text: //  Copyright (c) 2006 Xiaogang Zhang
  //  Use, modification and distribution are subject to the
  //  Boost Software License, Version 1.0. (See accompanying file
  //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
  
  #ifndef BOOST_MATH_BESSEL_J1_HPP
  #define BOOST_MATH_BESSEL_J1_HPP
  
  #ifdef _MSC_VER
  #pragma once
  #endif
  
  #include <boost/math/constants/constants.hpp>
  #include <boost/math/tools/rational.hpp>
  #include <boost/math/tools/big_constant.hpp>
  #include <boost/assert.hpp>
  
  // Bessel function of the first kind of order one
  // x <= 8, minimax rational approximations on root-bracketing intervals
  // x > 8, Hankel asymptotic expansion in Hart, Computer Approximations, 1968
  
  namespace boost { namespace math{  namespace detail{
  
  template <typename T>
  T bessel_j1(T x);
  
  template <class T>
  struct bessel_j1_initializer
  {
     struct init
     {
        init()
        {
           do_init();
        }
        static void do_init()
        {
           bessel_j1(T(1));
        }
        void force_instantiate()const{}
     };
     static const init initializer;
     static void force_instantiate()
     {
        initializer.force_instantiate();
     }
  };
  
  template <class T>
  const typename bessel_j1_initializer<T>::init bessel_j1_initializer<T>::initializer;
  
  template <typename T>
  T bessel_j1(T x)
  {
      bessel_j1_initializer<T>::force_instantiate();
  
      static const T P1[] = {
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4258509801366645672e+11)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6781041261492395835e+09)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1548696764841276794e+08)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.8062904098958257677e+05)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.4615792982775076130e+03)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0650724020080236441e+01)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0767857011487300348e-02))
      };
      static const T Q1[] = {
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1868604460820175290e+12)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.2091902282580133541e+10)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.0228375140097033958e+08)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.9117614494174794095e+05)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0742272239517380498e+03)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0))
      };
      static const T P2[] = {
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.7527881995806511112e+16)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.6608531731299018674e+15)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.6658018905416665164e+13)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5580665670910619166e+11)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8113931269860667829e+09)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.0793266148011179143e+06)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -7.5023342220781607561e+03)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.6179191852758252278e+00))
      };
      static const T Q2[] = {
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7253905888447681194e+18)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7128800897135812012e+16)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.4899346165481429307e+13)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.7622777286244082666e+11)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.4872502899596389593e+08)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1267125065029138050e+06)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3886978985861357615e+03)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0))
      };
      static const T PC[] = {
          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.4357578167941278571e+06)),
          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -9.9422465050776411957e+06)),
          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.6033732483649391093e+06)),
          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.5235293511811373833e+06)),
          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0982405543459346727e+05)),
          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.6116166443246101165e+03)),
          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0))
      };
      static const T QC[] = {
          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.4357578167941278568e+06)),
          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -9.9341243899345856590e+06)),
          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.5853394797230870728e+06)),
          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.5118095066341608816e+06)),
          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0726385991103820119e+05)),
          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4550094401904961825e+03)),
          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0))
      };
      static const T PS[] = {
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.3220913409857223519e+04)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.5145160675335701966e+04)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6178836581270835179e+04)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8494262873223866797e+04)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7063754290207680021e+03)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5265133846636032186e+01)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0))
      };
      static const T QS[] = {
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0871281941028743574e+05)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8194580422439972989e+06)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4194606696037208929e+06)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0029443582266975117e+05)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.7890229745772202641e+04)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.6383677696049909675e+02)),
           static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0))
      };
      static const T x1  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.8317059702075123156e+00)),
                     x2  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0155866698156187535e+00)),
                     x11 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.810e+02)),
                     x12 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.2527979248768438556e-04)),
                     x21 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7960e+03)),
                     x22 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.8330184381246462950e-05));
  
      T value, factor, r, rc, rs, w;
  
      BOOST_MATH_STD_USING
      using namespace boost::math::tools;
      using namespace boost::math::constants;
  
      w = abs(x);
      if (x == 0)
      {
          return static_cast<T>(0);
      }
      if (w <= 4)                       // w in (0, 4]
      {
          T y = x * x;
          BOOST_ASSERT(sizeof(P1) == sizeof(Q1));
          r = evaluate_rational(P1, Q1, y);
          factor = w * (w + x1) * ((w - x11/256) - x12);
          value = factor * r;
      }
      else if (w <= 8)                  // w in (4, 8]
      {
          T y = x * x;
          BOOST_ASSERT(sizeof(P2) == sizeof(Q2));
          r = evaluate_rational(P2, Q2, y);
          factor = w * (w + x2) * ((w - x21/256) - x22);
          value = factor * r;
      }
      else                                // w in (8, \infty)
      {
          T y = 8 / w;
          T y2 = y * y;
          BOOST_ASSERT(sizeof(PC) == sizeof(QC));
          BOOST_ASSERT(sizeof(PS) == sizeof(QS));
          rc = evaluate_rational(PC, QC, y2);
          rs = evaluate_rational(PS, QS, y2);
          factor = 1 / (sqrt(w) * constants::root_pi<T>());
          //
          // What follows is really just:
          //
          // T z = w - 0.75f * pi<T>();
          // value = factor * (rc * cos(z) - y * rs * sin(z));
          //
          // but using the sin/cos addition rules plus constants
          // for the values of sin/cos of 3PI/4 which then cancel
          // out with corresponding terms in "factor".
          //
          T sx = sin(x);
          T cx = cos(x);
          value = factor * (rc * (sx - cx) + y * rs * (sx + cx));
      }
  
      if (x < 0)
      {
          value *= -1;                 // odd function
      }
      return value;
  }
  
  }}} // namespaces
  
  #endif // BOOST_MATH_BESSEL_J1_HPP
  