| // Copyright John Maddock 2006. |
| // Copyright Paul A. Bristow 2007 |
| // Use, modification and distribution are subject to the |
| // Boost Software License, Version 1.0. (See accompanying file |
| // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) |
| |
| #ifndef BOOST_MATH_SPECIAL_FUNCTIONS_IBETA_INVERSE_HPP |
| #define BOOST_MATH_SPECIAL_FUNCTIONS_IBETA_INVERSE_HPP |
| |
| #ifdef _MSC_VER |
| #pragma once |
| #endif |
| |
| #include <boost/math/special_functions/beta.hpp> |
| #include <boost/math/special_functions/erf.hpp> |
| #include <boost/math/tools/roots.hpp> |
| #include <boost/math/special_functions/detail/t_distribution_inv.hpp> |
| |
| namespace boost{ namespace math{ namespace detail{ |
| |
| // |
| // Helper object used by root finding |
| // code to convert eta to x. |
| // |
| template <class T> |
| struct temme_root_finder |
| { |
| temme_root_finder(const T t_, const T a_) : t(t_), a(a_) {} |
| |
| boost::math::tuple<T, T> operator()(T x) |
| { |
| BOOST_MATH_STD_USING // ADL of std names |
| |
| T y = 1 - x; |
| if(y == 0) |
| { |
| T big = tools::max_value<T>() / 4; |
| return boost::math::make_tuple(-big, -big); |
| } |
| if(x == 0) |
| { |
| T big = tools::max_value<T>() / 4; |
| return boost::math::make_tuple(-big, big); |
| } |
| T f = log(x) + a * log(y) + t; |
| T f1 = (1 / x) - (a / (y)); |
| return boost::math::make_tuple(f, f1); |
| } |
| private: |
| T t, a; |
| }; |
| // |
| // See: |
| // "Asymptotic Inversion of the Incomplete Beta Function" |
| // N.M. Temme |
| // Journal of Computation and Applied Mathematics 41 (1992) 145-157. |
| // Section 2. |
| // |
| template <class T, class Policy> |
| T temme_method_1_ibeta_inverse(T a, T b, T z, const Policy& pol) |
| { |
| BOOST_MATH_STD_USING // ADL of std names |
| |
| const T r2 = sqrt(T(2)); |
| // |
| // get the first approximation for eta from the inverse |
| // error function (Eq: 2.9 and 2.10). |
| // |
| T eta0 = boost::math::erfc_inv(2 * z, pol); |
| eta0 /= -sqrt(a / 2); |
| |
| T terms[4] = { eta0 }; |
| T workspace[7]; |
| // |
| // calculate powers: |
| // |
| T B = b - a; |
| T B_2 = B * B; |
| T B_3 = B_2 * B; |
| // |
| // Calculate correction terms: |
| // |
| |
| // See eq following 2.15: |
| workspace[0] = -B * r2 / 2; |
| workspace[1] = (1 - 2 * B) / 8; |
| workspace[2] = -(B * r2 / 48); |
| workspace[3] = T(-1) / 192; |
| workspace[4] = -B * r2 / 3840; |
| terms[1] = tools::evaluate_polynomial(workspace, eta0, 5); |
| // Eq Following 2.17: |
| workspace[0] = B * r2 * (3 * B - 2) / 12; |
| workspace[1] = (20 * B_2 - 12 * B + 1) / 128; |
| workspace[2] = B * r2 * (20 * B - 1) / 960; |
| workspace[3] = (16 * B_2 + 30 * B - 15) / 4608; |
| workspace[4] = B * r2 * (21 * B + 32) / 53760; |
| workspace[5] = (-32 * B_2 + 63) / 368640; |
| workspace[6] = -B * r2 * (120 * B + 17) / 25804480; |
| terms[2] = tools::evaluate_polynomial(workspace, eta0, 7); |
| // Eq Following 2.17: |
| workspace[0] = B * r2 * (-75 * B_2 + 80 * B - 16) / 480; |
| workspace[1] = (-1080 * B_3 + 868 * B_2 - 90 * B - 45) / 9216; |
| workspace[2] = B * r2 * (-1190 * B_2 + 84 * B + 373) / 53760; |
| workspace[3] = (-2240 * B_3 - 2508 * B_2 + 2100 * B - 165) / 368640; |
| terms[3] = tools::evaluate_polynomial(workspace, eta0, 4); |
| // |
| // Bring them together to get a final estimate for eta: |
| // |
| T eta = tools::evaluate_polynomial(terms, T(1/a), 4); |
| // |
| // now we need to convert eta to x, by solving the appropriate |
| // quadratic equation: |
| // |
| T eta_2 = eta * eta; |
| T c = -exp(-eta_2 / 2); |
| T x; |
| if(eta_2 == 0) |
| x = 0.5; |
| else |
| x = (1 + eta * sqrt((1 + c) / eta_2)) / 2; |
| |
| BOOST_ASSERT(x >= 0); |
| BOOST_ASSERT(x <= 1); |
| BOOST_ASSERT(eta * (x - 0.5) >= 0); |
| #ifdef BOOST_INSTRUMENT |
| std::cout << "Estimating x with Temme method 1: " << x << std::endl; |
| #endif |
| return x; |
| } |
| // |
| // See: |
| // "Asymptotic Inversion of the Incomplete Beta Function" |
| // N.M. Temme |
| // Journal of Computation and Applied Mathematics 41 (1992) 145-157. |
| // Section 3. |
| // |
| template <class T, class Policy> |
| T temme_method_2_ibeta_inverse(T /*a*/, T /*b*/, T z, T r, T theta, const Policy& pol) |
| { |
| BOOST_MATH_STD_USING // ADL of std names |
| |
| // |
| // Get first estimate for eta, see Eq 3.9 and 3.10, |
| // but note there is a typo in Eq 3.10: |
| // |
| T eta0 = boost::math::erfc_inv(2 * z, pol); |
| eta0 /= -sqrt(r / 2); |
| |
| T s = sin(theta); |
| T c = cos(theta); |
| // |
| // Now we need to purturb eta0 to get eta, which we do by |
| // evaluating the polynomial in 1/r at the bottom of page 151, |
| // to do this we first need the error terms e1, e2 e3 |
| // which we'll fill into the array "terms". Since these |
| // terms are themselves polynomials, we'll need another |
| // array "workspace" to calculate those... |
| // |
| T terms[4] = { eta0 }; |
| T workspace[6]; |
| // |
| // some powers of sin(theta)cos(theta) that we'll need later: |
| // |
| T sc = s * c; |
| T sc_2 = sc * sc; |
| T sc_3 = sc_2 * sc; |
| T sc_4 = sc_2 * sc_2; |
| T sc_5 = sc_2 * sc_3; |
| T sc_6 = sc_3 * sc_3; |
| T sc_7 = sc_4 * sc_3; |
| // |
| // Calculate e1 and put it in terms[1], see the middle of page 151: |
| // |
| workspace[0] = (2 * s * s - 1) / (3 * s * c); |
| static const BOOST_MATH_INT_TABLE_TYPE(T, int) co1[] = { -1, -5, 5 }; |
| workspace[1] = -tools::evaluate_even_polynomial(co1, s, 3) / (36 * sc_2); |
| static const BOOST_MATH_INT_TABLE_TYPE(T, int) co2[] = { 1, 21, -69, 46 }; |
| workspace[2] = tools::evaluate_even_polynomial(co2, s, 4) / (1620 * sc_3); |
| static const BOOST_MATH_INT_TABLE_TYPE(T, int) co3[] = { 7, -2, 33, -62, 31 }; |
| workspace[3] = -tools::evaluate_even_polynomial(co3, s, 5) / (6480 * sc_4); |
| static const BOOST_MATH_INT_TABLE_TYPE(T, int) co4[] = { 25, -52, -17, 88, -115, 46 }; |
| workspace[4] = tools::evaluate_even_polynomial(co4, s, 6) / (90720 * sc_5); |
| terms[1] = tools::evaluate_polynomial(workspace, eta0, 5); |
| // |
| // Now evaluate e2 and put it in terms[2]: |
| // |
| static const BOOST_MATH_INT_TABLE_TYPE(T, int) co5[] = { 7, 12, -78, 52 }; |
| workspace[0] = -tools::evaluate_even_polynomial(co5, s, 4) / (405 * sc_3); |
| static const BOOST_MATH_INT_TABLE_TYPE(T, int) co6[] = { -7, 2, 183, -370, 185 }; |
| workspace[1] = tools::evaluate_even_polynomial(co6, s, 5) / (2592 * sc_4); |
| static const BOOST_MATH_INT_TABLE_TYPE(T, int) co7[] = { -533, 776, -1835, 10240, -13525, 5410 }; |
| workspace[2] = -tools::evaluate_even_polynomial(co7, s, 6) / (204120 * sc_5); |
| static const BOOST_MATH_INT_TABLE_TYPE(T, int) co8[] = { -1579, 3747, -3372, -15821, 45588, -45213, 15071 }; |
| workspace[3] = -tools::evaluate_even_polynomial(co8, s, 7) / (2099520 * sc_6); |
| terms[2] = tools::evaluate_polynomial(workspace, eta0, 4); |
| // |
| // And e3, and put it in terms[3]: |
| // |
| static const BOOST_MATH_INT_TABLE_TYPE(T, int) co9[] = {449, -1259, -769, 6686, -9260, 3704 }; |
| workspace[0] = tools::evaluate_even_polynomial(co9, s, 6) / (102060 * sc_5); |
| static const BOOST_MATH_INT_TABLE_TYPE(T, int) co10[] = { 63149, -151557, 140052, -727469, 2239932, -2251437, 750479 }; |
| workspace[1] = -tools::evaluate_even_polynomial(co10, s, 7) / (20995200 * sc_6); |
| static const BOOST_MATH_INT_TABLE_TYPE(T, int) co11[] = { 29233, -78755, 105222, 146879, -1602610, 3195183, -2554139, 729754 }; |
| workspace[2] = tools::evaluate_even_polynomial(co11, s, 8) / (36741600 * sc_7); |
| terms[3] = tools::evaluate_polynomial(workspace, eta0, 3); |
| // |
| // Bring the correction terms together to evaluate eta, |
| // this is the last equation on page 151: |
| // |
| T eta = tools::evaluate_polynomial(terms, T(1/r), 4); |
| // |
| // Now that we have eta we need to back solve for x, |
| // we seek the value of x that gives eta in Eq 3.2. |
| // The two methods used are described in section 5. |
| // |
| // Begin by defining a few variables we'll need later: |
| // |
| T x; |
| T s_2 = s * s; |
| T c_2 = c * c; |
| T alpha = c / s; |
| alpha *= alpha; |
| T lu = (-(eta * eta) / (2 * s_2) + log(s_2) + c_2 * log(c_2) / s_2); |
| // |
| // Temme doesn't specify what value to switch on here, |
| // but this seems to work pretty well: |
| // |
| if(fabs(eta) < 0.7) |
| { |
| // |
| // Small eta use the expansion Temme gives in the second equation |
| // of section 5, it's a polynomial in eta: |
| // |
| workspace[0] = s * s; |
| workspace[1] = s * c; |
| workspace[2] = (1 - 2 * workspace[0]) / 3; |
| static const BOOST_MATH_INT_TABLE_TYPE(T, int) co12[] = { 1, -13, 13 }; |
| workspace[3] = tools::evaluate_polynomial(co12, workspace[0], 3) / (36 * s * c); |
| static const BOOST_MATH_INT_TABLE_TYPE(T, int) co13[] = { 1, 21, -69, 46 }; |
| workspace[4] = tools::evaluate_polynomial(co13, workspace[0], 4) / (270 * workspace[0] * c * c); |
| x = tools::evaluate_polynomial(workspace, eta, 5); |
| #ifdef BOOST_INSTRUMENT |
| std::cout << "Estimating x with Temme method 2 (small eta): " << x << std::endl; |
| #endif |
| } |
| else |
| { |
| // |
| // If eta is large we need to solve Eq 3.2 more directly, |
| // begin by getting an initial approximation for x from |
| // the last equation on page 155, this is a polynomial in u: |
| // |
| T u = exp(lu); |
| workspace[0] = u; |
| workspace[1] = alpha; |
| workspace[2] = 0; |
| workspace[3] = 3 * alpha * (3 * alpha + 1) / 6; |
| workspace[4] = 4 * alpha * (4 * alpha + 1) * (4 * alpha + 2) / 24; |
| workspace[5] = 5 * alpha * (5 * alpha + 1) * (5 * alpha + 2) * (5 * alpha + 3) / 120; |
| x = tools::evaluate_polynomial(workspace, u, 6); |
| // |
| // At this point we may or may not have the right answer, Eq-3.2 has |
| // two solutions for x for any given eta, however the mapping in 3.2 |
| // is 1:1 with the sign of eta and x-sin^2(theta) being the same. |
| // So we can check if we have the right root of 3.2, and if not |
| // switch x for 1-x. This transformation is motivated by the fact |
| // that the distribution is *almost* symetric so 1-x will be in the right |
| // ball park for the solution: |
| // |
| if((x - s_2) * eta < 0) |
| x = 1 - x; |
| #ifdef BOOST_INSTRUMENT |
| std::cout << "Estimating x with Temme method 2 (large eta): " << x << std::endl; |
| #endif |
| } |
| // |
| // The final step is a few Newton-Raphson iterations to |
| // clean up our approximation for x, this is pretty cheap |
| // in general, and very cheap compared to an incomplete beta |
| // evaluation. The limits set on x come from the observation |
| // that the sign of eta and x-sin^2(theta) are the same. |
| // |
| T lower, upper; |
| if(eta < 0) |
| { |
| lower = 0; |
| upper = s_2; |
| } |
| else |
| { |
| lower = s_2; |
| upper = 1; |
| } |
| // |
| // If our initial approximation is out of bounds then bisect: |
| // |
| if((x < lower) || (x > upper)) |
| x = (lower+upper) / 2; |
| // |
| // And iterate: |
| // |
| x = tools::newton_raphson_iterate( |
| temme_root_finder<T>(-lu, alpha), x, lower, upper, policies::digits<T, Policy>() / 2); |
| |
| return x; |
| } |
| // |
| // See: |
| // "Asymptotic Inversion of the Incomplete Beta Function" |
| // N.M. Temme |
| // Journal of Computation and Applied Mathematics 41 (1992) 145-157. |
| // Section 4. |
| // |
| template <class T, class Policy> |
| T temme_method_3_ibeta_inverse(T a, T b, T p, T q, const Policy& pol) |
| { |
| BOOST_MATH_STD_USING // ADL of std names |
| |
| // |
| // Begin by getting an initial approximation for the quantity |
| // eta from the dominant part of the incomplete beta: |
| // |
| T eta0; |
| if(p < q) |
| eta0 = boost::math::gamma_q_inv(b, p, pol); |
| else |
| eta0 = boost::math::gamma_p_inv(b, q, pol); |
| eta0 /= a; |
| // |
| // Define the variables and powers we'll need later on: |
| // |
| T mu = b / a; |
| T w = sqrt(1 + mu); |
| T w_2 = w * w; |
| T w_3 = w_2 * w; |
| T w_4 = w_2 * w_2; |
| T w_5 = w_3 * w_2; |
| T w_6 = w_3 * w_3; |
| T w_7 = w_4 * w_3; |
| T w_8 = w_4 * w_4; |
| T w_9 = w_5 * w_4; |
| T w_10 = w_5 * w_5; |
| T d = eta0 - mu; |
| T d_2 = d * d; |
| T d_3 = d_2 * d; |
| T d_4 = d_2 * d_2; |
| T w1 = w + 1; |
| T w1_2 = w1 * w1; |
| T w1_3 = w1 * w1_2; |
| T w1_4 = w1_2 * w1_2; |
| // |
| // Now we need to compute the purturbation error terms that |
| // convert eta0 to eta, these are all polynomials of polynomials. |
| // Probably these should be re-written to use tabulated data |
| // (see examples above), but it's less of a win in this case as we |
| // need to calculate the individual powers for the denominator terms |
| // anyway, so we might as well use them for the numerator-polynomials |
| // as well.... |
| // |
| // Refer to p154-p155 for the details of these expansions: |
| // |
| T e1 = (w + 2) * (w - 1) / (3 * w); |
| e1 += (w_3 + 9 * w_2 + 21 * w + 5) * d / (36 * w_2 * w1); |
| e1 -= (w_4 - 13 * w_3 + 69 * w_2 + 167 * w + 46) * d_2 / (1620 * w1_2 * w_3); |
| e1 -= (7 * w_5 + 21 * w_4 + 70 * w_3 + 26 * w_2 - 93 * w - 31) * d_3 / (6480 * w1_3 * w_4); |
| e1 -= (75 * w_6 + 202 * w_5 + 188 * w_4 - 888 * w_3 - 1345 * w_2 + 118 * w + 138) * d_4 / (272160 * w1_4 * w_5); |
| |
| T e2 = (28 * w_4 + 131 * w_3 + 402 * w_2 + 581 * w + 208) * (w - 1) / (1620 * w1 * w_3); |
| e2 -= (35 * w_6 - 154 * w_5 - 623 * w_4 - 1636 * w_3 - 3983 * w_2 - 3514 * w - 925) * d / (12960 * w1_2 * w_4); |
| e2 -= (2132 * w_7 + 7915 * w_6 + 16821 * w_5 + 35066 * w_4 + 87490 * w_3 + 141183 * w_2 + 95993 * w + 21640) * d_2 / (816480 * w_5 * w1_3); |
| e2 -= (11053 * w_8 + 53308 * w_7 + 117010 * w_6 + 163924 * w_5 + 116188 * w_4 - 258428 * w_3 - 677042 * w_2 - 481940 * w - 105497) * d_3 / (14696640 * w1_4 * w_6); |
| |
| T e3 = -((3592 * w_7 + 8375 * w_6 - 1323 * w_5 - 29198 * w_4 - 89578 * w_3 - 154413 * w_2 - 116063 * w - 29632) * (w - 1)) / (816480 * w_5 * w1_2); |
| e3 -= (442043 * w_9 + 2054169 * w_8 + 3803094 * w_7 + 3470754 * w_6 + 2141568 * w_5 - 2393568 * w_4 - 19904934 * w_3 - 34714674 * w_2 - 23128299 * w - 5253353) * d / (146966400 * w_6 * w1_3); |
| e3 -= (116932 * w_10 + 819281 * w_9 + 2378172 * w_8 + 4341330 * w_7 + 6806004 * w_6 + 10622748 * w_5 + 18739500 * w_4 + 30651894 * w_3 + 30869976 * w_2 + 15431867 * w + 2919016) * d_2 / (146966400 * w1_4 * w_7); |
| // |
| // Combine eta0 and the error terms to compute eta (Second eqaution p155): |
| // |
| T eta = eta0 + e1 / a + e2 / (a * a) + e3 / (a * a * a); |
| // |
| // Now we need to solve Eq 4.2 to obtain x. For any given value of |
| // eta there are two solutions to this equation, and since the distribtion |
| // may be very skewed, these are not related by x ~ 1-x we used when |
| // implementing section 3 above. However we know that: |
| // |
| // cross < x <= 1 ; iff eta < mu |
| // x == cross ; iff eta == mu |
| // 0 <= x < cross ; iff eta > mu |
| // |
| // Where cross == 1 / (1 + mu) |
| // Many thanks to Prof Temme for clarifying this point. |
| // |
| // Therefore we'll just jump straight into Newton iterations |
| // to solve Eq 4.2 using these bounds, and simple bisection |
| // as the first guess, in practice this converges pretty quickly |
| // and we only need a few digits correct anyway: |
| // |
| if(eta <= 0) |
| eta = tools::min_value<T>(); |
| T u = eta - mu * log(eta) + (1 + mu) * log(1 + mu) - mu; |
| T cross = 1 / (1 + mu); |
| T lower = eta < mu ? cross : 0; |
| T upper = eta < mu ? 1 : cross; |
| T x = (lower + upper) / 2; |
| x = tools::newton_raphson_iterate( |
| temme_root_finder<T>(u, mu), x, lower, upper, policies::digits<T, Policy>() / 2); |
| #ifdef BOOST_INSTRUMENT |
| std::cout << "Estimating x with Temme method 3: " << x << std::endl; |
| #endif |
| return x; |
| } |
| |
| template <class T, class Policy> |
| struct ibeta_roots |
| { |
| ibeta_roots(T _a, T _b, T t, bool inv = false) |
| : a(_a), b(_b), target(t), invert(inv) {} |
| |
| boost::math::tuple<T, T, T> operator()(T x) |
| { |
| BOOST_MATH_STD_USING // ADL of std names |
| |
| BOOST_FPU_EXCEPTION_GUARD |
| |
| T f1; |
| T y = 1 - x; |
| T f = ibeta_imp(a, b, x, Policy(), invert, true, &f1) - target; |
| if(invert) |
| f1 = -f1; |
| if(y == 0) |
| y = tools::min_value<T>() * 64; |
| if(x == 0) |
| x = tools::min_value<T>() * 64; |
| |
| T f2 = f1 * (-y * a + (b - 2) * x + 1); |
| if(fabs(f2) < y * x * tools::max_value<T>()) |
| f2 /= (y * x); |
| if(invert) |
| f2 = -f2; |
| |
| // make sure we don't have a zero derivative: |
| if(f1 == 0) |
| f1 = (invert ? -1 : 1) * tools::min_value<T>() * 64; |
| |
| return boost::math::make_tuple(f, f1, f2); |
| } |
| private: |
| T a, b, target; |
| bool invert; |
| }; |
| |
| template <class T, class Policy> |
| T ibeta_inv_imp(T a, T b, T p, T q, const Policy& pol, T* py) |
| { |
| BOOST_MATH_STD_USING // For ADL of math functions. |
| |
| // |
| // The flag invert is set to true if we swap a for b and p for q, |
| // in which case the result has to be subtracted from 1: |
| // |
| bool invert = false; |
| // |
| // Depending upon which approximation method we use, we may end up |
| // calculating either x or y initially (where y = 1-x): |
| // |
| T x = 0; // Set to a safe zero to avoid a |
| // MSVC 2005 warning C4701: potentially uninitialized local variable 'x' used |
| // But code inspection appears to ensure that x IS assigned whatever the code path. |
| T y; |
| |
| // For some of the methods we can put tighter bounds |
| // on the result than simply [0,1]: |
| // |
| T lower = 0; |
| T upper = 1; |
| // |
| // Student's T with b = 0.5 gets handled as a special case, swap |
| // around if the arguments are in the "wrong" order: |
| // |
| if(a == 0.5f) |
| { |
| std::swap(a, b); |
| std::swap(p, q); |
| invert = !invert; |
| } |
| // |
| // Handle trivial cases first: |
| // |
| if(q == 0) |
| { |
| if(py) *py = 0; |
| return 1; |
| } |
| else if(p == 0) |
| { |
| if(py) *py = 1; |
| return 0; |
| } |
| else if((a == 1) && (b == 1)) |
| { |
| if(py) *py = 1 - p; |
| return p; |
| } |
| else if((b == 0.5f) && (a >= 0.5f)) |
| { |
| // |
| // We have a Student's T distribution: |
| x = find_ibeta_inv_from_t_dist(a, p, q, &y, pol); |
| } |
| else if(a + b > 5) |
| { |
| // |
| // When a+b is large then we can use one of Prof Temme's |
| // asymptotic expansions, begin by swapping things around |
| // so that p < 0.5, we do this to avoid cancellations errors |
| // when p is large. |
| // |
| if(p > 0.5) |
| { |
| std::swap(a, b); |
| std::swap(p, q); |
| invert = !invert; |
| } |
| T minv = (std::min)(a, b); |
| T maxv = (std::max)(a, b); |
| if((sqrt(minv) > (maxv - minv)) && (minv > 5)) |
| { |
| // |
| // When a and b differ by a small amount |
| // the curve is quite symmetrical and we can use an error |
| // function to approximate the inverse. This is the cheapest |
| // of the three Temme expantions, and the calculated value |
| // for x will never be much larger than p, so we don't have |
| // to worry about cancellation as long as p is small. |
| // |
| x = temme_method_1_ibeta_inverse(a, b, p, pol); |
| y = 1 - x; |
| } |
| else |
| { |
| T r = a + b; |
| T theta = asin(sqrt(a / r)); |
| T lambda = minv / r; |
| if((lambda >= 0.2) && (lambda <= 0.8) && (lambda >= 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 = true; |
| 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 = true; |
| } |
| // |
| // 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 = true; |
| } |
| if(pow(p, 1/a) < 0.5) |
| { |
| x = pow(p * a * boost::math::beta(a, b, pol), 1 / a); |
| if(x == 0) |
| x = boost::math::tools::min_value<T>(); |
| y = 1 - x; |
| } |
| else /*if(pow(q, 1/b) < 0.1)*/ |
| { |
| // model a distorted quarter circle: |
| y = pow(1 - pow(p, b * boost::math::beta(a, b, pol)), 1/b); |
| if(y == 0) |
| y = boost::math::tools::min_value<T>(); |
| x = 1 - y; |
| } |
| } |
| |
| // |
| // Now we have a guess for x (and for y) we can set things up for |
| // iteration. If x > 0.5 it pays to swap things round: |
| // |
| if(x > 0.5) |
| { |
| std::swap(a, b); |
| std::swap(p, q); |
| std::swap(x, y); |
| invert = !invert; |
| T l = 1 - upper; |
| T u = 1 - lower; |
| lower = l; |
| upper = u; |
| } |
| // |
| // lower bound for our search: |
| // |
| // We're not interested in denormalised answers as these tend to |
| // these tend to take up lots of iterations, given that we can't get |
| // accurate derivatives in this area (they tend to be infinite). |
| // |
| if(lower == 0) |
| { |
| if(invert && (py == 0)) |
| { |
| // |
| // We're not interested in answers smaller than machine epsilon: |
| // |
| lower = boost::math::tools::epsilon<T>(); |
| if(x < lower) |
| x = lower; |
| } |
| else |
| lower = boost::math::tools::min_value<T>(); |
| if(x < lower) |
| x = lower; |
| } |
| // |
| // Figure out how many digits to iterate towards: |
| // |
| int digits = boost::math::policies::digits<T, Policy>() / 2; |
| if((x < 1e-50) && ((a < 1) || (b < 1))) |
| { |
| // |
| // If we're in a region where the first derivative is very |
| // large, then we have to take care that the root-finder |
| // doesn't terminate prematurely. We'll bump the precision |
| // up to avoid this, but we have to take care not to set the |
| // precision too high or the last few iterations will just |
| // thrash around and convergence may be slow in this case. |
| // Try 3/4 of machine epsilon: |
| // |
| digits *= 3; |
| digits /= 2; |
| } |
| // |
| // Now iterate, we can use either p or q as the target here |
| // depending on which is smaller: |
| // |
| boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>(); |
| x = boost::math::tools::halley_iterate( |
| boost::math::detail::ibeta_roots<T, Policy>(a, b, (p < q ? p : q), (p < q ? false : true)), x, lower, upper, digits, max_iter); |
| policies::check_root_iterations("boost::math::ibeta<%1%>(%1%, %1%, %1%)", max_iter, pol); |
| // |
| // We don't really want these asserts here, but they are useful for sanity |
| // checking that we have the limits right, uncomment if you suspect bugs *only*. |
| // |
| //BOOST_ASSERT(x != upper); |
| //BOOST_ASSERT((x != lower) || (x == boost::math::tools::min_value<T>()) || (x == boost::math::tools::epsilon<T>())); |
| // |
| // Tidy up, if we "lower" was too high then zero is the best answer we have: |
| // |
| if(x == lower) |
| x = 0; |
| if(py) |
| *py = invert ? x : 1 - x; |
| return invert ? 1-x : x; |
| } |
| |
| } // namespace detail |
| |
| template <class T1, class T2, class T3, class T4, class Policy> |
| inline typename tools::promote_args<T1, T2, T3, T4>::type |
| ibeta_inv(T1 a, T2 b, T3 p, T4* py, const Policy& pol) |
| { |
| static const char* function = "boost::math::ibeta_inv<%1%>(%1%,%1%,%1%)"; |
| BOOST_FPU_EXCEPTION_GUARD |
| typedef typename tools::promote_args<T1, T2, T3, T4>::type result_type; |
| typedef typename policies::evaluation<result_type, Policy>::type value_type; |
| typedef typename policies::normalise< |
| Policy, |
| policies::promote_float<false>, |
| policies::promote_double<false>, |
| policies::discrete_quantile<>, |
| policies::assert_undefined<> >::type forwarding_policy; |
| |
| if(a <= 0) |
| return policies::raise_domain_error<result_type>(function, "The argument a to the incomplete beta function inverse must be greater than zero (got a=%1%).", a, pol); |
| if(b <= 0) |
| return policies::raise_domain_error<result_type>(function, "The argument b to the incomplete beta function inverse must be greater than zero (got b=%1%).", b, pol); |
| if((p < 0) || (p > 1)) |
| return policies::raise_domain_error<result_type>(function, "Argument p outside the range [0,1] in the incomplete beta function inverse (got p=%1%).", p, pol); |
| |
| value_type rx, ry; |
| |
| rx = detail::ibeta_inv_imp( |
| static_cast<value_type>(a), |
| static_cast<value_type>(b), |
| static_cast<value_type>(p), |
| static_cast<value_type>(1 - p), |
| forwarding_policy(), &ry); |
| |
| if(py) *py = policies::checked_narrowing_cast<T4, forwarding_policy>(ry, function); |
| return policies::checked_narrowing_cast<result_type, forwarding_policy>(rx, function); |
| } |
| |
| template <class T1, class T2, class T3, class T4> |
| inline typename tools::promote_args<T1, T2, T3, T4>::type |
| ibeta_inv(T1 a, T2 b, T3 p, T4* py) |
| { |
| return ibeta_inv(a, b, p, py, policies::policy<>()); |
| } |
| |
| template <class T1, class T2, class T3> |
| inline typename tools::promote_args<T1, T2, T3>::type |
| ibeta_inv(T1 a, T2 b, T3 p) |
| { |
| return ibeta_inv(a, b, p, static_cast<T1*>(0), policies::policy<>()); |
| } |
| |
| template <class T1, class T2, class T3, class Policy> |
| inline typename tools::promote_args<T1, T2, T3>::type |
| ibeta_inv(T1 a, T2 b, T3 p, const Policy& pol) |
| { |
| return ibeta_inv(a, b, p, static_cast<T1*>(0), pol); |
| } |
| |
| template <class T1, class T2, class T3, class T4, class Policy> |
| inline typename tools::promote_args<T1, T2, T3, T4>::type |
| ibetac_inv(T1 a, T2 b, T3 q, T4* py, const Policy& pol) |
| { |
| static const char* function = "boost::math::ibetac_inv<%1%>(%1%,%1%,%1%)"; |
| BOOST_FPU_EXCEPTION_GUARD |
| typedef typename tools::promote_args<T1, T2, T3, T4>::type result_type; |
| typedef typename policies::evaluation<result_type, Policy>::type value_type; |
| typedef typename policies::normalise< |
| Policy, |
| policies::promote_float<false>, |
| policies::promote_double<false>, |
| policies::discrete_quantile<>, |
| policies::assert_undefined<> >::type forwarding_policy; |
| |
| if(a <= 0) |
| policies::raise_domain_error<result_type>(function, "The argument a to the incomplete beta function inverse must be greater than zero (got a=%1%).", a, pol); |
| if(b <= 0) |
| policies::raise_domain_error<result_type>(function, "The argument b to the incomplete beta function inverse must be greater than zero (got b=%1%).", b, pol); |
| if((q < 0) || (q > 1)) |
| policies::raise_domain_error<result_type>(function, "Argument q outside the range [0,1] in the incomplete beta function inverse (got q=%1%).", q, pol); |
| |
| value_type rx, ry; |
| |
| rx = detail::ibeta_inv_imp( |
| static_cast<value_type>(a), |
| static_cast<value_type>(b), |
| static_cast<value_type>(1 - q), |
| static_cast<value_type>(q), |
| forwarding_policy(), &ry); |
| |
| if(py) *py = policies::checked_narrowing_cast<T4, forwarding_policy>(ry, function); |
| return policies::checked_narrowing_cast<result_type, forwarding_policy>(rx, function); |
| } |
| |
| template <class T1, class T2, class T3, class T4> |
| inline typename tools::promote_args<T1, T2, T3, T4>::type |
| ibetac_inv(T1 a, T2 b, T3 q, T4* py) |
| { |
| return ibetac_inv(a, b, q, py, policies::policy<>()); |
| } |
| |
| template <class RT1, class RT2, class RT3> |
| inline typename tools::promote_args<RT1, RT2, RT3>::type |
| ibetac_inv(RT1 a, RT2 b, RT3 q) |
| { |
| typedef typename remove_cv<RT1>::type dummy; |
| return ibetac_inv(a, b, q, static_cast<dummy*>(0), policies::policy<>()); |
| } |
| |
| template <class RT1, class RT2, class RT3, class Policy> |
| inline typename tools::promote_args<RT1, RT2, RT3>::type |
| ibetac_inv(RT1 a, RT2 b, RT3 q, const Policy& pol) |
| { |
| typedef typename remove_cv<RT1>::type dummy; |
| return ibetac_inv(a, b, q, static_cast<dummy*>(0), pol); |
| } |
| |
| } // namespace math |
| } // namespace boost |
| |
| #endif // BOOST_MATH_SPECIAL_FUNCTIONS_IGAMMA_INVERSE_HPP |
| |
| |
| |
| |