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Diffstat (limited to 'boost/math/special_functions/detail/airy_ai_bi_zero.hpp')
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diff --git a/boost/math/special_functions/detail/airy_ai_bi_zero.hpp b/boost/math/special_functions/detail/airy_ai_bi_zero.hpp new file mode 100644 index 0000000000..dbb7388dda --- /dev/null +++ b/boost/math/special_functions/detail/airy_ai_bi_zero.hpp @@ -0,0 +1,160 @@ +// 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 _AIRY_AI_BI_ZERO_2013_01_20_HPP_ + #define _AIRY_AI_BI_ZERO_2013_01_20_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> + T airy_ai_imp(T x, const Policy& pol); + template <class T, class Policy> + T airy_bi_imp(T x, const Policy& pol); + template <class T, class Policy> + T airy_ai_prime_imp(T x, const Policy& pol); + template <class T, class Policy> + T airy_bi_prime_imp(T x, const Policy& pol); + + namespace airy_zero + { + template<class T> + T equation_as_10_4_105(const T& z) + { + 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)); + 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> + T initial_guess(const int m) + { + 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); + break; + } + } + + return guess; + } + + template<class T, class Policy> + class function_object_ai_and_ai_prime + { + public: + function_object_ai_and_ai_prime(const Policy pol) : my_pol(pol) { } + + 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&); + }; + } // namespace airy_ai_zero_detail + + namespace airy_bi_zero_detail + { + template<class T> + T initial_guess(const int m) + { + 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); + break; + } + } + + return guess; + } + + template<class T, class Policy> + class function_object_bi_and_bi_prime + { + public: + function_object_bi_and_bi_prime(const Policy pol) : my_pol(pol) { } + + 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&); + }; + } // namespace airy_bi_zero_detail + } // namespace airy_zero + } // namespace detail + } // namespace math + } // namespaces boost + +#endif // _AIRY_AI_BI_ZERO_2013_01_20_HPP_ |