boost/math/special_functions/detail/bessel_j0.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_J0_HPP #define BOOST_MATH_BESSEL_J0_HPP #ifdef _MSC_VER #pragma once #endif #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 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> T bessel_j0(T x); template <typename T> T bessel_j0(T x) { #ifdef BOOST_MATH_INSTRUMENT static bool b = false; if (!b) { std::cout << "bessel_j0 called with " << typeid(x).name() << std::endl; std::cout << "double = " << typeid(double).name() << std::endl; std::cout << "long double = " << typeid(long double).name() << std::endl; b = true; } #endif static const T P1[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -4.1298668500990866786e+11)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.7282507878605942706e+10)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -6.2140700423540120665e+08)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.6302997904833794242e+06)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.6629814655107086448e+04)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0344222815443188943e+02)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2117036164593528341e-01)) }; static const T Q1[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.3883787996332290397e+12)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.6328198300859648632e+10)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3985097372263433271e+08)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.5612696224219938200e+05)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.3614022392337710626e+02)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0)) }; static const T P2[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.8319397969392084011e+03)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.2254078161378989535e+04)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -7.2879702464464618998e+03)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0341910641583726701e+04)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1725046279757103576e+04)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.4176707025325087628e+03)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.4321196680624245801e+02)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.8591703355916499363e+01)) }; static const T Q2[] = { static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.5783478026152301072e+05)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.4599102262586308984e+05)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -8.4055062591169562211e+04)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8680990008359188352e+04)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.9458766545509337327e+03)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.3307310774649071172e+02)), static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.5258076240801555057e+01)), 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, 2.4048255576957727686e+00)), x2 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.5200781102863106496e+00)), x11 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6.160e+02)), x12 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.42444230422723137837e-03)), x21 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4130e+03)), x22 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.46860286310649596604e-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); // reflection handled elsewhere. if (x == 0) { return static_cast<T>(1); } if (x <= 4) // x in (0, 4] { T y = x * x; BOOST_MATH_ASSERT(sizeof(P1) == sizeof(Q1)); r = evaluate_rational(P1, Q1, y); factor = (x + x1) * ((x - x11/256) - x12); value = factor * r; } else if (x <= 8.0) // x in (4, 8] { T y = 1 - (x * x)/64; BOOST_MATH_ASSERT(sizeof(P2) == sizeof(Q2)); r = evaluate_rational(P2, Q2, y); factor = (x + x2) * ((x - x21/256) - x22); value = factor * r; } else // x in (8, \infty) { T y = 8 / x; 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 = constants::one_div_root_pi<T>() / sqrt(x); // // What follows is really just: // // T z = x - pi/4; // value = factor * (rc * cos(z) - y * rs * sin(z)); // // But using the addition formulae for sin and cos, plus // the special values for sin/cos of pi/4. // T sx = sin(x); T cx = cos(x); BOOST_MATH_INSTRUMENT_VARIABLE(rc); BOOST_MATH_INSTRUMENT_VARIABLE(rs); BOOST_MATH_INSTRUMENT_VARIABLE(factor); BOOST_MATH_INSTRUMENT_VARIABLE(sx); BOOST_MATH_INSTRUMENT_VARIABLE(cx); value = factor * (rc * (cx + sx) - y * rs * (sx - cx)); } return value; } }}} // namespaces #endif // BOOST_MATH_BESSEL_J0_HPP