bessel_jn.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_JN_HPP
#define BOOST_MATH_BESSEL_JN_HPP

#ifdef _MSC_VER
#pragma once
#endif

#include <boost/math/special_functions/detail/bessel_j0.hpp>
#include <boost/math/special_functions/detail/bessel_j1.hpp>
#include <boost/math/special_functions/detail/bessel_jy.hpp>

// Bessel function of the first kind of integer order
// J_n(z) is the minimal solution
// n < abs(z), forward recurrence stable and usable
// n >= abs(z), forward recurrence unstable, use Miller's algorithm

namespace boost { namespace math { namespace detail{

template <typename T, typename Policy>
T bessel_jn(int n, T x, const Policy& pol)
{
    T value(0), factor, current, prev, next;

    BOOST_MATH_STD_USING

    if (n == 0)
    {
        return bessel_j0(x);
    }
    if (n == 1)
    {
        return bessel_j1(x);
    }
    if (n < 0)
    {
        factor = (n & 0x1) ? -1 : 1;  // J_{-n}(z) = (-1)^n J_n(z)
        n = -n;
    }
    else
    {
        factor = 1;
    }

    if (x == 0)                             // n >= 2
    {
        return static_cast<T>(0);
    }

    if (n < abs(x))                         // forward recurrence
    {
        prev = bessel_j0(x);
        current = bessel_j1(x);
        for (int k = 1; k < n; k++)
        {
            value = 2 * k * current / x - prev;
            prev = current;
            current = value;
        }
    }
    else                                    // backward recurrence
    {
        T fn; int s;                        // fn = J_(n+1) / J_n
        // |x| <= n, fast convergence for continued fraction CF1
        boost::math::detail::CF1_jy(static_cast<T>(n), x, &fn, &s, pol);
        // tiny initial value to prevent overflow
        T init = sqrt(tools::min_value<T>());
        prev = fn * init;
        current = init;
        for (int k = n; k > 0; k--)
        {
            next = 2 * k * current / x - prev;
            prev = current;
            current = next;
        }
        T ratio = init / current;           // scaling ratio
        value = bessel_j0(x) * ratio;       // normalization
    }
    value *= factor;

    return value;
}

}}} // namespaces

#endif // BOOST_MATH_BESSEL_JN_HPP