non_central_beta.hpp

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            T fb = fa;
            //
            // Set up invocation count:
            //
            boost::uintmax_t count = max_iter - 1;

            if((fa < 0) == (guess < 0 ? !rising : rising))
            {
               //
               // Zero is to the right of b, so walk upwards
               // until we find it:
               //
               while((boost::math::sign)(fb) == (boost::math::sign)(fa))
               {
                  if(count == 0)
                  {
                     b = policies::raise_evaluation_error(function, "Unable to bracket root, last nearest value was %1%", b, pol);
                     return std::make_pair(a, b);
                  }
                  //
                  // Heuristic: every 20 iterations we double the growth factor in case the
                  // initial guess was *really* bad !
                  //
                  if((max_iter - count) % 20 == 0)
                     factor *= 2;
                  //
                  // Now go ahead and move are guess by "factor",
                  // we do this by reducing 1-guess by factor:
                  //
                  a = b;
                  fa = fb;
                  b = 1 - ((1 - b) / factor);
                  fb = f(b);
                  --count;
                  BOOST_MATH_INSTRUMENT_CODE("a = " << a << " b = " << b << " fa = " << fa << " fb = " << fb << " count = " << count);
               }
            }
            else
            {
               //
               // Zero is to the left of a, so walk downwards
               // until we find it:
               //
               while((boost::math::sign)(fb) == (boost::math::sign)(fa))
               {
                  if(fabs(a) < tools::min_value<T>())
                  {
                     // Escape route just in case the answer is zero!
                     max_iter -= count;
                     max_iter += 1;
                     return a > 0 ? std::make_pair(T(0), T(a)) : std::make_pair(T(a), T(0)); 
                  }
                  if(count == 0)
                  {
                     a = policies::raise_evaluation_error(function, "Unable to bracket root, last nearest value was %1%", a, pol);
                     return std::make_pair(a, b);
                  }
                  //
                  // Heuristic: every 20 iterations we double the growth factor in case the
                  // initial guess was *really* bad !
                  //
                  if((max_iter - count) % 20 == 0)
                     factor *= 2;
                  //
                  // Now go ahead and move are guess by "factor":
                  //
                  b = a;
                  fb = fa;
                  a /= factor;
                  fa = f(a);
                  --count;
                  BOOST_MATH_INSTRUMENT_CODE("a = " << a << " b = " << b << " fa = " << fa << " fb = " << fb << " count = " << count);
               }
            }
            max_iter -= count;
            max_iter += 1;
            std::pair<T, T> r = toms748_solve(
               f, 
               (a < 0 ? b : a), 
               (a < 0 ? a : b), 
               (a < 0 ? fb : fa), 
               (a < 0 ? fa : fb), 
               tol, 
               count, 
               pol);
            max_iter += count;
            BOOST_MATH_INSTRUMENT_CODE("max_iter = " << max_iter << " count = " << count);
            return r;
         }

         template <class RealType, class Policy>
         RealType nc_beta_quantile(const non_central_beta_distribution<RealType, Policy>& dist, const RealType& p, bool comp)
         {
            static const char* function = "quantile(non_central_beta_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 a = dist.alpha();
            value_type b = dist.beta();
            value_type l = dist.non_centrality();
            value_type r;
            if(!beta_detail::check_alpha(
               function,
               a, &r, Policy())
               ||
            !beta_detail::check_beta(
               function,
               b, &r, Policy())
               ||            
            !detail::check_non_centrality(
               function,
               l,
               &r,
               Policy())
               ||
            !detail::check_probability(
               function,
               static_cast<value_type>(p),
               &r,
               Policy()))
                  return (RealType)r;
            //
            // Special cases first:
            //
            if(p == 0)
               return comp
               ? 1.0f
               : 0.0f;
            if(p == 1)
               return !comp
               ? 1.0f
               : 0.0f;

            value_type c = a + b + l / 2;
            value_type mean = 1 - (b / c) * (1 + l / (2 * c * c));
            /*
            //
            // Calculate a normal approximation to the quantile, 
            // uses mean and variance approximations from:
            // Algorithm AS 310: 
            // Computing the Non-Central Beta Distribution Function
            // R. Chattamvelli; R. Shanmugam
            // Applied Statistics, Vol. 46, No. 1. (1997), pp. 146-156.
            //
            // Unfortunately, when this is wrong it tends to be *very*
            // wrong, so it's disabled for now, even though it often
            // gets the initial guess quite close.  Probably we could
            // do much better by factoring in the skewness if only
            // we could calculate it....
            //
            value_type delta = l / 2;
            value_type delta2 = delta * delta;
            value_type delta3 = delta * delta2;
            value_type delta4 = delta2 * delta2;
            value_type G = c * (c + 1) + delta;
            value_type alpha = a + b;
            value_type alpha2 = alpha * alpha;
            value_type eta = (2 * alpha + 1) * (2 * alpha + 1) + 1;
            value_type H = 3 * alpha2 + 5 * alpha + 2;
            value_type F = alpha2 * (alpha + 1) + H * delta 
               + (2 * alpha + 4) * delta2 + delta3;
            value_type P = (3 * alpha + 1) * (9 * alpha + 17)
               + 2 * alpha * (3 * alpha + 2) * (3 * alpha + 4) + 15;
            value_type Q = 54 * alpha2 + 162 * alpha + 130;
            value_type R = 6 * (6 * alpha + 11);
            value_type D = delta 
               * (H * H + 2 * P * delta + Q * delta2 + R * delta3 + 9 * delta4);
            value_type variance = (b / G)
               * (1 + delta * (l * l + 3 * l + eta) / (G * G))
               - (b * b / F) * (1 + D / (F * F));
            value_type sd = sqrt(variance);

            value_type guess = comp 
               ? quantile(complement(normal_distribution<RealType, Policy>(static_cast<RealType>(mean), static_cast<RealType>(sd)), p))
               : quantile(normal_distribution<RealType, Policy>(static_cast<RealType>(mean), static_cast<RealType>(sd)), p);

            if(guess >= 1)
               guess = mean;
            if(guess <= tools::min_value<value_type>())
               guess = mean;
            */
            value_type guess = mean;
            detail::nc_beta_quantile_functor<value_type, Policy>
               f(non_central_beta_distribution<value_type, Policy>(a, b, l), p, comp);
            tools::eps_tolerance<value_type> tol(policies::digits<RealType, Policy>());
            boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();

            std::pair<value_type, value_type> ir 
               = bracket_and_solve_root_01(
                  f, guess, value_type(2.5), true, tol, 
                  max_iter, Policy());
            value_type 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:"
                  " either there is no answer to quantile of the non central beta distribution"
                  " or the answer is infinite.  Current best guess is %1%", 
                  policies::checked_narrowing_cast<RealType, forwarding_policy>(
                     result, 
                     function), Policy());
            }
            return policies::checked_narrowing_cast<RealType, forwarding_policy>(
               result, 
               function);
         }

         template <class T, class Policy>
         T non_central_beta_pdf(T a, T b, T lam, T x, T y, const Policy& pol)
         {
            BOOST_MATH_STD_USING
               using namespace boost::math;
            //
            // Variables come first:
            //
            boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
            T errtol = ldexp(1.0f, -boost::math::policies::digits<T, Policy>());
            T l2 = lam / 2;
            //
            // k is the starting point for iteration, and is the
            // maximum of the poisson weighting term:
            //
            int k = itrunc(l2);
            // Starting Poisson weight:
            T pois = gamma_p_derivative(T(k+1), l2, pol);
            // Starting beta term:
            T beta = x < y ? 
               ibeta_derivative(a + k, b, x, pol)
               : ibeta_derivative(b, a + k, y, pol);
            T sum = 0;
            T poisf(pois);
            T betaf(beta);

            //
            // Stable backwards recursion first:
            //
            boost::uintmax_t count = k;
            for(int i = k; i >= 0; --i)
            {
               T term = beta * pois;
               sum += term;
               if((fabs(term/sum) < errtol) || (term == 0))
               {
                  count = k - i;
                  break;
               }
               pois *= i / l2;
               beta *= (a + i - 1) / (x * (a + i + b - 1));
            }
            for(int i = k + 1; ; ++i)
            {
               poisf *= l2 / i;
               betaf *= x * (a + b + i - 1) / (a + i - 1);

               T term = poisf * betaf;
               sum += term;
               if((fabs(term/sum) < errtol) || (term == 0))
               {
                  break;
               }
               if(static_cast<boost::uintmax_t>(count + i - k) > max_iter)
               {
                  return policies::raise_evaluation_error(
                     "pdf(non_central_beta_distribution<%1%>, %1%)", 
                     "Series did not converge, closest value was %1%", sum, pol);
               }
            }
            return sum;
         }

         template <class RealType, class Policy>
         RealType nc_beta_pdf(const non_central_beta_distribution<RealType, Policy>& dist, const RealType& x)
         {
            BOOST_MATH_STD_USING
            static const char* function = "pdf(non_central_beta_distribution<%1%>, %1%)";
            typedef typename policies::evaluation<RealType, Policy>::type value_type;
            typedef typename policies::normalise<