boost/math/special_functions/detail/bessel_y1.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_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    {
  34       init()
  35       {
  36          do_init();
  37       }
  38       static void do_init()
  39       {
  40          bessel_y1(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_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