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f216854453
The functions are conventionally called i0e and i1e. The exponentially scaled version is more numerically stable. The standard Bessel functions can be obtained as i0(x) = exp(|x|) i0e(x) The code is ported from Cephes and tested against SciPy.
386 lines
18 KiB
C++
386 lines
18 KiB
C++
// This file is part of Eigen, a lightweight C++ template library
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// for linear algebra.
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//
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// Copyright (C) 2016 Gael Guennebaud <gael.guennebaud@inria.fr>
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//
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// This Source Code Form is subject to the terms of the Mozilla
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// Public License v. 2.0. If a copy of the MPL was not distributed
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// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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#include "main.h"
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#include "../Eigen/SpecialFunctions"
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template<typename X, typename Y>
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void verify_component_wise(const X& x, const Y& y)
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{
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for(Index i=0; i<x.size(); ++i)
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{
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if((numext::isfinite)(y(i)))
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VERIFY_IS_APPROX( x(i), y(i) );
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else if((numext::isnan)(y(i)))
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VERIFY((numext::isnan)(x(i)));
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else
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VERIFY_IS_EQUAL( x(i), y(i) );
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}
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}
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template<typename ArrayType> void array_special_functions()
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{
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using std::abs;
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using std::sqrt;
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typedef typename ArrayType::Scalar Scalar;
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typedef typename NumTraits<Scalar>::Real RealScalar;
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Scalar plusinf = std::numeric_limits<Scalar>::infinity();
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Scalar nan = std::numeric_limits<Scalar>::quiet_NaN();
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Index rows = internal::random<Index>(1,30);
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Index cols = 1;
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// API
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{
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ArrayType m1 = ArrayType::Random(rows,cols);
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#if EIGEN_HAS_C99_MATH
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VERIFY_IS_APPROX(m1.lgamma(), lgamma(m1));
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VERIFY_IS_APPROX(m1.digamma(), digamma(m1));
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VERIFY_IS_APPROX(m1.erf(), erf(m1));
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VERIFY_IS_APPROX(m1.erfc(), erfc(m1));
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#endif // EIGEN_HAS_C99_MATH
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}
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#if EIGEN_HAS_C99_MATH
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// check special functions (comparing against numpy implementation)
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if (!NumTraits<Scalar>::IsComplex)
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{
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{
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ArrayType m1 = ArrayType::Random(rows,cols);
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ArrayType m2 = ArrayType::Random(rows,cols);
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// Test various propreties of igamma & igammac. These are normalized
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// gamma integrals where
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// igammac(a, x) = Gamma(a, x) / Gamma(a)
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// igamma(a, x) = gamma(a, x) / Gamma(a)
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// where Gamma and gamma are considered the standard unnormalized
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// upper and lower incomplete gamma functions, respectively.
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ArrayType a = m1.abs() + 2;
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ArrayType x = m2.abs() + 2;
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ArrayType zero = ArrayType::Zero(rows, cols);
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ArrayType one = ArrayType::Constant(rows, cols, Scalar(1.0));
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ArrayType a_m1 = a - one;
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ArrayType Gamma_a_x = Eigen::igammac(a, x) * a.lgamma().exp();
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ArrayType Gamma_a_m1_x = Eigen::igammac(a_m1, x) * a_m1.lgamma().exp();
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ArrayType gamma_a_x = Eigen::igamma(a, x) * a.lgamma().exp();
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ArrayType gamma_a_m1_x = Eigen::igamma(a_m1, x) * a_m1.lgamma().exp();
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// Gamma(a, 0) == Gamma(a)
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VERIFY_IS_APPROX(Eigen::igammac(a, zero), one);
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// Gamma(a, x) + gamma(a, x) == Gamma(a)
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VERIFY_IS_APPROX(Gamma_a_x + gamma_a_x, a.lgamma().exp());
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// Gamma(a, x) == (a - 1) * Gamma(a-1, x) + x^(a-1) * exp(-x)
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VERIFY_IS_APPROX(Gamma_a_x, (a - 1) * Gamma_a_m1_x + x.pow(a-1) * (-x).exp());
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// gamma(a, x) == (a - 1) * gamma(a-1, x) - x^(a-1) * exp(-x)
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VERIFY_IS_APPROX(gamma_a_x, (a - 1) * gamma_a_m1_x - x.pow(a-1) * (-x).exp());
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}
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{
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// Check exact values of igamma and igammac against a third party calculation.
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Scalar a_s[] = {Scalar(0), Scalar(1), Scalar(1.5), Scalar(4), Scalar(0.0001), Scalar(1000.5)};
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Scalar x_s[] = {Scalar(0), Scalar(1), Scalar(1.5), Scalar(4), Scalar(0.0001), Scalar(1000.5)};
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// location i*6+j corresponds to a_s[i], x_s[j].
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Scalar igamma_s[][6] = {{0.0, nan, nan, nan, nan, nan},
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{0.0, 0.6321205588285578, 0.7768698398515702,
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0.9816843611112658, 9.999500016666262e-05, 1.0},
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{0.0, 0.4275932955291202, 0.608374823728911,
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0.9539882943107686, 7.522076445089201e-07, 1.0},
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{0.0, 0.01898815687615381, 0.06564245437845008,
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0.5665298796332909, 4.166333347221828e-18, 1.0},
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{0.0, 0.9999780593618628, 0.9999899967080838,
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0.9999996219837988, 0.9991370418689945, 1.0},
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{0.0, 0.0, 0.0, 0.0, 0.0, 0.5042041932513908}};
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Scalar igammac_s[][6] = {{nan, nan, nan, nan, nan, nan},
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{1.0, 0.36787944117144233, 0.22313016014842982,
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0.018315638888734182, 0.9999000049998333, 0.0},
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{1.0, 0.5724067044708798, 0.3916251762710878,
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0.04601170568923136, 0.9999992477923555, 0.0},
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{1.0, 0.9810118431238462, 0.9343575456215499,
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0.4334701203667089, 1.0, 0.0},
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{1.0, 2.1940638138146658e-05, 1.0003291916285e-05,
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3.7801620118431334e-07, 0.0008629581310054535,
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0.0},
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{1.0, 1.0, 1.0, 1.0, 1.0, 0.49579580674813944}};
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for (int i = 0; i < 6; ++i) {
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for (int j = 0; j < 6; ++j) {
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if ((std::isnan)(igamma_s[i][j])) {
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VERIFY((std::isnan)(numext::igamma(a_s[i], x_s[j])));
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} else {
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VERIFY_IS_APPROX(numext::igamma(a_s[i], x_s[j]), igamma_s[i][j]);
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}
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if ((std::isnan)(igammac_s[i][j])) {
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VERIFY((std::isnan)(numext::igammac(a_s[i], x_s[j])));
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} else {
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VERIFY_IS_APPROX(numext::igammac(a_s[i], x_s[j]), igammac_s[i][j]);
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}
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}
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}
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}
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}
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#endif // EIGEN_HAS_C99_MATH
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// Check the zeta function against scipy.special.zeta
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{
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ArrayType x(7), q(7), res(7), ref(7);
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x << 1.5, 4, 10.5, 10000.5, 3, 1, 0.9;
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q << 2, 1.5, 3, 1.0001, -2.5, 1.2345, 1.2345;
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ref << 1.61237534869, 0.234848505667, 1.03086757337e-5, 0.367879440865, 0.054102025820864097, plusinf, nan;
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CALL_SUBTEST( verify_component_wise(ref, ref); );
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CALL_SUBTEST( res = x.zeta(q); verify_component_wise(res, ref); );
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CALL_SUBTEST( res = zeta(x,q); verify_component_wise(res, ref); );
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}
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// digamma
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{
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ArrayType x(7), res(7), ref(7);
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x << 1, 1.5, 4, -10.5, 10000.5, 0, -1;
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ref << -0.5772156649015329, 0.03648997397857645, 1.2561176684318, 2.398239129535781, 9.210340372392849, plusinf, plusinf;
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CALL_SUBTEST( verify_component_wise(ref, ref); );
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CALL_SUBTEST( res = x.digamma(); verify_component_wise(res, ref); );
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CALL_SUBTEST( res = digamma(x); verify_component_wise(res, ref); );
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}
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#if EIGEN_HAS_C99_MATH
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{
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ArrayType n(11), x(11), res(11), ref(11);
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n << 1, 1, 1, 1.5, 17, 31, 28, 8, 42, 147, 170;
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x << 2, 3, 25.5, 1.5, 4.7, 11.8, 17.7, 30.2, 15.8, 54.1, 64;
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ref << 0.644934066848, 0.394934066848, 0.0399946696496, nan, 293.334565435, 0.445487887616, -2.47810300902e-07, -8.29668781082e-09, -0.434562276666, 0.567742190178, -0.0108615497927;
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CALL_SUBTEST( verify_component_wise(ref, ref); );
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if(sizeof(RealScalar)>=8) { // double
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// Reason for commented line: http://eigen.tuxfamily.org/bz/show_bug.cgi?id=1232
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// CALL_SUBTEST( res = x.polygamma(n); verify_component_wise(res, ref); );
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CALL_SUBTEST( res = polygamma(n,x); verify_component_wise(res, ref); );
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}
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else {
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// CALL_SUBTEST( res = x.polygamma(n); verify_component_wise(res.head(8), ref.head(8)); );
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CALL_SUBTEST( res = polygamma(n,x); verify_component_wise(res.head(8), ref.head(8)); );
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}
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}
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#endif
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#if EIGEN_HAS_C99_MATH
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{
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// Inputs and ground truth generated with scipy via:
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// a = np.logspace(-3, 3, 5) - 1e-3
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// b = np.logspace(-3, 3, 5) - 1e-3
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// x = np.linspace(-0.1, 1.1, 5)
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// (full_a, full_b, full_x) = np.vectorize(lambda a, b, x: (a, b, x))(*np.ix_(a, b, x))
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// full_a = full_a.flatten().tolist() # same for full_b, full_x
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// v = scipy.special.betainc(full_a, full_b, full_x).flatten().tolist()
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//
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// Note in Eigen, we call betainc with arguments in the order (x, a, b).
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ArrayType a(125);
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ArrayType b(125);
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ArrayType x(125);
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ArrayType v(125);
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ArrayType res(125);
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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,
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0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
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0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
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0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
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0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
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0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
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0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
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0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
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0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
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0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
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0.03062277660168379, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999,
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0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999,
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0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999, 0.999,
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31.62177660168379, 31.62177660168379, 31.62177660168379,
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31.62177660168379, 31.62177660168379, 31.62177660168379,
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31.62177660168379, 31.62177660168379, 31.62177660168379,
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31.62177660168379, 31.62177660168379, 31.62177660168379,
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31.62177660168379, 31.62177660168379, 31.62177660168379,
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31.62177660168379, 31.62177660168379, 31.62177660168379,
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31.62177660168379, 31.62177660168379, 31.62177660168379,
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31.62177660168379, 31.62177660168379, 31.62177660168379,
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31.62177660168379, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999,
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999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999,
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999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999, 999.999,
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999.999, 999.999, 999.999;
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b << 0.0, 0.0, 0.0, 0.0, 0.0, 0.03062277660168379, 0.03062277660168379,
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0.03062277660168379, 0.03062277660168379, 0.03062277660168379, 0.999,
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0.999, 0.999, 0.999, 0.999, 31.62177660168379, 31.62177660168379,
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31.62177660168379, 31.62177660168379, 31.62177660168379, 999.999,
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999.999, 999.999, 999.999, 999.999, 0.0, 0.0, 0.0, 0.0, 0.0,
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0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
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0.03062277660168379, 0.03062277660168379, 0.999, 0.999, 0.999, 0.999,
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0.999, 31.62177660168379, 31.62177660168379, 31.62177660168379,
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31.62177660168379, 31.62177660168379, 999.999, 999.999, 999.999,
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999.999, 999.999, 0.0, 0.0, 0.0, 0.0, 0.0, 0.03062277660168379,
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0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
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0.03062277660168379, 0.999, 0.999, 0.999, 0.999, 0.999,
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31.62177660168379, 31.62177660168379, 31.62177660168379,
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31.62177660168379, 31.62177660168379, 999.999, 999.999, 999.999,
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999.999, 999.999, 0.0, 0.0, 0.0, 0.0, 0.0, 0.03062277660168379,
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0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
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0.03062277660168379, 0.999, 0.999, 0.999, 0.999, 0.999,
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31.62177660168379, 31.62177660168379, 31.62177660168379,
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31.62177660168379, 31.62177660168379, 999.999, 999.999, 999.999,
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999.999, 999.999, 0.0, 0.0, 0.0, 0.0, 0.0, 0.03062277660168379,
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0.03062277660168379, 0.03062277660168379, 0.03062277660168379,
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0.03062277660168379, 0.999, 0.999, 0.999, 0.999, 0.999,
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31.62177660168379, 31.62177660168379, 31.62177660168379,
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31.62177660168379, 31.62177660168379, 999.999, 999.999, 999.999,
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999.999, 999.999;
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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,
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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,
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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,
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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,
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-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,
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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,
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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,
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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,
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0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5, 0.8, 1.1, -0.1, 0.2, 0.5,
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0.8, 1.1;
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v << nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
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nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan, nan,
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nan, nan, nan, 0.47972119876364683, 0.5, 0.5202788012363533, nan, nan,
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0.9518683957740043, 0.9789663010413743, 0.9931729188073435, nan, nan,
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0.999995949033062, 0.9999999999993698, 0.9999999999999999, nan, nan,
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0.9999999999999999, 0.9999999999999999, 0.9999999999999999, nan, nan,
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nan, nan, nan, nan, nan, 0.006827081192655869, 0.0210336989586256,
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0.04813160422599567, nan, nan, 0.20014344256217678, 0.5000000000000001,
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0.7998565574378232, nan, nan, 0.9991401428435834, 0.999999999698403,
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0.9999999999999999, nan, nan, 0.9999999999999999, 0.9999999999999999,
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0.9999999999999999, nan, nan, nan, nan, nan, nan, nan,
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1.0646600232370887e-25, 6.301722877826246e-13, 4.050966937974938e-06,
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nan, nan, 7.864342668429763e-23, 3.015969667594166e-10,
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0.0008598571564165444, nan, nan, 6.031987710123844e-08,
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0.5000000000000007, 0.9999999396801229, nan, nan, 0.9999999999999999,
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0.9999999999999999, 0.9999999999999999, nan, nan, nan, nan, nan, nan,
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nan, 0.0, 7.029920380986636e-306, 2.2450728208591345e-101, nan, nan,
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0.0, 9.275871147869727e-302, 1.2232913026152827e-97, nan, nan, 0.0,
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3.0891393081932924e-252, 2.9303043666183996e-60, nan, nan,
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2.248913486879199e-196, 0.5000000000004947, 0.9999999999999999, nan;
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CALL_SUBTEST(res = betainc(a, b, x);
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verify_component_wise(res, v););
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}
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// Test various properties of betainc
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{
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ArrayType m1 = ArrayType::Random(32);
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ArrayType m2 = ArrayType::Random(32);
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ArrayType m3 = ArrayType::Random(32);
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ArrayType one = ArrayType::Constant(32, Scalar(1.0));
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const Scalar eps = std::numeric_limits<Scalar>::epsilon();
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ArrayType a = (m1 * 4.0).exp();
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ArrayType b = (m2 * 4.0).exp();
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ArrayType x = m3.abs();
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// betainc(a, 1, x) == x**a
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CALL_SUBTEST(
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ArrayType test = betainc(a, one, x);
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ArrayType expected = x.pow(a);
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verify_component_wise(test, expected););
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// 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););
|
|
}
|
|
|
|
// Test Bessel function i0e. Reference results obtained with SciPy.
|
|
{
|
|
ArrayType x(21);
|
|
ArrayType expected(21);
|
|
ArrayType res(21);
|
|
|
|
x << -20.0, -18.0, -16.0, -14.0, -12.0, -10.0, -8.0, -6.0, -4.0, -2.0, 0.0,
|
|
2.0, 4.0, 6.0, 8.0, 10.0, 12.0, 14.0, 16.0, 18.0, 20.0;
|
|
|
|
expected << 0.0897803118848, 0.0947062952128, 0.100544127361,
|
|
0.107615251671, 0.116426221213, 0.127833337163, 0.143431781857,
|
|
0.16665743264, 0.207001921224, 0.308508322554, 1.0, 0.308508322554,
|
|
0.207001921224, 0.16665743264, 0.143431781857, 0.127833337163,
|
|
0.116426221213, 0.107615251671, 0.100544127361, 0.0947062952128,
|
|
0.0897803118848;
|
|
|
|
CALL_SUBTEST(res = i0e(x);
|
|
verify_component_wise(res, expected););
|
|
}
|
|
|
|
// Test Bessel function i1e. Reference results obtained with SciPy.
|
|
{
|
|
ArrayType x(21);
|
|
ArrayType expected(21);
|
|
ArrayType res(21);
|
|
|
|
x << -20.0, -18.0, -16.0, -14.0, -12.0, -10.0, -8.0, -6.0, -4.0, -2.0, 0.0,
|
|
2.0, 4.0, 6.0, 8.0, 10.0, 12.0, 14.0, 16.0, 18.0, 20.0;
|
|
|
|
expected << -0.0875062221833, -0.092036796872, -0.0973496147565,
|
|
-0.103697667463, -0.11146429929, -0.121262681384, -0.134142493293,
|
|
-0.152051459309, -0.178750839502, -0.215269289249, 0.0, 0.215269289249,
|
|
0.178750839502, 0.152051459309, 0.134142493293, 0.121262681384,
|
|
0.11146429929, 0.103697667463, 0.0973496147565, 0.092036796872,
|
|
0.0875062221833;
|
|
|
|
CALL_SUBTEST(res = i1e(x);
|
|
verify_component_wise(res, expected););
|
|
}
|
|
#endif
|
|
}
|
|
|
|
void test_special_functions()
|
|
{
|
|
CALL_SUBTEST_1(array_special_functions<ArrayXf>());
|
|
CALL_SUBTEST_2(array_special_functions<ArrayXd>());
|
|
}
|