--- /dev/null
+// Copyright (c) 2006 Xiaogang Zhang
+// 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)
+
+#ifndef BOOST_MATH_BESSEL_IK_HPP
+#define BOOST_MATH_BESSEL_IK_HPP
+
+#ifdef _MSC_VER
+#pragma once
+#endif
+
+#include <boost/math/special_functions/round.hpp>
+#include <boost/math/special_functions/gamma.hpp>
+#include <boost/math/special_functions/sin_pi.hpp>
+#include <boost/math/constants/constants.hpp>
+#include <boost/math/policies/error_handling.hpp>
+#include <boost/math/tools/config.hpp>
+
+// Modified Bessel functions of the first and second kind of fractional order
+
+namespace boost { namespace math {
+
+namespace detail {
+
+template <class T, class Policy>
+struct cyl_bessel_i_small_z
+{
+ typedef T result_type;
+
+ cyl_bessel_i_small_z(T v_, T z_) : k(0), v(v_), mult(z_*z_/4)
+ {
+ BOOST_MATH_STD_USING
+ term = 1;
+ }
+
+ T operator()()
+ {
+ T result = term;
+ ++k;
+ term *= mult / k;
+ term /= k + v;
+ return result;
+ }
+private:
+ unsigned k;
+ T v;
+ T term;
+ T mult;
+};
+
+template <class T, class Policy>
+inline T bessel_i_small_z_series(T v, T x, const Policy& pol)
+{
+ BOOST_MATH_STD_USING
+ T prefix;
+ if(v < max_factorial<T>::value)
+ {
+ prefix = pow(x / 2, v) / boost::math::tgamma(v + 1, pol);
+ }
+ else
+ {
+ prefix = v * log(x / 2) - boost::math::lgamma(v + 1, pol);
+ prefix = exp(prefix);
+ }
+ if(prefix == 0)
+ return prefix;
+
+ cyl_bessel_i_small_z<T, Policy> s(v, x);
+ boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
+#if BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582))
+ T zero = 0;
+ T result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter, zero);
+#else
+ T result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter);
+#endif
+ policies::check_series_iterations<T>("boost::math::bessel_j_small_z_series<%1%>(%1%,%1%)", max_iter, pol);
+ return prefix * result;
+}
+
+// Calculate K(v, x) and K(v+1, x) by method analogous to
+// Temme, Journal of Computational Physics, vol 21, 343 (1976)
+template <typename T, typename Policy>
+int temme_ik(T v, T x, T* K, T* K1, const Policy& pol)
+{
+ T f, h, p, q, coef, sum, sum1, tolerance;
+ T a, b, c, d, sigma, gamma1, gamma2;
+ unsigned long k;
+
+ BOOST_MATH_STD_USING
+ using namespace boost::math::tools;
+ using namespace boost::math::constants;
+
+
+ // |x| <= 2, Temme series converge rapidly
+ // |x| > 2, the larger the |x|, the slower the convergence
+ BOOST_ASSERT(abs(x) <= 2);
+ BOOST_ASSERT(abs(v) <= 0.5f);
+
+ T gp = boost::math::tgamma1pm1(v, pol);
+ T gm = boost::math::tgamma1pm1(-v, pol);
+
+ a = log(x / 2);
+ b = exp(v * a);
+ sigma = -a * v;
+ c = abs(v) < tools::epsilon<T>() ?
+ T(1) : T(boost::math::sin_pi(v) / (v * pi<T>()));
+ d = abs(sigma) < tools::epsilon<T>() ?
+ T(1) : T(sinh(sigma) / sigma);
+ gamma1 = abs(v) < tools::epsilon<T>() ?
+ T(-euler<T>()) : T((0.5f / v) * (gp - gm) * c);
+ gamma2 = (2 + gp + gm) * c / 2;
+
+ // initial values
+ p = (gp + 1) / (2 * b);
+ q = (1 + gm) * b / 2;
+ f = (cosh(sigma) * gamma1 + d * (-a) * gamma2) / c;
+ h = p;
+ coef = 1;
+ sum = coef * f;
+ sum1 = coef * h;
+
+ BOOST_MATH_INSTRUMENT_VARIABLE(p);
+ BOOST_MATH_INSTRUMENT_VARIABLE(q);
+ BOOST_MATH_INSTRUMENT_VARIABLE(f);
+ BOOST_MATH_INSTRUMENT_VARIABLE(sigma);
+ BOOST_MATH_INSTRUMENT_CODE(sinh(sigma));
+ BOOST_MATH_INSTRUMENT_VARIABLE(gamma1);
+ BOOST_MATH_INSTRUMENT_VARIABLE(gamma2);
+ BOOST_MATH_INSTRUMENT_VARIABLE(c);
+ BOOST_MATH_INSTRUMENT_VARIABLE(d);
+ BOOST_MATH_INSTRUMENT_VARIABLE(a);
+
+ // series summation
+ tolerance = tools::epsilon<T>();
+ for (k = 1; k < policies::get_max_series_iterations<Policy>(); k++)
+ {
+ f = (k * f + p + q) / (k*k - v*v);
+ p /= k - v;
+ q /= k + v;
+ h = p - k * f;
+ coef *= x * x / (4 * k);
+ sum += coef * f;
+ sum1 += coef * h;
+ if (abs(coef * f) < abs(sum) * tolerance)
+ {
+ break;
+ }
+ }
+ policies::check_series_iterations<T>("boost::math::bessel_ik<%1%>(%1%,%1%) in temme_ik", k, pol);
+
+ *K = sum;
+ *K1 = 2 * sum1 / x;
+
+ return 0;
+}
+
+// Evaluate continued fraction fv = I_(v+1) / I_v, derived from
+// Abramowitz and Stegun, Handbook of Mathematical Functions, 1972, 9.1.73
+template <typename T, typename Policy>
+int CF1_ik(T v, T x, T* fv, const Policy& pol)
+{
+ T C, D, f, a, b, delta, tiny, tolerance;
+ unsigned long k;
+
+ BOOST_MATH_STD_USING
+
+ // |x| <= |v|, CF1_ik converges rapidly
+ // |x| > |v|, CF1_ik needs O(|x|) iterations to converge
+
+ // modified Lentz's method, see
+ // Lentz, Applied Optics, vol 15, 668 (1976)
+ tolerance = 2 * tools::epsilon<T>();
+ BOOST_MATH_INSTRUMENT_VARIABLE(tolerance);
+ tiny = sqrt(tools::min_value<T>());
+ BOOST_MATH_INSTRUMENT_VARIABLE(tiny);
+ C = f = tiny; // b0 = 0, replace with tiny
+ D = 0;
+ for (k = 1; k < policies::get_max_series_iterations<Policy>(); k++)
+ {
+ a = 1;
+ b = 2 * (v + k) / x;
+ C = b + a / C;
+ D = b + a * D;
+ if (C == 0) { C = tiny; }
+ if (D == 0) { D = tiny; }
+ D = 1 / D;
+ delta = C * D;
+ f *= delta;
+ BOOST_MATH_INSTRUMENT_VARIABLE(delta-1);
+ if (abs(delta - 1) <= tolerance)
+ {
+ break;
+ }
+ }
+ BOOST_MATH_INSTRUMENT_VARIABLE(k);
+ policies::check_series_iterations<T>("boost::math::bessel_ik<%1%>(%1%,%1%) in CF1_ik", k, pol);
+
+ *fv = f;
+
+ return 0;
+}
+
+// Calculate K(v, x) and K(v+1, x) by evaluating continued fraction
+// z1 / z0 = U(v+1.5, 2v+1, 2x) / U(v+0.5, 2v+1, 2x), see
+// Thompson and Barnett, Computer Physics Communications, vol 47, 245 (1987)
+template <typename T, typename Policy>
+int CF2_ik(T v, T x, T* Kv, T* Kv1, const Policy& pol)
+{
+ BOOST_MATH_STD_USING
+ using namespace boost::math::constants;
+
+ T S, C, Q, D, f, a, b, q, delta, tolerance, current, prev;
+ unsigned long k;
+
+ // |x| >= |v|, CF2_ik converges rapidly
+ // |x| -> 0, CF2_ik fails to converge
+
+ BOOST_ASSERT(abs(x) > 1);
+
+ // Steed's algorithm, see Thompson and Barnett,
+ // Journal of Computational Physics, vol 64, 490 (1986)
+ tolerance = tools::epsilon<T>();
+ a = v * v - 0.25f;
+ b = 2 * (x + 1); // b1
+ D = 1 / b; // D1 = 1 / b1
+ f = delta = D; // f1 = delta1 = D1, coincidence
+ prev = 0; // q0
+ current = 1; // q1
+ Q = C = -a; // Q1 = C1 because q1 = 1
+ S = 1 + Q * delta; // S1
+ BOOST_MATH_INSTRUMENT_VARIABLE(tolerance);
+ BOOST_MATH_INSTRUMENT_VARIABLE(a);
+ BOOST_MATH_INSTRUMENT_VARIABLE(b);
+ BOOST_MATH_INSTRUMENT_VARIABLE(D);
+ BOOST_MATH_INSTRUMENT_VARIABLE(f);
+
+ for (k = 2; k < policies::get_max_series_iterations<Policy>(); k++) // starting from 2
+ {
+ // continued fraction f = z1 / z0
+ a -= 2 * (k - 1);
+ b += 2;
+ D = 1 / (b + a * D);
+ delta *= b * D - 1;
+ f += delta;
+
+ // series summation S = 1 + \sum_{n=1}^{\infty} C_n * z_n / z_0
+ q = (prev - (b - 2) * current) / a;
+ prev = current;
+ current = q; // forward recurrence for q
+ C *= -a / k;
+ Q += C * q;
+ S += Q * delta;
+ //
+ // Under some circumstances q can grow very small and C very
+ // large, leading to under/overflow. This is particularly an
+ // issue for types which have many digits precision but a narrow
+ // exponent range. A typical example being a "double double" type.
+ // To avoid this situation we can normalise q (and related prev/current)
+ // and C. All other variables remain unchanged in value. A typical
+ // test case occurs when x is close to 2, for example cyl_bessel_k(9.125, 2.125).
+ //
+ if(q < tools::epsilon<T>())
+ {
+ C *= q;
+ prev /= q;
+ current /= q;
+ q = 1;
+ }
+
+ // S converges slower than f
+ BOOST_MATH_INSTRUMENT_VARIABLE(Q * delta);
+ BOOST_MATH_INSTRUMENT_VARIABLE(abs(S) * tolerance);
+ if (abs(Q * delta) < abs(S) * tolerance)
+ {
+ break;
+ }
+ }
+ policies::check_series_iterations<T>("boost::math::bessel_ik<%1%>(%1%,%1%) in CF2_ik", k, pol);
+
+ *Kv = sqrt(pi<T>() / (2 * x)) * exp(-x) / S;
+ *Kv1 = *Kv * (0.5f + v + x + (v * v - 0.25f) * f) / x;
+ BOOST_MATH_INSTRUMENT_VARIABLE(*Kv);
+ BOOST_MATH_INSTRUMENT_VARIABLE(*Kv1);
+
+ return 0;
+}
+
+enum{
+ need_i = 1,
+ need_k = 2
+};
+
+// Compute I(v, x) and K(v, x) simultaneously by Temme's method, see
+// Temme, Journal of Computational Physics, vol 19, 324 (1975)
+template <typename T, typename Policy>
+int bessel_ik(T v, T x, T* I, T* K, int kind, const Policy& pol)
+{
+ // Kv1 = K_(v+1), fv = I_(v+1) / I_v
+ // Ku1 = K_(u+1), fu = I_(u+1) / I_u
+ T u, Iv, Kv, Kv1, Ku, Ku1, fv;
+ T W, current, prev, next;
+ bool reflect = false;
+ unsigned n, k;
+ int org_kind = kind;
+ BOOST_MATH_INSTRUMENT_VARIABLE(v);
+ BOOST_MATH_INSTRUMENT_VARIABLE(x);
+ BOOST_MATH_INSTRUMENT_VARIABLE(kind);
+
+ BOOST_MATH_STD_USING
+ using namespace boost::math::tools;
+ using namespace boost::math::constants;
+
+ static const char* function = "boost::math::bessel_ik<%1%>(%1%,%1%)";
+
+ if (v < 0)
+ {
+ reflect = true;
+ v = -v; // v is non-negative from here
+ kind |= need_k;
+ }
+ n = iround(v, pol);
+ u = v - n; // -1/2 <= u < 1/2
+ BOOST_MATH_INSTRUMENT_VARIABLE(n);
+ BOOST_MATH_INSTRUMENT_VARIABLE(u);
+
+ if (x < 0)
+ {
+ *I = *K = policies::raise_domain_error<T>(function,
+ "Got x = %1% but real argument x must be non-negative, complex number result not supported.", x, pol);
+ return 1;
+ }
+ if (x == 0)
+ {
+ Iv = (v == 0) ? static_cast<T>(1) : static_cast<T>(0);
+ if(kind & need_k)
+ {
+ Kv = policies::raise_overflow_error<T>(function, 0, pol);
+ }
+ else
+ {
+ Kv = std::numeric_limits<T>::quiet_NaN(); // any value will do
+ }
+
+ if(reflect && (kind & need_i))
+ {
+ T z = (u + n % 2);
+ Iv = boost::math::sin_pi(z, pol) == 0 ?
+ Iv :
+ policies::raise_overflow_error<T>(function, 0, pol); // reflection formula
+ }
+
+ *I = Iv;
+ *K = Kv;
+ return 0;
+ }
+
+ // x is positive until reflection
+ W = 1 / x; // Wronskian
+ if (x <= 2) // x in (0, 2]
+ {
+ temme_ik(u, x, &Ku, &Ku1, pol); // Temme series
+ }
+ else // x in (2, \infty)
+ {
+ CF2_ik(u, x, &Ku, &Ku1, pol); // continued fraction CF2_ik
+ }
+ BOOST_MATH_INSTRUMENT_VARIABLE(Ku);
+ BOOST_MATH_INSTRUMENT_VARIABLE(Ku1);
+ prev = Ku;
+ current = Ku1;
+ T scale = 1;
+ for (k = 1; k <= n; k++) // forward recurrence for K
+ {
+ T fact = 2 * (u + k) / x;
+ if((tools::max_value<T>() - fabs(prev)) / fact < fabs(current))
+ {
+ prev /= current;
+ scale /= current;
+ current = 1;
+ }
+ next = fact * current + prev;
+ prev = current;
+ current = next;
+ }
+ Kv = prev;
+ Kv1 = current;
+ BOOST_MATH_INSTRUMENT_VARIABLE(Kv);
+ BOOST_MATH_INSTRUMENT_VARIABLE(Kv1);
+ if(kind & need_i)
+ {
+ T lim = (4 * v * v + 10) / (8 * x);
+ lim *= lim;
+ lim *= lim;
+ lim /= 24;
+ if((lim < tools::epsilon<T>() * 10) && (x > 100))
+ {
+ // x is huge compared to v, CF1 may be very slow
+ // to converge so use asymptotic expansion for large
+ // x case instead. Note that the asymptotic expansion
+ // isn't very accurate - so it's deliberately very hard
+ // to get here - probably we're going to overflow:
+ Iv = asymptotic_bessel_i_large_x(v, x, pol);
+ }
+ else if((v > 0) && (x / v < 0.25))
+ {
+ Iv = bessel_i_small_z_series(v, x, pol);
+ }
+ else
+ {
+ CF1_ik(v, x, &fv, pol); // continued fraction CF1_ik
+ Iv = scale * W / (Kv * fv + Kv1); // Wronskian relation
+ }
+ }
+ else
+ Iv = std::numeric_limits<T>::quiet_NaN(); // any value will do
+
+ if (reflect)
+ {
+ T z = (u + n % 2);
+ T fact = (2 / pi<T>()) * (boost::math::sin_pi(z) * Kv);
+ if(fact == 0)
+ *I = Iv;
+ else if(tools::max_value<T>() * scale < fact)
+ *I = (org_kind & need_i) ? T(sign(fact) * sign(scale) * policies::raise_overflow_error<T>(function, 0, pol)) : T(0);
+ else
+ *I = Iv + fact / scale; // reflection formula
+ }
+ else
+ {
+ *I = Iv;
+ }
+ if(tools::max_value<T>() * scale < Kv)
+ *K = (org_kind & need_k) ? T(sign(Kv) * sign(scale) * policies::raise_overflow_error<T>(function, 0, pol)) : T(0);
+ else
+ *K = Kv / scale;
+ BOOST_MATH_INSTRUMENT_VARIABLE(*I);
+ BOOST_MATH_INSTRUMENT_VARIABLE(*K);
+ return 0;
+}
+
+}}} // namespaces
+
+#endif // BOOST_MATH_BESSEL_IK_HPP
+
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