_CONSTANT(T, 64, 1.368634935531158398439e+01), BOOST_MATH_BIG_CONSTANT(T, 64, 2.543310879400359967327e-01) }; if(x < tools::log_max_value<T>()) return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * exp(-x) / sqrt(x)); else { T ex = exp(-x / 2); return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * ex / sqrt(x)) * ex; } } } template <typename T> T bessel_k0_imp(const T& x, const mpl::int_<113>&) { BOOST_MATH_STD_USING if(x <= 1) { // Maximum Deviation Found: 5.682e-37 // Expected Error Term : 5.682e-37 // Maximum Relative Change in Control Points : 6.094e-04 // Max Error found at float128 precision = Poly : 5.338213e-35 static const T Y = 1.137250900268554687500000000000000000e+00f; static const T P[] = { BOOST_MATH_BIG_CONSTANT(T, 113, -1.372509002685546875000000000000000006e-01), BOOST_MATH_BIG_CONSTANT(T, 113, 2.556212905071072782462974351698081303e-01), BOOST_MATH_BIG_CONSTANT(T, 113, 1.742459135264203478530904179889103929e-02), BOOST_MATH_BIG_CONSTANT(T, 113, 8.077860530453688571555479526961318918e-04), BOOST_MATH_BIG_CONSTANT(T, 113, 1.868173911669241091399374307788635148e-05), BOOST_MATH_BIG_CONSTANT(T, 113, 2.496405768838992243478709145123306602e-07), BOOST_MATH_BIG_CONSTANT(T, 113, 1.752489221949580551692915881999762125e-09), BOOST_MATH_BIG_CONSTANT(T, 113, 5.243010555737173524710512824955368526e-12) }; static const T Q[] = { BOOST_MATH_BIG_CONSTANT(T, 113, 1.000000000000000000000000000000000000e+00), BOOST_MATH_BIG_CONSTANT(T, 113, -4.095631064064621099785696980653193721e-02), BOOST_MATH_BIG_CONSTANT(T, 113, 8.313880983725212151967078809725835532e-04), BOOST_MATH_BIG_CONSTANT(T, 113, -1.095229912293480063501285562382835142e-05), BOOST_MATH_BIG_CONSTANT(T, 113, 1.022828799511943141130509410251996277e-07), BOOST_MATH_BIG_CONSTANT(T, 113, -6.860874007419812445494782795829046836e-10), BOOST_MATH_BIG_CONSTANT(T, 113, 3.107297802344970725756092082686799037e-12), BOOST_MATH_BIG_CONSTANT(T, 113, -7.460529579244623559164763757787600944e-15) }; T a = x * x / 4; a = (tools::evaluate_rational(P, Q, a) + Y) * a + 1; // Maximum Deviation Found: 5.173e-38 // Expected Error Term : 5.105e-38 // Maximum Relative Change in Control Points : 9.734e-03 // Max Error found at float128 precision = Poly : 1.688806e-34 static const T P2[] = { BOOST_MATH_BIG_CONSTANT(T, 113, 1.159315156584124488107200313757741370e-01), BOOST_MATH_BIG_CONSTANT(T, 113, 2.789828789146031122026800078439435369e-01), BOOST_MATH_BIG_CONSTANT(T, 113, 2.524892993216269451266750049024628432e-02), BOOST_MATH_BIG_CONSTANT(T, 113, 8.460350907082229957222453839935101823e-04), BOOST_MATH_BIG_CONSTANT(T, 113, 1.491471929926042875260452849503857976e-05), BOOST_MATH_BIG_CONSTANT(T, 113, 1.627105610481598430816014719558896866e-07), BOOST_MATH_BIG_CONSTANT(T, 113, 1.208426165007797264194914898538250281e-09), BOOST_MATH_BIG_CONSTANT(T, 113, 6.508697838747354949164182457073784117e-12), BOOST_MATH_BIG_CONSTANT(T, 113, 2.659784680639805301101014383907273109e-14), BOOST_MATH_BIG_CONSTANT(T, 113, 8.531090131964391104248859415958109654e-17), BOOST_MATH_BIG_CONSTANT(T, 113, 2.205195117066478034260323124669936314e-19), BOOST_MATH_BIG_CONSTANT(T, 113, 4.692219280289030165761119775783115426e-22), BOOST_MATH_BIG_CONSTANT(T, 113, 8.362350161092532344171965861545860747e-25), BOOST_MATH_BIG_CONSTANT(T, 113, 1.277990623924628999539014980773738258e-27) }; return tools::evaluate_polynomial(P2, T(x * x)) - log(x) * a; } else { // Maximum Deviation Found: 1.462e-34 // Expected Error Term : 4.917e-40 // Maximum Relative Change in Control Points : 3.385e-01 // Max Error found at float128 precision = Poly : 1.567573e-34 static const T Y = 1; static const T P[] = { BOOST_MATH_BIG_CONSTANT(T, 113, 2.533141373155002512078826424055226265e-01), BOOST_MATH_BIG_CONSTANT(T, 113, 2.001949740768235770078339977110749204e+01), BOOST_MATH_BIG_CONSTANT(T, 113, 6.991516715983883248363351472378349986e+02), BOOST_MATH_BIG_CONSTANT(T, 113, 1.429587951594593159075690819360687720e+04), BOOST_MATH_BIG_CONSTANT(T, 113, 1.911933815201948768044660065771258450e+05), BOOST_MATH_BIG_CONSTANT(T, 113, 1.769943016204926614862175317962439875e+06), BOOST_MATH_BIG_CONSTANT(T, 113, 1.170866154649560750500954150401105606e+07), BOOST_MATH_BIG_CONSTANT(T, 113, 5.634687099724383996792011977705727661e+07), BOOST_MATH_BIG_CONSTANT(T, 113, 1.989524036456492581597607246664394014e+08), BOOST_MATH_BIG_CONSTANT(T, 113, 5.160394785715328062088529400178080360e+08), BOOST_MATH_BIG_CONSTANT(T, 113, 9.778173054417826368076483100902201433e+08), BOOST_MATH_BIG_CONSTANT(T, 113, 1.335667778588806892764139643950439733e+09), BOOST_MATH_BIG_CONSTANT(T, 113, 1.283635100080306980206494425043706838e+09), BOOST_MATH_BIG_CONSTANT(T, 113, 8.300616188213640626577036321085025855e+08), BOOST_MATH_BIG_CONSTANT(T, 113, 3.277591957076162984986406540894621482e+08), BOOST_MATH_BIG_CONSTANT(T, 113, 5.564360536834214058158565361486115932e+07), BOOST_MATH_BIG_CONSTANT(T, 113, -1.043505161612403359098596828115690596e+07), BOOST_MATH_BIG_CONSTANT(T, 113, -7.217035248223503605127967970903027314e+06), BOOST_MATH_BIG_CONSTANT(T, 113, -1.422938158797326748375799596769964430e+06), BOOST_MATH_BIG_CONSTANT(T, 113, -1.229125746200586805278634786674745210e+05), BOOST_MATH_BIG_CONSTANT(T, 113, -4.201632288615609937883545928660649813e+03), BOOST_MATH_BIG_CONSTANT(T, 113, -3.690820607338480548346746717311811406e+01) }; static const T Q[] = { BOOST_MATH_BIG_CONSTANT(T, 113, 1.000000000000000000000000000000000000e+00), BOOST_MATH_BIG_CONSTANT(T, 113, 7.964877874035741452203497983642653107e+01), BOOST_MATH_BIG_CONSTANT(T, 113, 2.808929943826193766839360018583294769e+03), BOOST_MATH_BIG_CONSTANT(T, 113, 5.814524004679994110944366890912384139e+04), BOOST_MATH_BIG_CONSTANT(T, 113, 7.897794522506725610540209610337355118e+05), BOOST_MATH_BIG_CONSTANT(T, 113, 7.456339470955813675629523617440433672e+06), BOOST_MATH_BIG_CONSTANT(T, 113, 5.057818717813969772198911392875127212e+07), BOOST_MATH_BIG_CONSTANT(T, 113, 2.513821619536852436424913886081133209e+08), BOOST_MATH_BIG_CONSTANT(T, 113, 9.255938846873380596038513316919990776e+08), BOOST_MATH_BIG_CONSTANT(T, 113, 2.537077551699028079347581816919572141e+09), BOOST_MATH_BIG_CONSTANT(T, 113, 5.176769339768120752974843214652367321e+09), BOOST_MATH_BIG_CONSTANT(T, 113, 7.828722317390455845253191337207432060e+09), BOOST_MATH_BIG_CONSTANT(T, 113, 8.698864296569996402006511705803675890e+09), BOOST_MATH_BIG_CONSTANT(T, 113, 7.007803261356636409943826918468544629e+09), BOOST_MATH_BIG_CONSTANT(T, 113, 4.016564631288740308993071395104715469e+09), BOOST_MATH_BIG_CONSTANT(T, 113, 1.595893010619754750655947035567624730e+09), BOOST_MATH_BIG_CONSTANT(T, 113, 4.241241839120481076862742189989406856e+08), BOOST_MATH_BIG_CONSTANT(T, 113, 7.168778094393076220871007550235840858e+07), BOOST_MATH_BIG_CONSTANT(T, 113, 7.156200301360388147635052029404211109e+06), BOOST_MATH_BIG_CONSTANT(T, 113, 3.752130382550379886741949463587008794e+05), BOOST_MATH_BIG_CONSTANT(T, 113, 8.370574966987293592457152146806662562e+03), BOOST_MATH_BIG_CONSTANT(T, 113, 4.871254714311063594080644835895740323e+01) }; if(x < tools::log_max_value<T>()) return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * exp(-x) / sqrt(x)); else { T ex = exp(-x / 2); return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * ex / sqrt(x)) * ex; } } } template <typename T> T bessel_k0_imp(const T& x, const mpl::int_<0>&) { if(boost::math::tools::digits<T>() <= 24) return bessel_k0_imp(x, mpl::int_<24>()); else if(boost::math::tools::digits<T>() <= 53) return bessel_k0_imp(x, mpl::int_<53>()); else if(boost::math::tools::digits<T>() <= 64) return bessel_k0_imp(x, mpl::int_<64>()); else if(boost::math::tools::digits<T>() <= 113) return bessel_k0_imp(x, mpl::int_<113>()); BOOST_ASSERT(0); return 0; } template <typename T> inline T bessel_k0(const T& x) { typedef mpl::int_< ((std::numeric_limits<T>::digits == 0) || (std::numeric_limits<T>::radix != 2)) ? 0 : std::numeric_limits<T>::digits <= 24 ? 24 : std::numeric_limits<T>::digits <= 53 ? 53 : std::numeric_limits<T>::digits <= 64 ? 64 : std::numeric_limits<T>::digits <= 113 ? 113 : -1 > tag_type; bessel_k0_initializer<T, tag_type>::force_instantiate(); return bessel_k0_imp(x, tag_type()); } }}} // namespaces #ifdef _MSC_VER #pragma warning(pop) #endif #endif // BOOST_MATH_BESSEL_K0_HPP