--- /dev/null
+// 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 = -expm1(log(p) / a);
+ }
+ else
+ {
+ x = pow(p, 1 / a);
+ y = 1 - x;
+ }
+ }
+ else
+ {
+ x = exp(log1p(-q) / a);
+ y = -expm1(log1p(-q) / a);
+ }
+ 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)
+ 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 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
+
+
+
+
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