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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_J0_HPP
0007 #define BOOST_MATH_BESSEL_J0_HPP
0008 
0009 #ifdef _MSC_VER
0010 #pragma once
0011 #endif
0012 
0013 #include <boost/math/tools/config.hpp>
0014 #include <boost/math/constants/constants.hpp>
0015 #include <boost/math/tools/rational.hpp>
0016 #include <boost/math/tools/big_constant.hpp>
0017 #include <boost/math/tools/assert.hpp>
0018 
0019 #if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
0020 //
0021 // This is the only way we can avoid
0022 // warning: non-standard suffix on floating constant [-Wpedantic]
0023 // when building with -Wall -pedantic.  Neither __extension__
0024 // nor #pragma diagnostic ignored work :(
0025 //
0026 #pragma GCC system_header
0027 #endif
0028 
0029 // Bessel function of the first kind of order zero
0030 // x <= 8, minimax rational approximations on root-bracketing intervals
0031 // x > 8, Hankel asymptotic expansion in Hart, Computer Approximations, 1968
0032 
0033 namespace boost { namespace math { namespace detail{
0034 
0035 template <typename T>
0036 BOOST_MATH_GPU_ENABLED T bessel_j0(T x);
0037 
0038 template <typename T>
0039 BOOST_MATH_GPU_ENABLED T bessel_j0(T x)
0040 {
0041 #ifdef BOOST_MATH_INSTRUMENT
0042     static bool b = false;
0043     if (!b)
0044     {
0045        std::cout << "bessel_j0 called with " << typeid(x).name() << std::endl;
0046        std::cout << "double      = " << typeid(double).name() << std::endl;
0047        std::cout << "long double = " << typeid(long double).name() << std::endl;
0048        b = true;
0049     }
0050 #endif
0051 
0052     BOOST_MATH_STATIC const T P1[] = {
0053          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.1298668500990866786e+11)),
0054          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.7282507878605942706e+10)),
0055          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.2140700423540120665e+08)),
0056          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6302997904833794242e+06)),
0057          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.6629814655107086448e+04)),
0058          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0344222815443188943e+02)),
0059          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2117036164593528341e-01))
0060     };
0061     BOOST_MATH_STATIC const T Q1[] = {
0062          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.3883787996332290397e+12)),
0063          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.6328198300859648632e+10)),
0064          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3985097372263433271e+08)),
0065          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.5612696224219938200e+05)),
0066          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.3614022392337710626e+02)),
0067          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)),
0068          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0))
0069     };
0070     BOOST_MATH_STATIC const T P2[] = {
0071          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8319397969392084011e+03)),
0072          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2254078161378989535e+04)),
0073          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -7.2879702464464618998e+03)),
0074          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0341910641583726701e+04)),
0075          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1725046279757103576e+04)),
0076          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.4176707025325087628e+03)),
0077          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.4321196680624245801e+02)),
0078          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.8591703355916499363e+01))
0079     };
0080     BOOST_MATH_STATIC const T Q2[] = {
0081          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.5783478026152301072e+05)),
0082          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.4599102262586308984e+05)),
0083          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.4055062591169562211e+04)),
0084          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8680990008359188352e+04)),
0085          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.9458766545509337327e+03)),
0086          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.3307310774649071172e+02)),
0087          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.5258076240801555057e+01)),
0088          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0))
0089     };
0090     BOOST_MATH_STATIC const T PC[] = {
0091          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684302e+04)),
0092          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1345386639580765797e+04)),
0093          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1170523380864944322e+04)),
0094          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4806486443249270347e+03)),
0095          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5376201909008354296e+02)),
0096          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.8961548424210455236e-01))
0097     };
0098     BOOST_MATH_STATIC const T QC[] = {
0099          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684318e+04)),
0100          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1370412495510416640e+04)),
0101          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1215350561880115730e+04)),
0102          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5028735138235608207e+03)),
0103          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5711159858080893649e+02)),
0104          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0))
0105     };
0106     BOOST_MATH_STATIC const T PS[] = {
0107         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.9226600200800094098e+01)),
0108         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8591953644342993800e+02)),
0109         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1183429920482737611e+02)),
0110         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2300261666214198472e+01)),
0111         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2441026745835638459e+00)),
0112         static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.8033303048680751817e-03))
0113     };
0114     BOOST_MATH_STATIC const T QS[] = {
0115          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.7105024128512061905e+03)),
0116          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1951131543434613647e+04)),
0117          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.2642780169211018836e+03)),
0118          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4887231232283756582e+03)),
0119          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.0593769594993125859e+01)),
0120          static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0))
0121     };
0122 
0123     BOOST_MATH_STATIC const T x1  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.4048255576957727686e+00));
0124     BOOST_MATH_STATIC const T x2  =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.5200781102863106496e+00));
0125     BOOST_MATH_STATIC const T x11 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.160e+02));
0126     BOOST_MATH_STATIC const T x12 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.42444230422723137837e-03));
0127     BOOST_MATH_STATIC const T x21 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4130e+03));
0128     BOOST_MATH_STATIC const T x22 =  static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.46860286310649596604e-04));
0129 
0130     T value, factor, r, rc, rs;
0131 
0132     BOOST_MATH_STD_USING
0133     using namespace boost::math::tools;
0134     using namespace boost::math::constants;
0135 
0136     BOOST_MATH_ASSERT(x >= 0); // reflection handled elsewhere.
0137 
0138     if (x == 0)
0139     {
0140         return static_cast<T>(1);
0141     }
0142     if (x <= 4)                       // x in (0, 4]
0143     {
0144         T y = x * x;
0145         BOOST_MATH_ASSERT(sizeof(P1) == sizeof(Q1));
0146         r = evaluate_rational(P1, Q1, y);
0147         factor = (x + x1) * ((x - x11/256) - x12);
0148         value = factor * r;
0149     }
0150     else if (x <= 8.0)                  // x in (4, 8]
0151     {
0152         T y = 1 - (x * x)/64;
0153         BOOST_MATH_ASSERT(sizeof(P2) == sizeof(Q2));
0154         r = evaluate_rational(P2, Q2, y);
0155         factor = (x + x2) * ((x - x21/256) - x22);
0156         value = factor * r;
0157     }
0158     else                                // x in (8, \infty)
0159     {
0160         T y = 8 / x;
0161         T y2 = y * y;
0162         BOOST_MATH_ASSERT(sizeof(PC) == sizeof(QC));
0163         BOOST_MATH_ASSERT(sizeof(PS) == sizeof(QS));
0164         rc = evaluate_rational(PC, QC, y2);
0165         rs = evaluate_rational(PS, QS, y2);
0166         factor = constants::one_div_root_pi<T>() / sqrt(x);
0167         //
0168         // What follows is really just:
0169         //
0170         // T z = x - pi/4;
0171         // value = factor * (rc * cos(z) - y * rs * sin(z));
0172         //
0173         // But using the addition formulae for sin and cos, plus
0174         // the special values for sin/cos of pi/4.
0175         //
0176         T sx = sin(x);
0177         T cx = cos(x);
0178         BOOST_MATH_INSTRUMENT_VARIABLE(rc);
0179         BOOST_MATH_INSTRUMENT_VARIABLE(rs);
0180         BOOST_MATH_INSTRUMENT_VARIABLE(factor);
0181         BOOST_MATH_INSTRUMENT_VARIABLE(sx);
0182         BOOST_MATH_INSTRUMENT_VARIABLE(cx);
0183         value = factor * (rc * (cx + sx) - y * rs * (sx - cx));
0184     }
0185 
0186     return value;
0187 }
0188 
0189 }}} // namespaces
0190 
0191 #endif // BOOST_MATH_BESSEL_J0_HPP
0192
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