bessel_j1.hpp

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//  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