ceph/src/boost/libs/math/include/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 #pragma warning(push)
  12 #pragma warning(disable:4702) // Unreachable code (release mode only warning)
  13 #endif
  14
  15 #include <boost/math/special_functions/detail/bessel_j0.hpp>
  16 #include <boost/math/constants/constants.hpp>
  17 #include <boost/math/tools/rational.hpp>
  18 #include <boost/math/tools/big_constant.hpp>
  19 #include <boost/math/policies/error_handling.hpp>
  20 #include <boost/assert.hpp>
  21
  22 // Bessel function of the second kind of order zero
  23 // x <= 8, minimax rational approximations on root-bracketing intervals
  24 // x > 8, Hankel asymptotic expansion in Hart, Computer Approximations, 1968
  25
  26 namespace boost { namespace math { namespace detail{
  27
  28 template <typename T, typename Policy>
  29 T bessel_y0(T x, const Policy&);
  30
  31 template <class T, class Policy>
  32 struct bessel_y0_initializer
  33 {
  34    struct init
  35    {
  36       init()
  37       {
  38          do_init();
  39       }
  40       static void do_init()
  41       {
  42          bessel_y0(T(1), Policy());
  43       }
  44       void force_instantiate()const{}
  45    };
  46    static const init initializer;
  47    static void force_instantiate()
  48    {
  49       initializer.force_instantiate();
  50    }
  51 };
  52
  53 template <class T, class Policy>
  54 const typename bessel_y0_initializer<T, Policy>::init bessel_y0_initializer<T, Policy>::initializer;
  55
  56 template <typename T, typename Policy>
  57 T bessel_y0(T x, const Policy& pol)
  58 {
  59     bessel_y0_initializer<T, Policy>::force_instantiate();
  60
  61     static const T P1[] = {
  62          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0723538782003176831e+11)),
  63         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.3716255451260504098e+09)),
  64          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.0422274357376619816e+08)),
  65         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.1287548474401797963e+06)),
  66          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0102532948020907590e+04)),
  67         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8402381979244993524e+01)),
  68     };
  69     static const T Q1[] = {
  70          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.8873865738997033405e+11)),
  71          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.1617187777290363573e+09)),
  72          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.5662956624278251596e+07)),
  73          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.3889393209447253406e+05)),
  74          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6475986689240190091e+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, -2.2213976967566192242e+13)),
  79         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.5107435206722644429e+11)),
  80          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3600098638603061642e+10)),
  81         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.9590439394619619534e+08)),
  82          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.6905288611678631510e+06)),
  83         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4566865832663635920e+04)),
  84          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7427031242901594547e+01)),
  85     };
  86     static const T Q2[] = {
  87          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3386146580707264428e+14)),
  88          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4266824419412347550e+12)),
  89          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4015103849971240096e+10)),
  90          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3960202770986831075e+08)),
  91          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0669982352539552018e+05)),
  92          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.3030857612070288823e+02)),
  93          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
  94     };
  95     static const T P3[] = {
  96         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.0728726905150210443e+15)),
  97          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.7016641869173237784e+14)),
  98         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2829912364088687306e+11)),
  99         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.9363051266772083678e+11)),
 100          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1958827170518100757e+09)),
 101         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0085539923498211426e+07)),
 102          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1363534169313901632e+04)),
 103         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.7439661319197499338e+01)),
 104     };
 105     static const T Q3[] = {
 106          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4563724628846457519e+17)),
 107          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9272425569640309819e+15)),
 108          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2598377924042897629e+13)),
 109          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.6926121104209825246e+10)),
 110          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.4727219475672302327e+08)),
 111          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.3924739209768057030e+05)),
 112          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.7903362168128450017e+02)),
 113          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
 114     };
 115     static const T PC[] = {
 116          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684302e+04)),
 117          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1345386639580765797e+04)),
 118          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1170523380864944322e+04)),
 119          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4806486443249270347e+03)),
 120          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5376201909008354296e+02)),
 121          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.8961548424210455236e-01)),
 122     };
 123     static const T QC[] = {
 124          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684318e+04)),
 125          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1370412495510416640e+04)),
 126          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1215350561880115730e+04)),
 127          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5028735138235608207e+03)),
 128          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5711159858080893649e+02)),
 129          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
 130     };
 131     static const T PS[] = {
 132         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.9226600200800094098e+01)),
 133         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8591953644342993800e+02)),
 134         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1183429920482737611e+02)),
 135         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2300261666214198472e+01)),
 136         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2441026745835638459e+00)),
 137         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.8033303048680751817e-03)),
 138     };
 139     static const T QS[] = {
 140          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.7105024128512061905e+03)),
 141          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1951131543434613647e+04)),
 142          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.2642780169211018836e+03)),
 143          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4887231232283756582e+03)),
 144          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.0593769594993125859e+01)),
 145          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
 146     };
 147     static const T x1  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.9357696627916752158e-01)),
 148                    x2  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9576784193148578684e+00)),
 149                    x3  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0860510603017726976e+00)),
 150                    x11 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.280e+02)),
 151                    x12 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.9519662791675215849e-03)),
 152                    x21 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0130e+03)),
 153                    x22 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.4716931485786837568e-04)),
 154                    x31 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8140e+03)),
 155                    x32 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1356030177269762362e-04))
 156     ;
 157     T value, factor, r, rc, rs;
 158
 159     BOOST_MATH_STD_USING
 160     using namespace boost::math::tools;
 161     using namespace boost::math::constants;
 162
 163     static const char* function = "boost::math::bessel_y0<%1%>(%1%,%1%)";
 164
 165     if (x < 0)
 166     {
 167        return policies::raise_domain_error<T>(function,
 168             "Got x = %1% but x must be non-negative, complex result not supported.", x, pol);
 169     }
 170     if (x == 0)
 171     {
 172        return -policies::raise_overflow_error<T>(function, 0, pol);
 173     }
 174     if (x <= 3)                       // x in (0, 3]
 175     {
 176         T y = x * x;
 177         T z = 2 * log(x/x1) * bessel_j0(x) / pi<T>();
 178         r = evaluate_rational(P1, Q1, y);
 179         factor = (x + x1) * ((x - x11/256) - x12);
 180         value = z + factor * r;
 181     }
 182     else if (x <= 5.5f)                  // x in (3, 5.5]
 183     {
 184         T y = x * x;
 185         T z = 2 * log(x/x2) * bessel_j0(x) / pi<T>();
 186         r = evaluate_rational(P2, Q2, y);
 187         factor = (x + x2) * ((x - x21/256) - x22);
 188         value = z + factor * r;
 189     }
 190     else if (x <= 8)                  // x in (5.5, 8]
 191     {
 192         T y = x * x;
 193         T z = 2 * log(x/x3) * bessel_j0(x) / pi<T>();
 194         r = evaluate_rational(P3, Q3, y);
 195         factor = (x + x3) * ((x - x31/256) - x32);
 196         value = z + factor * r;
 197     }
 198     else                                // x in (8, \infty)
 199     {
 200         T y = 8 / x;
 201         T y2 = y * y;
 202         rc = evaluate_rational(PC, QC, y2);
 203         rs = evaluate_rational(PS, QS, y2);
 204         factor = constants::one_div_root_pi<T>() / sqrt(x);
 205         //
 206         // The following code is really just:
 207         //
 208         // T z = x - 0.25f * pi<T>();
 209         // value = factor * (rc * sin(z) + y * rs * cos(z));
 210         //
 211         // But using the sin/cos addition formulae and constant values for
 212         // sin/cos of PI/4 which then cancel part of the "factor" term as they're all
 213         // 1 / sqrt(2):
 214         //
 215         T sx = sin(x);
 216         T cx = cos(x);
 217         value = factor * (rc * (sx - cx) + y * rs * (cx + sx));
 218     }
 219
 220     return value;
 221 }
 222
 223 }}} // namespaces
 224
 225 #ifdef _MSC_VER
 226 #pragma warning(pop)
 227 #endif
 228
 229 #endif // BOOST_MATH_BESSEL_Y0_HPP
 230