special_functions/detail/bessel_y0.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_Y0_HPP
0007 #define BOOST_MATH_BESSEL_Y0_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_j0.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 zero
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_y0(T x, const Policy&);
0040
0041 template <typename T, typename Policy>
0042 T bessel_y0(T x, const Policy&)
0043 {
0044     static const T P1[] = {
0045          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0723538782003176831e+11)),
0046         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.3716255451260504098e+09)),
0047          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.0422274357376619816e+08)),
0048         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.1287548474401797963e+06)),
0049          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0102532948020907590e+04)),
0050         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8402381979244993524e+01)),
0051     };
0052     static const T Q1[] = {
0053          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.8873865738997033405e+11)),
0054          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.1617187777290363573e+09)),
0055          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.5662956624278251596e+07)),
0056          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.3889393209447253406e+05)),
0057          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6475986689240190091e+02)),
0058          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
0059     };
0060     static const T P2[] = {
0061         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2213976967566192242e+13)),
0062         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.5107435206722644429e+11)),
0063          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3600098638603061642e+10)),
0064         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.9590439394619619534e+08)),
0065          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.6905288611678631510e+06)),
0066         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4566865832663635920e+04)),
0067          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7427031242901594547e+01)),
0068     };
0069     static const T Q2[] = {
0070          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3386146580707264428e+14)),
0071          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4266824419412347550e+12)),
0072          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4015103849971240096e+10)),
0073          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3960202770986831075e+08)),
0074          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0669982352539552018e+05)),
0075          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.3030857612070288823e+02)),
0076          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
0077     };
0078     static const T P3[] = {
0079         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.0728726905150210443e+15)),
0080          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.7016641869173237784e+14)),
0081         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2829912364088687306e+11)),
0082         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.9363051266772083678e+11)),
0083          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1958827170518100757e+09)),
0084         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0085539923498211426e+07)),
0085          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1363534169313901632e+04)),
0086         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.7439661319197499338e+01)),
0087     };
0088     static const T Q3[] = {
0089          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4563724628846457519e+17)),
0090          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9272425569640309819e+15)),
0091          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2598377924042897629e+13)),
0092          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.6926121104209825246e+10)),
0093          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.4727219475672302327e+08)),
0094          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.3924739209768057030e+05)),
0095          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.7903362168128450017e+02)),
0096          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
0097     };
0098     static const T PC[] = {
0099          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684302e+04)),
0100          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1345386639580765797e+04)),
0101          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1170523380864944322e+04)),
0102          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4806486443249270347e+03)),
0103          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5376201909008354296e+02)),
0104          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.8961548424210455236e-01)),
0105     };
0106     static const T QC[] = {
0107          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684318e+04)),
0108          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1370412495510416640e+04)),
0109          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1215350561880115730e+04)),
0110          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5028735138235608207e+03)),
0111          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5711159858080893649e+02)),
0112          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
0113     };
0114     static const T PS[] = {
0115         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.9226600200800094098e+01)),
0116         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8591953644342993800e+02)),
0117         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1183429920482737611e+02)),
0118         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2300261666214198472e+01)),
0119         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2441026745835638459e+00)),
0120         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.8033303048680751817e-03)),
0121     };
0122     static const T QS[] = {
0123          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.7105024128512061905e+03)),
0124          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1951131543434613647e+04)),
0125          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.2642780169211018836e+03)),
0126          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4887231232283756582e+03)),
0127          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.0593769594993125859e+01)),
0128          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
0129     };
0130     static const T x1  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.9357696627916752158e-01)),
0131                    x2  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9576784193148578684e+00)),
0132                    x3  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0860510603017726976e+00)),
0133                    x11 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.280e+02)),
0134                    x12 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.9519662791675215849e-03)),
0135                    x21 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0130e+03)),
0136                    x22 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.4716931485786837568e-04)),
0137                    x31 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8140e+03)),
0138                    x32 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1356030177269762362e-04))
0139     ;
0140     T value, factor, r, rc, rs;
0141
0142     BOOST_MATH_STD_USING
0143     using namespace boost::math::tools;
0144     using namespace boost::math::constants;
0145
0146     BOOST_MATH_ASSERT(x > 0);
0147
0148     if (x <= 3)                       // x in (0, 3]
0149     {
0150         T y = x * x;
0151         T z = 2 * log(x/x1) * bessel_j0(x) / pi<T>();
0152         r = evaluate_rational(P1, Q1, y);
0153         factor = (x + x1) * ((x - x11/256) - x12);
0154         value = z + factor * r;
0155     }
0156     else if (x <= 5.5f)                  // x in (3, 5.5]
0157     {
0158         T y = x * x;
0159         T z = 2 * log(x/x2) * bessel_j0(x) / pi<T>();
0160         r = evaluate_rational(P2, Q2, y);
0161         factor = (x + x2) * ((x - x21/256) - x22);
0162         value = z + factor * r;
0163     }
0164     else if (x <= 8)                  // x in (5.5, 8]
0165     {
0166         T y = x * x;
0167         T z = 2 * log(x/x3) * bessel_j0(x) / pi<T>();
0168         r = evaluate_rational(P3, Q3, y);
0169         factor = (x + x3) * ((x - x31/256) - x32);
0170         value = z + factor * r;
0171     }
0172     else                                // x in (8, \infty)
0173     {
0174         T y = 8 / x;
0175         T y2 = y * y;
0176         rc = evaluate_rational(PC, QC, y2);
0177         rs = evaluate_rational(PS, QS, y2);
0178         factor = constants::one_div_root_pi<T>() / sqrt(x);
0179         //
0180         // The following code is really just:
0181         //
0182         // T z = x - 0.25f * pi<T>();
0183         // value = factor * (rc * sin(z) + y * rs * cos(z));
0184         //
0185         // But using the sin/cos addition formulae and constant values for
0186         // sin/cos of PI/4 which then cancel part of the "factor" term as they're all
0187         // 1 / sqrt(2):
0188         //
0189         T sx = sin(x);
0190         T cx = cos(x);
0191         value = factor * (rc * (sx - cx) + y * rs * (cx + sx));
0192     }
0193
0194     return value;
0195 }
0196
0197 }}} // namespaces
0198
0199 #ifdef _MSC_VER
0200 #pragma warning(pop)
0201 #endif
0202
0203 #endif // BOOST_MATH_BESSEL_Y0_HPP
0204