comparison DEPENDENCIES/generic/include/boost/math/special_functions/bessel.hpp @ 16:2665513ce2d3

Add boost headers

+//  Copyright (c) 2007, 2013 John Maddock
+//  Copyright Christopher Kormanyos 2013.
+//  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)
+//
+// This header just defines the function entry points, and adds dispatch
+// to the right implementation method.  Most of the implementation details
+// are in separate headers and copyright Xiaogang Zhang.
+//
+#ifndef BOOST_MATH_BESSEL_HPP
+#define BOOST_MATH_BESSEL_HPP
+
+#ifdef _MSC_VER
+#  pragma once
+#endif
+
+#include <boost/math/special_functions/detail/bessel_jy.hpp>
+#include <boost/math/special_functions/detail/bessel_jn.hpp>
+#include <boost/math/special_functions/detail/bessel_yn.hpp>
+#include <boost/math/special_functions/detail/bessel_jy_zero.hpp>
+#include <boost/math/special_functions/detail/bessel_ik.hpp>
+#include <boost/math/special_functions/detail/bessel_i0.hpp>
+#include <boost/math/special_functions/detail/bessel_i1.hpp>
+#include <boost/math/special_functions/detail/bessel_kn.hpp>
+#include <boost/math/special_functions/detail/iconv.hpp>
+#include <boost/math/special_functions/sin_pi.hpp>
+#include <boost/math/special_functions/cos_pi.hpp>
+#include <boost/math/special_functions/sinc.hpp>
+#include <boost/math/special_functions/trunc.hpp>
+#include <boost/math/special_functions/round.hpp>
+#include <boost/math/tools/rational.hpp>
+#include <boost/math/tools/promotion.hpp>
+#include <boost/math/tools/series.hpp>
+#include <boost/math/tools/roots.hpp>
+
+namespace boost{ namespace math{
+
+namespace detail{
+
+template <class T, class Policy>
+struct sph_bessel_j_small_z_series_term
+{
+   typedef T result_type;
+
+   sph_bessel_j_small_z_series_term(unsigned v_, T x)
+      : N(0), v(v_)
+   {
+      BOOST_MATH_STD_USING
+      mult = x / 2;
+      if(v + 3 > max_factorial<T>::value)
+      {
+         term = v * log(mult) - boost::math::lgamma(v+1+T(0.5f), Policy());
+         term = exp(term);
+      }
+      else
+         term = pow(mult, T(v)) / boost::math::tgamma(v+1+T(0.5f), Policy());
+      mult *= -mult;
+   }
+   T operator()()
+   {
+      T r = term;
+      ++N;
+      term *= mult / (N * T(N + v + 0.5f));
+      return r;
+   }
+private:
+   unsigned N;
+   unsigned v;
+   T mult;
+   T term;
+};
+
+template <class T, class Policy>
+inline T sph_bessel_j_small_z_series(unsigned v, T x, const Policy& pol)
+{
+   BOOST_MATH_STD_USING // ADL of std names
+   sph_bessel_j_small_z_series_term<T, Policy> s(v, x);
+   boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
+#if BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582))
+   T zero = 0;
+   T result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter, zero);
+#else
+   T result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter);
+#endif
+   policies::check_series_iterations<T>("boost::math::sph_bessel_j_small_z_series<%1%>(%1%,%1%)", max_iter, pol);
+   return result * sqrt(constants::pi<T>() / 4);
+}
+
+template <class T, class Policy>
+T cyl_bessel_j_imp(T v, T x, const bessel_no_int_tag& t, const Policy& pol)
+{
+   BOOST_MATH_STD_USING
+   static const char* function = "boost::math::bessel_j<%1%>(%1%,%1%)";
+   if(x < 0)
+   {
+      // better have integer v:
+      if(floor(v) == v)
+      {
+         T r = cyl_bessel_j_imp(v, T(-x), t, pol);
+         if(iround(v, pol) & 1)
+            r = -r;
+         return r;
+      }
+      else
+         return policies::raise_domain_error<T>(
+            function,
+            "Got x = %1%, but we need x >= 0", x, pol);
+   }
+   
+   T j, y;
+   bessel_jy(v, x, &j, &y, need_j, pol);
+   return j;
+}
+
+template <class T, class Policy>
+inline T cyl_bessel_j_imp(T v, T x, const bessel_maybe_int_tag&, const Policy& pol)
+{
+   BOOST_MATH_STD_USING  // ADL of std names.
+   int ival = detail::iconv(v, pol);
+   // If v is an integer, use the integer recursion
+   // method, both that and Steeds method are O(v):
+   if((0 == v - ival))
+   {
+      return bessel_jn(ival, x, pol);
+   }
+   return cyl_bessel_j_imp(v, x, bessel_no_int_tag(), pol);
+}
+
+template <class T, class Policy>
+inline T cyl_bessel_j_imp(int v, T x, const bessel_int_tag&, const Policy& pol)
+{
+   BOOST_MATH_STD_USING
+   return bessel_jn(v, x, pol);
+}
+
+template <class T, class Policy>
+inline T sph_bessel_j_imp(unsigned n, T x, const Policy& pol)
+{
+   BOOST_MATH_STD_USING // ADL of std names
+   if(x < 0)
+      return policies::raise_domain_error<T>(
+         "boost::math::sph_bessel_j<%1%>(%1%,%1%)",
+         "Got x = %1%, but function requires x > 0.", x, pol);
+   //
+   // Special case, n == 0 resolves down to the sinus cardinal of x:
+   //
+   if(n == 0)
+      return boost::math::sinc_pi(x, pol);
+   //
+   // Special case for x == 0:
+   //
+   if(x == 0)
+      return 0;
+   //
+   // When x is small we may end up with 0/0, use series evaluation
+   // instead, especially as it converges rapidly:
+   //
+   if(x < 1)
+      return sph_bessel_j_small_z_series(n, x, pol);
+   //
+   // Default case is just a naive evaluation of the definition:
+   //
+   return sqrt(constants::pi<T>() / (2 * x)) 
+      * cyl_bessel_j_imp(T(T(n)+T(0.5f)), x, bessel_no_int_tag(), pol);
+}
+
+template <class T, class Policy>
+T cyl_bessel_i_imp(T v, T x, const Policy& pol)
+{
+   //
+   // This handles all the bessel I functions, note that we don't optimise
+   // for integer v, other than the v = 0 or 1 special cases, as Millers
+   // algorithm is at least as inefficient as the general case (the general
+   // case has better error handling too).
+   //
+   BOOST_MATH_STD_USING
+   if(x < 0)
+   {
+      // better have integer v:
+      if(floor(v) == v)
+      {
+         T r = cyl_bessel_i_imp(v, T(-x), pol);
+         if(iround(v, pol) & 1)
+            r = -r;
+         return r;
+      }
+      else
+         return policies::raise_domain_error<T>(
+         "boost::math::cyl_bessel_i<%1%>(%1%,%1%)",
+            "Got x = %1%, but we need x >= 0", x, pol);
+   }
+   if(x == 0)
+   {
+      return (v == 0) ? 1 : 0;
+   }
+   if(v == 0.5f)
+   {
+      // common special case, note try and avoid overflow in exp(x):
+      if(x >= tools::log_max_value<T>())
+      {
+         T e = exp(x / 2);
+         return e * (e / sqrt(2 * x * constants::pi<T>()));
+      }
+      return sqrt(2 / (x * constants::pi<T>())) * sinh(x);
+   }
+   if(policies::digits<T, Policy>() <= 64)
+   {
+      if(v == 0)
+      {
+         return bessel_i0(x);
+      }
+      if(v == 1)
+      {
+         return bessel_i1(x);
+      }
+   }
+   if((v > 0) && (x / v < 0.25))
+      return bessel_i_small_z_series(v, x, pol);
+   T I, K;
+   bessel_ik(v, x, &I, &K, need_i, pol);
+   return I;
+}
+
+template <class T, class Policy>
+inline T cyl_bessel_k_imp(T v, T x, const bessel_no_int_tag& /* t */, const Policy& pol)
+{
+   static const char* function = "boost::math::cyl_bessel_k<%1%>(%1%,%1%)";
+   BOOST_MATH_STD_USING
+   if(x < 0)
+   {
+      return policies::raise_domain_error<T>(
+         function,
+         "Got x = %1%, but we need x > 0", x, pol);
+   }
+   if(x == 0)
+   {
+      return (v == 0) ? policies::raise_overflow_error<T>(function, 0, pol)
+         : policies::raise_domain_error<T>(
+         function,
+         "Got x = %1%, but we need x > 0", x, pol);
+   }
+   T I, K;
+   bessel_ik(v, x, &I, &K, need_k, pol);
+   return K;
+}
+
+template <class T, class Policy>
+inline T cyl_bessel_k_imp(T v, T x, const bessel_maybe_int_tag&, const Policy& pol)
+{
+   BOOST_MATH_STD_USING
+   if((floor(v) == v))
+   {
+      return bessel_kn(itrunc(v), x, pol);
+   }
+   return cyl_bessel_k_imp(v, x, bessel_no_int_tag(), pol);
+}
+
+template <class T, class Policy>
+inline T cyl_bessel_k_imp(int v, T x, const bessel_int_tag&, const Policy& pol)
+{
+   return bessel_kn(v, x, pol);
+}
+
+template <class T, class Policy>
+inline T cyl_neumann_imp(T v, T x, const bessel_no_int_tag&, const Policy& pol)
+{
+   static const char* function = "boost::math::cyl_neumann<%1%>(%1%,%1%)";
+
+   BOOST_MATH_INSTRUMENT_VARIABLE(v);
+   BOOST_MATH_INSTRUMENT_VARIABLE(x);
+
+   if(x <= 0)
+   {
+      return (v == 0) && (x == 0) ?
+         policies::raise_overflow_error<T>(function, 0, pol)
+         : policies::raise_domain_error<T>(
+               function,
+               "Got x = %1%, but result is complex for x <= 0", x, pol);
+   }
+   T j, y;
+   bessel_jy(v, x, &j, &y, need_y, pol);
+   // 
+   // Post evaluation check for internal overflow during evaluation,
+   // can occur when x is small and v is large, in which case the result
+   // is -INF:
+   //
+   if(!(boost::math::isfinite)(y))
+      return -policies::raise_overflow_error<T>(function, 0, pol);
+   return y;
+}
+
+template <class T, class Policy>
+inline T cyl_neumann_imp(T v, T x, const bessel_maybe_int_tag&, const Policy& pol)
+{
+   BOOST_MATH_STD_USING
+
+   BOOST_MATH_INSTRUMENT_VARIABLE(v);
+   BOOST_MATH_INSTRUMENT_VARIABLE(x);
+
+   if(floor(v) == v)
+   {
+      if(asymptotic_bessel_large_x_limit(v, x))
+      {
+         T r = asymptotic_bessel_y_large_x_2(static_cast<T>(abs(v)), x);
+         if((v < 0) && (itrunc(v, pol) & 1))
+            r = -r;
+         BOOST_MATH_INSTRUMENT_VARIABLE(r);
+         return r;
+      }
+      else
+      {
+         T r = bessel_yn(itrunc(v, pol), x, pol);
+         BOOST_MATH_INSTRUMENT_VARIABLE(r);
+         return r;
+      }
+   }
+   T r = cyl_neumann_imp<T>(v, x, bessel_no_int_tag(), pol);
+   BOOST_MATH_INSTRUMENT_VARIABLE(r);
+   return r;
+}
+
+template <class T, class Policy>
+inline T cyl_neumann_imp(int v, T x, const bessel_int_tag&, const Policy& pol)
+{
+   BOOST_MATH_STD_USING
+
+   BOOST_MATH_INSTRUMENT_VARIABLE(v);
+   BOOST_MATH_INSTRUMENT_VARIABLE(x);
+
+   if(asymptotic_bessel_large_x_limit(T(v), x))
+   {
+      T r = asymptotic_bessel_y_large_x_2(static_cast<T>(abs(v)), x);
+      if((v < 0) && (v & 1))
+         r = -r;
+      return r;
+   }
+   else
+      return bessel_yn(v, x, pol);
+}
+
+template <class T, class Policy>
+inline T sph_neumann_imp(unsigned v, T x, const Policy& pol)
+{
+   BOOST_MATH_STD_USING // ADL of std names
+   static const char* function = "boost::math::sph_neumann<%1%>(%1%,%1%)";
+   //
+   // Nothing much to do here but check for errors, and
+   // evaluate the function's definition directly:
+   //
+   if(x < 0)
+      return policies::raise_domain_error<T>(
+         function,
+         "Got x = %1%, but function requires x > 0.", x, pol);
+
+   if(x < 2 * tools::min_value<T>())
+      return -policies::raise_overflow_error<T>(function, 0, pol);
+
+   T result = cyl_neumann_imp(T(T(v)+0.5f), x, bessel_no_int_tag(), pol);
+   T tx = sqrt(constants::pi<T>() / (2 * x));
+
+   if((tx > 1) && (tools::max_value<T>() / tx < result))
+      return -policies::raise_overflow_error<T>(function, 0, pol);
+
+   return result * tx;
+}
+
+template <class T, class Policy>
+inline T cyl_bessel_j_zero_imp(T v, int m, const Policy& pol)
+{
+   BOOST_MATH_STD_USING // ADL of std names, needed for floor.
+
+   static const char* function = "boost::math::cyl_bessel_j_zero<%1%>(%1%, int)";
+
+   const T half_epsilon(boost::math::tools::epsilon<T>() / 2U);
+
+   // Handle non-finite order.
+   if (!(boost::math::isfinite)(v) )
+   {
+     return policies::raise_domain_error<T>(function, "Order argument is %1%, but must be finite >= 0 !", v, pol);
+   }
+
+   // Handle negative rank.
+   if(m < 0)
+   {
+      // Zeros of Jv(x) with negative rank are not defined and requesting one raises a domain error.
+      return policies::raise_domain_error<T>(function, "Requested the %1%'th zero, but the rank must be positive !", m, pol);
+   }
+
+   // Get the absolute value of the order.
+   const bool order_is_negative = (v < 0);
+   const T vv((!order_is_negative) ? v : T(-v));
+
+   // Check if the order is very close to zero or very close to an integer.
+   const bool order_is_zero    = (vv < half_epsilon);
+   const bool order_is_integer = ((vv - floor(vv)) < half_epsilon);
+
+   if(m == 0)
+   {
+      if(order_is_zero)
+      {
+         // The zero'th zero of J0(x) is not defined and requesting it raises a domain error.
+         return policies::raise_domain_error<T>(function, "Requested the %1%'th zero of J0, but the rank must be > 0 !", m, pol);
+      }
+
+      // The zero'th zero of Jv(x) for v < 0 is not defined
+      // unless the order is a negative integer.
+      if(order_is_negative && (!order_is_integer))
+      {
+         // For non-integer, negative order, requesting the zero'th zero raises a domain error.
+         return policies::raise_domain_error<T>(function, "Requested the %1%'th zero of Jv for negative, non-integer order, but the rank must be > 0 !", m, pol);
+      }
+
+      // The zero'th zero does exist and its value is zero.
+      return T(0);
+   }
+
+   // Set up the initial guess for the upcoming root-finding.
+   // If the order is a negative integer, then use the corresponding
+   // positive integer for the order.
+   const T guess_root = boost::math::detail::bessel_zero::cyl_bessel_j_zero_detail::initial_guess<T, Policy>((order_is_integer ? vv : v), m, pol);
+
+   // Select the maximum allowed iterations from the policy.
+   boost::uintmax_t number_of_iterations = policies::get_max_root_iterations<Policy>();
+
+   // Select the desired number of binary digits of precision.
+   // Account for the radix of number representations having non-two radix!
+   const int my_digits2 = policies::digits<T, Policy>();
+
+   const T delta_lo = ((guess_root > 0.2F) ? T(0.2) : T(guess_root / 2U));
+
+   // Perform the root-finding using Newton-Raphson iteration from Boost.Math.
+   const T jvm =
+      boost::math::tools::newton_raphson_iterate(
+         boost::math::detail::bessel_zero::cyl_bessel_j_zero_detail::function_object_jv_and_jv_prime<T, Policy>((order_is_integer ? vv : v), order_is_zero, pol),
+         guess_root,
+         T(guess_root - delta_lo),
+         T(guess_root + 0.2F),
+         my_digits2,
+         number_of_iterations);
+
+   if(number_of_iterations >= policies::get_max_root_iterations<Policy>())
+   {
+      policies::raise_evaluation_error<T>(function, "Unable to locate root in a reasonable time:"
+         "  Current best guess is %1%", jvm, Policy());
+   }
+
+   return jvm;
+}
+
+template <class T, class Policy>
+inline T cyl_neumann_zero_imp(T v, int m, const Policy& pol)
+{
+   BOOST_MATH_STD_USING // ADL of std names, needed for floor.
+
+   static const char* function = "boost::math::cyl_neumann_zero<%1%>(%1%, int)";
+
+   // Handle non-finite order.
+   if (!(boost::math::isfinite)(v) )
+   {
+     return policies::raise_domain_error<T>(function, "Order argument is %1%, but must be finite >= 0 !", v, pol);
+   }
+
+   // Handle negative rank.
+   if(m < 0)
+   {
+      return policies::raise_domain_error<T>(function, "Requested the %1%'th zero, but the rank must be positive !", m, pol);
+   }
+
+   const T half_epsilon(boost::math::tools::epsilon<T>() / 2U);
+
+   // Get the absolute value of the order.
+   const bool order_is_negative = (v < 0);
+   const T vv((!order_is_negative) ? v : T(-v));
+
+   const bool order_is_integer = ((vv - floor(vv)) < half_epsilon);
+
+   // For negative integers, use reflection to positive integer order.
+   if(order_is_negative && order_is_integer)
+      return boost::math::detail::cyl_neumann_zero_imp(vv, m, pol);
+
+   // Check if the order is very close to a negative half-integer.
+   const T delta_half_integer(vv - (floor(vv) + 0.5F));
+
+   const bool order_is_negative_half_integer =
+      (order_is_negative && ((delta_half_integer > -half_epsilon) && (delta_half_integer < +half_epsilon)));
+
+   // The zero'th zero of Yv(x) for v < 0 is not defined
+   // unless the order is a negative integer.
+   if((m == 0) && (!order_is_negative_half_integer))
+   {
+      // For non-integer, negative order, requesting the zero'th zero raises a domain error.
+      return policies::raise_domain_error<T>(function, "Requested the %1%'th zero of Yv for negative, non-half-integer order, but the rank must be > 0 !", m, pol);
+   }
+
+   // For negative half-integers, use the corresponding
+   // spherical Bessel function of positive half-integer order.
+   if(order_is_negative_half_integer)
+      return boost::math::detail::cyl_bessel_j_zero_imp(vv, m, pol);
+
+   // Set up the initial guess for the upcoming root-finding.
+   // If the order is a negative integer, then use the corresponding
+   // positive integer for the order.
+   const T guess_root = boost::math::detail::bessel_zero::cyl_neumann_zero_detail::initial_guess<T, Policy>(v, m, pol);
+
+   // Select the maximum allowed iterations from the policy.
+   boost::uintmax_t number_of_iterations = policies::get_max_root_iterations<Policy>();
+
+   // Select the desired number of binary digits of precision.
+   // Account for the radix of number representations having non-two radix!
+   const int my_digits2 = policies::digits<T, Policy>();
+
+   const T delta_lo = ((guess_root > 0.2F) ? T(0.2) : T(guess_root / 2U));
+
+   // Perform the root-finding using Newton-Raphson iteration from Boost.Math.
+   const T yvm =
+      boost::math::tools::newton_raphson_iterate(
+         boost::math::detail::bessel_zero::cyl_neumann_zero_detail::function_object_yv_and_yv_prime<T, Policy>(v, pol),
+         guess_root,
+         T(guess_root - delta_lo),
+         T(guess_root + 0.2F),
+         my_digits2,
+         number_of_iterations);
+
+   if(number_of_iterations >= policies::get_max_root_iterations<Policy>())
+   {
+      policies::raise_evaluation_error<T>(function, "Unable to locate root in a reasonable time:"
+         "  Current best guess is %1%", yvm, Policy());
+   }
+
+   return yvm;
+}
+
+} // namespace detail
+
+template <class T1, class T2, class Policy>
+inline typename detail::bessel_traits<T1, T2, Policy>::result_type cyl_bessel_j(T1 v, T2 x, const Policy& /* pol */)
+{
+   BOOST_FPU_EXCEPTION_GUARD
+   typedef typename detail::bessel_traits<T1, T2, Policy>::result_type result_type;
+   typedef typename detail::bessel_traits<T1, T2, Policy>::optimisation_tag tag_type;
+   typedef typename policies::evaluation<result_type, Policy>::type value_type;
+   typedef typename policies::normalise<
+      Policy, 
+      policies::promote_float<false>, 
+      policies::promote_double<false>, 
+      policies::discrete_quantile<>,
+      policies::assert_undefined<> >::type forwarding_policy;
+   return policies::checked_narrowing_cast<result_type, Policy>(detail::cyl_bessel_j_imp<value_type>(v, static_cast<value_type>(x), tag_type(), forwarding_policy()), "boost::math::cyl_bessel_j<%1%>(%1%,%1%)");
+}
+
+template <class T1, class T2>
+inline typename detail::bessel_traits<T1, T2, policies::policy<> >::result_type cyl_bessel_j(T1 v, T2 x)
+{
+   return cyl_bessel_j(v, x, policies::policy<>());
+}
+
+template <class T, class Policy>
+inline typename detail::bessel_traits<T, T, Policy>::result_type sph_bessel(unsigned v, T x, const Policy& /* pol */)
+{
+   BOOST_FPU_EXCEPTION_GUARD
+   typedef typename detail::bessel_traits<T, T, Policy>::result_type result_type;
+   typedef typename policies::evaluation<result_type, Policy>::type value_type;
+   typedef typename policies::normalise<
+      Policy, 
+      policies::promote_float<false>, 
+      policies::promote_double<false>, 
+      policies::discrete_quantile<>,
+      policies::assert_undefined<> >::type forwarding_policy;
+   return policies::checked_narrowing_cast<result_type, Policy>(detail::sph_bessel_j_imp<value_type>(v, static_cast<value_type>(x), forwarding_policy()), "boost::math::sph_bessel<%1%>(%1%,%1%)");
+}
+
+template <class T>
+inline typename detail::bessel_traits<T, T, policies::policy<> >::result_type sph_bessel(unsigned v, T x)
+{
+   return sph_bessel(v, x, policies::policy<>());
+}
+
+template <class T1, class T2, class Policy>
+inline typename detail::bessel_traits<T1, T2, Policy>::result_type cyl_bessel_i(T1 v, T2 x, const Policy& /* pol */)
+{
+   BOOST_FPU_EXCEPTION_GUARD
+   typedef typename detail::bessel_traits<T1, T2, Policy>::result_type result_type;
+   typedef typename policies::evaluation<result_type, Policy>::type value_type;
+   typedef typename policies::normalise<
+      Policy, 
+      policies::promote_float<false>, 
+      policies::promote_double<false>, 
+      policies::discrete_quantile<>,
+      policies::assert_undefined<> >::type forwarding_policy;
+   return policies::checked_narrowing_cast<result_type, Policy>(detail::cyl_bessel_i_imp<value_type>(v, static_cast<value_type>(x), forwarding_policy()), "boost::math::cyl_bessel_i<%1%>(%1%,%1%)");
+}
+
+template <class T1, class T2>
+inline typename detail::bessel_traits<T1, T2, policies::policy<> >::result_type cyl_bessel_i(T1 v, T2 x)
+{
+   return cyl_bessel_i(v, x, policies::policy<>());
+}
+
+template <class T1, class T2, class Policy>
+inline typename detail::bessel_traits<T1, T2, Policy>::result_type cyl_bessel_k(T1 v, T2 x, const Policy& /* pol */)
+{
+   BOOST_FPU_EXCEPTION_GUARD
+   typedef typename detail::bessel_traits<T1, T2, Policy>::result_type result_type;
+   typedef typename detail::bessel_traits<T1, T2, Policy>::optimisation_tag tag_type;
+   typedef typename policies::evaluation<result_type, Policy>::type value_type;
+   typedef typename policies::normalise<
+      Policy, 
+      policies::promote_float<false>, 
+      policies::promote_double<false>, 
+      policies::discrete_quantile<>,
+      policies::assert_undefined<> >::type forwarding_policy;
+   return policies::checked_narrowing_cast<result_type, Policy>(detail::cyl_bessel_k_imp<value_type>(v, static_cast<value_type>(x), tag_type(), forwarding_policy()), "boost::math::cyl_bessel_k<%1%>(%1%,%1%)");
+}
+
+template <class T1, class T2>
+inline typename detail::bessel_traits<T1, T2, policies::policy<> >::result_type cyl_bessel_k(T1 v, T2 x)
+{
+   return cyl_bessel_k(v, x, policies::policy<>());
+}
+
+template <class T1, class T2, class Policy>
+inline typename detail::bessel_traits<T1, T2, Policy>::result_type cyl_neumann(T1 v, T2 x, const Policy& /* pol */)
+{
+   BOOST_FPU_EXCEPTION_GUARD
+   typedef typename detail::bessel_traits<T1, T2, Policy>::result_type result_type;
+   typedef typename detail::bessel_traits<T1, T2, Policy>::optimisation_tag tag_type;
+   typedef typename policies::evaluation<result_type, Policy>::type value_type;
+   typedef typename policies::normalise<
+      Policy, 
+      policies::promote_float<false>, 
+      policies::promote_double<false>, 
+      policies::discrete_quantile<>,
+      policies::assert_undefined<> >::type forwarding_policy;
+   return policies::checked_narrowing_cast<result_type, Policy>(detail::cyl_neumann_imp<value_type>(v, static_cast<value_type>(x), tag_type(), forwarding_policy()), "boost::math::cyl_neumann<%1%>(%1%,%1%)");
+}
+
+template <class T1, class T2>
+inline typename detail::bessel_traits<T1, T2, policies::policy<> >::result_type cyl_neumann(T1 v, T2 x)
+{
+   return cyl_neumann(v, x, policies::policy<>());
+}
+
+template <class T, class Policy>
+inline typename detail::bessel_traits<T, T, Policy>::result_type sph_neumann(unsigned v, T x, const Policy& /* pol */)
+{
+   BOOST_FPU_EXCEPTION_GUARD
+   typedef typename detail::bessel_traits<T, T, Policy>::result_type result_type;
+   typedef typename policies::evaluation<result_type, Policy>::type value_type;
+   typedef typename policies::normalise<
+      Policy, 
+      policies::promote_float<false>, 
+      policies::promote_double<false>, 
+      policies::discrete_quantile<>,
+      policies::assert_undefined<> >::type forwarding_policy;
+   return policies::checked_narrowing_cast<result_type, Policy>(detail::sph_neumann_imp<value_type>(v, static_cast<value_type>(x), forwarding_policy()), "boost::math::sph_neumann<%1%>(%1%,%1%)");
+}
+
+template <class T>
+inline typename detail::bessel_traits<T, T, policies::policy<> >::result_type sph_neumann(unsigned v, T x)
+{
+   return sph_neumann(v, x, policies::policy<>());
+}
+
+template <class T, class Policy>
+inline typename detail::bessel_traits<T, T, Policy>::result_type cyl_bessel_j_zero(T v, int m, const Policy& /* pol */)
+{
+   BOOST_FPU_EXCEPTION_GUARD
+   typedef typename detail::bessel_traits<T, T, Policy>::result_type result_type;
+   typedef typename policies::evaluation<result_type, Policy>::type value_type;
+   typedef typename policies::normalise<
+      Policy, 
+      policies::promote_float<false>, 
+      policies::promote_double<false>, 
+      policies::discrete_quantile<>,
+      policies::assert_undefined<> >::type forwarding_policy;
+   BOOST_STATIC_ASSERT_MSG(false == std::numeric_limits<value_type>::is_integer, "Order must be a floating-point type.");
+   return policies::checked_narrowing_cast<result_type, Policy>(detail::cyl_bessel_j_zero_imp<value_type>(v, m, forwarding_policy()), "boost::math::cyl_bessel_j_zero<%1%>(%1%,%1%)");
+}
+
+template <class T>
+inline typename detail::bessel_traits<T, T, policies::policy<> >::result_type cyl_bessel_j_zero(T v, int m)
+{
+   BOOST_STATIC_ASSERT_MSG(false == std::numeric_limits<T>::is_integer, "Order must be a floating-point type.");
+   return cyl_bessel_j_zero<T, policies::policy<> >(v, m, policies::policy<>());
+}
+
+template <class T, class OutputIterator, class Policy>
+inline OutputIterator cyl_bessel_j_zero(T v,
+                              int start_index,
+                              unsigned number_of_zeros,
+                              OutputIterator out_it,
+                              const Policy& pol)
+{
+   BOOST_STATIC_ASSERT_MSG(false == std::numeric_limits<T>::is_integer, "Order must be a floating-point type.");
+   for(unsigned i = 0; i < number_of_zeros; ++i)
+   {
+      *out_it = boost::math::cyl_bessel_j_zero(v, start_index + i, pol);
+      ++out_it;
+   }
+   return out_it;
+}
+
+template <class T, class OutputIterator>
+inline OutputIterator cyl_bessel_j_zero(T v,
+                              int start_index,
+                              unsigned number_of_zeros,
+                              OutputIterator out_it)
+{
+   return cyl_bessel_j_zero(v, start_index, number_of_zeros, out_it, policies::policy<>());
+}
+
+template <class T, class Policy>
+inline typename detail::bessel_traits<T, T, Policy>::result_type cyl_neumann_zero(T v, int m, const Policy& /* pol */)
+{
+   BOOST_FPU_EXCEPTION_GUARD
+   typedef typename detail::bessel_traits<T, T, Policy>::result_type result_type;
+   typedef typename policies::evaluation<result_type, Policy>::type value_type;
+   typedef typename policies::normalise<
+      Policy, 
+      policies::promote_float<false>, 
+      policies::promote_double<false>, 
+      policies::discrete_quantile<>,
+      policies::assert_undefined<> >::type forwarding_policy;
+   BOOST_STATIC_ASSERT_MSG(false == std::numeric_limits<value_type>::is_integer, "Order must be a floating-point type.");
+   return policies::checked_narrowing_cast<result_type, Policy>(detail::cyl_neumann_zero_imp<value_type>(v, m, forwarding_policy()), "boost::math::cyl_neumann_zero<%1%>(%1%,%1%)");
+}
+
+template <class T>
+inline typename detail::bessel_traits<T, T, policies::policy<> >::result_type cyl_neumann_zero(T v, int m)
+{
+   BOOST_STATIC_ASSERT_MSG(false == std::numeric_limits<T>::is_integer, "Order must be a floating-point type.");
+   return cyl_neumann_zero<T, policies::policy<> >(v, m, policies::policy<>());
+}
+
+template <class T, class OutputIterator, class Policy>
+inline OutputIterator cyl_neumann_zero(T v,
+                             int start_index,
+                             unsigned number_of_zeros,
+                             OutputIterator out_it,
+                             const Policy& pol)
+{
+   BOOST_STATIC_ASSERT_MSG(false == std::numeric_limits<T>::is_integer, "Order must be a floating-point type.");
+   for(unsigned i = 0; i < number_of_zeros; ++i)
+   {
+      *out_it = boost::math::cyl_neumann_zero(v, start_index + i, pol);
+      ++out_it;
+   }
+   return out_it;
+}
+
+template <class T, class OutputIterator>
+inline OutputIterator cyl_neumann_zero(T v,
+                             int start_index,
+                             unsigned number_of_zeros,
+                             OutputIterator out_it)
+{
+   return cyl_neumann_zero(v, start_index, number_of_zeros, out_it, policies::policy<>());
+}
+
+} // namespace math
+} // namespace boost
+
+#endif // BOOST_MATH_BESSEL_HPP
+
+