// 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/assert.hpp> | |
// 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) | |
{ | |
static const T P1[] = { | |
static_cast<T>(-4.1298668500990866786e+11L), | |
static_cast<T>(2.7282507878605942706e+10L), | |
static_cast<T>(-6.2140700423540120665e+08L), | |
static_cast<T>(6.6302997904833794242e+06L), | |
static_cast<T>(-3.6629814655107086448e+04L), | |
static_cast<T>(1.0344222815443188943e+02L), | |
static_cast<T>(-1.2117036164593528341e-01L) | |
}; | |
static const T Q1[] = { | |
static_cast<T>(2.3883787996332290397e+12L), | |
static_cast<T>(2.6328198300859648632e+10L), | |
static_cast<T>(1.3985097372263433271e+08L), | |
static_cast<T>(4.5612696224219938200e+05L), | |
static_cast<T>(9.3614022392337710626e+02L), | |
static_cast<T>(1.0L), | |
static_cast<T>(0.0L) | |
}; | |
static const T P2[] = { | |
static_cast<T>(-1.8319397969392084011e+03L), | |
static_cast<T>(-1.2254078161378989535e+04L), | |
static_cast<T>(-7.2879702464464618998e+03L), | |
static_cast<T>(1.0341910641583726701e+04L), | |
static_cast<T>(1.1725046279757103576e+04L), | |
static_cast<T>(4.4176707025325087628e+03L), | |
static_cast<T>(7.4321196680624245801e+02L), | |
static_cast<T>(4.8591703355916499363e+01L) | |
}; | |
static const T Q2[] = { | |
static_cast<T>(-3.5783478026152301072e+05L), | |
static_cast<T>(2.4599102262586308984e+05L), | |
static_cast<T>(-8.4055062591169562211e+04L), | |
static_cast<T>(1.8680990008359188352e+04L), | |
static_cast<T>(-2.9458766545509337327e+03L), | |
static_cast<T>(3.3307310774649071172e+02L), | |
static_cast<T>(-2.5258076240801555057e+01L), | |
static_cast<T>(1.0L) | |
}; | |
static const T PC[] = { | |
static_cast<T>(2.2779090197304684302e+04L), | |
static_cast<T>(4.1345386639580765797e+04L), | |
static_cast<T>(2.1170523380864944322e+04L), | |
static_cast<T>(3.4806486443249270347e+03L), | |
static_cast<T>(1.5376201909008354296e+02L), | |
static_cast<T>(8.8961548424210455236e-01L) | |
}; | |
static const T QC[] = { | |
static_cast<T>(2.2779090197304684318e+04L), | |
static_cast<T>(4.1370412495510416640e+04L), | |
static_cast<T>(2.1215350561880115730e+04L), | |
static_cast<T>(3.5028735138235608207e+03L), | |
static_cast<T>(1.5711159858080893649e+02L), | |
static_cast<T>(1.0L) | |
}; | |
static const T PS[] = { | |
static_cast<T>(-8.9226600200800094098e+01L), | |
static_cast<T>(-1.8591953644342993800e+02L), | |
static_cast<T>(-1.1183429920482737611e+02L), | |
static_cast<T>(-2.2300261666214198472e+01L), | |
static_cast<T>(-1.2441026745835638459e+00L), | |
static_cast<T>(-8.8033303048680751817e-03L) | |
}; | |
static const T QS[] = { | |
static_cast<T>(5.7105024128512061905e+03L), | |
static_cast<T>(1.1951131543434613647e+04L), | |
static_cast<T>(7.2642780169211018836e+03L), | |
static_cast<T>(1.4887231232283756582e+03L), | |
static_cast<T>(9.0593769594993125859e+01L), | |
static_cast<T>(1.0L) | |
}; | |
static const T x1 = static_cast<T>(2.4048255576957727686e+00L), | |
x2 = static_cast<T>(5.5200781102863106496e+00L), | |
x11 = static_cast<T>(6.160e+02L), | |
x12 = static_cast<T>(-1.42444230422723137837e-03L), | |
x21 = static_cast<T>(1.4130e+03L), | |
x22 = static_cast<T>(5.46860286310649596604e-04L); | |
T value, factor, r, rc, rs; | |
BOOST_MATH_STD_USING | |
using namespace boost::math::tools; | |
using namespace boost::math::constants; | |
if (x < 0) | |
{ | |
x = -x; // even function | |
} | |
if (x == 0) | |
{ | |
return static_cast<T>(1); | |
} | |
if (x <= 4) // x in (0, 4] | |
{ | |
T y = x * x; | |
BOOST_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_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; | |
T z = x - 0.25f * pi<T>(); | |
BOOST_ASSERT(sizeof(PC) == sizeof(QC)); | |
BOOST_ASSERT(sizeof(PS) == sizeof(QS)); | |
rc = evaluate_rational(PC, QC, y2); | |
rs = evaluate_rational(PS, QS, y2); | |
factor = sqrt(2 / (x * pi<T>())); | |
value = factor * (rc * cos(z) - y * rs * sin(z)); | |
} | |
return value; | |
} | |
}}} // namespaces | |
#endif // BOOST_MATH_BESSEL_J0_HPP | |