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diff --git a/src/third_party/boost-1.60.0/boost/math/special_functions/detail/ibeta_inverse.hpp b/src/third_party/boost-1.60.0/boost/math/special_functions/detail/ibeta_inverse.hpp new file mode 100644 index 00000000000..a9fe8cd49cb --- /dev/null +++ b/src/third_party/boost-1.60.0/boost/math/special_functions/detail/ibeta_inverse.hpp @@ -0,0 +1,993 @@ +// Copyright John Maddock 2006. +// Copyright Paul A. Bristow 2007 +// 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_FUNCTIONS_IBETA_INVERSE_HPP +#define BOOST_MATH_SPECIAL_FUNCTIONS_IBETA_INVERSE_HPP + +#ifdef _MSC_VER +#pragma once +#endif + +#include <boost/math/special_functions/beta.hpp> +#include <boost/math/special_functions/erf.hpp> +#include <boost/math/tools/roots.hpp> +#include <boost/math/special_functions/detail/t_distribution_inv.hpp> + +namespace boost{ namespace math{ namespace detail{ + +// +// Helper object used by root finding +// code to convert eta to x. +// +template <class T> +struct temme_root_finder +{ + temme_root_finder(const T t_, const T a_) : t(t_), a(a_) {} + + boost::math::tuple<T, T> operator()(T x) + { + BOOST_MATH_STD_USING // ADL of std names + + T y = 1 - x; + if(y == 0) + { + T big = tools::max_value<T>() / 4; + return boost::math::make_tuple(static_cast<T>(-big), static_cast<T>(-big)); + } + if(x == 0) + { + T big = tools::max_value<T>() / 4; + return boost::math::make_tuple(static_cast<T>(-big), big); + } + T f = log(x) + a * log(y) + t; + T f1 = (1 / x) - (a / (y)); + return boost::math::make_tuple(f, f1); + } +private: + T t, a; +}; +// +// See: +// "Asymptotic Inversion of the Incomplete Beta Function" +// N.M. Temme +// Journal of Computation and Applied Mathematics 41 (1992) 145-157. +// Section 2. +// +template <class T, class Policy> +T temme_method_1_ibeta_inverse(T a, T b, T z, const Policy& pol) +{ + BOOST_MATH_STD_USING // ADL of std names + + const T r2 = sqrt(T(2)); + // + // get the first approximation for eta from the inverse + // error function (Eq: 2.9 and 2.10). + // + T eta0 = boost::math::erfc_inv(2 * z, pol); + eta0 /= -sqrt(a / 2); + + T terms[4] = { eta0 }; + T workspace[7]; + // + // calculate powers: + // + T B = b - a; + T B_2 = B * B; + T B_3 = B_2 * B; + // + // Calculate correction terms: + // + + // See eq following 2.15: + workspace[0] = -B * r2 / 2; + workspace[1] = (1 - 2 * B) / 8; + workspace[2] = -(B * r2 / 48); + workspace[3] = T(-1) / 192; + workspace[4] = -B * r2 / 3840; + terms[1] = tools::evaluate_polynomial(workspace, eta0, 5); + // Eq Following 2.17: + workspace[0] = B * r2 * (3 * B - 2) / 12; + workspace[1] = (20 * B_2 - 12 * B + 1) / 128; + workspace[2] = B * r2 * (20 * B - 1) / 960; + workspace[3] = (16 * B_2 + 30 * B - 15) / 4608; + workspace[4] = B * r2 * (21 * B + 32) / 53760; + workspace[5] = (-32 * B_2 + 63) / 368640; + workspace[6] = -B * r2 * (120 * B + 17) / 25804480; + terms[2] = tools::evaluate_polynomial(workspace, eta0, 7); + // Eq Following 2.17: + workspace[0] = B * r2 * (-75 * B_2 + 80 * B - 16) / 480; + workspace[1] = (-1080 * B_3 + 868 * B_2 - 90 * B - 45) / 9216; + workspace[2] = B * r2 * (-1190 * B_2 + 84 * B + 373) / 53760; + workspace[3] = (-2240 * B_3 - 2508 * B_2 + 2100 * B - 165) / 368640; + terms[3] = tools::evaluate_polynomial(workspace, eta0, 4); + // + // Bring them together to get a final estimate for eta: + // + T eta = tools::evaluate_polynomial(terms, T(1/a), 4); + // + // now we need to convert eta to x, by solving the appropriate + // quadratic equation: + // + T eta_2 = eta * eta; + T c = -exp(-eta_2 / 2); + T x; + if(eta_2 == 0) + x = 0.5; + else + x = (1 + eta * sqrt((1 + c) / eta_2)) / 2; + + BOOST_ASSERT(x >= 0); + BOOST_ASSERT(x <= 1); + BOOST_ASSERT(eta * (x - 0.5) >= 0); +#ifdef BOOST_INSTRUMENT + std::cout << "Estimating x with Temme method 1: " << x << std::endl; +#endif + return x; +} +// +// See: +// "Asymptotic Inversion of the Incomplete Beta Function" +// N.M. Temme +// Journal of Computation and Applied Mathematics 41 (1992) 145-157. +// Section 3. +// +template <class T, class Policy> +T temme_method_2_ibeta_inverse(T /*a*/, T /*b*/, T z, T r, T theta, const Policy& pol) +{ + BOOST_MATH_STD_USING // ADL of std names + + // + // Get first estimate for eta, see Eq 3.9 and 3.10, + // but note there is a typo in Eq 3.10: + // + T eta0 = boost::math::erfc_inv(2 * z, pol); + eta0 /= -sqrt(r / 2); + + T s = sin(theta); + T c = cos(theta); + // + // Now we need to purturb eta0 to get eta, which we do by + // evaluating the polynomial in 1/r at the bottom of page 151, + // to do this we first need the error terms e1, e2 e3 + // which we'll fill into the array "terms". Since these + // terms are themselves polynomials, we'll need another + // array "workspace" to calculate those... + // + T terms[4] = { eta0 }; + T workspace[6]; + // + // some powers of sin(theta)cos(theta) that we'll need later: + // + T sc = s * c; + T sc_2 = sc * sc; + T sc_3 = sc_2 * sc; + T sc_4 = sc_2 * sc_2; + T sc_5 = sc_2 * sc_3; + T sc_6 = sc_3 * sc_3; + T sc_7 = sc_4 * sc_3; + // + // Calculate e1 and put it in terms[1], see the middle of page 151: + // + workspace[0] = (2 * s * s - 1) / (3 * s * c); + static const BOOST_MATH_INT_TABLE_TYPE(T, int) co1[] = { -1, -5, 5 }; + workspace[1] = -tools::evaluate_even_polynomial(co1, s, 3) / (36 * sc_2); + static const BOOST_MATH_INT_TABLE_TYPE(T, int) co2[] = { 1, 21, -69, 46 }; + workspace[2] = tools::evaluate_even_polynomial(co2, s, 4) / (1620 * sc_3); + static const BOOST_MATH_INT_TABLE_TYPE(T, int) co3[] = { 7, -2, 33, -62, 31 }; + workspace[3] = -tools::evaluate_even_polynomial(co3, s, 5) / (6480 * sc_4); + static const BOOST_MATH_INT_TABLE_TYPE(T, int) co4[] = { 25, -52, -17, 88, -115, 46 }; + workspace[4] = tools::evaluate_even_polynomial(co4, s, 6) / (90720 * sc_5); + terms[1] = tools::evaluate_polynomial(workspace, eta0, 5); + // + // Now evaluate e2 and put it in terms[2]: + // + static const BOOST_MATH_INT_TABLE_TYPE(T, int) co5[] = { 7, 12, -78, 52 }; + workspace[0] = -tools::evaluate_even_polynomial(co5, s, 4) / (405 * sc_3); + static const BOOST_MATH_INT_TABLE_TYPE(T, int) co6[] = { -7, 2, 183, -370, 185 }; + workspace[1] = tools::evaluate_even_polynomial(co6, s, 5) / (2592 * sc_4); + static const BOOST_MATH_INT_TABLE_TYPE(T, int) co7[] = { -533, 776, -1835, 10240, -13525, 5410 }; + workspace[2] = -tools::evaluate_even_polynomial(co7, s, 6) / (204120 * sc_5); + static const BOOST_MATH_INT_TABLE_TYPE(T, int) co8[] = { -1579, 3747, -3372, -15821, 45588, -45213, 15071 }; + workspace[3] = -tools::evaluate_even_polynomial(co8, s, 7) / (2099520 * sc_6); + terms[2] = tools::evaluate_polynomial(workspace, eta0, 4); + // + // And e3, and put it in terms[3]: + // + static const BOOST_MATH_INT_TABLE_TYPE(T, int) co9[] = {449, -1259, -769, 6686, -9260, 3704 }; + workspace[0] = tools::evaluate_even_polynomial(co9, s, 6) / (102060 * sc_5); + static const BOOST_MATH_INT_TABLE_TYPE(T, int) co10[] = { 63149, -151557, 140052, -727469, 2239932, -2251437, 750479 }; + workspace[1] = -tools::evaluate_even_polynomial(co10, s, 7) / (20995200 * sc_6); + static const BOOST_MATH_INT_TABLE_TYPE(T, int) co11[] = { 29233, -78755, 105222, 146879, -1602610, 3195183, -2554139, 729754 }; + workspace[2] = tools::evaluate_even_polynomial(co11, s, 8) / (36741600 * sc_7); + terms[3] = tools::evaluate_polynomial(workspace, eta0, 3); + // + // Bring the correction terms together to evaluate eta, + // this is the last equation on page 151: + // + T eta = tools::evaluate_polynomial(terms, T(1/r), 4); + // + // Now that we have eta we need to back solve for x, + // we seek the value of x that gives eta in Eq 3.2. + // The two methods used are described in section 5. + // + // Begin by defining a few variables we'll need later: + // + T x; + T s_2 = s * s; + T c_2 = c * c; + T alpha = c / s; + alpha *= alpha; + T lu = (-(eta * eta) / (2 * s_2) + log(s_2) + c_2 * log(c_2) / s_2); + // + // Temme doesn't specify what value to switch on here, + // but this seems to work pretty well: + // + if(fabs(eta) < 0.7) + { + // + // Small eta use the expansion Temme gives in the second equation + // of section 5, it's a polynomial in eta: + // + workspace[0] = s * s; + workspace[1] = s * c; + workspace[2] = (1 - 2 * workspace[0]) / 3; + static const BOOST_MATH_INT_TABLE_TYPE(T, int) co12[] = { 1, -13, 13 }; + workspace[3] = tools::evaluate_polynomial(co12, workspace[0], 3) / (36 * s * c); + static const BOOST_MATH_INT_TABLE_TYPE(T, int) co13[] = { 1, 21, -69, 46 }; + workspace[4] = tools::evaluate_polynomial(co13, workspace[0], 4) / (270 * workspace[0] * c * c); + x = tools::evaluate_polynomial(workspace, eta, 5); +#ifdef BOOST_INSTRUMENT + std::cout << "Estimating x with Temme method 2 (small eta): " << x << std::endl; +#endif + } + else + { + // + // If eta is large we need to solve Eq 3.2 more directly, + // begin by getting an initial approximation for x from + // the last equation on page 155, this is a polynomial in u: + // + T u = exp(lu); + workspace[0] = u; + workspace[1] = alpha; + workspace[2] = 0; + workspace[3] = 3 * alpha * (3 * alpha + 1) / 6; + workspace[4] = 4 * alpha * (4 * alpha + 1) * (4 * alpha + 2) / 24; + workspace[5] = 5 * alpha * (5 * alpha + 1) * (5 * alpha + 2) * (5 * alpha + 3) / 120; + x = tools::evaluate_polynomial(workspace, u, 6); + // + // At this point we may or may not have the right answer, Eq-3.2 has + // two solutions for x for any given eta, however the mapping in 3.2 + // is 1:1 with the sign of eta and x-sin^2(theta) being the same. + // So we can check if we have the right root of 3.2, and if not + // switch x for 1-x. This transformation is motivated by the fact + // that the distribution is *almost* symetric so 1-x will be in the right + // ball park for the solution: + // + if((x - s_2) * eta < 0) + x = 1 - x; +#ifdef BOOST_INSTRUMENT + std::cout << "Estimating x with Temme method 2 (large eta): " << x << std::endl; +#endif + } + // + // The final step is a few Newton-Raphson iterations to + // clean up our approximation for x, this is pretty cheap + // in general, and very cheap compared to an incomplete beta + // evaluation. The limits set on x come from the observation + // that the sign of eta and x-sin^2(theta) are the same. + // + T lower, upper; + if(eta < 0) + { + lower = 0; + upper = s_2; + } + else + { + lower = s_2; + upper = 1; + } + // + // If our initial approximation is out of bounds then bisect: + // + if((x < lower) || (x > upper)) + x = (lower+upper) / 2; + // + // And iterate: + // + x = tools::newton_raphson_iterate( + temme_root_finder<T>(-lu, alpha), x, lower, upper, policies::digits<T, Policy>() / 2); + + return x; +} +// +// See: +// "Asymptotic Inversion of the Incomplete Beta Function" +// N.M. Temme +// Journal of Computation and Applied Mathematics 41 (1992) 145-157. +// Section 4. +// +template <class T, class Policy> +T temme_method_3_ibeta_inverse(T a, T b, T p, T q, const Policy& pol) +{ + BOOST_MATH_STD_USING // ADL of std names + + // + // Begin by getting an initial approximation for the quantity + // eta from the dominant part of the incomplete beta: + // + T eta0; + if(p < q) + eta0 = boost::math::gamma_q_inv(b, p, pol); + else + eta0 = boost::math::gamma_p_inv(b, q, pol); + eta0 /= a; + // + // Define the variables and powers we'll need later on: + // + T mu = b / a; + T w = sqrt(1 + mu); + T w_2 = w * w; + T w_3 = w_2 * w; + T w_4 = w_2 * w_2; + T w_5 = w_3 * w_2; + T w_6 = w_3 * w_3; + T w_7 = w_4 * w_3; + T w_8 = w_4 * w_4; + T w_9 = w_5 * w_4; + T w_10 = w_5 * w_5; + T d = eta0 - mu; + T d_2 = d * d; + T d_3 = d_2 * d; + T d_4 = d_2 * d_2; + T w1 = w + 1; + T w1_2 = w1 * w1; + T w1_3 = w1 * w1_2; + T w1_4 = w1_2 * w1_2; + // + // Now we need to compute the purturbation error terms that + // convert eta0 to eta, these are all polynomials of polynomials. + // Probably these should be re-written to use tabulated data + // (see examples above), but it's less of a win in this case as we + // need to calculate the individual powers for the denominator terms + // anyway, so we might as well use them for the numerator-polynomials + // as well.... + // + // Refer to p154-p155 for the details of these expansions: + // + T e1 = (w + 2) * (w - 1) / (3 * w); + e1 += (w_3 + 9 * w_2 + 21 * w + 5) * d / (36 * w_2 * w1); + e1 -= (w_4 - 13 * w_3 + 69 * w_2 + 167 * w + 46) * d_2 / (1620 * w1_2 * w_3); + e1 -= (7 * w_5 + 21 * w_4 + 70 * w_3 + 26 * w_2 - 93 * w - 31) * d_3 / (6480 * w1_3 * w_4); + e1 -= (75 * w_6 + 202 * w_5 + 188 * w_4 - 888 * w_3 - 1345 * w_2 + 118 * w + 138) * d_4 / (272160 * w1_4 * w_5); + + T e2 = (28 * w_4 + 131 * w_3 + 402 * w_2 + 581 * w + 208) * (w - 1) / (1620 * w1 * w_3); + e2 -= (35 * w_6 - 154 * w_5 - 623 * w_4 - 1636 * w_3 - 3983 * w_2 - 3514 * w - 925) * d / (12960 * w1_2 * w_4); + e2 -= (2132 * w_7 + 7915 * w_6 + 16821 * w_5 + 35066 * w_4 + 87490 * w_3 + 141183 * w_2 + 95993 * w + 21640) * d_2 / (816480 * w_5 * w1_3); + e2 -= (11053 * w_8 + 53308 * w_7 + 117010 * w_6 + 163924 * w_5 + 116188 * w_4 - 258428 * w_3 - 677042 * w_2 - 481940 * w - 105497) * d_3 / (14696640 * w1_4 * w_6); + + T e3 = -((3592 * w_7 + 8375 * w_6 - 1323 * w_5 - 29198 * w_4 - 89578 * w_3 - 154413 * w_2 - 116063 * w - 29632) * (w - 1)) / (816480 * w_5 * w1_2); + e3 -= (442043 * w_9 + 2054169 * w_8 + 3803094 * w_7 + 3470754 * w_6 + 2141568 * w_5 - 2393568 * w_4 - 19904934 * w_3 - 34714674 * w_2 - 23128299 * w - 5253353) * d / (146966400 * w_6 * w1_3); + e3 -= (116932 * w_10 + 819281 * w_9 + 2378172 * w_8 + 4341330 * w_7 + 6806004 * w_6 + 10622748 * w_5 + 18739500 * w_4 + 30651894 * w_3 + 30869976 * w_2 + 15431867 * w + 2919016) * d_2 / (146966400 * w1_4 * w_7); + // + // Combine eta0 and the error terms to compute eta (Second eqaution p155): + // + T eta = eta0 + e1 / a + e2 / (a * a) + e3 / (a * a * a); + // + // Now we need to solve Eq 4.2 to obtain x. For any given value of + // eta there are two solutions to this equation, and since the distribtion + // may be very skewed, these are not related by x ~ 1-x we used when + // implementing section 3 above. However we know that: + // + // cross < x <= 1 ; iff eta < mu + // x == cross ; iff eta == mu + // 0 <= x < cross ; iff eta > mu + // + // Where cross == 1 / (1 + mu) + // Many thanks to Prof Temme for clarifying this point. + // + // Therefore we'll just jump straight into Newton iterations + // to solve Eq 4.2 using these bounds, and simple bisection + // as the first guess, in practice this converges pretty quickly + // and we only need a few digits correct anyway: + // + if(eta <= 0) + eta = tools::min_value<T>(); + T u = eta - mu * log(eta) + (1 + mu) * log(1 + mu) - mu; + T cross = 1 / (1 + mu); + T lower = eta < mu ? cross : 0; + T upper = eta < mu ? 1 : cross; + T x = (lower + upper) / 2; + x = tools::newton_raphson_iterate( + temme_root_finder<T>(u, mu), x, lower, upper, policies::digits<T, Policy>() / 2); +#ifdef BOOST_INSTRUMENT + std::cout << "Estimating x with Temme method 3: " << x << std::endl; +#endif + return x; +} + +template <class T, class Policy> +struct ibeta_roots +{ + ibeta_roots(T _a, T _b, T t, bool inv = false) + : a(_a), b(_b), target(t), invert(inv) {} + + boost::math::tuple<T, T, T> operator()(T x) + { + BOOST_MATH_STD_USING // ADL of std names + + BOOST_FPU_EXCEPTION_GUARD + + T f1; + T y = 1 - x; + T f = ibeta_imp(a, b, x, Policy(), invert, true, &f1) - target; + if(invert) + f1 = -f1; + if(y == 0) + y = tools::min_value<T>() * 64; + if(x == 0) + x = tools::min_value<T>() * 64; + + T f2 = f1 * (-y * a + (b - 2) * x + 1); + if(fabs(f2) < y * x * tools::max_value<T>()) + f2 /= (y * x); + if(invert) + f2 = -f2; + + // make sure we don't have a zero derivative: + if(f1 == 0) + f1 = (invert ? -1 : 1) * tools::min_value<T>() * 64; + + return boost::math::make_tuple(f, f1, f2); + } +private: + T a, b, target; + bool invert; +}; + +template <class T, class Policy> +T ibeta_inv_imp(T a, T b, T p, T q, const Policy& pol, T* py) +{ + BOOST_MATH_STD_USING // For ADL of math functions. + + // + // The flag invert is set to true if we swap a for b and p for q, + // in which case the result has to be subtracted from 1: + // + bool invert = false; + // + // Handle trivial cases first: + // + if(q == 0) + { + if(py) *py = 0; + return 1; + } + else if(p == 0) + { + if(py) *py = 1; + return 0; + } + else if(a == 1) + { + if(b == 1) + { + if(py) *py = 1 - p; + return p; + } + // Change things around so we can handle as b == 1 special case below: + std::swap(a, b); + std::swap(p, q); + invert = true; + } + // + // Depending upon which approximation method we use, we may end up + // calculating either x or y initially (where y = 1-x): + // + T x = 0; // Set to a safe zero to avoid a + // MSVC 2005 warning C4701: potentially uninitialized local variable 'x' used + // But code inspection appears to ensure that x IS assigned whatever the code path. + T y; + + // For some of the methods we can put tighter bounds + // on the result than simply [0,1]: + // + T lower = 0; + T upper = 1; + // + // Student's T with b = 0.5 gets handled as a special case, swap + // around if the arguments are in the "wrong" order: + // + if(a == 0.5f) + { + if(b == 0.5f) + { + x = sin(p * constants::half_pi<T>()); + x *= x; + if(py) + { + *py = sin(q * constants::half_pi<T>()); + *py *= *py; + } + return x; + } + else if(b > 0.5f) + { + std::swap(a, b); + std::swap(p, q); + invert = !invert; + } + } + // + // Select calculation method for the initial estimate: + // + if((b == 0.5f) && (a >= 0.5f) && (p != 1)) + { + // + // We have a Student's T distribution: + x = find_ibeta_inv_from_t_dist(a, p, q, &y, pol); + } + else if(b == 1) + { + if(p < q) + { + if(a > 1) + { + x = pow(p, 1 / a); + y = -boost::math::expm1(log(p) / a, pol); + } + else + { + x = pow(p, 1 / a); + y = 1 - x; + } + } + else + { + x = exp(boost::math::log1p(-q, pol) / a); + y = -boost::math::expm1(boost::math::log1p(-q, pol) / a, pol); + } + if(invert) + std::swap(x, y); + if(py) + *py = y; + return x; + } + else if(a + b > 5) + { + // + // When a+b is large then we can use one of Prof Temme's + // asymptotic expansions, begin by swapping things around + // so that p < 0.5, we do this to avoid cancellations errors + // when p is large. + // + if(p > 0.5) + { + std::swap(a, b); + std::swap(p, q); + invert = !invert; + } + T minv = (std::min)(a, b); + T maxv = (std::max)(a, b); + if((sqrt(minv) > (maxv - minv)) && (minv > 5)) + { + // + // When a and b differ by a small amount + // the curve is quite symmetrical and we can use an error + // function to approximate the inverse. This is the cheapest + // of the three Temme expantions, and the calculated value + // for x will never be much larger than p, so we don't have + // to worry about cancellation as long as p is small. + // + x = temme_method_1_ibeta_inverse(a, b, p, pol); + y = 1 - x; + } + else + { + T r = a + b; + T theta = asin(sqrt(a / r)); + T lambda = minv / r; + if((lambda >= 0.2) && (lambda <= 0.8) && (r >= 10)) + { + // + // The second error function case is the next cheapest + // to use, it brakes down when the result is likely to be + // very small, if a+b is also small, but we can use a + // cheaper expansion there in any case. As before x won't + // be much larger than p, so as long as p is small we should + // be free of cancellation error. + // + T ppa = pow(p, 1/a); + if((ppa < 0.0025) && (a + b < 200)) + { + x = ppa * pow(a * boost::math::beta(a, b, pol), 1/a); + } + else + x = temme_method_2_ibeta_inverse(a, b, p, r, theta, pol); + y = 1 - x; + } + else + { + // + // If we get here then a and b are very different in magnitude + // and we need to use the third of Temme's methods which + // involves inverting the incomplete gamma. This is much more + // expensive than the other methods. We also can only use this + // method when a > b, which can lead to cancellation errors + // if we really want y (as we will when x is close to 1), so + // a different expansion is used in that case. + // + if(a < b) + { + std::swap(a, b); + std::swap(p, q); + invert = !invert; + } + // + // Try and compute the easy way first: + // + T bet = 0; + if(b < 2) + bet = boost::math::beta(a, b, pol); + if(bet != 0) + { + y = pow(b * q * bet, 1/b); + x = 1 - y; + } + else + y = 1; + if(y > 1e-5) + { + x = temme_method_3_ibeta_inverse(a, b, p, q, pol); + y = 1 - x; + } + } + } + } + else if((a < 1) && (b < 1)) + { + // + // Both a and b less than 1, + // there is a point of inflection at xs: + // + T xs = (1 - a) / (2 - a - b); + // + // Now we need to ensure that we start our iteration from the + // right side of the inflection point: + // + T fs = boost::math::ibeta(a, b, xs, pol) - p; + if(fabs(fs) / p < tools::epsilon<T>() * 3) + { + // The result is at the point of inflection, best just return it: + *py = invert ? xs : 1 - xs; + return invert ? 1-xs : xs; + } + if(fs < 0) + { + std::swap(a, b); + std::swap(p, q); + invert = !invert; + xs = 1 - xs; + } + T xg = pow(a * p * boost::math::beta(a, b, pol), 1/a); + x = xg / (1 + xg); + y = 1 / (1 + xg); + // + // And finally we know that our result is below the inflection + // point, so set an upper limit on our search: + // + if(x > xs) + x = xs; + upper = xs; + } + else if((a > 1) && (b > 1)) + { + // + // Small a and b, both greater than 1, + // there is a point of inflection at xs, + // and it's complement is xs2, we must always + // start our iteration from the right side of the + // point of inflection. + // + T xs = (a - 1) / (a + b - 2); + T xs2 = (b - 1) / (a + b - 2); + T ps = boost::math::ibeta(a, b, xs, pol) - p; + + if(ps < 0) + { + std::swap(a, b); + std::swap(p, q); + std::swap(xs, xs2); + invert = !invert; + } + // + // Estimate x and y, using expm1 to get a good estimate + // for y when it's very small: + // + T lx = log(p * a * boost::math::beta(a, b, pol)) / a; + x = exp(lx); + y = x < 0.9 ? T(1 - x) : (T)(-boost::math::expm1(lx, pol)); + + if((b < a) && (x < 0.2)) + { + // + // Under a limited range of circumstances we can improve + // our estimate for x, frankly it's clear if this has much effect! + // + T ap1 = a - 1; + T bm1 = b - 1; + T a_2 = a * a; + T a_3 = a * a_2; + T b_2 = b * b; + T terms[5] = { 0, 1 }; + terms[2] = bm1 / ap1; + ap1 *= ap1; + terms[3] = bm1 * (3 * a * b + 5 * b + a_2 - a - 4) / (2 * (a + 2) * ap1); + ap1 *= (a + 1); + terms[4] = bm1 * (33 * a * b_2 + 31 * b_2 + 8 * a_2 * b_2 - 30 * a * b - 47 * b + 11 * a_2 * b + 6 * a_3 * b + 18 + 4 * a - a_3 + a_2 * a_2 - 10 * a_2) + / (3 * (a + 3) * (a + 2) * ap1); + x = tools::evaluate_polynomial(terms, x, 5); + } + // + // And finally we know that our result is below the inflection + // point, so set an upper limit on our search: + // + if(x > xs) + x = xs; + upper = xs; + } + else /*if((a <= 1) != (b <= 1))*/ + { + // + // If all else fails we get here, only one of a and b + // is above 1, and a+b is small. Start by swapping + // things around so that we have a concave curve with b > a + // and no points of inflection in [0,1]. As long as we expect + // x to be small then we can use the simple (and cheap) power + // term to estimate x, but when we expect x to be large then + // this greatly underestimates x and leaves us trying to + // iterate "round the corner" which may take almost forever... + // + // We could use Temme's inverse gamma function case in that case, + // this works really rather well (albeit expensively) even though + // strictly speaking we're outside it's defined range. + // + // However it's expensive to compute, and an alternative approach + // which models the curve as a distorted quarter circle is much + // cheaper to compute, and still keeps the number of iterations + // required down to a reasonable level. With thanks to Prof Temme + // for this suggestion. + // + if(b < a) + { + std::swap(a, b); + std::swap(p, q); + invert = !invert; + } + if(pow(p, 1/a) < 0.5) + { + x = pow(p * a * boost::math::beta(a, b, pol), 1 / a); + if(x == 0) + x = boost::math::tools::min_value<T>(); + y = 1 - x; + } + else /*if(pow(q, 1/b) < 0.1)*/ + { + // model a distorted quarter circle: + y = pow(1 - pow(p, b * boost::math::beta(a, b, pol)), 1/b); + if(y == 0) + y = boost::math::tools::min_value<T>(); + x = 1 - y; + } + } + + // + // Now we have a guess for x (and for y) we can set things up for + // iteration. If x > 0.5 it pays to swap things round: + // + if(x > 0.5) + { + std::swap(a, b); + std::swap(p, q); + std::swap(x, y); + invert = !invert; + T l = 1 - upper; + T u = 1 - lower; + lower = l; + upper = u; + } + // + // lower bound for our search: + // + // We're not interested in denormalised answers as these tend to + // these tend to take up lots of iterations, given that we can't get + // accurate derivatives in this area (they tend to be infinite). + // + if(lower == 0) + { + if(invert && (py == 0)) + { + // + // We're not interested in answers smaller than machine epsilon: + // + lower = boost::math::tools::epsilon<T>(); + if(x < lower) + x = lower; + } + else + lower = boost::math::tools::min_value<T>(); + if(x < lower) + x = lower; + } + // + // Figure out how many digits to iterate towards: + // + int digits = boost::math::policies::digits<T, Policy>() / 2; + if((x < 1e-50) && ((a < 1) || (b < 1))) + { + // + // If we're in a region where the first derivative is very + // large, then we have to take care that the root-finder + // doesn't terminate prematurely. We'll bump the precision + // up to avoid this, but we have to take care not to set the + // precision too high or the last few iterations will just + // thrash around and convergence may be slow in this case. + // Try 3/4 of machine epsilon: + // + digits *= 3; + digits /= 2; + } + // + // Now iterate, we can use either p or q as the target here + // depending on which is smaller: + // + boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>(); + x = boost::math::tools::halley_iterate( + boost::math::detail::ibeta_roots<T, Policy>(a, b, (p < q ? p : q), (p < q ? false : true)), x, lower, upper, digits, max_iter); + policies::check_root_iterations<T>("boost::math::ibeta<%1%>(%1%, %1%, %1%)", max_iter, pol); + // + // We don't really want these asserts here, but they are useful for sanity + // checking that we have the limits right, uncomment if you suspect bugs *only*. + // + //BOOST_ASSERT(x != upper); + //BOOST_ASSERT((x != lower) || (x == boost::math::tools::min_value<T>()) || (x == boost::math::tools::epsilon<T>())); + // + // Tidy up, if we "lower" was too high then zero is the best answer we have: + // + if(x == lower) + x = 0; + if(py) + *py = invert ? x : 1 - x; + return invert ? 1-x : x; +} + +} // namespace detail + +template <class T1, class T2, class T3, class T4, class Policy> +inline typename tools::promote_args<T1, T2, T3, T4>::type + ibeta_inv(T1 a, T2 b, T3 p, T4* py, const Policy& pol) +{ + static const char* function = "boost::math::ibeta_inv<%1%>(%1%,%1%,%1%)"; + BOOST_FPU_EXCEPTION_GUARD + typedef typename tools::promote_args<T1, T2, T3, T4>::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(a <= 0) + return policies::raise_domain_error<result_type>(function, "The argument a to the incomplete beta function inverse must be greater than zero (got a=%1%).", a, pol); + if(b <= 0) + return policies::raise_domain_error<result_type>(function, "The argument b to the incomplete beta function inverse must be greater than zero (got b=%1%).", b, pol); + if((p < 0) || (p > 1)) + return policies::raise_domain_error<result_type>(function, "Argument p outside the range [0,1] in the incomplete beta function inverse (got p=%1%).", p, pol); + + value_type rx, ry; + + rx = detail::ibeta_inv_imp( + static_cast<value_type>(a), + static_cast<value_type>(b), + static_cast<value_type>(p), + static_cast<value_type>(1 - p), + forwarding_policy(), &ry); + + if(py) *py = policies::checked_narrowing_cast<T4, forwarding_policy>(ry, function); + return policies::checked_narrowing_cast<result_type, forwarding_policy>(rx, function); +} + +template <class T1, class T2, class T3, class T4> +inline typename tools::promote_args<T1, T2, T3, T4>::type + ibeta_inv(T1 a, T2 b, T3 p, T4* py) +{ + return ibeta_inv(a, b, p, py, policies::policy<>()); +} + +template <class T1, class T2, class T3> +inline typename tools::promote_args<T1, T2, T3>::type + ibeta_inv(T1 a, T2 b, T3 p) +{ + typedef typename tools::promote_args<T1, T2, T3>::type result_type; + return ibeta_inv(a, b, p, static_cast<result_type*>(0), policies::policy<>()); +} + +template <class T1, class T2, class T3, class Policy> +inline typename tools::promote_args<T1, T2, T3>::type + ibeta_inv(T1 a, T2 b, T3 p, const Policy& pol) +{ + typedef typename tools::promote_args<T1, T2, T3>::type result_type; + return ibeta_inv(a, b, p, static_cast<result_type*>(0), pol); +} + +template <class T1, class T2, class T3, class T4, class Policy> +inline typename tools::promote_args<T1, T2, T3, T4>::type + ibetac_inv(T1 a, T2 b, T3 q, T4* py, const Policy& pol) +{ + static const char* function = "boost::math::ibetac_inv<%1%>(%1%,%1%,%1%)"; + BOOST_FPU_EXCEPTION_GUARD + typedef typename tools::promote_args<T1, T2, T3, T4>::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(a <= 0) + return policies::raise_domain_error<result_type>(function, "The argument a to the incomplete beta function inverse must be greater than zero (got a=%1%).", a, pol); + if(b <= 0) + return policies::raise_domain_error<result_type>(function, "The argument b to the incomplete beta function inverse must be greater than zero (got b=%1%).", b, pol); + if((q < 0) || (q > 1)) + return policies::raise_domain_error<result_type>(function, "Argument q outside the range [0,1] in the incomplete beta function inverse (got q=%1%).", q, pol); + + value_type rx, ry; + + rx = detail::ibeta_inv_imp( + static_cast<value_type>(a), + static_cast<value_type>(b), + static_cast<value_type>(1 - q), + static_cast<value_type>(q), + forwarding_policy(), &ry); + + if(py) *py = policies::checked_narrowing_cast<T4, forwarding_policy>(ry, function); + return policies::checked_narrowing_cast<result_type, forwarding_policy>(rx, function); +} + +template <class T1, class T2, class T3, class T4> +inline typename tools::promote_args<T1, T2, T3, T4>::type + ibetac_inv(T1 a, T2 b, T3 q, T4* py) +{ + return ibetac_inv(a, b, q, py, policies::policy<>()); +} + +template <class RT1, class RT2, class RT3> +inline typename tools::promote_args<RT1, RT2, RT3>::type + ibetac_inv(RT1 a, RT2 b, RT3 q) +{ + typedef typename tools::promote_args<RT1, RT2, RT3>::type result_type; + return ibetac_inv(a, b, q, static_cast<result_type*>(0), policies::policy<>()); +} + +template <class RT1, class RT2, class RT3, class Policy> +inline typename tools::promote_args<RT1, RT2, RT3>::type + ibetac_inv(RT1 a, RT2 b, RT3 q, const Policy& pol) +{ + typedef typename tools::promote_args<RT1, RT2, RT3>::type result_type; + return ibetac_inv(a, b, q, static_cast<result_type*>(0), pol); +} + +} // namespace math +} // namespace boost + +#endif // BOOST_MATH_SPECIAL_FUNCTIONS_IGAMMA_INVERSE_HPP + + + + |