| // (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 L, class Policy> |
| T beta_imp(T a, T b, const L&, 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 + L::g() - T(0.5); |
| T bgh = b + L::g() - T(0.5); |
| T cgh = c + L::g() - T(0.5); |
| result = L::lanczos_sum_expG_scaled(a) * L::lanczos_sum_expG_scaled(b) / L::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 L> beta_imp(T a, T b, const L&) |
| |
| // |
| // 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), 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 L, class Policy> |
| T ibeta_power_terms(T a, |
| T b, |
| T x, |
| T y, |
| const L&, |
| 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 + L::g() - T(0.5); |
| T bgh = b + L::g() - T(0.5); |
| T cgh = c + L::g() - T(0.5); |
| result = L::lanczos_sum_expG_scaled(c) / (L::lanczos_sum_expG_scaled(a) * L::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); |
| 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; |
| |
| T prefix = 1; |
| 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 L, class Policy> |
| T ibeta_series(T a, T b, T x, T s0, const L&, 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 + L::g() - T(0.5); |
| T bgh = b + L::g() - T(0.5); |
| T cgh = c + L::g() - T(0.5); |
| result = L::lanczos_sum_expG_scaled(c) / (L::lanczos_sum_expG_scaled(a) * L::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("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 prefix = 1; |
| 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("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_) : a(a_), b(b_), x(x_), 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 - (a + b) * x + 1 + m *(2 - x))) / (a + 2*m + 1); |
| |
| ++m; |
| |
| return std::make_pair(aN, bN); |
| } |
| |
| private: |
| T a, b, x; |
| 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); |
| T fract = boost::math::tools::continued_fraction_b(f, boost::math::policies::get_epsilon<T, Policy>()); |
| 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 L>T beta_small_b_large_a_series(T a, T b, T x, T y, T s0, T mult, const L& 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((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)) |
| { |
| // 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 L>T ibeta_imp(T a, T b, T x, const L& 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 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::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 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::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 |
| |
| |
| |
| |
| |