1 // Copyright (c) 2006 Xiaogang Zhang 2 // Use, modification and distribution are subject to the 3 // Boost Software License, Version 1.0. (See accompanying file 4 // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) 5 6 #ifndef BOOST_MATH_BESSEL_Y1_HPP 7 #define BOOST_MATH_BESSEL_Y1_HPP 8 9 #ifdef _MSC_VER 10 #pragma once 11 #endif 12 13 #include <boost/math/special_functions/detail/bessel_j1.hpp> 14 #include <boost/math/constants/constants.hpp> 15 #include <boost/math/tools/rational.hpp> 16 #include <boost/math/tools/big_constant.hpp> 17 #include <boost/math/policies/error_handling.hpp> 18 #include <boost/assert.hpp> 19 20 // Bessel function of the second kind of order one 21 // x <= 8, minimax rational approximations on root-bracketing intervals 22 // x > 8, Hankel asymptotic expansion in Hart, Computer Approximations, 1968 23 24 namespace boost { namespace math { namespace detail{ 25 26 template <typename T, typename Policy> 27 T bessel_y1(T x, const Policy&); 28 29 template <class T, class Policy> 30 struct bessel_y1_initializer 31 { 32 struct init 33 { initboost::math::detail::bessel_y1_initializer::init34 init() 35 { 36 do_init(); 37 } do_initboost::math::detail::bessel_y1_initializer::init38 static void do_init() 39 { 40 bessel_y1(T(1), Policy()); 41 } force_instantiateboost::math::detail::bessel_y1_initializer::init42 void force_instantiate()const{} 43 }; 44 static const init initializer; force_instantiateboost::math::detail::bessel_y1_initializer45 static void force_instantiate() 46 { 47 initializer.force_instantiate(); 48 } 49 }; 50 51 template <class T, class Policy> 52 const typename bessel_y1_initializer<T, Policy>::init bessel_y1_initializer<T, Policy>::initializer; 53 54 template <typename T, typename Policy> 55 T bessel_y1(T x, const Policy& pol) 56 { 57 bessel_y1_initializer<T, Policy>::force_instantiate(); 58 59 static const T P1[] = { 60 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0535726612579544093e+13)), 61 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4708611716525426053e+12)), 62 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.7595974497819597599e+11)), 63 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.2144548214502560419e+09)), 64 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.9157479997408395984e+07)), 65 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2157953222280260820e+05)), 66 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.1714424660046133456e+02)), 67 }; 68 static const T Q1[] = { 69 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.0737873921079286084e+14)), 70 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1272286200406461981e+12)), 71 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.7800352738690585613e+10)), 72 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.2250435122182963220e+08)), 73 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.8136470753052572164e+05)), 74 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.2079908168393867438e+02)), 75 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)), 76 }; 77 static const T P2[] = { 78 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1514276357909013326e+19)), 79 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.6808094574724204577e+18)), 80 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.3638408497043134724e+16)), 81 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0686275289804744814e+15)), 82 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.9530713129741981618e+13)), 83 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.7453673962438488783e+11)), 84 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1957961912070617006e+09)), 85 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.9153806858264202986e+06)), 86 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2337180442012953128e+03)), 87 }; 88 static const T Q2[] = { 89 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.3321844313316185697e+20)), 90 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.6968198822857178911e+18)), 91 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.0837179548112881950e+16)), 92 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1187010065856971027e+14)), 93 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.0221766852960403645e+11)), 94 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.3550318087088919566e+08)), 95 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0453748201934079734e+06)), 96 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.2855164849321609336e+03)), 97 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)), 98 }; 99 static const T PC[] = { 100 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.4357578167941278571e+06)), 101 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -9.9422465050776411957e+06)), 102 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.6033732483649391093e+06)), 103 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.5235293511811373833e+06)), 104 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0982405543459346727e+05)), 105 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.6116166443246101165e+03)), 106 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0)), 107 }; 108 static const T QC[] = { 109 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.4357578167941278568e+06)), 110 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -9.9341243899345856590e+06)), 111 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.5853394797230870728e+06)), 112 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.5118095066341608816e+06)), 113 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0726385991103820119e+05)), 114 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4550094401904961825e+03)), 115 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)), 116 }; 117 static const T PS[] = { 118 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.3220913409857223519e+04)), 119 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.5145160675335701966e+04)), 120 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6178836581270835179e+04)), 121 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8494262873223866797e+04)), 122 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7063754290207680021e+03)), 123 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5265133846636032186e+01)), 124 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0)), 125 }; 126 static const T QS[] = { 127 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0871281941028743574e+05)), 128 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8194580422439972989e+06)), 129 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4194606696037208929e+06)), 130 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0029443582266975117e+05)), 131 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.7890229745772202641e+04)), 132 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.6383677696049909675e+02)), 133 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)), 134 }; 135 static const T x1 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1971413260310170351e+00)), 136 x2 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4296810407941351328e+00)), 137 x11 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.620e+02)), 138 x12 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8288260310170351490e-03)), 139 x21 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3900e+03)), 140 x22 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.4592058648672279948e-06)) 141 ; 142 T value, factor, r, rc, rs; 143 144 BOOST_MATH_STD_USING 145 using namespace boost::math::tools; 146 using namespace boost::math::constants; 147 148 if (x <= 0) 149 { 150 return policies::raise_domain_error<T>("bost::math::bessel_y1<%1%>(%1%,%1%)", 151 "Got x == %1%, but x must be > 0, complex result not supported.", x, pol); 152 } 153 if (x <= 4) // x in (0, 4] 154 { 155 T y = x * x; 156 T z = 2 * log(x/x1) * bessel_j1(x) / pi<T>(); 157 r = evaluate_rational(P1, Q1, y); 158 factor = (x + x1) * ((x - x11/256) - x12) / x; 159 value = z + factor * r; 160 } 161 else if (x <= 8) // x in (4, 8] 162 { 163 T y = x * x; 164 T z = 2 * log(x/x2) * bessel_j1(x) / pi<T>(); 165 r = evaluate_rational(P2, Q2, y); 166 factor = (x + x2) * ((x - x21/256) - x22) / x; 167 value = z + factor * r; 168 } 169 else // x in (8, \infty) 170 { 171 T y = 8 / x; 172 T y2 = y * y; 173 rc = evaluate_rational(PC, QC, y2); 174 rs = evaluate_rational(PS, QS, y2); 175 factor = 1 / (sqrt(x) * root_pi<T>()); 176 // 177 // This code is really just: 178 // 179 // T z = x - 0.75f * pi<T>(); 180 // value = factor * (rc * sin(z) + y * rs * cos(z)); 181 // 182 // But using the sin/cos addition rules, plus constants for sin/cos of 3PI/4 183 // which then cancel out with corresponding terms in "factor". 184 // 185 T sx = sin(x); 186 T cx = cos(x); 187 value = factor * (y * rs * (sx - cx) - rc * (sx + cx)); 188 } 189 190 return value; 191 } 192 193 }}} // namespaces 194 195 #endif // BOOST_MATH_BESSEL_Y1_HPP 196 197