boost/math/special_functions/detail/airy_ai_bi_zero.hpp
// Copyright (c) 2013 Christopher Kormanyos // 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 work is based on an earlier work: // "Algorithm 910: A Portable C++ Multiple-Precision System for Special-Function Calculations", // in ACM TOMS, {VOL 37, ISSUE 4, (February 2011)} (C) ACM, 2011. http://doi.acm.org/10.1145/1916461.1916469 // // This header contains implementation details for estimating the zeros // of the Airy functions airy_ai and airy_bi on the negative real axis. // #ifndef BOOST_MATH_AIRY_AI_BI_ZERO_2013_01_20_HPP_ #define BOOST_MATH_AIRY_AI_BI_ZERO_2013_01_20_HPP_ #include <boost/math/tools/config.hpp> #include <boost/math/tools/tuple.hpp> #include <boost/math/constants/constants.hpp> #include <boost/math/special_functions/cbrt.hpp> namespace boost { namespace math { namespace detail { // Forward declarations of the needed Airy function implementations. template <class T, class Policy> BOOST_MATH_GPU_ENABLED T airy_ai_imp(T x, const Policy& pol); template <class T, class Policy> BOOST_MATH_GPU_ENABLED T airy_bi_imp(T x, const Policy& pol); template <class T, class Policy> BOOST_MATH_GPU_ENABLED T airy_ai_prime_imp(T x, const Policy& pol); template <class T, class Policy> BOOST_MATH_GPU_ENABLED T airy_bi_prime_imp(T x, const Policy& pol); namespace airy_zero { template<class T, class Policy> BOOST_MATH_GPU_ENABLED T equation_as_10_4_105(const T& z, const Policy& pol) { const T one_over_z (T(1) / z); const T one_over_z_squared(one_over_z * one_over_z); const T z_pow_third (boost::math::cbrt(z, pol)); const T z_pow_two_thirds(z_pow_third * z_pow_third); // Implement the top line of Eq. 10.4.105. const T fz(z_pow_two_thirds * ((((( + (T(162375596875.0) / 334430208UL) * one_over_z_squared - ( T(108056875.0) / 6967296UL)) * one_over_z_squared + ( T(77125UL) / 82944UL)) * one_over_z_squared - ( T(5U) / 36U)) * one_over_z_squared + ( T(5U) / 48U)) * one_over_z_squared + 1)); return fz; } namespace airy_ai_zero_detail { template<class T, class Policy> BOOST_MATH_GPU_ENABLED T initial_guess(const int m, const Policy& pol) { T guess; switch(m) { case 0: guess = T(0); break; case 1: guess = T(-2.33810741045976703849); break; case 2: guess = T(-4.08794944413097061664); break; case 3: guess = T(-5.52055982809555105913); break; case 4: guess = T(-6.78670809007175899878); break; case 5: guess = T(-7.94413358712085312314); break; case 6: guess = T(-9.02265085334098038016); break; case 7: guess = T(-10.0401743415580859306); break; case 8: guess = T(-11.0085243037332628932); break; case 9: guess = T(-11.9360155632362625170); break; case 10: guess = T(-12.8287767528657572004); break; default: const T t(((boost::math::constants::pi<T>() * 3) * ((T(m) * 4) - 1)) / 8); guess = -boost::math::detail::airy_zero::equation_as_10_4_105(t, pol); break; } return guess; } template<class T, class Policy> class function_object_ai_and_ai_prime { public: BOOST_MATH_GPU_ENABLED explicit function_object_ai_and_ai_prime(const Policy& pol) : my_pol(pol) { } #ifdef BOOST_MATH_ENABLE_CUDA # pragma nv_diag_suppress 20012 #endif BOOST_MATH_GPU_ENABLED function_object_ai_and_ai_prime(const function_object_ai_and_ai_prime&) = default; #ifdef BOOST_MATH_ENABLE_CUDA # pragma nv_diag_default 20012 #endif BOOST_MATH_GPU_ENABLED boost::math::tuple<T, T> operator()(const T& x) const { // Return a tuple containing both Ai(x) and Ai'(x). return boost::math::make_tuple( boost::math::detail::airy_ai_imp (x, my_pol), boost::math::detail::airy_ai_prime_imp(x, my_pol)); } private: const Policy& my_pol; const function_object_ai_and_ai_prime& operator=(const function_object_ai_and_ai_prime&) = delete; }; } // namespace airy_ai_zero_detail namespace airy_bi_zero_detail { template<class T, class Policy> BOOST_MATH_GPU_ENABLED T initial_guess(const int m, const Policy& pol) { T guess; switch(m) { case 0: guess = T(0); break; case 1: guess = T(-1.17371322270912792492); break; case 2: guess = T(-3.27109330283635271568); break; case 3: guess = T(-4.83073784166201593267); break; case 4: guess = T(-6.16985212831025125983); break; case 5: guess = T(-7.37676207936776371360); break; case 6: guess = T(-8.49194884650938801345); break; case 7: guess = T(-9.53819437934623888663); break; case 8: guess = T(-10.5299135067053579244); break; case 9: guess = T(-11.4769535512787794379); break; case 10: guess = T(-12.3864171385827387456); break; default: const T t(((boost::math::constants::pi<T>() * 3) * ((T(m) * 4) - 3)) / 8); guess = -boost::math::detail::airy_zero::equation_as_10_4_105(t, pol); break; } return guess; } template<class T, class Policy> class function_object_bi_and_bi_prime { public: BOOST_MATH_GPU_ENABLED explicit function_object_bi_and_bi_prime(const Policy& pol) : my_pol(pol) { } #ifdef BOOST_MATH_ENABLE_CUDA # pragma nv_diag_suppress 20012 #endif BOOST_MATH_GPU_ENABLED function_object_bi_and_bi_prime(const function_object_bi_and_bi_prime&) = default; #ifdef BOOST_MATH_ENABLE_CUDA # pragma nv_diag_default 20012 #endif BOOST_MATH_GPU_ENABLED boost::math::tuple<T, T> operator()(const T& x) const { // Return a tuple containing both Bi(x) and Bi'(x). return boost::math::make_tuple( boost::math::detail::airy_bi_imp (x, my_pol), boost::math::detail::airy_bi_prime_imp(x, my_pol)); } private: const Policy& my_pol; const function_object_bi_and_bi_prime& operator=(const function_object_bi_and_bi_prime&) = delete; }; } // namespace airy_bi_zero_detail } // namespace airy_zero } // namespace detail } // namespace math } // namespaces boost #endif // BOOST_MATH_AIRY_AI_BI_ZERO_2013_01_20_HPP_