special_functions/detail/bessel_y1.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_Y1_HPP
0007 #define BOOST_MATH_BESSEL_Y1_HPP
0008
0009 #ifdef _MSC_VER
0010 #pragma once
0011 #pragma warning(push)
0012 #pragma warning(disable:4702) // Unreachable code (release mode only warning)
0013 #endif
0014
0015 #include <boost/math/special_functions/detail/bessel_j1.hpp>
0016 #include <boost/math/constants/constants.hpp>
0017 #include <boost/math/tools/rational.hpp>
0018 #include <boost/math/tools/big_constant.hpp>
0019 #include <boost/math/policies/error_handling.hpp>
0020 #include <boost/math/tools/assert.hpp>
0021
0022 #if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
0023 //
0024 // This is the only way we can avoid
0025 // warning: non-standard suffix on floating constant [-Wpedantic]
0026 // when building with -Wall -pedantic.  Neither __extension__
0027 // nor #pragma diagnostic ignored work :(
0028 //
0029 #pragma GCC system_header
0030 #endif
0031
0032 // Bessel function of the second kind of order one
0033 // x <= 8, minimax rational approximations on root-bracketing intervals
0034 // x > 8, Hankel asymptotic expansion in Hart, Computer Approximations, 1968
0035
0036 namespace boost { namespace math { namespace detail{
0037
0038 template <typename T, typename Policy>
0039 T bessel_y1(T x, const Policy&);
0040
0041 template <class T, class Policy>
0042 struct bessel_y1_initializer
0043 {
0044    struct init
0045    {
0046       init()
0047       {
0048          do_init();
0049       }
0050       static void do_init()
0051       {
0052          bessel_y1(T(1), Policy());
0053       }
0054       void force_instantiate()const{}
0055    };
0056    static const init initializer;
0057    static void force_instantiate()
0058    {
0059       initializer.force_instantiate();
0060    }
0061 };
0062
0063 template <class T, class Policy>
0064 const typename bessel_y1_initializer<T, Policy>::init bessel_y1_initializer<T, Policy>::initializer;
0065
0066 template <typename T, typename Policy>
0067 T bessel_y1(T x, const Policy& pol)
0068 {
0069     bessel_y1_initializer<T, Policy>::force_instantiate();
0070
0071     static const T P1[] = {
0072          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0535726612579544093e+13)),
0073          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4708611716525426053e+12)),
0074         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.7595974497819597599e+11)),
0075          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.2144548214502560419e+09)),
0076         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.9157479997408395984e+07)),
0077          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2157953222280260820e+05)),
0078         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.1714424660046133456e+02)),
0079     };
0080     static const T Q1[] = {
0081          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.0737873921079286084e+14)),
0082          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1272286200406461981e+12)),
0083          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.7800352738690585613e+10)),
0084          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.2250435122182963220e+08)),
0085          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.8136470753052572164e+05)),
0086          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.2079908168393867438e+02)),
0087          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
0088     };
0089     static const T P2[] = {
0090          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1514276357909013326e+19)),
0091         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.6808094574724204577e+18)),
0092         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.3638408497043134724e+16)),
0093          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0686275289804744814e+15)),
0094         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.9530713129741981618e+13)),
0095          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.7453673962438488783e+11)),
0096         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1957961912070617006e+09)),
0097          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.9153806858264202986e+06)),
0098         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2337180442012953128e+03)),
0099     };
0100     static const T Q2[] = {
0101          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.3321844313316185697e+20)),
0102          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.6968198822857178911e+18)),
0103          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.0837179548112881950e+16)),
0104          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1187010065856971027e+14)),
0105          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.0221766852960403645e+11)),
0106          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.3550318087088919566e+08)),
0107          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0453748201934079734e+06)),
0108          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.2855164849321609336e+03)),
0109          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
0110     };
0111     static const T PC[] = {
0112         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.4357578167941278571e+06)),
0113         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -9.9422465050776411957e+06)),
0114         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.6033732483649391093e+06)),
0115         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.5235293511811373833e+06)),
0116         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0982405543459346727e+05)),
0117         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.6116166443246101165e+03)),
0118          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0)),
0119     };
0120     static const T QC[] = {
0121         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.4357578167941278568e+06)),
0122         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -9.9341243899345856590e+06)),
0123         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.5853394797230870728e+06)),
0124         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.5118095066341608816e+06)),
0125         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0726385991103820119e+05)),
0126         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4550094401904961825e+03)),
0127          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
0128     };
0129     static const T PS[] = {
0130          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.3220913409857223519e+04)),
0131          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.5145160675335701966e+04)),
0132          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6178836581270835179e+04)),
0133          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8494262873223866797e+04)),
0134          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7063754290207680021e+03)),
0135          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5265133846636032186e+01)),
0136          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0)),
0137     };
0138     static const T QS[] = {
0139          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0871281941028743574e+05)),
0140          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8194580422439972989e+06)),
0141          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4194606696037208929e+06)),
0142          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0029443582266975117e+05)),
0143          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.7890229745772202641e+04)),
0144          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.6383677696049909675e+02)),
0145          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
0146     };
0147     static const T x1  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1971413260310170351e+00)),
0148                    x2  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4296810407941351328e+00)),
0149                    x11 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.620e+02)),
0150                    x12 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8288260310170351490e-03)),
0151                    x21 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3900e+03)),
0152                    x22 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.4592058648672279948e-06))
0153     ;
0154     T value, factor, r, rc, rs;
0155
0156     BOOST_MATH_STD_USING
0157     using namespace boost::math::tools;
0158     using namespace boost::math::constants;
0159
0160     if (x <= 0)
0161     {
0162        return policies::raise_domain_error<T>("boost::math::bessel_y1<%1%>(%1%,%1%)",
0163             "Got x == %1%, but x must be > 0, complex result not supported.", x, pol);
0164     }
0165     if (x <= 4)                       // x in (0, 4]
0166     {
0167         T y = x * x;
0168         T z = 2 * log(x/x1) * bessel_j1(x) / pi<T>();
0169         r = evaluate_rational(P1, Q1, y);
0170         factor = (x + x1) * ((x - x11/256) - x12) / x;
0171         value = z + factor * r;
0172     }
0173     else if (x <= 8)                  // x in (4, 8]
0174     {
0175         T y = x * x;
0176         T z = 2 * log(x/x2) * bessel_j1(x) / pi<T>();
0177         r = evaluate_rational(P2, Q2, y);
0178         factor = (x + x2) * ((x - x21/256) - x22) / x;
0179         value = z + factor * r;
0180     }
0181     else                                // x in (8, \infty)
0182     {
0183         T y = 8 / x;
0184         T y2 = y * y;
0185         rc = evaluate_rational(PC, QC, y2);
0186         rs = evaluate_rational(PS, QS, y2);
0187         factor = 1 / (sqrt(x) * root_pi<T>());
0188         //
0189         // This code is really just:
0190         //
0191         // T z = x - 0.75f * pi<T>();
0192         // value = factor * (rc * sin(z) + y * rs * cos(z));
0193         //
0194         // But using the sin/cos addition rules, plus constants for sin/cos of 3PI/4
0195         // which then cancel out with corresponding terms in "factor".
0196         //
0197         T sx = sin(x);
0198         T cx = cos(x);
0199         value = factor * (y * rs * (sx - cx) - rc * (sx + cx));
0200     }
0201
0202     return value;
0203 }
0204
0205 }}} // namespaces
0206
0207 #ifdef _MSC_VER
0208 #pragma warning(pop)
0209 #endif
0210
0211 #endif // BOOST_MATH_BESSEL_Y1_HPP
0212