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// Copyright (c) 2013 Anton Bikineev
// 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 is a partial header, do not include on it's own!!!
//
// Contains asymptotic expansions for derivatives of Bessel J(v,x) and Y(v,x)
// functions, as x -> INF.
#ifndef BOOST_MATH_SF_DETAIL_BESSEL_JY_DERIVATIVES_ASYM_HPP
#define BOOST_MATH_SF_DETAIL_BESSEL_JY_DERIVATIVES_ASYM_HPP
#ifdef _MSC_VER
#pragma once
#endif
namespace boost{ namespace math{ namespace detail{
template <class T>
inline T asymptotic_bessel_derivative_amplitude(T v, T x)
{
// Calculate the amplitude for J'(v,x) and I'(v,x)
// for large x: see A&S 9.2.30.
BOOST_MATH_STD_USING
T s = 1;
const T mu = 4 * v * v;
T txq = 2 * x;
txq *= txq;
s -= (mu - 3) / (2 * txq);
s -= ((mu - 1) * (mu - 45)) / (txq * txq * 8);
return sqrt(s * 2 / (boost::math::constants::pi<T>() * x));
}
template <class T>
inline T asymptotic_bessel_derivative_phase_mx(T v, T x)
{
// Calculate the phase of J'(v, x) and Y'(v, x) for large x.
// See A&S 9.2.31.
// Note that the result returned is the phase less (x - PI(v/2 - 1/4))
// which we'll factor in later when we calculate the sines/cosines of the result:
const T mu = 4 * v * v;
const T mu2 = mu * mu;
const T mu3 = mu2 * mu;
T denom = 4 * x;
T denom_mult = denom * denom;
T s = 0;
s += (mu + 3) / (2 * denom);
denom *= denom_mult;
s += (mu2 + (46 * mu) - 63) / (6 * denom);
denom *= denom_mult;
s += (mu3 + (185 * mu2) - (2053 * mu) + 1899) / (5 * denom);
return s;
}
template <class T>
inline T asymptotic_bessel_y_derivative_large_x_2(T v, T x)
{
// See A&S 9.2.20.
BOOST_MATH_STD_USING
// Get the phase and amplitude:
const T ampl = asymptotic_bessel_derivative_amplitude(v, x);
const T phase = asymptotic_bessel_derivative_phase_mx(v, x);
BOOST_MATH_INSTRUMENT_VARIABLE(ampl);
BOOST_MATH_INSTRUMENT_VARIABLE(phase);
//
// Calculate the sine of the phase, using
// sine/cosine addition rules to factor in
// the x - PI(v/2 - 1/4) term not added to the
// phase when we calculated it.
//
const T cx = cos(x);
const T sx = sin(x);
const T vd2shifted = (v / 2) - 0.25f;
const T ci = cos_pi(vd2shifted);
const T si = sin_pi(vd2shifted);
const T sin_phase = sin(phase) * (cx * ci + sx * si) + cos(phase) * (sx * ci - cx * si);
BOOST_MATH_INSTRUMENT_CODE(sin(phase));
BOOST_MATH_INSTRUMENT_CODE(cos(x));
BOOST_MATH_INSTRUMENT_CODE(cos(phase));
BOOST_MATH_INSTRUMENT_CODE(sin(x));
return sin_phase * ampl;
}
template <class T>
inline T asymptotic_bessel_j_derivative_large_x_2(T v, T x)
{
// See A&S 9.2.20.
BOOST_MATH_STD_USING
// Get the phase and amplitude:
const T ampl = asymptotic_bessel_derivative_amplitude(v, x);
const T phase = asymptotic_bessel_derivative_phase_mx(v, x);
BOOST_MATH_INSTRUMENT_VARIABLE(ampl);
BOOST_MATH_INSTRUMENT_VARIABLE(phase);
//
// Calculate the sine of the phase, using
// sine/cosine addition rules to factor in
// the x - PI(v/2 - 1/4) term not added to the
// phase when we calculated it.
//
BOOST_MATH_INSTRUMENT_CODE(cos(phase));
BOOST_MATH_INSTRUMENT_CODE(cos(x));
BOOST_MATH_INSTRUMENT_CODE(sin(phase));
BOOST_MATH_INSTRUMENT_CODE(sin(x));
const T cx = cos(x);
const T sx = sin(x);
const T vd2shifted = (v / 2) - 0.25f;
const T ci = cos_pi(vd2shifted);
const T si = sin_pi(vd2shifted);
const T sin_phase = cos(phase) * (cx * ci + sx * si) - sin(phase) * (sx * ci - cx * si);
BOOST_MATH_INSTRUMENT_VARIABLE(sin_phase);
return sin_phase * ampl;
}
template <class T>
inline bool asymptotic_bessel_derivative_large_x_limit(const T& v, const T& x)
{
BOOST_MATH_STD_USING
//
// This function is the copy of math::asymptotic_bessel_large_x_limit
// It means that we use the same rules for determining how x is large
// compared to v.
//
// Determines if x is large enough compared to v to take the asymptotic
// forms above. From A&S 9.2.28 we require:
// v < x * eps^1/8
// and from A&S 9.2.29 we require:
// v^12/10 < 1.5 * x * eps^1/10
// using the former seems to work OK in practice with broadly similar
// error rates either side of the divide for v < 10000.
// At double precision eps^1/8 ~= 0.01.
//
return (std::max)(T(fabs(v)), T(1)) < x * sqrt(boost::math::tools::forth_root_epsilon<T>());
}
}}} // namespaces
#endif // BOOST_MATH_SF_DETAIL_BESSEL_JY_DERIVATIVES_ASYM_HPP
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