contrib/test/autodiff.cpp

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009 Gael Guennebaud <g.gael@free.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/.
// SPDX-License-Identifier: MPL-2.0

#include "main.h"
#include <Eigen/Jacobi>
#include <Eigen/QR>
#include <Eigen/SVD>
#include <contrib/Eigen/AutoDiff>
#include <contrib/Eigen/Splines>

template <typename Scalar>
EIGEN_DONT_INLINE Scalar foo(const Scalar& x, const Scalar& y) {
  using namespace std;
  //   return x+std::sin(y);
  EIGEN_ASM_COMMENT("mybegin");
  // pow(float, int) promotes to pow(double, double)
  return x * 2 - 1 + static_cast<Scalar>(pow(1 + x, 2)) + 2 * sqrt(y * y + 0) - 4 * sin(0 + x) + 2 * cos(y + 0) -
         exp(Scalar(-0.5) * x * x + 0);
  // return x+2*y*x;//x*2 -std::pow(x,2);//(2*y/x);// - y*2;
  EIGEN_ASM_COMMENT("myend");
}

template <typename Vector>
EIGEN_DONT_INLINE typename Vector::Scalar foo(const Vector& p) {
  typedef typename Vector::Scalar Scalar;
  return (p - Vector(Scalar(-1), Scalar(1.))).norm() + (p.array() * p.array()).sum() + p.dot(p);
}

template <typename Scalar_, int NX = Dynamic, int NY = Dynamic>
struct TestFunc1 {
  typedef Scalar_ Scalar;
  enum { InputsAtCompileTime = NX, ValuesAtCompileTime = NY };
  typedef Matrix<Scalar, InputsAtCompileTime, 1> InputType;
  typedef Matrix<Scalar, ValuesAtCompileTime, 1> ValueType;
  typedef Matrix<Scalar, ValuesAtCompileTime, InputsAtCompileTime> JacobianType;

  int m_inputs, m_values;

  TestFunc1() : m_inputs(InputsAtCompileTime), m_values(ValuesAtCompileTime) {}
  TestFunc1(int inputs_, int values_) : m_inputs(inputs_), m_values(values_) {}

  int inputs() const { return InputsAtCompileTime == Dynamic ? m_inputs : InputsAtCompileTime; }
  int values() const { return ValuesAtCompileTime == Dynamic ? m_values : ValuesAtCompileTime; }

  template <typename T>
  void operator()(const Matrix<T, InputsAtCompileTime, 1>& x, Matrix<T, ValuesAtCompileTime, 1>* _v) const {
    Matrix<T, ValuesAtCompileTime, 1>& v = *_v;

    v[0] = 2 * x[0] * x[0] + x[0] * x[1];
    v[1] = 3 * x[1] * x[0] + 0.5 * x[1] * x[1];
    if (inputs() > 2) {
      v[0] += 0.5 * x[2];
      v[1] += x[2];
    }
    if (values() > 2) {
      v[2] = 3 * x[1] * x[0] * x[0];
    }
    if (inputs() > 2 && values() > 2) v[2] *= x[2];
  }

  void operator()(const InputType& x, ValueType* v, JacobianType* _j) const {
    (*this)(x, v);

    if (_j) {
      JacobianType& j = *_j;

      j(0, 0) = 4 * x[0] + x[1];
      j(1, 0) = 3 * x[1];

      j(0, 1) = x[0];
      j(1, 1) = 3 * x[0] + 2 * 0.5 * x[1];

      if (inputs() > 2) {
        j(0, 2) = 0.5;
        j(1, 2) = 1;
      }
      if (values() > 2) {
        j(2, 0) = 3 * x[1] * 2 * x[0];
        j(2, 1) = 3 * x[0] * x[0];
      }
      if (inputs() > 2 && values() > 2) {
        j(2, 0) *= x[2];
        j(2, 1) *= x[2];

        j(2, 2) = 3 * x[1] * x[0] * x[0];
      }
    }
  }
};

template <typename Scalar>
struct integratorFunctor {
  typedef Matrix<Scalar, 2, 1> InputType;
  typedef Matrix<Scalar, 2, 1> ValueType;

  /*
   * Implementation starts here.
   */
  integratorFunctor(const Scalar gain) : _gain(gain) {}
  integratorFunctor(const integratorFunctor& f) : _gain(f._gain) {}
  const Scalar _gain;

  template <typename T1, typename T2>
  void operator()(const T1& input, T2* output, const Scalar dt) const {
    T2& o = *output;

    /* Integrator to test the AD. */
    o[0] = input[0] + input[1] * dt * _gain;
    o[1] = input[1] * _gain;
  }

  /* Only needed for the test */
  template <typename T1, typename T2, typename T3>
  void operator()(const T1& input, T2* output, T3* jacobian, const Scalar dt) const {
    T2& o = *output;

    /* Integrator to test the AD. */
    o[0] = input[0] + input[1] * dt * _gain;
    o[1] = input[1] * _gain;

    if (jacobian) {
      T3& j = *jacobian;

      j(0, 0) = 1;
      j(0, 1) = dt * _gain;
      j(1, 0) = 0;
      j(1, 1) = _gain;
    }
  }
};

template <typename Func>
void forward_jacobian_with_scalar_arg(const Func& f) {
  typedef typename Func::ValueType::Scalar Scalar;
  typedef typename Func::ValueType ValueType;
  typedef typename Func::InputType InputType;
  typedef typename AutoDiffJacobian<Func>::JacobianType JacobianType;

  InputType x = InputType::Random(InputType::RowsAtCompileTime);
  ValueType y, yref;
  JacobianType j, jref;

  const Scalar dt = internal::random<double>();

  jref.setZero();
  yref.setZero();
  f(x, &yref, &jref, dt);

  // std::cerr << "y, yref, jref: " << "\n";
  // std::cerr << y.transpose() << "\n\n";
  // std::cerr << yref << "\n\n";
  // std::cerr << jref << "\n\n";

  AutoDiffJacobian<Func> autoj(f);
  autoj(x, &y, &j, dt);

  // std::cerr << "y j (via autodiff): " << "\n";
  // std::cerr << y.transpose() << "\n\n";
  // std::cerr << j << "\n\n";

  VERIFY_IS_APPROX(y, yref);
  VERIFY_IS_APPROX(j, jref);
}

template <typename Func>
void forward_jacobian(const Func& f) {
  typename Func::InputType x = Func::InputType::Random(f.inputs());
  typename Func::ValueType y(f.values()), yref(f.values());
  typename Func::JacobianType j(f.values(), f.inputs()), jref(f.values(), f.inputs());

  jref.setZero();
  yref.setZero();
  f(x, &yref, &jref);

  j.setZero();
  y.setZero();
  AutoDiffJacobian<Func> autoj(f);
  autoj(x, &y, &j);

  VERIFY_IS_APPROX(y, yref);
  VERIFY_IS_APPROX(j, jref);
}

// TODO also check actual derivatives!
template <int>
void test_autodiff_scalar() {
  Vector2f p = Vector2f::Random();
  typedef AutoDiffScalar<Vector2f> AD;
  AD ax(p.x(), Vector2f::UnitX());
  AD ay(p.y(), Vector2f::UnitY());
  AD res = foo<AD>(ax, ay);
  VERIFY_IS_APPROX(res.value(), foo(p.x(), p.y()));
}

template <int>
void test_autodiff_vector() {
  Vector2f p = Vector2f::Random();
  typedef AutoDiffScalar<Vector2f> AD;
  typedef Matrix<AD, 2, 1> VectorAD;
  VectorAD ap = p.cast<AD>();
  ap.x().derivatives() = Vector2f::UnitX();
  ap.y().derivatives() = Vector2f::UnitY();

  AD res = foo<VectorAD>(ap);
  VERIFY_IS_APPROX(res.value(), foo(p));
}

// foo(p) = |p - (-1, 1)| + 2 |p|^2, so grad foo(p) = (p - (-1, 1)) / |p - (-1, 1)| + 4 p.
inline void check_autodiff_vector_derivative(const Vector2f& p, float expected_value,
                                             const Vector2f& expected_gradient) {
  using AD = AutoDiffScalar<Vector2f>;
  using VectorAD = Matrix<AD, 2, 1>;
  VectorAD ap = p.cast<AD>();
  ap.x().derivatives() = Vector2f::UnitX();
  ap.y().derivatives() = Vector2f::UnitY();
  const AD res = foo<VectorAD>(ap);
  VERIFY_IS_APPROX(res.value(), expected_value);
  VERIFY_IS_APPROX(res.derivatives(), expected_gradient);
}

template <int>
void test_autodiff_vector_derivative() {
  Vector2f p = Vector2f::Random();
  // Stay away from the kink of the norm term at (-1, 1).
  if (numext::abs(p.x() + 1.0f) < 0.01f) p.x() = -0.9f;
  const float x = p.x(), y = p.y();
  const float norm = std::sqrt((x + 1.0f) * (x + 1.0f) + (y - 1.0f) * (y - 1.0f));
  check_autodiff_vector_derivative(p, norm + 2.0f * (x * x + y * y),
                                   Vector2f((x + 1.0f) / norm + 4.0f * x, (y - 1.0f) / norm + 4.0f * y));
}

template <int>
void test_autodiff_vector_derivative_specific_values() {
  check_autodiff_vector_derivative(Vector2f(2.0f, 3.0f), 29.605551f, Vector2f(8.832050f, 12.554700f));
  check_autodiff_vector_derivative(Vector2f(-3.4f, 7.2f), 133.448308f, Vector2f(-13.960994f, 29.732568f));
}

template <int>
void test_autodiff_jacobian() {
  CALL_SUBTEST((forward_jacobian(TestFunc1<double, 2, 2>())));
  CALL_SUBTEST((forward_jacobian(TestFunc1<double, 2, 3>())));
  CALL_SUBTEST((forward_jacobian(TestFunc1<double, 3, 2>())));
  CALL_SUBTEST((forward_jacobian(TestFunc1<double, 3, 3>())));
  CALL_SUBTEST((forward_jacobian(TestFunc1<double>(2, 2))));
  CALL_SUBTEST((forward_jacobian(TestFunc1<double>(2, 3))));
  CALL_SUBTEST((forward_jacobian(TestFunc1<double>(3, 2))));
  CALL_SUBTEST((forward_jacobian(TestFunc1<double>(3, 3))));
  CALL_SUBTEST((forward_jacobian_with_scalar_arg(integratorFunctor<double>(10))));
}

template <int>
void test_autodiff_hessian() {
  typedef AutoDiffScalar<VectorXd> AD;
  typedef Matrix<AD, Eigen::Dynamic, 1> VectorAD;
  typedef AutoDiffScalar<VectorAD> ADD;
  typedef Matrix<ADD, Eigen::Dynamic, 1> VectorADD;
  VectorADD x(2);
  double s1 = internal::random<double>(), s2 = internal::random<double>(), s3 = internal::random<double>(),
         s4 = internal::random<double>();
  x(0).value() = s1;
  x(1).value() = s2;

  // set unit vectors for the derivative directions (partial derivatives of the input vector)
  x(0).derivatives().resize(2);
  x(0).derivatives().setZero();
  x(0).derivatives()(0) = 1;
  x(1).derivatives().resize(2);
  x(1).derivatives().setZero();
  x(1).derivatives()(1) = 1;

  // repeat partial derivatives for the inner AutoDiffScalar
  x(0).value().derivatives() = VectorXd::Unit(2, 0);
  x(1).value().derivatives() = VectorXd::Unit(2, 1);

  // set the hessian matrix to zero
  for (int idx = 0; idx < 2; idx++) {
    x(0).derivatives()(idx).derivatives() = VectorXd::Zero(2);
    x(1).derivatives()(idx).derivatives() = VectorXd::Zero(2);
  }

  ADD y = sin(AD(s3) * x(0) + AD(s4) * x(1));

  VERIFY_IS_APPROX(y.value().derivatives()(0), y.derivatives()(0).value());
  VERIFY_IS_APPROX(y.value().derivatives()(1), y.derivatives()(1).value());
  VERIFY_IS_APPROX(y.value().derivatives()(0), s3 * std::cos(s1 * s3 + s2 * s4));
  VERIFY_IS_APPROX(y.value().derivatives()(1), s4 * std::cos(s1 * s3 + s2 * s4));
  VERIFY_IS_APPROX(y.derivatives()(0).derivatives(), -std::sin(s1 * s3 + s2 * s4) * Vector2d(s3 * s3, s4 * s3));
  VERIFY_IS_APPROX(y.derivatives()(1).derivatives(), -std::sin(s1 * s3 + s2 * s4) * Vector2d(s3 * s4, s4 * s4));

  ADD z = x(0) * x(1);
  VERIFY_IS_APPROX(z.derivatives()(0).derivatives(), Vector2d(0, 1));
  VERIFY_IS_APPROX(z.derivatives()(1).derivatives(), Vector2d(1, 0));
}

double bug_1222() {
  typedef Eigen::AutoDiffScalar<Eigen::Vector3d> AD;
  const double _cv1_3 = 1.0;
  const AD chi_3 = 1.0;
  // this line did not work, because operator+ returns ADS<DerType&>, which then cannot be converted to ADS<DerType>
  const AD denom = chi_3 + _cv1_3;
  return denom.value();
}

double bug_1223() {
  using std::min;
  typedef Eigen::AutoDiffScalar<Eigen::Vector3d> AD;

  const double _cv1_3 = 1.0;
  const AD chi_3 = 1.0;
  const AD denom = 1.0;

// failed because implementation of min attempts to construct ADS<DerType&> via constructor AutoDiffScalar(const Real&
// value) without initializing m_derivatives (which is a reference in this case)
#define EIGEN_TEST_SPACE
  const AD t = min EIGEN_TEST_SPACE(denom / chi_3, 1.0);

  const AD t2 = min EIGEN_TEST_SPACE(denom / (chi_3 * _cv1_3), 1.0);

  return t.value() + t2.value();
}

// regression test for some compilation issues with specializations of ScalarBinaryOpTraits
void bug_1260() {
  Matrix4d A = Matrix4d::Ones();
  Vector4d v = Vector4d::Ones();
  A* v;
}

// check a compilation issue with numext::max
double bug_1261() {
  typedef AutoDiffScalar<Matrix2d> AD;
  typedef Matrix<AD, 2, 1> VectorAD;

  VectorAD v(0., 0.);
  const AD maxVal = v.maxCoeff();
  const AD minVal = v.minCoeff();
  return maxVal.value() + minVal.value();
}

double bug_1264() {
  typedef AutoDiffScalar<Vector2d> AD;
  const AD s = 0.;
  const Matrix<AD, 3, 1> v1(0., 0., 0.);
  const Matrix<AD, 3, 1> v2 = (s + 3.0) * v1;
  return v2(0).value();
}

// check with expressions on constants
double bug_1281() {
  int n = 2;
  typedef AutoDiffScalar<VectorXd> AD;
  const AD c = 1.;
  AD x0(2, n, 0);
  AD y1 = (AD(c) + AD(c)) * x0;
  y1 = x0 * (AD(c) + AD(c));
  AD y2 = (-AD(c)) + x0;
  y2 = x0 + (-AD(c));
  AD y3 = (AD(c) * (-AD(c)) + AD(c)) * x0;
  y3 = x0 * (AD(c) * (-AD(c)) + AD(c));
  return (y1 + y2 + y3).value();
}

// atan2 must pad the empty derivatives of a constant, as the arithmetic operators do.
void test_autodiff_atan2_constant() {
  typedef AutoDiffScalar<VectorXd> AD;
  const AD x(1.0, 2, 0), c(2.0);
  VERIFY_IS_APPROX(atan2(x, c).derivatives(), Vector2d(0.4, 0));
  VERIFY_IS_APPROX(atan2(c, x).derivatives(), Vector2d(-0.4, 0));
}

// The zero-padded derivatives of a sum must read column 0 of a non-plain derivative expression.
void test_autodiff_padded_block_derivatives() {
  using ColMajorMatrixXd = Matrix<double, Dynamic, Dynamic, ColMajor>;
  ColMajorMatrixXd store(3, 2);
  store << 1, 10, 2, 20, 3, 30;
  AutoDiffScalar<Block<ColMajorMatrixXd, Dynamic, 1, true>> a(1.0, store.col(0));
  AutoDiffScalar<VectorXd> b(2.0);
  MatrixXd jac = MatrixXd::Zero(3, 2);
  jac.topRows(3).leftCols(1) = (a + b).derivatives();
  VERIFY_IS_EQUAL(jac.col(0), store.col(0));
}

// LLT stores the 1-norm of its input for rcond(); the sum of AutoDiffScalars is an expression, so
// the norm must materialize its column totals before comparing them.
void test_autodiff_selfadjoint_l1norm() {
  typedef AutoDiffScalar<Vector2d> AD;
  Matrix<AD, 8, 8> m = Matrix<AD, 8, 8>::Identity();
  m(3, 3) = AD(2, Vector2d(1, 0));
  AD norm = m.selfadjointView<Lower>().l1Norm();
  VERIFY_IS_EQUAL(norm.value(), 2.0);
  VERIFY_IS_EQUAL(norm.derivatives()(0), 1.0);
  LLT<Matrix<AD, 8, 8>> llt(m);
  VERIFY(llt.info() == Success);
}

template <typename Derivatives>
void test_autodiff_makegivens() {
  using AD = AutoDiffScalar<Derivatives>;
  // Exercise both ratio branches and compare analytical first derivatives.
  for (int swap = 0; swap < 2; ++swap) {
    const double x = swap ? 4.0 : 3.0;
    const double y = swap ? 3.0 : 4.0;
    const AD p(x, Derivatives(Vector2d::UnitX()));
    const AD q(y, Derivatives(Vector2d::UnitY()));
    AD r;
    JacobiRotation<AD> rotation;
    rotation.makeGivens(p, q, &r);
    VERIFY_IS_APPROX(rotation.c().value(), x / 5.0);
    VERIFY_IS_APPROX(rotation.s().value(), -y / 5.0);
    VERIFY_IS_APPROX(r.value(), 5.0);
    VERIFY_IS_APPROX(r.derivatives(), Vector2d(x / 5.0, y / 5.0));
    VERIFY_IS_APPROX(rotation.c().derivatives(), Vector2d(y * y / 125.0, -x * y / 125.0));
    VERIFY_IS_APPROX(rotation.s().derivatives(), Vector2d(x * y / 125.0, -x * x / 125.0));
    JacobiRotation<AD> without_norm;
    without_norm.makeGivens(p, q);
    VERIFY_IS_APPROX(without_norm.c().derivatives(), rotation.c().derivatives());
    VERIFY_IS_APPROX(without_norm.s().derivatives(), rotation.s().derivatives());
  }
}

// Householder and Jacobi kernels must materialize AutoDiffScalar expressions before numext::sqrt and need
// isnan/isinf overloads.
template <typename Derivatives>
void test_autodiff_householder_qr() {
  using AD = AutoDiffScalar<Derivatives>;
  Matrix<AD, 2, 2> a;
  for (int k = 0; k < 4; ++k) a(k / 2, k % 2) = AD(double(k + 1), Derivatives(Vector4d::Unit(k)));
  HouseholderQR<Matrix<AD, 2, 2>> qr(a);
  const AD r00 = qr.matrixQR()(0, 0);
  const double n = std::sqrt(10.0);
  VERIFY_IS_APPROX(numext::abs(r00.value()), n);
  VERIFY_IS_APPROX(r00.derivatives(), Vector4d(1.0 / n, 0, 3.0 / n, 0) * numext::sign(r00.value()));
  const Matrix<AD, 2, 2> q = qr.householderQ();
  const Matrix<AD, 2, 2> r = qr.matrixQR().template triangularView<Upper>();
  const Matrix<AD, 2, 2> qra = q * r;
  for (int k = 0; k < 4; ++k) {
    VERIFY_IS_APPROX(qra(k / 2, k % 2).value(), a(k / 2, k % 2).value());
    VERIFY_IS_APPROX(qra(k / 2, k % 2).derivatives(), Derivatives(Vector4d::Unit(k)));
  }
}

template <typename Derivatives>
void test_autodiff_jacobi_svd() {
  using AD = AutoDiffScalar<Derivatives>;
  Matrix<AD, 2, 2> a;
  for (int k = 0; k < 4; ++k) a(k / 2, k % 2) = AD(double(k + 1), Derivatives(Vector4d::Unit(k)));
  JacobiSVD<Matrix<AD, 2, 2>, ComputeFullU | ComputeFullV> svd(a);
  const Matrix<AD, 2, 2> usv = svd.matrixU() * svd.singularValues().asDiagonal() * svd.matrixV().transpose();
  for (int k = 0; k < 4; ++k) {
    VERIFY_IS_APPROX(usv(k / 2, k % 2).value(), a(k / 2, k % 2).value());
    VERIFY_IS_APPROX(usv(k / 2, k % 2).derivatives(), Derivatives(Vector4d::Unit(k)));
  }
}

// SplineFitting::Interpolate solves its collocation system with HouseholderQR.
template <typename Derivatives>
void test_autodiff_spline_interpolate() {
  using AD = AutoDiffScalar<Derivatives>;
  using Spline1 = Spline<AD, 1>;
  const Vector4d values(1, 2, 5, 10);
  Matrix<AD, 1, 4> points;
  for (int k = 0; k < 4; ++k) points(k) = AD(values(k), Derivatives(Vector4d::Unit(k)));
  Matrix<AD, 1, 4> parameters;
  parameters << AD(0.0), AD(1.0 / 3), AD(2.0 / 3), AD(1.0);
  const Spline1 spline = SplineFitting<Spline1>::Interpolate(points, 2, parameters);
  for (int k = 0; k < 4; ++k) {
    const AD y = spline(parameters(k))(0);
    VERIFY_IS_APPROX(y.value(), values(k));
    VERIFY_IS_APPROX(y.derivatives(), Derivatives(Vector4d::Unit(k)));
  }
  // With fixed parameters the interpolant is linear in the points: y(u) = sum_k (dy/dp_k) p_k.
  const AD y = spline(AD(0.5))(0);
  VERIFY_IS_APPROX(y.value(), Vector4d(y.derivatives()).dot(values));
  VERIFY_IS_APPROX(Vector4d(y.derivatives()).sum(), 1.0);

  // Chord-length parameters depend on the points; the total derivative of y(u_k(p), p) is still e_k.
  Matrix<AD, 2, 4> points2;
  for (int k = 0; k < 4; ++k) {
    points2(0, k) = AD(double(k), Derivatives(Vector4d::Zero()));
    points2(1, k) = points(k);
  }
  using Spline2 = Spline<AD, 2>;
  typename Spline2::KnotVectorType chordLengths;
  ChordLengths(points2, chordLengths);
  const Spline2 spline2 = SplineFitting<Spline2>::Interpolate(points2, 2, chordLengths);
  for (int k = 0; k < 4; ++k) {
    const AD y2 = spline2(chordLengths(k))(1);
    VERIFY_IS_APPROX(y2.value(), values(k));
    VERIFY_IS_APPROX(y2.derivatives(), Derivatives(Vector4d::Unit(k)));
  }
}

EIGEN_DECLARE_TEST(autodiff) {
  for (int i = 0; i < g_repeat; i++) {
    CALL_SUBTEST_1(test_autodiff_scalar<1>());
    CALL_SUBTEST_2(test_autodiff_vector<1>());
    CALL_SUBTEST_2(test_autodiff_vector_derivative<1>());
    CALL_SUBTEST_3(test_autodiff_jacobian<1>());
    CALL_SUBTEST_4(test_autodiff_hessian<1>());
  }
  CALL_SUBTEST_2(test_autodiff_vector_derivative_specific_values<1>());

  CALL_SUBTEST_5(bug_1222());
  CALL_SUBTEST_5(bug_1223());
  CALL_SUBTEST_5(bug_1260());
  CALL_SUBTEST_5(bug_1261());
  CALL_SUBTEST_5(bug_1281());
  CALL_SUBTEST_5(test_autodiff_selfadjoint_l1norm());
  CALL_SUBTEST_5(test_autodiff_padded_block_derivatives());
  CALL_SUBTEST_5(test_autodiff_atan2_constant());
  CALL_SUBTEST_6(test_autodiff_makegivens<Vector2d>());
  CALL_SUBTEST_6(test_autodiff_makegivens<VectorXd>());
  CALL_SUBTEST_7(test_autodiff_householder_qr<Vector4d>());
  CALL_SUBTEST_7(test_autodiff_householder_qr<VectorXd>());
  CALL_SUBTEST_7(test_autodiff_jacobi_svd<Vector4d>());
  CALL_SUBTEST_7(test_autodiff_jacobi_svd<VectorXd>());
  CALL_SUBTEST_7(test_autodiff_spline_interpolate<Vector4d>());
  CALL_SUBTEST_7(test_autodiff_spline_interpolate<VectorXd>());
}