142 lines
4.7 KiB
Plaintext
142 lines
4.7 KiB
Plaintext
// Copyright (c) 2013 Anton Bikineev
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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//
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// This is a partial header, do not include on it's own!!!
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//
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// Contains asymptotic expansions for derivatives of Bessel J(v,x) and Y(v,x)
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// functions, as x -> INF.
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#ifndef BOOST_MATH_SF_DETAIL_BESSEL_JY_DERIVATIVES_ASYM_HPP
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#define BOOST_MATH_SF_DETAIL_BESSEL_JY_DERIVATIVES_ASYM_HPP
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#ifdef _MSC_VER
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#pragma once
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#endif
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namespace boost{ namespace math{ namespace detail{
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template <class T>
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inline T asymptotic_bessel_derivative_amplitude(T v, T x)
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{
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// Calculate the amplitude for J'(v,x) and I'(v,x)
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// for large x: see A&S 9.2.30.
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BOOST_MATH_STD_USING
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T s = 1;
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const T mu = 4 * v * v;
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T txq = 2 * x;
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txq *= txq;
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s -= (mu - 3) / (2 * txq);
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s -= ((mu - 1) * (mu - 45)) / (txq * txq * 8);
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return sqrt(s * 2 / (boost::math::constants::pi<T>() * x));
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}
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template <class T>
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inline T asymptotic_bessel_derivative_phase_mx(T v, T x)
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{
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// Calculate the phase of J'(v, x) and Y'(v, x) for large x.
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// See A&S 9.2.31.
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// Note that the result returned is the phase less (x - PI(v/2 - 1/4))
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// which we'll factor in later when we calculate the sines/cosines of the result:
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const T mu = 4 * v * v;
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const T mu2 = mu * mu;
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const T mu3 = mu2 * mu;
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T denom = 4 * x;
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T denom_mult = denom * denom;
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T s = 0;
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s += (mu + 3) / (2 * denom);
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denom *= denom_mult;
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s += (mu2 + (46 * mu) - 63) / (6 * denom);
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denom *= denom_mult;
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s += (mu3 + (185 * mu2) - (2053 * mu) + 1899) / (5 * denom);
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return s;
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}
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template <class T>
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inline T asymptotic_bessel_y_derivative_large_x_2(T v, T x)
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{
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// See A&S 9.2.20.
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BOOST_MATH_STD_USING
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// Get the phase and amplitude:
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const T ampl = asymptotic_bessel_derivative_amplitude(v, x);
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const T phase = asymptotic_bessel_derivative_phase_mx(v, x);
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BOOST_MATH_INSTRUMENT_VARIABLE(ampl);
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BOOST_MATH_INSTRUMENT_VARIABLE(phase);
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//
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// Calculate the sine of the phase, using
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// sine/cosine addition rules to factor in
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// the x - PI(v/2 - 1/4) term not added to the
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// phase when we calculated it.
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//
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const T cx = cos(x);
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const T sx = sin(x);
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const T vd2shifted = (v / 2) - 0.25f;
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const T ci = cos_pi(vd2shifted);
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const T si = sin_pi(vd2shifted);
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const T sin_phase = sin(phase) * (cx * ci + sx * si) + cos(phase) * (sx * ci - cx * si);
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BOOST_MATH_INSTRUMENT_CODE(sin(phase));
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BOOST_MATH_INSTRUMENT_CODE(cos(x));
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BOOST_MATH_INSTRUMENT_CODE(cos(phase));
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BOOST_MATH_INSTRUMENT_CODE(sin(x));
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return sin_phase * ampl;
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}
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template <class T>
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inline T asymptotic_bessel_j_derivative_large_x_2(T v, T x)
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{
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// See A&S 9.2.20.
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BOOST_MATH_STD_USING
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// Get the phase and amplitude:
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const T ampl = asymptotic_bessel_derivative_amplitude(v, x);
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const T phase = asymptotic_bessel_derivative_phase_mx(v, x);
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BOOST_MATH_INSTRUMENT_VARIABLE(ampl);
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BOOST_MATH_INSTRUMENT_VARIABLE(phase);
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//
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// Calculate the sine of the phase, using
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// sine/cosine addition rules to factor in
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// the x - PI(v/2 - 1/4) term not added to the
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// phase when we calculated it.
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//
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BOOST_MATH_INSTRUMENT_CODE(cos(phase));
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BOOST_MATH_INSTRUMENT_CODE(cos(x));
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BOOST_MATH_INSTRUMENT_CODE(sin(phase));
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BOOST_MATH_INSTRUMENT_CODE(sin(x));
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const T cx = cos(x);
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const T sx = sin(x);
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const T vd2shifted = (v / 2) - 0.25f;
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const T ci = cos_pi(vd2shifted);
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const T si = sin_pi(vd2shifted);
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const T sin_phase = cos(phase) * (cx * ci + sx * si) - sin(phase) * (sx * ci - cx * si);
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BOOST_MATH_INSTRUMENT_VARIABLE(sin_phase);
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return sin_phase * ampl;
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}
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template <class T>
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inline bool asymptotic_bessel_derivative_large_x_limit(const T& v, const T& x)
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{
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BOOST_MATH_STD_USING
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//
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// This function is the copy of math::asymptotic_bessel_large_x_limit
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// It means that we use the same rules for determining how x is large
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// compared to v.
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//
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// Determines if x is large enough compared to v to take the asymptotic
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// forms above. From A&S 9.2.28 we require:
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// v < x * eps^1/8
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// and from A&S 9.2.29 we require:
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// v^12/10 < 1.5 * x * eps^1/10
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// using the former seems to work OK in practice with broadly similar
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// error rates either side of the divide for v < 10000.
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// At double precision eps^1/8 ~= 0.01.
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//
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return (std::max)(T(fabs(v)), T(1)) < x * sqrt(boost::math::tools::forth_root_epsilon<T>());
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}
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}}} // namespaces
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#endif // BOOST_MATH_SF_DETAIL_BESSEL_JY_DERIVATIVES_ASYM_HPP
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