// Copyright (c) 2006 Xiaogang Zhang
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_MATH_BESSEL_JN_HPP
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#define BOOST_MATH_BESSEL_JN_HPP
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#ifdef _MSC_VER
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#pragma once
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#endif
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#include <boost/math/special_functions/detail/bessel_j0.hpp>
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#include <boost/math/special_functions/detail/bessel_j1.hpp>
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#include <boost/math/special_functions/detail/bessel_jy.hpp>
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#include <boost/math/special_functions/detail/bessel_jy_asym.hpp>
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#include <boost/math/special_functions/detail/bessel_jy_series.hpp>
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// Bessel function of the first kind of integer order
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// J_n(z) is the minimal solution
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// n < abs(z), forward recurrence stable and usable
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// n >= abs(z), forward recurrence unstable, use Miller's algorithm
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namespace boost { namespace math { namespace detail{
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template <typename T, typename Policy>
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T bessel_jn(int n, T x, const Policy& pol)
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{
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T value(0), factor, current, prev, next;
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BOOST_MATH_STD_USING
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//
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// Reflection has to come first:
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//
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if (n < 0)
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{
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factor = static_cast<T>((n & 0x1) ? -1 : 1); // J_{-n}(z) = (-1)^n J_n(z)
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n = -n;
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}
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else
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{
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factor = 1;
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}
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if(x < 0)
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{
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factor *= (n & 0x1) ? -1 : 1; // J_{n}(-z) = (-1)^n J_n(z)
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x = -x;
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}
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//
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// Special cases:
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//
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if(asymptotic_bessel_large_x_limit(T(n), x))
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return factor * asymptotic_bessel_j_large_x_2<T>(T(n), x, pol);
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if (n == 0)
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{
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return factor * bessel_j0(x);
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}
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if (n == 1)
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{
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return factor * bessel_j1(x);
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}
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if (x == 0) // n >= 2
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{
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return static_cast<T>(0);
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}
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BOOST_ASSERT(n > 1);
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T scale = 1;
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if (n < abs(x)) // forward recurrence
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{
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prev = bessel_j0(x);
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current = bessel_j1(x);
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policies::check_series_iterations<T>("boost::math::bessel_j_n<%1%>(%1%,%1%)", n, pol);
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for (int k = 1; k < n; k++)
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{
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T fact = 2 * k / x;
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//
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// rescale if we would overflow or underflow:
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//
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if((fabs(fact) > 1) && ((tools::max_value<T>() - fabs(prev)) / fabs(fact) < fabs(current)))
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{
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scale /= current;
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prev /= current;
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current = 1;
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}
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value = fact * current - prev;
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prev = current;
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current = value;
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}
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}
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else if((x < 1) || (n > x * x / 4) || (x < 5))
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{
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return factor * bessel_j_small_z_series(T(n), x, pol);
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}
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else // backward recurrence
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{
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T fn; int s; // fn = J_(n+1) / J_n
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// |x| <= n, fast convergence for continued fraction CF1
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boost::math::detail::CF1_jy(static_cast<T>(n), x, &fn, &s, pol);
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prev = fn;
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current = 1;
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// Check recursion won't go on too far:
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policies::check_series_iterations<T>("boost::math::bessel_j_n<%1%>(%1%,%1%)", n, pol);
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for (int k = n; k > 0; k--)
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{
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T fact = 2 * k / x;
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if((fabs(fact) > 1) && ((tools::max_value<T>() - fabs(prev)) / fabs(fact) < fabs(current)))
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{
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prev /= current;
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scale /= current;
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current = 1;
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}
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next = fact * current - prev;
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prev = current;
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current = next;
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}
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value = bessel_j0(x) / current; // normalization
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scale = 1 / scale;
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}
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value *= factor;
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if(tools::max_value<T>() * scale < fabs(value))
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return policies::raise_overflow_error<T>("boost::math::bessel_jn<%1%>(%1%,%1%)", 0, pol);
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return value / scale;
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}
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}}} // namespaces
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#endif // BOOST_MATH_BESSEL_JN_HPP
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