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annotate DEPENDENCIES/generic/include/boost/math/special_functions/detail/bessel_y0.hpp @ 125:34e428693f5d vext

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Vext -> Repoint
author Chris Cannam
date Thu, 14 Jun 2018 11:15:39 +0100
parents 2665513ce2d3
children
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Chris@16 1 // Copyright (c) 2006 Xiaogang Zhang
Chris@16 2 // Use, modification and distribution are subject to the
Chris@16 3 // Boost Software License, Version 1.0. (See accompanying file
Chris@16 4 // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
Chris@16 5
Chris@16 6 #ifndef BOOST_MATH_BESSEL_Y0_HPP
Chris@16 7 #define BOOST_MATH_BESSEL_Y0_HPP
Chris@16 8
Chris@16 9 #ifdef _MSC_VER
Chris@16 10 #pragma once
Chris@16 11 #endif
Chris@16 12
Chris@16 13 #include <boost/math/special_functions/detail/bessel_j0.hpp>
Chris@16 14 #include <boost/math/constants/constants.hpp>
Chris@16 15 #include <boost/math/tools/rational.hpp>
Chris@16 16 #include <boost/math/tools/big_constant.hpp>
Chris@16 17 #include <boost/math/policies/error_handling.hpp>
Chris@16 18 #include <boost/assert.hpp>
Chris@16 19
Chris@16 20 // Bessel function of the second kind of order zero
Chris@16 21 // x <= 8, minimax rational approximations on root-bracketing intervals
Chris@16 22 // x > 8, Hankel asymptotic expansion in Hart, Computer Approximations, 1968
Chris@16 23
Chris@16 24 namespace boost { namespace math { namespace detail{
Chris@16 25
Chris@16 26 template <typename T, typename Policy>
Chris@16 27 T bessel_y0(T x, const Policy&);
Chris@16 28
Chris@16 29 template <class T, class Policy>
Chris@16 30 struct bessel_y0_initializer
Chris@16 31 {
Chris@16 32 struct init
Chris@16 33 {
Chris@16 34 init()
Chris@16 35 {
Chris@16 36 do_init();
Chris@16 37 }
Chris@16 38 static void do_init()
Chris@16 39 {
Chris@16 40 bessel_y0(T(1), Policy());
Chris@16 41 }
Chris@16 42 void force_instantiate()const{}
Chris@16 43 };
Chris@16 44 static const init initializer;
Chris@16 45 static void force_instantiate()
Chris@16 46 {
Chris@16 47 initializer.force_instantiate();
Chris@16 48 }
Chris@16 49 };
Chris@16 50
Chris@16 51 template <class T, class Policy>
Chris@16 52 const typename bessel_y0_initializer<T, Policy>::init bessel_y0_initializer<T, Policy>::initializer;
Chris@16 53
Chris@16 54 template <typename T, typename Policy>
Chris@16 55 T bessel_y0(T x, const Policy& pol)
Chris@16 56 {
Chris@16 57 bessel_y0_initializer<T, Policy>::force_instantiate();
Chris@16 58
Chris@16 59 static const T P1[] = {
Chris@16 60 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0723538782003176831e+11)),
Chris@16 61 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.3716255451260504098e+09)),
Chris@16 62 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.0422274357376619816e+08)),
Chris@16 63 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.1287548474401797963e+06)),
Chris@16 64 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0102532948020907590e+04)),
Chris@16 65 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8402381979244993524e+01)),
Chris@16 66 };
Chris@16 67 static const T Q1[] = {
Chris@16 68 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.8873865738997033405e+11)),
Chris@16 69 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.1617187777290363573e+09)),
Chris@16 70 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.5662956624278251596e+07)),
Chris@16 71 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.3889393209447253406e+05)),
Chris@16 72 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6475986689240190091e+02)),
Chris@16 73 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
Chris@16 74 };
Chris@16 75 static const T P2[] = {
Chris@16 76 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2213976967566192242e+13)),
Chris@16 77 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.5107435206722644429e+11)),
Chris@16 78 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3600098638603061642e+10)),
Chris@16 79 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.9590439394619619534e+08)),
Chris@16 80 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.6905288611678631510e+06)),
Chris@16 81 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4566865832663635920e+04)),
Chris@16 82 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7427031242901594547e+01)),
Chris@16 83 };
Chris@16 84 static const T Q2[] = {
Chris@16 85 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3386146580707264428e+14)),
Chris@16 86 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4266824419412347550e+12)),
Chris@16 87 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4015103849971240096e+10)),
Chris@16 88 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3960202770986831075e+08)),
Chris@16 89 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0669982352539552018e+05)),
Chris@16 90 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.3030857612070288823e+02)),
Chris@16 91 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
Chris@16 92 };
Chris@16 93 static const T P3[] = {
Chris@16 94 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.0728726905150210443e+15)),
Chris@16 95 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.7016641869173237784e+14)),
Chris@16 96 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2829912364088687306e+11)),
Chris@16 97 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.9363051266772083678e+11)),
Chris@16 98 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1958827170518100757e+09)),
Chris@16 99 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0085539923498211426e+07)),
Chris@16 100 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1363534169313901632e+04)),
Chris@16 101 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.7439661319197499338e+01)),
Chris@16 102 };
Chris@16 103 static const T Q3[] = {
Chris@16 104 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4563724628846457519e+17)),
Chris@16 105 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9272425569640309819e+15)),
Chris@16 106 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2598377924042897629e+13)),
Chris@16 107 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.6926121104209825246e+10)),
Chris@16 108 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.4727219475672302327e+08)),
Chris@16 109 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.3924739209768057030e+05)),
Chris@16 110 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.7903362168128450017e+02)),
Chris@16 111 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
Chris@16 112 };
Chris@16 113 static const T PC[] = {
Chris@16 114 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684302e+04)),
Chris@16 115 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1345386639580765797e+04)),
Chris@16 116 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1170523380864944322e+04)),
Chris@16 117 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4806486443249270347e+03)),
Chris@16 118 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5376201909008354296e+02)),
Chris@16 119 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.8961548424210455236e-01)),
Chris@16 120 };
Chris@16 121 static const T QC[] = {
Chris@16 122 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684318e+04)),
Chris@16 123 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1370412495510416640e+04)),
Chris@16 124 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1215350561880115730e+04)),
Chris@16 125 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5028735138235608207e+03)),
Chris@16 126 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5711159858080893649e+02)),
Chris@16 127 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
Chris@16 128 };
Chris@16 129 static const T PS[] = {
Chris@16 130 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.9226600200800094098e+01)),
Chris@16 131 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8591953644342993800e+02)),
Chris@16 132 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1183429920482737611e+02)),
Chris@16 133 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2300261666214198472e+01)),
Chris@16 134 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2441026745835638459e+00)),
Chris@16 135 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.8033303048680751817e-03)),
Chris@16 136 };
Chris@16 137 static const T QS[] = {
Chris@16 138 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.7105024128512061905e+03)),
Chris@16 139 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1951131543434613647e+04)),
Chris@16 140 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.2642780169211018836e+03)),
Chris@16 141 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4887231232283756582e+03)),
Chris@16 142 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.0593769594993125859e+01)),
Chris@16 143 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
Chris@16 144 };
Chris@16 145 static const T x1 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.9357696627916752158e-01)),
Chris@16 146 x2 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9576784193148578684e+00)),
Chris@16 147 x3 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0860510603017726976e+00)),
Chris@16 148 x11 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.280e+02)),
Chris@16 149 x12 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.9519662791675215849e-03)),
Chris@16 150 x21 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0130e+03)),
Chris@16 151 x22 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.4716931485786837568e-04)),
Chris@16 152 x31 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8140e+03)),
Chris@16 153 x32 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1356030177269762362e-04))
Chris@16 154 ;
Chris@16 155 T value, factor, r, rc, rs;
Chris@16 156
Chris@16 157 BOOST_MATH_STD_USING
Chris@16 158 using namespace boost::math::tools;
Chris@16 159 using namespace boost::math::constants;
Chris@16 160
Chris@16 161 static const char* function = "boost::math::bessel_y0<%1%>(%1%,%1%)";
Chris@16 162
Chris@16 163 if (x < 0)
Chris@16 164 {
Chris@16 165 return policies::raise_domain_error<T>(function,
Chris@16 166 "Got x = %1% but x must be non-negative, complex result not supported.", x, pol);
Chris@16 167 }
Chris@16 168 if (x == 0)
Chris@16 169 {
Chris@16 170 return -policies::raise_overflow_error<T>(function, 0, pol);
Chris@16 171 }
Chris@16 172 if (x <= 3) // x in (0, 3]
Chris@16 173 {
Chris@16 174 T y = x * x;
Chris@16 175 T z = 2 * log(x/x1) * bessel_j0(x) / pi<T>();
Chris@16 176 r = evaluate_rational(P1, Q1, y);
Chris@16 177 factor = (x + x1) * ((x - x11/256) - x12);
Chris@16 178 value = z + factor * r;
Chris@16 179 }
Chris@16 180 else if (x <= 5.5f) // x in (3, 5.5]
Chris@16 181 {
Chris@16 182 T y = x * x;
Chris@16 183 T z = 2 * log(x/x2) * bessel_j0(x) / pi<T>();
Chris@16 184 r = evaluate_rational(P2, Q2, y);
Chris@16 185 factor = (x + x2) * ((x - x21/256) - x22);
Chris@16 186 value = z + factor * r;
Chris@16 187 }
Chris@16 188 else if (x <= 8) // x in (5.5, 8]
Chris@16 189 {
Chris@16 190 T y = x * x;
Chris@16 191 T z = 2 * log(x/x3) * bessel_j0(x) / pi<T>();
Chris@16 192 r = evaluate_rational(P3, Q3, y);
Chris@16 193 factor = (x + x3) * ((x - x31/256) - x32);
Chris@16 194 value = z + factor * r;
Chris@16 195 }
Chris@16 196 else // x in (8, \infty)
Chris@16 197 {
Chris@16 198 T y = 8 / x;
Chris@16 199 T y2 = y * y;
Chris@16 200 rc = evaluate_rational(PC, QC, y2);
Chris@16 201 rs = evaluate_rational(PS, QS, y2);
Chris@16 202 factor = constants::one_div_root_pi<T>() / sqrt(x);
Chris@16 203 //
Chris@16 204 // The following code is really just:
Chris@16 205 //
Chris@16 206 // T z = x - 0.25f * pi<T>();
Chris@16 207 // value = factor * (rc * sin(z) + y * rs * cos(z));
Chris@16 208 //
Chris@16 209 // But using the sin/cos addition formulae and constant values for
Chris@16 210 // sin/cos of PI/4 which then cancel part of the "factor" term as they're all
Chris@16 211 // 1 / sqrt(2):
Chris@16 212 //
Chris@16 213 T sx = sin(x);
Chris@16 214 T cx = cos(x);
Chris@16 215 value = factor * (rc * (sx - cx) + y * rs * (cx + sx));
Chris@16 216 }
Chris@16 217
Chris@16 218 return value;
Chris@16 219 }
Chris@16 220
Chris@16 221 }}} // namespaces
Chris@16 222
Chris@16 223 #endif // BOOST_MATH_BESSEL_Y0_HPP
Chris@16 224