eigen/unsupported/test/special_functions.cpp

// 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  // EIGEN_HAS_C99_MATH

  // 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););
  }

  /* Code to generate the data for the following two test cases.
    N = 5
    np.random.seed(3)

    a = np.logspace(-2, 3, 6)
    a = np.ravel(np.tile(np.reshape(a, [-1, 1]), [1, N]))
    x = np.random.gamma(a, 1.0)
    x = np.maximum(x, np.finfo(np.float32).tiny)

    def igamma(a, x):
      return mpmath.gammainc(a, 0, x, regularized=True)

    def igamma_der_a(a, x):
      res = mpmath.diff(lambda a_prime: igamma(a_prime, x), a)
      return np.float64(res)

    def gamma_sample_der_alpha(a, x):
      igamma_x = igamma(a, x)
      def igammainv_of_igamma(a_prime):
        return mpmath.findroot(lambda x_prime: igamma(a_prime, x_prime) -
            igamma_x, x, solver='newton')
      return np.float64(mpmath.diff(igammainv_of_igamma, a))

    v_igamma_der_a = np.vectorize(igamma_der_a)(a, x)
    v_gamma_sample_der_alpha = np.vectorize(gamma_sample_der_alpha)(a, x)
  */

#if EIGEN_HAS_C99_MATH
  // Test igamma_der_a
  {
    ArrayType a(30);
    ArrayType x(30);
    ArrayType res(30);
    ArrayType v(30);

    a << 0.01, 0.01, 0.01, 0.01, 0.01, 0.1, 0.1, 0.1, 0.1, 0.1, 1.0, 1.0, 1.0,
        1.0, 1.0, 10.0, 10.0, 10.0, 10.0, 10.0, 100.0, 100.0, 100.0, 100.0,
        100.0, 1000.0, 1000.0, 1000.0, 1000.0, 1000.0;

    x << 1.25668890405e-26, 1.17549435082e-38, 1.20938905072e-05,
        1.17549435082e-38, 1.17549435082e-38, 5.66572070696e-16,
        0.0132865061065, 0.0200034203853, 6.29263709118e-17, 1.37160367764e-06,
        0.333412038288, 1.18135687766, 0.580629033777, 0.170631439426,
        0.786686768458, 7.63873279537, 13.1944344379, 11.896042354,
        10.5830172417, 10.5020942233, 92.8918587747, 95.003720371,
        86.3715926467, 96.0330217672, 82.6389930677, 968.702906754,
        969.463546828, 1001.79726022, 955.047416547, 1044.27458568;

    v << -32.7256441441, -36.4394150514, -9.66467612263, -36.4394150514,
        -36.4394150514, -1.0891900302, -2.66351229645, -2.48666868596,
        -0.929700494428, -3.56327722764, -0.455320135314, -0.391437214323,
        -0.491352055991, -0.350454834292, -0.471773162921, -0.104084440522,
        -0.0723646747909, -0.0992828975532, -0.121638215446, -0.122619605294,
        -0.0317670267286, -0.0359974812869, -0.0154359225363, -0.0375775365921,
        -0.00794899153653, -0.00777303219211, -0.00796085782042,
        -0.0125850719397, -0.00455500206958, -0.00476436993148;

    CALL_SUBTEST(res = igamma_der_a(a, x); verify_component_wise(res, v););
  }

  // Test gamma_sample_der_alpha
  {
    ArrayType alpha(30);
    ArrayType sample(30);
    ArrayType res(30);
    ArrayType v(30);

    alpha << 0.01, 0.01, 0.01, 0.01, 0.01, 0.1, 0.1, 0.1, 0.1, 0.1, 1.0, 1.0,
        1.0, 1.0, 1.0, 10.0, 10.0, 10.0, 10.0, 10.0, 100.0, 100.0, 100.0, 100.0,
        100.0, 1000.0, 1000.0, 1000.0, 1000.0, 1000.0;

    sample << 1.25668890405e-26, 1.17549435082e-38, 1.20938905072e-05,
        1.17549435082e-38, 1.17549435082e-38, 5.66572070696e-16,
        0.0132865061065, 0.0200034203853, 6.29263709118e-17, 1.37160367764e-06,
        0.333412038288, 1.18135687766, 0.580629033777, 0.170631439426,
        0.786686768458, 7.63873279537, 13.1944344379, 11.896042354,
        10.5830172417, 10.5020942233, 92.8918587747, 95.003720371,
        86.3715926467, 96.0330217672, 82.6389930677, 968.702906754,
        969.463546828, 1001.79726022, 955.047416547, 1044.27458568;

    v << 7.42424742367e-23, 1.02004297287e-34, 0.0130155240738,
        1.02004297287e-34, 1.02004297287e-34, 1.96505168277e-13, 0.525575786243,
        0.713903991771, 2.32077561808e-14, 0.000179348049886, 0.635500453302,
        1.27561284917, 0.878125852156, 0.41565819538, 1.03606488534,
        0.885964824887, 1.16424049334, 1.10764479598, 1.04590810812,
        1.04193666963, 0.965193152414, 0.976217589464, 0.93008035061,
        0.98153216096, 0.909196397698, 0.98434963993, 0.984738050206,
        1.00106492525, 0.97734200649, 1.02198794179;

    CALL_SUBTEST(res = gamma_sample_der_alpha(alpha, sample);
                 verify_component_wise(res, v););
  }
#endif  // EIGEN_HAS_C99_MATH
}

EIGEN_DECLARE_TEST(special_functions)
{
  CALL_SUBTEST_1(array_special_functions<ArrayXf>());
  CALL_SUBTEST_2(array_special_functions<ArrayXd>());
}