// 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_J1_HPP | |
#define BOOST_MATH_BESSEL_J1_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 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> | |
T bessel_j1(T x) | |
{ | |
static const T P1[] = { | |
static_cast<T>(-1.4258509801366645672e+11L), | |
static_cast<T>(6.6781041261492395835e+09L), | |
static_cast<T>(-1.1548696764841276794e+08L), | |
static_cast<T>(9.8062904098958257677e+05L), | |
static_cast<T>(-4.4615792982775076130e+03L), | |
static_cast<T>(1.0650724020080236441e+01L), | |
static_cast<T>(-1.0767857011487300348e-02L) | |
}; | |
static const T Q1[] = { | |
static_cast<T>(4.1868604460820175290e+12L), | |
static_cast<T>(4.2091902282580133541e+10L), | |
static_cast<T>(2.0228375140097033958e+08L), | |
static_cast<T>(5.9117614494174794095e+05L), | |
static_cast<T>(1.0742272239517380498e+03L), | |
static_cast<T>(1.0L), | |
static_cast<T>(0.0L) | |
}; | |
static const T P2[] = { | |
static_cast<T>(-1.7527881995806511112e+16L), | |
static_cast<T>(1.6608531731299018674e+15L), | |
static_cast<T>(-3.6658018905416665164e+13L), | |
static_cast<T>(3.5580665670910619166e+11L), | |
static_cast<T>(-1.8113931269860667829e+09L), | |
static_cast<T>(5.0793266148011179143e+06L), | |
static_cast<T>(-7.5023342220781607561e+03L), | |
static_cast<T>(4.6179191852758252278e+00L) | |
}; | |
static const T Q2[] = { | |
static_cast<T>(1.7253905888447681194e+18L), | |
static_cast<T>(1.7128800897135812012e+16L), | |
static_cast<T>(8.4899346165481429307e+13L), | |
static_cast<T>(2.7622777286244082666e+11L), | |
static_cast<T>(6.4872502899596389593e+08L), | |
static_cast<T>(1.1267125065029138050e+06L), | |
static_cast<T>(1.3886978985861357615e+03L), | |
static_cast<T>(1.0L) | |
}; | |
static const T PC[] = { | |
static_cast<T>(-4.4357578167941278571e+06L), | |
static_cast<T>(-9.9422465050776411957e+06L), | |
static_cast<T>(-6.6033732483649391093e+06L), | |
static_cast<T>(-1.5235293511811373833e+06L), | |
static_cast<T>(-1.0982405543459346727e+05L), | |
static_cast<T>(-1.6116166443246101165e+03L), | |
static_cast<T>(0.0L) | |
}; | |
static const T QC[] = { | |
static_cast<T>(-4.4357578167941278568e+06L), | |
static_cast<T>(-9.9341243899345856590e+06L), | |
static_cast<T>(-6.5853394797230870728e+06L), | |
static_cast<T>(-1.5118095066341608816e+06L), | |
static_cast<T>(-1.0726385991103820119e+05L), | |
static_cast<T>(-1.4550094401904961825e+03L), | |
static_cast<T>(1.0L) | |
}; | |
static const T PS[] = { | |
static_cast<T>(3.3220913409857223519e+04L), | |
static_cast<T>(8.5145160675335701966e+04L), | |
static_cast<T>(6.6178836581270835179e+04L), | |
static_cast<T>(1.8494262873223866797e+04L), | |
static_cast<T>(1.7063754290207680021e+03L), | |
static_cast<T>(3.5265133846636032186e+01L), | |
static_cast<T>(0.0L) | |
}; | |
static const T QS[] = { | |
static_cast<T>(7.0871281941028743574e+05L), | |
static_cast<T>(1.8194580422439972989e+06L), | |
static_cast<T>(1.4194606696037208929e+06L), | |
static_cast<T>(4.0029443582266975117e+05L), | |
static_cast<T>(3.7890229745772202641e+04L), | |
static_cast<T>(8.6383677696049909675e+02L), | |
static_cast<T>(1.0L) | |
}; | |
static const T x1 = static_cast<T>(3.8317059702075123156e+00L), | |
x2 = static_cast<T>(7.0155866698156187535e+00L), | |
x11 = static_cast<T>(9.810e+02L), | |
x12 = static_cast<T>(-3.2527979248768438556e-04L), | |
x21 = static_cast<T>(1.7960e+03L), | |
x22 = static_cast<T>(-3.8330184381246462950e-05L); | |
T value, factor, r, rc, rs, w; | |
BOOST_MATH_STD_USING | |
using namespace boost::math::tools; | |
using namespace boost::math::constants; | |
w = abs(x); | |
if (x == 0) | |
{ | |
return static_cast<T>(0); | |
} | |
if (w <= 4) // w in (0, 4] | |
{ | |
T y = x * x; | |
BOOST_ASSERT(sizeof(P1) == sizeof(Q1)); | |
r = evaluate_rational(P1, Q1, y); | |
factor = w * (w + x1) * ((w - x11/256) - x12); | |
value = factor * r; | |
} | |
else if (w <= 8) // w in (4, 8] | |
{ | |
T y = x * x; | |
BOOST_ASSERT(sizeof(P2) == sizeof(Q2)); | |
r = evaluate_rational(P2, Q2, y); | |
factor = w * (w + x2) * ((w - x21/256) - x22); | |
value = factor * r; | |
} | |
else // w in (8, \infty) | |
{ | |
T y = 8 / w; | |
T y2 = y * y; | |
T z = w - 0.75f * 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 / (w * pi<T>())); | |
value = factor * (rc * cos(z) - y * rs * sin(z)); | |
} | |
if (x < 0) | |
{ | |
value *= -1; // odd function | |
} | |
return value; | |
} | |
}}} // namespaces | |
#endif // BOOST_MATH_BESSEL_J1_HPP | |