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469 lines
15 KiB
469 lines
15 KiB
// Copyright (c) 2007 John Maddock |
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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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// |
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// This header just defines the function entry points, and adds dispatch |
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// to the right implementation method. Most of the implementation details |
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// are in separate headers and copyright Xiaogang Zhang. |
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// |
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#ifndef BOOST_MATH_BESSEL_HPP |
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#define BOOST_MATH_BESSEL_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_jy.hpp> |
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#include <boost/math/special_functions/detail/bessel_jn.hpp> |
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#include <boost/math/special_functions/detail/bessel_yn.hpp> |
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#include <boost/math/special_functions/detail/bessel_ik.hpp> |
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#include <boost/math/special_functions/detail/bessel_i0.hpp> |
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#include <boost/math/special_functions/detail/bessel_i1.hpp> |
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#include <boost/math/special_functions/detail/bessel_kn.hpp> |
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#include <boost/math/special_functions/detail/iconv.hpp> |
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#include <boost/math/special_functions/sin_pi.hpp> |
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#include <boost/math/special_functions/cos_pi.hpp> |
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#include <boost/math/special_functions/sinc.hpp> |
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#include <boost/math/special_functions/trunc.hpp> |
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#include <boost/math/special_functions/round.hpp> |
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#include <boost/math/tools/rational.hpp> |
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#include <boost/math/tools/promotion.hpp> |
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#include <boost/math/tools/series.hpp> |
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namespace boost{ namespace math{ |
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namespace detail{ |
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template <class T, class Policy> |
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struct sph_bessel_j_small_z_series_term |
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{ |
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typedef T result_type; |
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sph_bessel_j_small_z_series_term(unsigned v_, T x) |
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: N(0), v(v_) |
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{ |
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BOOST_MATH_STD_USING |
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mult = x / 2; |
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term = pow(mult, T(v)) / boost::math::tgamma(v+1+T(0.5f), Policy()); |
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mult *= -mult; |
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} |
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T operator()() |
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{ |
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T r = term; |
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++N; |
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term *= mult / (N * T(N + v + 0.5f)); |
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return r; |
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} |
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private: |
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unsigned N; |
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unsigned v; |
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T mult; |
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T term; |
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}; |
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template <class T, class Policy> |
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inline T sph_bessel_j_small_z_series(unsigned v, T x, const Policy& pol) |
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{ |
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BOOST_MATH_STD_USING // ADL of std names |
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sph_bessel_j_small_z_series_term<T, Policy> s(v, x); |
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boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>(); |
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#if BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) |
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T zero = 0; |
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T result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter, zero); |
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#else |
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T result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter); |
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#endif |
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policies::check_series_iterations<T>("boost::math::sph_bessel_j_small_z_series<%1%>(%1%,%1%)", max_iter, pol); |
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return result * sqrt(constants::pi<T>() / 4); |
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} |
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template <class T, class Policy> |
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T cyl_bessel_j_imp(T v, T x, const bessel_no_int_tag& t, const Policy& pol) |
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{ |
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BOOST_MATH_STD_USING |
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static const char* function = "boost::math::bessel_j<%1%>(%1%,%1%)"; |
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if(x < 0) |
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{ |
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// better have integer v: |
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if(floor(v) == v) |
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{ |
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T r = cyl_bessel_j_imp(v, T(-x), t, pol); |
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if(iround(v, pol) & 1) |
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r = -r; |
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return r; |
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} |
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else |
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return policies::raise_domain_error<T>( |
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function, |
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"Got x = %1%, but we need x >= 0", x, pol); |
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} |
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if(x == 0) |
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return (v == 0) ? 1 : (v > 0) ? 0 : |
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policies::raise_domain_error<T>( |
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function, |
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"Got v = %1%, but require v >= 0 or a negative integer: the result would be complex.", v, pol); |
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if((v >= 0) && ((x < 1) || (v > x * x / 4) || (x < 5))) |
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{ |
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// |
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// This series will actually converge rapidly for all small |
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// x - say up to x < 20 - but the first few terms are large |
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// and divergent which leads to large errors :-( |
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// |
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return bessel_j_small_z_series(v, x, pol); |
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} |
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T j, y; |
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bessel_jy(v, x, &j, &y, need_j, pol); |
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return j; |
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} |
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template <class T, class Policy> |
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inline T cyl_bessel_j_imp(T v, T x, const bessel_maybe_int_tag&, const Policy& pol) |
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{ |
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BOOST_MATH_STD_USING // ADL of std names. |
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int ival = detail::iconv(v, pol); |
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if((abs(ival) < 200) && (0 == v - ival)) |
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{ |
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return bessel_jn(ival/*iround(v, pol)*/, x, pol); |
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} |
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return cyl_bessel_j_imp(v, x, bessel_no_int_tag(), pol); |
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} |
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template <class T, class Policy> |
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inline T cyl_bessel_j_imp(int v, T x, const bessel_int_tag&, const Policy& pol) |
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{ |
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BOOST_MATH_STD_USING |
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return bessel_jn(v, x, pol); |
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} |
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template <class T, class Policy> |
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inline T sph_bessel_j_imp(unsigned n, T x, const Policy& pol) |
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{ |
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BOOST_MATH_STD_USING // ADL of std names |
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if(x < 0) |
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return policies::raise_domain_error<T>( |
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"boost::math::sph_bessel_j<%1%>(%1%,%1%)", |
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"Got x = %1%, but function requires x > 0.", x, pol); |
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// |
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// Special case, n == 0 resolves down to the sinus cardinal of x: |
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// |
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if(n == 0) |
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return boost::math::sinc_pi(x, pol); |
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// |
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// When x is small we may end up with 0/0, use series evaluation |
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// instead, especially as it converges rapidly: |
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// |
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if(x < 1) |
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return sph_bessel_j_small_z_series(n, x, pol); |
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// |
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// Default case is just a naive evaluation of the definition: |
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// |
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return sqrt(constants::pi<T>() / (2 * x)) |
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* cyl_bessel_j_imp(T(T(n)+T(0.5f)), x, bessel_no_int_tag(), pol); |
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} |
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template <class T, class Policy> |
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T cyl_bessel_i_imp(T v, T x, const Policy& pol) |
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{ |
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// |
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// This handles all the bessel I functions, note that we don't optimise |
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// for integer v, other than the v = 0 or 1 special cases, as Millers |
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// algorithm is at least as inefficient as the general case (the general |
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// case has better error handling too). |
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// |
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BOOST_MATH_STD_USING |
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if(x < 0) |
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{ |
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// better have integer v: |
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if(floor(v) == v) |
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{ |
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T r = cyl_bessel_i_imp(v, T(-x), pol); |
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if(iround(v, pol) & 1) |
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r = -r; |
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return r; |
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} |
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else |
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return policies::raise_domain_error<T>( |
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"boost::math::cyl_bessel_i<%1%>(%1%,%1%)", |
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"Got x = %1%, but we need x >= 0", x, pol); |
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} |
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if(x == 0) |
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{ |
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return (v == 0) ? 1 : 0; |
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} |
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if(v == 0.5f) |
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{ |
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// common special case, note try and avoid overflow in exp(x): |
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if(x >= tools::log_max_value<T>()) |
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{ |
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T e = exp(x / 2); |
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return e * (e / sqrt(2 * x * constants::pi<T>())); |
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} |
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return sqrt(2 / (x * constants::pi<T>())) * sinh(x); |
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} |
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if(policies::digits<T, Policy>() <= 64) |
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{ |
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if(v == 0) |
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{ |
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return bessel_i0(x); |
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} |
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if(v == 1) |
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{ |
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return bessel_i1(x); |
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} |
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} |
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if((x / v < 0.25) && (v > 0)) |
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return bessel_i_small_z_series(v, x, pol); |
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T I, K; |
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bessel_ik(v, x, &I, &K, need_i, pol); |
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return I; |
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} |
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template <class T, class Policy> |
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inline T cyl_bessel_k_imp(T v, T x, const bessel_no_int_tag& /* t */, const Policy& pol) |
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{ |
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static const char* function = "boost::math::cyl_bessel_k<%1%>(%1%,%1%)"; |
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BOOST_MATH_STD_USING |
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if(x < 0) |
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{ |
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return policies::raise_domain_error<T>( |
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function, |
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"Got x = %1%, but we need x > 0", x, pol); |
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} |
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if(x == 0) |
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{ |
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return (v == 0) ? policies::raise_overflow_error<T>(function, 0, pol) |
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: policies::raise_domain_error<T>( |
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function, |
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"Got x = %1%, but we need x > 0", x, pol); |
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} |
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T I, K; |
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bessel_ik(v, x, &I, &K, need_k, pol); |
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return K; |
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} |
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template <class T, class Policy> |
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inline T cyl_bessel_k_imp(T v, T x, const bessel_maybe_int_tag&, const Policy& pol) |
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{ |
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BOOST_MATH_STD_USING |
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if((floor(v) == v)) |
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{ |
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return bessel_kn(itrunc(v), x, pol); |
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} |
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return cyl_bessel_k_imp(v, x, bessel_no_int_tag(), pol); |
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} |
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template <class T, class Policy> |
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inline T cyl_bessel_k_imp(int v, T x, const bessel_int_tag&, const Policy& pol) |
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{ |
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return bessel_kn(v, x, pol); |
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} |
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template <class T, class Policy> |
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inline T cyl_neumann_imp(T v, T x, const bessel_no_int_tag&, const Policy& pol) |
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{ |
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static const char* function = "boost::math::cyl_neumann<%1%>(%1%,%1%)"; |
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BOOST_MATH_INSTRUMENT_VARIABLE(v); |
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BOOST_MATH_INSTRUMENT_VARIABLE(x); |
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if(x <= 0) |
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{ |
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return (v == 0) && (x == 0) ? |
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policies::raise_overflow_error<T>(function, 0, pol) |
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: policies::raise_domain_error<T>( |
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function, |
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"Got x = %1%, but result is complex for x <= 0", x, pol); |
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} |
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T j, y; |
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bessel_jy(v, x, &j, &y, need_y, pol); |
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// |
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// Post evaluation check for internal overflow during evaluation, |
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// can occur when x is small and v is large, in which case the result |
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// is -INF: |
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// |
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if(!(boost::math::isfinite)(y)) |
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return -policies::raise_overflow_error<T>(function, 0, pol); |
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return y; |
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} |
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template <class T, class Policy> |
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inline T cyl_neumann_imp(T v, T x, const bessel_maybe_int_tag&, const Policy& pol) |
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{ |
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BOOST_MATH_STD_USING |
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typedef typename bessel_asymptotic_tag<T, Policy>::type tag_type; |
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BOOST_MATH_INSTRUMENT_VARIABLE(v); |
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BOOST_MATH_INSTRUMENT_VARIABLE(x); |
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if(floor(v) == v) |
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{ |
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if((fabs(x) > asymptotic_bessel_y_limit<T>(tag_type())) && (fabs(x) > 5 * abs(v))) |
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{ |
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T r = asymptotic_bessel_y_large_x_2(static_cast<T>(abs(v)), x); |
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if((v < 0) && (itrunc(v, pol) & 1)) |
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r = -r; |
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BOOST_MATH_INSTRUMENT_VARIABLE(r); |
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return r; |
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} |
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else |
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{ |
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T r = bessel_yn(itrunc(v, pol), x, pol); |
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BOOST_MATH_INSTRUMENT_VARIABLE(r); |
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return r; |
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} |
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} |
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T r = cyl_neumann_imp<T>(v, x, bessel_no_int_tag(), pol); |
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BOOST_MATH_INSTRUMENT_VARIABLE(r); |
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return r; |
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} |
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template <class T, class Policy> |
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inline T cyl_neumann_imp(int v, T x, const bessel_int_tag&, const Policy& pol) |
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{ |
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BOOST_MATH_STD_USING |
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typedef typename bessel_asymptotic_tag<T, Policy>::type tag_type; |
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BOOST_MATH_INSTRUMENT_VARIABLE(v); |
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BOOST_MATH_INSTRUMENT_VARIABLE(x); |
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if((fabs(x) > asymptotic_bessel_y_limit<T>(tag_type())) && (fabs(x) > 5 * abs(v))) |
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{ |
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T r = asymptotic_bessel_y_large_x_2(static_cast<T>(abs(v)), x); |
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if((v < 0) && (v & 1)) |
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r = -r; |
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return r; |
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} |
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else |
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return bessel_yn(v, x, pol); |
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} |
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template <class T, class Policy> |
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inline T sph_neumann_imp(unsigned v, T x, const Policy& pol) |
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{ |
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BOOST_MATH_STD_USING // ADL of std names |
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static const char* function = "boost::math::sph_neumann<%1%>(%1%,%1%)"; |
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// |
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// Nothing much to do here but check for errors, and |
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// evaluate the function's definition directly: |
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// |
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if(x < 0) |
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return policies::raise_domain_error<T>( |
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function, |
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"Got x = %1%, but function requires x > 0.", x, pol); |
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if(x < 2 * tools::min_value<T>()) |
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return -policies::raise_overflow_error<T>(function, 0, pol); |
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T result = cyl_neumann_imp(T(T(v)+0.5f), x, bessel_no_int_tag(), pol); |
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T tx = sqrt(constants::pi<T>() / (2 * x)); |
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if((tx > 1) && (tools::max_value<T>() / tx < result)) |
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return -policies::raise_overflow_error<T>(function, 0, pol); |
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return result * tx; |
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} |
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} // namespace detail |
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template <class T1, class T2, class Policy> |
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inline typename detail::bessel_traits<T1, T2, Policy>::result_type cyl_bessel_j(T1 v, T2 x, const Policy& pol) |
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{ |
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BOOST_FPU_EXCEPTION_GUARD |
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typedef typename detail::bessel_traits<T1, T2, Policy>::result_type result_type; |
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typedef typename detail::bessel_traits<T1, T2, Policy>::optimisation_tag tag_type; |
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typedef typename policies::evaluation<result_type, Policy>::type value_type; |
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return policies::checked_narrowing_cast<result_type, Policy>(detail::cyl_bessel_j_imp<value_type>(v, static_cast<value_type>(x), tag_type(), pol), "boost::math::cyl_bessel_j<%1%>(%1%,%1%)"); |
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} |
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template <class T1, class T2> |
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inline typename detail::bessel_traits<T1, T2, policies::policy<> >::result_type cyl_bessel_j(T1 v, T2 x) |
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{ |
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return cyl_bessel_j(v, x, policies::policy<>()); |
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} |
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template <class T, class Policy> |
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inline typename detail::bessel_traits<T, T, Policy>::result_type sph_bessel(unsigned v, T x, const Policy& pol) |
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{ |
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BOOST_FPU_EXCEPTION_GUARD |
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typedef typename detail::bessel_traits<T, T, Policy>::result_type result_type; |
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typedef typename policies::evaluation<result_type, Policy>::type value_type; |
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return policies::checked_narrowing_cast<result_type, Policy>(detail::sph_bessel_j_imp<value_type>(v, static_cast<value_type>(x), pol), "boost::math::sph_bessel<%1%>(%1%,%1%)"); |
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} |
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template <class T> |
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inline typename detail::bessel_traits<T, T, policies::policy<> >::result_type sph_bessel(unsigned v, T x) |
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{ |
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return sph_bessel(v, x, policies::policy<>()); |
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} |
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template <class T1, class T2, class Policy> |
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inline typename detail::bessel_traits<T1, T2, Policy>::result_type cyl_bessel_i(T1 v, T2 x, const Policy& pol) |
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{ |
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BOOST_FPU_EXCEPTION_GUARD |
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typedef typename detail::bessel_traits<T1, T2, Policy>::result_type result_type; |
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typedef typename policies::evaluation<result_type, Policy>::type value_type; |
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return policies::checked_narrowing_cast<result_type, Policy>(detail::cyl_bessel_i_imp<value_type>(v, static_cast<value_type>(x), pol), "boost::math::cyl_bessel_i<%1%>(%1%,%1%)"); |
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} |
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template <class T1, class T2> |
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inline typename detail::bessel_traits<T1, T2, policies::policy<> >::result_type cyl_bessel_i(T1 v, T2 x) |
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{ |
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return cyl_bessel_i(v, x, policies::policy<>()); |
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} |
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template <class T1, class T2, class Policy> |
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inline typename detail::bessel_traits<T1, T2, Policy>::result_type cyl_bessel_k(T1 v, T2 x, const Policy& pol) |
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{ |
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BOOST_FPU_EXCEPTION_GUARD |
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typedef typename detail::bessel_traits<T1, T2, Policy>::result_type result_type; |
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typedef typename detail::bessel_traits<T1, T2, Policy>::optimisation_tag tag_type; |
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typedef typename policies::evaluation<result_type, Policy>::type value_type; |
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return policies::checked_narrowing_cast<result_type, Policy>(detail::cyl_bessel_k_imp<value_type>(v, static_cast<value_type>(x), tag_type(), pol), "boost::math::cyl_bessel_k<%1%>(%1%,%1%)"); |
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} |
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template <class T1, class T2> |
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inline typename detail::bessel_traits<T1, T2, policies::policy<> >::result_type cyl_bessel_k(T1 v, T2 x) |
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{ |
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return cyl_bessel_k(v, x, policies::policy<>()); |
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} |
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template <class T1, class T2, class Policy> |
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inline typename detail::bessel_traits<T1, T2, Policy>::result_type cyl_neumann(T1 v, T2 x, const Policy& pol) |
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{ |
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BOOST_FPU_EXCEPTION_GUARD |
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typedef typename detail::bessel_traits<T1, T2, Policy>::result_type result_type; |
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typedef typename detail::bessel_traits<T1, T2, Policy>::optimisation_tag tag_type; |
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typedef typename policies::evaluation<result_type, Policy>::type value_type; |
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return policies::checked_narrowing_cast<result_type, Policy>(detail::cyl_neumann_imp<value_type>(v, static_cast<value_type>(x), tag_type(), pol), "boost::math::cyl_neumann<%1%>(%1%,%1%)"); |
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} |
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template <class T1, class T2> |
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inline typename detail::bessel_traits<T1, T2, policies::policy<> >::result_type cyl_neumann(T1 v, T2 x) |
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{ |
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return cyl_neumann(v, x, policies::policy<>()); |
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} |
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template <class T, class Policy> |
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inline typename detail::bessel_traits<T, T, Policy>::result_type sph_neumann(unsigned v, T x, const Policy& pol) |
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{ |
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BOOST_FPU_EXCEPTION_GUARD |
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typedef typename detail::bessel_traits<T, T, Policy>::result_type result_type; |
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typedef typename policies::evaluation<result_type, Policy>::type value_type; |
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return policies::checked_narrowing_cast<result_type, Policy>(detail::sph_neumann_imp<value_type>(v, static_cast<value_type>(x), pol), "boost::math::sph_neumann<%1%>(%1%,%1%)"); |
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} |
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template <class T> |
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inline typename detail::bessel_traits<T, T, policies::policy<> >::result_type sph_neumann(unsigned v, T x) |
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{ |
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return sph_neumann(v, x, policies::policy<>()); |
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} |
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} // namespace math |
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} // namespace boost |
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#endif // BOOST_MATH_BESSEL_HPP |
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