Chris@16: //  Copyright (c) 2006 Xiaogang Zhang
Chris@16: //  Use, modification and distribution are subject to the
Chris@16: //  Boost Software License, Version 1.0. (See accompanying file
Chris@16: //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
Chris@16: 
Chris@16: #ifndef BOOST_MATH_BESSEL_JY_HPP
Chris@16: #define BOOST_MATH_BESSEL_JY_HPP
Chris@16: 
Chris@16: #ifdef _MSC_VER
Chris@16: #pragma once
Chris@16: #endif
Chris@16: 
Chris@16: #include <boost/math/tools/config.hpp>
Chris@16: #include <boost/math/special_functions/gamma.hpp>
Chris@16: #include <boost/math/special_functions/sign.hpp>
Chris@16: #include <boost/math/special_functions/hypot.hpp>
Chris@16: #include <boost/math/special_functions/sin_pi.hpp>
Chris@16: #include <boost/math/special_functions/cos_pi.hpp>
Chris@16: #include <boost/math/special_functions/detail/bessel_jy_asym.hpp>
Chris@16: #include <boost/math/special_functions/detail/bessel_jy_series.hpp>
Chris@16: #include <boost/math/constants/constants.hpp>
Chris@16: #include <boost/math/policies/error_handling.hpp>
Chris@16: #include <boost/mpl/if.hpp>
Chris@16: #include <boost/type_traits/is_floating_point.hpp>
Chris@16: #include <complex>
Chris@16: 
Chris@16: // Bessel functions of the first and second kind of fractional order
Chris@16: 
Chris@16: namespace boost { namespace math {
Chris@16: 
Chris@16:    namespace detail {
Chris@16: 
Chris@16:       //
Chris@16:       // Simultaneous calculation of A&S 9.2.9 and 9.2.10
Chris@16:       // for use in A&S 9.2.5 and 9.2.6.
Chris@16:       // This series is quick to evaluate, but divergent unless
Chris@16:       // x is very large, in fact it's pretty hard to figure out
Chris@16:       // with any degree of precision when this series actually 
Chris@16:       // *will* converge!!  Consequently, we may just have to
Chris@16:       // try it and see...
Chris@16:       //
Chris@16:       template <class T, class Policy>
Chris@16:       bool hankel_PQ(T v, T x, T* p, T* q, const Policy& )
Chris@16:       {
Chris@16:          BOOST_MATH_STD_USING
Chris@16:             T tolerance = 2 * policies::get_epsilon<T, Policy>();
Chris@16:          *p = 1;
Chris@16:          *q = 0;
Chris@16:          T k = 1;
Chris@16:          T z8 = 8 * x;
Chris@16:          T sq = 1;
Chris@16:          T mu = 4 * v * v;
Chris@16:          T term = 1;
Chris@16:          bool ok = true;
Chris@16:          do
Chris@16:          {
Chris@16:             term *= (mu - sq * sq) / (k * z8);
Chris@16:             *q += term;
Chris@16:             k += 1;
Chris@16:             sq += 2;
Chris@16:             T mult = (sq * sq - mu) / (k * z8);
Chris@16:             ok = fabs(mult) < 0.5f;
Chris@16:             term *= mult;
Chris@16:             *p += term;
Chris@16:             k += 1;
Chris@16:             sq += 2;
Chris@16:          }
Chris@16:          while((fabs(term) > tolerance * *p) && ok);
Chris@16:          return ok;
Chris@16:       }
Chris@16: 
Chris@16:       // Calculate Y(v, x) and Y(v+1, x) by Temme's method, see
Chris@16:       // Temme, Journal of Computational Physics, vol 21, 343 (1976)
Chris@16:       template <typename T, typename Policy>
Chris@16:       int temme_jy(T v, T x, T* Y, T* Y1, const Policy& pol)
Chris@16:       {
Chris@16:          T g, h, p, q, f, coef, sum, sum1, tolerance;
Chris@16:          T a, d, e, sigma;
Chris@16:          unsigned long k;
Chris@16: 
Chris@16:          BOOST_MATH_STD_USING
Chris@16:             using namespace boost::math::tools;
Chris@16:          using namespace boost::math::constants;
Chris@16: 
Chris@16:          BOOST_ASSERT(fabs(v) <= 0.5f);  // precondition for using this routine
Chris@16: 
Chris@16:          T gp = boost::math::tgamma1pm1(v, pol);
Chris@16:          T gm = boost::math::tgamma1pm1(-v, pol);
Chris@16:          T spv = boost::math::sin_pi(v, pol);
Chris@16:          T spv2 = boost::math::sin_pi(v/2, pol);
Chris@16:          T xp = pow(x/2, v);
Chris@16: 
Chris@16:          a = log(x / 2);
Chris@16:          sigma = -a * v;
Chris@16:          d = abs(sigma) < tools::epsilon<T>() ?
Chris@16:             T(1) : sinh(sigma) / sigma;
Chris@16:          e = abs(v) < tools::epsilon<T>() ? T(v*pi<T>()*pi<T>() / 2)
Chris@16:             : T(2 * spv2 * spv2 / v);
Chris@16: 
Chris@16:          T g1 = (v == 0) ? T(-euler<T>()) : T((gp - gm) / ((1 + gp) * (1 + gm) * 2 * v));
Chris@16:          T g2 = (2 + gp + gm) / ((1 + gp) * (1 + gm) * 2);
Chris@16:          T vspv = (fabs(v) < tools::epsilon<T>()) ? T(1/constants::pi<T>()) : T(v / spv);
Chris@16:          f = (g1 * cosh(sigma) - g2 * a * d) * 2 * vspv;
Chris@16: 
Chris@16:          p = vspv / (xp * (1 + gm));
Chris@16:          q = vspv * xp / (1 + gp);
Chris@16: 
Chris@16:          g = f + e * q;
Chris@16:          h = p;
Chris@16:          coef = 1;
Chris@16:          sum = coef * g;
Chris@16:          sum1 = coef * h;
Chris@16: 
Chris@16:          T v2 = v * v;
Chris@16:          T coef_mult = -x * x / 4;
Chris@16: 
Chris@16:          // series summation
Chris@16:          tolerance = policies::get_epsilon<T, Policy>();
Chris@16:          for (k = 1; k < policies::get_max_series_iterations<Policy>(); k++)
Chris@16:          {
Chris@16:             f = (k * f + p + q) / (k*k - v2);
Chris@16:             p /= k - v;
Chris@16:             q /= k + v;
Chris@16:             g = f + e * q;
Chris@16:             h = p - k * g;
Chris@16:             coef *= coef_mult / k;
Chris@16:             sum += coef * g;
Chris@16:             sum1 += coef * h;
Chris@16:             if (abs(coef * g) < abs(sum) * tolerance) 
Chris@16:             { 
Chris@16:                break; 
Chris@16:             }
Chris@16:          }
Chris@16:          policies::check_series_iterations<T>("boost::math::bessel_jy<%1%>(%1%,%1%) in temme_jy", k, pol);
Chris@16:          *Y = -sum;
Chris@16:          *Y1 = -2 * sum1 / x;
Chris@16: 
Chris@16:          return 0;
Chris@16:       }
Chris@16: 
Chris@16:       // Evaluate continued fraction fv = J_(v+1) / J_v, see
Chris@16:       // Abramowitz and Stegun, Handbook of Mathematical Functions, 1972, 9.1.73
Chris@16:       template <typename T, typename Policy>
Chris@16:       int CF1_jy(T v, T x, T* fv, int* sign, const Policy& pol)
Chris@16:       {
Chris@16:          T C, D, f, a, b, delta, tiny, tolerance;
Chris@16:          unsigned long k;
Chris@16:          int s = 1;
Chris@16: 
Chris@16:          BOOST_MATH_STD_USING
Chris@16: 
Chris@16:             // |x| <= |v|, CF1_jy converges rapidly
Chris@16:             // |x| > |v|, CF1_jy needs O(|x|) iterations to converge
Chris@16: 
Chris@16:             // modified Lentz's method, see
Chris@16:             // Lentz, Applied Optics, vol 15, 668 (1976)
Chris@16:             tolerance = 2 * policies::get_epsilon<T, Policy>();;
Chris@16:          tiny = sqrt(tools::min_value<T>());
Chris@16:          C = f = tiny;                           // b0 = 0, replace with tiny
Chris@16:          D = 0;
Chris@16:          for (k = 1; k < policies::get_max_series_iterations<Policy>() * 100; k++)
Chris@16:          {
Chris@16:             a = -1;
Chris@16:             b = 2 * (v + k) / x;
Chris@16:             C = b + a / C;
Chris@16:             D = b + a * D;
Chris@16:             if (C == 0) { C = tiny; }
Chris@16:             if (D == 0) { D = tiny; }
Chris@16:             D = 1 / D;
Chris@16:             delta = C * D;
Chris@16:             f *= delta;
Chris@16:             if (D < 0) { s = -s; }
Chris@16:             if (abs(delta - 1) < tolerance) 
Chris@16:             { break; }
Chris@16:          }
Chris@16:          policies::check_series_iterations<T>("boost::math::bessel_jy<%1%>(%1%,%1%) in CF1_jy", k / 100, pol);
Chris@16:          *fv = -f;
Chris@16:          *sign = s;                              // sign of denominator
Chris@16: 
Chris@16:          return 0;
Chris@16:       }
Chris@16:       //
Chris@16:       // This algorithm was originally written by Xiaogang Zhang
Chris@16:       // using std::complex to perform the complex arithmetic.
Chris@16:       // However, that turns out to 10x or more slower than using
Chris@16:       // all real-valued arithmetic, so it's been rewritten using
Chris@16:       // real values only.
Chris@16:       //
Chris@16:       template <typename T, typename Policy>
Chris@16:       int CF2_jy(T v, T x, T* p, T* q, const Policy& pol)
Chris@16:       {
Chris@16:          BOOST_MATH_STD_USING
Chris@16: 
Chris@16:             T Cr, Ci, Dr, Di, fr, fi, a, br, bi, delta_r, delta_i, temp;
Chris@16:          T tiny;
Chris@16:          unsigned long k;
Chris@16: 
Chris@16:          // |x| >= |v|, CF2_jy converges rapidly
Chris@16:          // |x| -> 0, CF2_jy fails to converge
Chris@16:          BOOST_ASSERT(fabs(x) > 1);
Chris@16: 
Chris@16:          // modified Lentz's method, complex numbers involved, see
Chris@16:          // Lentz, Applied Optics, vol 15, 668 (1976)
Chris@16:          T tolerance = 2 * policies::get_epsilon<T, Policy>();
Chris@16:          tiny = sqrt(tools::min_value<T>());
Chris@16:          Cr = fr = -0.5f / x;
Chris@16:          Ci = fi = 1;
Chris@16:          //Dr = Di = 0;
Chris@16:          T v2 = v * v;
Chris@16:          a = (0.25f - v2) / x; // Note complex this one time only!
Chris@16:          br = 2 * x;
Chris@16:          bi = 2;
Chris@16:          temp = Cr * Cr + 1;
Chris@16:          Ci = bi + a * Cr / temp;
Chris@16:          Cr = br + a / temp;
Chris@16:          Dr = br;
Chris@16:          Di = bi;
Chris@16:          if (fabs(Cr) + fabs(Ci) < tiny) { Cr = tiny; }
Chris@16:          if (fabs(Dr) + fabs(Di) < tiny) { Dr = tiny; }
Chris@16:          temp = Dr * Dr + Di * Di;
Chris@16:          Dr = Dr / temp;
Chris@16:          Di = -Di / temp;
Chris@16:          delta_r = Cr * Dr - Ci * Di;
Chris@16:          delta_i = Ci * Dr + Cr * Di;
Chris@16:          temp = fr;
Chris@16:          fr = temp * delta_r - fi * delta_i;
Chris@16:          fi = temp * delta_i + fi * delta_r;
Chris@16:          for (k = 2; k < policies::get_max_series_iterations<Policy>(); k++)
Chris@16:          {
Chris@16:             a = k - 0.5f;
Chris@16:             a *= a;
Chris@16:             a -= v2;
Chris@16:             bi += 2;
Chris@16:             temp = Cr * Cr + Ci * Ci;
Chris@16:             Cr = br + a * Cr / temp;
Chris@16:             Ci = bi - a * Ci / temp;
Chris@16:             Dr = br + a * Dr;
Chris@16:             Di = bi + a * Di;
Chris@16:             if (fabs(Cr) + fabs(Ci) < tiny) { Cr = tiny; }
Chris@16:             if (fabs(Dr) + fabs(Di) < tiny) { Dr = tiny; }
Chris@16:             temp = Dr * Dr + Di * Di;
Chris@16:             Dr = Dr / temp;
Chris@16:             Di = -Di / temp;
Chris@16:             delta_r = Cr * Dr - Ci * Di;
Chris@16:             delta_i = Ci * Dr + Cr * Di;
Chris@16:             temp = fr;
Chris@16:             fr = temp * delta_r - fi * delta_i;
Chris@16:             fi = temp * delta_i + fi * delta_r;
Chris@16:             if (fabs(delta_r - 1) + fabs(delta_i) < tolerance)
Chris@16:                break;
Chris@16:          }
Chris@16:          policies::check_series_iterations<T>("boost::math::bessel_jy<%1%>(%1%,%1%) in CF2_jy", k, pol);
Chris@16:          *p = fr;
Chris@16:          *q = fi;
Chris@16: 
Chris@16:          return 0;
Chris@16:       }
Chris@16: 
Chris@16:       static const int need_j = 1;
Chris@16:       static const int need_y = 2;
Chris@16: 
Chris@16:       // Compute J(v, x) and Y(v, x) simultaneously by Steed's method, see
Chris@16:       // Barnett et al, Computer Physics Communications, vol 8, 377 (1974)
Chris@16:       template <typename T, typename Policy>
Chris@16:       int bessel_jy(T v, T x, T* J, T* Y, int kind, const Policy& pol)
Chris@16:       {
Chris@16:          BOOST_ASSERT(x >= 0);
Chris@16: 
Chris@16:          T u, Jv, Ju, Yv, Yv1, Yu, Yu1(0), fv, fu;
Chris@16:          T W, p, q, gamma, current, prev, next;
Chris@16:          bool reflect = false;
Chris@16:          unsigned n, k;
Chris@16:          int s;
Chris@16:          int org_kind = kind;
Chris@16:          T cp = 0;
Chris@16:          T sp = 0;
Chris@16: 
Chris@16:          static const char* function = "boost::math::bessel_jy<%1%>(%1%,%1%)";
Chris@16: 
Chris@16:          BOOST_MATH_STD_USING
Chris@16:             using namespace boost::math::tools;
Chris@16:          using namespace boost::math::constants;
Chris@16: 
Chris@16:          if (v < 0)
Chris@16:          {
Chris@16:             reflect = true;
Chris@16:             v = -v;                             // v is non-negative from here
Chris@16:          }
Chris@101:          if (v > static_cast<T>((std::numeric_limits<int>::max)()))
Chris@101:          {
Chris@101:             *J = *Y = policies::raise_evaluation_error<T>(function, "Order of Bessel function is too large to evaluate: got %1%", v, pol);
Chris@101:             return 1;
Chris@101:          }
Chris@16:          n = iround(v, pol);
Chris@16:          u = v - n;                              // -1/2 <= u < 1/2
Chris@16: 
Chris@16:          if(reflect)
Chris@16:          {
Chris@16:             T z = (u + n % 2);
Chris@16:             cp = boost::math::cos_pi(z, pol);
Chris@16:             sp = boost::math::sin_pi(z, pol);
Chris@16:             if(u != 0)
Chris@16:                kind = need_j|need_y;               // need both for reflection formula
Chris@16:          }
Chris@16: 
Chris@16:          if(x == 0)
Chris@16:          {
Chris@16:             if(v == 0)
Chris@16:                *J = 1;
Chris@16:             else if((u == 0) || !reflect)
Chris@16:                *J = 0;
Chris@16:             else if(kind & need_j)
Chris@16:                *J = policies::raise_domain_error<T>(function, "Value of Bessel J_v(x) is complex-infinity at %1%", x, pol); // complex infinity
Chris@16:             else
Chris@16:                *J = std::numeric_limits<T>::quiet_NaN();  // any value will do, not using J.
Chris@16: 
Chris@16:             if((kind & need_y) == 0)
Chris@16:                *Y = std::numeric_limits<T>::quiet_NaN();  // any value will do, not using Y.
Chris@16:             else if(v == 0)
Chris@16:                *Y = -policies::raise_overflow_error<T>(function, 0, pol);
Chris@16:             else
Chris@16:                *Y = policies::raise_domain_error<T>(function, "Value of Bessel Y_v(x) is complex-infinity at %1%", x, pol); // complex infinity
Chris@16:             return 1;
Chris@16:          }
Chris@16: 
Chris@16:          // x is positive until reflection
Chris@16:          W = T(2) / (x * pi<T>());               // Wronskian
Chris@16:          T Yv_scale = 1;
Chris@16:          if(((kind & need_y) == 0) && ((x < 1) || (v > x * x / 4) || (x < 5)))
Chris@16:          {
Chris@16:             //
Chris@16:             // This series will actually converge rapidly for all small
Chris@16:             // x - say up to x < 20 - but the first few terms are large
Chris@16:             // and divergent which leads to large errors :-(
Chris@16:             //
Chris@16:             Jv = bessel_j_small_z_series(v, x, pol);
Chris@16:             Yv = std::numeric_limits<T>::quiet_NaN();
Chris@16:          }
Chris@16:          else if((x < 1) && (u != 0) && (log(policies::get_epsilon<T, Policy>() / 2) > v * log((x/2) * (x/2) / v)))
Chris@16:          {
Chris@16:             // Evaluate using series representations.
Chris@16:             // This is particularly important for x << v as in this
Chris@16:             // area temme_jy may be slow to converge, if it converges at all.
Chris@16:             // Requires x is not an integer.
Chris@16:             if(kind&need_j)
Chris@16:                Jv = bessel_j_small_z_series(v, x, pol);
Chris@16:             else
Chris@16:                Jv = std::numeric_limits<T>::quiet_NaN();
Chris@16:             if((org_kind&need_y && (!reflect || (cp != 0))) 
Chris@16:                || (org_kind & need_j && (reflect && (sp != 0))))
Chris@16:             {
Chris@16:                // Only calculate if we need it, and if the reflection formula will actually use it:
Chris@16:                Yv = bessel_y_small_z_series(v, x, &Yv_scale, pol);
Chris@16:             }
Chris@16:             else
Chris@16:                Yv = std::numeric_limits<T>::quiet_NaN();
Chris@16:          }
Chris@16:          else if((u == 0) && (x < policies::get_epsilon<T, Policy>()))
Chris@16:          {
Chris@16:             // Truncated series evaluation for small x and v an integer,
Chris@16:             // much quicker in this area than temme_jy below.
Chris@16:             if(kind&need_j)
Chris@16:                Jv = bessel_j_small_z_series(v, x, pol);
Chris@16:             else
Chris@16:                Jv = std::numeric_limits<T>::quiet_NaN();
Chris@16:             if((org_kind&need_y && (!reflect || (cp != 0))) 
Chris@16:                || (org_kind & need_j && (reflect && (sp != 0))))
Chris@16:             {
Chris@16:                // Only calculate if we need it, and if the reflection formula will actually use it:
Chris@16:                Yv = bessel_yn_small_z(n, x, &Yv_scale, pol);
Chris@16:             }
Chris@16:             else
Chris@16:                Yv = std::numeric_limits<T>::quiet_NaN();
Chris@16:          }
Chris@16:          else if(asymptotic_bessel_large_x_limit(v, x))
Chris@16:          {
Chris@16:             if(kind&need_y)
Chris@16:             {
Chris@16:                Yv = asymptotic_bessel_y_large_x_2(v, x);
Chris@16:             }
Chris@16:             else
Chris@16:                Yv = std::numeric_limits<T>::quiet_NaN(); // any value will do, we're not using it.
Chris@16:             if(kind&need_j)
Chris@16:             {
Chris@16:                Jv = asymptotic_bessel_j_large_x_2(v, x);
Chris@16:             }
Chris@16:             else
Chris@16:                Jv = std::numeric_limits<T>::quiet_NaN(); // any value will do, we're not using it.
Chris@16:          }
Chris@16:          else if((x > 8) && hankel_PQ(v, x, &p, &q, pol))
Chris@16:          {
Chris@16:             //
Chris@16:             // Hankel approximation: note that this method works best when x 
Chris@16:             // is large, but in that case we end up calculating sines and cosines
Chris@16:             // of large values, with horrendous resulting accuracy.  It is fast though
Chris@16:             // when it works....
Chris@16:             //
Chris@16:             // Normally we calculate sin/cos(chi) where:
Chris@16:             //
Chris@16:             // chi = x - fmod(T(v / 2 + 0.25f), T(2)) * boost::math::constants::pi<T>();
Chris@16:             //
Chris@16:             // But this introduces large errors, so use sin/cos addition formulae to
Chris@16:             // improve accuracy:
Chris@16:             //
Chris@16:             T mod_v = fmod(T(v / 2 + 0.25f), T(2));
Chris@16:             T sx = sin(x);
Chris@16:             T cx = cos(x);
Chris@16:             T sv = sin_pi(mod_v);
Chris@16:             T cv = cos_pi(mod_v);
Chris@16: 
Chris@16:             T sc = sx * cv - sv * cx; // == sin(chi);
Chris@16:             T cc = cx * cv + sx * sv; // == cos(chi);
Chris@16:             T chi = boost::math::constants::root_two<T>() / (boost::math::constants::root_pi<T>() * sqrt(x)); //sqrt(2 / (boost::math::constants::pi<T>() * x));
Chris@16:             Yv = chi * (p * sc + q * cc);
Chris@16:             Jv = chi * (p * cc - q * sc);
Chris@16:          }
Chris@16:          else if (x <= 2)                           // x in (0, 2]
Chris@16:          {
Chris@16:             if(temme_jy(u, x, &Yu, &Yu1, pol))             // Temme series
Chris@16:             {
Chris@16:                // domain error:
Chris@16:                *J = *Y = Yu;
Chris@16:                return 1;
Chris@16:             }
Chris@16:             prev = Yu;
Chris@16:             current = Yu1;
Chris@16:             T scale = 1;
Chris@16:             policies::check_series_iterations<T>(function, n, pol);
Chris@16:             for (k = 1; k <= n; k++)            // forward recurrence for Y
Chris@16:             {
Chris@16:                T fact = 2 * (u + k) / x;
Chris@16:                if((tools::max_value<T>() - fabs(prev)) / fact < fabs(current))
Chris@16:                {
Chris@16:                   scale /= current;
Chris@16:                   prev /= current;
Chris@16:                   current = 1;
Chris@16:                }
Chris@16:                next = fact * current - prev;
Chris@16:                prev = current;
Chris@16:                current = next;
Chris@16:             }
Chris@16:             Yv = prev;
Chris@16:             Yv1 = current;
Chris@16:             if(kind&need_j)
Chris@16:             {
Chris@16:                CF1_jy(v, x, &fv, &s, pol);                 // continued fraction CF1_jy
Chris@16:                Jv = scale * W / (Yv * fv - Yv1);           // Wronskian relation
Chris@16:             }
Chris@16:             else
Chris@16:                Jv = std::numeric_limits<T>::quiet_NaN(); // any value will do, we're not using it.
Chris@16:             Yv_scale = scale;
Chris@16:          }
Chris@16:          else                                    // x in (2, \infty)
Chris@16:          {
Chris@16:             // Get Y(u, x):
Chris@16: 
Chris@16:             T ratio;
Chris@16:             CF1_jy(v, x, &fv, &s, pol);
Chris@16:             // tiny initial value to prevent overflow
Chris@16:             T init = sqrt(tools::min_value<T>());
Chris@101:             BOOST_MATH_INSTRUMENT_VARIABLE(init);
Chris@16:             prev = fv * s * init;
Chris@16:             current = s * init;
Chris@16:             if(v < max_factorial<T>::value)
Chris@16:             {
Chris@16:                policies::check_series_iterations<T>(function, n, pol);
Chris@16:                for (k = n; k > 0; k--)             // backward recurrence for J
Chris@16:                {
Chris@16:                   next = 2 * (u + k) * current / x - prev;
Chris@16:                   prev = current;
Chris@16:                   current = next;
Chris@16:                }
Chris@16:                ratio = (s * init) / current;     // scaling ratio
Chris@16:                // can also call CF1_jy() to get fu, not much difference in precision
Chris@16:                fu = prev / current;
Chris@16:             }
Chris@16:             else
Chris@16:             {
Chris@16:                //
Chris@16:                // When v is large we may get overflow in this calculation
Chris@16:                // leading to NaN's and other nasty surprises:
Chris@16:                //
Chris@16:                policies::check_series_iterations<T>(function, n, pol);
Chris@16:                bool over = false;
Chris@16:                for (k = n; k > 0; k--)             // backward recurrence for J
Chris@16:                {
Chris@16:                   T t = 2 * (u + k) / x;
Chris@16:                   if((t > 1) && (tools::max_value<T>() / t < current))
Chris@16:                   {
Chris@16:                      over = true;
Chris@16:                      break;
Chris@16:                   }
Chris@16:                   next = t * current - prev;
Chris@16:                   prev = current;
Chris@16:                   current = next;
Chris@16:                }
Chris@16:                if(!over)
Chris@16:                {
Chris@16:                   ratio = (s * init) / current;     // scaling ratio
Chris@16:                   // can also call CF1_jy() to get fu, not much difference in precision
Chris@16:                   fu = prev / current;
Chris@16:                }
Chris@16:                else
Chris@16:                {
Chris@16:                   ratio = 0;
Chris@16:                   fu = 1;
Chris@16:                }
Chris@16:             }
Chris@16:             CF2_jy(u, x, &p, &q, pol);                  // continued fraction CF2_jy
Chris@16:             T t = u / x - fu;                   // t = J'/J
Chris@16:             gamma = (p - t) / q;
Chris@16:             //
Chris@16:             // We can't allow gamma to cancel out to zero competely as it messes up
Chris@16:             // the subsequent logic.  So pretend that one bit didn't cancel out
Chris@16:             // and set to a suitably small value.  The only test case we've been able to
Chris@16:             // find for this, is when v = 8.5 and x = 4*PI.
Chris@16:             //
Chris@16:             if(gamma == 0)
Chris@16:             {
Chris@16:                gamma = u * tools::epsilon<T>() / x;
Chris@16:             }
Chris@101:             BOOST_MATH_INSTRUMENT_VARIABLE(current);
Chris@101:             BOOST_MATH_INSTRUMENT_VARIABLE(W);
Chris@101:             BOOST_MATH_INSTRUMENT_VARIABLE(q);
Chris@101:             BOOST_MATH_INSTRUMENT_VARIABLE(gamma);
Chris@101:             BOOST_MATH_INSTRUMENT_VARIABLE(p);
Chris@101:             BOOST_MATH_INSTRUMENT_VARIABLE(t);
Chris@16:             Ju = sign(current) * sqrt(W / (q + gamma * (p - t)));
Chris@101:             BOOST_MATH_INSTRUMENT_VARIABLE(Ju);
Chris@16: 
Chris@16:             Jv = Ju * ratio;                    // normalization
Chris@16: 
Chris@16:             Yu = gamma * Ju;
Chris@16:             Yu1 = Yu * (u/x - p - q/gamma);
Chris@16: 
Chris@16:             if(kind&need_y)
Chris@16:             {
Chris@16:                // compute Y:
Chris@16:                prev = Yu;
Chris@16:                current = Yu1;
Chris@16:                policies::check_series_iterations<T>(function, n, pol);
Chris@16:                for (k = 1; k <= n; k++)            // forward recurrence for Y
Chris@16:                {
Chris@16:                   T fact = 2 * (u + k) / x;
Chris@16:                   if((tools::max_value<T>() - fabs(prev)) / fact < fabs(current))
Chris@16:                   {
Chris@16:                      prev /= current;
Chris@16:                      Yv_scale /= current;
Chris@16:                      current = 1;
Chris@16:                   }
Chris@16:                   next = fact * current - prev;
Chris@16:                   prev = current;
Chris@16:                   current = next;
Chris@16:                }
Chris@16:                Yv = prev;
Chris@16:             }
Chris@16:             else
Chris@16:                Yv = std::numeric_limits<T>::quiet_NaN(); // any value will do, we're not using it.
Chris@16:          }
Chris@16: 
Chris@16:          if (reflect)
Chris@16:          {
Chris@16:             if((sp != 0) && (tools::max_value<T>() * fabs(Yv_scale) < fabs(sp * Yv)))
Chris@16:                *J = org_kind & need_j ? T(-sign(sp) * sign(Yv) * sign(Yv_scale) * policies::raise_overflow_error<T>(function, 0, pol)) : T(0);
Chris@16:             else
Chris@16:                *J = cp * Jv - (sp == 0 ? T(0) : T((sp * Yv) / Yv_scale));     // reflection formula
Chris@16:             if((cp != 0) && (tools::max_value<T>() * fabs(Yv_scale) < fabs(cp * Yv)))
Chris@16:                *Y = org_kind & need_y ? T(-sign(cp) * sign(Yv) * sign(Yv_scale) * policies::raise_overflow_error<T>(function, 0, pol)) : T(0);
Chris@16:             else
Chris@16:                *Y = (sp != 0 ? sp * Jv : T(0)) + (cp == 0 ? T(0) : T((cp * Yv) / Yv_scale));
Chris@16:          }
Chris@16:          else
Chris@16:          {
Chris@16:             *J = Jv;
Chris@16:             if(tools::max_value<T>() * fabs(Yv_scale) < fabs(Yv))
Chris@16:                *Y = org_kind & need_y ? T(sign(Yv) * sign(Yv_scale) * policies::raise_overflow_error<T>(function, 0, pol)) : T(0);
Chris@16:             else
Chris@16:                *Y = Yv / Yv_scale;
Chris@16:          }
Chris@16: 
Chris@16:          return 0;
Chris@16:       }
Chris@16: 
Chris@16:    } // namespace detail
Chris@16: 
Chris@16: }} // namespaces
Chris@16: 
Chris@16: #endif // BOOST_MATH_BESSEL_JY_HPP
Chris@16: