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-// 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
-
-
-
-
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