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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)
//
// This is not a complete header file, it is included by gamma.hpp
// after it has defined it's definitions. This inverts the incomplete
// gamma functions P and Q on the first parameter "a" using a generic
// root finding algorithm (TOMS Algorithm 748).
//
#ifndef BOOST_MATH_SP_DETAIL_GAMMA_INVA
#define BOOST_MATH_SP_DETAIL_GAMMA_INVA
#ifdef _MSC_VER
#pragma once
#endif
#include <boost/math/tools/toms748_solve.hpp>
#include <boost/cstdint.hpp>
namespace boost{ namespace math{ namespace detail{
template <class T, class Policy>
struct gamma_inva_t
{
gamma_inva_t(T z_, T p_, bool invert_) : z(z_), p(p_), invert(invert_) {}
T operator()(T a)
{
return invert ? p - boost::math::gamma_q(a, z, Policy()) : boost::math::gamma_p(a, z, Policy()) - p;
}
private:
T z, p;
bool invert;
};
template <class T, class Policy>
T inverse_poisson_cornish_fisher(T lambda, T p, T q, const Policy& pol)
{
BOOST_MATH_STD_USING
// mean:
T m = lambda;
// standard deviation:
T sigma = sqrt(lambda);
// skewness
T sk = 1 / sigma;
// kurtosis:
// T k = 1/lambda;
// Get the inverse of a std normal distribution:
T x = boost::math::erfc_inv(p > q ? 2 * q : 2 * p, pol) * constants::root_two<T>();
// Set the sign:
if(p < 0.5)
x = -x;
T x2 = x * x;
// w is correction term due to skewness
T w = x + sk * (x2 - 1) / 6;
/*
// Add on correction due to kurtosis.
// Disabled for now, seems to make things worse?
//
if(lambda >= 10)
w += k * x * (x2 - 3) / 24 + sk * sk * x * (2 * x2 - 5) / -36;
*/
w = m + sigma * w;
return w > tools::min_value<T>() ? w : tools::min_value<T>();
}
template <class T, class Policy>
T gamma_inva_imp(const T& z, const T& p, const T& q, const Policy& pol)
{
BOOST_MATH_STD_USING // for ADL of std lib math functions
//
// Special cases first:
//
if(p == 0)
{
return tools::max_value<T>();
}
if(q == 0)
{
return tools::min_value<T>();
}
//
// Function object, this is the functor whose root
// we have to solve:
//
gamma_inva_t<T, Policy> f(z, (p < q) ? p : q, (p < q) ? false : true);
//
// Tolerance: full precision.
//
tools::eps_tolerance<T> tol(policies::digits<T, Policy>());
//
// Now figure out a starting guess for what a may be,
// we'll start out with a value that'll put p or q
// right bang in the middle of their range, the functions
// are quite sensitive so we should need too many steps
// to bracket the root from there:
//
T guess;
T factor = 8;
if(z >= 1)
{
//
// We can use the relationship between the incomplete
// gamma function and the poisson distribution to
// calculate an approximate inverse, for large z
// this is actually pretty accurate, but it fails badly
// when z is very small. Also set our step-factor according
// to how accurate we think the result is likely to be:
//
guess = 1 + inverse_poisson_cornish_fisher(z, q, p, pol);
if(z > 5)
{
if(z > 1000)
factor = 1.01f;
else if(z > 50)
factor = 1.1f;
else if(guess > 10)
factor = 1.25f;
else
factor = 2;
if(guess < 1.1)
factor = 8;
}
}
else if(z > 0.5)
{
guess = z * 1.2f;
}
else
{
guess = -0.4f / log(z);
}
//
// Max iterations permitted:
//
boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();
//
// Use our generic derivative-free root finding procedure.
// We could use Newton steps here, taking the PDF of the
// Poisson distribution as our derivative, but that's
// even worse performance-wise than the generic method :-(
//
std::pair<T, T> r = bracket_and_solve_root(f, guess, factor, false, tol, max_iter, pol);
if(max_iter >= policies::get_max_root_iterations<Policy>())
policies::raise_evaluation_error<T>("boost::math::gamma_p_inva<%1%>(%1%, %1%)", "Unable to locate the root within a reasonable number of iterations, closest approximation so far was %1%", r.first, pol);
return (r.first + r.second) / 2;
}
} // namespace detail
template <class T1, class T2, class Policy>
inline typename tools::promote_args<T1, T2>::type
gamma_p_inva(T1 x, T2 p, const Policy& pol)
{
typedef typename tools::promote_args<T1, T2>::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;
if(p == 0)
{
return tools::max_value<result_type>();
}
if(p == 1)
{
return tools::min_value<result_type>();
}
return policies::checked_narrowing_cast<result_type, forwarding_policy>(
detail::gamma_inva_imp(
static_cast<value_type>(x),
static_cast<value_type>(p),
static_cast<value_type>(1 - static_cast<value_type>(p)),
pol), "boost::math::gamma_p_inva<%1%>(%1%, %1%)");
}
template <class T1, class T2, class Policy>
inline typename tools::promote_args<T1, T2>::type
gamma_q_inva(T1 x, T2 q, const Policy& pol)
{
typedef typename tools::promote_args<T1, T2>::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;
if(q == 1)
{
return tools::max_value<result_type>();
}
if(q == 0)
{
return tools::min_value<result_type>();
}
return policies::checked_narrowing_cast<result_type, forwarding_policy>(
detail::gamma_inva_imp(
static_cast<value_type>(x),
static_cast<value_type>(1 - static_cast<value_type>(q)),
static_cast<value_type>(q),
pol), "boost::math::gamma_q_inva<%1%>(%1%, %1%)");
}
template <class T1, class T2>
inline typename tools::promote_args<T1, T2>::type
gamma_p_inva(T1 x, T2 p)
{
return boost::math::gamma_p_inva(x, p, policies::policy<>());
}
template <class T1, class T2>
inline typename tools::promote_args<T1, T2>::type
gamma_q_inva(T1 x, T2 q)
{
return boost::math::gamma_q_inva(x, q, policies::policy<>());
}
} // namespace math
} // namespace boost
#endif // BOOST_MATH_SP_DETAIL_GAMMA_INVA
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