boost/math/special_functions/detail/bessel_j0.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_J0_HPP
   7 #define BOOST_MATH_BESSEL_J0_HPP
   8
   9 #ifdef _MSC_VER
  10 #pragma once
  11 #endif
  12
  13 #include <boost/math/constants/constants.hpp>
  14 #include <boost/math/tools/rational.hpp>
  15 #include <boost/math/tools/big_constant.hpp>
  16 #include <boost/assert.hpp>
  17
  18 // Bessel function of the first kind of order zero
  19 // x <= 8, minimax rational approximations on root-bracketing intervals
  20 // x > 8, Hankel asymptotic expansion in Hart, Computer Approximations, 1968
  21
  22 namespace boost { namespace math { namespace detail{
  23
  24 template <typename T>
  25 T bessel_j0(T x);
  26
  27 template <class T>
  28 struct bessel_j0_initializer
  29 {
  30    struct init
  31    {
  32       init()
  33       {
  34          do_init();
  35       }
  36       static void do_init()
  37       {
  38          bessel_j0(T(1));
  39       }
  40       void force_instantiate()const{}
  41    };
  42    static const init initializer;
  43    static void force_instantiate()
  44    {
  45       initializer.force_instantiate();
  46    }
  47 };
  48
  49 template <class T>
  50 const typename bessel_j0_initializer<T>::init bessel_j0_initializer<T>::initializer;
  51
  52 template <typename T>
  53 T bessel_j0(T x)
  54 {
  55     bessel_j0_initializer<T>::force_instantiate();
  56
  57     static const T P1[] = {
  58          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.1298668500990866786e+11)),
  59          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.7282507878605942706e+10)),
  60          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.2140700423540120665e+08)),
  61          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6302997904833794242e+06)),
  62          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.6629814655107086448e+04)),
  63          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0344222815443188943e+02)),
  64          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2117036164593528341e-01))
  65     };
  66     static const T Q1[] = {
  67          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.3883787996332290397e+12)),
  68          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.6328198300859648632e+10)),
  69          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3985097372263433271e+08)),
  70          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.5612696224219938200e+05)),
  71          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.3614022392337710626e+02)),
  72          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
  73          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0))
  74     };
  75     static const T P2[] = {
  76          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8319397969392084011e+03)),
  77          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2254078161378989535e+04)),
  78          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -7.2879702464464618998e+03)),
  79          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0341910641583726701e+04)),
  80          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1725046279757103576e+04)),
  81          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.4176707025325087628e+03)),
  82          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.4321196680624245801e+02)),
  83          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.8591703355916499363e+01))
  84     };
  85     static const T Q2[] = {
  86          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.5783478026152301072e+05)),
  87          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.4599102262586308984e+05)),
  88          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.4055062591169562211e+04)),
  89          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8680990008359188352e+04)),
  90          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.9458766545509337327e+03)),
  91          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.3307310774649071172e+02)),
  92          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.5258076240801555057e+01)),
  93          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0))
  94     };
  95     static const T PC[] = {
  96          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684302e+04)),
  97          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1345386639580765797e+04)),
  98          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1170523380864944322e+04)),
  99          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4806486443249270347e+03)),
 100          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5376201909008354296e+02)),
 101          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.8961548424210455236e-01))
 102     };
 103     static const T QC[] = {
 104          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684318e+04)),
 105          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1370412495510416640e+04)),
 106          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1215350561880115730e+04)),
 107          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5028735138235608207e+03)),
 108          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5711159858080893649e+02)),
 109          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0))
 110     };
 111     static const T PS[] = {
 112         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.9226600200800094098e+01)),
 113         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8591953644342993800e+02)),
 114         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1183429920482737611e+02)),
 115         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2300261666214198472e+01)),
 116         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2441026745835638459e+00)),
 117         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.8033303048680751817e-03))
 118     };
 119     static const T QS[] = {
 120          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.7105024128512061905e+03)),
 121          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1951131543434613647e+04)),
 122          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.2642780169211018836e+03)),
 123          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4887231232283756582e+03)),
 124          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.0593769594993125859e+01)),
 125          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0))
 126     };
 127     static const T x1  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.4048255576957727686e+00)),
 128                    x2  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.5200781102863106496e+00)),
 129                    x11 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.160e+02)),
 130                    x12 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.42444230422723137837e-03)),
 131                    x21 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4130e+03)),
 132                    x22 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.46860286310649596604e-04));
 133
 134     T value, factor, r, rc, rs;
 135
 136     BOOST_MATH_STD_USING
 137     using namespace boost::math::tools;
 138     using namespace boost::math::constants;
 139
 140     if (x < 0)
 141     {
 142         x = -x;                         // even function
 143     }
 144     if (x == 0)
 145     {
 146         return static_cast<T>(1);
 147     }
 148     if (x <= 4)                       // x in (0, 4]
 149     {
 150         T y = x * x;
 151         BOOST_ASSERT(sizeof(P1) == sizeof(Q1));
 152         r = evaluate_rational(P1, Q1, y);
 153         factor = (x + x1) * ((x - x11/256) - x12);
 154         value = factor * r;
 155     }
 156     else if (x <= 8.0)                  // x in (4, 8]
 157     {
 158         T y = 1 - (x * x)/64;
 159         BOOST_ASSERT(sizeof(P2) == sizeof(Q2));
 160         r = evaluate_rational(P2, Q2, y);
 161         factor = (x + x2) * ((x - x21/256) - x22);
 162         value = factor * r;
 163     }
 164     else                                // x in (8, \infty)
 165     {
 166         T y = 8 / x;
 167         T y2 = y * y;
 168         BOOST_ASSERT(sizeof(PC) == sizeof(QC));
 169         BOOST_ASSERT(sizeof(PS) == sizeof(QS));
 170         rc = evaluate_rational(PC, QC, y2);
 171         rs = evaluate_rational(PS, QS, y2);
 172         factor = constants::one_div_root_pi<T>() / sqrt(x);
 173         //
 174         // What follows is really just:
 175         //
 176         // T z = x - pi/4;
 177         // value = factor * (rc * cos(z) - y * rs * sin(z));
 178         //
 179         // But using the addition formulae for sin and cos, plus
 180         // the special values for sin/cos of pi/4.
 181         //
 182         T sx = sin(x);
 183         T cx = cos(x);
 184         value = factor * (rc * (cx + sx) - y * rs * (sx - cx));
 185     }
 186
 187     return value;
 188 }
 189
 190 }}} // namespaces
 191
 192 #endif // BOOST_MATH_BESSEL_J0_HPP
 193