special_functions/detail/bessel_j1.hpp

0001 //  Copyright (c) 2006 Xiaogang Zhang
0002 //  Use, modification and distribution are subject to the
0003 //  Boost Software License, Version 1.0. (See accompanying file
0004 //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
0005
0006 #ifndef BOOST_MATH_BESSEL_J1_HPP
0007 #define BOOST_MATH_BESSEL_J1_HPP
0008
0009 #ifdef _MSC_VER
0010 #pragma once
0011 #endif
0012
0013 #include <boost/math/constants/constants.hpp>
0014 #include <boost/math/tools/rational.hpp>
0015 #include <boost/math/tools/big_constant.hpp>
0016 #include <boost/math/tools/assert.hpp>
0017
0018 #if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
0019 //
0020 // This is the only way we can avoid
0021 // warning: non-standard suffix on floating constant [-Wpedantic]
0022 // when building with -Wall -pedantic.  Neither __extension__
0023 // nor #pragma diagnostic ignored work :(
0024 //
0025 #pragma GCC system_header
0026 #endif
0027
0028 // Bessel function of the first kind of order one
0029 // x <= 8, minimax rational approximations on root-bracketing intervals
0030 // x > 8, Hankel asymptotic expansion in Hart, Computer Approximations, 1968
0031
0032 namespace boost { namespace math{  namespace detail{
0033
0034 template <typename T>
0035 T bessel_j1(T x);
0036
0037 template <class T>
0038 struct bessel_j1_initializer
0039 {
0040    struct init
0041    {
0042       init()
0043       {
0044          do_init();
0045       }
0046       static void do_init()
0047       {
0048          bessel_j1(T(1));
0049       }
0050       void force_instantiate()const{}
0051    };
0052    static const init initializer;
0053    static void force_instantiate()
0054    {
0055       initializer.force_instantiate();
0056    }
0057 };
0058
0059 template <class T>
0060 const typename bessel_j1_initializer<T>::init bessel_j1_initializer<T>::initializer;
0061
0062 template <typename T>
0063 T bessel_j1(T x)
0064 {
0065     bessel_j1_initializer<T>::force_instantiate();
0066
0067     static const T P1[] = {
0068          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4258509801366645672e+11)),
0069          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6781041261492395835e+09)),
0070          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1548696764841276794e+08)),
0071          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.8062904098958257677e+05)),
0072          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.4615792982775076130e+03)),
0073          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0650724020080236441e+01)),
0074          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0767857011487300348e-02))
0075     };
0076     static const T Q1[] = {
0077          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1868604460820175290e+12)),
0078          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.2091902282580133541e+10)),
0079          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.0228375140097033958e+08)),
0080          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.9117614494174794095e+05)),
0081          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0742272239517380498e+03)),
0082          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
0083          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0))
0084     };
0085     static const T P2[] = {
0086          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.7527881995806511112e+16)),
0087          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.6608531731299018674e+15)),
0088          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.6658018905416665164e+13)),
0089          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5580665670910619166e+11)),
0090          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8113931269860667829e+09)),
0091          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.0793266148011179143e+06)),
0092          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -7.5023342220781607561e+03)),
0093          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.6179191852758252278e+00))
0094     };
0095     static const T Q2[] = {
0096          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7253905888447681194e+18)),
0097          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7128800897135812012e+16)),
0098          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.4899346165481429307e+13)),
0099          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.7622777286244082666e+11)),
0100          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.4872502899596389593e+08)),
0101          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1267125065029138050e+06)),
0102          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3886978985861357615e+03)),
0103          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0))
0104     };
0105     static const T PC[] = {
0106         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.4357578167941278571e+06)),
0107         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -9.9422465050776411957e+06)),
0108         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.6033732483649391093e+06)),
0109         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.5235293511811373833e+06)),
0110         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0982405543459346727e+05)),
0111         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.6116166443246101165e+03)),
0112         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0))
0113     };
0114     static const T QC[] = {
0115         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.4357578167941278568e+06)),
0116         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -9.9341243899345856590e+06)),
0117         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.5853394797230870728e+06)),
0118         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.5118095066341608816e+06)),
0119         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0726385991103820119e+05)),
0120         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4550094401904961825e+03)),
0121         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0))
0122     };
0123     static const T PS[] = {
0124          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.3220913409857223519e+04)),
0125          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.5145160675335701966e+04)),
0126          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6178836581270835179e+04)),
0127          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8494262873223866797e+04)),
0128          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7063754290207680021e+03)),
0129          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5265133846636032186e+01)),
0130          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0))
0131     };
0132     static const T QS[] = {
0133          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0871281941028743574e+05)),
0134          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8194580422439972989e+06)),
0135          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4194606696037208929e+06)),
0136          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0029443582266975117e+05)),
0137          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.7890229745772202641e+04)),
0138          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.6383677696049909675e+02)),
0139          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0))
0140     };
0141     static const T x1  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.8317059702075123156e+00)),
0142                    x2  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0155866698156187535e+00)),
0143                    x11 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.810e+02)),
0144                    x12 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.2527979248768438556e-04)),
0145                    x21 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7960e+03)),
0146                    x22 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.8330184381246462950e-05));
0147
0148     T value, factor, r, rc, rs, w;
0149
0150     BOOST_MATH_STD_USING
0151     using namespace boost::math::tools;
0152     using namespace boost::math::constants;
0153
0154     w = abs(x);
0155     if (x == 0)
0156     {
0157         return static_cast<T>(0);
0158     }
0159     if (w <= 4)                       // w in (0, 4]
0160     {
0161         T y = x * x;
0162         BOOST_MATH_ASSERT(sizeof(P1) == sizeof(Q1));
0163         r = evaluate_rational(P1, Q1, y);
0164         factor = w * (w + x1) * ((w - x11/256) - x12);
0165         value = factor * r;
0166     }
0167     else if (w <= 8)                  // w in (4, 8]
0168     {
0169         T y = x * x;
0170         BOOST_MATH_ASSERT(sizeof(P2) == sizeof(Q2));
0171         r = evaluate_rational(P2, Q2, y);
0172         factor = w * (w + x2) * ((w - x21/256) - x22);
0173         value = factor * r;
0174     }
0175     else                                // w in (8, \infty)
0176     {
0177         T y = 8 / w;
0178         T y2 = y * y;
0179         BOOST_MATH_ASSERT(sizeof(PC) == sizeof(QC));
0180         BOOST_MATH_ASSERT(sizeof(PS) == sizeof(QS));
0181         rc = evaluate_rational(PC, QC, y2);
0182         rs = evaluate_rational(PS, QS, y2);
0183         factor = 1 / (sqrt(w) * constants::root_pi<T>());
0184         //
0185         // What follows is really just:
0186         //
0187         // T z = w - 0.75f * pi<T>();
0188         // value = factor * (rc * cos(z) - y * rs * sin(z));
0189         //
0190         // but using the sin/cos addition rules plus constants
0191         // for the values of sin/cos of 3PI/4 which then cancel
0192         // out with corresponding terms in "factor".
0193         //
0194         T sx = sin(x);
0195         T cx = cos(x);
0196         value = factor * (rc * (sx - cx) + y * rs * (sx + cx));
0197     }
0198
0199     if (x < 0)
0200     {
0201         value *= -1;                 // odd function
0202     }
0203     return value;
0204 }
0205
0206 }}} // namespaces
0207
0208 #endif // BOOST_MATH_BESSEL_J1_HPP
0209