Chris@16: //  Copyright John Maddock 2006.
Chris@16: //  Copyright Paul A. Bristow 2007
Chris@16: //  Use, modification and distribution are subject to the
Chris@16: //  Boost Software License, Version 1.0. (See accompanying file
Chris@16: //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
Chris@16: 
Chris@16: #ifndef BOOST_MATH_SPECIAL_FUNCTIONS_IBETA_INVERSE_HPP
Chris@16: #define BOOST_MATH_SPECIAL_FUNCTIONS_IBETA_INVERSE_HPP
Chris@16: 
Chris@16: #ifdef _MSC_VER
Chris@16: #pragma once
Chris@16: #endif
Chris@16: 
Chris@16: #include <boost/math/special_functions/beta.hpp>
Chris@16: #include <boost/math/special_functions/erf.hpp>
Chris@16: #include <boost/math/tools/roots.hpp>
Chris@16: #include <boost/math/special_functions/detail/t_distribution_inv.hpp>
Chris@16: 
Chris@16: namespace boost{ namespace math{ namespace detail{
Chris@16: 
Chris@16: //
Chris@16: // Helper object used by root finding
Chris@16: // code to convert eta to x.
Chris@16: //
Chris@16: template <class T>
Chris@16: struct temme_root_finder
Chris@16: {
Chris@16:    temme_root_finder(const T t_, const T a_) : t(t_), a(a_) {}
Chris@16: 
Chris@16:    boost::math::tuple<T, T> operator()(T x)
Chris@16:    {
Chris@16:       BOOST_MATH_STD_USING // ADL of std names
Chris@16: 
Chris@16:       T y = 1 - x;
Chris@16:       if(y == 0)
Chris@16:       {
Chris@16:          T big = tools::max_value<T>() / 4;
Chris@16:          return boost::math::make_tuple(static_cast<T>(-big), static_cast<T>(-big));
Chris@16:       }
Chris@16:       if(x == 0)
Chris@16:       {
Chris@16:          T big = tools::max_value<T>() / 4;
Chris@16:          return boost::math::make_tuple(static_cast<T>(-big), big);
Chris@16:       }
Chris@16:       T f = log(x) + a * log(y) + t;
Chris@16:       T f1 = (1 / x) - (a / (y));
Chris@16:       return boost::math::make_tuple(f, f1);
Chris@16:    }
Chris@16: private:
Chris@16:    T t, a;
Chris@16: };
Chris@16: //
Chris@16: // See:
Chris@16: // "Asymptotic Inversion of the Incomplete Beta Function"
Chris@16: // N.M. Temme
Chris@16: // Journal of Computation and Applied Mathematics 41 (1992) 145-157.
Chris@16: // Section 2.
Chris@16: //
Chris@16: template <class T, class Policy>
Chris@16: T temme_method_1_ibeta_inverse(T a, T b, T z, const Policy& pol)
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING // ADL of std names
Chris@16: 
Chris@16:    const T r2 = sqrt(T(2));
Chris@16:    //
Chris@16:    // get the first approximation for eta from the inverse
Chris@16:    // error function (Eq: 2.9 and 2.10).
Chris@16:    //
Chris@16:    T eta0 = boost::math::erfc_inv(2 * z, pol);
Chris@16:    eta0 /= -sqrt(a / 2);
Chris@16: 
Chris@16:    T terms[4] = { eta0 };
Chris@16:    T workspace[7];
Chris@16:    //
Chris@16:    // calculate powers:
Chris@16:    //
Chris@16:    T B = b - a;
Chris@16:    T B_2 = B * B;
Chris@16:    T B_3 = B_2 * B;
Chris@16:    //
Chris@16:    // Calculate correction terms:
Chris@16:    //
Chris@16: 
Chris@16:    // See eq following 2.15:
Chris@16:    workspace[0] = -B * r2 / 2;
Chris@16:    workspace[1] = (1 - 2 * B) / 8;
Chris@16:    workspace[2] = -(B * r2 / 48);
Chris@16:    workspace[3] = T(-1) / 192;
Chris@16:    workspace[4] = -B * r2 / 3840;
Chris@16:    terms[1] = tools::evaluate_polynomial(workspace, eta0, 5);
Chris@16:    // Eq Following 2.17:
Chris@16:    workspace[0] = B * r2 * (3 * B - 2) / 12;
Chris@16:    workspace[1] = (20 * B_2 - 12 * B + 1) / 128;
Chris@16:    workspace[2] = B * r2 * (20 * B - 1) / 960;
Chris@16:    workspace[3] = (16 * B_2 + 30 * B - 15) / 4608;
Chris@16:    workspace[4] = B * r2 * (21 * B + 32) / 53760;
Chris@16:    workspace[5] = (-32 * B_2 + 63) / 368640;
Chris@16:    workspace[6] = -B * r2 * (120 * B + 17) / 25804480;
Chris@16:    terms[2] = tools::evaluate_polynomial(workspace, eta0, 7);
Chris@16:    // Eq Following 2.17:
Chris@16:    workspace[0] = B * r2 * (-75 * B_2 + 80 * B - 16) / 480;
Chris@16:    workspace[1] = (-1080 * B_3 + 868 * B_2 - 90 * B - 45) / 9216;
Chris@16:    workspace[2] = B * r2 * (-1190 * B_2 + 84 * B + 373) / 53760;
Chris@16:    workspace[3] = (-2240 * B_3 - 2508 * B_2 + 2100 * B - 165) / 368640;
Chris@16:    terms[3] = tools::evaluate_polynomial(workspace, eta0, 4);
Chris@16:    //
Chris@16:    // Bring them together to get a final estimate for eta:
Chris@16:    //
Chris@16:    T eta = tools::evaluate_polynomial(terms, T(1/a), 4);
Chris@16:    //
Chris@16:    // now we need to convert eta to x, by solving the appropriate
Chris@16:    // quadratic equation:
Chris@16:    //
Chris@16:    T eta_2 = eta * eta;
Chris@16:    T c = -exp(-eta_2 / 2);
Chris@16:    T x;
Chris@16:    if(eta_2 == 0)
Chris@16:       x = 0.5;
Chris@16:    else
Chris@16:       x = (1 + eta * sqrt((1 + c) / eta_2)) / 2;
Chris@16: 
Chris@16:    BOOST_ASSERT(x >= 0);
Chris@16:    BOOST_ASSERT(x <= 1);
Chris@16:    BOOST_ASSERT(eta * (x - 0.5) >= 0);
Chris@16: #ifdef BOOST_INSTRUMENT
Chris@16:    std::cout << "Estimating x with Temme method 1: " << x << std::endl;
Chris@16: #endif
Chris@16:    return x;
Chris@16: }
Chris@16: //
Chris@16: // See:
Chris@16: // "Asymptotic Inversion of the Incomplete Beta Function"
Chris@16: // N.M. Temme
Chris@16: // Journal of Computation and Applied Mathematics 41 (1992) 145-157.
Chris@16: // Section 3.
Chris@16: //
Chris@16: template <class T, class Policy>
Chris@16: T temme_method_2_ibeta_inverse(T /*a*/, T /*b*/, T z, T r, T theta, const Policy& pol)
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING // ADL of std names
Chris@16: 
Chris@16:    //
Chris@16:    // Get first estimate for eta, see Eq 3.9 and 3.10,
Chris@16:    // but note there is a typo in Eq 3.10:
Chris@16:    //
Chris@16:    T eta0 = boost::math::erfc_inv(2 * z, pol);
Chris@16:    eta0 /= -sqrt(r / 2);
Chris@16: 
Chris@16:    T s = sin(theta);
Chris@16:    T c = cos(theta);
Chris@16:    //
Chris@16:    // Now we need to purturb eta0 to get eta, which we do by
Chris@16:    // evaluating the polynomial in 1/r at the bottom of page 151,
Chris@16:    // to do this we first need the error terms e1, e2 e3
Chris@16:    // which we'll fill into the array "terms".  Since these
Chris@16:    // terms are themselves polynomials, we'll need another
Chris@16:    // array "workspace" to calculate those...
Chris@16:    //
Chris@16:    T terms[4] = { eta0 };
Chris@16:    T workspace[6];
Chris@16:    //
Chris@16:    // some powers of sin(theta)cos(theta) that we'll need later:
Chris@16:    //
Chris@16:    T sc = s * c;
Chris@16:    T sc_2 = sc * sc;
Chris@16:    T sc_3 = sc_2 * sc;
Chris@16:    T sc_4 = sc_2 * sc_2;
Chris@16:    T sc_5 = sc_2 * sc_3;
Chris@16:    T sc_6 = sc_3 * sc_3;
Chris@16:    T sc_7 = sc_4 * sc_3;
Chris@16:    //
Chris@16:    // Calculate e1 and put it in terms[1], see the middle of page 151:
Chris@16:    //
Chris@16:    workspace[0] = (2 * s * s - 1) / (3 * s * c);
Chris@16:    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co1[] = { -1, -5, 5 };
Chris@16:    workspace[1] = -tools::evaluate_even_polynomial(co1, s, 3) / (36 * sc_2);
Chris@16:    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co2[] = { 1, 21, -69, 46 };
Chris@16:    workspace[2] = tools::evaluate_even_polynomial(co2, s, 4) / (1620 * sc_3);
Chris@16:    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co3[] = { 7, -2, 33, -62, 31 };
Chris@16:    workspace[3] = -tools::evaluate_even_polynomial(co3, s, 5) / (6480 * sc_4);
Chris@16:    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co4[] = { 25, -52, -17, 88, -115, 46 };
Chris@16:    workspace[4] = tools::evaluate_even_polynomial(co4, s, 6) / (90720 * sc_5);
Chris@16:    terms[1] = tools::evaluate_polynomial(workspace, eta0, 5);
Chris@16:    //
Chris@16:    // Now evaluate e2 and put it in terms[2]:
Chris@16:    //
Chris@16:    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co5[] = { 7, 12, -78, 52 };
Chris@16:    workspace[0] = -tools::evaluate_even_polynomial(co5, s, 4) / (405 * sc_3);
Chris@16:    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co6[] = { -7, 2, 183, -370, 185 };
Chris@16:    workspace[1] = tools::evaluate_even_polynomial(co6, s, 5) / (2592 * sc_4);
Chris@16:    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co7[] = { -533, 776, -1835, 10240, -13525, 5410 };
Chris@16:    workspace[2] = -tools::evaluate_even_polynomial(co7, s, 6) / (204120 * sc_5);
Chris@16:    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co8[] = { -1579, 3747, -3372, -15821, 45588, -45213, 15071 };
Chris@16:    workspace[3] = -tools::evaluate_even_polynomial(co8, s, 7) / (2099520 * sc_6);
Chris@16:    terms[2] = tools::evaluate_polynomial(workspace, eta0, 4);
Chris@16:    //
Chris@16:    // And e3, and put it in terms[3]:
Chris@16:    //
Chris@16:    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co9[] = {449, -1259, -769, 6686, -9260, 3704 };
Chris@16:    workspace[0] = tools::evaluate_even_polynomial(co9, s, 6) / (102060 * sc_5);
Chris@16:    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co10[] = { 63149, -151557, 140052, -727469, 2239932, -2251437, 750479 };
Chris@16:    workspace[1] = -tools::evaluate_even_polynomial(co10, s, 7) / (20995200 * sc_6);
Chris@16:    static const BOOST_MATH_INT_TABLE_TYPE(T, int) co11[] = { 29233, -78755, 105222, 146879, -1602610, 3195183, -2554139, 729754 };
Chris@16:    workspace[2] = tools::evaluate_even_polynomial(co11, s, 8) / (36741600 * sc_7);
Chris@16:    terms[3] = tools::evaluate_polynomial(workspace, eta0, 3);
Chris@16:    //
Chris@16:    // Bring the correction terms together to evaluate eta,
Chris@16:    // this is the last equation on page 151:
Chris@16:    //
Chris@16:    T eta = tools::evaluate_polynomial(terms, T(1/r), 4);
Chris@16:    //
Chris@16:    // Now that we have eta we need to back solve for x,
Chris@16:    // we seek the value of x that gives eta in Eq 3.2.
Chris@16:    // The two methods used are described in section 5.
Chris@16:    //
Chris@16:    // Begin by defining a few variables we'll need later:
Chris@16:    //
Chris@16:    T x;
Chris@16:    T s_2 = s * s;
Chris@16:    T c_2 = c * c;
Chris@16:    T alpha = c / s;
Chris@16:    alpha *= alpha;
Chris@16:    T lu = (-(eta * eta) / (2 * s_2) + log(s_2) + c_2 * log(c_2) / s_2);
Chris@16:    //
Chris@16:    // Temme doesn't specify what value to switch on here,
Chris@16:    // but this seems to work pretty well:
Chris@16:    //
Chris@16:    if(fabs(eta) < 0.7)
Chris@16:    {
Chris@16:       //
Chris@16:       // Small eta use the expansion Temme gives in the second equation
Chris@16:       // of section 5, it's a polynomial in eta:
Chris@16:       //
Chris@16:       workspace[0] = s * s;
Chris@16:       workspace[1] = s * c;
Chris@16:       workspace[2] = (1 - 2 * workspace[0]) / 3;
Chris@16:       static const BOOST_MATH_INT_TABLE_TYPE(T, int) co12[] = { 1, -13, 13 };
Chris@16:       workspace[3] = tools::evaluate_polynomial(co12, workspace[0], 3) / (36 * s * c);
Chris@16:       static const BOOST_MATH_INT_TABLE_TYPE(T, int) co13[] = { 1, 21, -69, 46 };
Chris@16:       workspace[4] = tools::evaluate_polynomial(co13, workspace[0], 4) / (270 * workspace[0] * c * c);
Chris@16:       x = tools::evaluate_polynomial(workspace, eta, 5);
Chris@16: #ifdef BOOST_INSTRUMENT
Chris@16:       std::cout << "Estimating x with Temme method 2 (small eta): " << x << std::endl;
Chris@16: #endif
Chris@16:    }
Chris@16:    else
Chris@16:    {
Chris@16:       //
Chris@16:       // If eta is large we need to solve Eq 3.2 more directly,
Chris@16:       // begin by getting an initial approximation for x from
Chris@16:       // the last equation on page 155, this is a polynomial in u:
Chris@16:       //
Chris@16:       T u = exp(lu);
Chris@16:       workspace[0] = u;
Chris@16:       workspace[1] = alpha;
Chris@16:       workspace[2] = 0;
Chris@16:       workspace[3] = 3 * alpha * (3 * alpha + 1) / 6;
Chris@16:       workspace[4] = 4 * alpha * (4 * alpha + 1) * (4 * alpha + 2) / 24;
Chris@16:       workspace[5] = 5 * alpha * (5 * alpha + 1) * (5 * alpha + 2) * (5 * alpha + 3) / 120;
Chris@16:       x = tools::evaluate_polynomial(workspace, u, 6);
Chris@16:       //
Chris@16:       // At this point we may or may not have the right answer, Eq-3.2 has
Chris@16:       // two solutions for x for any given eta, however the mapping in 3.2
Chris@16:       // is 1:1 with the sign of eta and x-sin^2(theta) being the same.
Chris@16:       // So we can check if we have the right root of 3.2, and if not
Chris@16:       // switch x for 1-x.  This transformation is motivated by the fact
Chris@16:       // that the distribution is *almost* symetric so 1-x will be in the right
Chris@16:       // ball park for the solution:
Chris@16:       //
Chris@16:       if((x - s_2) * eta < 0)
Chris@16:          x = 1 - x;
Chris@16: #ifdef BOOST_INSTRUMENT
Chris@16:       std::cout << "Estimating x with Temme method 2 (large eta): " << x << std::endl;
Chris@16: #endif
Chris@16:    }
Chris@16:    //
Chris@16:    // The final step is a few Newton-Raphson iterations to
Chris@16:    // clean up our approximation for x, this is pretty cheap
Chris@16:    // in general, and very cheap compared to an incomplete beta
Chris@16:    // evaluation.  The limits set on x come from the observation
Chris@16:    // that the sign of eta and x-sin^2(theta) are the same.
Chris@16:    //
Chris@16:    T lower, upper;
Chris@16:    if(eta < 0)
Chris@16:    {
Chris@16:       lower = 0;
Chris@16:       upper = s_2;
Chris@16:    }
Chris@16:    else
Chris@16:    {
Chris@16:       lower = s_2;
Chris@16:       upper = 1;
Chris@16:    }
Chris@16:    //
Chris@16:    // If our initial approximation is out of bounds then bisect:
Chris@16:    //
Chris@16:    if((x < lower) || (x > upper))
Chris@16:       x = (lower+upper) / 2;
Chris@16:    //
Chris@16:    // And iterate:
Chris@16:    //
Chris@16:    x = tools::newton_raphson_iterate(
Chris@16:       temme_root_finder<T>(-lu, alpha), x, lower, upper, policies::digits<T, Policy>() / 2);
Chris@16: 
Chris@16:    return x;
Chris@16: }
Chris@16: //
Chris@16: // See:
Chris@16: // "Asymptotic Inversion of the Incomplete Beta Function"
Chris@16: // N.M. Temme
Chris@16: // Journal of Computation and Applied Mathematics 41 (1992) 145-157.
Chris@16: // Section 4.
Chris@16: //
Chris@16: template <class T, class Policy>
Chris@16: T temme_method_3_ibeta_inverse(T a, T b, T p, T q, const Policy& pol)
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING // ADL of std names
Chris@16: 
Chris@16:    //
Chris@16:    // Begin by getting an initial approximation for the quantity
Chris@16:    // eta from the dominant part of the incomplete beta:
Chris@16:    //
Chris@16:    T eta0;
Chris@16:    if(p < q)
Chris@16:       eta0 = boost::math::gamma_q_inv(b, p, pol);
Chris@16:    else
Chris@16:       eta0 = boost::math::gamma_p_inv(b, q, pol);
Chris@16:    eta0 /= a;
Chris@16:    //
Chris@16:    // Define the variables and powers we'll need later on:
Chris@16:    //
Chris@16:    T mu = b / a;
Chris@16:    T w = sqrt(1 + mu);
Chris@16:    T w_2 = w * w;
Chris@16:    T w_3 = w_2 * w;
Chris@16:    T w_4 = w_2 * w_2;
Chris@16:    T w_5 = w_3 * w_2;
Chris@16:    T w_6 = w_3 * w_3;
Chris@16:    T w_7 = w_4 * w_3;
Chris@16:    T w_8 = w_4 * w_4;
Chris@16:    T w_9 = w_5 * w_4;
Chris@16:    T w_10 = w_5 * w_5;
Chris@16:    T d = eta0 - mu;
Chris@16:    T d_2 = d * d;
Chris@16:    T d_3 = d_2 * d;
Chris@16:    T d_4 = d_2 * d_2;
Chris@16:    T w1 = w + 1;
Chris@16:    T w1_2 = w1 * w1;
Chris@16:    T w1_3 = w1 * w1_2;
Chris@16:    T w1_4 = w1_2 * w1_2;
Chris@16:    //
Chris@16:    // Now we need to compute the purturbation error terms that
Chris@16:    // convert eta0 to eta, these are all polynomials of polynomials.
Chris@16:    // Probably these should be re-written to use tabulated data
Chris@16:    // (see examples above), but it's less of a win in this case as we
Chris@16:    // need to calculate the individual powers for the denominator terms
Chris@16:    // anyway, so we might as well use them for the numerator-polynomials
Chris@16:    // as well....
Chris@16:    //
Chris@16:    // Refer to p154-p155 for the details of these expansions:
Chris@16:    //
Chris@16:    T e1 = (w + 2) * (w - 1) / (3 * w);
Chris@16:    e1 += (w_3 + 9 * w_2 + 21 * w + 5) * d / (36 * w_2 * w1);
Chris@16:    e1 -= (w_4 - 13 * w_3 + 69 * w_2 + 167 * w + 46) * d_2 / (1620 * w1_2 * w_3);
Chris@16:    e1 -= (7 * w_5 + 21 * w_4 + 70 * w_3 + 26 * w_2 - 93 * w - 31) * d_3 / (6480 * w1_3 * w_4);
Chris@16:    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);
Chris@16: 
Chris@16:    T e2 = (28 * w_4 + 131 * w_3 + 402 * w_2 + 581 * w + 208) * (w - 1) / (1620 * w1 * w_3);
Chris@16:    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);
Chris@16:    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);
Chris@16:    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);
Chris@16: 
Chris@16:    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);
Chris@16:    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);
Chris@16:    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);
Chris@16:    //
Chris@16:    // Combine eta0 and the error terms to compute eta (Second eqaution p155):
Chris@16:    //
Chris@16:    T eta = eta0 + e1 / a + e2 / (a * a) + e3 / (a * a * a);
Chris@16:    //
Chris@16:    // Now we need to solve Eq 4.2 to obtain x.  For any given value of
Chris@16:    // eta there are two solutions to this equation, and since the distribtion
Chris@16:    // may be very skewed, these are not related by x ~ 1-x we used when
Chris@16:    // implementing section 3 above.  However we know that:
Chris@16:    //
Chris@16:    //  cross < x <= 1       ; iff eta < mu
Chris@16:    //          x == cross   ; iff eta == mu
Chris@16:    //     0 <= x < cross    ; iff eta > mu
Chris@16:    //
Chris@16:    // Where cross == 1 / (1 + mu)
Chris@16:    // Many thanks to Prof Temme for clarifying this point.
Chris@16:    //
Chris@16:    // Therefore we'll just jump straight into Newton iterations
Chris@16:    // to solve Eq 4.2 using these bounds, and simple bisection
Chris@16:    // as the first guess, in practice this converges pretty quickly
Chris@16:    // and we only need a few digits correct anyway:
Chris@16:    //
Chris@16:    if(eta <= 0)
Chris@16:       eta = tools::min_value<T>();
Chris@16:    T u = eta - mu * log(eta) + (1 + mu) * log(1 + mu) - mu;
Chris@16:    T cross = 1 / (1 + mu);
Chris@16:    T lower = eta < mu ? cross : 0;
Chris@16:    T upper = eta < mu ? 1 : cross;
Chris@16:    T x = (lower + upper) / 2;
Chris@16:    x = tools::newton_raphson_iterate(
Chris@16:       temme_root_finder<T>(u, mu), x, lower, upper, policies::digits<T, Policy>() / 2);
Chris@16: #ifdef BOOST_INSTRUMENT
Chris@16:    std::cout << "Estimating x with Temme method 3: " << x << std::endl;
Chris@16: #endif
Chris@16:    return x;
Chris@16: }
Chris@16: 
Chris@16: template <class T, class Policy>
Chris@16: struct ibeta_roots
Chris@16: {
Chris@16:    ibeta_roots(T _a, T _b, T t, bool inv = false)
Chris@16:       : a(_a), b(_b), target(t), invert(inv) {}
Chris@16: 
Chris@16:    boost::math::tuple<T, T, T> operator()(T x)
Chris@16:    {
Chris@16:       BOOST_MATH_STD_USING // ADL of std names
Chris@16: 
Chris@16:       BOOST_FPU_EXCEPTION_GUARD
Chris@16:       
Chris@16:       T f1;
Chris@16:       T y = 1 - x;
Chris@16:       T f = ibeta_imp(a, b, x, Policy(), invert, true, &f1) - target;
Chris@16:       if(invert)
Chris@16:          f1 = -f1;
Chris@16:       if(y == 0)
Chris@16:          y = tools::min_value<T>() * 64;
Chris@16:       if(x == 0)
Chris@16:          x = tools::min_value<T>() * 64;
Chris@16: 
Chris@16:       T f2 = f1 * (-y * a + (b - 2) * x + 1);
Chris@16:       if(fabs(f2) < y * x * tools::max_value<T>())
Chris@16:          f2 /= (y * x);
Chris@16:       if(invert)
Chris@16:          f2 = -f2;
Chris@16: 
Chris@16:       // make sure we don't have a zero derivative:
Chris@16:       if(f1 == 0)
Chris@16:          f1 = (invert ? -1 : 1) * tools::min_value<T>() * 64;
Chris@16: 
Chris@16:       return boost::math::make_tuple(f, f1, f2);
Chris@16:    }
Chris@16: private:
Chris@16:    T a, b, target;
Chris@16:    bool invert;
Chris@16: };
Chris@16: 
Chris@16: template <class T, class Policy>
Chris@16: T ibeta_inv_imp(T a, T b, T p, T q, const Policy& pol, T* py)
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING  // For ADL of math functions.
Chris@16: 
Chris@16:    //
Chris@16:    // The flag invert is set to true if we swap a for b and p for q,
Chris@16:    // in which case the result has to be subtracted from 1:
Chris@16:    //
Chris@16:    bool invert = false;
Chris@16:    //
Chris@16:    // Handle trivial cases first:
Chris@16:    //
Chris@16:    if(q == 0)
Chris@16:    {
Chris@16:       if(py) *py = 0;
Chris@16:       return 1;
Chris@16:    }
Chris@16:    else if(p == 0)
Chris@16:    {
Chris@16:       if(py) *py = 1;
Chris@16:       return 0;
Chris@16:    }
Chris@16:    else if(a == 1)
Chris@16:    {
Chris@16:       if(b == 1)
Chris@16:       {
Chris@16:          if(py) *py = 1 - p;
Chris@16:          return p;
Chris@16:       }
Chris@16:       // Change things around so we can handle as b == 1 special case below:
Chris@16:       std::swap(a, b);
Chris@16:       std::swap(p, q);
Chris@16:       invert = true;
Chris@16:    }
Chris@16:    //
Chris@16:    // Depending upon which approximation method we use, we may end up
Chris@16:    // calculating either x or y initially (where y = 1-x):
Chris@16:    //
Chris@16:    T x = 0; // Set to a safe zero to avoid a
Chris@16:    // MSVC 2005 warning C4701: potentially uninitialized local variable 'x' used
Chris@16:    // But code inspection appears to ensure that x IS assigned whatever the code path.
Chris@16:    T y; 
Chris@16: 
Chris@16:    // For some of the methods we can put tighter bounds
Chris@16:    // on the result than simply [0,1]:
Chris@16:    //
Chris@16:    T lower = 0;
Chris@16:    T upper = 1;
Chris@16:    //
Chris@16:    // Student's T with b = 0.5 gets handled as a special case, swap
Chris@16:    // around if the arguments are in the "wrong" order:
Chris@16:    //
Chris@16:    if(a == 0.5f)
Chris@16:    {
Chris@16:       if(b == 0.5f)
Chris@16:       {
Chris@16:          x = sin(p * constants::half_pi<T>());
Chris@16:          x *= x;
Chris@16:          if(py)
Chris@16:          {
Chris@16:             *py = sin(q * constants::half_pi<T>());
Chris@16:             *py *= *py;
Chris@16:          }
Chris@16:          return x;
Chris@16:       }
Chris@16:       else if(b > 0.5f)
Chris@16:       {
Chris@16:          std::swap(a, b);
Chris@16:          std::swap(p, q);
Chris@16:          invert = !invert;
Chris@16:       }
Chris@16:    }
Chris@16:    //
Chris@16:    // Select calculation method for the initial estimate:
Chris@16:    //
Chris@16:    if((b == 0.5f) && (a >= 0.5f) && (p != 1))
Chris@16:    {
Chris@16:       //
Chris@16:       // We have a Student's T distribution:
Chris@16:       x = find_ibeta_inv_from_t_dist(a, p, q, &y, pol);
Chris@16:    }
Chris@16:    else if(b == 1)
Chris@16:    {
Chris@16:       if(p < q)
Chris@16:       {
Chris@16:          if(a > 1)
Chris@16:          {
Chris@16:             x = pow(p, 1 / a);
Chris@101:             y = -boost::math::expm1(log(p) / a, pol);
Chris@16:          }
Chris@16:          else
Chris@16:          {
Chris@16:             x = pow(p, 1 / a);
Chris@16:             y = 1 - x;
Chris@16:          }
Chris@16:       }
Chris@16:       else
Chris@16:       {
Chris@101:          x = exp(boost::math::log1p(-q, pol) / a);
Chris@101:          y = -boost::math::expm1(boost::math::log1p(-q, pol) / a, pol);
Chris@16:       }
Chris@16:       if(invert)
Chris@16:          std::swap(x, y);
Chris@16:       if(py)
Chris@16:          *py = y;
Chris@16:       return x;
Chris@16:    }
Chris@16:    else if(a + b > 5)
Chris@16:    {
Chris@16:       //
Chris@16:       // When a+b is large then we can use one of Prof Temme's
Chris@16:       // asymptotic expansions, begin by swapping things around
Chris@16:       // so that p < 0.5, we do this to avoid cancellations errors
Chris@16:       // when p is large.
Chris@16:       //
Chris@16:       if(p > 0.5)
Chris@16:       {
Chris@16:          std::swap(a, b);
Chris@16:          std::swap(p, q);
Chris@16:          invert = !invert;
Chris@16:       }
Chris@16:       T minv = (std::min)(a, b);
Chris@16:       T maxv = (std::max)(a, b);
Chris@16:       if((sqrt(minv) > (maxv - minv)) && (minv > 5))
Chris@16:       {
Chris@16:          //
Chris@16:          // When a and b differ by a small amount
Chris@16:          // the curve is quite symmetrical and we can use an error
Chris@16:          // function to approximate the inverse. This is the cheapest
Chris@16:          // of the three Temme expantions, and the calculated value
Chris@16:          // for x will never be much larger than p, so we don't have
Chris@16:          // to worry about cancellation as long as p is small.
Chris@16:          //
Chris@16:          x = temme_method_1_ibeta_inverse(a, b, p, pol);
Chris@16:          y = 1 - x;
Chris@16:       }
Chris@16:       else
Chris@16:       {
Chris@16:          T r = a + b;
Chris@16:          T theta = asin(sqrt(a / r));
Chris@16:          T lambda = minv / r;
Chris@16:          if((lambda >= 0.2) && (lambda <= 0.8) && (r >= 10))
Chris@16:          {
Chris@16:             //
Chris@16:             // The second error function case is the next cheapest
Chris@16:             // to use, it brakes down when the result is likely to be
Chris@16:             // very small, if a+b is also small, but we can use a
Chris@16:             // cheaper expansion there in any case.  As before x won't
Chris@16:             // be much larger than p, so as long as p is small we should
Chris@16:             // be free of cancellation error.
Chris@16:             //
Chris@16:             T ppa = pow(p, 1/a);
Chris@16:             if((ppa < 0.0025) && (a + b < 200))
Chris@16:             {
Chris@16:                x = ppa * pow(a * boost::math::beta(a, b, pol), 1/a);
Chris@16:             }
Chris@16:             else
Chris@16:                x = temme_method_2_ibeta_inverse(a, b, p, r, theta, pol);
Chris@16:             y = 1 - x;
Chris@16:          }
Chris@16:          else
Chris@16:          {
Chris@16:             //
Chris@16:             // If we get here then a and b are very different in magnitude
Chris@16:             // and we need to use the third of Temme's methods which
Chris@16:             // involves inverting the incomplete gamma.  This is much more
Chris@16:             // expensive than the other methods.  We also can only use this
Chris@16:             // method when a > b, which can lead to cancellation errors
Chris@16:             // if we really want y (as we will when x is close to 1), so
Chris@16:             // a different expansion is used in that case.
Chris@16:             //
Chris@16:             if(a < b)
Chris@16:             {
Chris@16:                std::swap(a, b);
Chris@16:                std::swap(p, q);
Chris@16:                invert = !invert;
Chris@16:             }
Chris@16:             //
Chris@16:             // Try and compute the easy way first:
Chris@16:             //
Chris@16:             T bet = 0;
Chris@16:             if(b < 2)
Chris@16:                bet = boost::math::beta(a, b, pol);
Chris@16:             if(bet != 0)
Chris@16:             {
Chris@16:                y = pow(b * q * bet, 1/b);
Chris@16:                x = 1 - y;
Chris@16:             }
Chris@16:             else 
Chris@16:                y = 1;
Chris@16:             if(y > 1e-5)
Chris@16:             {
Chris@16:                x = temme_method_3_ibeta_inverse(a, b, p, q, pol);
Chris@16:                y = 1 - x;
Chris@16:             }
Chris@16:          }
Chris@16:       }
Chris@16:    }
Chris@16:    else if((a < 1) && (b < 1))
Chris@16:    {
Chris@16:       //
Chris@16:       // Both a and b less than 1,
Chris@16:       // there is a point of inflection at xs:
Chris@16:       //
Chris@16:       T xs = (1 - a) / (2 - a - b);
Chris@16:       //
Chris@16:       // Now we need to ensure that we start our iteration from the
Chris@16:       // right side of the inflection point:
Chris@16:       //
Chris@16:       T fs = boost::math::ibeta(a, b, xs, pol) - p;
Chris@16:       if(fabs(fs) / p < tools::epsilon<T>() * 3)
Chris@16:       {
Chris@16:          // The result is at the point of inflection, best just return it:
Chris@16:          *py = invert ? xs : 1 - xs;
Chris@16:          return invert ? 1-xs : xs;
Chris@16:       }
Chris@16:       if(fs < 0)
Chris@16:       {
Chris@16:          std::swap(a, b);
Chris@16:          std::swap(p, q);
Chris@16:          invert = !invert;
Chris@16:          xs = 1 - xs;
Chris@16:       }
Chris@16:       T xg = pow(a * p * boost::math::beta(a, b, pol), 1/a);
Chris@16:       x = xg / (1 + xg);
Chris@16:       y = 1 / (1 + xg);
Chris@16:       //
Chris@16:       // And finally we know that our result is below the inflection
Chris@16:       // point, so set an upper limit on our search:
Chris@16:       //
Chris@16:       if(x > xs)
Chris@16:          x = xs;
Chris@16:       upper = xs;
Chris@16:    }
Chris@16:    else if((a > 1) && (b > 1))
Chris@16:    {
Chris@16:       //
Chris@16:       // Small a and b, both greater than 1,
Chris@16:       // there is a point of inflection at xs,
Chris@16:       // and it's complement is xs2, we must always
Chris@16:       // start our iteration from the right side of the
Chris@16:       // point of inflection.
Chris@16:       //
Chris@16:       T xs = (a - 1) / (a + b - 2);
Chris@16:       T xs2 = (b - 1) / (a + b - 2);
Chris@16:       T ps = boost::math::ibeta(a, b, xs, pol) - p;
Chris@16: 
Chris@16:       if(ps < 0)
Chris@16:       {
Chris@16:          std::swap(a, b);
Chris@16:          std::swap(p, q);
Chris@16:          std::swap(xs, xs2);
Chris@16:          invert = !invert;
Chris@16:       }
Chris@16:       //
Chris@16:       // Estimate x and y, using expm1 to get a good estimate
Chris@16:       // for y when it's very small:
Chris@16:       //
Chris@16:       T lx = log(p * a * boost::math::beta(a, b, pol)) / a;
Chris@16:       x = exp(lx);
Chris@16:       y = x < 0.9 ? T(1 - x) : (T)(-boost::math::expm1(lx, pol));
Chris@16: 
Chris@16:       if((b < a) && (x < 0.2))
Chris@16:       {
Chris@16:          //
Chris@16:          // Under a limited range of circumstances we can improve
Chris@16:          // our estimate for x, frankly it's clear if this has much effect!
Chris@16:          //
Chris@16:          T ap1 = a - 1;
Chris@16:          T bm1 = b - 1;
Chris@16:          T a_2 = a * a;
Chris@16:          T a_3 = a * a_2;
Chris@16:          T b_2 = b * b;
Chris@16:          T terms[5] = { 0, 1 };
Chris@16:          terms[2] = bm1 / ap1;
Chris@16:          ap1 *= ap1;
Chris@16:          terms[3] = bm1 * (3 * a * b + 5 * b + a_2 - a - 4) / (2 * (a + 2) * ap1);
Chris@16:          ap1 *= (a + 1);
Chris@16:          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)
Chris@16:                     / (3 * (a + 3) * (a + 2) * ap1);
Chris@16:          x = tools::evaluate_polynomial(terms, x, 5);
Chris@16:       }
Chris@16:       //
Chris@16:       // And finally we know that our result is below the inflection
Chris@16:       // point, so set an upper limit on our search:
Chris@16:       //
Chris@16:       if(x > xs)
Chris@16:          x = xs;
Chris@16:       upper = xs;
Chris@16:    }
Chris@16:    else /*if((a <= 1) != (b <= 1))*/
Chris@16:    {
Chris@16:       //
Chris@16:       // If all else fails we get here, only one of a and b
Chris@16:       // is above 1, and a+b is small.  Start by swapping
Chris@16:       // things around so that we have a concave curve with b > a
Chris@16:       // and no points of inflection in [0,1].  As long as we expect
Chris@16:       // x to be small then we can use the simple (and cheap) power
Chris@16:       // term to estimate x, but when we expect x to be large then
Chris@16:       // this greatly underestimates x and leaves us trying to
Chris@16:       // iterate "round the corner" which may take almost forever...
Chris@16:       //
Chris@16:       // We could use Temme's inverse gamma function case in that case,
Chris@16:       // this works really rather well (albeit expensively) even though
Chris@16:       // strictly speaking we're outside it's defined range.
Chris@16:       //
Chris@16:       // However it's expensive to compute, and an alternative approach
Chris@16:       // which models the curve as a distorted quarter circle is much
Chris@16:       // cheaper to compute, and still keeps the number of iterations
Chris@16:       // required down to a reasonable level.  With thanks to Prof Temme
Chris@16:       // for this suggestion.
Chris@16:       //
Chris@16:       if(b < a)
Chris@16:       {
Chris@16:          std::swap(a, b);
Chris@16:          std::swap(p, q);
Chris@16:          invert = !invert;
Chris@16:       }
Chris@16:       if(pow(p, 1/a) < 0.5)
Chris@16:       {
Chris@16:          x = pow(p * a * boost::math::beta(a, b, pol), 1 / a);
Chris@16:          if(x == 0)
Chris@16:             x = boost::math::tools::min_value<T>();
Chris@16:          y = 1 - x;
Chris@16:       }
Chris@16:       else /*if(pow(q, 1/b) < 0.1)*/
Chris@16:       {
Chris@16:          // model a distorted quarter circle:
Chris@16:          y = pow(1 - pow(p, b * boost::math::beta(a, b, pol)), 1/b);
Chris@16:          if(y == 0)
Chris@16:             y = boost::math::tools::min_value<T>();
Chris@16:          x = 1 - y;
Chris@16:       }
Chris@16:    }
Chris@16: 
Chris@16:    //
Chris@16:    // Now we have a guess for x (and for y) we can set things up for
Chris@16:    // iteration.  If x > 0.5 it pays to swap things round:
Chris@16:    //
Chris@16:    if(x > 0.5)
Chris@16:    {
Chris@16:       std::swap(a, b);
Chris@16:       std::swap(p, q);
Chris@16:       std::swap(x, y);
Chris@16:       invert = !invert;
Chris@16:       T l = 1 - upper;
Chris@16:       T u = 1 - lower;
Chris@16:       lower = l;
Chris@16:       upper = u;
Chris@16:    }
Chris@16:    //
Chris@16:    // lower bound for our search:
Chris@16:    //
Chris@16:    // We're not interested in denormalised answers as these tend to
Chris@16:    // these tend to take up lots of iterations, given that we can't get
Chris@16:    // accurate derivatives in this area (they tend to be infinite).
Chris@16:    //
Chris@16:    if(lower == 0)
Chris@16:    {
Chris@16:       if(invert && (py == 0))
Chris@16:       {
Chris@16:          //
Chris@16:          // We're not interested in answers smaller than machine epsilon:
Chris@16:          //
Chris@16:          lower = boost::math::tools::epsilon<T>();
Chris@16:          if(x < lower)
Chris@16:             x = lower;
Chris@16:       }
Chris@16:       else
Chris@16:          lower = boost::math::tools::min_value<T>();
Chris@16:       if(x < lower)
Chris@16:          x = lower;
Chris@16:    }
Chris@16:    //
Chris@16:    // Figure out how many digits to iterate towards:
Chris@16:    //
Chris@16:    int digits = boost::math::policies::digits<T, Policy>() / 2;
Chris@16:    if((x < 1e-50) && ((a < 1) || (b < 1)))
Chris@16:    {
Chris@16:       //
Chris@16:       // If we're in a region where the first derivative is very
Chris@16:       // large, then we have to take care that the root-finder
Chris@16:       // doesn't terminate prematurely.  We'll bump the precision
Chris@16:       // up to avoid this, but we have to take care not to set the
Chris@16:       // precision too high or the last few iterations will just
Chris@16:       // thrash around and convergence may be slow in this case.
Chris@16:       // Try 3/4 of machine epsilon:
Chris@16:       //
Chris@16:       digits *= 3;  
Chris@16:       digits /= 2;
Chris@16:    }
Chris@16:    //
Chris@16:    // Now iterate, we can use either p or q as the target here
Chris@16:    // depending on which is smaller:
Chris@16:    //
Chris@16:    boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();
Chris@16:    x = boost::math::tools::halley_iterate(
Chris@16:       boost::math::detail::ibeta_roots<T, Policy>(a, b, (p < q ? p : q), (p < q ? false : true)), x, lower, upper, digits, max_iter);
Chris@16:    policies::check_root_iterations<T>("boost::math::ibeta<%1%>(%1%, %1%, %1%)", max_iter, pol);
Chris@16:    //
Chris@16:    // We don't really want these asserts here, but they are useful for sanity
Chris@16:    // checking that we have the limits right, uncomment if you suspect bugs *only*.
Chris@16:    //
Chris@16:    //BOOST_ASSERT(x != upper);
Chris@16:    //BOOST_ASSERT((x != lower) || (x == boost::math::tools::min_value<T>()) || (x == boost::math::tools::epsilon<T>()));
Chris@16:    //
Chris@16:    // Tidy up, if we "lower" was too high then zero is the best answer we have:
Chris@16:    //
Chris@16:    if(x == lower)
Chris@16:       x = 0;
Chris@16:    if(py)
Chris@16:       *py = invert ? x : 1 - x;
Chris@16:    return invert ? 1-x : x;
Chris@16: }
Chris@16: 
Chris@16: } // namespace detail
Chris@16: 
Chris@16: template <class T1, class T2, class T3, class T4, class Policy>
Chris@16: inline typename tools::promote_args<T1, T2, T3, T4>::type  
Chris@16:    ibeta_inv(T1 a, T2 b, T3 p, T4* py, const Policy& pol)
Chris@16: {
Chris@16:    static const char* function = "boost::math::ibeta_inv<%1%>(%1%,%1%,%1%)";
Chris@16:    BOOST_FPU_EXCEPTION_GUARD
Chris@16:    typedef typename tools::promote_args<T1, T2, T3, T4>::type result_type;
Chris@16:    typedef typename policies::evaluation<result_type, Policy>::type value_type;
Chris@16:    typedef typename policies::normalise<
Chris@16:       Policy, 
Chris@16:       policies::promote_float<false>, 
Chris@16:       policies::promote_double<false>, 
Chris@16:       policies::discrete_quantile<>,
Chris@16:       policies::assert_undefined<> >::type forwarding_policy;
Chris@16: 
Chris@16:    if(a <= 0)
Chris@16:       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);
Chris@16:    if(b <= 0)
Chris@16:       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);
Chris@16:    if((p < 0) || (p > 1))
Chris@16:       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);
Chris@16: 
Chris@16:    value_type rx, ry;
Chris@16: 
Chris@16:    rx = detail::ibeta_inv_imp(
Chris@16:          static_cast<value_type>(a),
Chris@16:          static_cast<value_type>(b),
Chris@16:          static_cast<value_type>(p),
Chris@16:          static_cast<value_type>(1 - p),
Chris@16:          forwarding_policy(), &ry);
Chris@16: 
Chris@16:    if(py) *py = policies::checked_narrowing_cast<T4, forwarding_policy>(ry, function);
Chris@16:    return policies::checked_narrowing_cast<result_type, forwarding_policy>(rx, function);
Chris@16: }
Chris@16: 
Chris@16: template <class T1, class T2, class T3, class T4>
Chris@16: inline typename tools::promote_args<T1, T2, T3, T4>::type  
Chris@16:    ibeta_inv(T1 a, T2 b, T3 p, T4* py)
Chris@16: {
Chris@16:    return ibeta_inv(a, b, p, py, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: template <class T1, class T2, class T3>
Chris@16: inline typename tools::promote_args<T1, T2, T3>::type 
Chris@16:    ibeta_inv(T1 a, T2 b, T3 p)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<T1, T2, T3>::type result_type;
Chris@16:    return ibeta_inv(a, b, p, static_cast<result_type*>(0), policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: template <class T1, class T2, class T3, class Policy>
Chris@16: inline typename tools::promote_args<T1, T2, T3>::type 
Chris@16:    ibeta_inv(T1 a, T2 b, T3 p, const Policy& pol)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<T1, T2, T3>::type result_type;
Chris@16:    return ibeta_inv(a, b, p, static_cast<result_type*>(0), pol);
Chris@16: }
Chris@16: 
Chris@16: template <class T1, class T2, class T3, class T4, class Policy>
Chris@16: inline typename tools::promote_args<T1, T2, T3, T4>::type 
Chris@16:    ibetac_inv(T1 a, T2 b, T3 q, T4* py, const Policy& pol)
Chris@16: {
Chris@16:    static const char* function = "boost::math::ibetac_inv<%1%>(%1%,%1%,%1%)";
Chris@16:    BOOST_FPU_EXCEPTION_GUARD
Chris@16:    typedef typename tools::promote_args<T1, T2, T3, T4>::type result_type;
Chris@16:    typedef typename policies::evaluation<result_type, Policy>::type value_type;
Chris@16:    typedef typename policies::normalise<
Chris@16:       Policy, 
Chris@16:       policies::promote_float<false>, 
Chris@16:       policies::promote_double<false>, 
Chris@16:       policies::discrete_quantile<>,
Chris@16:       policies::assert_undefined<> >::type forwarding_policy;
Chris@16: 
Chris@16:    if(a <= 0)
Chris@101:       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);
Chris@16:    if(b <= 0)
Chris@101:       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);
Chris@16:    if((q < 0) || (q > 1))
Chris@101:       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);
Chris@16: 
Chris@16:    value_type rx, ry;
Chris@16: 
Chris@16:    rx = detail::ibeta_inv_imp(
Chris@16:          static_cast<value_type>(a),
Chris@16:          static_cast<value_type>(b),
Chris@16:          static_cast<value_type>(1 - q),
Chris@16:          static_cast<value_type>(q),
Chris@16:          forwarding_policy(), &ry);
Chris@16: 
Chris@16:    if(py) *py = policies::checked_narrowing_cast<T4, forwarding_policy>(ry, function);
Chris@16:    return policies::checked_narrowing_cast<result_type, forwarding_policy>(rx, function);
Chris@16: }
Chris@16: 
Chris@16: template <class T1, class T2, class T3, class T4>
Chris@16: inline typename tools::promote_args<T1, T2, T3, T4>::type 
Chris@16:    ibetac_inv(T1 a, T2 b, T3 q, T4* py)
Chris@16: {
Chris@16:    return ibetac_inv(a, b, q, py, policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: template <class RT1, class RT2, class RT3>
Chris@16: inline typename tools::promote_args<RT1, RT2, RT3>::type 
Chris@16:    ibetac_inv(RT1 a, RT2 b, RT3 q)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<RT1, RT2, RT3>::type result_type;
Chris@16:    return ibetac_inv(a, b, q, static_cast<result_type*>(0), policies::policy<>());
Chris@16: }
Chris@16: 
Chris@16: template <class RT1, class RT2, class RT3, class Policy>
Chris@16: inline typename tools::promote_args<RT1, RT2, RT3>::type
Chris@16:    ibetac_inv(RT1 a, RT2 b, RT3 q, const Policy& pol)
Chris@16: {
Chris@16:    typedef typename tools::promote_args<RT1, RT2, RT3>::type result_type;
Chris@16:    return ibetac_inv(a, b, q, static_cast<result_type*>(0), pol);
Chris@16: }
Chris@16: 
Chris@16: } // namespace math
Chris@16: } // namespace boost
Chris@16: 
Chris@16: #endif // BOOST_MATH_SPECIAL_FUNCTIONS_IGAMMA_INVERSE_HPP
Chris@16: 
Chris@16: 
Chris@16: 
Chris@16: