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glm/test/external/boost/math/special_functions/beta.hpp

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// (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), T(a+b));
// calculate the fraction parts:
T sa = detail::lower_gamma_series(a, la, pol) / a;
sa += detail::upper_gamma_fraction(a, la, ::boost::math::policies::get_epsilon<T, Policy>());
T sb = detail::lower_gamma_series(b, lb, pol) / b;
sb += detail::upper_gamma_fraction(b, lb, ::boost::math::policies::get_epsilon<T, Policy>());
T sc = detail::lower_gamma_series(c, lc, pol) / c;
sc += detail::upper_gamma_fraction(c, lc, ::boost::math::policies::get_epsilon<T, Policy>());
// and the exponent part:
result = exp(lc - la - lb) * pow(la/lc, a) * pow(lb/lc, b);
// and combine:
result *= sa * sb / sc;
// if a and b were originally less than 1 we need to scale the result:
result *= prefix;
return result;
} // template <class T>T beta_imp(T a, T b, const lanczos::undefined_lanczos& l)
//
// Compute the leading power terms in the incomplete Beta:
//
// (x^a)(y^b)/Beta(a,b) when normalised, and
// (x^a)(y^b) otherwise.
//
// Almost all of the error in the incomplete beta comes from this
// function: particularly when a and b are large. Computing large
// powers are *hard* though, and using logarithms just leads to
// horrendous cancellation errors.
//
template <class T, class 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);
BOOST_MATH_INSTRUMENT_VARIABLE(b1);
BOOST_MATH_INSTRUMENT_VARIABLE(b2);
BOOST_MATH_INSTRUMENT_VARIABLE(l1);
BOOST_MATH_INSTRUMENT_VARIABLE(l2);
if((l1 >= tools::log_max_value<T>())
|| (l1 <= tools::log_min_value<T>())
|| (l2 >= tools::log_max_value<T>())
|| (l2 <= tools::log_min_value<T>())
)
{
// Oops, overflow, sidestep:
if(a < b)
result *= pow(pow(b2, b/a) * b1, a);
else
result *= pow(pow(b1, a/b) * b2, b);
BOOST_MATH_INSTRUMENT_VARIABLE(result);
}
else
{
// finally the normal case:
result *= pow(b1, a) * pow(b2, b);
BOOST_MATH_INSTRUMENT_VARIABLE(result);
}
}
// combine with the leftover terms from the Lanczos approximation:
result *= sqrt(bgh / boost::math::constants::e<T>());
result *= sqrt(agh / cgh);
result *= prefix;
BOOST_MATH_INSTRUMENT_VARIABLE(result);
return result;
}
//
// Compute the leading power terms in the incomplete Beta:
//
// (x^a)(y^b)/Beta(a,b) when normalised, and
// (x^a)(y^b) otherwise.
//
// Almost all of the error in the incomplete beta comes from this
// function: particularly when a and b are large. Computing large
// powers are *hard* though, and using logarithms just leads to
// horrendous cancellation errors.
//
// This version is generic, slow, and does not use the Lanczos approximation.
//
template <class T, class Policy>
T ibeta_power_terms(T a,
T b,
T x,
T y,
const boost::math::lanczos::undefined_lanczos&,
bool normalised,
const Policy& pol)
{
BOOST_MATH_STD_USING
if(!normalised)
{
return pow(x, a) * pow(y, b);
}
T result= 0; // assignment here silences warnings later
T c = a + b;
// integration limits for the gamma functions:
//T la = (std::max)(T(10), a);
//T lb = (std::max)(T(10), b);
//T lc = (std::max)(T(10), a+b);
T la = a + 5;
T lb = b + 5;
T lc = a + b + 5;
// gamma function partials:
T sa = detail::lower_gamma_series(a, la, pol) / a;
sa += detail::upper_gamma_fraction(a, la, ::boost::math::policies::get_epsilon<T, Policy>());
T sb = detail::lower_gamma_series(b, lb, pol) / b;
sb += detail::upper_gamma_fraction(b, lb, ::boost::math::policies::get_epsilon<T, Policy>());
T sc = detail::lower_gamma_series(c, lc, pol) / c;
sc += detail::upper_gamma_fraction(c, lc, ::boost::math::policies::get_epsilon<T, Policy>());
// gamma function powers combined with incomplete beta powers:
T b1 = (x * lc) / la;
T b2 = (y * lc) / lb;
T e1 = lc - la - lb;
T lb1 = a * log(b1);
T lb2 = b * log(b2);
if((lb1 >= tools::log_max_value<T>())
|| (lb1 <= tools::log_min_value<T>())
|| (lb2 >= tools::log_max_value<T>())
|| (lb2 <= tools::log_min_value<T>())
|| (e1 >= tools::log_max_value<T>())
|| (e1 <= tools::log_min_value<T>())
)
{
result = exp(lb1 + lb2 - e1);
}
else
{
T p1, p2;
if((fabs(b1 - 1) * a < 10) && (a > 1))
p1 = exp(a * boost::math::log1p((x * b - y * la) / la, pol));
else
p1 = pow(b1, a);
if((fabs(b2 - 1) * b < 10) && (b > 1))
p2 = exp(b * boost::math::log1p((y * a - x * lb) / lb, pol));
else
p2 = pow(b2, b);
T p3 = exp(e1);
result = p1 * p2 / p3;
}
// and combine with the remaining gamma function components:
result /= sa * sb / sc;
return result;
}
//
// Series approximation to the incomplete beta:
//
template <class T>
struct ibeta_series_t
{
typedef T result_type;
ibeta_series_t(T a_, T b_, T x_, T mult) : result(mult), x(x_), apn(a_), poch(1-b_), n(1) {}
T operator()()
{
T r = result / apn;
apn += 1;
result *= poch * x / n;
++n;
poch += 1;
return r;
}
private:
T result, x, apn, poch;
int n;
};
template <class T, class 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<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_) : 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