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1// Copyright (c) 2006 Xiaogang Zhang2// Use, modification and distribution are subject to the3// Boost Software License, Version 1.0. (See accompanying file4// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)5 6#ifndef BOOST_MATH_BESSEL_Y0_HPP7#define BOOST_MATH_BESSEL_Y0_HPP8 9#ifdef _MSC_VER10#pragma once11#pragma warning(push)12#pragma warning(disable:4702) // Unreachable code (release mode only warning)13#endif14 15#include <boost/math/tools/config.hpp>16#include <boost/math/special_functions/detail/bessel_j0.hpp>17#include <boost/math/constants/constants.hpp>18#include <boost/math/tools/rational.hpp>19#include <boost/math/tools/big_constant.hpp>20#include <boost/math/policies/error_handling.hpp>21#include <boost/math/tools/assert.hpp>22 23#if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)24//25// This is the only way we can avoid26// warning: non-standard suffix on floating constant [-Wpedantic]27// when building with -Wall -pedantic. Neither __extension__28// nor #pragma diagnostic ignored work :(29//30#pragma GCC system_header31#endif32 33// Bessel function of the second kind of order zero34// x <= 8, minimax rational approximations on root-bracketing intervals35// x > 8, Hankel asymptotic expansion in Hart, Computer Approximations, 196836 37namespace boost { namespace math { namespace detail{38 39template <typename T, typename Policy>40BOOST_MATH_GPU_ENABLED T bessel_y0(T x, const Policy&);41 42template <typename T, typename Policy>43BOOST_MATH_GPU_ENABLED T bessel_y0(T x, const Policy&)44{45 BOOST_MATH_STATIC const T P1[] = {46 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0723538782003176831e+11)),47 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.3716255451260504098e+09)),48 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.0422274357376619816e+08)),49 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.1287548474401797963e+06)),50 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0102532948020907590e+04)),51 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8402381979244993524e+01)),52 };53 BOOST_MATH_STATIC const T Q1[] = {54 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.8873865738997033405e+11)),55 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.1617187777290363573e+09)),56 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.5662956624278251596e+07)),57 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.3889393209447253406e+05)),58 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6475986689240190091e+02)),59 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),60 };61 BOOST_MATH_STATIC const T P2[] = {62 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2213976967566192242e+13)),63 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.5107435206722644429e+11)),64 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3600098638603061642e+10)),65 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.9590439394619619534e+08)),66 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.6905288611678631510e+06)),67 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4566865832663635920e+04)),68 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7427031242901594547e+01)),69 };70 BOOST_MATH_STATIC const T Q2[] = {71 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3386146580707264428e+14)),72 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4266824419412347550e+12)),73 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4015103849971240096e+10)),74 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3960202770986831075e+08)),75 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0669982352539552018e+05)),76 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.3030857612070288823e+02)),77 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),78 };79 BOOST_MATH_STATIC const T P3[] = {80 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.0728726905150210443e+15)),81 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.7016641869173237784e+14)),82 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2829912364088687306e+11)),83 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.9363051266772083678e+11)),84 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1958827170518100757e+09)),85 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0085539923498211426e+07)),86 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1363534169313901632e+04)),87 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.7439661319197499338e+01)),88 };89 BOOST_MATH_STATIC const T Q3[] = {90 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4563724628846457519e+17)),91 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9272425569640309819e+15)),92 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2598377924042897629e+13)),93 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.6926121104209825246e+10)),94 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.4727219475672302327e+08)),95 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.3924739209768057030e+05)),96 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.7903362168128450017e+02)),97 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),98 };99 BOOST_MATH_STATIC const T PC[] = {100 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684302e+04)),101 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1345386639580765797e+04)),102 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1170523380864944322e+04)),103 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4806486443249270347e+03)),104 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5376201909008354296e+02)),105 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.8961548424210455236e-01)),106 };107 BOOST_MATH_STATIC const T QC[] = {108 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684318e+04)),109 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1370412495510416640e+04)),110 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1215350561880115730e+04)),111 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5028735138235608207e+03)),112 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5711159858080893649e+02)),113 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),114 };115 BOOST_MATH_STATIC const T PS[] = {116 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.9226600200800094098e+01)),117 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8591953644342993800e+02)),118 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1183429920482737611e+02)),119 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2300261666214198472e+01)),120 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2441026745835638459e+00)),121 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.8033303048680751817e-03)),122 };123 BOOST_MATH_STATIC const T QS[] = {124 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.7105024128512061905e+03)),125 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1951131543434613647e+04)),126 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.2642780169211018836e+03)),127 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4887231232283756582e+03)),128 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.0593769594993125859e+01)),129 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),130 };131 BOOST_MATH_STATIC const T x1 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.9357696627916752158e-01)),132 x2 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9576784193148578684e+00)),133 x3 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0860510603017726976e+00)),134 x11 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.280e+02)),135 x12 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.9519662791675215849e-03)),136 x21 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0130e+03)),137 x22 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.4716931485786837568e-04)),138 x31 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8140e+03)),139 x32 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1356030177269762362e-04))140 ;141 T value, factor, r, rc, rs;142 143 BOOST_MATH_STD_USING144 using namespace boost::math::tools;145 using namespace boost::math::constants;146 147 BOOST_MATH_ASSERT(x > 0);148 149 if (x <= 3) // x in (0, 3]150 {151 T y = x * x;152 T z = 2 * log(x/x1) * bessel_j0(x) / pi<T>();153 r = evaluate_rational(P1, Q1, y);154 factor = (x + x1) * ((x - x11/256) - x12);155 value = z + factor * r;156 }157 else if (x <= 5.5f) // x in (3, 5.5]158 {159 T y = x * x;160 T z = 2 * log(x/x2) * bessel_j0(x) / pi<T>();161 r = evaluate_rational(P2, Q2, y);162 factor = (x + x2) * ((x - x21/256) - x22);163 value = z + factor * r;164 }165 else if (x <= 8) // x in (5.5, 8]166 {167 T y = x * x;168 T z = 2 * log(x/x3) * bessel_j0(x) / pi<T>();169 r = evaluate_rational(P3, Q3, y);170 factor = (x + x3) * ((x - x31/256) - x32);171 value = z + factor * r;172 }173 else // x in (8, \infty)174 {175 T y = 8 / x;176 T y2 = y * y;177 rc = evaluate_rational(PC, QC, y2);178 rs = evaluate_rational(PS, QS, y2);179 factor = constants::one_div_root_pi<T>() / sqrt(x);180 //181 // The following code is really just:182 //183 // T z = x - 0.25f * pi<T>();184 // value = factor * (rc * sin(z) + y * rs * cos(z));185 //186 // But using the sin/cos addition formulae and constant values for187 // sin/cos of PI/4 which then cancel part of the "factor" term as they're all188 // 1 / sqrt(2):189 //190 T sx = sin(x);191 T cx = cos(x);192 value = factor * (rc * (sx - cx) + y * rs * (cx + sx));193 }194 195 return value;196}197 198}}} // namespaces199 200#ifdef _MSC_VER201#pragma warning(pop)202#endif203 204#endif // BOOST_MATH_BESSEL_Y0_HPP205 206