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// Copyright John Maddock 2012. // Copyright Matt Borland 2024. // 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 #include #include #include #include #include #include #include #include #include #include namespace boost{ namespace math{ namespace detail{ template BOOST_MATH_GPU_ENABLED 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 tg = boost::math::tgamma(constants::twothirds(), pol); T ai = 1 / (pow(T(3), constants::twothirds()) * 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(); //sqrt(x) * (j1 - j2) / 3; //T bi = sqrt(x / 3) * (j1 + j2); return ai; } } template BOOST_MATH_GPU_ENABLED 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 tg = boost::math::tgamma(constants::twothirds(), pol); //T ai = 1 / (pow(T(3), constants::twothirds()) * tg); T bi = 1 / (sqrt(boost::math::cbrt(T(3), pol)) * 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 BOOST_MATH_GPU_ENABLED 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 tg = boost::math::tgamma(constants::third(), pol); T aip = 1 / (boost::math::cbrt(T(3), pol) * 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() * boost::math::constants::pi()); return aip; } } template BOOST_MATH_GPU_ENABLED 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(); return aip; } else if(fabs(x * x) / 2 < tools::epsilon()) { T tg = boost::math::tgamma(constants::third(), pol); T bip = sqrt(boost::math::cbrt(T(3), pol)) / 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(); return aip; } } template BOOST_MATH_GPU_ENABLED T airy_ai_zero_imp(int m, const Policy& pol) { BOOST_MATH_STD_USING // ADL of std names, needed for log, sqrt. // Handle cases when a negative zero (negative rank) is requested. if(m < 0) { return policies::raise_domain_error("boost::math::airy_ai_zero<%1%>(%1%, int)", "Requested the %1%'th zero, but the rank must be 1 or more !", static_cast(m), pol); } // Handle case when the zero'th zero is requested. if(m == 0U) { return policies::raise_domain_error("boost::math::airy_ai_zero<%1%>(%1%,%1%)", "The requested rank of the zero is %1%, but must be 1 or more !", static_cast(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(m, pol); // Select the maximum allowed iterations based on the number // of decimal digits in the numeric type T, being at least 12. const int my_digits10 = static_cast(static_cast(policies::digits() * 0.301F)); const std::uintmax_t iterations_allowed = static_cast(BOOST_MATH_GPU_SAFE_MAX(12, my_digits10 * 2)); std::uintmax_t iterations_used = iterations_allowed; // Use a dynamic tolerance because the roots get closer the higher m gets. T tolerance; // LCOV_EXCL_LINE if (m <= 10) { tolerance = T(0.3F); } else if(m <= 100) { tolerance = T(0.1F); } else if(m <= 1000) { 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(pol), guess_root, T(guess_root - tolerance), T(guess_root + tolerance), policies::digits(), iterations_used); static_cast(iterations_used); return am; } template BOOST_MATH_GPU_ENABLED T airy_bi_zero_imp(int m, const Policy& pol) { BOOST_MATH_STD_USING // ADL of std names, needed for log, sqrt. // Handle cases when a negative zero (negative rank) is requested. if(m < 0) { return policies::raise_domain_error("boost::math::airy_bi_zero<%1%>(%1%, int)", "Requested the %1%'th zero, but the rank must 1 or more !", static_cast(m), pol); } // Handle case when the zero'th zero is requested. if(m == 0U) { return policies::raise_domain_error("boost::math::airy_bi_zero<%1%>(%1%,%1%)", "The requested rank of the zero is %1%, but must be 1 or more !", static_cast(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(m, pol); // Select the maximum allowed iterations based on the number // of decimal digits in the numeric type T, being at least 12. const int my_digits10 = static_cast(static_cast(policies::digits() * 0.301F)); const std::uintmax_t iterations_allowed = static_cast(BOOST_MATH_GPU_SAFE_MAX(12, my_digits10 * 2)); std::uintmax_t iterations_used = iterations_allowed; // Use a dynamic tolerance because the roots get closer the higher m gets. T tolerance; // LCOV_EXCL_LINE if (m <= 10) { tolerance = T(0.3F); } else if(m <= 100) { tolerance = T(0.1F); } else if(m <= 1000) { 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(pol), guess_root, T(guess_root - tolerance), T(guess_root + tolerance), policies::digits(), iterations_used); static_cast(iterations_used); return bm; } } // namespace detail template BOOST_MATH_GPU_ENABLED inline typename tools::promote_args::type airy_ai(T x, const Policy&) { BOOST_FPU_EXCEPTION_GUARD typedef typename tools::promote_args::type result_type; typedef typename policies::evaluation::type value_type; typedef typename policies::normalise< Policy, policies::promote_float, policies::promote_double, policies::discrete_quantile<>, policies::assert_undefined<> >::type forwarding_policy; return policies::checked_narrowing_cast(detail::airy_ai_imp(static_cast(x), forwarding_policy()), "boost::math::airy<%1%>(%1%)"); } template BOOST_MATH_GPU_ENABLED inline typename tools::promote_args::type airy_ai(T x) { return airy_ai(x, policies::policy<>()); } template BOOST_MATH_GPU_ENABLED inline typename tools::promote_args::type airy_bi(T x, const Policy&) { BOOST_FPU_EXCEPTION_GUARD typedef typename tools::promote_args::type result_type; typedef typename policies::evaluation::type value_type; typedef typename policies::normalise< Policy, policies::promote_float, policies::promote_double, policies::discrete_quantile<>, policies::assert_undefined<> >::type forwarding_policy; return policies::checked_narrowing_cast(detail::airy_bi_imp(static_cast(x), forwarding_policy()), "boost::math::airy<%1%>(%1%)"); } template BOOST_MATH_GPU_ENABLED inline typename tools::promote_args::type airy_bi(T x) { return airy_bi(x, policies::policy<>()); } template BOOST_MATH_GPU_ENABLED inline typename tools::promote_args::type airy_ai_prime(T x, const Policy&) { BOOST_FPU_EXCEPTION_GUARD typedef typename tools::promote_args::type result_type; typedef typename policies::evaluation::type value_type; typedef typename policies::normalise< Policy, policies::promote_float, policies::promote_double, policies::discrete_quantile<>, policies::assert_undefined<> >::type forwarding_policy; return policies::checked_narrowing_cast(detail::airy_ai_prime_imp(static_cast(x), forwarding_policy()), "boost::math::airy<%1%>(%1%)"); } template BOOST_MATH_GPU_ENABLED inline typename tools::promote_args::type airy_ai_prime(T x) { return airy_ai_prime(x, policies::policy<>()); } template BOOST_MATH_GPU_ENABLED inline typename tools::promote_args::type airy_bi_prime(T x, const Policy&) { BOOST_FPU_EXCEPTION_GUARD typedef typename tools::promote_args::type result_type; typedef typename policies::evaluation::type value_type; typedef typename policies::normalise< Policy, policies::promote_float, policies::promote_double, policies::discrete_quantile<>, policies::assert_undefined<> >::type forwarding_policy; return policies::checked_narrowing_cast(detail::airy_bi_prime_imp(static_cast(x), forwarding_policy()), "boost::math::airy<%1%>(%1%)"); } template BOOST_MATH_GPU_ENABLED inline typename tools::promote_args::type airy_bi_prime(T x) { return airy_bi_prime(x, policies::policy<>()); } template BOOST_MATH_GPU_ENABLED inline T airy_ai_zero(int m, const Policy& /*pol*/) { BOOST_FPU_EXCEPTION_GUARD typedef typename policies::evaluation::type value_type; typedef typename policies::normalise< Policy, policies::promote_float, policies::promote_double, policies::discrete_quantile<>, policies::assert_undefined<> >::type forwarding_policy; static_assert( false == std::numeric_limits::is_specialized || ( true == std::numeric_limits::is_specialized && false == std::numeric_limits::is_integer), "Airy value type must be a floating-point type."); return policies::checked_narrowing_cast(detail::airy_ai_zero_imp(m, forwarding_policy()), "boost::math::airy_ai_zero<%1%>(unsigned)"); } template BOOST_MATH_GPU_ENABLED inline T airy_ai_zero(int m) { return airy_ai_zero(m, policies::policy<>()); } template BOOST_MATH_GPU_ENABLED inline OutputIterator airy_ai_zero( int start_index, unsigned number_of_zeros, OutputIterator out_it, const Policy& pol) { typedef T result_type; static_assert( false == std::numeric_limits::is_specialized || ( true == std::numeric_limits::is_specialized && false == std::numeric_limits::is_integer), "Airy value type must be a floating-point type."); for(unsigned i = 0; i < number_of_zeros; ++i) { *out_it = boost::math::airy_ai_zero(start_index + i, pol); ++out_it; } return out_it; } template BOOST_MATH_GPU_ENABLED inline OutputIterator airy_ai_zero( int start_index, unsigned number_of_zeros, OutputIterator out_it) { return airy_ai_zero(start_index, number_of_zeros, out_it, policies::policy<>()); } template BOOST_MATH_GPU_ENABLED inline T airy_bi_zero(int m, const Policy& /*pol*/) { BOOST_FPU_EXCEPTION_GUARD typedef typename policies::evaluation::type value_type; typedef typename policies::normalise< Policy, policies::promote_float, policies::promote_double, policies::discrete_quantile<>, policies::assert_undefined<> >::type forwarding_policy; static_assert( false == std::numeric_limits::is_specialized || ( true == std::numeric_limits::is_specialized && false == std::numeric_limits::is_integer), "Airy value type must be a floating-point type."); return policies::checked_narrowing_cast(detail::airy_bi_zero_imp(m, forwarding_policy()), "boost::math::airy_bi_zero<%1%>(unsigned)"); } template BOOST_MATH_GPU_ENABLED inline T airy_bi_zero(int m) { return airy_bi_zero(m, policies::policy<>()); } template BOOST_MATH_GPU_ENABLED inline OutputIterator airy_bi_zero( int start_index, unsigned number_of_zeros, OutputIterator out_it, const Policy& pol) { typedef T result_type; static_assert( false == std::numeric_limits::is_specialized || ( true == std::numeric_limits::is_specialized && false == std::numeric_limits::is_integer), "Airy value type must be a floating-point type."); for(unsigned i = 0; i < number_of_zeros; ++i) { *out_it = boost::math::airy_bi_zero(start_index + i, pol); ++out_it; } return out_it; } template BOOST_MATH_GPU_ENABLED inline OutputIterator airy_bi_zero( int start_index, unsigned number_of_zeros, OutputIterator out_it) { return airy_bi_zero(start_index, number_of_zeros, out_it, policies::policy<>()); } }} // namespaces #endif // BOOST_MATH_AIRY_HPP