590 lines
22 KiB
Plaintext
590 lines
22 KiB
Plaintext
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// Copyright (c) 2006 Xiaogang Zhang
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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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#ifndef BOOST_MATH_BESSEL_JY_HPP
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#define BOOST_MATH_BESSEL_JY_HPP
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#ifdef _MSC_VER
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#pragma once
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#endif
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#include <boost/math/tools/config.hpp>
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#include <boost/math/special_functions/gamma.hpp>
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#include <boost/math/special_functions/sign.hpp>
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#include <boost/math/special_functions/hypot.hpp>
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#include <boost/math/special_functions/sin_pi.hpp>
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#include <boost/math/special_functions/cos_pi.hpp>
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#include <boost/math/special_functions/detail/bessel_jy_asym.hpp>
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#include <boost/math/special_functions/detail/bessel_jy_series.hpp>
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#include <boost/math/constants/constants.hpp>
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#include <boost/math/policies/error_handling.hpp>
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#include <boost/mpl/if.hpp>
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#include <boost/type_traits/is_floating_point.hpp>
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#include <complex>
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// Bessel functions of the first and second kind of fractional order
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namespace boost { namespace math {
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namespace detail {
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//
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// Simultaneous calculation of A&S 9.2.9 and 9.2.10
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// for use in A&S 9.2.5 and 9.2.6.
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// This series is quick to evaluate, but divergent unless
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// x is very large, in fact it's pretty hard to figure out
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// with any degree of precision when this series actually
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// *will* converge!! Consequently, we may just have to
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// try it and see...
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//
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template <class T, class Policy>
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bool hankel_PQ(T v, T x, T* p, T* q, const Policy& )
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{
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BOOST_MATH_STD_USING
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T tolerance = 2 * policies::get_epsilon<T, Policy>();
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*p = 1;
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*q = 0;
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T k = 1;
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T z8 = 8 * x;
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T sq = 1;
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T mu = 4 * v * v;
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T term = 1;
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bool ok = true;
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do
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{
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term *= (mu - sq * sq) / (k * z8);
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*q += term;
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k += 1;
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sq += 2;
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T mult = (sq * sq - mu) / (k * z8);
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ok = fabs(mult) < 0.5f;
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term *= mult;
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*p += term;
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k += 1;
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sq += 2;
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}
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while((fabs(term) > tolerance * *p) && ok);
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return ok;
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}
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// Calculate Y(v, x) and Y(v+1, x) by Temme's method, see
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// Temme, Journal of Computational Physics, vol 21, 343 (1976)
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template <typename T, typename Policy>
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int temme_jy(T v, T x, T* Y, T* Y1, const Policy& pol)
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{
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T g, h, p, q, f, coef, sum, sum1, tolerance;
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T a, d, e, sigma;
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unsigned long k;
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BOOST_MATH_STD_USING
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using namespace boost::math::tools;
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using namespace boost::math::constants;
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BOOST_ASSERT(fabs(v) <= 0.5f); // precondition for using this routine
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T gp = boost::math::tgamma1pm1(v, pol);
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T gm = boost::math::tgamma1pm1(-v, pol);
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T spv = boost::math::sin_pi(v, pol);
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T spv2 = boost::math::sin_pi(v/2, pol);
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T xp = pow(x/2, v);
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a = log(x / 2);
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sigma = -a * v;
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d = abs(sigma) < tools::epsilon<T>() ?
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T(1) : sinh(sigma) / sigma;
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e = abs(v) < tools::epsilon<T>() ? T(v*pi<T>()*pi<T>() / 2)
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: T(2 * spv2 * spv2 / v);
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T g1 = (v == 0) ? T(-euler<T>()) : T((gp - gm) / ((1 + gp) * (1 + gm) * 2 * v));
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T g2 = (2 + gp + gm) / ((1 + gp) * (1 + gm) * 2);
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T vspv = (fabs(v) < tools::epsilon<T>()) ? T(1/constants::pi<T>()) : T(v / spv);
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f = (g1 * cosh(sigma) - g2 * a * d) * 2 * vspv;
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p = vspv / (xp * (1 + gm));
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q = vspv * xp / (1 + gp);
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g = f + e * q;
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h = p;
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coef = 1;
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sum = coef * g;
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sum1 = coef * h;
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T v2 = v * v;
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T coef_mult = -x * x / 4;
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// series summation
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tolerance = policies::get_epsilon<T, Policy>();
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for (k = 1; k < policies::get_max_series_iterations<Policy>(); k++)
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{
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f = (k * f + p + q) / (k*k - v2);
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p /= k - v;
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q /= k + v;
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g = f + e * q;
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h = p - k * g;
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coef *= coef_mult / k;
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sum += coef * g;
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sum1 += coef * h;
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if (abs(coef * g) < abs(sum) * tolerance)
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{
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break;
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}
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}
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policies::check_series_iterations<T>("boost::math::bessel_jy<%1%>(%1%,%1%) in temme_jy", k, pol);
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*Y = -sum;
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*Y1 = -2 * sum1 / x;
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return 0;
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}
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// Evaluate continued fraction fv = J_(v+1) / J_v, see
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// Abramowitz and Stegun, Handbook of Mathematical Functions, 1972, 9.1.73
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template <typename T, typename Policy>
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int CF1_jy(T v, T x, T* fv, int* sign, const Policy& pol)
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{
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T C, D, f, a, b, delta, tiny, tolerance;
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unsigned long k;
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int s = 1;
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BOOST_MATH_STD_USING
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// |x| <= |v|, CF1_jy converges rapidly
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// |x| > |v|, CF1_jy needs O(|x|) iterations to converge
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// modified Lentz's method, see
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// Lentz, Applied Optics, vol 15, 668 (1976)
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tolerance = 2 * policies::get_epsilon<T, Policy>();;
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tiny = sqrt(tools::min_value<T>());
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C = f = tiny; // b0 = 0, replace with tiny
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D = 0;
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for (k = 1; k < policies::get_max_series_iterations<Policy>() * 100; k++)
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{
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a = -1;
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b = 2 * (v + k) / x;
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C = b + a / C;
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D = b + a * D;
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if (C == 0) { C = tiny; }
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if (D == 0) { D = tiny; }
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D = 1 / D;
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delta = C * D;
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f *= delta;
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if (D < 0) { s = -s; }
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if (abs(delta - 1) < tolerance)
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{ break; }
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}
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policies::check_series_iterations<T>("boost::math::bessel_jy<%1%>(%1%,%1%) in CF1_jy", k / 100, pol);
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*fv = -f;
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*sign = s; // sign of denominator
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return 0;
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}
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//
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// This algorithm was originally written by Xiaogang Zhang
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// using std::complex to perform the complex arithmetic.
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// However, that turns out to 10x or more slower than using
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// all real-valued arithmetic, so it's been rewritten using
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// real values only.
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//
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template <typename T, typename Policy>
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int CF2_jy(T v, T x, T* p, T* q, const Policy& pol)
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{
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BOOST_MATH_STD_USING
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T Cr, Ci, Dr, Di, fr, fi, a, br, bi, delta_r, delta_i, temp;
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T tiny;
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unsigned long k;
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// |x| >= |v|, CF2_jy converges rapidly
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// |x| -> 0, CF2_jy fails to converge
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BOOST_ASSERT(fabs(x) > 1);
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// modified Lentz's method, complex numbers involved, see
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// Lentz, Applied Optics, vol 15, 668 (1976)
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T tolerance = 2 * policies::get_epsilon<T, Policy>();
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tiny = sqrt(tools::min_value<T>());
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Cr = fr = -0.5f / x;
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Ci = fi = 1;
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//Dr = Di = 0;
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T v2 = v * v;
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a = (0.25f - v2) / x; // Note complex this one time only!
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br = 2 * x;
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bi = 2;
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temp = Cr * Cr + 1;
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Ci = bi + a * Cr / temp;
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Cr = br + a / temp;
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Dr = br;
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Di = bi;
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if (fabs(Cr) + fabs(Ci) < tiny) { Cr = tiny; }
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if (fabs(Dr) + fabs(Di) < tiny) { Dr = tiny; }
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temp = Dr * Dr + Di * Di;
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Dr = Dr / temp;
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Di = -Di / temp;
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delta_r = Cr * Dr - Ci * Di;
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delta_i = Ci * Dr + Cr * Di;
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temp = fr;
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fr = temp * delta_r - fi * delta_i;
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fi = temp * delta_i + fi * delta_r;
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for (k = 2; k < policies::get_max_series_iterations<Policy>(); k++)
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{
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a = k - 0.5f;
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a *= a;
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a -= v2;
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bi += 2;
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temp = Cr * Cr + Ci * Ci;
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Cr = br + a * Cr / temp;
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Ci = bi - a * Ci / temp;
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Dr = br + a * Dr;
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Di = bi + a * Di;
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if (fabs(Cr) + fabs(Ci) < tiny) { Cr = tiny; }
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if (fabs(Dr) + fabs(Di) < tiny) { Dr = tiny; }
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temp = Dr * Dr + Di * Di;
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Dr = Dr / temp;
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Di = -Di / temp;
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delta_r = Cr * Dr - Ci * Di;
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delta_i = Ci * Dr + Cr * Di;
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temp = fr;
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fr = temp * delta_r - fi * delta_i;
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fi = temp * delta_i + fi * delta_r;
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if (fabs(delta_r - 1) + fabs(delta_i) < tolerance)
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break;
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}
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policies::check_series_iterations<T>("boost::math::bessel_jy<%1%>(%1%,%1%) in CF2_jy", k, pol);
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*p = fr;
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*q = fi;
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return 0;
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}
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static const int need_j = 1;
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static const int need_y = 2;
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// Compute J(v, x) and Y(v, x) simultaneously by Steed's method, see
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// Barnett et al, Computer Physics Communications, vol 8, 377 (1974)
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template <typename T, typename Policy>
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int bessel_jy(T v, T x, T* J, T* Y, int kind, const Policy& pol)
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{
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BOOST_ASSERT(x >= 0);
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T u, Jv, Ju, Yv, Yv1, Yu, Yu1(0), fv, fu;
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T W, p, q, gamma, current, prev, next;
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bool reflect = false;
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unsigned n, k;
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int s;
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int org_kind = kind;
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T cp = 0;
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T sp = 0;
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static const char* function = "boost::math::bessel_jy<%1%>(%1%,%1%)";
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BOOST_MATH_STD_USING
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using namespace boost::math::tools;
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using namespace boost::math::constants;
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if (v < 0)
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{
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reflect = true;
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v = -v; // v is non-negative from here
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}
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if (v > static_cast<T>((std::numeric_limits<int>::max)()))
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{
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*J = *Y = policies::raise_evaluation_error<T>(function, "Order of Bessel function is too large to evaluate: got %1%", v, pol);
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return 1;
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}
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n = iround(v, pol);
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u = v - n; // -1/2 <= u < 1/2
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if(reflect)
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{
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T z = (u + n % 2);
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cp = boost::math::cos_pi(z, pol);
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sp = boost::math::sin_pi(z, pol);
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if(u != 0)
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kind = need_j|need_y; // need both for reflection formula
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}
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if(x == 0)
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{
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if(v == 0)
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*J = 1;
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else if((u == 0) || !reflect)
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*J = 0;
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else if(kind & need_j)
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*J = policies::raise_domain_error<T>(function, "Value of Bessel J_v(x) is complex-infinity at %1%", x, pol); // complex infinity
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else
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*J = std::numeric_limits<T>::quiet_NaN(); // any value will do, not using J.
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if((kind & need_y) == 0)
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*Y = std::numeric_limits<T>::quiet_NaN(); // any value will do, not using Y.
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else if(v == 0)
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*Y = -policies::raise_overflow_error<T>(function, 0, pol);
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else
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*Y = policies::raise_domain_error<T>(function, "Value of Bessel Y_v(x) is complex-infinity at %1%", x, pol); // complex infinity
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return 1;
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}
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// x is positive until reflection
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W = T(2) / (x * pi<T>()); // Wronskian
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T Yv_scale = 1;
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if(((kind & need_y) == 0) && ((x < 1) || (v > x * x / 4) || (x < 5)))
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{
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//
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// This series will actually converge rapidly for all small
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// x - say up to x < 20 - but the first few terms are large
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// and divergent which leads to large errors :-(
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//
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Jv = bessel_j_small_z_series(v, x, pol);
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Yv = std::numeric_limits<T>::quiet_NaN();
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}
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else if((x < 1) && (u != 0) && (log(policies::get_epsilon<T, Policy>() / 2) > v * log((x/2) * (x/2) / v)))
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{
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// Evaluate using series representations.
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// This is particularly important for x << v as in this
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// area temme_jy may be slow to converge, if it converges at all.
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// Requires x is not an integer.
|
||
|
|
if(kind&need_j)
|
||
|
|
Jv = bessel_j_small_z_series(v, x, pol);
|
||
|
|
else
|
||
|
|
Jv = std::numeric_limits<T>::quiet_NaN();
|
||
|
|
if((org_kind&need_y && (!reflect || (cp != 0)))
|
||
|
|
|| (org_kind & need_j && (reflect && (sp != 0))))
|
||
|
|
{
|
||
|
|
// Only calculate if we need it, and if the reflection formula will actually use it:
|
||
|
|
Yv = bessel_y_small_z_series(v, x, &Yv_scale, pol);
|
||
|
|
}
|
||
|
|
else
|
||
|
|
Yv = std::numeric_limits<T>::quiet_NaN();
|
||
|
|
}
|
||
|
|
else if((u == 0) && (x < policies::get_epsilon<T, Policy>()))
|
||
|
|
{
|
||
|
|
// Truncated series evaluation for small x and v an integer,
|
||
|
|
// much quicker in this area than temme_jy below.
|
||
|
|
if(kind&need_j)
|
||
|
|
Jv = bessel_j_small_z_series(v, x, pol);
|
||
|
|
else
|
||
|
|
Jv = std::numeric_limits<T>::quiet_NaN();
|
||
|
|
if((org_kind&need_y && (!reflect || (cp != 0)))
|
||
|
|
|| (org_kind & need_j && (reflect && (sp != 0))))
|
||
|
|
{
|
||
|
|
// Only calculate if we need it, and if the reflection formula will actually use it:
|
||
|
|
Yv = bessel_yn_small_z(n, x, &Yv_scale, pol);
|
||
|
|
}
|
||
|
|
else
|
||
|
|
Yv = std::numeric_limits<T>::quiet_NaN();
|
||
|
|
}
|
||
|
|
else if(asymptotic_bessel_large_x_limit(v, x))
|
||
|
|
{
|
||
|
|
if(kind&need_y)
|
||
|
|
{
|
||
|
|
Yv = asymptotic_bessel_y_large_x_2(v, x);
|
||
|
|
}
|
||
|
|
else
|
||
|
|
Yv = std::numeric_limits<T>::quiet_NaN(); // any value will do, we're not using it.
|
||
|
|
if(kind&need_j)
|
||
|
|
{
|
||
|
|
Jv = asymptotic_bessel_j_large_x_2(v, x);
|
||
|
|
}
|
||
|
|
else
|
||
|
|
Jv = std::numeric_limits<T>::quiet_NaN(); // any value will do, we're not using it.
|
||
|
|
}
|
||
|
|
else if((x > 8) && hankel_PQ(v, x, &p, &q, pol))
|
||
|
|
{
|
||
|
|
//
|
||
|
|
// Hankel approximation: note that this method works best when x
|
||
|
|
// is large, but in that case we end up calculating sines and cosines
|
||
|
|
// of large values, with horrendous resulting accuracy. It is fast though
|
||
|
|
// when it works....
|
||
|
|
//
|
||
|
|
// Normally we calculate sin/cos(chi) where:
|
||
|
|
//
|
||
|
|
// chi = x - fmod(T(v / 2 + 0.25f), T(2)) * boost::math::constants::pi<T>();
|
||
|
|
//
|
||
|
|
// But this introduces large errors, so use sin/cos addition formulae to
|
||
|
|
// improve accuracy:
|
||
|
|
//
|
||
|
|
T mod_v = fmod(T(v / 2 + 0.25f), T(2));
|
||
|
|
T sx = sin(x);
|
||
|
|
T cx = cos(x);
|
||
|
|
T sv = sin_pi(mod_v);
|
||
|
|
T cv = cos_pi(mod_v);
|
||
|
|
|
||
|
|
T sc = sx * cv - sv * cx; // == sin(chi);
|
||
|
|
T cc = cx * cv + sx * sv; // == cos(chi);
|
||
|
|
T chi = boost::math::constants::root_two<T>() / (boost::math::constants::root_pi<T>() * sqrt(x)); //sqrt(2 / (boost::math::constants::pi<T>() * x));
|
||
|
|
Yv = chi * (p * sc + q * cc);
|
||
|
|
Jv = chi * (p * cc - q * sc);
|
||
|
|
}
|
||
|
|
else if (x <= 2) // x in (0, 2]
|
||
|
|
{
|
||
|
|
if(temme_jy(u, x, &Yu, &Yu1, pol)) // Temme series
|
||
|
|
{
|
||
|
|
// domain error:
|
||
|
|
*J = *Y = Yu;
|
||
|
|
return 1;
|
||
|
|
}
|
||
|
|
prev = Yu;
|
||
|
|
current = Yu1;
|
||
|
|
T scale = 1;
|
||
|
|
policies::check_series_iterations<T>(function, n, pol);
|
||
|
|
for (k = 1; k <= n; k++) // forward recurrence for Y
|
||
|
|
{
|
||
|
|
T fact = 2 * (u + k) / x;
|
||
|
|
if((tools::max_value<T>() - fabs(prev)) / fact < fabs(current))
|
||
|
|
{
|
||
|
|
scale /= current;
|
||
|
|
prev /= current;
|
||
|
|
current = 1;
|
||
|
|
}
|
||
|
|
next = fact * current - prev;
|
||
|
|
prev = current;
|
||
|
|
current = next;
|
||
|
|
}
|
||
|
|
Yv = prev;
|
||
|
|
Yv1 = current;
|
||
|
|
if(kind&need_j)
|
||
|
|
{
|
||
|
|
CF1_jy(v, x, &fv, &s, pol); // continued fraction CF1_jy
|
||
|
|
Jv = scale * W / (Yv * fv - Yv1); // Wronskian relation
|
||
|
|
}
|
||
|
|
else
|
||
|
|
Jv = std::numeric_limits<T>::quiet_NaN(); // any value will do, we're not using it.
|
||
|
|
Yv_scale = scale;
|
||
|
|
}
|
||
|
|
else // x in (2, \infty)
|
||
|
|
{
|
||
|
|
// Get Y(u, x):
|
||
|
|
|
||
|
|
T ratio;
|
||
|
|
CF1_jy(v, x, &fv, &s, pol);
|
||
|
|
// tiny initial value to prevent overflow
|
||
|
|
T init = sqrt(tools::min_value<T>());
|
||
|
|
BOOST_MATH_INSTRUMENT_VARIABLE(init);
|
||
|
|
prev = fv * s * init;
|
||
|
|
current = s * init;
|
||
|
|
if(v < max_factorial<T>::value)
|
||
|
|
{
|
||
|
|
policies::check_series_iterations<T>(function, n, pol);
|
||
|
|
for (k = n; k > 0; k--) // backward recurrence for J
|
||
|
|
{
|
||
|
|
next = 2 * (u + k) * current / x - prev;
|
||
|
|
prev = current;
|
||
|
|
current = next;
|
||
|
|
}
|
||
|
|
ratio = (s * init) / current; // scaling ratio
|
||
|
|
// can also call CF1_jy() to get fu, not much difference in precision
|
||
|
|
fu = prev / current;
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
//
|
||
|
|
// When v is large we may get overflow in this calculation
|
||
|
|
// leading to NaN's and other nasty surprises:
|
||
|
|
//
|
||
|
|
policies::check_series_iterations<T>(function, n, pol);
|
||
|
|
bool over = false;
|
||
|
|
for (k = n; k > 0; k--) // backward recurrence for J
|
||
|
|
{
|
||
|
|
T t = 2 * (u + k) / x;
|
||
|
|
if((t > 1) && (tools::max_value<T>() / t < current))
|
||
|
|
{
|
||
|
|
over = true;
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
next = t * current - prev;
|
||
|
|
prev = current;
|
||
|
|
current = next;
|
||
|
|
}
|
||
|
|
if(!over)
|
||
|
|
{
|
||
|
|
ratio = (s * init) / current; // scaling ratio
|
||
|
|
// can also call CF1_jy() to get fu, not much difference in precision
|
||
|
|
fu = prev / current;
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
ratio = 0;
|
||
|
|
fu = 1;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
CF2_jy(u, x, &p, &q, pol); // continued fraction CF2_jy
|
||
|
|
T t = u / x - fu; // t = J'/J
|
||
|
|
gamma = (p - t) / q;
|
||
|
|
//
|
||
|
|
// We can't allow gamma to cancel out to zero competely as it messes up
|
||
|
|
// the subsequent logic. So pretend that one bit didn't cancel out
|
||
|
|
// and set to a suitably small value. The only test case we've been able to
|
||
|
|
// find for this, is when v = 8.5 and x = 4*PI.
|
||
|
|
//
|
||
|
|
if(gamma == 0)
|
||
|
|
{
|
||
|
|
gamma = u * tools::epsilon<T>() / x;
|
||
|
|
}
|
||
|
|
BOOST_MATH_INSTRUMENT_VARIABLE(current);
|
||
|
|
BOOST_MATH_INSTRUMENT_VARIABLE(W);
|
||
|
|
BOOST_MATH_INSTRUMENT_VARIABLE(q);
|
||
|
|
BOOST_MATH_INSTRUMENT_VARIABLE(gamma);
|
||
|
|
BOOST_MATH_INSTRUMENT_VARIABLE(p);
|
||
|
|
BOOST_MATH_INSTRUMENT_VARIABLE(t);
|
||
|
|
Ju = sign(current) * sqrt(W / (q + gamma * (p - t)));
|
||
|
|
BOOST_MATH_INSTRUMENT_VARIABLE(Ju);
|
||
|
|
|
||
|
|
Jv = Ju * ratio; // normalization
|
||
|
|
|
||
|
|
Yu = gamma * Ju;
|
||
|
|
Yu1 = Yu * (u/x - p - q/gamma);
|
||
|
|
|
||
|
|
if(kind&need_y)
|
||
|
|
{
|
||
|
|
// compute Y:
|
||
|
|
prev = Yu;
|
||
|
|
current = Yu1;
|
||
|
|
policies::check_series_iterations<T>(function, n, pol);
|
||
|
|
for (k = 1; k <= n; k++) // forward recurrence for Y
|
||
|
|
{
|
||
|
|
T fact = 2 * (u + k) / x;
|
||
|
|
if((tools::max_value<T>() - fabs(prev)) / fact < fabs(current))
|
||
|
|
{
|
||
|
|
prev /= current;
|
||
|
|
Yv_scale /= current;
|
||
|
|
current = 1;
|
||
|
|
}
|
||
|
|
next = fact * current - prev;
|
||
|
|
prev = current;
|
||
|
|
current = next;
|
||
|
|
}
|
||
|
|
Yv = prev;
|
||
|
|
}
|
||
|
|
else
|
||
|
|
Yv = std::numeric_limits<T>::quiet_NaN(); // any value will do, we're not using it.
|
||
|
|
}
|
||
|
|
|
||
|
|
if (reflect)
|
||
|
|
{
|
||
|
|
if((sp != 0) && (tools::max_value<T>() * fabs(Yv_scale) < fabs(sp * Yv)))
|
||
|
|
*J = org_kind & need_j ? T(-sign(sp) * sign(Yv) * sign(Yv_scale) * policies::raise_overflow_error<T>(function, 0, pol)) : T(0);
|
||
|
|
else
|
||
|
|
*J = cp * Jv - (sp == 0 ? T(0) : T((sp * Yv) / Yv_scale)); // reflection formula
|
||
|
|
if((cp != 0) && (tools::max_value<T>() * fabs(Yv_scale) < fabs(cp * Yv)))
|
||
|
|
*Y = org_kind & need_y ? T(-sign(cp) * sign(Yv) * sign(Yv_scale) * policies::raise_overflow_error<T>(function, 0, pol)) : T(0);
|
||
|
|
else
|
||
|
|
*Y = (sp != 0 ? sp * Jv : T(0)) + (cp == 0 ? T(0) : T((cp * Yv) / Yv_scale));
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
*J = Jv;
|
||
|
|
if(tools::max_value<T>() * fabs(Yv_scale) < fabs(Yv))
|
||
|
|
*Y = org_kind & need_y ? T(sign(Yv) * sign(Yv_scale) * policies::raise_overflow_error<T>(function, 0, pol)) : T(0);
|
||
|
|
else
|
||
|
|
*Y = Yv / Yv_scale;
|
||
|
|
}
|
||
|
|
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
|
||
|
|
} // namespace detail
|
||
|
|
|
||
|
|
}} // namespaces
|
||
|
|
|
||
|
|
#endif // BOOST_MATH_BESSEL_JY_HPP
|
||
|
|
|