1597 lines
53 KiB
Plaintext
1597 lines
53 KiB
Plaintext
// (C) Copyright John Maddock 2006.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_MATH_SPECIAL_BETA_HPP
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#define BOOST_MATH_SPECIAL_BETA_HPP
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#ifdef _MSC_VER
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#pragma once
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#endif
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#include <boost/math/special_functions/math_fwd.hpp>
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#include <boost/math/tools/config.hpp>
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#include <boost/math/special_functions/gamma.hpp>
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#include <boost/math/special_functions/binomial.hpp>
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#include <boost/math/special_functions/factorials.hpp>
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#include <boost/math/special_functions/erf.hpp>
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#include <boost/math/special_functions/log1p.hpp>
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#include <boost/math/special_functions/expm1.hpp>
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#include <boost/math/special_functions/trunc.hpp>
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#include <boost/math/tools/roots.hpp>
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#include <boost/static_assert.hpp>
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#include <boost/config/no_tr1/cmath.hpp>
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namespace boost{ namespace math{
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namespace detail{
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//
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// Implementation of Beta(a,b) using the Lanczos approximation:
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//
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template <class T, class Lanczos, class Policy>
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T beta_imp(T a, T b, const Lanczos&, const Policy& pol)
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{
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BOOST_MATH_STD_USING // for ADL of std names
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if(a <= 0)
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return 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);
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if(b <= 0)
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return 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);
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T result;
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T prefix = 1;
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T c = a + b;
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// Special cases:
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if((c == a) && (b < tools::epsilon<T>()))
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return 1 / b;
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else if((c == b) && (a < tools::epsilon<T>()))
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return 1 / a;
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if(b == 1)
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return 1/a;
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else if(a == 1)
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return 1/b;
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else if(c < tools::epsilon<T>())
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{
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result = c / a;
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result /= b;
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return result;
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}
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/*
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//
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// This code appears to be no longer necessary: it was
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// used to offset errors introduced from the Lanczos
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// approximation, but the current Lanczos approximations
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// are sufficiently accurate for all z that we can ditch
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// this. It remains in the file for future reference...
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//
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// If a or b are less than 1, shift to greater than 1:
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if(a < 1)
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{
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prefix *= c / a;
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c += 1;
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a += 1;
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}
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if(b < 1)
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{
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prefix *= c / b;
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c += 1;
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b += 1;
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}
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*/
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if(a < b)
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std::swap(a, b);
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// Lanczos calculation:
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T agh = static_cast<T>(a + Lanczos::g() - 0.5f);
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T bgh = static_cast<T>(b + Lanczos::g() - 0.5f);
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T cgh = static_cast<T>(c + Lanczos::g() - 0.5f);
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result = Lanczos::lanczos_sum_expG_scaled(a) * (Lanczos::lanczos_sum_expG_scaled(b) / Lanczos::lanczos_sum_expG_scaled(c));
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T ambh = a - 0.5f - b;
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if((fabs(b * ambh) < (cgh * 100)) && (a > 100))
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{
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// Special case where the base of the power term is close to 1
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// compute (1+x)^y instead:
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result *= exp(ambh * boost::math::log1p(-b / cgh, pol));
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}
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else
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{
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result *= pow(agh / cgh, a - T(0.5) - b);
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}
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if(cgh > 1e10f)
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// this avoids possible overflow, but appears to be marginally less accurate:
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result *= pow((agh / cgh) * (bgh / cgh), b);
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else
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result *= pow((agh * bgh) / (cgh * cgh), b);
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result *= sqrt(boost::math::constants::e<T>() / bgh);
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// If a and b were originally less than 1 we need to scale the result:
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result *= prefix;
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return result;
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} // template <class T, class Lanczos> beta_imp(T a, T b, const Lanczos&)
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//
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// Generic implementation of Beta(a,b) without Lanczos approximation support
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// (Caution this is slow!!!):
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//
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template <class T, class Policy>
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T beta_imp(T a, T b, const lanczos::undefined_lanczos& /* l */, const Policy& pol)
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{
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BOOST_MATH_STD_USING
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if(a <= 0)
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return 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);
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if(b <= 0)
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return 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);
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T result;
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T prefix = 1;
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T c = a + b;
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// special cases:
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if((c == a) && (b < tools::epsilon<T>()))
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return boost::math::tgamma(b, pol);
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else if((c == b) && (a < tools::epsilon<T>()))
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return boost::math::tgamma(a, pol);
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if(b == 1)
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return 1/a;
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else if(a == 1)
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return 1/b;
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// shift to a and b > 1 if required:
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if(a < 1)
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{
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prefix *= c / a;
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c += 1;
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a += 1;
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}
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if(b < 1)
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{
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prefix *= c / b;
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c += 1;
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b += 1;
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}
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if(a < b)
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std::swap(a, b);
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// set integration limits:
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T la = (std::max)(T(10), a);
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T lb = (std::max)(T(10), b);
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T lc = (std::max)(T(10), T(a+b));
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// calculate the fraction parts:
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T sa = detail::lower_gamma_series(a, la, pol) / a;
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sa += detail::upper_gamma_fraction(a, la, ::boost::math::policies::get_epsilon<T, Policy>());
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T sb = detail::lower_gamma_series(b, lb, pol) / b;
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sb += detail::upper_gamma_fraction(b, lb, ::boost::math::policies::get_epsilon<T, Policy>());
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T sc = detail::lower_gamma_series(c, lc, pol) / c;
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sc += detail::upper_gamma_fraction(c, lc, ::boost::math::policies::get_epsilon<T, Policy>());
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// and the exponent part:
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result = exp(lc - la - lb) * pow(la/lc, a) * pow(lb/lc, b);
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// and combine:
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result *= sa * sb / sc;
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// if a and b were originally less than 1 we need to scale the result:
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result *= prefix;
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return result;
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} // template <class T>T beta_imp(T a, T b, const lanczos::undefined_lanczos& l)
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//
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// Compute the leading power terms in the incomplete Beta:
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//
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// (x^a)(y^b)/Beta(a,b) when normalised, and
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// (x^a)(y^b) otherwise.
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//
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// Almost all of the error in the incomplete beta comes from this
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// function: particularly when a and b are large. Computing large
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// powers are *hard* though, and using logarithms just leads to
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// horrendous cancellation errors.
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//
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template <class T, class Lanczos, class Policy>
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T ibeta_power_terms(T a,
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T b,
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T x,
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T y,
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const Lanczos&,
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bool normalised,
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const Policy& pol,
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T prefix = 1,
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const char* function = "boost::math::ibeta<%1%>(%1%, %1%, %1%)")
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{
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BOOST_MATH_STD_USING
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if(!normalised)
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{
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// can we do better here?
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return pow(x, a) * pow(y, b);
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}
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T result;
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T c = a + b;
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// combine power terms with Lanczos approximation:
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T agh = static_cast<T>(a + Lanczos::g() - 0.5f);
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T bgh = static_cast<T>(b + Lanczos::g() - 0.5f);
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T cgh = static_cast<T>(c + Lanczos::g() - 0.5f);
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result = Lanczos::lanczos_sum_expG_scaled(c) / (Lanczos::lanczos_sum_expG_scaled(a) * Lanczos::lanczos_sum_expG_scaled(b));
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result *= prefix;
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// combine with the leftover terms from the Lanczos approximation:
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result *= sqrt(bgh / boost::math::constants::e<T>());
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result *= sqrt(agh / cgh);
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// l1 and l2 are the base of the exponents minus one:
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T l1 = (x * b - y * agh) / agh;
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T l2 = (y * a - x * bgh) / bgh;
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if(((std::min)(fabs(l1), fabs(l2)) < 0.2))
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{
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// when the base of the exponent is very near 1 we get really
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// gross errors unless extra care is taken:
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if((l1 * l2 > 0) || ((std::min)(a, b) < 1))
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{
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//
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// This first branch handles the simple cases where either:
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//
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// * The two power terms both go in the same direction
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// (towards zero or towards infinity). In this case if either
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// term overflows or underflows, then the product of the two must
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// do so also.
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// *Alternatively if one exponent is less than one, then we
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// can't productively use it to eliminate overflow or underflow
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// from the other term. Problems with spurious overflow/underflow
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// can't be ruled out in this case, but it is *very* unlikely
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// since one of the power terms will evaluate to a number close to 1.
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//
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if(fabs(l1) < 0.1)
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{
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result *= exp(a * boost::math::log1p(l1, pol));
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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else
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{
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result *= pow((x * cgh) / agh, a);
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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if(fabs(l2) < 0.1)
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{
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result *= exp(b * boost::math::log1p(l2, pol));
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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else
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{
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result *= pow((y * cgh) / bgh, b);
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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}
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else if((std::max)(fabs(l1), fabs(l2)) < 0.5)
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{
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//
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// Both exponents are near one and both the exponents are
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// greater than one and further these two
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// power terms tend in opposite directions (one towards zero,
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// the other towards infinity), so we have to combine the terms
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// to avoid any risk of overflow or underflow.
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//
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// We do this by moving one power term inside the other, we have:
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//
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// (1 + l1)^a * (1 + l2)^b
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// = ((1 + l1)*(1 + l2)^(b/a))^a
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// = (1 + l1 + l3 + l1*l3)^a ; l3 = (1 + l2)^(b/a) - 1
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// = exp((b/a) * log(1 + l2)) - 1
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//
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// The tricky bit is deciding which term to move inside :-)
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// By preference we move the larger term inside, so that the
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// size of the largest exponent is reduced. However, that can
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// only be done as long as l3 (see above) is also small.
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//
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bool small_a = a < b;
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T ratio = b / a;
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if((small_a && (ratio * l2 < 0.1)) || (!small_a && (l1 / ratio > 0.1)))
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{
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T l3 = boost::math::expm1(ratio * boost::math::log1p(l2, pol), pol);
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l3 = l1 + l3 + l3 * l1;
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l3 = a * boost::math::log1p(l3, pol);
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result *= exp(l3);
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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else
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{
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T l3 = boost::math::expm1(boost::math::log1p(l1, pol) / ratio, pol);
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l3 = l2 + l3 + l3 * l2;
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l3 = b * boost::math::log1p(l3, pol);
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result *= exp(l3);
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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}
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else if(fabs(l1) < fabs(l2))
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{
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// First base near 1 only:
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T l = a * boost::math::log1p(l1, pol)
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+ b * log((y * cgh) / bgh);
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if((l <= tools::log_min_value<T>()) || (l >= tools::log_max_value<T>()))
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{
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l += log(result);
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if(l >= tools::log_max_value<T>())
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return policies::raise_overflow_error<T>(function, 0, pol);
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result = exp(l);
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}
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else
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result *= exp(l);
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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else
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{
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// Second base near 1 only:
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T l = b * boost::math::log1p(l2, pol)
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+ a * log((x * cgh) / agh);
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if((l <= tools::log_min_value<T>()) || (l >= tools::log_max_value<T>()))
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{
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l += log(result);
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if(l >= tools::log_max_value<T>())
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return policies::raise_overflow_error<T>(function, 0, pol);
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result = exp(l);
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}
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else
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result *= exp(l);
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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}
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else
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{
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// general case:
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T b1 = (x * cgh) / agh;
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T b2 = (y * cgh) / bgh;
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l1 = a * log(b1);
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l2 = b * log(b2);
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BOOST_MATH_INSTRUMENT_VARIABLE(b1);
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BOOST_MATH_INSTRUMENT_VARIABLE(b2);
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BOOST_MATH_INSTRUMENT_VARIABLE(l1);
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BOOST_MATH_INSTRUMENT_VARIABLE(l2);
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if((l1 >= tools::log_max_value<T>())
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|| (l1 <= tools::log_min_value<T>())
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|| (l2 >= tools::log_max_value<T>())
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|| (l2 <= tools::log_min_value<T>())
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)
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{
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// Oops, under/overflow, sidestep if we can:
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if(a < b)
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{
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T p1 = pow(b2, b / a);
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T l3 = a * (log(b1) + log(p1));
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if((l3 < tools::log_max_value<T>())
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&& (l3 > tools::log_min_value<T>()))
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{
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result *= pow(p1 * b1, a);
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}
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else
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{
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l2 += l1 + log(result);
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if(l2 >= tools::log_max_value<T>())
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return policies::raise_overflow_error<T>(function, 0, pol);
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result = exp(l2);
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}
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}
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else
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{
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T p1 = pow(b1, a / b);
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T l3 = (log(p1) + log(b2)) * b;
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if((l3 < tools::log_max_value<T>())
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&& (l3 > tools::log_min_value<T>()))
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{
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result *= pow(p1 * b2, b);
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}
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else
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{
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l2 += l1 + log(result);
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if(l2 >= tools::log_max_value<T>())
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return policies::raise_overflow_error<T>(function, 0, pol);
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result = exp(l2);
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}
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}
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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else
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{
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// finally the normal case:
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result *= pow(b1, a) * pow(b2, b);
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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}
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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return result;
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}
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//
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// Compute the leading power terms in the incomplete Beta:
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//
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// (x^a)(y^b)/Beta(a,b) when normalised, and
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// (x^a)(y^b) otherwise.
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//
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// Almost all of the error in the incomplete beta comes from this
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// function: particularly when a and b are large. Computing large
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// powers are *hard* though, and using logarithms just leads to
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// horrendous cancellation errors.
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//
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// This version is generic, slow, and does not use the Lanczos approximation.
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//
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template <class T, class Policy>
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T ibeta_power_terms(T a,
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T b,
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T x,
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T y,
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const boost::math::lanczos::undefined_lanczos&,
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bool normalised,
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const Policy& pol,
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T prefix = 1,
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const char* = "boost::math::ibeta<%1%>(%1%, %1%, %1%)")
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{
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BOOST_MATH_STD_USING
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if(!normalised)
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{
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return pow(x, a) * pow(y, b);
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}
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T result= 0; // assignment here silences warnings later
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T c = a + b;
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// integration limits for the gamma functions:
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//T la = (std::max)(T(10), a);
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//T lb = (std::max)(T(10), b);
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//T lc = (std::max)(T(10), a+b);
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T la = a + 5;
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T lb = b + 5;
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T lc = a + b + 5;
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// gamma function partials:
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T sa = detail::lower_gamma_series(a, la, pol) / a;
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sa += detail::upper_gamma_fraction(a, la, ::boost::math::policies::get_epsilon<T, Policy>());
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T sb = detail::lower_gamma_series(b, lb, pol) / b;
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sb += detail::upper_gamma_fraction(b, lb, ::boost::math::policies::get_epsilon<T, Policy>());
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T sc = detail::lower_gamma_series(c, lc, pol) / c;
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sc += detail::upper_gamma_fraction(c, lc, ::boost::math::policies::get_epsilon<T, Policy>());
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// gamma function powers combined with incomplete beta powers:
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T b1 = (x * lc) / la;
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T b2 = (y * lc) / lb;
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T e1 = -5; // lc - la - lb;
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T lb1 = a * log(b1);
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T lb2 = b * log(b2);
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|
|
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 + log(prefix));
|
|
}
|
|
else
|
|
{
|
|
T p1, p2;
|
|
p1 = (x * b - y * la) / la;
|
|
if(fabs(p1) < 0.5f)
|
|
p1 = exp(a * boost::math::log1p(p1, pol));
|
|
else
|
|
p1 = pow(b1, a);
|
|
p2 = (y * a - x * lb) / lb;
|
|
if(fabs(p2) < 0.5f)
|
|
p2 = exp(b * boost::math::log1p(p2, pol));
|
|
else
|
|
p2 = pow(b2, b);
|
|
T p3 = exp(e1);
|
|
result = prefix * 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 = static_cast<T>(a + Lanczos::g() - 0.5f);
|
|
T bgh = static_cast<T>(b + Lanczos::g() - 0.5f);
|
|
T cgh = static_cast<T>(c + Lanczos::g() - 0.5f);
|
|
result = Lanczos::lanczos_sum_expG_scaled(c) / (Lanczos::lanczos_sum_expG_scaled(a) * Lanczos::lanczos_sum_expG_scaled(b));
|
|
|
|
T l1 = log(cgh / bgh) * (b - 0.5f);
|
|
T l2 = log(x * cgh / agh) * a;
|
|
//
|
|
// Check for over/underflow in the power terms:
|
|
//
|
|
if((l1 > tools::log_min_value<T>())
|
|
&& (l1 < tools::log_max_value<T>())
|
|
&& (l2 > tools::log_min_value<T>())
|
|
&& (l2 < tools::log_max_value<T>()))
|
|
{
|
|
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
|
|
{
|
|
//
|
|
// Oh dear, we need logs, and this *will* cancel:
|
|
//
|
|
result = log(result) + l1 + l2 + (log(agh) - 1) / 2;
|
|
if(p_derivative)
|
|
*p_derivative = exp(result + b * log(y));
|
|
result = exp(result);
|
|
}
|
|
}
|
|
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 = static_cast<T>(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>
|
|
T binomial_ccdf(T n, T k, T x, T y)
|
|
{
|
|
BOOST_MATH_STD_USING // ADL of std names
|
|
|
|
T result = pow(x, n);
|
|
|
|
if(result > tools::min_value<T>())
|
|
{
|
|
T term = result;
|
|
for(unsigned i = itrunc(T(n - 1)); i > k; --i)
|
|
{
|
|
term *= ((i + 1) * y) / ((n - i) * x);
|
|
result += term;
|
|
}
|
|
}
|
|
else
|
|
{
|
|
// First term underflows so we need to start at the mode of the
|
|
// distribution and work outwards:
|
|
int start = itrunc(n * x);
|
|
if(start <= k + 1)
|
|
start = itrunc(k + 2);
|
|
result = pow(x, start) * pow(y, n - start) * boost::math::binomial_coefficient<T>(itrunc(n), itrunc(start));
|
|
if(result == 0)
|
|
{
|
|
// OK, starting slightly above the mode didn't work,
|
|
// we'll have to sum the terms the old fashioned way:
|
|
for(unsigned i = start - 1; i > k; --i)
|
|
{
|
|
result += pow(x, (int)i) * pow(y, n - i) * boost::math::binomial_coefficient<T>(itrunc(n), itrunc(i));
|
|
}
|
|
}
|
|
else
|
|
{
|
|
T term = result;
|
|
T start_term = result;
|
|
for(unsigned i = start - 1; i > k; --i)
|
|
{
|
|
term *= ((i + 1) * y) / ((n - i) * x);
|
|
result += term;
|
|
}
|
|
term = start_term;
|
|
for(unsigned i = start + 1; i <= n; ++i)
|
|
{
|
|
term *= (n - i + 1) * x / (i * y);
|
|
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))
|
|
return 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)
|
|
return 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)
|
|
return 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)
|
|
return 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 static_cast<T>(inv ? 0 : 1);
|
|
}
|
|
else if(b == 0)
|
|
{
|
|
if(a > 0)
|
|
return static_cast<T>(inv ? 1 : 0);
|
|
}
|
|
}
|
|
else
|
|
{
|
|
if(a <= 0)
|
|
return 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)
|
|
return 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 = 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 = 1;
|
|
return invert ? y : x;
|
|
}
|
|
|
|
if(p_derivative)
|
|
{
|
|
*p_derivative = a * pow(x, a - 1);
|
|
}
|
|
T p;
|
|
if(y < 0.5)
|
|
p = invert ? T(-boost::math::expm1(a * boost::math::log1p(-y, pol), pol)) : T(exp(a * boost::math::log1p(-y, pol)));
|
|
else
|
|
p = invert ? T(-boost::math::powm1(x, a, pol)) : 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) && (y != 1))
|
|
{
|
|
// 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 and only use this in the normalized case....
|
|
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 -= 1; // Note this line would need changing if we ever enable this branch in non-normalized case
|
|
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)
|
|
return 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)
|
|
return 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))
|
|
return 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 y = (1 - x) * x;
|
|
T f1 = ibeta_power_terms<T>(a, b, x, 1 - x, lanczos_type(), true, pol, 1 / y, function);
|
|
return f1;
|
|
}
|
|
//
|
|
// Some forwarding functions that dis-ambiguate the third argument type:
|
|
//
|
|
template <class RT1, class RT2, class Policy>
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|
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%)");
|
|
}
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|
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
|
|
|
|
|
|
|
|
|
|
|