+// Copyright John Maddock 2012.
+// 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_AIRY_HPP
+#define BOOST_MATH_AIRY_HPP
+
+#include <boost/math/special_functions/bessel.hpp>
+#include <boost/math/special_functions/cbrt.hpp>
+#include <boost/math/special_functions/detail/airy_ai_bi_zero.hpp>
+#include <boost/math/tools/roots.hpp>
+
+namespace boost{ namespace math{
+
+namespace detail{
+
+template <class T, class Policy>
+T airy_ai_imp(T x, const Policy& pol)
+{
+ BOOST_MATH_STD_USING
+
+ if(x < 0)
+ {
+ T p = (-x * sqrt(-x) * 2) / 3;
+ T v = T(1) / 3;
+ T j1 = boost::math::cyl_bessel_j(v, p, pol);
+ T j2 = boost::math::cyl_bessel_j(-v, p, pol);
+ T ai = sqrt(-x) * (j1 + j2) / 3;
+ //T bi = sqrt(-x / 3) * (j2 - j1);
+ return ai;
+ }
+ else if(fabs(x * x * x) / 6 < tools::epsilon<T>())
+ {
+ T tg = boost::math::tgamma(constants::twothirds<T>(), pol);
+ T ai = 1 / (pow(T(3), constants::twothirds<T>()) * tg);
+ //T bi = 1 / (sqrt(boost::math::cbrt(T(3))) * tg);
+ return ai;
+ }
+ else
+ {
+ T p = 2 * x * sqrt(x) / 3;
+ T v = T(1) / 3;
+ //T j1 = boost::math::cyl_bessel_i(-v, p, pol);
+ //T j2 = boost::math::cyl_bessel_i(v, p, pol);
+ //
+ // Note that although we can calculate ai from j1 and j2, the accuracy is horrible
+ // as we're subtracting two very large values, so use the Bessel K relation instead:
+ //
+ T ai = cyl_bessel_k(v, p, pol) * sqrt(x / 3) / boost::math::constants::pi<T>(); //sqrt(x) * (j1 - j2) / 3;
+ //T bi = sqrt(x / 3) * (j1 + j2);
+ return ai;
+ }
+}
+
+template <class T, class Policy>
+T airy_bi_imp(T x, const Policy& pol)
+{
+ BOOST_MATH_STD_USING
+
+ if(x < 0)
+ {
+ T p = (-x * sqrt(-x) * 2) / 3;
+ T v = T(1) / 3;
+ T j1 = boost::math::cyl_bessel_j(v, p, pol);
+ T j2 = boost::math::cyl_bessel_j(-v, p, pol);
+ //T ai = sqrt(-x) * (j1 + j2) / 3;
+ T bi = sqrt(-x / 3) * (j2 - j1);
+ return bi;
+ }
+ else if(fabs(x * x * x) / 6 < tools::epsilon<T>())
+ {
+ T tg = boost::math::tgamma(constants::twothirds<T>(), pol);
+ //T ai = 1 / (pow(T(3), constants::twothirds<T>()) * tg);
+ T bi = 1 / (sqrt(boost::math::cbrt(T(3))) * tg);
+ return bi;
+ }
+ else
+ {
+ T p = 2 * x * sqrt(x) / 3;
+ T v = T(1) / 3;
+ T j1 = boost::math::cyl_bessel_i(-v, p, pol);
+ T j2 = boost::math::cyl_bessel_i(v, p, pol);
+ T bi = sqrt(x / 3) * (j1 + j2);
+ return bi;
+ }
+}
+
+template <class T, class Policy>
+T airy_ai_prime_imp(T x, const Policy& pol)
+{
+ BOOST_MATH_STD_USING
+
+ if(x < 0)
+ {
+ T p = (-x * sqrt(-x) * 2) / 3;
+ T v = T(2) / 3;
+ T j1 = boost::math::cyl_bessel_j(v, p, pol);
+ T j2 = boost::math::cyl_bessel_j(-v, p, pol);
+ T aip = -x * (j1 - j2) / 3;
+ return aip;
+ }
+ else if(fabs(x * x) / 2 < tools::epsilon<T>())
+ {
+ T tg = boost::math::tgamma(constants::third<T>(), pol);
+ T aip = 1 / (boost::math::cbrt(T(3)) * tg);
+ return -aip;
+ }
+ else
+ {
+ T p = 2 * x * sqrt(x) / 3;
+ T v = T(2) / 3;
+ //T j1 = boost::math::cyl_bessel_i(-v, p, pol);
+ //T j2 = boost::math::cyl_bessel_i(v, p, pol);
+ //
+ // Note that although we can calculate ai from j1 and j2, the accuracy is horrible
+ // as we're subtracting two very large values, so use the Bessel K relation instead:
+ //
+ T aip = -cyl_bessel_k(v, p, pol) * x / (boost::math::constants::root_three<T>() * boost::math::constants::pi<T>());
+ return aip;
+ }
+}
+
+template <class T, class Policy>
+T airy_bi_prime_imp(T x, const Policy& pol)
+{
+ BOOST_MATH_STD_USING
+
+ if(x < 0)
+ {
+ T p = (-x * sqrt(-x) * 2) / 3;
+ T v = T(2) / 3;
+ T j1 = boost::math::cyl_bessel_j(v, p, pol);
+ T j2 = boost::math::cyl_bessel_j(-v, p, pol);
+ T aip = -x * (j1 + j2) / constants::root_three<T>();
+ return aip;
+ }
+ else if(fabs(x * x) / 2 < tools::epsilon<T>())
+ {
+ T tg = boost::math::tgamma(constants::third<T>(), pol);
+ T bip = sqrt(boost::math::cbrt(T(3))) / tg;
+ return bip;
+ }
+ else
+ {
+ T p = 2 * x * sqrt(x) / 3;
+ T v = T(2) / 3;
+ T j1 = boost::math::cyl_bessel_i(-v, p, pol);
+ T j2 = boost::math::cyl_bessel_i(v, p, pol);
+ T aip = x * (j1 + j2) / boost::math::constants::root_three<T>();
+ return aip;
+ }
+}
+
+template <class T, class Policy>
+T airy_ai_zero_imp(unsigned m, const Policy& pol)
+{
+ BOOST_MATH_STD_USING // ADL of std names, needed for log, sqrt.
+
+ // Handle case when the zero'th zero is requested.
+ if(m == 0U)
+ {
+ return policies::raise_domain_error<T>("boost::math::airy_ai_zero<%1%>(%1%,%1%)",
+ "The requested rank of the zero is %1%, but must be 1 or more !", static_cast<T>(m), pol);
+ }
+
+ // Set up the initial guess for the upcoming root-finding.
+ const T guess_root = boost::math::detail::airy_zero::airy_ai_zero_detail::initial_guess<T>(m);
+
+ // Select the maximum allowed iterations, being at least 24.
+ boost::uintmax_t number_of_iterations = (std::max)(24, int(std::numeric_limits<T>::digits10));
+
+ // Select the desired number of binary digits of precision.
+ // Account for the radix of number representations having non-two radix!
+ const int my_digits2 = int(float(std::numeric_limits<T>::digits)
+ * ( log(float(std::numeric_limits<T>::radix))
+ / log(2.0F)));
+
+ // Use a dynamic tolerance because the roots get closer the higher m gets.
+ T tolerance;
+
+ if (m <= 10U) { tolerance = T(0.3F); }
+ else if(m <= 100U) { tolerance = T(0.1F); }
+ else if(m <= 1000U) { tolerance = T(0.05F); }
+ else { tolerance = T(1) / sqrt(T(m)); }
+
+ // Perform the root-finding using Newton-Raphson iteration from Boost.Math.
+ const T am =
+ boost::math::tools::newton_raphson_iterate(
+ boost::math::detail::airy_zero::airy_ai_zero_detail::function_object_ai_and_ai_prime<T, Policy>(pol),
+ guess_root,
+ T(guess_root - tolerance),
+ T(guess_root + tolerance),
+ my_digits2,
+ number_of_iterations);
+
+ static_cast<void>(number_of_iterations);
+
+ return am;
+}
+
+template <class T, class Policy>
+T airy_bi_zero_imp(unsigned m, const Policy& pol)
+{
+ BOOST_MATH_STD_USING // ADL of std names, needed for log, sqrt.
+
+ // Handle case when the zero'th zero is requested.
+ if(m == 0U)
+ {
+ return policies::raise_domain_error<T>("boost::math::airy_bi_zero<%1%>(%1%,%1%)",
+ "The requested rank of the zero is %1%, but must be 1 or more !", static_cast<T>(m), pol);
+ }
+ // Set up the initial guess for the upcoming root-finding.
+ const T guess_root = boost::math::detail::airy_zero::airy_bi_zero_detail::initial_guess<T>(m);
+
+ // Select the maximum allowed iterations, being at least 24.
+ boost::uintmax_t number_of_iterations = (std::max)(24, int(std::numeric_limits<T>::digits10));
+
+ // Select the desired number of binary digits of precision.
+ // Account for the radix of number representations having non-two radix!
+ const int my_digits2 = int(float(std::numeric_limits<T>::digits)
+ * ( log(float(std::numeric_limits<T>::radix))
+ / log(2.0F)));
+
+ // Use a dynamic tolerance because the roots get closer the higher m gets.
+ T tolerance;
+
+ if (m <= 10U) { tolerance = T(0.3F); }
+ else if(m <= 100U) { tolerance = T(0.1F); }
+ else if(m <= 1000U) { tolerance = T(0.05F); }
+ else { tolerance = T(1) / sqrt(T(m)); }
+
+ // Perform the root-finding using Newton-Raphson iteration from Boost.Math.
+ const T bm =
+ boost::math::tools::newton_raphson_iterate(
+ boost::math::detail::airy_zero::airy_bi_zero_detail::function_object_bi_and_bi_prime<T, Policy>(pol),
+ guess_root,
+ T(guess_root - tolerance),
+ T(guess_root + tolerance),
+ my_digits2,
+ number_of_iterations);
+
+ static_cast<void>(number_of_iterations);
+
+ return bm;
+}
+
+} // namespace detail
+
+template <class T, class Policy>
+inline typename tools::promote_args<T>::type airy_ai(T x, const Policy&)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ typedef typename tools::promote_args<T>::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::airy_ai_imp<value_type>(static_cast<value_type>(x), forwarding_policy()), "boost::math::airy<%1%>(%1%)");
+}
+
+template <class T>
+inline typename tools::promote_args<T>::type airy_ai(T x)
+{
+ return airy_ai(x, policies::policy<>());
+}
+
+template <class T, class Policy>
+inline typename tools::promote_args<T>::type airy_bi(T x, const Policy&)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ typedef typename tools::promote_args<T>::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::airy_bi_imp<value_type>(static_cast<value_type>(x), forwarding_policy()), "boost::math::airy<%1%>(%1%)");
+}
+
+template <class T>
+inline typename tools::promote_args<T>::type airy_bi(T x)
+{
+ return airy_bi(x, policies::policy<>());
+}
+
+template <class T, class Policy>
+inline typename tools::promote_args<T>::type airy_ai_prime(T x, const Policy&)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ typedef typename tools::promote_args<T>::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::airy_ai_prime_imp<value_type>(static_cast<value_type>(x), forwarding_policy()), "boost::math::airy<%1%>(%1%)");
+}
+
+template <class T>
+inline typename tools::promote_args<T>::type airy_ai_prime(T x)
+{
+ return airy_ai_prime(x, policies::policy<>());
+}
+
+template <class T, class Policy>
+inline typename tools::promote_args<T>::type airy_bi_prime(T x, const Policy&)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ typedef typename tools::promote_args<T>::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::airy_bi_prime_imp<value_type>(static_cast<value_type>(x), forwarding_policy()), "boost::math::airy<%1%>(%1%)");
+}
+
+template <class T>
+inline typename tools::promote_args<T>::type airy_bi_prime(T x)
+{
+ return airy_bi_prime(x, policies::policy<>());
+}
+
+template <class T, class Policy>
+inline T airy_ai_zero(unsigned m, const Policy& /*pol*/)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ typedef typename policies::evaluation<T, 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<T>::is_integer, "Airy return type must be a floating-point type.");
+ return policies::checked_narrowing_cast<T, Policy>(detail::airy_ai_zero_imp<value_type>(m, forwarding_policy()), "boost::math::airy_ai_zero<%1%>(unsigned)");
+}
+
+template <class T>
+inline T airy_ai_zero(unsigned m)
+{
+ return airy_ai_zero<T>(m, policies::policy<>());
+}
+
+template <class T, class OutputIterator, class Policy>
+inline OutputIterator airy_ai_zero(
+ unsigned start_index,
+ unsigned number_of_zeros,
+ OutputIterator out_it,
+ const Policy& pol)
+{
+ typedef T result_type;
+ BOOST_STATIC_ASSERT_MSG(false == std::numeric_limits<result_type>::is_integer, "Airy return type must be a floating-point type.");
+
+ for(unsigned i = 0; i < number_of_zeros; ++i)
+ {
+ *out_it = boost::math::airy_ai_zero<result_type>(start_index + i, pol);
+ ++out_it;
+ }
+ return out_it;
+}
+
+template <class T, class OutputIterator>
+inline OutputIterator airy_ai_zero(
+ unsigned start_index,
+ unsigned number_of_zeros,
+ OutputIterator out_it)
+{
+ return airy_ai_zero<T>(start_index, number_of_zeros, out_it, policies::policy<>());
+}
+
+template <class T, class Policy>
+inline T airy_bi_zero(unsigned m, const Policy& /*pol*/)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ typedef typename policies::evaluation<T, 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<T>::is_integer, "Airy return type must be a floating-point type.");
+ return policies::checked_narrowing_cast<T, Policy>(detail::airy_bi_zero_imp<value_type>(m, forwarding_policy()), "boost::math::airy_bi_zero<%1%>(unsigned)");
+}
+
+template <typename T>
+inline T airy_bi_zero(unsigned m)
+{
+ return airy_bi_zero<T>(m, policies::policy<>());
+}
+
+template <class T, class OutputIterator, class Policy>
+inline OutputIterator airy_bi_zero(
+ unsigned start_index,
+ unsigned number_of_zeros,
+ OutputIterator out_it,
+ const Policy& pol)
+{
+ typedef T result_type;
+ BOOST_STATIC_ASSERT_MSG(false == std::numeric_limits<result_type>::is_integer, "Airy return type must be a floating-point type.");
+
+ for(unsigned i = 0; i < number_of_zeros; ++i)
+ {
+ *out_it = boost::math::airy_bi_zero<result_type>(start_index + i, pol);
+ ++out_it;
+ }
+ return out_it;
+}
+
+template <class T, class OutputIterator>
+inline OutputIterator airy_bi_zero(
+ unsigned start_index,
+ unsigned number_of_zeros,
+ OutputIterator out_it)
+{
+ return airy_bi_zero<T>(start_index, number_of_zeros, out_it, policies::policy<>());
+}
+
+}} // namespaces
+
+#endif // BOOST_MATH_AIRY_HPP
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