161 lines
6.3 KiB
Plaintext
161 lines
6.3 KiB
Plaintext
|
// Copyright (c) 2013 Christopher Kormanyos
|
||
|
|
// 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)
|
||
|
|
//
|
||
|
|
// This work is based on an earlier work:
|
||
|
|
// "Algorithm 910: A Portable C++ Multiple-Precision System for Special-Function Calculations",
|
||
|
|
// in ACM TOMS, {VOL 37, ISSUE 4, (February 2011)} (C) ACM, 2011. http://doi.acm.org/10.1145/1916461.1916469
|
||
|
|
//
|
||
|
|
// This header contains implementation details for estimating the zeros
|
||
|
|
// of the Airy functions airy_ai and airy_bi on the negative real axis.
|
||
|
|
//
|
||
|
|
#ifndef _AIRY_AI_BI_ZERO_2013_01_20_HPP_
|
||
|
|
#define _AIRY_AI_BI_ZERO_2013_01_20_HPP_
|
||
|
|
|
||
|
|
#include <boost/math/constants/constants.hpp>
|
||
|
|
#include <boost/math/special_functions/cbrt.hpp>
|
||
|
|
|
||
|
|
namespace boost { namespace math {
|
||
|
|
namespace detail
|
||
|
|
{
|
||
|
|
// Forward declarations of the needed Airy function implementations.
|
||
|
|
template <class T, class Policy>
|
||
|
|
T airy_ai_imp(T x, const Policy& pol);
|
||
|
|
template <class T, class Policy>
|
||
|
|
T airy_bi_imp(T x, const Policy& pol);
|
||
|
|
template <class T, class Policy>
|
||
|
|
T airy_ai_prime_imp(T x, const Policy& pol);
|
||
|
|
template <class T, class Policy>
|
||
|
|
T airy_bi_prime_imp(T x, const Policy& pol);
|
||
|
|
|
||
|
|
namespace airy_zero
|
||
|
|
{
|
||
|
|
template<class T>
|
||
|
|
T equation_as_10_4_105(const T& z)
|
||
|
|
{
|
||
|
|
const T one_over_z (T(1) / z);
|
||
|
|
const T one_over_z_squared(one_over_z * one_over_z);
|
||
|
|
|
||
|
|
const T z_pow_third (boost::math::cbrt(z));
|
||
|
|
const T z_pow_two_thirds(z_pow_third * z_pow_third);
|
||
|
|
|
||
|
|
// Implement the top line of Eq. 10.4.105.
|
||
|
|
const T fz(z_pow_two_thirds * ((((( + (T(162375596875.0) / 334430208UL)
|
||
|
|
* one_over_z_squared - ( T(108056875.0) / 6967296UL))
|
||
|
|
* one_over_z_squared + ( T(77125UL) / 82944UL))
|
||
|
|
* one_over_z_squared - ( T(5U) / 36U))
|
||
|
|
* one_over_z_squared + ( T(5U) / 48U))
|
||
|
|
* one_over_z_squared + (1)));
|
||
|
|
|
||
|
|
return fz;
|
||
|
|
}
|
||
|
|
|
||
|
|
namespace airy_ai_zero_detail
|
||
|
|
{
|
||
|
|
template<class T>
|
||
|
|
T initial_guess(const int m)
|
||
|
|
{
|
||
|
|
T guess;
|
||
|
|
|
||
|
|
switch(m)
|
||
|
|
{
|
||
|
|
case 0: { guess = T(0); break; }
|
||
|
|
case 1: { guess = T(-2.33810741045976703849); break; }
|
||
|
|
case 2: { guess = T(-4.08794944413097061664); break; }
|
||
|
|
case 3: { guess = T(-5.52055982809555105913); break; }
|
||
|
|
case 4: { guess = T(-6.78670809007175899878); break; }
|
||
|
|
case 5: { guess = T(-7.94413358712085312314); break; }
|
||
|
|
case 6: { guess = T(-9.02265085334098038016); break; }
|
||
|
|
case 7: { guess = T(-10.0401743415580859306); break; }
|
||
|
|
case 8: { guess = T(-11.0085243037332628932); break; }
|
||
|
|
case 9: { guess = T(-11.9360155632362625170); break; }
|
||
|
|
case 10:{ guess = T(-12.8287767528657572004); break; }
|
||
|
|
default:
|
||
|
|
{
|
||
|
|
const T t(((boost::math::constants::pi<T>() * 3) * ((T(m) * 4) - 1)) / 8);
|
||
|
|
guess = -boost::math::detail::airy_zero::equation_as_10_4_105(t);
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
return guess;
|
||
|
|
}
|
||
|
|
|
||
|
|
template<class T, class Policy>
|
||
|
|
class function_object_ai_and_ai_prime
|
||
|
|
{
|
||
|
|
public:
|
||
|
|
function_object_ai_and_ai_prime(const Policy pol) : my_pol(pol) { }
|
||
|
|
|
||
|
|
boost::math::tuple<T, T> operator()(const T& x) const
|
||
|
|
{
|
||
|
|
// Return a tuple containing both Ai(x) and Ai'(x).
|
||
|
|
return boost::math::make_tuple(
|
||
|
|
boost::math::detail::airy_ai_imp (x, my_pol),
|
||
|
|
boost::math::detail::airy_ai_prime_imp(x, my_pol));
|
||
|
|
}
|
||
|
|
|
||
|
|
private:
|
||
|
|
const Policy& my_pol;
|
||
|
|
const function_object_ai_and_ai_prime& operator=(const function_object_ai_and_ai_prime&);
|
||
|
|
};
|
||
|
|
} // namespace airy_ai_zero_detail
|
||
|
|
|
||
|
|
namespace airy_bi_zero_detail
|
||
|
|
{
|
||
|
|
template<class T>
|
||
|
|
T initial_guess(const int m)
|
||
|
|
{
|
||
|
|
T guess;
|
||
|
|
|
||
|
|
switch(m)
|
||
|
|
{
|
||
|
|
case 0: { guess = T(0); break; }
|
||
|
|
case 1: { guess = T(-1.17371322270912792492); break; }
|
||
|
|
case 2: { guess = T(-3.27109330283635271568); break; }
|
||
|
|
case 3: { guess = T(-4.83073784166201593267); break; }
|
||
|
|
case 4: { guess = T(-6.16985212831025125983); break; }
|
||
|
|
case 5: { guess = T(-7.37676207936776371360); break; }
|
||
|
|
case 6: { guess = T(-8.49194884650938801345); break; }
|
||
|
|
case 7: { guess = T(-9.53819437934623888663); break; }
|
||
|
|
case 8: { guess = T(-10.5299135067053579244); break; }
|
||
|
|
case 9: { guess = T(-11.4769535512787794379); break; }
|
||
|
|
case 10: { guess = T(-12.3864171385827387456); break; }
|
||
|
|
default:
|
||
|
|
{
|
||
|
|
const T t(((boost::math::constants::pi<T>() * 3) * ((T(m) * 4) - 3)) / 8);
|
||
|
|
guess = -boost::math::detail::airy_zero::equation_as_10_4_105(t);
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
return guess;
|
||
|
|
}
|
||
|
|
|
||
|
|
template<class T, class Policy>
|
||
|
|
class function_object_bi_and_bi_prime
|
||
|
|
{
|
||
|
|
public:
|
||
|
|
function_object_bi_and_bi_prime(const Policy pol) : my_pol(pol) { }
|
||
|
|
|
||
|
|
boost::math::tuple<T, T> operator()(const T& x) const
|
||
|
|
{
|
||
|
|
// Return a tuple containing both Bi(x) and Bi'(x).
|
||
|
|
return boost::math::make_tuple(
|
||
|
|
boost::math::detail::airy_bi_imp (x, my_pol),
|
||
|
|
boost::math::detail::airy_bi_prime_imp(x, my_pol));
|
||
|
|
}
|
||
|
|
|
||
|
|
private:
|
||
|
|
const Policy& my_pol;
|
||
|
|
const function_object_bi_and_bi_prime& operator=(const function_object_bi_and_bi_prime&);
|
||
|
|
};
|
||
|
|
} // namespace airy_bi_zero_detail
|
||
|
|
} // namespace airy_zero
|
||
|
|
} // namespace detail
|
||
|
|
} // namespace math
|
||
|
|
} // namespaces boost
|
||
|
|
|
||
|
|
#endif // _AIRY_AI_BI_ZERO_2013_01_20_HPP_
|