// This file is part of Eigen, a lightweight C++ template library // for linear algebra. // // Copyright (C) 2016 Gael Guennebaud <gael.guennebaud@inria.fr> // // This Source Code Form is subject to the terms of the Mozilla // Public License v. 2.0. If a copy of the MPL was not distributed // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. #include "main.h" #include "../Eigen/SpecialFunctions" template<typename X, typename Y> void verify_component_wise(const X& x, const Y& y) { for(Index i=0; i<x.size(); ++i) { if((numext::isfinite)(y(i))) VERIFY_IS_APPROX( x(i), y(i) ); else if((numext::isnan)(y(i))) VERIFY((numext::isnan)(x(i))); else VERIFY_IS_EQUAL( x(i), y(i) ); } } template<typename ArrayType> void array_special_functions() { using std::abs; using std::sqrt; typedef typename ArrayType::Scalar Scalar; typedef typename NumTraits<Scalar>::Real RealScalar; Scalar plusinf = std::numeric_limits<Scalar>::infinity(); Scalar nan = std::numeric_limits<Scalar>::quiet_NaN(); Index rows = internal::random<Index>(1,30); Index cols = 1; // API { ArrayType m1 = ArrayType::Random(rows,cols); #if EIGEN_HAS_C99_MATH VERIFY_IS_APPROX(m1.lgamma(), lgamma(m1)); VERIFY_IS_APPROX(m1.digamma(), digamma(m1)); VERIFY_IS_APPROX(m1.erf(), erf(m1)); VERIFY_IS_APPROX(m1.erfc(), erfc(m1)); #endif // EIGEN_HAS_C99_MATH } #if EIGEN_HAS_C99_MATH // check special functions (comparing against numpy implementation) if (!NumTraits<Scalar>::IsComplex) { { ArrayType m1 = ArrayType::Random(rows,cols); ArrayType m2 = ArrayType::Random(rows,cols); // Test various propreties of igamma & igammac. These are normalized // gamma integrals where // igammac(a, x) = Gamma(a, x) / Gamma(a) // igamma(a, x) = gamma(a, x) / Gamma(a) // where Gamma and gamma are considered the standard unnormalized // upper and lower incomplete gamma functions, respectively. ArrayType a = m1.abs() + 2; ArrayType x = m2.abs() + 2; ArrayType zero = ArrayType::Zero(rows, cols); ArrayType one = ArrayType::Constant(rows, cols, Scalar(1.0)); ArrayType a_m1 = a - one; ArrayType Gamma_a_x = Eigen::igammac(a, x) * a.lgamma().exp(); ArrayType Gamma_a_m1_x = Eigen::igammac(a_m1, x) * a_m1.lgamma().exp(); ArrayType gamma_a_x = Eigen::igamma(a, x) * a.lgamma().exp(); ArrayType gamma_a_m1_x = Eigen::igamma(a_m1, x) * a_m1.lgamma().exp(); // Gamma(a, 0) == Gamma(a) VERIFY_IS_APPROX(Eigen::igammac(a, zero), one); // Gamma(a, x) + gamma(a, x) == Gamma(a) VERIFY_IS_APPROX(Gamma_a_x + gamma_a_x, a.lgamma().exp()); // Gamma(a, x) == (a - 1) * Gamma(a-1, x) + x^(a-1) * exp(-x) VERIFY_IS_APPROX(Gamma_a_x, (a - 1) * Gamma_a_m1_x + x.pow(a-1) * (-x).exp()); // gamma(a, x) == (a - 1) * gamma(a-1, x) - x^(a-1) * exp(-x) VERIFY_IS_APPROX(gamma_a_x, (a - 1) * gamma_a_m1_x - x.pow(a-1) * (-x).exp()); } { // Check exact values of igamma and igammac against a third party calculation. Scalar a_s[] = {Scalar(0), Scalar(1), Scalar(1.5), Scalar(4), Scalar(0.0001), Scalar(1000.5)}; Scalar x_s[] = {Scalar(0), Scalar(1), Scalar(1.5), Scalar(4), Scalar(0.0001), Scalar(1000.5)}; // location i*6+j corresponds to a_s[i], x_s[j]. Scalar igamma_s[][6] = {{0.0, nan, nan, nan, nan, nan}, {0.0, 0.6321205588285578, 0.7768698398515702, 0.9816843611112658, 9.999500016666262e-05, 1.0}, {0.0, 0.4275932955291202, 0.608374823728911, 0.9539882943107686, 7.522076445089201e-07, 1.0}, {0.0, 0.01898815687615381, 0.06564245437845008, 0.5665298796332909, 4.166333347221828e-18, 1.0}, {0.0, 0.9999780593618628, 0.9999899967080838, 0.9999996219837988, 0.9991370418689945, 1.0}, {0.0, 0.0, 0.0, 0.0, 0.0, 0.5042041932513908}}; Scalar igammac_s[][6] = {{nan, nan, nan, nan, nan, nan}, {1.0, 0.36787944117144233, 0.22313016014842982, 0.018315638888734182, 0.9999000049998333, 0.0}, {1.0, 0.5724067044708798, 0.3916251762710878, 0.04601170568923136, 0.9999992477923555, 0.0}, {1.0, 0.9810118431238462, 0.9343575456215499, 0.4334701203667089, 1.0, 0.0}, {1.0, 2.1940638138146658e-05, 1.0003291916285e-05, 3.7801620118431334e-07, 0.0008629581310054535, 0.0}, {1.0, 1.0, 1.0, 1.0, 1.0, 0.49579580674813944}}; for (int i = 0; i < 6; ++i) { for (int j = 0; j < 6; ++j) { if ((std::isnan)(igamma_s[i][j])) { VERIFY((std::isnan)(numext::igamma(a_s[i], x_s[j]))); } else { VERIFY_IS_APPROX(numext::igamma(a_s[i], x_s[j]), igamma_s[i][j]); } if ((std::isnan)(igammac_s[i][j])) { VERIFY((std::isnan)(numext::igammac(a_s[i], x_s[j]))); } else { VERIFY_IS_APPROX(numext::igammac(a_s[i], x_s[j]), igammac_s[i][j]); } } } } } #endif // EIGEN_HAS_C99_MATH // Check the zeta function against scipy.special.zeta { ArrayType x(7), q(7), res(7), ref(7); x << 1.5, 4, 10.5, 10000.5, 3, 1, 0.9; q << 2, 1.5, 3, 1.0001, -2.5, 1.2345, 1.2345; ref << 1.61237534869, 0.234848505667, 1.03086757337e-5, 0.367879440865, 0.054102025820864097, plusinf, nan; CALL_SUBTEST( verify_component_wise(ref, ref); ); CALL_SUBTEST( res = x.zeta(q); verify_component_wise(res, ref); ); CALL_SUBTEST( res = zeta(x,q); verify_component_wise(res, ref); ); } // digamma { ArrayType x(7), res(7), ref(7); x << 1, 1.5, 4, -10.5, 10000.5, 0, -1; ref << -0.5772156649015329, 0.03648997397857645, 1.2561176684318, 2.398239129535781, 9.210340372392849, plusinf, plusinf; CALL_SUBTEST( verify_component_wise(ref, ref); ); CALL_SUBTEST( res = x.digamma(); verify_component_wise(res, ref); ); CALL_SUBTEST( res = digamma(x); verify_component_wise(res, ref); ); } #if EIGEN_HAS_C99_MATH { ArrayType n(11), x(11), res(11), ref(11); n << 1, 1, 1, 1.5, 17, 31, 28, 8, 42, 147, 170; x << 2, 3, 25.5, 1.5, 4.7, 11.8, 17.7, 30.2, 15.8, 54.1, 64; ref << 0.644934066848, 0.394934066848, 0.0399946696496, nan, 293.334565435, 0.445487887616, -2.47810300902e-07, -8.29668781082e-09, -0.434562276666, 0.567742190178, -0.0108615497927; CALL_SUBTEST( verify_component_wise(ref, ref); ); if(sizeof(RealScalar)>=8) { // double // Reason for commented line: http://eigen.tuxfamily.org/bz/show_bug.cgi?id=1232 // CALL_SUBTEST( res = x.polygamma(n); verify_component_wise(res, ref); ); CALL_SUBTEST( res = polygamma(n,x); verify_component_wise(res, ref); ); } else { // CALL_SUBTEST( res = x.polygamma(n); verify_component_wise(res.head(8), ref.head(8)); ); CALL_SUBTEST( res = polygamma(n,x); verify_component_wise(res.head(8), ref.head(8)); ); } } #endif #if EIGEN_HAS_C99_MATH { // Inputs and ground truth generated with scipy via: // a = np.logspace(-3, 3, 5) - 1e-3 // b = np.logspace(-3, 3, 5) - 1e-3 // x = np.linspace(-0.1, 1.1, 5) // (full_a, full_b, full_x) = np.vectorize(lambda a, b, x: (a, b, x))(*np.ix_(a, b, x)) // full_a = full_a.flatten().tolist() # same for full_b, full_x // v = scipy.special.betainc(full_a, full_b, full_x).flatten().tolist() // // Note in Eigen, we call betainc with arguments in the order (x, a, b). ArrayType a(125); ArrayType b(125); ArrayType x(125); ArrayType v(125); ArrayType res(125); a << 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999; b << 0.0, 0.0, 0.0, 0.0, 0.0, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.999, 0.999, 0.999, 0.999, 0.999, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 999.999, 999.999, 999.999, 999.999, 999.999, 0.0, 0.0, 0.0, 0.0, 0.0, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.999, 0.999, 0.999, 0.999, 0.999, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 999.999, 999.999, 999.999, 999.999, 999.999, 0.0, 0.0, 0.0, 0.0, 0.0, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.999, 0.999, 0.999, 0.999, 0.999, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 999.999, 999.999, 999.999, 999.999, 999.999, 0.0, 0.0, 0.0, 0.0, 0.0, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.999, 0.999, 0.999, 0.999, 0.999, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 999.999, 999.999, 999.999, 999.999, 999.999, 0.0, 0.0, 0.0, 0.0, 0.0, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.999, 0.999, 0.999, 0.999, 0.999, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 31.62177660168379, 999.999, 999.999, 999.999, 999.999, 999.999; x << -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1; v << nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, 0.47972119876364683, 0.5, 0.5202788012363533, nan, nan, 0.9518683957740043, 0.9789663010413743, 0.9931729188073435, nan, nan, 0.999995949033062, 0.9999999999993698, 0.9999999999999999, nan, nan, 0.9999999999999999, 0.9999999999999999, 0.9999999999999999, nan, nan, nan, nan, nan, nan, nan, 0.006827081192655869, 0.0210336989586256, 0.04813160422599567, nan, nan, 0.20014344256217678, 0.5000000000000001, 0.7998565574378232, nan, nan, 0.9991401428435834, 0.999999999698403, 0.9999999999999999, nan, nan, 0.9999999999999999, 0.9999999999999999, 0.9999999999999999, nan, nan, nan, nan, nan, nan, nan, 1.0646600232370887e-25, 6.301722877826246e-13, 4.050966937974938e-06, nan, nan, 7.864342668429763e-23, 3.015969667594166e-10, 0.0008598571564165444, nan, nan, 6.031987710123844e-08, 0.5000000000000007, 0.9999999396801229, nan, nan, 0.9999999999999999, 0.9999999999999999, 0.9999999999999999, nan, nan, nan, nan, nan, nan, nan, 0.0, 7.029920380986636e-306, 2.2450728208591345e-101, nan, nan, 0.0, 9.275871147869727e-302, 1.2232913026152827e-97, nan, nan, 0.0, 3.0891393081932924e-252, 2.9303043666183996e-60, nan, nan, 2.248913486879199e-196, 0.5000000000004947, 0.9999999999999999, nan; CALL_SUBTEST(res = betainc(a, b, x); verify_component_wise(res, v);); } // Test various properties of betainc { ArrayType m1 = ArrayType::Random(32); ArrayType m2 = ArrayType::Random(32); ArrayType m3 = ArrayType::Random(32); ArrayType one = ArrayType::Constant(32, Scalar(1.0)); const Scalar eps = std::numeric_limits<Scalar>::epsilon(); ArrayType a = (m1 * 4.0).exp(); ArrayType b = (m2 * 4.0).exp(); ArrayType x = m3.abs(); // betainc(a, 1, x) == x**a CALL_SUBTEST( ArrayType test = betainc(a, one, x); ArrayType expected = x.pow(a); verify_component_wise(test, expected);); // betainc(1, b, x) == 1 - (1 - x)**b CALL_SUBTEST( ArrayType test = betainc(one, b, x); ArrayType expected = one - (one - x).pow(b); verify_component_wise(test, expected);); // betainc(a, b, x) == 1 - betainc(b, a, 1-x) CALL_SUBTEST( ArrayType test = betainc(a, b, x) + betainc(b, a, one - x); ArrayType expected = one; verify_component_wise(test, expected);); // betainc(a+1, b, x) = betainc(a, b, x) - x**a * (1 - x)**b / (a * beta(a, b)) CALL_SUBTEST( ArrayType num = x.pow(a) * (one - x).pow(b); ArrayType denom = a * (a.lgamma() + b.lgamma() - (a + b).lgamma()).exp(); // Add eps to rhs and lhs so that component-wise test doesn't result in // nans when both outputs are zeros. ArrayType expected = betainc(a, b, x) - num / denom + eps; ArrayType test = betainc(a + one, b, x) + eps; if (sizeof(Scalar) >= 8) { // double verify_component_wise(test, expected); } else { // Reason for limited test: http://eigen.tuxfamily.org/bz/show_bug.cgi?id=1232 verify_component_wise(test.head(8), expected.head(8)); }); // betainc(a, b+1, x) = betainc(a, b, x) + x**a * (1 - x)**b / (b * beta(a, b)) CALL_SUBTEST( // Add eps to rhs and lhs so that component-wise test doesn't result in // nans when both outputs are zeros. ArrayType num = x.pow(a) * (one - x).pow(b); ArrayType denom = b * (a.lgamma() + b.lgamma() - (a + b).lgamma()).exp(); ArrayType expected = betainc(a, b, x) + num / denom + eps; ArrayType test = betainc(a, b + one, x) + eps; verify_component_wise(test, expected);); } #endif } void test_special_functions() { CALL_SUBTEST_1(array_special_functions<ArrayXf>()); CALL_SUBTEST_2(array_special_functions<ArrayXd>()); }