| // 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 |
| #endif |
| |
| #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/assert.hpp> |
| |
| // 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> |
| T bessel_y1(T x, const Policy&); |
| |
| template <class T, class Policy> |
| struct bessel_y1_initializer |
| { |
| struct init |
| { |
| init() |
| { |
| do_init(); |
| } |
| static void do_init() |
| { |
| bessel_y1(T(1), Policy()); |
| } |
| void force_instantiate()const{} |
| }; |
| static const init initializer; |
| static void force_instantiate() |
| { |
| initializer.force_instantiate(); |
| } |
| }; |
| |
| template <class T, class Policy> |
| const typename bessel_y1_initializer<T, Policy>::init bessel_y1_initializer<T, Policy>::initializer; |
| |
| template <typename T, typename Policy> |
| T bessel_y1(T x, const Policy& pol) |
| { |
| bessel_y1_initializer<T, Policy>::force_instantiate(); |
| |
| 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)), |
| }; |
| 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)), |
| }; |
| 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)), |
| }; |
| 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)), |
| }; |
| 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)), |
| }; |
| 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)), |
| }; |
| 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)), |
| }; |
| 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)), |
| }; |
| 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; |
| |
| if (x <= 0) |
| { |
| return policies::raise_domain_error<T>("bost::math::bessel_y1<%1%>(%1%,%1%)", |
| "Got x == %1%, but x must be > 0, complex result not supported.", x, pol); |
| } |
| 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 |
| |
| #endif // BOOST_MATH_BESSEL_Y1_HPP |
| |
