| // 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> |
| #include <boost/math/special_functions/detail/bessel_jy_asym.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 |
| |
| // |
| // Reflection has to come first: |
| // |
| if (n < 0) |
| { |
| factor = (n & 0x1) ? -1 : 1; // J_{-n}(z) = (-1)^n J_n(z) |
| n = -n; |
| } |
| else |
| { |
| factor = 1; |
| } |
| // |
| // Special cases: |
| // |
| if (n == 0) |
| { |
| return factor * bessel_j0(x); |
| } |
| if (n == 1) |
| { |
| return factor * bessel_j1(x); |
| } |
| |
| if (x == 0) // n >= 2 |
| { |
| return static_cast<T>(0); |
| } |
| |
| typedef typename bessel_asymptotic_tag<T, Policy>::type tag_type; |
| if(fabs(x) > asymptotic_bessel_j_limit<T>(n, tag_type())) |
| return factor * asymptotic_bessel_j_large_x_2<T>(n, x); |
| |
| BOOST_ASSERT(n > 1); |
| 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 |
| |