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 //  Copyright (c) 2013 Christopher Kormanyos
 //  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 work is based on an earlier work:
 // "Algorithm 910: A Portable C++ Multiple-Precision System for Special-Function Calculations",
 // in ACM TOMS, {VOL 37, ISSUE 4, (February 2011)} (C) ACM, 2011. http://doi.acm.org/10.1145/1916461.1916469
 //
 // This header contains implementation details for estimating the zeros
 // of cylindrical Bessel and Neumann functions on the positive real axis.
 // Support is included for both positive as well as negative order.
 // Various methods are used to estimate the roots. These include
 // empirical curve fitting and McMahon's asymptotic approximation
 // for small order, uniform asymptotic expansion for large order,
 // and iteration and root interlacing for negative order.
 //
 #ifndef BOOST_MATH_BESSEL_JY_ZERO_2013_01_18_HPP_
   #define BOOST_MATH_BESSEL_JY_ZERO_2013_01_18_HPP_
 
   #include <algorithm>
   #include <boost/math/constants/constants.hpp>
   #include <boost/math/special_functions/math_fwd.hpp>
   #include <boost/math/special_functions/cbrt.hpp>
   #include <boost/math/special_functions/detail/airy_ai_bi_zero.hpp>
 
   namespace boost { namespace math {
   namespace detail
   {
     namespace bessel_zero
     {
       template<class T>
       T equation_nist_10_21_19(const T& v, const T& a)
       {
         // Get the initial estimate of the m'th root of Jv or Yv.
         // This subroutine is used for the order m with m > 1.
         // The order m has been used to create the input parameter a.
 
         // This is Eq. 10.21.19 in the NIST Handbook.
         const T mu                  = (v * v) * 4U;
         const T mu_minus_one        = mu - T(1);
         const T eight_a_inv         = T(1) / (a * 8U);
         const T eight_a_inv_squared = eight_a_inv * eight_a_inv;
 
         const T term3 = ((mu_minus_one *  4U) *     ((mu *    7U) -     T(31U) )) / 3U;
         const T term5 = ((mu_minus_one * 32U) *   ((((mu *   83U) -    T(982U) ) * mu) +    T(3779U) )) / 15U;
         const T term7 = ((mu_minus_one * 64U) * ((((((mu * 6949U) - T(153855UL)) * mu) + T(1585743UL)) * mu) - T(6277237UL))) / 105U;
 
         return a + ((((                      - term7
                        * eight_a_inv_squared - term5)
                        * eight_a_inv_squared - term3)
                        * eight_a_inv_squared - mu_minus_one)
                        * eight_a_inv);
       }
 
       template<typename T>
       class equation_as_9_3_39_and_its_derivative
       {
       public:
         explicit equation_as_9_3_39_and_its_derivative(const T& zt) : zeta(zt) { }
 
         equation_as_9_3_39_and_its_derivative(const equation_as_9_3_39_and_its_derivative&) = default;
 
         boost::math::tuple<T, T> operator()(const T& z) const
         {
           BOOST_MATH_STD_USING // ADL of std names, needed for acos, sqrt.
 
           // Return the function of zeta that is implicitly defined
           // in A&S Eq. 9.3.39 as a function of z. The function is
           // returned along with its derivative with respect to z.
 
           const T zsq_minus_one_sqrt = sqrt((z * z) - T(1));
 
           const T the_function(
               zsq_minus_one_sqrt
             - (  acos(T(1) / z) + ((T(2) / 3U) * (zeta * sqrt(zeta)))));
 
           const T its_derivative(zsq_minus_one_sqrt / z);
 
           return boost::math::tuple<T, T>(the_function, its_derivative);
         }
 
       private:
         const equation_as_9_3_39_and_its_derivative& operator=(const equation_as_9_3_39_and_its_derivative&) = delete;
         const T zeta;
       };
 
       template<class T, class Policy>
       static T equation_as_9_5_26(const T& v, const T& ai_bi_root, const Policy& pol)
       {
         BOOST_MATH_STD_USING // ADL of std names, needed for log, sqrt.
 
         // Obtain the estimate of the m'th zero of Jv or Yv.
         // The order m has been used to create the input parameter ai_bi_root.
         // Here, v is larger than about 2.2. The estimate is computed
         // from Abramowitz and Stegun Eqs. 9.5.22 and 9.5.26, page 371.
         //
         // The inversion of z as a function of zeta is mentioned in the text
         // following A&S Eq. 9.5.26. Here, we accomplish the inversion by
         // performing a Taylor expansion of Eq. 9.3.39 for large z to order 2
         // and solving the resulting quadratic equation, thereby taking
         // the positive root of the quadratic.
         // In other words: (2/3)(-zeta)^(3/2) approx = z + 1/(2z) - pi/2.
         // This leads to: z^2 - [(2/3)(-zeta)^(3/2) + pi/2]z + 1/2 = 0.
         //
         // With this initial estimate, Newton-Raphson iteration is used
         // to refine the value of the estimate of the root of z
         // as a function of zeta.
 
         const T v_pow_third(boost::math::cbrt(v, pol));
         const T v_pow_minus_two_thirds(T(1) / (v_pow_third * v_pow_third));
 
         // Obtain zeta using the order v combined with the m'th root of
         // an airy function, as shown in  A&S Eq. 9.5.22.
         const T zeta = v_pow_minus_two_thirds * (-ai_bi_root);
 
         const T zeta_sqrt = sqrt(zeta);
 
         // Set up a quadratic equation based on the Taylor series
         // expansion mentioned above.
         const T b = -((((zeta * zeta_sqrt) * 2U) / 3U) + boost::math::constants::half_pi<T>());
 
         // Solve the quadratic equation, taking the positive root.
         const T z_estimate = (-b + sqrt((b * b) - T(2))) / 2U;
 
         // Establish the range, the digits, and the iteration limit
         // for the upcoming root-finding.
         const T range_zmin = (std::max<T>)(z_estimate - T(1), T(1));
         const T range_zmax = z_estimate + T(1);
 
         const auto my_digits10 = static_cast<int>(static_cast<float>(boost::math::tools::digits<T>() * 0.301F));
 
         // Select the maximum allowed iterations based on the number
         // of decimal digits in the numeric type T, being at least 12.
         const auto iterations_allowed = static_cast<std::uintmax_t>((std::max)(12, my_digits10 * 2));
 
         std::uintmax_t iterations_used = iterations_allowed;
 
         // Calculate the root of z as a function of zeta.
         const T z = boost::math::tools::newton_raphson_iterate(
           boost::math::detail::bessel_zero::equation_as_9_3_39_and_its_derivative<T>(zeta),
           z_estimate,
           range_zmin,
           range_zmax,
           (std::min)(boost::math::tools::digits<T>(), boost::math::tools::digits<float>()),
           iterations_used);
 
         static_cast<void>(iterations_used);
 
         // Continue with the implementation of A&S Eq. 9.3.39.
         const T zsq_minus_one      = (z * z) - T(1);
         const T zsq_minus_one_sqrt = sqrt(zsq_minus_one);
 
         // This is A&S Eq. 9.3.42.
         const T b0_term_5_24 = T(5) / ((zsq_minus_one * zsq_minus_one_sqrt) * 24U);
         const T b0_term_1_8  = T(1) / ( zsq_minus_one_sqrt * 8U);
         const T b0_term_5_48 = T(5) / ((zeta * zeta) * 48U);
 
         const T b0 = -b0_term_5_48 + ((b0_term_5_24 + b0_term_1_8) / zeta_sqrt);
 
         // This is the second line of A&S Eq. 9.5.26 for f_k with k = 1.
         const T f1 = ((z * zeta_sqrt) * b0) / zsq_minus_one_sqrt;
 
         // This is A&S Eq. 9.5.22 expanded to k = 1 (i.e., one term in the series).
         return (v * z) + (f1 / v);
       }
 
       namespace cyl_bessel_j_zero_detail
       {
         template<class T, class Policy>
         T equation_nist_10_21_40_a(const T& v, const Policy& pol)
         {
           const T v_pow_third(boost::math::cbrt(v, pol));
           const T v_pow_minus_two_thirds(T(1) / (v_pow_third * v_pow_third));
 
           return v * (((((                         + T(0.043)
                           * v_pow_minus_two_thirds - T(0.0908))
                           * v_pow_minus_two_thirds - T(0.00397))
                           * v_pow_minus_two_thirds + T(1.033150))
                           * v_pow_minus_two_thirds + T(1.8557571))
                           * v_pow_minus_two_thirds + T(1));
         }
 
         template<class T, class Policy>
         class function_object_jv
         {
         public:
           function_object_jv(const T& v,
                              const Policy& pol) : my_v(v),
                                                   my_pol(pol) { }
 
           function_object_jv(const function_object_jv&) = default;
 
           T operator()(const T& x) const
           {
             return boost::math::cyl_bessel_j(my_v, x, my_pol);
           }
 
         private:
           const T my_v;
           const Policy& my_pol;
           const function_object_jv& operator=(const function_object_jv&) = delete;
         };
 
         template<class T, class Policy>
         class function_object_jv_and_jv_prime
         {
         public:
           function_object_jv_and_jv_prime(const T& v,
                                           const bool order_is_zero,
                                           const Policy& pol) : my_v(v),
                                                                my_order_is_zero(order_is_zero),
                                                                my_pol(pol) { }
 
           function_object_jv_and_jv_prime(const function_object_jv_and_jv_prime&) = default;
 
           boost::math::tuple<T, T> operator()(const T& x) const
           {
             // Obtain Jv(x) and Jv'(x).
             // Chris's original code called the Bessel function implementation layer direct, 
             // but that circumvented optimizations for integer-orders.  Call the documented
             // top level functions instead, and let them sort out which implementation to use.
             T j_v;
             T j_v_prime;
 
             if(my_order_is_zero)
             {
               j_v       =  boost::math::cyl_bessel_j(0, x, my_pol);
               j_v_prime = -boost::math::cyl_bessel_j(1, x, my_pol);
             }
             else
             {
                       j_v       = boost::math::cyl_bessel_j(  my_v,      x, my_pol);
               const T j_v_m1     (boost::math::cyl_bessel_j(T(my_v - 1), x, my_pol));
                       j_v_prime = j_v_m1 - ((my_v * j_v) / x);
             }
 
             // Return a tuple containing both Jv(x) and Jv'(x).
             return boost::math::make_tuple(j_v, j_v_prime);
           }
 
         private:
           const T my_v;
           const bool my_order_is_zero;
           const Policy& my_pol;
           const function_object_jv_and_jv_prime& operator=(const function_object_jv_and_jv_prime&) = delete;
         };
 
         template<class T> bool my_bisection_unreachable_tolerance(const T&, const T&) { return false; }
 
         template<class T, class Policy>
         T initial_guess(const T& v, const int m, const Policy& pol)
         {
           BOOST_MATH_STD_USING // ADL of std names, needed for floor.
 
           // Compute an estimate of the m'th root of cyl_bessel_j.
 
           T guess;
 
           // There is special handling for negative order.
           if(v < 0)
           {
             if((m == 1) && (v > -0.5F))
             {
               // For small, negative v, use the results of empirical curve fitting.
               // Mathematica(R) session for the coefficients:
               //  Table[{n, BesselJZero[n, 1]}, {n, -(1/2), 0, 1/10}]
               //  N[%, 20]
               //  Fit[%, {n^0, n^1, n^2, n^3, n^4, n^5, n^6}, n]
               guess = (((((    - T(0.2321156900729)
                            * v - T(0.1493247777488))
                            * v - T(0.15205419167239))
                            * v + T(0.07814930561249))
                            * v - T(0.17757573537688))
                            * v + T(1.542805677045663))
                            * v + T(2.40482555769577277);
 
               return guess;
             }
 
             // Create the positive order and extract its positive floor integer part.
             const T vv(-v);
             const T vv_floor(floor(vv));
 
             // The to-be-found root is bracketed by the roots of the
             // Bessel function whose reflected, positive integer order
             // is less than, but nearest to vv.
 
             T root_hi = boost::math::detail::bessel_zero::cyl_bessel_j_zero_detail::initial_guess(vv_floor, m, pol);
             T root_lo;
 
             if(m == 1)
             {
               // The estimate of the first root for negative order is found using
               // an adaptive range-searching algorithm.
               root_lo = T(root_hi - 0.1F);
 
               const bool hi_end_of_bracket_is_negative = (boost::math::cyl_bessel_j(v, root_hi, pol) < 0);
 
               while((root_lo > boost::math::tools::epsilon<T>()))
               {
                 const bool lo_end_of_bracket_is_negative = (boost::math::cyl_bessel_j(v, root_lo, pol) < 0);
 
                 if(hi_end_of_bracket_is_negative != lo_end_of_bracket_is_negative)
                 {
                   break;
                 }
 
                 root_hi = root_lo;
 
                 // Decrease the lower end of the bracket using an adaptive algorithm.
                 if(root_lo > 0.5F)
                 {
                   root_lo -= 0.5F;
                 }
                 else
                 {
                   root_lo *= 0.75F;
                 }
               }
             }
             else
             {
               root_lo = boost::math::detail::bessel_zero::cyl_bessel_j_zero_detail::initial_guess(vv_floor, m - 1, pol);
             }
 
             // Perform several steps of bisection iteration to refine the guess.
             std::uintmax_t number_of_iterations(12U);
 
             // Do the bisection iteration.
             const boost::math::tuple<T, T> guess_pair =
                boost::math::tools::bisect(
                   boost::math::detail::bessel_zero::cyl_bessel_j_zero_detail::function_object_jv<T, Policy>(v, pol),
                   root_lo,
                   root_hi,
                   boost::math::detail::bessel_zero::cyl_bessel_j_zero_detail::my_bisection_unreachable_tolerance<T>,
                   number_of_iterations);
 
             return (boost::math::get<0>(guess_pair) + boost::math::get<1>(guess_pair)) / 2U;
           }
 
           if(m == 1U)
           {
             // Get the initial estimate of the first root.
 
             if(v < 2.2F)
             {
               // For small v, use the results of empirical curve fitting.
               // Mathematica(R) session for the coefficients:
               //  Table[{n, BesselJZero[n, 1]}, {n, 0, 22/10, 1/10}]
               //  N[%, 20]
               //  Fit[%, {n^0, n^1, n^2, n^3, n^4, n^5, n^6}, n]
               guess = (((((    - T(0.0008342379046010)
                            * v + T(0.007590035637410))
                            * v - T(0.030640914772013))
                            * v + T(0.078232088020106))
                            * v - T(0.169668712590620))
                            * v + T(1.542187960073750))
                            * v + T(2.4048359915254634);
             }
             else
             {
               // For larger v, use the first line of Eqs. 10.21.40 in the NIST Handbook.
               guess = boost::math::detail::bessel_zero::cyl_bessel_j_zero_detail::equation_nist_10_21_40_a(v, pol);
             }
           }
           else
           {
             if(v < 2.2F)
             {
               // Use Eq. 10.21.19 in the NIST Handbook.
               const T a(((v + T(m * 2U)) - T(0.5)) * boost::math::constants::half_pi<T>());
 
               guess = boost::math::detail::bessel_zero::equation_nist_10_21_19(v, a);
             }
             else
             {
               // Get an estimate of the m'th root of airy_ai.
               const T airy_ai_root(boost::math::detail::airy_zero::airy_ai_zero_detail::initial_guess<T>(m, pol));
 
               // Use Eq. 9.5.26 in the A&S Handbook.
               guess = boost::math::detail::bessel_zero::equation_as_9_5_26(v, airy_ai_root, pol);
             }
           }
 
           return guess;
         }
       } // namespace cyl_bessel_j_zero_detail
 
       namespace cyl_neumann_zero_detail
       {
         template<class T, class Policy>
         T equation_nist_10_21_40_b(const T& v, const Policy& pol)
         {
           const T v_pow_third(boost::math::cbrt(v, pol));
           const T v_pow_minus_two_thirds(T(1) / (v_pow_third * v_pow_third));
 
           return v * (((((                         - T(0.001)
                           * v_pow_minus_two_thirds - T(0.0060))
                           * v_pow_minus_two_thirds + T(0.01198))
                           * v_pow_minus_two_thirds + T(0.260351))
                           * v_pow_minus_two_thirds + T(0.9315768))
                           * v_pow_minus_two_thirds + T(1));
         }
 
         template<class T, class Policy>
         class function_object_yv
         {
         public:
           function_object_yv(const T& v,
                              const Policy& pol) : my_v(v),
                                                   my_pol(pol) { }
 
           function_object_yv(const function_object_yv&) = default;
 
           T operator()(const T& x) const
           {
             return boost::math::cyl_neumann(my_v, x, my_pol);
           }
 
         private:
           const T my_v;
           const Policy& my_pol;
           const function_object_yv& operator=(const function_object_yv&) = delete;
         };
 
         template<class T, class Policy>
         class function_object_yv_and_yv_prime
         {
         public:
           function_object_yv_and_yv_prime(const T& v,
                                           const Policy& pol) : my_v(v),
                                                                my_pol(pol) { }
 
           function_object_yv_and_yv_prime(const function_object_yv_and_yv_prime&) = default;
 
           boost::math::tuple<T, T> operator()(const T& x) const
           {
             const T half_epsilon(boost::math::tools::epsilon<T>() / 2U);
 
             const bool order_is_zero = ((my_v > -half_epsilon) && (my_v < +half_epsilon));
 
             // Obtain Yv(x) and Yv'(x).
             // Chris's original code called the Bessel function implementation layer direct, 
             // but that circumvented optimizations for integer-orders.  Call the documented
             // top level functions instead, and let them sort out which implementation to use.
             T y_v;
             T y_v_prime;
 
             if(order_is_zero)
             {
               y_v       =  boost::math::cyl_neumann(0, x, my_pol);
               y_v_prime = -boost::math::cyl_neumann(1, x, my_pol);
             }
             else
             {
                       y_v       = boost::math::cyl_neumann(  my_v,      x, my_pol);
               const T y_v_m1     (boost::math::cyl_neumann(T(my_v - 1), x, my_pol));
                       y_v_prime = y_v_m1 - ((my_v * y_v) / x);
             }
 
             // Return a tuple containing both Yv(x) and Yv'(x).
             return boost::math::make_tuple(y_v, y_v_prime);
           }
 
         private:
           const T my_v;
           const Policy& my_pol;
           const function_object_yv_and_yv_prime& operator=(const function_object_yv_and_yv_prime&) = delete;
         };
 
         template<class T> bool my_bisection_unreachable_tolerance(const T&, const T&) { return false; }
 
         template<class T, class Policy>
         T initial_guess(const T& v, const int m, const Policy& pol)
         {
           BOOST_MATH_STD_USING // ADL of std names, needed for floor.
 
           // Compute an estimate of the m'th root of cyl_neumann.
 
           T guess;
 
           // There is special handling for negative order.
           if(v < 0)
           {
             // Create the positive order and extract its positive floor and ceiling integer parts.
             const T vv(-v);
             const T vv_floor(floor(vv));
 
             // The to-be-found root is bracketed by the roots of the
             // Bessel function whose reflected, positive integer order
             // is less than, but nearest to vv.
 
             // The special case of negative, half-integer order uses
             // the relation between Yv and spherical Bessel functions
             // in order to obtain the bracket for the root.
             // In these special cases, cyl_neumann(-n/2, x) = sph_bessel_j(+n/2, x)
             // for v = -n/2.
 
             T root_hi;
             T root_lo;
 
             if(m == 1)
             {
               // The estimate of the first root for negative order is found using
               // an adaptive range-searching algorithm.
               // Take special precautions for the discontinuity at negative,
               // half-integer orders and use different brackets above and below these.
               if(T(vv - vv_floor) < 0.5F)
               {
                 root_hi = boost::math::detail::bessel_zero::cyl_neumann_zero_detail::initial_guess(vv_floor, m, pol);
               }
               else
               {
                 root_hi = boost::math::detail::bessel_zero::cyl_bessel_j_zero_detail::initial_guess(T(vv_floor + 0.5F), m, pol);
               }
 
               root_lo = T(root_hi - 0.1F);
 
               const bool hi_end_of_bracket_is_negative = (boost::math::cyl_neumann(v, root_hi, pol) < 0);
 
               while((root_lo > boost::math::tools::epsilon<T>()))
               {
                 const bool lo_end_of_bracket_is_negative = (boost::math::cyl_neumann(v, root_lo, pol) < 0);
 
                 if(hi_end_of_bracket_is_negative != lo_end_of_bracket_is_negative)
                 {
                   break;
                 }
 
                 root_hi = root_lo;
 
                 // Decrease the lower end of the bracket using an adaptive algorithm.
                 if(root_lo > 0.5F)
                 {
                   root_lo -= 0.5F;
                 }
                 else
                 {
                   root_lo *= 0.75F;
                 }
               }
             }
             else
             {
               if(T(vv - vv_floor) < 0.5F)
               {
                 root_lo  = boost::math::detail::bessel_zero::cyl_neumann_zero_detail::initial_guess(vv_floor, m - 1, pol);
                 root_hi = boost::math::detail::bessel_zero::cyl_neumann_zero_detail::initial_guess(vv_floor, m, pol);
                 root_lo += 0.01F;
                 root_hi += 0.01F;
               }
               else
               {
                 root_lo = boost::math::detail::bessel_zero::cyl_bessel_j_zero_detail::initial_guess(T(vv_floor + 0.5F), m - 1, pol);
                 root_hi = boost::math::detail::bessel_zero::cyl_bessel_j_zero_detail::initial_guess(T(vv_floor + 0.5F), m, pol);
                 root_lo += 0.01F;
                 root_hi += 0.01F;
               }
             }
 
             // Perform several steps of bisection iteration to refine the guess.
             std::uintmax_t number_of_iterations(12U);
 
             // Do the bisection iteration.
             const boost::math::tuple<T, T> guess_pair =
                boost::math::tools::bisect(
                   boost::math::detail::bessel_zero::cyl_neumann_zero_detail::function_object_yv<T, Policy>(v, pol),
                   root_lo,
                   root_hi,
                   boost::math::detail::bessel_zero::cyl_neumann_zero_detail::my_bisection_unreachable_tolerance<T>,
                   number_of_iterations);
 
             return (boost::math::get<0>(guess_pair) + boost::math::get<1>(guess_pair)) / 2U;
           }
 
           if(m == 1U)
           {
             // Get the initial estimate of the first root.
 
             if(v < 2.2F)
             {
               // For small v, use the results of empirical curve fitting.
               // Mathematica(R) session for the coefficients:
               //  Table[{n, BesselYZero[n, 1]}, {n, 0, 22/10, 1/10}]
               //  N[%, 20]
               //  Fit[%, {n^0, n^1, n^2, n^3, n^4, n^5, n^6}, n]
               guess = (((((    - T(0.0025095909235652)
                            * v + T(0.021291887049053))
                            * v - T(0.076487785486526))
                            * v + T(0.159110268115362))
                            * v - T(0.241681668765196))
                            * v + T(1.4437846310885244))
                            * v + T(0.89362115190200490);
             }
             else
             {
               // For larger v, use the second line of Eqs. 10.21.40 in the NIST Handbook.
               guess = boost::math::detail::bessel_zero::cyl_neumann_zero_detail::equation_nist_10_21_40_b(v, pol);
             }
           }
           else
           {
             if(v < 2.2F)
             {
               // Use Eq. 10.21.19 in the NIST Handbook.
               const T a(((v + T(m * 2U)) - T(1.5)) * boost::math::constants::half_pi<T>());
 
               guess = boost::math::detail::bessel_zero::equation_nist_10_21_19(v, a);
             }
             else
             {
               // Get an estimate of the m'th root of airy_bi.
               const T airy_bi_root(boost::math::detail::airy_zero::airy_bi_zero_detail::initial_guess<T>(m, pol));
 
               // Use Eq. 9.5.26 in the A&S Handbook.
               guess = boost::math::detail::bessel_zero::equation_as_9_5_26(v, airy_bi_root, pol);
             }
           }
 
           return guess;
         }
       } // namespace cyl_neumann_zero_detail
     } // namespace bessel_zero
   } } } // namespace boost::math::detail
 
 #endif // BOOST_MATH_BESSEL_JY_ZERO_2013_01_18_HPP_

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