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/tools/config.hpp>
0016 #include <boost/math/special_functions/detail/bessel_j0.hpp>
0017 #include <boost/math/constants/constants.hpp>
0018 #include <boost/math/tools/rational.hpp>
0019 #include <boost/math/tools/big_constant.hpp>
0020 #include <boost/math/policies/error_handling.hpp>
0021 #include <boost/math/tools/assert.hpp>
0022
0023 #if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
0024 //
0025 // This is the only way we can avoid
0026 // warning: non-standard suffix on floating constant [-Wpedantic]
0027 // when building with -Wall -pedantic.  Neither __extension__
0028 // nor #pragma diagnostic ignored work :(
0029 //
0030 #pragma GCC system_header
0031 #endif
0032
0033 // Bessel function of the second kind of order zero
0034 // x <= 8, minimax rational approximations on root-bracketing intervals
0035 // x > 8, Hankel asymptotic expansion in Hart, Computer Approximations, 1968
0036
0037 namespace boost { namespace math { namespace detail{
0038
0039 template <typename T, typename Policy>
0040 BOOST_MATH_GPU_ENABLED T bessel_y0(T x, const Policy&);
0041
0042 template <typename T, typename Policy>
0043 BOOST_MATH_GPU_ENABLED T bessel_y0(T x, const Policy&)
0044 {
0045     BOOST_MATH_STATIC const T P1[] = {
0046          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0723538782003176831e+11)),
0047         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.3716255451260504098e+09)),
0048          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.0422274357376619816e+08)),
0049         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.1287548474401797963e+06)),
0050          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0102532948020907590e+04)),
0051         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8402381979244993524e+01)),
0052     };
0053     BOOST_MATH_STATIC const T Q1[] = {
0054          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.8873865738997033405e+11)),
0055          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.1617187777290363573e+09)),
0056          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.5662956624278251596e+07)),
0057          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.3889393209447253406e+05)),
0058          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6475986689240190091e+02)),
0059          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
0060     };
0061     BOOST_MATH_STATIC const T P2[] = {
0062         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2213976967566192242e+13)),
0063         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.5107435206722644429e+11)),
0064          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3600098638603061642e+10)),
0065         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.9590439394619619534e+08)),
0066          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.6905288611678631510e+06)),
0067         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4566865832663635920e+04)),
0068          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7427031242901594547e+01)),
0069     };
0070     BOOST_MATH_STATIC const T Q2[] = {
0071          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3386146580707264428e+14)),
0072          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4266824419412347550e+12)),
0073          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4015103849971240096e+10)),
0074          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3960202770986831075e+08)),
0075          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0669982352539552018e+05)),
0076          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.3030857612070288823e+02)),
0077          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
0078     };
0079     BOOST_MATH_STATIC const T P3[] = {
0080         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.0728726905150210443e+15)),
0081          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.7016641869173237784e+14)),
0082         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2829912364088687306e+11)),
0083         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.9363051266772083678e+11)),
0084          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1958827170518100757e+09)),
0085         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0085539923498211426e+07)),
0086          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1363534169313901632e+04)),
0087         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.7439661319197499338e+01)),
0088     };
0089     BOOST_MATH_STATIC const T Q3[] = {
0090          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4563724628846457519e+17)),
0091          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9272425569640309819e+15)),
0092          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2598377924042897629e+13)),
0093          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.6926121104209825246e+10)),
0094          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.4727219475672302327e+08)),
0095          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.3924739209768057030e+05)),
0096          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.7903362168128450017e+02)),
0097          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
0098     };
0099     BOOST_MATH_STATIC const T PC[] = {
0100          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684302e+04)),
0101          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1345386639580765797e+04)),
0102          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1170523380864944322e+04)),
0103          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4806486443249270347e+03)),
0104          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5376201909008354296e+02)),
0105          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.8961548424210455236e-01)),
0106     };
0107     BOOST_MATH_STATIC const T QC[] = {
0108          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684318e+04)),
0109          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1370412495510416640e+04)),
0110          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1215350561880115730e+04)),
0111          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5028735138235608207e+03)),
0112          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5711159858080893649e+02)),
0113          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
0114     };
0115     BOOST_MATH_STATIC const T PS[] = {
0116         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.9226600200800094098e+01)),
0117         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8591953644342993800e+02)),
0118         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1183429920482737611e+02)),
0119         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2300261666214198472e+01)),
0120         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2441026745835638459e+00)),
0121         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.8033303048680751817e-03)),
0122     };
0123     BOOST_MATH_STATIC const T QS[] = {
0124          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.7105024128512061905e+03)),
0125          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1951131543434613647e+04)),
0126          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.2642780169211018836e+03)),
0127          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4887231232283756582e+03)),
0128          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.0593769594993125859e+01)),
0129          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
0130     };
0131     BOOST_MATH_STATIC const T x1  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.9357696627916752158e-01)),
0132                    x2  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9576784193148578684e+00)),
0133                    x3  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0860510603017726976e+00)),
0134                    x11 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.280e+02)),
0135                    x12 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.9519662791675215849e-03)),
0136                    x21 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0130e+03)),
0137                    x22 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.4716931485786837568e-04)),
0138                    x31 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8140e+03)),
0139                    x32 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1356030177269762362e-04))
0140     ;
0141     T value, factor, r, rc, rs;
0142
0143     BOOST_MATH_STD_USING
0144     using namespace boost::math::tools;
0145     using namespace boost::math::constants;
0146
0147     BOOST_MATH_ASSERT(x > 0);
0148
0149     if (x <= 3)                       // x in (0, 3]
0150     {
0151         T y = x * x;
0152         T z = 2 * log(x/x1) * bessel_j0(x) / pi<T>();
0153         r = evaluate_rational(P1, Q1, y);
0154         factor = (x + x1) * ((x - x11/256) - x12);
0155         value = z + factor * r;
0156     }
0157     else if (x <= 5.5f)                  // x in (3, 5.5]
0158     {
0159         T y = x * x;
0160         T z = 2 * log(x/x2) * bessel_j0(x) / pi<T>();
0161         r = evaluate_rational(P2, Q2, y);
0162         factor = (x + x2) * ((x - x21/256) - x22);
0163         value = z + factor * r;
0164     }
0165     else if (x <= 8)                  // x in (5.5, 8]
0166     {
0167         T y = x * x;
0168         T z = 2 * log(x/x3) * bessel_j0(x) / pi<T>();
0169         r = evaluate_rational(P3, Q3, y);
0170         factor = (x + x3) * ((x - x31/256) - x32);
0171         value = z + factor * r;
0172     }
0173     else                                // x in (8, \infty)
0174     {
0175         T y = 8 / x;
0176         T y2 = y * y;
0177         rc = evaluate_rational(PC, QC, y2);
0178         rs = evaluate_rational(PS, QS, y2);
0179         factor = constants::one_div_root_pi<T>() / sqrt(x);
0180         //
0181         // The following code is really just:
0182         //
0183         // T z = x - 0.25f * pi<T>();
0184         // value = factor * (rc * sin(z) + y * rs * cos(z));
0185         //
0186         // But using the sin/cos addition formulae and constant values for
0187         // sin/cos of PI/4 which then cancel part of the "factor" term as they're all
0188         // 1 / sqrt(2):
0189         //
0190         T sx = sin(x);
0191         T cx = cos(x);
0192         value = factor * (rc * (sx - cx) + y * rs * (cx + sx));
0193     }
0194
0195     return value;
0196 }
0197
0198 }}} // namespaces
0199
0200 #ifdef _MSC_VER
0201 #pragma warning(pop)
0202 #endif
0203
0204 #endif // BOOST_MATH_BESSEL_Y0_HPP
0205