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usr/include/boost/math/special_functions/airy.hpp 0000644 00000037647 15126373232 0016271 0 ustar 00 // 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), 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 <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), 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<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), 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<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 !", static_cast<T>(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, 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<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 !", static_cast<T>(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, 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<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 usr/include/boost/geometry/srs/projections/proj/airy.hpp 0000644 00000021660 15126723227 0017572 0 ustar 00 // Boost.Geometry - gis-projections (based on PROJ4) // Copyright (c) 2008-2015 Barend Gehrels, Amsterdam, the Netherlands. // This file was modified by Oracle on 2017, 2018, 2019. // Modifications copyright (c) 2017-2019, Oracle and/or its affiliates. // Contributed and/or modified by Adam Wulkiewicz, on behalf of Oracle. // Use, modification and distribution is 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 file is converted from PROJ4, http://trac.osgeo.org/proj // PROJ4 is originally written by Gerald Evenden (then of the USGS) // PROJ4 is maintained by Frank Warmerdam // PROJ4 is converted to Boost.Geometry by Barend Gehrels // Last updated version of proj: 5.0.0 // Original copyright notice: // Purpose: Implementation of the airy (Airy) projection. // Author: Gerald Evenden (1995) // Thomas Knudsen (2016) - revise/add regression tests // Copyright (c) 1995, Gerald Evenden // Permission is hereby granted, free of charge, to any person obtaining a // copy of this software and associated documentation files (the "Software"), // to deal in the Software without restriction, including without limitation // the rights to use, copy, modify, merge, publish, distribute, sublicense, // and/or sell copies of the Software, and to permit persons to whom the // Software is furnished to do so, subject to the following conditions: // The above copyright notice and this permission notice shall be included // in all copies or substantial portions of the Software. // THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS // OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, // FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL // THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER // LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING // FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER // DEALINGS IN THE SOFTWARE. #ifndef BOOST_GEOMETRY_PROJECTIONS_AIRY_HPP #define BOOST_GEOMETRY_PROJECTIONS_AIRY_HPP #include <boost/geometry/srs/projections/impl/base_static.hpp> #include <boost/geometry/srs/projections/impl/base_dynamic.hpp> #include <boost/geometry/srs/projections/impl/factory_entry.hpp> #include <boost/geometry/srs/projections/impl/pj_param.hpp> #include <boost/geometry/srs/projections/impl/projects.hpp> #include <boost/geometry/util/math.hpp> namespace boost { namespace geometry { namespace projections { #ifndef DOXYGEN_NO_DETAIL namespace detail { namespace airy { static const double epsilon = 1.e-10; enum mode_type { n_pole = 0, s_pole = 1, equit = 2, obliq = 3 }; template <typename T> struct par_airy { T p_halfpi; T sinph0; T cosph0; T Cb; mode_type mode; bool no_cut; /* do not cut at hemisphere limit */ }; template <typename T, typename Parameters> struct base_airy_spheroid { par_airy<T> m_proj_parm; // FORWARD(s_forward) spheroid // Project coordinates from geographic (lon, lat) to cartesian (x, y) inline void fwd(Parameters const& , T const& lp_lon, T lp_lat, T& xy_x, T& xy_y) const { static const T half_pi = detail::half_pi<T>(); T sinlam, coslam, cosphi, sinphi, t, s, Krho, cosz; sinlam = sin(lp_lon); coslam = cos(lp_lon); switch (this->m_proj_parm.mode) { case equit: case obliq: sinphi = sin(lp_lat); cosphi = cos(lp_lat); cosz = cosphi * coslam; if (this->m_proj_parm.mode == obliq) cosz = this->m_proj_parm.sinph0 * sinphi + this->m_proj_parm.cosph0 * cosz; if (!this->m_proj_parm.no_cut && cosz < -epsilon) { BOOST_THROW_EXCEPTION( projection_exception(error_tolerance_condition) ); } if (fabs(s = 1. - cosz) > epsilon) { t = 0.5 * (1. + cosz); Krho = -log(t)/s - this->m_proj_parm.Cb / t; } else Krho = 0.5 - this->m_proj_parm.Cb; xy_x = Krho * cosphi * sinlam; if (this->m_proj_parm.mode == obliq) xy_y = Krho * (this->m_proj_parm.cosph0 * sinphi - this->m_proj_parm.sinph0 * cosphi * coslam); else xy_y = Krho * sinphi; break; case s_pole: case n_pole: lp_lat = fabs(this->m_proj_parm.p_halfpi - lp_lat); if (!this->m_proj_parm.no_cut && (lp_lat - epsilon) > half_pi) BOOST_THROW_EXCEPTION( projection_exception(error_tolerance_condition) ); if ((lp_lat *= 0.5) > epsilon) { t = tan(lp_lat); Krho = -2.*(log(cos(lp_lat)) / t + t * this->m_proj_parm.Cb); xy_x = Krho * sinlam; xy_y = Krho * coslam; if (this->m_proj_parm.mode == n_pole) xy_y = -xy_y; } else xy_x = xy_y = 0.; } } static inline std::string get_name() { return "airy_spheroid"; } }; // Airy template <typename Params, typename Parameters, typename T> inline void setup_airy(Params const& params, Parameters& par, par_airy<T>& proj_parm) { static const T half_pi = detail::half_pi<T>(); T beta; proj_parm.no_cut = pj_get_param_b<srs::spar::no_cut>(params, "no_cut", srs::dpar::no_cut); beta = 0.5 * (half_pi - pj_get_param_r<T, srs::spar::lat_b>(params, "lat_b", srs::dpar::lat_b)); if (fabs(beta) < epsilon) proj_parm.Cb = -0.5; else { proj_parm.Cb = 1./tan(beta); proj_parm.Cb *= proj_parm.Cb * log(cos(beta)); } if (fabs(fabs(par.phi0) - half_pi) < epsilon) if (par.phi0 < 0.) { proj_parm.p_halfpi = -half_pi; proj_parm.mode = s_pole; } else { proj_parm.p_halfpi = half_pi; proj_parm.mode = n_pole; } else { if (fabs(par.phi0) < epsilon) proj_parm.mode = equit; else { proj_parm.mode = obliq; proj_parm.sinph0 = sin(par.phi0); proj_parm.cosph0 = cos(par.phi0); } } par.es = 0.; } }} // namespace detail::airy #endif // doxygen /*! \brief Airy projection \ingroup projections \tparam Geographic latlong point type \tparam Cartesian xy point type \tparam Parameters parameter type \par Projection characteristics - Miscellaneous - Spheroid - no inverse \par Projection parameters - no_cut: Do not cut at hemisphere limit (boolean) - lat_b (degrees) \par Example \image html ex_airy.gif */ template <typename T, typename Parameters> struct airy_spheroid : public detail::airy::base_airy_spheroid<T, Parameters> { template <typename Params> inline airy_spheroid(Params const& params, Parameters & par) { detail::airy::setup_airy(params, par, this->m_proj_parm); } }; #ifndef DOXYGEN_NO_DETAIL namespace detail { // Static projection BOOST_GEOMETRY_PROJECTIONS_DETAIL_STATIC_PROJECTION_F(srs::spar::proj_airy, airy_spheroid) // Factory entry(s) BOOST_GEOMETRY_PROJECTIONS_DETAIL_FACTORY_ENTRY_F(airy_entry, airy_spheroid) BOOST_GEOMETRY_PROJECTIONS_DETAIL_FACTORY_INIT_BEGIN(airy_init) { BOOST_GEOMETRY_PROJECTIONS_DETAIL_FACTORY_INIT_ENTRY(airy, airy_entry) } } // namespace detail #endif // doxygen } // namespace projections }} // namespace boost::geometry #endif // BOOST_GEOMETRY_PROJECTIONS_AIRY_HPP
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