boost/math/special_functions/detail/bessel_y0.hpp

   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_Y0_HPP
   7 #define BOOST_MATH_BESSEL_Y0_HPP
   8
   9 #ifdef _MSC_VER
  10 #pragma once
  11 #endif
  12
  13 #include <boost/math/special_functions/detail/bessel_j0.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 zero
  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_y0(T x, const Policy&);
  28
  29 template <class T, class Policy>
  30 struct bessel_y0_initializer
  31 {
  32    struct init
  33    {
  34       init()
  35       {
  36          do_init();
  37       }
  38       static void do_init()
  39       {
  40          bessel_y0(T(1), Policy());
  41       }
  42       void force_instantiate()const{}
  43    };
  44    static const init initializer;
  45    static void force_instantiate()
  46    {
  47       initializer.force_instantiate();
  48    }
  49 };
  50
  51 template <class T, class Policy>
  52 const typename bessel_y0_initializer<T, Policy>::init bessel_y0_initializer<T, Policy>::initializer;
  53
  54 template <typename T, typename Policy>
  55 T bessel_y0(T x, const Policy& pol)
  56 {
  57     bessel_y0_initializer<T, Policy>::force_instantiate();
  58
  59     static const T P1[] = {
  60          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0723538782003176831e+11)),
  61         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.3716255451260504098e+09)),
  62          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.0422274357376619816e+08)),
  63         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.1287548474401797963e+06)),
  64          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0102532948020907590e+04)),
  65         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8402381979244993524e+01)),
  66     };
  67     static const T Q1[] = {
  68          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.8873865738997033405e+11)),
  69          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.1617187777290363573e+09)),
  70          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.5662956624278251596e+07)),
  71          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.3889393209447253406e+05)),
  72          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6475986689240190091e+02)),
  73          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
  74     };
  75     static const T P2[] = {
  76         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2213976967566192242e+13)),
  77         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.5107435206722644429e+11)),
  78          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3600098638603061642e+10)),
  79         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.9590439394619619534e+08)),
  80          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.6905288611678631510e+06)),
  81         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4566865832663635920e+04)),
  82          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7427031242901594547e+01)),
  83     };
  84     static const T Q2[] = {
  85          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3386146580707264428e+14)),
  86          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4266824419412347550e+12)),
  87          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4015103849971240096e+10)),
  88          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3960202770986831075e+08)),
  89          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0669982352539552018e+05)),
  90          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.3030857612070288823e+02)),
  91          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
  92     };
  93     static const T P3[] = {
  94         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.0728726905150210443e+15)),
  95          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.7016641869173237784e+14)),
  96         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2829912364088687306e+11)),
  97         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.9363051266772083678e+11)),
  98          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1958827170518100757e+09)),
  99         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0085539923498211426e+07)),
 100          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1363534169313901632e+04)),
 101         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.7439661319197499338e+01)),
 102     };
 103     static const T Q3[] = {
 104          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4563724628846457519e+17)),
 105          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9272425569640309819e+15)),
 106          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2598377924042897629e+13)),
 107          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.6926121104209825246e+10)),
 108          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.4727219475672302327e+08)),
 109          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.3924739209768057030e+05)),
 110          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.7903362168128450017e+02)),
 111          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
 112     };
 113     static const T PC[] = {
 114          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684302e+04)),
 115          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1345386639580765797e+04)),
 116          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1170523380864944322e+04)),
 117          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4806486443249270347e+03)),
 118          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5376201909008354296e+02)),
 119          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.8961548424210455236e-01)),
 120     };
 121     static const T QC[] = {
 122          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684318e+04)),
 123          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1370412495510416640e+04)),
 124          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1215350561880115730e+04)),
 125          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5028735138235608207e+03)),
 126          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5711159858080893649e+02)),
 127          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
 128     };
 129     static const T PS[] = {
 130         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.9226600200800094098e+01)),
 131         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8591953644342993800e+02)),
 132         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1183429920482737611e+02)),
 133         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2300261666214198472e+01)),
 134         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2441026745835638459e+00)),
 135         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.8033303048680751817e-03)),
 136     };
 137     static const T QS[] = {
 138          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.7105024128512061905e+03)),
 139          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1951131543434613647e+04)),
 140          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.2642780169211018836e+03)),
 141          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4887231232283756582e+03)),
 142          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.0593769594993125859e+01)),
 143          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
 144     };
 145     static const T x1  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.9357696627916752158e-01)),
 146                    x2  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9576784193148578684e+00)),
 147                    x3  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0860510603017726976e+00)),
 148                    x11 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.280e+02)),
 149                    x12 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.9519662791675215849e-03)),
 150                    x21 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0130e+03)),
 151                    x22 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.4716931485786837568e-04)),
 152                    x31 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8140e+03)),
 153                    x32 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1356030177269762362e-04))
 154     ;
 155     T value, factor, r, rc, rs;
 156
 157     BOOST_MATH_STD_USING
 158     using namespace boost::math::tools;
 159     using namespace boost::math::constants;
 160
 161     static const char* function = "boost::math::bessel_y0<%1%>(%1%,%1%)";
 162
 163     if (x < 0)
 164     {
 165        return policies::raise_domain_error<T>(function,
 166             "Got x = %1% but x must be non-negative, complex result not supported.", x, pol);
 167     }
 168     if (x == 0)
 169     {
 170        return -policies::raise_overflow_error<T>(function, 0, pol);
 171     }
 172     if (x <= 3)                       // x in (0, 3]
 173     {
 174         T y = x * x;
 175         T z = 2 * log(x/x1) * bessel_j0(x) / pi<T>();
 176         r = evaluate_rational(P1, Q1, y);
 177         factor = (x + x1) * ((x - x11/256) - x12);
 178         value = z + factor * r;
 179     }
 180     else if (x <= 5.5f)                  // x in (3, 5.5]
 181     {
 182         T y = x * x;
 183         T z = 2 * log(x/x2) * bessel_j0(x) / pi<T>();
 184         r = evaluate_rational(P2, Q2, y);
 185         factor = (x + x2) * ((x - x21/256) - x22);
 186         value = z + factor * r;
 187     }
 188     else if (x <= 8)                  // x in (5.5, 8]
 189     {
 190         T y = x * x;
 191         T z = 2 * log(x/x3) * bessel_j0(x) / pi<T>();
 192         r = evaluate_rational(P3, Q3, y);
 193         factor = (x + x3) * ((x - x31/256) - x32);
 194         value = z + factor * r;
 195     }
 196     else                                // x in (8, \infty)
 197     {
 198         T y = 8 / x;
 199         T y2 = y * y;
 200         rc = evaluate_rational(PC, QC, y2);
 201         rs = evaluate_rational(PS, QS, y2);
 202         factor = constants::one_div_root_pi<T>() / sqrt(x);
 203         //
 204         // The following code is really just:
 205         //
 206         // T z = x - 0.25f * pi<T>();
 207         // value = factor * (rc * sin(z) + y * rs * cos(z));
 208         //
 209         // But using the sin/cos addition formulae and constant values for
 210         // sin/cos of PI/4 which then cancel part of the "factor" term as they're all
 211         // 1 / sqrt(2):
 212         //
 213         T sx = sin(x);
 214         T cx = cos(x);
 215         value = factor * (rc * (sx - cx) + y * rs * (cx + sx));
 216     }
 217
 218     return value;
 219 }
 220
 221 }}} // namespaces
 222
 223 #endif // BOOST_MATH_BESSEL_Y0_HPP
 224