470 lines
16 KiB
Plaintext
470 lines
16 KiB
Plaintext
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// Copyright John Maddock 2012.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0.
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// (See accompanying file LICENSE_1_0.txt
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// or copy at http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_MATH_AIRY_HPP
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#define BOOST_MATH_AIRY_HPP
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#include <limits>
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#include <boost/math/special_functions/math_fwd.hpp>
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#include <boost/math/special_functions/bessel.hpp>
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#include <boost/math/special_functions/cbrt.hpp>
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#include <boost/math/special_functions/detail/airy_ai_bi_zero.hpp>
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#include <boost/math/tools/roots.hpp>
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namespace boost{ namespace math{
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namespace detail{
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template <class T, class Policy>
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T airy_ai_imp(T x, const Policy& pol)
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{
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BOOST_MATH_STD_USING
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if(x < 0)
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{
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T p = (-x * sqrt(-x) * 2) / 3;
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T v = T(1) / 3;
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T j1 = boost::math::cyl_bessel_j(v, p, pol);
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T j2 = boost::math::cyl_bessel_j(-v, p, pol);
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T ai = sqrt(-x) * (j1 + j2) / 3;
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//T bi = sqrt(-x / 3) * (j2 - j1);
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return ai;
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}
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else if(fabs(x * x * x) / 6 < tools::epsilon<T>())
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{
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T tg = boost::math::tgamma(constants::twothirds<T>(), pol);
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T ai = 1 / (pow(T(3), constants::twothirds<T>()) * tg);
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//T bi = 1 / (sqrt(boost::math::cbrt(T(3))) * tg);
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return ai;
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}
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else
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{
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T p = 2 * x * sqrt(x) / 3;
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T v = T(1) / 3;
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//T j1 = boost::math::cyl_bessel_i(-v, p, pol);
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//T j2 = boost::math::cyl_bessel_i(v, p, pol);
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//
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// Note that although we can calculate ai from j1 and j2, the accuracy is horrible
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// as we're subtracting two very large values, so use the Bessel K relation instead:
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//
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T ai = cyl_bessel_k(v, p, pol) * sqrt(x / 3) / boost::math::constants::pi<T>(); //sqrt(x) * (j1 - j2) / 3;
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//T bi = sqrt(x / 3) * (j1 + j2);
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return ai;
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}
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}
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template <class T, class Policy>
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T airy_bi_imp(T x, const Policy& pol)
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{
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BOOST_MATH_STD_USING
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if(x < 0)
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{
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T p = (-x * sqrt(-x) * 2) / 3;
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T v = T(1) / 3;
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T j1 = boost::math::cyl_bessel_j(v, p, pol);
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T j2 = boost::math::cyl_bessel_j(-v, p, pol);
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//T ai = sqrt(-x) * (j1 + j2) / 3;
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T bi = sqrt(-x / 3) * (j2 - j1);
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return bi;
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}
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else if(fabs(x * x * x) / 6 < tools::epsilon<T>())
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{
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T tg = boost::math::tgamma(constants::twothirds<T>(), pol);
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//T ai = 1 / (pow(T(3), constants::twothirds<T>()) * tg);
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T bi = 1 / (sqrt(boost::math::cbrt(T(3))) * tg);
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return bi;
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}
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else
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{
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T p = 2 * x * sqrt(x) / 3;
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T v = T(1) / 3;
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T j1 = boost::math::cyl_bessel_i(-v, p, pol);
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T j2 = boost::math::cyl_bessel_i(v, p, pol);
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T bi = sqrt(x / 3) * (j1 + j2);
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return bi;
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}
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}
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template <class T, class Policy>
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T airy_ai_prime_imp(T x, const Policy& pol)
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{
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BOOST_MATH_STD_USING
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if(x < 0)
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{
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T p = (-x * sqrt(-x) * 2) / 3;
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T v = T(2) / 3;
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T j1 = boost::math::cyl_bessel_j(v, p, pol);
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T j2 = boost::math::cyl_bessel_j(-v, p, pol);
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T aip = -x * (j1 - j2) / 3;
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return aip;
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}
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else if(fabs(x * x) / 2 < tools::epsilon<T>())
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{
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T tg = boost::math::tgamma(constants::third<T>(), pol);
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T aip = 1 / (boost::math::cbrt(T(3)) * tg);
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return -aip;
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}
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else
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{
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T p = 2 * x * sqrt(x) / 3;
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T v = T(2) / 3;
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//T j1 = boost::math::cyl_bessel_i(-v, p, pol);
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//T j2 = boost::math::cyl_bessel_i(v, p, pol);
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//
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// Note that although we can calculate ai from j1 and j2, the accuracy is horrible
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// as we're subtracting two very large values, so use the Bessel K relation instead:
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//
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T aip = -cyl_bessel_k(v, p, pol) * x / (boost::math::constants::root_three<T>() * boost::math::constants::pi<T>());
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return aip;
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}
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}
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template <class T, class Policy>
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T airy_bi_prime_imp(T x, const Policy& pol)
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{
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BOOST_MATH_STD_USING
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if(x < 0)
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{
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T p = (-x * sqrt(-x) * 2) / 3;
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T v = T(2) / 3;
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T j1 = boost::math::cyl_bessel_j(v, p, pol);
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T j2 = boost::math::cyl_bessel_j(-v, p, pol);
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T aip = -x * (j1 + j2) / constants::root_three<T>();
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return aip;
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}
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else if(fabs(x * x) / 2 < tools::epsilon<T>())
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{
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T tg = boost::math::tgamma(constants::third<T>(), pol);
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T bip = sqrt(boost::math::cbrt(T(3))) / tg;
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return bip;
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}
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else
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{
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T p = 2 * x * sqrt(x) / 3;
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T v = T(2) / 3;
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T j1 = boost::math::cyl_bessel_i(-v, p, pol);
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T j2 = boost::math::cyl_bessel_i(v, p, pol);
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T aip = x * (j1 + j2) / boost::math::constants::root_three<T>();
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return aip;
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}
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}
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template <class T, class Policy>
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T airy_ai_zero_imp(int m, const Policy& pol)
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{
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BOOST_MATH_STD_USING // ADL of std names, needed for log, sqrt.
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// Handle cases when a negative zero (negative rank) is requested.
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if(m < 0)
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{
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return policies::raise_domain_error<T>("boost::math::airy_ai_zero<%1%>(%1%, int)",
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"Requested the %1%'th zero, but the rank must be 1 or more !", static_cast<T>(m), pol);
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}
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// Handle case when the zero'th zero is requested.
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if(m == 0U)
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{
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return policies::raise_domain_error<T>("boost::math::airy_ai_zero<%1%>(%1%,%1%)",
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"The requested rank of the zero is %1%, but must be 1 or more !", static_cast<T>(m), pol);
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}
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// Set up the initial guess for the upcoming root-finding.
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const T guess_root = boost::math::detail::airy_zero::airy_ai_zero_detail::initial_guess<T>(m);
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// Select the maximum allowed iterations based on the number
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// of decimal digits in the numeric type T, being at least 12.
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const int my_digits10 = static_cast<int>(static_cast<float>(policies::digits<T, Policy>() * 0.301F));
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const boost::uintmax_t iterations_allowed = static_cast<boost::uintmax_t>((std::max)(12, my_digits10 * 2));
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boost::uintmax_t iterations_used = iterations_allowed;
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// Use a dynamic tolerance because the roots get closer the higher m gets.
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T tolerance;
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if (m <= 10) { tolerance = T(0.3F); }
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else if(m <= 100) { tolerance = T(0.1F); }
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else if(m <= 1000) { tolerance = T(0.05F); }
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else { tolerance = T(1) / sqrt(T(m)); }
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// Perform the root-finding using Newton-Raphson iteration from Boost.Math.
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const T am =
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boost::math::tools::newton_raphson_iterate(
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boost::math::detail::airy_zero::airy_ai_zero_detail::function_object_ai_and_ai_prime<T, Policy>(pol),
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guess_root,
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T(guess_root - tolerance),
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T(guess_root + tolerance),
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policies::digits<T, Policy>(),
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iterations_used);
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static_cast<void>(iterations_used);
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return am;
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}
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template <class T, class Policy>
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T airy_bi_zero_imp(int m, const Policy& pol)
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{
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BOOST_MATH_STD_USING // ADL of std names, needed for log, sqrt.
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// Handle cases when a negative zero (negative rank) is requested.
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if(m < 0)
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{
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return policies::raise_domain_error<T>("boost::math::airy_bi_zero<%1%>(%1%, int)",
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"Requested the %1%'th zero, but the rank must 1 or more !", static_cast<T>(m), pol);
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}
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// Handle case when the zero'th zero is requested.
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if(m == 0U)
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{
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return policies::raise_domain_error<T>("boost::math::airy_bi_zero<%1%>(%1%,%1%)",
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"The requested rank of the zero is %1%, but must be 1 or more !", static_cast<T>(m), pol);
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}
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// Set up the initial guess for the upcoming root-finding.
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const T guess_root = boost::math::detail::airy_zero::airy_bi_zero_detail::initial_guess<T>(m);
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// Select the maximum allowed iterations based on the number
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// of decimal digits in the numeric type T, being at least 12.
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const int my_digits10 = static_cast<int>(static_cast<float>(policies::digits<T, Policy>() * 0.301F));
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const boost::uintmax_t iterations_allowed = static_cast<boost::uintmax_t>((std::max)(12, my_digits10 * 2));
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boost::uintmax_t iterations_used = iterations_allowed;
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// Use a dynamic tolerance because the roots get closer the higher m gets.
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T tolerance;
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if (m <= 10) { tolerance = T(0.3F); }
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else if(m <= 100) { tolerance = T(0.1F); }
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else if(m <= 1000) { tolerance = T(0.05F); }
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else { tolerance = T(1) / sqrt(T(m)); }
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// Perform the root-finding using Newton-Raphson iteration from Boost.Math.
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const T bm =
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boost::math::tools::newton_raphson_iterate(
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boost::math::detail::airy_zero::airy_bi_zero_detail::function_object_bi_and_bi_prime<T, Policy>(pol),
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guess_root,
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T(guess_root - tolerance),
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T(guess_root + tolerance),
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policies::digits<T, Policy>(),
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iterations_used);
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static_cast<void>(iterations_used);
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return bm;
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}
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} // namespace detail
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template <class T, class Policy>
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inline typename tools::promote_args<T>::type airy_ai(T x, const Policy&)
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{
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BOOST_FPU_EXCEPTION_GUARD
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typedef typename tools::promote_args<T>::type result_type;
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typedef typename policies::evaluation<result_type, Policy>::type value_type;
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typedef typename policies::normalise<
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Policy,
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policies::promote_float<false>,
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policies::promote_double<false>,
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policies::discrete_quantile<>,
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policies::assert_undefined<> >::type forwarding_policy;
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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%)");
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}
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template <class T>
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inline typename tools::promote_args<T>::type airy_ai(T x)
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{
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return airy_ai(x, policies::policy<>());
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}
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template <class T, class Policy>
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inline typename tools::promote_args<T>::type airy_bi(T x, const Policy&)
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{
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BOOST_FPU_EXCEPTION_GUARD
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typedef typename tools::promote_args<T>::type result_type;
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typedef typename policies::evaluation<result_type, Policy>::type value_type;
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typedef typename policies::normalise<
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Policy,
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policies::promote_float<false>,
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policies::promote_double<false>,
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policies::discrete_quantile<>,
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policies::assert_undefined<> >::type forwarding_policy;
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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%)");
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}
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template <class T>
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inline typename tools::promote_args<T>::type airy_bi(T x)
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{
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return airy_bi(x, policies::policy<>());
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}
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template <class T, class Policy>
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inline typename tools::promote_args<T>::type airy_ai_prime(T x, const Policy&)
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{
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BOOST_FPU_EXCEPTION_GUARD
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typedef typename tools::promote_args<T>::type result_type;
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typedef typename policies::evaluation<result_type, Policy>::type value_type;
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typedef typename policies::normalise<
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Policy,
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policies::promote_float<false>,
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policies::promote_double<false>,
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policies::discrete_quantile<>,
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policies::assert_undefined<> >::type forwarding_policy;
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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%)");
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}
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template <class T>
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inline typename tools::promote_args<T>::type airy_ai_prime(T x)
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{
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return airy_ai_prime(x, policies::policy<>());
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}
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template <class T, class Policy>
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inline typename tools::promote_args<T>::type airy_bi_prime(T x, const Policy&)
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{
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BOOST_FPU_EXCEPTION_GUARD
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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
|