boost/math/special_functions/detail/ibeta_inverse.hpp
// 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(-big, -big); } if(x == 0) { T big = tools::max_value<T>() / 4; return boost::math::make_tuple(-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. // // 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) && (b == 1)) { if(py) *py = 1 - p; return p; } // // 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; // // 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) && (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)) { // // We have a Student's T distribution: x = find_ibeta_inv_from_t_dist(a, p, q, &y, pol); } 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("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) { return ibeta_inv(a, b, p, static_cast<T1*>(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) { return ibeta_inv(a, b, p, static_cast<T1*>(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) 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) 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)) 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 remove_cv<RT1>::type dummy; return ibetac_inv(a, b, q, static_cast<dummy*>(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 remove_cv<RT1>::type dummy; return ibetac_inv(a, b, q, static_cast<dummy*>(0), pol); } } // namespace math } // namespace boost #endif // BOOST_MATH_SPECIAL_FUNCTIONS_IGAMMA_INVERSE_HPP