--- /dev/null
+// (C) Copyright John Maddock 2006.
+// 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_BETA_HPP
+#define BOOST_MATH_SPECIAL_BETA_HPP
+
+#ifdef _MSC_VER
+#pragma once
+#endif
+
+#include <boost/math/special_functions/math_fwd.hpp>
+#include <boost/math/tools/config.hpp>
+#include <boost/math/special_functions/gamma.hpp>
+#include <boost/math/special_functions/factorials.hpp>
+#include <boost/math/special_functions/erf.hpp>
+#include <boost/math/special_functions/log1p.hpp>
+#include <boost/math/special_functions/expm1.hpp>
+#include <boost/math/special_functions/trunc.hpp>
+#include <boost/math/tools/roots.hpp>
+#include <boost/static_assert.hpp>
+#include <boost/config/no_tr1/cmath.hpp>
+
+namespace boost{ namespace math{
+
+namespace detail{
+
+//
+// Implementation of Beta(a,b) using the Lanczos approximation:
+//
+template <class T, class Lanczos, class Policy>
+T beta_imp(T a, T b, const Lanczos&, const Policy& pol)
+{
+ BOOST_MATH_STD_USING // for ADL of std names
+
+ if(a <= 0)
+ policies::raise_domain_error<T>("boost::math::beta<%1%>(%1%,%1%)", "The arguments to the beta function must be greater than zero (got a=%1%).", a, pol);
+ if(b <= 0)
+ policies::raise_domain_error<T>("boost::math::beta<%1%>(%1%,%1%)", "The arguments to the beta function must be greater than zero (got b=%1%).", b, pol);
+
+ T result;
+
+ T prefix = 1;
+ T c = a + b;
+
+ // Special cases:
+ if((c == a) && (b < tools::epsilon<T>()))
+ return boost::math::tgamma(b, pol);
+ else if((c == b) && (a < tools::epsilon<T>()))
+ return boost::math::tgamma(a, pol);
+ if(b == 1)
+ return 1/a;
+ else if(a == 1)
+ return 1/b;
+
+ /*
+ //
+ // This code appears to be no longer necessary: it was
+ // used to offset errors introduced from the Lanczos
+ // approximation, but the current Lanczos approximations
+ // are sufficiently accurate for all z that we can ditch
+ // this. It remains in the file for future reference...
+ //
+ // If a or b are less than 1, shift to greater than 1:
+ if(a < 1)
+ {
+ prefix *= c / a;
+ c += 1;
+ a += 1;
+ }
+ if(b < 1)
+ {
+ prefix *= c / b;
+ c += 1;
+ b += 1;
+ }
+ */
+
+ if(a < b)
+ std::swap(a, b);
+
+ // Lanczos calculation:
+ T agh = a + Lanczos::g() - T(0.5);
+ T bgh = b + Lanczos::g() - T(0.5);
+ T cgh = c + Lanczos::g() - T(0.5);
+ result = Lanczos::lanczos_sum_expG_scaled(a) * Lanczos::lanczos_sum_expG_scaled(b) / Lanczos::lanczos_sum_expG_scaled(c);
+ T ambh = a - T(0.5) - b;
+ if((fabs(b * ambh) < (cgh * 100)) && (a > 100))
+ {
+ // Special case where the base of the power term is close to 1
+ // compute (1+x)^y instead:
+ result *= exp(ambh * boost::math::log1p(-b / cgh, pol));
+ }
+ else
+ {
+ result *= pow(agh / cgh, a - T(0.5) - b);
+ }
+ if(cgh > 1e10f)
+ // this avoids possible overflow, but appears to be marginally less accurate:
+ result *= pow((agh / cgh) * (bgh / cgh), b);
+ else
+ result *= pow((agh * bgh) / (cgh * cgh), b);
+ result *= sqrt(boost::math::constants::e<T>() / bgh);
+
+ // If a and b were originally less than 1 we need to scale the result:
+ result *= prefix;
+
+ return result;
+} // template <class T, class Lanczos> beta_imp(T a, T b, const Lanczos&)
+
+//
+// Generic implementation of Beta(a,b) without Lanczos approximation support
+// (Caution this is slow!!!):
+//
+template <class T, class Policy>
+T beta_imp(T a, T b, const lanczos::undefined_lanczos& /* l */, const Policy& pol)
+{
+ BOOST_MATH_STD_USING
+
+ if(a <= 0)
+ policies::raise_domain_error<T>("boost::math::beta<%1%>(%1%,%1%)", "The arguments to the beta function must be greater than zero (got a=%1%).", a, pol);
+ if(b <= 0)
+ policies::raise_domain_error<T>("boost::math::beta<%1%>(%1%,%1%)", "The arguments to the beta function must be greater than zero (got b=%1%).", b, pol);
+
+ T result;
+
+ T prefix = 1;
+ T c = a + b;
+
+ // special cases:
+ if((c == a) && (b < tools::epsilon<T>()))
+ return boost::math::tgamma(b, pol);
+ else if((c == b) && (a < tools::epsilon<T>()))
+ return boost::math::tgamma(a, pol);
+ if(b == 1)
+ return 1/a;
+ else if(a == 1)
+ return 1/b;
+
+ // shift to a and b > 1 if required:
+ if(a < 1)
+ {
+ prefix *= c / a;
+ c += 1;
+ a += 1;
+ }
+ if(b < 1)
+ {
+ prefix *= c / b;
+ c += 1;
+ b += 1;
+ }
+ if(a < b)
+ std::swap(a, b);
+
+ // set integration limits:
+ T la = (std::max)(T(10), a);
+ T lb = (std::max)(T(10), b);
+ T lc = (std::max)(T(10), T(a+b));
+
+ // calculate the fraction parts:
+ T sa = detail::lower_gamma_series(a, la, pol) / a;
+ sa += detail::upper_gamma_fraction(a, la, ::boost::math::policies::get_epsilon<T, Policy>());
+ T sb = detail::lower_gamma_series(b, lb, pol) / b;
+ sb += detail::upper_gamma_fraction(b, lb, ::boost::math::policies::get_epsilon<T, Policy>());
+ T sc = detail::lower_gamma_series(c, lc, pol) / c;
+ sc += detail::upper_gamma_fraction(c, lc, ::boost::math::policies::get_epsilon<T, Policy>());
+
+ // and the exponent part:
+ result = exp(lc - la - lb) * pow(la/lc, a) * pow(lb/lc, b);
+
+ // and combine:
+ result *= sa * sb / sc;
+
+ // if a and b were originally less than 1 we need to scale the result:
+ result *= prefix;
+
+ return result;
+} // template <class T>T beta_imp(T a, T b, const lanczos::undefined_lanczos& l)
+
+
+//
+// Compute the leading power terms in the incomplete Beta:
+//
+// (x^a)(y^b)/Beta(a,b) when normalised, and
+// (x^a)(y^b) otherwise.
+//
+// Almost all of the error in the incomplete beta comes from this
+// function: particularly when a and b are large. Computing large
+// powers are *hard* though, and using logarithms just leads to
+// horrendous cancellation errors.
+//
+template <class T, class Lanczos, class Policy>
+T ibeta_power_terms(T a,
+ T b,
+ T x,
+ T y,
+ const Lanczos&,
+ bool normalised,
+ const Policy& pol)
+{
+ BOOST_MATH_STD_USING
+
+ if(!normalised)
+ {
+ // can we do better here?
+ return pow(x, a) * pow(y, b);
+ }
+
+ T result;
+
+ T prefix = 1;
+ T c = a + b;
+
+ // combine power terms with Lanczos approximation:
+ T agh = a + Lanczos::g() - T(0.5);
+ T bgh = b + Lanczos::g() - T(0.5);
+ T cgh = c + Lanczos::g() - T(0.5);
+ result = Lanczos::lanczos_sum_expG_scaled(c) / (Lanczos::lanczos_sum_expG_scaled(a) * Lanczos::lanczos_sum_expG_scaled(b));
+
+ // l1 and l2 are the base of the exponents minus one:
+ T l1 = (x * b - y * agh) / agh;
+ T l2 = (y * a - x * bgh) / bgh;
+ if(((std::min)(fabs(l1), fabs(l2)) < 0.2))
+ {
+ // when the base of the exponent is very near 1 we get really
+ // gross errors unless extra care is taken:
+ if((l1 * l2 > 0) || ((std::min)(a, b) < 1))
+ {
+ //
+ // This first branch handles the simple cases where either:
+ //
+ // * The two power terms both go in the same direction
+ // (towards zero or towards infinity). In this case if either
+ // term overflows or underflows, then the product of the two must
+ // do so also.
+ // *Alternatively if one exponent is less than one, then we
+ // can't productively use it to eliminate overflow or underflow
+ // from the other term. Problems with spurious overflow/underflow
+ // can't be ruled out in this case, but it is *very* unlikely
+ // since one of the power terms will evaluate to a number close to 1.
+ //
+ if(fabs(l1) < 0.1)
+ {
+ result *= exp(a * boost::math::log1p(l1, pol));
+ BOOST_MATH_INSTRUMENT_VARIABLE(result);
+ }
+ else
+ {
+ result *= pow((x * cgh) / agh, a);
+ BOOST_MATH_INSTRUMENT_VARIABLE(result);
+ }
+ if(fabs(l2) < 0.1)
+ {
+ result *= exp(b * boost::math::log1p(l2, pol));
+ BOOST_MATH_INSTRUMENT_VARIABLE(result);
+ }
+ else
+ {
+ result *= pow((y * cgh) / bgh, b);
+ BOOST_MATH_INSTRUMENT_VARIABLE(result);
+ }
+ }
+ else if((std::max)(fabs(l1), fabs(l2)) < 0.5)
+ {
+ //
+ // Both exponents are near one and both the exponents are
+ // greater than one and further these two
+ // power terms tend in opposite directions (one towards zero,
+ // the other towards infinity), so we have to combine the terms
+ // to avoid any risk of overflow or underflow.
+ //
+ // We do this by moving one power term inside the other, we have:
+ //
+ // (1 + l1)^a * (1 + l2)^b
+ // = ((1 + l1)*(1 + l2)^(b/a))^a
+ // = (1 + l1 + l3 + l1*l3)^a ; l3 = (1 + l2)^(b/a) - 1
+ // = exp((b/a) * log(1 + l2)) - 1
+ //
+ // The tricky bit is deciding which term to move inside :-)
+ // By preference we move the larger term inside, so that the
+ // size of the largest exponent is reduced. However, that can
+ // only be done as long as l3 (see above) is also small.
+ //
+ bool small_a = a < b;
+ T ratio = b / a;
+ if((small_a && (ratio * l2 < 0.1)) || (!small_a && (l1 / ratio > 0.1)))
+ {
+ T l3 = boost::math::expm1(ratio * boost::math::log1p(l2, pol), pol);
+ l3 = l1 + l3 + l3 * l1;
+ l3 = a * boost::math::log1p(l3, pol);
+ result *= exp(l3);
+ BOOST_MATH_INSTRUMENT_VARIABLE(result);
+ }
+ else
+ {
+ T l3 = boost::math::expm1(boost::math::log1p(l1, pol) / ratio, pol);
+ l3 = l2 + l3 + l3 * l2;
+ l3 = b * boost::math::log1p(l3, pol);
+ result *= exp(l3);
+ BOOST_MATH_INSTRUMENT_VARIABLE(result);
+ }
+ }
+ else if(fabs(l1) < fabs(l2))
+ {
+ // First base near 1 only:
+ T l = a * boost::math::log1p(l1, pol)
+ + b * log((y * cgh) / bgh);
+ result *= exp(l);
+ BOOST_MATH_INSTRUMENT_VARIABLE(result);
+ }
+ else
+ {
+ // Second base near 1 only:
+ T l = b * boost::math::log1p(l2, pol)
+ + a * log((x * cgh) / agh);
+ result *= exp(l);
+ BOOST_MATH_INSTRUMENT_VARIABLE(result);
+ }
+ }
+ else
+ {
+ // general case:
+ T b1 = (x * cgh) / agh;
+ T b2 = (y * cgh) / bgh;
+ l1 = a * log(b1);
+ l2 = b * log(b2);
+ BOOST_MATH_INSTRUMENT_VARIABLE(b1);
+ BOOST_MATH_INSTRUMENT_VARIABLE(b2);
+ BOOST_MATH_INSTRUMENT_VARIABLE(l1);
+ BOOST_MATH_INSTRUMENT_VARIABLE(l2);
+ if((l1 >= tools::log_max_value<T>())
+ || (l1 <= tools::log_min_value<T>())
+ || (l2 >= tools::log_max_value<T>())
+ || (l2 <= tools::log_min_value<T>())
+ )
+ {
+ // Oops, overflow, sidestep:
+ if(a < b)
+ result *= pow(pow(b2, b/a) * b1, a);
+ else
+ result *= pow(pow(b1, a/b) * b2, b);
+ BOOST_MATH_INSTRUMENT_VARIABLE(result);
+ }
+ else
+ {
+ // finally the normal case:
+ result *= pow(b1, a) * pow(b2, b);
+ BOOST_MATH_INSTRUMENT_VARIABLE(result);
+ }
+ }
+ // combine with the leftover terms from the Lanczos approximation:
+ result *= sqrt(bgh / boost::math::constants::e<T>());
+ result *= sqrt(agh / cgh);
+ result *= prefix;
+
+ BOOST_MATH_INSTRUMENT_VARIABLE(result);
+
+ return result;
+}
+//
+// Compute the leading power terms in the incomplete Beta:
+//
+// (x^a)(y^b)/Beta(a,b) when normalised, and
+// (x^a)(y^b) otherwise.
+//
+// Almost all of the error in the incomplete beta comes from this
+// function: particularly when a and b are large. Computing large
+// powers are *hard* though, and using logarithms just leads to
+// horrendous cancellation errors.
+//
+// This version is generic, slow, and does not use the Lanczos approximation.
+//
+template <class T, class Policy>
+T ibeta_power_terms(T a,
+ T b,
+ T x,
+ T y,
+ const boost::math::lanczos::undefined_lanczos&,
+ bool normalised,
+ const Policy& pol)
+{
+ BOOST_MATH_STD_USING
+
+ if(!normalised)
+ {
+ return pow(x, a) * pow(y, b);
+ }
+
+ T result= 0; // assignment here silences warnings later
+
+ T c = a + b;
+
+ // integration limits for the gamma functions:
+ //T la = (std::max)(T(10), a);
+ //T lb = (std::max)(T(10), b);
+ //T lc = (std::max)(T(10), a+b);
+ T la = a + 5;
+ T lb = b + 5;
+ T lc = a + b + 5;
+ // gamma function partials:
+ T sa = detail::lower_gamma_series(a, la, pol) / a;
+ sa += detail::upper_gamma_fraction(a, la, ::boost::math::policies::get_epsilon<T, Policy>());
+ T sb = detail::lower_gamma_series(b, lb, pol) / b;
+ sb += detail::upper_gamma_fraction(b, lb, ::boost::math::policies::get_epsilon<T, Policy>());
+ T sc = detail::lower_gamma_series(c, lc, pol) / c;
+ sc += detail::upper_gamma_fraction(c, lc, ::boost::math::policies::get_epsilon<T, Policy>());
+ // gamma function powers combined with incomplete beta powers:
+
+ T b1 = (x * lc) / la;
+ T b2 = (y * lc) / lb;
+ T e1 = lc - la - lb;
+ T lb1 = a * log(b1);
+ T lb2 = b * log(b2);
+
+ if((lb1 >= tools::log_max_value<T>())
+ || (lb1 <= tools::log_min_value<T>())
+ || (lb2 >= tools::log_max_value<T>())
+ || (lb2 <= tools::log_min_value<T>())
+ || (e1 >= tools::log_max_value<T>())
+ || (e1 <= tools::log_min_value<T>())
+ )
+ {
+ result = exp(lb1 + lb2 - e1);
+ }
+ else
+ {
+ T p1, p2;
+ if((fabs(b1 - 1) * a < 10) && (a > 1))
+ p1 = exp(a * boost::math::log1p((x * b - y * la) / la, pol));
+ else
+ p1 = pow(b1, a);
+ if((fabs(b2 - 1) * b < 10) && (b > 1))
+ p2 = exp(b * boost::math::log1p((y * a - x * lb) / lb, pol));
+ else
+ p2 = pow(b2, b);
+ T p3 = exp(e1);
+ result = p1 * p2 / p3;
+ }
+ // and combine with the remaining gamma function components:
+ result /= sa * sb / sc;
+
+ return result;
+}
+//
+// Series approximation to the incomplete beta:
+//
+template <class T>
+struct ibeta_series_t
+{
+ typedef T result_type;
+ ibeta_series_t(T a_, T b_, T x_, T mult) : result(mult), x(x_), apn(a_), poch(1-b_), n(1) {}
+ T operator()()
+ {
+ T r = result / apn;
+ apn += 1;
+ result *= poch * x / n;
+ ++n;
+ poch += 1;
+ return r;
+ }
+private:
+ T result, x, apn, poch;
+ int n;
+};
+
+template <class T, class Lanczos, class Policy>
+T ibeta_series(T a, T b, T x, T s0, const Lanczos&, bool normalised, T* p_derivative, T y, const Policy& pol)
+{
+ BOOST_MATH_STD_USING
+
+ T result;
+
+ BOOST_ASSERT((p_derivative == 0) || normalised);
+
+ if(normalised)
+ {
+ T c = a + b;
+
+ // incomplete beta power term, combined with the Lanczos approximation:
+ T agh = a + Lanczos::g() - T(0.5);
+ T bgh = b + Lanczos::g() - T(0.5);
+ T cgh = c + Lanczos::g() - T(0.5);
+ result = Lanczos::lanczos_sum_expG_scaled(c) / (Lanczos::lanczos_sum_expG_scaled(a) * Lanczos::lanczos_sum_expG_scaled(b));
+ if(a * b < bgh * 10)
+ result *= exp((b - 0.5f) * boost::math::log1p(a / bgh, pol));
+ else
+ result *= pow(cgh / bgh, b - 0.5f);
+ result *= pow(x * cgh / agh, a);
+ result *= sqrt(agh / boost::math::constants::e<T>());
+
+ if(p_derivative)
+ {
+ *p_derivative = result * pow(y, b);
+ BOOST_ASSERT(*p_derivative >= 0);
+ }
+ }
+ else
+ {
+ // Non-normalised, just compute the power:
+ result = pow(x, a);
+ }
+ if(result < tools::min_value<T>())
+ return s0; // Safeguard: series can't cope with denorms.
+ ibeta_series_t<T> s(a, b, x, result);
+ boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
+ result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter, s0);
+ policies::check_series_iterations<T>("boost::math::ibeta<%1%>(%1%, %1%, %1%) in ibeta_series (with lanczos)", max_iter, pol);
+ return result;
+}
+//
+// Incomplete Beta series again, this time without Lanczos support:
+//
+template <class T, class Policy>
+T ibeta_series(T a, T b, T x, T s0, const boost::math::lanczos::undefined_lanczos&, bool normalised, T* p_derivative, T y, const Policy& pol)
+{
+ BOOST_MATH_STD_USING
+
+ T result;
+ BOOST_ASSERT((p_derivative == 0) || normalised);
+
+ if(normalised)
+ {
+ T c = a + b;
+
+ // figure out integration limits for the gamma function:
+ //T la = (std::max)(T(10), a);
+ //T lb = (std::max)(T(10), b);
+ //T lc = (std::max)(T(10), a+b);
+ T la = a + 5;
+ T lb = b + 5;
+ T lc = a + b + 5;
+
+ // calculate the gamma parts:
+ T sa = detail::lower_gamma_series(a, la, pol) / a;
+ sa += detail::upper_gamma_fraction(a, la, ::boost::math::policies::get_epsilon<T, Policy>());
+ T sb = detail::lower_gamma_series(b, lb, pol) / b;
+ sb += detail::upper_gamma_fraction(b, lb, ::boost::math::policies::get_epsilon<T, Policy>());
+ T sc = detail::lower_gamma_series(c, lc, pol) / c;
+ sc += detail::upper_gamma_fraction(c, lc, ::boost::math::policies::get_epsilon<T, Policy>());
+
+ // and their combined power-terms:
+ T b1 = (x * lc) / la;
+ T b2 = lc/lb;
+ T e1 = lc - la - lb;
+ T lb1 = a * log(b1);
+ T lb2 = b * log(b2);
+
+ if((lb1 >= tools::log_max_value<T>())
+ || (lb1 <= tools::log_min_value<T>())
+ || (lb2 >= tools::log_max_value<T>())
+ || (lb2 <= tools::log_min_value<T>())
+ || (e1 >= tools::log_max_value<T>())
+ || (e1 <= tools::log_min_value<T>()) )
+ {
+ T p = lb1 + lb2 - e1;
+ result = exp(p);
+ }
+ else
+ {
+ result = pow(b1, a);
+ if(a * b < lb * 10)
+ result *= exp(b * boost::math::log1p(a / lb, pol));
+ else
+ result *= pow(b2, b);
+ result /= exp(e1);
+ }
+ // and combine the results:
+ result /= sa * sb / sc;
+
+ if(p_derivative)
+ {
+ *p_derivative = result * pow(y, b);
+ BOOST_ASSERT(*p_derivative >= 0);
+ }
+ }
+ else
+ {
+ // Non-normalised, just compute the power:
+ result = pow(x, a);
+ }
+ if(result < tools::min_value<T>())
+ return s0; // Safeguard: series can't cope with denorms.
+ ibeta_series_t<T> s(a, b, x, result);
+ boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
+ result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<T, Policy>(), max_iter, s0);
+ policies::check_series_iterations<T>("boost::math::ibeta<%1%>(%1%, %1%, %1%) in ibeta_series (without lanczos)", max_iter, pol);
+ return result;
+}
+
+//
+// Continued fraction for the incomplete beta:
+//
+template <class T>
+struct ibeta_fraction2_t
+{
+ typedef std::pair<T, T> result_type;
+
+ ibeta_fraction2_t(T a_, T b_, T x_, T y_) : a(a_), b(b_), x(x_), y(y_), m(0) {}
+
+ result_type operator()()
+ {
+ T aN = (a + m - 1) * (a + b + m - 1) * m * (b - m) * x * x;
+ T denom = (a + 2 * m - 1);
+ aN /= denom * denom;
+
+ T bN = m;
+ bN += (m * (b - m) * x) / (a + 2*m - 1);
+ bN += ((a + m) * (a * y - b * x + 1 + m *(2 - x))) / (a + 2*m + 1);
+
+ ++m;
+
+ return std::make_pair(aN, bN);
+ }
+
+private:
+ T a, b, x, y;
+ int m;
+};
+//
+// Evaluate the incomplete beta via the continued fraction representation:
+//
+template <class T, class Policy>
+inline T ibeta_fraction2(T a, T b, T x, T y, const Policy& pol, bool normalised, T* p_derivative)
+{
+ typedef typename lanczos::lanczos<T, Policy>::type lanczos_type;
+ BOOST_MATH_STD_USING
+ T result = ibeta_power_terms(a, b, x, y, lanczos_type(), normalised, pol);
+ if(p_derivative)
+ {
+ *p_derivative = result;
+ BOOST_ASSERT(*p_derivative >= 0);
+ }
+ if(result == 0)
+ return result;
+
+ ibeta_fraction2_t<T> f(a, b, x, y);
+ T fract = boost::math::tools::continued_fraction_b(f, boost::math::policies::get_epsilon<T, Policy>());
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ BOOST_MATH_INSTRUMENT_VARIABLE(result);
+ return result / fract;
+}
+//
+// Computes the difference between ibeta(a,b,x) and ibeta(a+k,b,x):
+//
+template <class T, class Policy>
+T ibeta_a_step(T a, T b, T x, T y, int k, const Policy& pol, bool normalised, T* p_derivative)
+{
+ typedef typename lanczos::lanczos<T, Policy>::type lanczos_type;
+
+ BOOST_MATH_INSTRUMENT_VARIABLE(k);
+
+ T prefix = ibeta_power_terms(a, b, x, y, lanczos_type(), normalised, pol);
+ if(p_derivative)
+ {
+ *p_derivative = prefix;
+ BOOST_ASSERT(*p_derivative >= 0);
+ }
+ prefix /= a;
+ if(prefix == 0)
+ return prefix;
+ T sum = 1;
+ T term = 1;
+ // series summation from 0 to k-1:
+ for(int i = 0; i < k-1; ++i)
+ {
+ term *= (a+b+i) * x / (a+i+1);
+ sum += term;
+ }
+ prefix *= sum;
+
+ return prefix;
+}
+//
+// This function is only needed for the non-regular incomplete beta,
+// it computes the delta in:
+// beta(a,b,x) = prefix + delta * beta(a+k,b,x)
+// it is currently only called for small k.
+//
+template <class T>
+inline T rising_factorial_ratio(T a, T b, int k)
+{
+ // calculate:
+ // (a)(a+1)(a+2)...(a+k-1)
+ // _______________________
+ // (b)(b+1)(b+2)...(b+k-1)
+
+ // This is only called with small k, for large k
+ // it is grossly inefficient, do not use outside it's
+ // intended purpose!!!
+ BOOST_MATH_INSTRUMENT_VARIABLE(k);
+ if(k == 0)
+ return 1;
+ T result = 1;
+ for(int i = 0; i < k; ++i)
+ result *= (a+i) / (b+i);
+ return result;
+}
+//
+// Routine for a > 15, b < 1
+//
+// Begin by figuring out how large our table of Pn's should be,
+// quoted accuracies are "guestimates" based on empiracal observation.
+// Note that the table size should never exceed the size of our
+// tables of factorials.
+//
+template <class T>
+struct Pn_size
+{
+ // This is likely to be enough for ~35-50 digit accuracy
+ // but it's hard to quantify exactly:
+ BOOST_STATIC_CONSTANT(unsigned, value = 50);
+ BOOST_STATIC_ASSERT(::boost::math::max_factorial<T>::value >= 100);
+};
+template <>
+struct Pn_size<float>
+{
+ BOOST_STATIC_CONSTANT(unsigned, value = 15); // ~8-15 digit accuracy
+ BOOST_STATIC_ASSERT(::boost::math::max_factorial<float>::value >= 30);
+};
+template <>
+struct Pn_size<double>
+{
+ BOOST_STATIC_CONSTANT(unsigned, value = 30); // 16-20 digit accuracy
+ BOOST_STATIC_ASSERT(::boost::math::max_factorial<double>::value >= 60);
+};
+template <>
+struct Pn_size<long double>
+{
+ BOOST_STATIC_CONSTANT(unsigned, value = 50); // ~35-50 digit accuracy
+ BOOST_STATIC_ASSERT(::boost::math::max_factorial<long double>::value >= 100);
+};
+
+template <class T, class Policy>
+T beta_small_b_large_a_series(T a, T b, T x, T y, T s0, T mult, const Policy& pol, bool normalised)
+{
+ typedef typename lanczos::lanczos<T, Policy>::type lanczos_type;
+ BOOST_MATH_STD_USING
+ //
+ // This is DiDonato and Morris's BGRAT routine, see Eq's 9 through 9.6.
+ //
+ // Some values we'll need later, these are Eq 9.1:
+ //
+ T bm1 = b - 1;
+ T t = a + bm1 / 2;
+ T lx, u;
+ if(y < 0.35)
+ lx = boost::math::log1p(-y, pol);
+ else
+ lx = log(x);
+ u = -t * lx;
+ // and from from 9.2:
+ T prefix;
+ T h = regularised_gamma_prefix(b, u, pol, lanczos_type());
+ if(h <= tools::min_value<T>())
+ return s0;
+ if(normalised)
+ {
+ prefix = h / boost::math::tgamma_delta_ratio(a, b, pol);
+ prefix /= pow(t, b);
+ }
+ else
+ {
+ prefix = full_igamma_prefix(b, u, pol) / pow(t, b);
+ }
+ prefix *= mult;
+ //
+ // now we need the quantity Pn, unfortunatately this is computed
+ // recursively, and requires a full history of all the previous values
+ // so no choice but to declare a big table and hope it's big enough...
+ //
+ T p[ ::boost::math::detail::Pn_size<T>::value ] = { 1 }; // see 9.3.
+ //
+ // Now an initial value for J, see 9.6:
+ //
+ T j = boost::math::gamma_q(b, u, pol) / h;
+ //
+ // Now we can start to pull things together and evaluate the sum in Eq 9:
+ //
+ T sum = s0 + prefix * j; // Value at N = 0
+ // some variables we'll need:
+ unsigned tnp1 = 1; // 2*N+1
+ T lx2 = lx / 2;
+ lx2 *= lx2;
+ T lxp = 1;
+ T t4 = 4 * t * t;
+ T b2n = b;
+
+ for(unsigned n = 1; n < sizeof(p)/sizeof(p[0]); ++n)
+ {
+ /*
+ // debugging code, enable this if you want to determine whether
+ // the table of Pn's is large enough...
+ //
+ static int max_count = 2;
+ if(n > max_count)
+ {
+ max_count = n;
+ std::cerr << "Max iterations in BGRAT was " << n << std::endl;
+ }
+ */
+ //
+ // begin by evaluating the next Pn from Eq 9.4:
+ //
+ tnp1 += 2;
+ p[n] = 0;
+ T mbn = b - n;
+ unsigned tmp1 = 3;
+ for(unsigned m = 1; m < n; ++m)
+ {
+ mbn = m * b - n;
+ p[n] += mbn * p[n-m] / boost::math::unchecked_factorial<T>(tmp1);
+ tmp1 += 2;
+ }
+ p[n] /= n;
+ p[n] += bm1 / boost::math::unchecked_factorial<T>(tnp1);
+ //
+ // Now we want Jn from Jn-1 using Eq 9.6:
+ //
+ j = (b2n * (b2n + 1) * j + (u + b2n + 1) * lxp) / t4;
+ lxp *= lx2;
+ b2n += 2;
+ //
+ // pull it together with Eq 9:
+ //
+ T r = prefix * p[n] * j;
+ sum += r;
+ if(r > 1)
+ {
+ if(fabs(r) < fabs(tools::epsilon<T>() * sum))
+ break;
+ }
+ else
+ {
+ if(fabs(r / tools::epsilon<T>()) < fabs(sum))
+ break;
+ }
+ }
+ return sum;
+} // template <class T, class Lanczos>T beta_small_b_large_a_series(T a, T b, T x, T y, T s0, T mult, const Lanczos& l, bool normalised)
+
+//
+// For integer arguments we can relate the incomplete beta to the
+// complement of the binomial distribution cdf and use this finite sum.
+//
+template <class T>
+inline T binomial_ccdf(T n, T k, T x, T y)
+{
+ BOOST_MATH_STD_USING // ADL of std names
+ T result = pow(x, n);
+ T term = result;
+ for(unsigned i = itrunc(T(n - 1)); i > k; --i)
+ {
+ term *= ((i + 1) * y) / ((n - i) * x) ;
+ result += term;
+ }
+
+ return result;
+}
+
+
+//
+// The incomplete beta function implementation:
+// This is just a big bunch of spagetti code to divide up the
+// input range and select the right implementation method for
+// each domain:
+//
+template <class T, class Policy>
+T ibeta_imp(T a, T b, T x, const Policy& pol, bool inv, bool normalised, T* p_derivative)
+{
+ static const char* function = "boost::math::ibeta<%1%>(%1%, %1%, %1%)";
+ typedef typename lanczos::lanczos<T, Policy>::type lanczos_type;
+ BOOST_MATH_STD_USING // for ADL of std math functions.
+
+ BOOST_MATH_INSTRUMENT_VARIABLE(a);
+ BOOST_MATH_INSTRUMENT_VARIABLE(b);
+ BOOST_MATH_INSTRUMENT_VARIABLE(x);
+ BOOST_MATH_INSTRUMENT_VARIABLE(inv);
+ BOOST_MATH_INSTRUMENT_VARIABLE(normalised);
+
+ bool invert = inv;
+ T fract;
+ T y = 1 - x;
+
+ BOOST_ASSERT((p_derivative == 0) || normalised);
+
+ if(p_derivative)
+ *p_derivative = -1; // value not set.
+
+ if((x < 0) || (x > 1))
+ policies::raise_domain_error<T>(function, "Parameter x outside the range [0,1] in the incomplete beta function (got x=%1%).", x, pol);
+
+ if(normalised)
+ {
+ if(a < 0)
+ policies::raise_domain_error<T>(function, "The argument a to the incomplete beta function must be >= zero (got a=%1%).", a, pol);
+ if(b < 0)
+ policies::raise_domain_error<T>(function, "The argument b to the incomplete beta function must be >= zero (got b=%1%).", b, pol);
+ // extend to a few very special cases:
+ if(a == 0)
+ {
+ if(b == 0)
+ policies::raise_domain_error<T>(function, "The arguments a and b to the incomplete beta function cannot both be zero, with x=%1%.", x, pol);
+ if(b > 0)
+ return inv ? 0 : 1;
+ }
+ else if(b == 0)
+ {
+ if(a > 0)
+ return inv ? 1 : 0;
+ }
+ }
+ else
+ {
+ if(a <= 0)
+ policies::raise_domain_error<T>(function, "The argument a to the incomplete beta function must be greater than zero (got a=%1%).", a, pol);
+ if(b <= 0)
+ policies::raise_domain_error<T>(function, "The argument b to the incomplete beta function must be greater than zero (got b=%1%).", b, pol);
+ }
+
+ if(x == 0)
+ {
+ if(p_derivative)
+ {
+ *p_derivative = (a == 1) ? (T)1 : (a < 1) ? T(tools::max_value<T>() / 2) : T(tools::min_value<T>() * 2);
+ }
+ return (invert ? (normalised ? T(1) : boost::math::beta(a, b, pol)) : T(0));
+ }
+ if(x == 1)
+ {
+ if(p_derivative)
+ {
+ *p_derivative = (b == 1) ? T(1) : (b < 1) ? T(tools::max_value<T>() / 2) : T(tools::min_value<T>() * 2);
+ }
+ return (invert == 0 ? (normalised ? 1 : boost::math::beta(a, b, pol)) : 0);
+ }
+ if((a == 0.5f) && (b == 0.5f))
+ {
+ // We have an arcsine distribution:
+ if(p_derivative)
+ {
+ *p_derivative = (invert ? -1 : 1) / constants::pi<T>() * sqrt(y * x);
+ }
+ T p = invert ? asin(sqrt(y)) / constants::half_pi<T>() : asin(sqrt(x)) / constants::half_pi<T>();
+ if(!normalised)
+ p *= constants::pi<T>();
+ return p;
+ }
+ if(a == 1)
+ {
+ std::swap(a, b);
+ std::swap(x, y);
+ invert = !invert;
+ }
+ if(b == 1)
+ {
+ //
+ // Special case see: http://functions.wolfram.com/GammaBetaErf/BetaRegularized/03/01/01/
+ //
+ if(a == 1)
+ {
+ if(p_derivative)
+ *p_derivative = invert ? -1 : 1;
+ return invert ? y : x;
+ }
+
+ if(p_derivative)
+ {
+ *p_derivative = (invert ? -1 : 1) * a * pow(x, a - 1);
+ }
+ T p;
+ if(y < 0.5)
+ p = invert ? T(-expm1(a * log1p(-y))) : T(exp(a * log1p(-y)));
+ else
+ p = invert ? T(-powm1(x, a)) : T(pow(x, a));
+ if(!normalised)
+ p /= a;
+ return p;
+ }
+
+ if((std::min)(a, b) <= 1)
+ {
+ if(x > 0.5)
+ {
+ std::swap(a, b);
+ std::swap(x, y);
+ invert = !invert;
+ BOOST_MATH_INSTRUMENT_VARIABLE(invert);
+ }
+ if((std::max)(a, b) <= 1)
+ {
+ // Both a,b < 1:
+ if((a >= (std::min)(T(0.2), b)) || (pow(x, a) <= 0.9))
+ {
+ if(!invert)
+ {
+ fract = ibeta_series(a, b, x, T(0), lanczos_type(), normalised, p_derivative, y, pol);
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ }
+ else
+ {
+ fract = -(normalised ? 1 : boost::math::beta(a, b, pol));
+ invert = false;
+ fract = -ibeta_series(a, b, x, fract, lanczos_type(), normalised, p_derivative, y, pol);
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ }
+ }
+ else
+ {
+ std::swap(a, b);
+ std::swap(x, y);
+ invert = !invert;
+ if(y >= 0.3)
+ {
+ if(!invert)
+ {
+ fract = ibeta_series(a, b, x, T(0), lanczos_type(), normalised, p_derivative, y, pol);
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ }
+ else
+ {
+ fract = -(normalised ? 1 : boost::math::beta(a, b, pol));
+ invert = false;
+ fract = -ibeta_series(a, b, x, fract, lanczos_type(), normalised, p_derivative, y, pol);
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ }
+ }
+ else
+ {
+ // Sidestep on a, and then use the series representation:
+ T prefix;
+ if(!normalised)
+ {
+ prefix = rising_factorial_ratio(T(a+b), a, 20);
+ }
+ else
+ {
+ prefix = 1;
+ }
+ fract = ibeta_a_step(a, b, x, y, 20, pol, normalised, p_derivative);
+ if(!invert)
+ {
+ fract = beta_small_b_large_a_series(T(a + 20), b, x, y, fract, prefix, pol, normalised);
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ }
+ else
+ {
+ fract -= (normalised ? 1 : boost::math::beta(a, b, pol));
+ invert = false;
+ fract = -beta_small_b_large_a_series(T(a + 20), b, x, y, fract, prefix, pol, normalised);
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ }
+ }
+ }
+ }
+ else
+ {
+ // One of a, b < 1 only:
+ if((b <= 1) || ((x < 0.1) && (pow(b * x, a) <= 0.7)))
+ {
+ if(!invert)
+ {
+ fract = ibeta_series(a, b, x, T(0), lanczos_type(), normalised, p_derivative, y, pol);
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ }
+ else
+ {
+ fract = -(normalised ? 1 : boost::math::beta(a, b, pol));
+ invert = false;
+ fract = -ibeta_series(a, b, x, fract, lanczos_type(), normalised, p_derivative, y, pol);
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ }
+ }
+ else
+ {
+ std::swap(a, b);
+ std::swap(x, y);
+ invert = !invert;
+
+ if(y >= 0.3)
+ {
+ if(!invert)
+ {
+ fract = ibeta_series(a, b, x, T(0), lanczos_type(), normalised, p_derivative, y, pol);
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ }
+ else
+ {
+ fract = -(normalised ? 1 : boost::math::beta(a, b, pol));
+ invert = false;
+ fract = -ibeta_series(a, b, x, fract, lanczos_type(), normalised, p_derivative, y, pol);
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ }
+ }
+ else if(a >= 15)
+ {
+ if(!invert)
+ {
+ fract = beta_small_b_large_a_series(a, b, x, y, T(0), T(1), pol, normalised);
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ }
+ else
+ {
+ fract = -(normalised ? 1 : boost::math::beta(a, b, pol));
+ invert = false;
+ fract = -beta_small_b_large_a_series(a, b, x, y, fract, T(1), pol, normalised);
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ }
+ }
+ else
+ {
+ // Sidestep to improve errors:
+ T prefix;
+ if(!normalised)
+ {
+ prefix = rising_factorial_ratio(T(a+b), a, 20);
+ }
+ else
+ {
+ prefix = 1;
+ }
+ fract = ibeta_a_step(a, b, x, y, 20, pol, normalised, p_derivative);
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ if(!invert)
+ {
+ fract = beta_small_b_large_a_series(T(a + 20), b, x, y, fract, prefix, pol, normalised);
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ }
+ else
+ {
+ fract -= (normalised ? 1 : boost::math::beta(a, b, pol));
+ invert = false;
+ fract = -beta_small_b_large_a_series(T(a + 20), b, x, y, fract, prefix, pol, normalised);
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ }
+ }
+ }
+ }
+ }
+ else
+ {
+ // Both a,b >= 1:
+ T lambda;
+ if(a < b)
+ {
+ lambda = a - (a + b) * x;
+ }
+ else
+ {
+ lambda = (a + b) * y - b;
+ }
+ if(lambda < 0)
+ {
+ std::swap(a, b);
+ std::swap(x, y);
+ invert = !invert;
+ BOOST_MATH_INSTRUMENT_VARIABLE(invert);
+ }
+
+ if(b < 40)
+ {
+ if((floor(a) == a) && (floor(b) == b) && (a < (std::numeric_limits<int>::max)() - 100))
+ {
+ // relate to the binomial distribution and use a finite sum:
+ T k = a - 1;
+ T n = b + k;
+ fract = binomial_ccdf(n, k, x, y);
+ if(!normalised)
+ fract *= boost::math::beta(a, b, pol);
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ }
+ else if(b * x <= 0.7)
+ {
+ if(!invert)
+ {
+ fract = ibeta_series(a, b, x, T(0), lanczos_type(), normalised, p_derivative, y, pol);
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ }
+ else
+ {
+ fract = -(normalised ? 1 : boost::math::beta(a, b, pol));
+ invert = false;
+ fract = -ibeta_series(a, b, x, fract, lanczos_type(), normalised, p_derivative, y, pol);
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ }
+ }
+ else if(a > 15)
+ {
+ // sidestep so we can use the series representation:
+ int n = itrunc(T(floor(b)), pol);
+ if(n == b)
+ --n;
+ T bbar = b - n;
+ T prefix;
+ if(!normalised)
+ {
+ prefix = rising_factorial_ratio(T(a+bbar), bbar, n);
+ }
+ else
+ {
+ prefix = 1;
+ }
+ fract = ibeta_a_step(bbar, a, y, x, n, pol, normalised, static_cast<T*>(0));
+ fract = beta_small_b_large_a_series(a, bbar, x, y, fract, T(1), pol, normalised);
+ fract /= prefix;
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ }
+ else if(normalised)
+ {
+ // the formula here for the non-normalised case is tricky to figure
+ // out (for me!!), and requires two pochhammer calculations rather
+ // than one, so leave it for now....
+ int n = itrunc(T(floor(b)), pol);
+ T bbar = b - n;
+ if(bbar <= 0)
+ {
+ --n;
+ bbar += 1;
+ }
+ fract = ibeta_a_step(bbar, a, y, x, n, pol, normalised, static_cast<T*>(0));
+ fract += ibeta_a_step(a, bbar, x, y, 20, pol, normalised, static_cast<T*>(0));
+ if(invert)
+ fract -= (normalised ? 1 : boost::math::beta(a, b, pol));
+ //fract = ibeta_series(a+20, bbar, x, fract, l, normalised, p_derivative, y);
+ fract = beta_small_b_large_a_series(T(a+20), bbar, x, y, fract, T(1), pol, normalised);
+ if(invert)
+ {
+ fract = -fract;
+ invert = false;
+ }
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ }
+ else
+ {
+ fract = ibeta_fraction2(a, b, x, y, pol, normalised, p_derivative);
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ }
+ }
+ else
+ {
+ fract = ibeta_fraction2(a, b, x, y, pol, normalised, p_derivative);
+ BOOST_MATH_INSTRUMENT_VARIABLE(fract);
+ }
+ }
+ if(p_derivative)
+ {
+ if(*p_derivative < 0)
+ {
+ *p_derivative = ibeta_power_terms(a, b, x, y, lanczos_type(), true, pol);
+ }
+ T div = y * x;
+
+ if(*p_derivative != 0)
+ {
+ if((tools::max_value<T>() * div < *p_derivative))
+ {
+ // overflow, return an arbitarily large value:
+ *p_derivative = tools::max_value<T>() / 2;
+ }
+ else
+ {
+ *p_derivative /= div;
+ }
+ }
+ }
+ return invert ? (normalised ? 1 : boost::math::beta(a, b, pol)) - fract : fract;
+} // template <class T, class Lanczos>T ibeta_imp(T a, T b, T x, const Lanczos& l, bool inv, bool normalised)
+
+template <class T, class Policy>
+inline T ibeta_imp(T a, T b, T x, const Policy& pol, bool inv, bool normalised)
+{
+ return ibeta_imp(a, b, x, pol, inv, normalised, static_cast<T*>(0));
+}
+
+template <class T, class Policy>
+T ibeta_derivative_imp(T a, T b, T x, const Policy& pol)
+{
+ static const char* function = "ibeta_derivative<%1%>(%1%,%1%,%1%)";
+ //
+ // start with the usual error checks:
+ //
+ if(a <= 0)
+ policies::raise_domain_error<T>(function, "The argument a to the incomplete beta function must be greater than zero (got a=%1%).", a, pol);
+ if(b <= 0)
+ policies::raise_domain_error<T>(function, "The argument b to the incomplete beta function must be greater than zero (got b=%1%).", b, pol);
+ if((x < 0) || (x > 1))
+ policies::raise_domain_error<T>(function, "Parameter x outside the range [0,1] in the incomplete beta function (got x=%1%).", x, pol);
+ //
+ // Now the corner cases:
+ //
+ if(x == 0)
+ {
+ return (a > 1) ? 0 :
+ (a == 1) ? 1 / boost::math::beta(a, b, pol) : policies::raise_overflow_error<T>(function, 0, pol);
+ }
+ else if(x == 1)
+ {
+ return (b > 1) ? 0 :
+ (b == 1) ? 1 / boost::math::beta(a, b, pol) : policies::raise_overflow_error<T>(function, 0, pol);
+ }
+ //
+ // Now the regular cases:
+ //
+ typedef typename lanczos::lanczos<T, Policy>::type lanczos_type;
+ T f1 = ibeta_power_terms<T>(a, b, x, 1 - x, lanczos_type(), true, pol);
+ T y = (1 - x) * x;
+
+ if(f1 == 0)
+ return 0;
+
+ if((tools::max_value<T>() * y < f1))
+ {
+ // overflow:
+ return policies::raise_overflow_error<T>(function, 0, pol);
+ }
+
+ f1 /= y;
+
+ return f1;
+}
+//
+// Some forwarding functions that dis-ambiguate the third argument type:
+//
+template <class RT1, class RT2, class Policy>
+inline typename tools::promote_args<RT1, RT2>::type
+ beta(RT1 a, RT2 b, const Policy&, const mpl::true_*)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ typedef typename tools::promote_args<RT1, RT2>::type result_type;
+ typedef typename policies::evaluation<result_type, Policy>::type value_type;
+ typedef typename lanczos::lanczos<value_type, Policy>::type evaluation_type;
+ typedef typename policies::normalise<
+ Policy,
+ policies::promote_float<false>,
+ policies::promote_double<false>,
+ policies::discrete_quantile<>,
+ policies::assert_undefined<> >::type forwarding_policy;
+
+ return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::beta_imp(static_cast<value_type>(a), static_cast<value_type>(b), evaluation_type(), forwarding_policy()), "boost::math::beta<%1%>(%1%,%1%)");
+}
+template <class RT1, class RT2, class RT3>
+inline typename tools::promote_args<RT1, RT2, RT3>::type
+ beta(RT1 a, RT2 b, RT3 x, const mpl::false_*)
+{
+ return boost::math::beta(a, b, x, policies::policy<>());
+}
+} // namespace detail
+
+//
+// The actual function entry-points now follow, these just figure out
+// which Lanczos approximation to use
+// and forward to the implementation functions:
+//
+template <class RT1, class RT2, class A>
+inline typename tools::promote_args<RT1, RT2, A>::type
+ beta(RT1 a, RT2 b, A arg)
+{
+ typedef typename policies::is_policy<A>::type tag;
+ return boost::math::detail::beta(a, b, arg, static_cast<tag*>(0));
+}
+
+template <class RT1, class RT2>
+inline typename tools::promote_args<RT1, RT2>::type
+ beta(RT1 a, RT2 b)
+{
+ return boost::math::beta(a, b, policies::policy<>());
+}
+
+template <class RT1, class RT2, class RT3, class Policy>
+inline typename tools::promote_args<RT1, RT2, RT3>::type
+ beta(RT1 a, RT2 b, RT3 x, const Policy&)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ typedef typename tools::promote_args<RT1, RT2, RT3>::type result_type;
+ typedef typename policies::evaluation<result_type, 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;
+
+ return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::ibeta_imp(static_cast<value_type>(a), static_cast<value_type>(b), static_cast<value_type>(x), forwarding_policy(), false, false), "boost::math::beta<%1%>(%1%,%1%,%1%)");
+}
+
+template <class RT1, class RT2, class RT3, class Policy>
+inline typename tools::promote_args<RT1, RT2, RT3>::type
+ betac(RT1 a, RT2 b, RT3 x, const Policy&)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ typedef typename tools::promote_args<RT1, RT2, RT3>::type result_type;
+ typedef typename policies::evaluation<result_type, 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;
+
+ return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::ibeta_imp(static_cast<value_type>(a), static_cast<value_type>(b), static_cast<value_type>(x), forwarding_policy(), true, false), "boost::math::betac<%1%>(%1%,%1%,%1%)");
+}
+template <class RT1, class RT2, class RT3>
+inline typename tools::promote_args<RT1, RT2, RT3>::type
+ betac(RT1 a, RT2 b, RT3 x)
+{
+ return boost::math::betac(a, b, x, policies::policy<>());
+}
+
+template <class RT1, class RT2, class RT3, class Policy>
+inline typename tools::promote_args<RT1, RT2, RT3>::type
+ ibeta(RT1 a, RT2 b, RT3 x, const Policy&)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ typedef typename tools::promote_args<RT1, RT2, RT3>::type result_type;
+ typedef typename policies::evaluation<result_type, 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;
+
+ return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::ibeta_imp(static_cast<value_type>(a), static_cast<value_type>(b), static_cast<value_type>(x), forwarding_policy(), false, true), "boost::math::ibeta<%1%>(%1%,%1%,%1%)");
+}
+template <class RT1, class RT2, class RT3>
+inline typename tools::promote_args<RT1, RT2, RT3>::type
+ ibeta(RT1 a, RT2 b, RT3 x)
+{
+ return boost::math::ibeta(a, b, x, policies::policy<>());
+}
+
+template <class RT1, class RT2, class RT3, class Policy>
+inline typename tools::promote_args<RT1, RT2, RT3>::type
+ ibetac(RT1 a, RT2 b, RT3 x, const Policy&)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ typedef typename tools::promote_args<RT1, RT2, RT3>::type result_type;
+ typedef typename policies::evaluation<result_type, 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;
+
+ return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::ibeta_imp(static_cast<value_type>(a), static_cast<value_type>(b), static_cast<value_type>(x), forwarding_policy(), true, true), "boost::math::ibetac<%1%>(%1%,%1%,%1%)");
+}
+template <class RT1, class RT2, class RT3>
+inline typename tools::promote_args<RT1, RT2, RT3>::type
+ ibetac(RT1 a, RT2 b, RT3 x)
+{
+ return boost::math::ibetac(a, b, x, policies::policy<>());
+}
+
+template <class RT1, class RT2, class RT3, class Policy>
+inline typename tools::promote_args<RT1, RT2, RT3>::type
+ ibeta_derivative(RT1 a, RT2 b, RT3 x, const Policy&)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ typedef typename tools::promote_args<RT1, RT2, RT3>::type result_type;
+ typedef typename policies::evaluation<result_type, 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;
+
+ return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::ibeta_derivative_imp(static_cast<value_type>(a), static_cast<value_type>(b), static_cast<value_type>(x), forwarding_policy()), "boost::math::ibeta_derivative<%1%>(%1%,%1%,%1%)");
+}
+template <class RT1, class RT2, class RT3>
+inline typename tools::promote_args<RT1, RT2, RT3>::type
+ ibeta_derivative(RT1 a, RT2 b, RT3 x)
+{
+ return boost::math::ibeta_derivative(a, b, x, policies::policy<>());
+}
+
+} // namespace math
+} // namespace boost
+
+#include <boost/math/special_functions/detail/ibeta_inverse.hpp>
+#include <boost/math/special_functions/detail/ibeta_inv_ab.hpp>
+
+#endif // BOOST_MATH_SPECIAL_BETA_HPP
+
+
+
+
+
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