491 lines
20 KiB
C++
491 lines
20 KiB
C++
/* ----------------------------------------------------------------------------
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* GTSAM Copyright 2010, Georgia Tech Research Corporation,
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* Atlanta, Georgia 30332-0415
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* All Rights Reserved
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* Authors: Frank Dellaert, et al. (see THANKS for the full author list)
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* See LICENSE for the license information
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* -------------------------------------------------------------------------- */
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/**
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* @file GncHelpers.h
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* @brief Helper functions for use with the GncOptimizer
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* @author Varun Agrawal
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*/
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#pragma once
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#include <algorithm>
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namespace gtsam {
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/// Template type for numeric limits
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template <class T>
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using LIM = std::numeric_limits<T>;
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template <typename T>
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using return_t =
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typename std::conditional<std::is_integral<T>::value, double, T>::type;
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template <typename... T>
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using common_t = typename std::common_type<T...>::type;
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template <typename... T>
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using common_return_t = return_t<common_t<T...>>;
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/// Check if integer is odd
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constexpr bool is_odd(const long long int x) noexcept { return (x & 1U) != 0; }
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/// Templated check for NaN
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template <typename T>
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constexpr bool is_nan(const T x) noexcept {
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return x != x;
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}
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/// @brief Gauss-Legendre quadrature: 50 points
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static const long double gauss_legendre_50_points[50] = {
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-0.03109833832718887611232898966595L, 0.03109833832718887611232898966595L,
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-0.09317470156008614085445037763960L, 0.09317470156008614085445037763960L,
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-0.15489058999814590207162862094111L, 0.15489058999814590207162862094111L,
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-0.21600723687604175684728453261710L, 0.21600723687604175684728453261710L,
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-0.27628819377953199032764527852113L, 0.27628819377953199032764527852113L,
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-0.33550024541943735683698825729107L, 0.33550024541943735683698825729107L,
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-0.39341431189756512739422925382382L, 0.39341431189756512739422925382382L,
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-0.44980633497403878914713146777838L, 0.44980633497403878914713146777838L,
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-0.50445814490746420165145913184914L, 0.50445814490746420165145913184914L,
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-0.55715830451465005431552290962580L, 0.55715830451465005431552290962580L,
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-0.60770292718495023918038179639183L, 0.60770292718495023918038179639183L,
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-0.65589646568543936078162486400368L, 0.65589646568543936078162486400368L,
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-0.70155246870682225108954625788366L, 0.70155246870682225108954625788366L,
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-0.74449430222606853826053625268219L, 0.74449430222606853826053625268219L,
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-0.78455583290039926390530519634099L, 0.78455583290039926390530519634099L,
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-0.82158207085933594835625411087394L, 0.82158207085933594835625411087394L,
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-0.85542976942994608461136264393476L, 0.85542976942994608461136264393476L,
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-0.88596797952361304863754098246675L, 0.88596797952361304863754098246675L,
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-0.91307855665579189308973564277166L, 0.91307855665579189308973564277166L,
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-0.93665661894487793378087494727250L, 0.93665661894487793378087494727250L,
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-0.95661095524280794299774564415662L, 0.95661095524280794299774564415662L,
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-0.97286438510669207371334410460625L, 0.97286438510669207371334410460625L,
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-0.98535408404800588230900962563249L, 0.98535408404800588230900962563249L,
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-0.99403196943209071258510820042069L, 0.99403196943209071258510820042069L,
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-0.99886640442007105018545944497422L, 0.99886640442007105018545944497422L};
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/// @brief Gauss-Legendre quadrature: 50 weights
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static const long double gauss_legendre_50_weights[50] = {
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0.06217661665534726232103310736061L, 0.06217661665534726232103310736061L,
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0.06193606742068324338408750978083L, 0.06193606742068324338408750978083L,
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0.06145589959031666375640678608392L, 0.06145589959031666375640678608392L,
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0.06073797084177021603175001538481L, 0.06073797084177021603175001538481L,
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0.05978505870426545750957640531259L, 0.05978505870426545750957640531259L,
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0.05860084981322244583512243663085L, 0.05860084981322244583512243663085L,
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0.05718992564772838372302931506599L, 0.05718992564772838372302931506599L,
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0.05555774480621251762356742561227L, 0.05555774480621251762356742561227L,
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0.05371062188899624652345879725566L, 0.05371062188899624652345879725566L,
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0.05165570306958113848990529584010L, 0.05165570306958113848990529584010L,
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0.04940093844946631492124358075143L, 0.04940093844946631492124358075143L,
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0.04695505130394843296563301363499L, 0.04695505130394843296563301363499L,
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0.04432750433880327549202228683039L, 0.04432750433880327549202228683039L,
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0.04152846309014769742241197896407L, 0.04152846309014769742241197896407L,
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0.03856875661258767524477015023639L, 0.03856875661258767524477015023639L,
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0.03545983561514615416073461100098L, 0.03545983561514615416073461100098L,
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0.03221372822357801664816582732300L, 0.03221372822357801664816582732300L,
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0.02884299358053519802990637311323L, 0.02884299358053519802990637311323L,
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0.02536067357001239044019487838544L, 0.02536067357001239044019487838544L,
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0.02178024317012479298159206906269L, 0.02178024317012479298159206906269L,
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0.01811556071348939035125994342235L, 0.01811556071348939035125994342235L,
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0.01438082276148557441937890892732L, 0.01438082276148557441937890892732L,
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0.01059054838365096926356968149924L, 0.01059054838365096926356968149924L,
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0.00675979919574540150277887817799L, 0.00675979919574540150277887817799L,
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0.00290862255315514095840072434286L, 0.00290862255315514095840072434286L};
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namespace internal {
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/// 50 point Gauss-Legendre quadrature
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template <typename T>
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constexpr T incomplete_gamma_quad_inp_vals(const T lb, const T ub,
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const int counter) noexcept {
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return (ub - lb) * gauss_legendre_50_points[counter] / T(2) +
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(ub + lb) / T(2);
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}
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template <typename T>
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constexpr T incomplete_gamma_quad_weight_vals(const T lb, const T ub,
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const int counter) noexcept {
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return (ub - lb) * gauss_legendre_50_weights[counter] / T(2);
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}
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template <typename T>
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constexpr T incomplete_gamma_quad_fn(const T x, const T a,
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const T lg_term) noexcept {
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return exp(-x + (a - T(1)) * log(x) - lg_term);
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}
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template <typename T>
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constexpr T incomplete_gamma_quad_recur(const T lb, const T ub, const T a,
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const T lg_term,
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const int counter) noexcept {
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return (counter < 49 ? // if
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incomplete_gamma_quad_fn(
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incomplete_gamma_quad_inp_vals(lb, ub, counter), a, lg_term) *
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incomplete_gamma_quad_weight_vals(lb, ub, counter) +
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incomplete_gamma_quad_recur(lb, ub, a, lg_term, counter + 1)
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:
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// else
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incomplete_gamma_quad_fn(
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incomplete_gamma_quad_inp_vals(lb, ub, counter), a, lg_term) *
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incomplete_gamma_quad_weight_vals(lb, ub, counter));
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}
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template <typename T>
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constexpr T incomplete_gamma_quad_lb(const T a, const T z) noexcept {
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// break integration into ranges
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return (a > T(1000) ? std::max(T(0), std::min(z, a) - 11 * sqrt(a))
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: a > T(800) ? std::max(T(0), std::min(z, a) - 11 * sqrt(a))
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: a > T(500) ? std::max(T(0), std::min(z, a) - 10 * sqrt(a))
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: a > T(300) ? std::max(T(0), std::min(z, a) - 10 * sqrt(a))
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: a > T(100) ? std::max(T(0), std::min(z, a) - 9 * sqrt(a))
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: a > T(90) ? std::max(T(0), std::min(z, a) - 9 * sqrt(a))
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: a > T(70) ? std::max(T(0), std::min(z, a) - 8 * sqrt(a))
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: a > T(50) ? std::max(T(0), std::min(z, a) - 7 * sqrt(a))
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: a > T(40) ? std::max(T(0), std::min(z, a) - 6 * sqrt(a))
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: a > T(30) ? std::max(T(0), std::min(z, a) - 5 * sqrt(a))
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: std::max(T(0), std::min(z, a) - 4 * sqrt(a)));
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}
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template <typename T>
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constexpr T incomplete_gamma_quad_ub(const T a, const T z) noexcept {
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return (a > T(1000) ? std::min(z, a + 10 * sqrt(a))
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: a > T(800) ? std::min(z, a + 10 * sqrt(a))
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: a > T(500) ? std::min(z, a + 9 * sqrt(a))
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: a > T(300) ? std::min(z, a + 9 * sqrt(a))
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: a > T(100) ? std::min(z, a + 8 * sqrt(a))
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: a > T(90) ? std::min(z, a + 8 * sqrt(a))
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: a > T(70) ? std::min(z, a + 7 * sqrt(a))
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: a > T(50) ? std::min(z, a + 6 * sqrt(a))
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: std::min(z, a + 5 * sqrt(a)));
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}
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template <typename T>
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constexpr T incomplete_gamma_quad(const T a, const T z) noexcept {
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return incomplete_gamma_quad_recur(incomplete_gamma_quad_lb(a, z),
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incomplete_gamma_quad_ub(a, z), a,
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lgamma(a), 0);
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}
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// reverse cf expansion
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// see: https://functions.wolfram.com/GammaBetaErf/Gamma2/10/0003/
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template <typename T>
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constexpr T incomplete_gamma_cf_2_recur(const T a, const T z,
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const int depth) noexcept {
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return (depth < 100 ? (1 + (depth - 1) * 2 - a + z) +
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depth * (a - depth) /
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incomplete_gamma_cf_2_recur(a, z, depth + 1)
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: (1 + (depth - 1) * 2 - a + z));
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}
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template <typename T>
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constexpr T incomplete_gamma_cf_2(
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const T a,
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const T z) noexcept { // lower (regularized) incomplete gamma function
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return (T(1.0) - exp(a * log(z) - z - lgamma(a)) /
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incomplete_gamma_cf_2_recur(a, z, 1));
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}
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// cf expansion
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// see: http://functions.wolfram.com/GammaBetaErf/Gamma2/10/0009/
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template <typename T>
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constexpr T incomplete_gamma_cf_1_coef(const T a, const T z,
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const int depth) noexcept {
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return (is_odd(depth) ? -(a - 1 + T(depth + 1) / T(2)) * z
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: T(depth) / T(2) * z);
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}
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template <typename T>
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constexpr T incomplete_gamma_cf_1_recur(const T a, const T z,
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const int depth) noexcept {
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return (depth < 55 ? // if
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(a + depth - 1) + incomplete_gamma_cf_1_coef(a, z, depth) /
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incomplete_gamma_cf_1_recur(a, z, depth + 1)
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:
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// else
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(a + depth - 1));
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}
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template <typename T>
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constexpr T incomplete_gamma_cf_1(
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const T a,
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const T z) noexcept { // lower (regularized) incomplete gamma function
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return (exp(a * log(z) - z - lgamma(a)) /
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incomplete_gamma_cf_1_recur(a, z, 1));
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}
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//
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template <typename T>
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constexpr T incomplete_gamma_check(const T a, const T z) noexcept {
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return ( // NaN check
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(is_nan(a) || is_nan(z)) ? LIM<T>::quiet_NaN() :
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//
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a < T(0) ? LIM<T>::quiet_NaN()
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:
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//
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LIM<T>::min() > z ? T(0)
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:
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//
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LIM<T>::min() > a ? T(1)
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:
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// cf or quadrature
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(a < T(10)) && (z - a < T(10)) ? incomplete_gamma_cf_1(a, z)
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: (a < T(10)) || (z / a > T(3)) ? incomplete_gamma_cf_2(a, z)
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:
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// else
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incomplete_gamma_quad(a, z));
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}
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template <typename T1, typename T2, typename TC = common_return_t<T1, T2>>
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constexpr TC incomplete_gamma_type_check(const T1 a, const T2 p) noexcept {
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return incomplete_gamma_check(static_cast<TC>(a), static_cast<TC>(p));
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}
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} // namespace internal
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/**
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* Compile-time regularized lower incomplete gamma function
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*
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* @param a a real-valued, non-negative input.
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* @param x a real-valued, non-negative input.
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*
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* @return the regularized lower incomplete gamma function evaluated at (\c a,
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* \c x), \f[ \frac{\gamma(a,x)}{\Gamma(a)} = \frac{1}{\Gamma(a)} \int_0^x
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* t^{a-1} \exp(-t) dt \f] When \c a is not too large, the value is computed
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* using the continued fraction representation of the upper incomplete gamma
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* function, \f$ \Gamma(a,x) \f$, using \f[ \Gamma(a,x) = \Gamma(a) -
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* \dfrac{x^a\exp(-x)}{a - \dfrac{ax}{a + 1 + \dfrac{x}{a + 2 - \dfrac{(a+1)x}{a
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* + 3 + \dfrac{2x}{a + 4 - \ddots}}}}} \f] where \f$ \gamma(a,x) \f$ and \f$
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* \Gamma(a,x) \f$ are connected via \f[ \frac{\gamma(a,x)}{\Gamma(a)} +
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* \frac{\Gamma(a,x)}{\Gamma(a)} = 1 \f] When \f$ a > 10 \f$, a 50-point
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* Gauss-Legendre quadrature scheme is employed.
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*/
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template <typename T1, typename T2>
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constexpr common_return_t<T1, T2> incomplete_gamma(const T1 a,
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const T2 x) noexcept {
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return internal::incomplete_gamma_type_check(a, x);
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}
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namespace internal {
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template <typename T>
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constexpr T incomplete_gamma_inv_decision(const T value, const T a, const T p,
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const T direc, const T lg_val,
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const int iter_count) noexcept;
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//
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// initial value for Halley's method
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template <typename T>
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constexpr T incomplete_gamma_inv_t_val_1(const T p) noexcept { // a > 1.0
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return (p > T(0.5) ? sqrt(-2 * log(T(1) - p)) : sqrt(-2 * log(p)));
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}
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template <typename T>
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constexpr T incomplete_gamma_inv_t_val_2(const T a) noexcept { // a <= 1.0
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return (T(1) - T(0.253) * a - T(0.12) * a * a);
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}
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//
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template <typename T>
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constexpr T incomplete_gamma_inv_initial_val_1_int_begin(
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const T t_val) noexcept { // internal for a > 1.0
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return (t_val -
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(T(2.515517L) + T(0.802853L) * t_val + T(0.010328L) * t_val * t_val) /
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(T(1) + T(1.432788L) * t_val + T(0.189269L) * t_val * t_val +
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T(0.001308L) * t_val * t_val * t_val));
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}
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template <typename T>
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constexpr T incomplete_gamma_inv_initial_val_1_int_end(
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const T value_inp, const T a) noexcept { // internal for a > 1.0
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return std::max(
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T(1E-04), a * pow(T(1) - T(1) / (9 * a) - value_inp / (3 * sqrt(a)), 3));
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}
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template <typename T>
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constexpr T incomplete_gamma_inv_initial_val_1(
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const T a, const T t_val, const T sgn_term) noexcept { // a > 1.0
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return incomplete_gamma_inv_initial_val_1_int_end(
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sgn_term * incomplete_gamma_inv_initial_val_1_int_begin(t_val), a);
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}
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template <typename T>
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constexpr T incomplete_gamma_inv_initial_val_2(
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const T a, const T p, const T t_val) noexcept { // a <= 1.0
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return (p < t_val ? // if
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pow(p / t_val, T(1) / a)
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:
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// else
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T(1) - log(T(1) - (p - t_val) / (T(1) - t_val)));
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}
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// initial value
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template <typename T>
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constexpr T incomplete_gamma_inv_initial_val(const T a, const T p) noexcept {
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return (a > T(1) ? // if
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incomplete_gamma_inv_initial_val_1(
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a, incomplete_gamma_inv_t_val_1(p), p > T(0.5) ? T(-1) : T(1))
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:
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// else
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incomplete_gamma_inv_initial_val_2(
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a, p, incomplete_gamma_inv_t_val_2(a)));
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}
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//
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// Halley recursion
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template <typename T>
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constexpr T incomplete_gamma_inv_err_val(
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const T value, const T a, const T p) noexcept { // err_val = f(x)
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return (incomplete_gamma(a, value) - p);
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}
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template <typename T>
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constexpr T incomplete_gamma_inv_deriv_1(
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const T value, const T a,
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const T lg_val) noexcept { // derivative of the incomplete gamma function
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// w.r.t. x
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return (exp(-value + (a - T(1)) * log(value) - lg_val));
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}
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template <typename T>
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constexpr T incomplete_gamma_inv_deriv_2(
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const T value, const T a,
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const T deriv_1) noexcept { // second derivative of the incomplete gamma
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// function w.r.t. x
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return (deriv_1 * ((a - T(1)) / value - T(1)));
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}
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template <typename T>
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constexpr T incomplete_gamma_inv_ratio_val_1(const T value, const T a,
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const T p,
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const T deriv_1) noexcept {
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return (incomplete_gamma_inv_err_val(value, a, p) / deriv_1);
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}
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template <typename T>
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constexpr T incomplete_gamma_inv_ratio_val_2(const T value, const T a,
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const T deriv_1) noexcept {
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return (incomplete_gamma_inv_deriv_2(value, a, deriv_1) / deriv_1);
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}
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template <typename T>
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constexpr T incomplete_gamma_inv_halley(const T ratio_val_1,
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const T ratio_val_2) noexcept {
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return (ratio_val_1 /
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std::max(T(0.8), std::min(T(1.2), T(1) - T(0.5) * ratio_val_1 *
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ratio_val_2)));
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}
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template <typename T>
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constexpr T incomplete_gamma_inv_recur(const T value, const T a, const T p,
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const T deriv_1, const T lg_val,
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const int iter_count) noexcept {
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return incomplete_gamma_inv_decision(
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value, a, p,
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incomplete_gamma_inv_halley(
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incomplete_gamma_inv_ratio_val_1(value, a, p, deriv_1),
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incomplete_gamma_inv_ratio_val_2(value, a, deriv_1)),
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lg_val, iter_count);
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}
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|
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template <typename T>
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constexpr T incomplete_gamma_inv_decision(const T value, const T a, const T p,
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const T direc, const T lg_val,
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|
const int iter_count) noexcept {
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|
// return( abs(direc) > GCEM_INCML_GAMMA_INV_TOL ?
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// incomplete_gamma_inv_recur(value - direc, a, p,
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// incomplete_gamma_inv_deriv_1(value,a,lg_val), lg_val) : value - direc );
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|
#define INCML_GAMMA_INV_MAX_ITER 35
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return (iter_count <= INCML_GAMMA_INV_MAX_ITER ? // if
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|
incomplete_gamma_inv_recur(
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|
value - direc, a, p,
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|
incomplete_gamma_inv_deriv_1(value, a, lg_val), lg_val,
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|
iter_count + 1)
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|
:
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|
// else
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|
value - direc);
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|
}
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|
|
|
template <typename T>
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|
constexpr T incomplete_gamma_inv_begin(const T initial_val, const T a,
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|
const T p, const T lg_val) noexcept {
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|
return incomplete_gamma_inv_recur(
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|
initial_val, a, p, incomplete_gamma_inv_deriv_1(initial_val, a, lg_val),
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|
lg_val, 1);
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|
}
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|
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|
template <typename T>
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|
constexpr T incomplete_gamma_inv_check(const T a, const T p) noexcept {
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|
return ( // NaN check
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|
(is_nan(a) || is_nan(p)) ? LIM<T>::quiet_NaN() :
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|
//
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|
LIM<T>::min() > p ? T(0)
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|
: p > T(1) ? LIM<T>::quiet_NaN()
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|
: LIM<T>::min() > abs(T(1) - p) ? LIM<T>::infinity()
|
|
:
|
|
//
|
|
LIM<T>::min() > a ? T(0)
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|
:
|
|
// else
|
|
incomplete_gamma_inv_begin(incomplete_gamma_inv_initial_val(a, p), a,
|
|
p, lgamma(a)));
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|
}
|
|
|
|
template <typename T1, typename T2, typename TC = common_return_t<T1, T2>>
|
|
constexpr TC incomplete_gamma_inv_type_check(const T1 a, const T2 p) noexcept {
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|
return incomplete_gamma_inv_check(static_cast<TC>(a), static_cast<TC>(p));
|
|
}
|
|
|
|
} // namespace internal
|
|
|
|
/**
|
|
* Compile-time inverse incomplete gamma function
|
|
*
|
|
* @param a a real-valued, non-negative input.
|
|
* @param p a real-valued input with values in the unit-interval.
|
|
*
|
|
* @return Computes the inverse incomplete gamma function, a value \f$ x \f$
|
|
* such that \f[ f(x) := \frac{\gamma(a,x)}{\Gamma(a)} - p \f] equal to zero,
|
|
* for a given \c p. GCE-Math finds this root using Halley's method: \f[ x_{n+1}
|
|
* = x_n - \frac{f(x_n)/f'(x_n)}{1 - 0.5 \frac{f(x_n)}{f'(x_n)}
|
|
* \frac{f''(x_n)}{f'(x_n)} } \f] where \f[ \frac{\partial}{\partial x}
|
|
* \left(\frac{\gamma(a,x)}{\Gamma(a)}\right) = \frac{1}{\Gamma(a)} x^{a-1}
|
|
* \exp(-x) \f] \f[ \frac{\partial^2}{\partial x^2}
|
|
* \left(\frac{\gamma(a,x)}{\Gamma(a)}\right) = \frac{1}{\Gamma(a)} x^{a-1}
|
|
* \exp(-x) \left( \frac{a-1}{x} - 1 \right) \f]
|
|
*/
|
|
|
|
template <typename T1, typename T2>
|
|
constexpr common_return_t<T1, T2> incomplete_gamma_inv(const T1 a,
|
|
const T2 p) noexcept {
|
|
return internal::incomplete_gamma_inv_type_check(a, p);
|
|
}
|
|
|
|
/**
|
|
* @brief Compute the quantile function of the Chi squared distribution.
|
|
*
|
|
* @param dofs Degrees of freedom
|
|
* @param alpha Quantile value
|
|
* @return constexpr double
|
|
*/
|
|
constexpr double chi_squared_quantile(const size_t dofs, const double alpha) {
|
|
// The quantile function of the Chi-squared distribution is the quantile of
|
|
// the specific (inverse) incomplete Gamma distribution
|
|
return 2 * incomplete_gamma_inv(dofs * 0.5, alpha);
|
|
}
|
|
|
|
} // namespace gtsam
|