boost/math/special_functions/detail/bessel_y1.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_Y1_HPP #define BOOST_MATH_BESSEL_Y1_HPP #ifdef _MSC_VER #pragma once #pragma warning(push) #pragma warning(disable:4702) // Unreachable code (release mode only warning) #endif #include <boost/math/tools/config.hpp> #include <boost/math/special_functions/detail/bessel_j1.hpp> #include <boost/math/constants/constants.hpp> #include <boost/math/tools/rational.hpp> #include <boost/math/tools/big_constant.hpp> #include <boost/math/policies/error_handling.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 second 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, typename Policy> BOOST_MATH_GPU_ENABLED T bessel_y1(T x, const Policy&); template <typename T, typename Policy> BOOST_MATH_GPU_ENABLED T bessel_y1(T x, const Policy&) { BOOST_MATH_STATIC const T P1[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0535726612579544093e+13)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4708611716525426053e+12)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.7595974497819597599e+11)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.2144548214502560419e+09)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.9157479997408395984e+07)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2157953222280260820e+05)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.1714424660046133456e+02)), }; BOOST_MATH_STATIC const T Q1[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.0737873921079286084e+14)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1272286200406461981e+12)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.7800352738690585613e+10)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.2250435122182963220e+08)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.8136470753052572164e+05)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.2079908168393867438e+02)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)), }; BOOST_MATH_STATIC const T P2[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1514276357909013326e+19)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.6808094574724204577e+18)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.3638408497043134724e+16)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0686275289804744814e+15)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.9530713129741981618e+13)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.7453673962438488783e+11)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1957961912070617006e+09)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.9153806858264202986e+06)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2337180442012953128e+03)), }; BOOST_MATH_STATIC const T Q2[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.3321844313316185697e+20)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.6968198822857178911e+18)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.0837179548112881950e+16)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1187010065856971027e+14)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.0221766852960403645e+11)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.3550318087088919566e+08)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0453748201934079734e+06)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.2855164849321609336e+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, 2.1971413260310170351e+00)), x2 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4296810407941351328e+00)), x11 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.620e+02)), x12 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8288260310170351490e-03)), x21 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3900e+03)), x22 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.4592058648672279948e-06)) ; T value, factor, r, rc, rs; BOOST_MATH_STD_USING using namespace boost::math::tools; using namespace boost::math::constants; BOOST_MATH_ASSERT(x > 0); if (x <= 4) // x in (0, 4] { T y = x * x; T z = 2 * log(x/x1) * bessel_j1(x) / pi<T>(); r = evaluate_rational(P1, Q1, y); factor = (x + x1) * ((x - x11/256) - x12) / x; value = z + factor * r; } else if (x <= 8) // x in (4, 8] { T y = x * x; T z = 2 * log(x/x2) * bessel_j1(x) / pi<T>(); r = evaluate_rational(P2, Q2, y); factor = (x + x2) * ((x - x21/256) - x22) / x; value = z + factor * r; } else // x in (8, \infty) { T y = 8 / x; T y2 = y * y; rc = evaluate_rational(PC, QC, y2); rs = evaluate_rational(PS, QS, y2); factor = 1 / (sqrt(x) * root_pi<T>()); // // This code is really just: // // T z = x - 0.75f * pi<T>(); // value = factor * (rc * sin(z) + y * rs * cos(z)); // // But using the sin/cos addition rules, plus constants for sin/cos of 3PI/4 // which then cancel out with corresponding terms in "factor". // T sx = sin(x); T cx = cos(x); value = factor * (y * rs * (sx - cx) - rc * (sx + cx)); } return value; } }}} // namespaces #ifdef _MSC_VER #pragma warning(pop) #endif #endif // BOOST_MATH_BESSEL_Y1_HPP