boost/math/special_functions/detail/bessel_y0.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_Y0_HPP #define BOOST_MATH_BESSEL_Y0_HPP #ifdef _MSC_VER #pragma once #pragma warning(push) #pragma warning(disable:4702) // Unreachable code (release mode only warning) #endif #include <boost/math/special_functions/detail/bessel_j0.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 zero // 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> T bessel_y0(T x, const Policy&); template <typename T, typename Policy> T bessel_y0(T x, const Policy&) { static const T P1[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0723538782003176831e+11)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.3716255451260504098e+09)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.0422274357376619816e+08)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.1287548474401797963e+06)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0102532948020907590e+04)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8402381979244993524e+01)), }; static const T Q1[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.8873865738997033405e+11)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.1617187777290363573e+09)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.5662956624278251596e+07)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.3889393209447253406e+05)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6475986689240190091e+02)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)), }; static const T P2[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2213976967566192242e+13)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -5.5107435206722644429e+11)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3600098638603061642e+10)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.9590439394619619534e+08)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.6905288611678631510e+06)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.4566865832663635920e+04)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7427031242901594547e+01)), }; static const T Q2[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.3386146580707264428e+14)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.4266824419412347550e+12)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4015103849971240096e+10)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3960202770986831075e+08)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.0669982352539552018e+05)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.3030857612070288823e+02)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)), }; static const T P3[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.0728726905150210443e+15)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.7016641869173237784e+14)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2829912364088687306e+11)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.9363051266772083678e+11)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1958827170518100757e+09)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.0085539923498211426e+07)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1363534169313901632e+04)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.7439661319197499338e+01)), }; static const T Q3[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4563724628846457519e+17)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9272425569640309819e+15)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2598377924042897629e+13)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.6926121104209825246e+10)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.4727219475672302327e+08)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.3924739209768057030e+05)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.7903362168128450017e+02)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)), }; static const T PC[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684302e+04)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1345386639580765797e+04)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1170523380864944322e+04)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.4806486443249270347e+03)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5376201909008354296e+02)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.8961548424210455236e-01)), }; static const T QC[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.2779090197304684318e+04)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.1370412495510416640e+04)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1215350561880115730e+04)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.5028735138235608207e+03)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5711159858080893649e+02)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)), }; static const T PS[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.9226600200800094098e+01)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8591953644342993800e+02)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.1183429920482737611e+02)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.2300261666214198472e+01)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2441026745835638459e+00)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.8033303048680751817e-03)), }; static const T QS[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.7105024128512061905e+03)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1951131543434613647e+04)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.2642780169211018836e+03)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4887231232283756582e+03)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.0593769594993125859e+01)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)), }; static const T x1 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 8.9357696627916752158e-01)), x2 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.9576784193148578684e+00)), x3 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.0860510603017726976e+00)), x11 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.280e+02)), x12 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.9519662791675215849e-03)), x21 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0130e+03)), x22 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.4716931485786837568e-04)), x31 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8140e+03)), x32 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1356030177269762362e-04)) ; 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 <= 3) // x in (0, 3] { T y = x * x; T z = 2 * log(x/x1) * bessel_j0(x) / pi<T>(); r = evaluate_rational(P1, Q1, y); factor = (x + x1) * ((x - x11/256) - x12); value = z + factor * r; } else if (x <= 5.5f) // x in (3, 5.5] { T y = x * x; T z = 2 * log(x/x2) * bessel_j0(x) / pi<T>(); r = evaluate_rational(P2, Q2, y); factor = (x + x2) * ((x - x21/256) - x22); value = z + factor * r; } else if (x <= 8) // x in (5.5, 8] { T y = x * x; T z = 2 * log(x/x3) * bessel_j0(x) / pi<T>(); r = evaluate_rational(P3, Q3, y); factor = (x + x3) * ((x - x31/256) - x32); 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 = constants::one_div_root_pi<T>() / sqrt(x); // // The following code is really just: // // T z = x - 0.25f * pi<T>(); // value = factor * (rc * sin(z) + y * rs * cos(z)); // // But using the sin/cos addition formulae and constant values for // sin/cos of PI/4 which then cancel part of the "factor" term as they're all // 1 / sqrt(2): // T sx = sin(x); T cx = cos(x); value = factor * (rc * (sx - cx) + y * rs * (cx + sx)); } return value; } }}} // namespaces #ifdef _MSC_VER #pragma warning(pop) #endif #endif // BOOST_MATH_BESSEL_Y0_HPP