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Diffstat (limited to 'boost/math/special_functions/airy.hpp')
-rw-r--r-- | boost/math/special_functions/airy.hpp | 469 |
1 files changed, 469 insertions, 0 deletions
diff --git a/boost/math/special_functions/airy.hpp b/boost/math/special_functions/airy.hpp new file mode 100644 index 0000000000..82167dc5f0 --- /dev/null +++ b/boost/math/special_functions/airy.hpp @@ -0,0 +1,469 @@ +// 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 <limits> +#include <boost/math/special_functions/math_fwd.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(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<T>("boost::math::airy_ai_zero<%1%>(%1%, int)", + "Requested the %1%'th zero, but the rank must be 1 or more !", m, pol); + } + + // 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 based on the number + // of decimal digits in the numeric type T, being at least 12. + const int my_digits10 = static_cast<int>(static_cast<float>(policies::digits<T, Policy>() * 0.301F)); + + const boost::uintmax_t iterations_allowed = static_cast<boost::uintmax_t>((std::max)(12, my_digits10 * 2)); + + boost::uintmax_t iterations_used = iterations_allowed; + + // Use a dynamic tolerance because the roots get closer the higher m gets. + T tolerance; + + 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<T, Policy>(pol), + guess_root, + T(guess_root - tolerance), + T(guess_root + tolerance), + policies::digits<T, Policy>(), + iterations_used); + + static_cast<void>(iterations_used); + + return am; +} + +template <class T, class Policy> +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<T>("boost::math::airy_bi_zero<%1%>(%1%, int)", + "Requested the %1%'th zero, but the rank must 1 or more !", m, pol); + } + + // 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 based on the number + // of decimal digits in the numeric type T, being at least 12. + const int my_digits10 = static_cast<int>(static_cast<float>(policies::digits<T, Policy>() * 0.301F)); + + const boost::uintmax_t iterations_allowed = static_cast<boost::uintmax_t>((std::max)(12, my_digits10 * 2)); + + boost::uintmax_t iterations_used = iterations_allowed; + + // Use a dynamic tolerance because the roots get closer the higher m gets. + T tolerance; + + 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<T, Policy>(pol), + guess_root, + T(guess_root - tolerance), + T(guess_root + tolerance), + policies::digits<T, Policy>(), + iterations_used); + + static_cast<void>(iterations_used); + + 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(int 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_specialized + || ( true == std::numeric_limits<T>::is_specialized + && false == std::numeric_limits<T>::is_integer), + "Airy value 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(int m) +{ + return airy_ai_zero<T>(m, policies::policy<>()); +} + +template <class T, class OutputIterator, class Policy> +inline OutputIterator airy_ai_zero( + int start_index, + unsigned number_of_zeros, + OutputIterator out_it, + const Policy& pol) +{ + typedef T result_type; + + BOOST_STATIC_ASSERT_MSG( false == std::numeric_limits<T>::is_specialized + || ( true == std::numeric_limits<T>::is_specialized + && false == std::numeric_limits<T>::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<result_type>(start_index + i, pol); + ++out_it; + } + return out_it; +} + +template <class T, class OutputIterator> +inline OutputIterator airy_ai_zero( + int 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(int 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_specialized + || ( true == std::numeric_limits<T>::is_specialized + && false == std::numeric_limits<T>::is_integer), + "Airy value 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(int m) +{ + return airy_bi_zero<T>(m, policies::policy<>()); +} + +template <class T, class OutputIterator, class Policy> +inline OutputIterator airy_bi_zero( + int start_index, + unsigned number_of_zeros, + OutputIterator out_it, + const Policy& pol) +{ + typedef T result_type; + + BOOST_STATIC_ASSERT_MSG( false == std::numeric_limits<T>::is_specialized + || ( true == std::numeric_limits<T>::is_specialized + && false == std::numeric_limits<T>::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<result_type>(start_index + i, pol); + ++out_it; + } + return out_it; +} + +template <class T, class OutputIterator> +inline OutputIterator airy_bi_zero( + int 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 |