beta.hpp File Reference

#include <concepts>
#include <type_traits>
#include <randlib/math/constants.hpp>
#include <randlib/math/functions.hpp>
#include <randlib/math/gamma.hpp>
#include <randlib/math/quadrature.hpp>
#include <randlib/math/roots.hpp>
#include <randlib/math/optimize.hpp>
#include <randlib/math/special_functions.hpp>

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Namespaces
namespace	randlib
namespace	randlib::math
	Beta-related functions.
namespace	randlib::math::detail

Functions
template<std::floating_point Real>
Real	randlib::math::logBeta (Real a, std::type_identity_t< Real > b)
	Calculate logarithm of beta function.
template<typename Left , typename Right > requires (std::is_arithmetic_v<Left> && std::is_arithmetic_v<Right> && (std::integral<Left> || std::integral<Right> || !std::same_as<Left, Right>))
auto	randlib::math::logBeta (Left a, Right b) -> std::common_type_t< double, Left, Right >
template<std::floating_point Real>
Real	randlib::math::ibeta (Real x, std::type_identity_t< Real > a, std::type_identity_t< Real > b, std::type_identity_t< Real > logBetaFun, std::type_identity_t< Real > logX, std::type_identity_t< Real > log1mX)
	Fast calculation of regularized beta function, using precalculated values.
template<std::floating_point Real>
Real	randlib::math::ibeta (Real x, std::type_identity_t< Real > a, std::type_identity_t< Real > b)
	Calculate regularized beta function.
template<std::floating_point Real>
Real	randlib::math::ibetaPowerSeries1 (Real x, Real a, Real b, Real logBetaFun, Real logX, Real log1mX)
template<std::floating_point Real>
Real	randlib::math::ibetacontinuedFraction1 (Real x, Real a, Real b, int number, Real logBetaFun, Real logX, Real log1mX)
template<std::floating_point Real>
Real	randlib::math::ibetaPowerSeries2 (Real x, Real a, Real b, Real logBetaFun, Real logX, Real log1mX)
template<std::floating_point Real>
Real	randlib::math::ibetaContinuedFraction2 (Real x, Real a, Real b, int number, Real logBetaFun, Real logX, Real log1mX)
template<std::floating_point Real>
Real	randlib::math::ibetaContinuedFractionLentz (Real x, Real a, Real b, Real logBetaFun, Real logX, Real log1mX)
	Continued fraction of I(x, a, b) (DiDonato and Morris, also Numerical Recipes betacf) by the modified Lentz method; it converges for x below (a + 1) / (a + b + 2) in far fewer terms than b for large shapes.
template<std::floating_point Real>
Real	randlib::math::ibetaLowerTail (Real x, Real a, Real b, Real logBetaFun, Real logX, Real log1mX)
	I(x, a, b) for x not above the mean a / (a + b) and shapes up to IBETA_LARGEST_FINITE_SHAPE.

Variables
constexpr double	randlib::math::detail::IBETA_LARGEST_FINITE_SHAPE = 4096
	Beyond this size of a shape the finite continued fractions of ibetaLowerTail accumulate rounding errors, and its power series converge at the rate x close to 1.