470 lines
16 KiB
Plaintext
470 lines
16 KiB
Plaintext
// 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 !", 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);
|
|
|
|
// 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);
|
|
|
|
// 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
|