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mariadb-columnstore-engine/utils/boost_idb/boost/math/special_functions/airy.hpp
david hill f6afc42dd0 the begginning
2016-01-06 14:08:59 -06:00

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// 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 <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(unsigned m, const Policy& pol)
{
BOOST_MATH_STD_USING // ADL of std names, needed for log, sqrt.
// 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, being at least 24.
boost::uintmax_t number_of_iterations = (std::max)(24, int(std::numeric_limits<T>::digits10));
// Select the desired number of binary digits of precision.
// Account for the radix of number representations having non-two radix!
const int my_digits2 = int(float(std::numeric_limits<T>::digits)
* ( log(float(std::numeric_limits<T>::radix))
/ log(2.0F)));
// Use a dynamic tolerance because the roots get closer the higher m gets.
T tolerance;
if (m <= 10U) { tolerance = T(0.3F); }
else if(m <= 100U) { tolerance = T(0.1F); }
else if(m <= 1000U) { 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),
my_digits2,
number_of_iterations);
static_cast<void>(number_of_iterations);
return am;
}
template <class T, class Policy>
T airy_bi_zero_imp(unsigned m, const Policy& pol)
{
BOOST_MATH_STD_USING // ADL of std names, needed for log, sqrt.
// 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, being at least 24.
boost::uintmax_t number_of_iterations = (std::max)(24, int(std::numeric_limits<T>::digits10));
// Select the desired number of binary digits of precision.
// Account for the radix of number representations having non-two radix!
const int my_digits2 = int(float(std::numeric_limits<T>::digits)
* ( log(float(std::numeric_limits<T>::radix))
/ log(2.0F)));
// Use a dynamic tolerance because the roots get closer the higher m gets.
T tolerance;
if (m <= 10U) { tolerance = T(0.3F); }
else if(m <= 100U) { tolerance = T(0.1F); }
else if(m <= 1000U) { 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),
my_digits2,
number_of_iterations);
static_cast<void>(number_of_iterations);
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(unsigned 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_integer, "Airy return 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(unsigned m)
{
return airy_ai_zero<T>(m, policies::policy<>());
}
template <class T, class OutputIterator, class Policy>
inline OutputIterator airy_ai_zero(
unsigned start_index,
unsigned number_of_zeros,
OutputIterator out_it,
const Policy& pol)
{
typedef T result_type;
BOOST_STATIC_ASSERT_MSG(false == std::numeric_limits<result_type>::is_integer, "Airy return 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(
unsigned 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(unsigned 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_integer, "Airy return 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(unsigned m)
{
return airy_bi_zero<T>(m, policies::policy<>());
}
template <class T, class OutputIterator, class Policy>
inline OutputIterator airy_bi_zero(
unsigned start_index,
unsigned number_of_zeros,
OutputIterator out_it,
const Policy& pol)
{
typedef T result_type;
BOOST_STATIC_ASSERT_MSG(false == std::numeric_limits<result_type>::is_integer, "Airy return 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(
unsigned 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
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