boost/math/special_functions/detail/bessel_j1.hpp
// Copyright (c) 2006 Xiaogang Zhang // Use, modification and distribution are subject to the // Boost Software License, Version 1.0. (See accompanying file // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) #ifndef BOOST_MATH_BESSEL_J1_HPP #define BOOST_MATH_BESSEL_J1_HPP #ifdef _MSC_VER #pragma once #endif #include <boost/math/tools/config.hpp> #include <boost/math/constants/constants.hpp> #include <boost/math/tools/rational.hpp> #include <boost/math/tools/big_constant.hpp> #include <boost/math/tools/assert.hpp> #if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128) // // This is the only way we can avoid // warning: non-standard suffix on floating constant [-Wpedantic] // when building with -Wall -pedantic. Neither __extension__ // nor #pragma diagnostic ignored work :( // #pragma GCC system_header #endif // Bessel function of the first kind of order one // x <= 8, minimax rational approximations on root-bracketing intervals // x > 8, Hankel asymptotic expansion in Hart, Computer Approximations, 1968 namespace boost { namespace math{ namespace detail{ template <typename T> BOOST_MATH_GPU_ENABLED T bessel_j1(T x); template <class T> struct bessel_j1_initializer { struct init { BOOST_MATH_GPU_ENABLED init() { do_init(); } BOOST_MATH_GPU_ENABLED static void do_init() { bessel_j1(T(1)); } BOOST_MATH_GPU_ENABLED void force_instantiate()const{} }; BOOST_MATH_STATIC const init initializer; BOOST_MATH_GPU_ENABLED static void force_instantiate() { #ifndef BOOST_MATH_HAS_GPU_SUPPORT initializer.force_instantiate(); #endif } }; template <class T> const typename bessel_j1_initializer<T>::init bessel_j1_initializer<T>::initializer; template <typename T> BOOST_MATH_GPU_ENABLED T bessel_j1(T x) { bessel_j1_initializer<T>::force_instantiate(); BOOST_MATH_STATIC const T P1[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4258509801366645672e+11)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6781041261492395835e+09)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1548696764841276794e+08)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.8062904098958257677e+05)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.4615792982775076130e+03)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0650724020080236441e+01)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0767857011487300348e-02)) }; BOOST_MATH_STATIC const T Q1[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1868604460820175290e+12)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.2091902282580133541e+10)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.0228375140097033958e+08)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.9117614494174794095e+05)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0742272239517380498e+03)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0)) }; BOOST_MATH_STATIC const T P2[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.7527881995806511112e+16)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.6608531731299018674e+15)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.6658018905416665164e+13)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5580665670910619166e+11)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8113931269860667829e+09)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.0793266148011179143e+06)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -7.5023342220781607561e+03)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.6179191852758252278e+00)) }; BOOST_MATH_STATIC const T Q2[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7253905888447681194e+18)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7128800897135812012e+16)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.4899346165481429307e+13)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.7622777286244082666e+11)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.4872502899596389593e+08)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1267125065029138050e+06)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3886978985861357615e+03)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)) }; BOOST_MATH_STATIC const T PC[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.4357578167941278571e+06)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -9.9422465050776411957e+06)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.6033732483649391093e+06)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.5235293511811373833e+06)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0982405543459346727e+05)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.6116166443246101165e+03)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0)) }; BOOST_MATH_STATIC const T QC[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.4357578167941278568e+06)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -9.9341243899345856590e+06)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.5853394797230870728e+06)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.5118095066341608816e+06)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0726385991103820119e+05)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4550094401904961825e+03)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)) }; BOOST_MATH_STATIC const T PS[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.3220913409857223519e+04)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.5145160675335701966e+04)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6178836581270835179e+04)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8494262873223866797e+04)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7063754290207680021e+03)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5265133846636032186e+01)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0)) }; BOOST_MATH_STATIC const T QS[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0871281941028743574e+05)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8194580422439972989e+06)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4194606696037208929e+06)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0029443582266975117e+05)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.7890229745772202641e+04)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.6383677696049909675e+02)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)) }; BOOST_MATH_STATIC const T x1 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.8317059702075123156e+00)); BOOST_MATH_STATIC const T x2 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0155866698156187535e+00)); BOOST_MATH_STATIC const T x11 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.810e+02)); BOOST_MATH_STATIC const T x12 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.2527979248768438556e-04)); BOOST_MATH_STATIC const T x21 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7960e+03)); BOOST_MATH_STATIC const T x22 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.8330184381246462950e-05)); T value, factor, r, rc, rs, w; BOOST_MATH_STD_USING using namespace boost::math::tools; using namespace boost::math::constants; w = abs(x); if (x == 0) { return static_cast<T>(0); } if (w <= 4) // w in (0, 4] { T y = x * x; BOOST_MATH_ASSERT(sizeof(P1) == sizeof(Q1)); r = evaluate_rational(P1, Q1, y); factor = w * (w + x1) * ((w - x11/256) - x12); value = factor * r; } else if (w <= 8) // w in (4, 8] { T y = x * x; BOOST_MATH_ASSERT(sizeof(P2) == sizeof(Q2)); r = evaluate_rational(P2, Q2, y); factor = w * (w + x2) * ((w - x21/256) - x22); value = factor * r; } else // w in (8, \infty) { T y = 8 / w; T y2 = y * y; BOOST_MATH_ASSERT(sizeof(PC) == sizeof(QC)); BOOST_MATH_ASSERT(sizeof(PS) == sizeof(QS)); rc = evaluate_rational(PC, QC, y2); rs = evaluate_rational(PS, QS, y2); factor = 1 / (sqrt(w) * constants::root_pi<T>()); // // What follows is really just: // // T z = w - 0.75f * pi<T>(); // value = factor * (rc * cos(z) - y * rs * sin(z)); // // but using the sin/cos addition rules plus constants // for the values of sin/cos of 3PI/4 which then cancel // out with corresponding terms in "factor". // T sx = sin(x); T cx = cos(x); value = factor * (rc * (sx - cx) + y * rs * (sx + cx)); } BOOST_MATH_ASSERT(x >= 0); // Negative values handled by the caller. return value; } }}} // namespaces #endif // BOOST_MATH_BESSEL_J1_HPP