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 // boost\math\distributions\non_central_chi_squared.hpp
 
 // Copyright John Maddock 2008.
 
 // 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_SPECIAL_NON_CENTRAL_CHI_SQUARE_HPP
 #define BOOST_MATH_SPECIAL_NON_CENTRAL_CHI_SQUARE_HPP
 
 #include <boost/math/distributions/fwd.hpp>
 #include <boost/math/special_functions/gamma.hpp> // for incomplete gamma. gamma_q
 #include <boost/math/special_functions/bessel.hpp> // for cyl_bessel_i
 #include <boost/math/special_functions/round.hpp> // for llround
 #include <boost/math/distributions/complement.hpp> // complements
 #include <boost/math/distributions/chi_squared.hpp> // central distribution
 #include <boost/math/distributions/detail/common_error_handling.hpp> // error checks
 #include <boost/math/special_functions/fpclassify.hpp> // isnan.
 #include <boost/math/tools/roots.hpp> // for root finding.
 #include <boost/math/distributions/detail/generic_mode.hpp>
 #include <boost/math/distributions/detail/generic_quantile.hpp>
 
 namespace boost
 {
    namespace math
    {
 
       template <class RealType, class Policy>
       class non_central_chi_squared_distribution;
 
       namespace detail{
 
          template <class T, class Policy>
          T non_central_chi_square_q(T x, T f, T theta, const Policy& pol, T init_sum = 0)
          {
             //
             // Computes the complement of the Non-Central Chi-Square
             // Distribution CDF by summing a weighted sum of complements
             // of the central-distributions.  The weighting factor is
             // a Poisson Distribution.
             //
             // This is an application of the technique described in:
             //
             // Computing discrete mixtures of continuous
             // distributions: noncentral chisquare, noncentral t
             // and the distribution of the square of the sample
             // multiple correlation coefficient.
             // D. Benton, K. Krishnamoorthy.
             // Computational Statistics & Data Analysis 43 (2003) 249 - 267
             //
             BOOST_MATH_STD_USING
 
             // Special case:
             if(x == 0)
                return 1;
 
             //
             // Initialize the variables we'll be using:
             //
             T lambda = theta / 2;
             T del = f / 2;
             T y = x / 2;
             std::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
             T errtol = boost::math::policies::get_epsilon<T, Policy>();
             T sum = init_sum;
             //
             // k is the starting location for iteration, we'll
             // move both forwards and backwards from this point.
             // k is chosen as the peek of the Poisson weights, which
             // will occur *before* the largest term.
             //
             long long k = llround(lambda, pol);
             // Forwards and backwards Poisson weights:
             T poisf = boost::math::gamma_p_derivative(static_cast<T>(1 + k), lambda, pol);
             T poisb = poisf * k / lambda;
             // Initial forwards central chi squared term:
             T gamf = boost::math::gamma_q(del + k, y, pol);
             // Forwards and backwards recursion terms on the central chi squared:
             T xtermf = boost::math::gamma_p_derivative(del + 1 + k, y, pol);
             T xtermb = xtermf * (del + k) / y;
             // Initial backwards central chi squared term:
             T gamb = gamf - xtermb;
 
             //
             // Forwards iteration first, this is the
             // stable direction for the gamma function
             // recurrences:
             //
             long long i;
             for(i = k; static_cast<std::uintmax_t>(i-k) < max_iter; ++i)
             {
                T term = poisf * gamf;
                sum += term;
                poisf *= lambda / (i + 1);
                gamf += xtermf;
                xtermf *= y / (del + i + 1);
                if(((sum == 0) || (fabs(term / sum) < errtol)) && (term >= poisf * gamf))
                   break;
             }
             //Error check:
             if(static_cast<std::uintmax_t>(i-k) >= max_iter)
                return policies::raise_evaluation_error("cdf(non_central_chi_squared_distribution<%1%>, %1%)", "Series did not converge, closest value was %1%", sum, pol); // LCOV_EXCL_LINE
             //
             // Now backwards iteration: the gamma
             // function recurrences are unstable in this
             // direction, we rely on the terms diminishing in size
             // faster than we introduce cancellation errors.
             // For this reason it's very important that we start
             // *before* the largest term so that backwards iteration
             // is strictly converging.
             //
             for(i = k - 1; i >= 0; --i)
             {
                T term = poisb * gamb;
                sum += term;
                poisb *= i / lambda;
                xtermb *= (del + i) / y;
                gamb -= xtermb;
                if((sum == 0) || (fabs(term / sum) < errtol))
                   break;
             }
 
             return sum;
          }
 
          template <class T, class Policy>
          T non_central_chi_square_p_ding(T x, T f, T theta, const Policy& pol, T init_sum = 0)
          {
             //
             // This is an implementation of:
             //
             // Algorithm AS 275:
             // Computing the Non-Central #2 Distribution Function
             // Cherng G. Ding
             // Applied Statistics, Vol. 41, No. 2. (1992), pp. 478-482.
             //
             // This uses a stable forward iteration to sum the
             // CDF, unfortunately this can not be used for large
             // values of the non-centrality parameter because:
             // * The first term may underflow to zero.
             // * We may need an extra-ordinary number of terms
             //   before we reach the first *significant* term.
             //
             BOOST_MATH_STD_USING
             // Special case:
             if(x == 0)
                return 0;
             T tk = boost::math::gamma_p_derivative(f/2 + 1, x/2, pol);
             T lambda = theta / 2;
             T vk = exp(-lambda);
             T uk = vk;
             T sum = init_sum + tk * vk;
             if(sum == 0)
                return sum;
 
             std::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
             T errtol = boost::math::policies::get_epsilon<T, Policy>();
 
             int i;
             T lterm(0), term(0);
             for(i = 1; static_cast<std::uintmax_t>(i) < max_iter; ++i)
             {
                tk = tk * x / (f + 2 * i);
                uk = uk * lambda / i;
                vk = vk + uk;
                lterm = term;
                term = vk * tk;
                sum += term;
                if((fabs(term / sum) < errtol) && (term <= lterm))
                   break;
             }
             //Error check:
             if(static_cast<std::uintmax_t>(i) >= max_iter)
                return policies::raise_evaluation_error("cdf(non_central_chi_squared_distribution<%1%>, %1%)", "Series did not converge, closest value was %1%", sum, pol); // LCOV_EXCL_LINE
             return sum;
          }
 
 
          template <class T, class Policy>
          T non_central_chi_square_p(T y, T n, T lambda, const Policy& pol, T init_sum)
          {
             //
             // This is taken more or less directly from:
             //
             // Computing discrete mixtures of continuous
             // distributions: noncentral chisquare, noncentral t
             // and the distribution of the square of the sample
             // multiple correlation coefficient.
             // D. Benton, K. Krishnamoorthy.
             // Computational Statistics & Data Analysis 43 (2003) 249 - 267
             //
             // We're summing a Poisson weighting term multiplied by
             // a central chi squared distribution.
             //
             BOOST_MATH_STD_USING
             // Special case:
             if(y == 0)
                return 0;
             std::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
             T errtol = boost::math::policies::get_epsilon<T, Policy>();
             T errorf(0), errorb(0);
 
             T x = y / 2;
             T del = lambda / 2;
             //
             // Starting location for the iteration, we'll iterate
             // both forwards and backwards from this point.  The
             // location chosen is the maximum of the Poisson weight
             // function, which ocurrs *after* the largest term in the
             // sum.
             //
             long long k = llround(del, pol);
             T a = n / 2 + k;
             // Central chi squared term for forward iteration:
             T gamkf = boost::math::gamma_p(a, x, pol);
 
             if(lambda == 0)
                return gamkf;
             // Central chi squared term for backward iteration:
             T gamkb = gamkf;
             // Forwards Poisson weight:
             T poiskf = gamma_p_derivative(static_cast<T>(k+1), del, pol);
             // Backwards Poisson weight:
             T poiskb = poiskf;
             // Forwards gamma function recursion term:
             T xtermf = boost::math::gamma_p_derivative(a, x, pol);
             // Backwards gamma function recursion term:
             T xtermb = xtermf * x / a;
             T sum = init_sum + poiskf * gamkf;
             if(sum == 0)
                return sum;
             int i = 1;
             //
             // Backwards recursion first, this is the stable
             // direction for gamma function recurrences:
             //
             while(i <= k)
             {
                xtermb *= (a - i + 1) / x;
                gamkb += xtermb;
                poiskb = poiskb * (k - i + 1) / del;
                errorf = errorb;
                errorb = gamkb * poiskb;
                sum += errorb;
                if((fabs(errorb / sum) < errtol) && (errorb <= errorf))
                   break;
                ++i;
             }
             i = 1;
             //
             // Now forwards recursion, the gamma function
             // recurrence relation is unstable in this direction,
             // so we rely on the magnitude of successive terms
             // decreasing faster than we introduce cancellation error.
             // For this reason it's vital that k is chosen to be *after*
             // the largest term, so that successive forward iterations
             // are strictly (and rapidly) converging.
             //
             do
             {
                xtermf = xtermf * x / (a + i - 1);
                gamkf = gamkf - xtermf;
                poiskf = poiskf * del / (k + i);
                errorf = poiskf * gamkf;
                sum += errorf;
                ++i;
             }while((fabs(errorf / sum) > errtol) && (static_cast<std::uintmax_t>(i) < max_iter));
 
             //Error check:
             if(static_cast<std::uintmax_t>(i) >= max_iter)
                return policies::raise_evaluation_error("cdf(non_central_chi_squared_distribution<%1%>, %1%)", "Series did not converge, closest value was %1%", sum, pol); // LCOV_EXCL_LINE
 
             return sum;
          }
 
          template <class T, class Policy>
          T non_central_chi_square_pdf(T x, T n, T lambda, const Policy& pol)
          {
             //
             // As above but for the PDF:
             //
             BOOST_MATH_STD_USING
             std::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
             T errtol = boost::math::policies::get_epsilon<T, Policy>();
             T x2 = x / 2;
             T n2 = n / 2;
             T l2 = lambda / 2;
             T sum = 0;
             long long k = lltrunc(l2);
             T pois = gamma_p_derivative(static_cast<T>(k + 1), l2, pol) * gamma_p_derivative(static_cast<T>(n2 + k), x2);
             if(pois == 0)
                return 0;
             T poisb = pois;
             for(long long i = k; ; ++i)
             {
                sum += pois;
                if(pois / sum < errtol)
                   break;
                if(static_cast<std::uintmax_t>(i - k) >= max_iter)
                   return policies::raise_evaluation_error("pdf(non_central_chi_squared_distribution<%1%>, %1%)", "Series did not converge, closest value was %1%", sum, pol); // LCOV_EXCL_LINE
                pois *= l2 * x2 / ((i + 1) * (n2 + i));
             }
             for(long long i = k - 1; i >= 0; --i)
             {
                poisb *= (i + 1) * (n2 + i) / (l2 * x2);
                sum += poisb;
                if(poisb / sum < errtol)
                   break;
             }
             return sum / 2;
          }
 
          template <class RealType, class Policy>
          inline RealType non_central_chi_squared_cdf(RealType x, RealType k, RealType l, bool invert, const Policy&)
          {
             typedef typename policies::evaluation<RealType, Policy>::type value_type;
             typedef typename policies::normalise<
                Policy,
                policies::promote_float<false>,
                policies::promote_double<false>,
                policies::discrete_quantile<>,
                policies::assert_undefined<> >::type forwarding_policy;
 
             BOOST_MATH_STD_USING
             value_type result;
             if(l == 0)
               return invert == false ? cdf(boost::math::chi_squared_distribution<RealType, Policy>(k), x) : cdf(complement(boost::math::chi_squared_distribution<RealType, Policy>(k), x));
             else if(x > k + l)
             {
                // Complement is the smaller of the two:
                result = detail::non_central_chi_square_q(
                   static_cast<value_type>(x),
                   static_cast<value_type>(k),
                   static_cast<value_type>(l),
                   forwarding_policy(),
                   static_cast<value_type>(invert ? 0 : -1));
                invert = !invert;
             }
             else if(l < 200)
             {
                // For small values of the non-centrality parameter
                // we can use Ding's method:
                result = detail::non_central_chi_square_p_ding(
                   static_cast<value_type>(x),
                   static_cast<value_type>(k),
                   static_cast<value_type>(l),
                   forwarding_policy(),
                   static_cast<value_type>(invert ? -1 : 0));
             }
             else
             {
                // For largers values of the non-centrality
                // parameter Ding's method will consume an
                // extra-ordinary number of terms, and worse
                // may return zero when the result is in fact
                // finite, use Krishnamoorthy's method instead:
                result = detail::non_central_chi_square_p(
                   static_cast<value_type>(x),
                   static_cast<value_type>(k),
                   static_cast<value_type>(l),
                   forwarding_policy(),
                   static_cast<value_type>(invert ? -1 : 0));
             }
             if(invert)
                result = -result;
             return policies::checked_narrowing_cast<RealType, forwarding_policy>(
                result,
                "boost::math::non_central_chi_squared_cdf<%1%>(%1%, %1%, %1%)");
          }
 
          template <class T, class Policy>
          struct nccs_quantile_functor
          {
             nccs_quantile_functor(const non_central_chi_squared_distribution<T,Policy>& d, T t, bool c)
                : dist(d), target(t), comp(c) {}
 
             T operator()(const T& x)
             {
                return comp ?
                   target - cdf(complement(dist, x))
                   : cdf(dist, x) - target;
             }
 
          private:
             non_central_chi_squared_distribution<T,Policy> dist;
             T target;
             bool comp;
          };
 
          template <class RealType, class Policy>
          RealType nccs_quantile(const non_central_chi_squared_distribution<RealType, Policy>& dist, const RealType& p, bool comp)
          {
             BOOST_MATH_STD_USING
             static const char* function = "quantile(non_central_chi_squared_distribution<%1%>, %1%)";
             typedef typename policies::evaluation<RealType, Policy>::type value_type;
             typedef typename policies::normalise<
                Policy,
                policies::promote_float<false>,
                policies::promote_double<false>,
                policies::discrete_quantile<>,
                policies::assert_undefined<> >::type forwarding_policy;
 
             value_type k = dist.degrees_of_freedom();
             value_type l = dist.non_centrality();
             value_type r;
             if(!detail::check_df(
                function,
                k, &r, Policy())
                ||
             !detail::check_non_centrality(
                function,
                l,
                &r,
                Policy())
                ||
             !detail::check_probability(
                function,
                static_cast<value_type>(p),
                &r,
                Policy()))
                   return static_cast<RealType>(r);
             //
             // Special cases get short-circuited first:
             //
             if(p == 0)
                return comp ? policies::raise_overflow_error<RealType>(function, 0, Policy()) : 0;
             if(p == 1)
                return comp ? 0 : policies::raise_overflow_error<RealType>(function, 0, Policy());
             //
             // This is Pearson's approximation to the quantile, see
             // Pearson, E. S. (1959) "Note on an approximation to the distribution of
             // noncentral chi squared", Biometrika 46: 364.
             // See also:
             // "A comparison of approximations to percentiles of the noncentral chi2-distribution",
             // Hardeo Sahai and Mario Miguel Ojeda, Revista de Matematica: Teoria y Aplicaciones 2003 10(1-2) : 57-76.
             // Note that the latter reference refers to an approximation of the CDF, when they really mean the quantile.
             //
             value_type b = -(l * l) / (k + 3 * l);
             value_type c = (k + 3 * l) / (k + 2 * l);
             value_type ff = (k + 2 * l) / (c * c);
             value_type guess;
             if(comp)
             {
                guess = b + c * quantile(complement(chi_squared_distribution<value_type, forwarding_policy>(ff), p));
             }
             else
             {
                guess = b + c * quantile(chi_squared_distribution<value_type, forwarding_policy>(ff), p);
             }
             //
             // Sometimes guess goes very small or negative, in that case we have
             // to do something else for the initial guess, this approximation
             // was provided in a private communication from Thomas Luu, PhD candidate,
             // University College London.  It's an asymptotic expansion for the
             // quantile which usually gets us within an order of magnitude of the
             // correct answer.
             // Fast and accurate parallel computation of quantile functions for random number generation,
             // Thomas LuuDoctorial Thesis 2016
             // http://discovery.ucl.ac.uk/1482128/
             //
             if(guess < 0.005)
             {
                value_type pp = comp ? 1 - p : p;
                //guess = pow(pow(value_type(2), (k / 2 - 1)) * exp(l / 2) * pp * k, 2 / k);
                guess = pow(pow(value_type(2), (k / 2 - 1)) * exp(l / 2) * pp * k * boost::math::tgamma(k / 2, forwarding_policy()), (2 / k));
                if(guess == 0)
                   guess = tools::min_value<value_type>();
             }
             value_type result = detail::generic_quantile(
                non_central_chi_squared_distribution<value_type, forwarding_policy>(k, l),
                p,
                guess,
                comp,
                function);
 
             return policies::checked_narrowing_cast<RealType, forwarding_policy>(
                result,
                function);
          }
 
          template <class RealType, class Policy>
          RealType nccs_pdf(const non_central_chi_squared_distribution<RealType, Policy>& dist, const RealType& x)
          {
             BOOST_MATH_STD_USING
             static const char* function = "pdf(non_central_chi_squared_distribution<%1%>, %1%)";
             typedef typename policies::evaluation<RealType, Policy>::type value_type;
             typedef typename policies::normalise<
                Policy,
                policies::promote_float<false>,
                policies::promote_double<false>,
                policies::discrete_quantile<>,
                policies::assert_undefined<> >::type forwarding_policy;
 
             value_type k = dist.degrees_of_freedom();
             value_type l = dist.non_centrality();
             value_type r;
             if(!detail::check_df(
                function,
                k, &r, Policy())
                ||
             !detail::check_non_centrality(
                function,
                l,
                &r,
                Policy())
                ||
             !detail::check_positive_x(
                function,
                (value_type)x,
                &r,
                Policy()))
                   return static_cast<RealType>(r);
 
          if(l == 0)
             return pdf(boost::math::chi_squared_distribution<RealType, forwarding_policy>(dist.degrees_of_freedom()), x);
 
          // Special case:
          if(x == 0)
             return 0;
          if(l > 50)
          {
             r = non_central_chi_square_pdf(static_cast<value_type>(x), k, l, forwarding_policy());
          }
          else
          {
             r = log(x / l) * (k / 4 - 0.5f) - (x + l) / 2;
             if(fabs(r) >= tools::log_max_value<RealType>() / 4)
             {
                r = non_central_chi_square_pdf(static_cast<value_type>(x), k, l, forwarding_policy());
             }
             else
             {
                r = exp(r);
                r = 0.5f * r
                   * boost::math::cyl_bessel_i(k/2 - 1, sqrt(l * x), forwarding_policy());
             }
          }
          return policies::checked_narrowing_cast<RealType, forwarding_policy>(
                r,
                function);
          }
 
          template <class RealType, class Policy>
          struct degrees_of_freedom_finder
          {
             degrees_of_freedom_finder(
                RealType lam_, RealType x_, RealType p_, bool c)
                : lam(lam_), x(x_), p(p_), comp(c) {}
 
             RealType operator()(const RealType& v)
             {
                non_central_chi_squared_distribution<RealType, Policy> d(v, lam);
                return comp ?
                   RealType(p - cdf(complement(d, x)))
                   : RealType(cdf(d, x) - p);
             }
          private:
             RealType lam;
             RealType x;
             RealType p;
             bool comp;
          };
 
          template <class RealType, class Policy>
          inline RealType find_degrees_of_freedom(
             RealType lam, RealType x, RealType p, RealType q, const Policy& pol)
          {
             const char* function = "non_central_chi_squared<%1%>::find_degrees_of_freedom";
             if((p == 0) || (q == 0))
             {
                //
                // Can't a thing if one of p and q is zero:
                //
                return policies::raise_evaluation_error<RealType>(function, "Can't find degrees of freedom when the probability is 0 or 1, only possible answer is %1%", // LCOV_EXCL_LINE
                   RealType(std::numeric_limits<RealType>::quiet_NaN()), Policy()); // LCOV_EXCL_LINE
             }
             degrees_of_freedom_finder<RealType, Policy> f(lam, x, p < q ? p : q, p < q ? false : true);
             tools::eps_tolerance<RealType> tol(policies::digits<RealType, Policy>());
             std::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();
             //
             // Pick an initial guess that we know will give us a probability
             // right around 0.5.
             //
             RealType guess = x - lam;
             if(guess < 1)
                guess = 1;
             std::pair<RealType, RealType> ir = tools::bracket_and_solve_root(
                f, guess, RealType(2), false, tol, max_iter, pol);
             RealType result = ir.first + (ir.second - ir.first) / 2;
             if(max_iter >= policies::get_max_root_iterations<Policy>())
             {
                return policies::raise_evaluation_error<RealType>(function, "Unable to locate solution in a reasonable time:" // LCOV_EXCL_LINE
                   " or there is no answer to problem.  Current best guess is %1%", result, Policy()); // LCOV_EXCL_LINE
             }
             return result;
          }
 
          template <class RealType, class Policy>
          struct non_centrality_finder
          {
             non_centrality_finder(
                RealType v_, RealType x_, RealType p_, bool c)
                : v(v_), x(x_), p(p_), comp(c) {}
 
             RealType operator()(const RealType& lam)
             {
                non_central_chi_squared_distribution<RealType, Policy> d(v, lam);
                return comp ?
                   RealType(p - cdf(complement(d, x)))
                   : RealType(cdf(d, x) - p);
             }
          private:
             RealType v;
             RealType x;
             RealType p;
             bool comp;
          };
 
          template <class RealType, class Policy>
          inline RealType find_non_centrality(
             RealType v, RealType x, RealType p, RealType q, const Policy& pol)
          {
             const char* function = "non_central_chi_squared<%1%>::find_non_centrality";
             if((p == 0) || (q == 0))
             {
                //
                // Can't do a thing if one of p and q is zero:
                //
                return policies::raise_evaluation_error<RealType>(function, "Can't find non centrality parameter when the probability is 0 or 1, only possible answer is %1%", // LCOV_EXCL_LINE
                   RealType(std::numeric_limits<RealType>::quiet_NaN()), Policy()); // LCOV_EXCL_LINE
             }
             non_centrality_finder<RealType, Policy> f(v, x, p < q ? p : q, p < q ? false : true);
             tools::eps_tolerance<RealType> tol(policies::digits<RealType, Policy>());
             std::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();
             //
             // Pick an initial guess that we know will give us a probability
             // right around 0.5.
             //
             RealType guess = x - v;
             if(guess < 1)
                guess = 1;
             std::pair<RealType, RealType> ir = tools::bracket_and_solve_root(
                f, guess, RealType(2), false, tol, max_iter, pol);
             RealType result = ir.first + (ir.second - ir.first) / 2;
             if(max_iter >= policies::get_max_root_iterations<Policy>())
             {
                return policies::raise_evaluation_error<RealType>(function, "Unable to locate solution in a reasonable time:" // LCOV_EXCL_LINE
                   " or there is no answer to problem.  Current best guess is %1%", result, Policy()); // LCOV_EXCL_LINE
             }
             return result;
          }
 
       }
 
       template <class RealType = double, class Policy = policies::policy<> >
       class non_central_chi_squared_distribution
       {
       public:
          typedef RealType value_type;
          typedef Policy policy_type;
 
          non_central_chi_squared_distribution(RealType df_, RealType lambda) : df(df_), ncp(lambda)
          {
             const char* function = "boost::math::non_central_chi_squared_distribution<%1%>::non_central_chi_squared_distribution(%1%,%1%)";
             RealType r;
             detail::check_df(
                function,
                df, &r, Policy());
             detail::check_non_centrality(
                function,
                ncp,
                &r,
                Policy());
          } // non_central_chi_squared_distribution constructor.
 
          RealType degrees_of_freedom() const
          { // Private data getter function.
             return df;
          }
          RealType non_centrality() const
          { // Private data getter function.
             return ncp;
          }
          static RealType find_degrees_of_freedom(RealType lam, RealType x, RealType p)
          {
             const char* function = "non_central_chi_squared<%1%>::find_degrees_of_freedom";
             typedef typename policies::evaluation<RealType, Policy>::type eval_type;
             typedef typename policies::normalise<
                Policy,
                policies::promote_float<false>,
                policies::promote_double<false>,
                policies::discrete_quantile<>,
                policies::assert_undefined<> >::type forwarding_policy;
             eval_type result = detail::find_degrees_of_freedom(
                static_cast<eval_type>(lam),
                static_cast<eval_type>(x),
                static_cast<eval_type>(p),
                static_cast<eval_type>(1-p),
                forwarding_policy());
             return policies::checked_narrowing_cast<RealType, forwarding_policy>(
                result,
                function);
          }
          template <class A, class B, class C>
          static RealType find_degrees_of_freedom(const complemented3_type<A,B,C>& c)
          {
             const char* function = "non_central_chi_squared<%1%>::find_degrees_of_freedom";
             typedef typename policies::evaluation<RealType, Policy>::type eval_type;
             typedef typename policies::normalise<
                Policy,
                policies::promote_float<false>,
                policies::promote_double<false>,
                policies::discrete_quantile<>,
                policies::assert_undefined<> >::type forwarding_policy;
             eval_type result = detail::find_degrees_of_freedom(
                static_cast<eval_type>(c.dist),
                static_cast<eval_type>(c.param1),
                static_cast<eval_type>(1-c.param2),
                static_cast<eval_type>(c.param2),
                forwarding_policy());
             return policies::checked_narrowing_cast<RealType, forwarding_policy>(
                result,
                function);
          }
          static RealType find_non_centrality(RealType v, RealType x, RealType p)
          {
             const char* function = "non_central_chi_squared<%1%>::find_non_centrality";
             typedef typename policies::evaluation<RealType, Policy>::type eval_type;
             typedef typename policies::normalise<
                Policy,
                policies::promote_float<false>,
                policies::promote_double<false>,
                policies::discrete_quantile<>,
                policies::assert_undefined<> >::type forwarding_policy;
             eval_type result = detail::find_non_centrality(
                static_cast<eval_type>(v),
                static_cast<eval_type>(x),
                static_cast<eval_type>(p),
                static_cast<eval_type>(1-p),
                forwarding_policy());
             return policies::checked_narrowing_cast<RealType, forwarding_policy>(
                result,
                function);
          }
          template <class A, class B, class C>
          static RealType find_non_centrality(const complemented3_type<A,B,C>& c)
          {
             const char* function = "non_central_chi_squared<%1%>::find_non_centrality";
             typedef typename policies::evaluation<RealType, Policy>::type eval_type;
             typedef typename policies::normalise<
                Policy,
                policies::promote_float<false>,
                policies::promote_double<false>,
                policies::discrete_quantile<>,
                policies::assert_undefined<> >::type forwarding_policy;
             eval_type result = detail::find_non_centrality(
                static_cast<eval_type>(c.dist),
                static_cast<eval_type>(c.param1),
                static_cast<eval_type>(1-c.param2),
                static_cast<eval_type>(c.param2),
                forwarding_policy());
             return policies::checked_narrowing_cast<RealType, forwarding_policy>(
                result,
                function);
          }
       private:
          // Data member, initialized by constructor.
          RealType df; // degrees of freedom.
          RealType ncp; // non-centrality parameter
       }; // template <class RealType, class Policy> class non_central_chi_squared_distribution
 
       typedef non_central_chi_squared_distribution<double> non_central_chi_squared; // Reserved name of type double.
 
       #ifdef __cpp_deduction_guides
       template <class RealType>
       non_central_chi_squared_distribution(RealType,RealType)->non_central_chi_squared_distribution<typename boost::math::tools::promote_args<RealType>::type>;
       #endif
 
       // Non-member functions to give properties of the distribution.
 
       template <class RealType, class Policy>
       inline const std::pair<RealType, RealType> range(const non_central_chi_squared_distribution<RealType, Policy>& /* dist */)
       { // Range of permissible values for random variable k.
          using boost::math::tools::max_value;
          return std::pair<RealType, RealType>(static_cast<RealType>(0), max_value<RealType>()); // Max integer?
       }
 
       template <class RealType, class Policy>
       inline const std::pair<RealType, RealType> support(const non_central_chi_squared_distribution<RealType, Policy>& /* dist */)
       { // Range of supported values for random variable k.
          // This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.
          using boost::math::tools::max_value;
          return std::pair<RealType, RealType>(static_cast<RealType>(0),  max_value<RealType>());
       }
 
       template <class RealType, class Policy>
       inline RealType mean(const non_central_chi_squared_distribution<RealType, Policy>& dist)
       { // Mean of poisson distribution = lambda.
          const char* function = "boost::math::non_central_chi_squared_distribution<%1%>::mean()";
          RealType k = dist.degrees_of_freedom();
          RealType l = dist.non_centrality();
          RealType r;
          if(!detail::check_df(
             function,
             k, &r, Policy())
             ||
          !detail::check_non_centrality(
             function,
             l,
             &r,
             Policy()))
                return static_cast<RealType>(r);
          return k + l;
       } // mean
 
       template <class RealType, class Policy>
       inline RealType mode(const non_central_chi_squared_distribution<RealType, Policy>& dist)
       { // mode.
          static const char* function = "mode(non_central_chi_squared_distribution<%1%> const&)";
 
          RealType k = dist.degrees_of_freedom();
          RealType l = dist.non_centrality();
          RealType r;
          if(!detail::check_df(
             function,
             k, &r, Policy())
             ||
          !detail::check_non_centrality(
             function,
             l,
             &r,
             Policy()))
                return static_cast<RealType>(r);
          bool asymptotic_mode = k < l/4;
          RealType starting_point = asymptotic_mode ? k + l - RealType(3) : RealType(1) + k;
          return detail::generic_find_mode(dist, starting_point, function);
       }
 
       template <class RealType, class Policy>
       inline RealType variance(const non_central_chi_squared_distribution<RealType, Policy>& dist)
       { // variance.
          const char* function = "boost::math::non_central_chi_squared_distribution<%1%>::variance()";
          RealType k = dist.degrees_of_freedom();
          RealType l = dist.non_centrality();
          RealType r;
          if(!detail::check_df(
             function,
             k, &r, Policy())
             ||
          !detail::check_non_centrality(
             function,
             l,
             &r,
             Policy()))
                return static_cast<RealType>(r);
          return 2 * (2 * l + k);
       }
 
       // RealType standard_deviation(const non_central_chi_squared_distribution<RealType, Policy>& dist)
       // standard_deviation provided by derived accessors.
 
       template <class RealType, class Policy>
       inline RealType skewness(const non_central_chi_squared_distribution<RealType, Policy>& dist)
       { // skewness = sqrt(l).
          const char* function = "boost::math::non_central_chi_squared_distribution<%1%>::skewness()";
          RealType k = dist.degrees_of_freedom();
          RealType l = dist.non_centrality();
          RealType r;
          if(!detail::check_df(
             function,
             k, &r, Policy())
             ||
          !detail::check_non_centrality(
             function,
             l,
             &r,
             Policy()))
                return static_cast<RealType>(r);
          BOOST_MATH_STD_USING
             return pow(2 / (k + 2 * l), RealType(3)/2) * (k + 3 * l);
       }
 
       template <class RealType, class Policy>
       inline RealType kurtosis_excess(const non_central_chi_squared_distribution<RealType, Policy>& dist)
       {
          const char* function = "boost::math::non_central_chi_squared_distribution<%1%>::kurtosis_excess()";
          RealType k = dist.degrees_of_freedom();
          RealType l = dist.non_centrality();
          RealType r;
          if(!detail::check_df(
             function,
             k, &r, Policy())
             ||
          !detail::check_non_centrality(
             function,
             l,
             &r,
             Policy()))
                return static_cast<RealType>(r);
          return 12 * (k + 4 * l) / ((k + 2 * l) * (k + 2 * l));
       } // kurtosis_excess
 
       template <class RealType, class Policy>
       inline RealType kurtosis(const non_central_chi_squared_distribution<RealType, Policy>& dist)
       {
          return kurtosis_excess(dist) + 3;
       }
 
       template <class RealType, class Policy>
       inline RealType pdf(const non_central_chi_squared_distribution<RealType, Policy>& dist, const RealType& x)
       { // Probability Density/Mass Function.
          return detail::nccs_pdf(dist, x);
       } // pdf
 
       template <class RealType, class Policy>
       RealType cdf(const non_central_chi_squared_distribution<RealType, Policy>& dist, const RealType& x)
       {
          const char* function = "boost::math::non_central_chi_squared_distribution<%1%>::cdf(%1%)";
          RealType k = dist.degrees_of_freedom();
          RealType l = dist.non_centrality();
          RealType r;
          if(!detail::check_df(
             function,
             k, &r, Policy())
             ||
          !detail::check_non_centrality(
             function,
             l,
             &r,
             Policy())
             ||
          !detail::check_positive_x(
             function,
             x,
             &r,
             Policy()))
                return static_cast<RealType>(r);
 
          return detail::non_central_chi_squared_cdf(x, k, l, false, Policy());
       } // cdf
 
       template <class RealType, class Policy>
       RealType cdf(const complemented2_type<non_central_chi_squared_distribution<RealType, Policy>, RealType>& c)
       { // Complemented Cumulative Distribution Function
          const char* function = "boost::math::non_central_chi_squared_distribution<%1%>::cdf(%1%)";
          non_central_chi_squared_distribution<RealType, Policy> const& dist = c.dist;
          RealType x = c.param;
          RealType k = dist.degrees_of_freedom();
          RealType l = dist.non_centrality();
          RealType r;
          if(!detail::check_df(
             function,
             k, &r, Policy())
             ||
          !detail::check_non_centrality(
             function,
             l,
             &r,
             Policy())
             ||
          !detail::check_positive_x(
             function,
             x,
             &r,
             Policy()))
                return static_cast<RealType>(r);
 
          return detail::non_central_chi_squared_cdf(x, k, l, true, Policy());
       } // ccdf
 
       template <class RealType, class Policy>
       inline RealType quantile(const non_central_chi_squared_distribution<RealType, Policy>& dist, const RealType& p)
       { // Quantile (or Percent Point) function.
          return detail::nccs_quantile(dist, p, false);
       } // quantile
 
       template <class RealType, class Policy>
       inline RealType quantile(const complemented2_type<non_central_chi_squared_distribution<RealType, Policy>, RealType>& c)
       { // Quantile (or Percent Point) function.
          return detail::nccs_quantile(c.dist, c.param, true);
       } // quantile complement.
 
    } // namespace math
 } // namespace boost
 
 // This include must be at the end, *after* the accessors
 // for this distribution have been defined, in order to
 // keep compilers that support two-phase lookup happy.
 #include <boost/math/distributions/detail/derived_accessors.hpp>
 
 #endif // BOOST_MATH_SPECIAL_NON_CENTRAL_CHI_SQUARE_HPP

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