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// 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_
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