bessel_i0.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_I0_HPP
#define BOOST_MATH_BESSEL_I0_HPP

#ifdef _MSC_VER
#pragma once
#endif

#include <boost/math/tools/rational.hpp>
#include <boost/assert.hpp>

// Modified Bessel function of the first kind of order zero
// minimax rational approximations on intervals, see
// Blair and Edwards, Chalk River Report AECL-4928, 1974

namespace boost { namespace math { namespace detail{

template <typename T>
T bessel_i0(T x)
{
    static const T P1[] = {
        static_cast<T>(-2.2335582639474375249e+15L),
        static_cast<T>(-5.5050369673018427753e+14L),
        static_cast<T>(-3.2940087627407749166e+13L),
        static_cast<T>(-8.4925101247114157499e+11L),
        static_cast<T>(-1.1912746104985237192e+10L),
        static_cast<T>(-1.0313066708737980747e+08L),
        static_cast<T>(-5.9545626019847898221e+05L),
        static_cast<T>(-2.4125195876041896775e+03L),
        static_cast<T>(-7.0935347449210549190e+00L),
        static_cast<T>(-1.5453977791786851041e-02L),
        static_cast<T>(-2.5172644670688975051e-05L),
        static_cast<T>(-3.0517226450451067446e-08L),
        static_cast<T>(-2.6843448573468483278e-11L),
        static_cast<T>(-1.5982226675653184646e-14L),
        static_cast<T>(-5.2487866627945699800e-18L),
    };
    static const T Q1[] = {
        static_cast<T>(-2.2335582639474375245e+15L),
        static_cast<T>(7.8858692566751002988e+12L),
        static_cast<T>(-1.2207067397808979846e+10L),
        static_cast<T>(1.0377081058062166144e+07L),
        static_cast<T>(-4.8527560179962773045e+03L),
        static_cast<T>(1.0L),
    };
    static const T P2[] = {
        static_cast<T>(-2.2210262233306573296e-04L),
        static_cast<T>(1.3067392038106924055e-02L),
        static_cast<T>(-4.4700805721174453923e-01L),
        static_cast<T>(5.5674518371240761397e+00L),
        static_cast<T>(-2.3517945679239481621e+01L),
        static_cast<T>(3.1611322818701131207e+01L),
        static_cast<T>(-9.6090021968656180000e+00L),
    };
    static const T Q2[] = {
        static_cast<T>(-5.5194330231005480228e-04L),
        static_cast<T>(3.2547697594819615062e-02L),
        static_cast<T>(-1.1151759188741312645e+00L),
        static_cast<T>(1.3982595353892851542e+01L),
        static_cast<T>(-6.0228002066743340583e+01L),
        static_cast<T>(8.5539563258012929600e+01L),
        static_cast<T>(-3.1446690275135491500e+01L),
        static_cast<T>(1.0L),
    };
    T value, factor, r;

    BOOST_MATH_STD_USING
    using namespace boost::math::tools;

    if (x < 0)
    {
        x = -x;                         // even function
    }
    if (x == 0)
    {
        return static_cast<T>(1);
    }
    if (x <= 15)                        // x in (0, 15]
    {
        T y = x * x;
        value = evaluate_polynomial(P1, y) / evaluate_polynomial(Q1, y);
    }
    else                                // x in (15, \infty)
    {
        T y = 1 / x - T(1) / 15;
        r = evaluate_polynomial(P2, y) / evaluate_polynomial(Q2, y);
        factor = exp(x) / sqrt(x);
        value = factor * r;
    }

    return value;
}

}}} // namespaces

#endif // BOOST_MATH_BESSEL_I0_HPP