test/skew_symmetric_matrix3.cpp

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// 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-FileCopyrightText: The Eigen Authors
// SPDX-License-Identifier: MPL-2.0

#include "main.h"
#include <Eigen/LU>

namespace {
template <typename Scalar>
void constructors() {
  typedef Matrix<Scalar, 3, 1> Vector;
  const Vector v = Vector::Random();
  // l-value
  const SkewSymmetricMatrix3<Scalar> s1(v);
  const Vector& v1 = s1.vector();
  VERIFY_IS_APPROX(v1, v);
  VERIFY(s1.cols() == 3);
  VERIFY(s1.rows() == 3);

  // r-value
  const SkewSymmetricMatrix3<Scalar> s2(std::move(v));
  VERIFY_IS_APPROX(v1, s2.vector());
  VERIFY_IS_APPROX(s1.toDenseMatrix(), s2.toDenseMatrix());

  // from scalars
  SkewSymmetricMatrix3<Scalar> s4(v1(0), v1(1), v1(2));
  VERIFY_IS_APPROX(v1, s4.vector());

  // zero
  const SkewSymmetricMatrix3<Scalar> s5 = SkewSymmetricMatrix3<Scalar>::Zero();
  VERIFY(s5.vector().isZero(0));
  VERIFY(s5.toDenseMatrix().isZero(0));

  // constructors with four vectors do not compile
  // Matrix<Scalar, 4, 1> vector4 = Matrix<Scalar, 4, 1>::Random();
  // SkewSymmetricMatrix3<Scalar> s5(vector4);
}

template <typename Scalar>
void assignments() {
  typedef Matrix<Scalar, 3, 1> Vector;
  typedef Matrix<Scalar, 3, 3> SquareMatrix;

  const Vector v = Vector::Random();

  // assign to square matrix
  SquareMatrix sq;
  sq = v.asSkewSymmetric();
  VERIFY(sq.isSkewSymmetric());

  // assign to skew symmetric matrix
  SkewSymmetricMatrix3<Scalar> sk;
  sk = v.asSkewSymmetric();
  VERIFY_IS_APPROX(v, sk.vector());

  // the vector is read from the destination
  SquareMatrix aliased = SquareMatrix::Random();
  const SquareMatrix expected = Vector(aliased.col(0)).asSkewSymmetric();
  aliased = aliased.col(0).asSkewSymmetric();
  VERIFY_IS_EQUAL(aliased, expected);
}

template <typename Scalar>
void plusMinus() {
  typedef Matrix<Scalar, 3, 1> Vector;
  typedef Matrix<Scalar, 3, 3> SquareMatrix;

  const Vector v1 = Vector::Random();
  const Vector v2 = Vector::Random();

  SquareMatrix sq1;
  sq1 = v1.asSkewSymmetric();
  SquareMatrix sq2;
  sq2 = v2.asSkewSymmetric();

  SkewSymmetricMatrix3<Scalar> sk1;
  sk1 = v1.asSkewSymmetric();
  SkewSymmetricMatrix3<Scalar> sk2;
  sk2 = v2.asSkewSymmetric();

  VERIFY_IS_APPROX((sk1 + sk2).toDenseMatrix(), sq1 + sq2);
  VERIFY_IS_APPROX((sk1 - sk2).toDenseMatrix(), sq1 - sq2);

  SquareMatrix sq3 = v1.asSkewSymmetric();
  VERIFY_IS_APPROX(sq3 = v1.asSkewSymmetric() + v2.asSkewSymmetric(), sq1 + sq2);
  VERIFY_IS_APPROX(sq3 = v1.asSkewSymmetric() - v2.asSkewSymmetric(), sq1 - sq2);
  VERIFY_IS_APPROX(sq3 = v1.asSkewSymmetric() - 2 * v2.asSkewSymmetric() + v1.asSkewSymmetric(), sq1 - 2 * sq2 + sq1);

  VERIFY_IS_APPROX((sk1 + sk1).vector(), 2 * v1);
  VERIFY((sk1 - sk1).vector().isZero());
  VERIFY((sk1 - sk1).toDenseMatrix().isZero());

  // dense ?= skew
  {
    const SquareMatrix A = SquareMatrix::Random();
    SquareMatrix acc = A;
    acc += v1.asSkewSymmetric();
    VERIFY_IS_APPROX(acc, A + sq1);
    acc -= sk2;
    VERIFY_IS_APPROX(acc, A + sq1 - sq2);
    Matrix<Scalar, Dynamic, Dynamic> dyn = A;
    dyn -= sk1;
    VERIFY_IS_APPROX(dyn, A - sq1);
    dyn.resize(2, 2);
    VERIFY_RAISES_ASSERT(dyn += sk1);
  }

  // Mixed dense / skew-symmetric +/- operators (see issue #2925, MR !2308).
  // Cases c1/c2/c3 below are exactly the three expressions from the issue.
  {
    const SquareMatrix A = SquareMatrix::Random();
    const Vector b = Vector::Random();

    const SquareMatrix skA = (A * b).asSkewSymmetric().toDenseMatrix();
    const SquareMatrix skB = v1.asSkewSymmetric().toDenseMatrix();
    const SquareMatrix skB_times_A = v1.asSkewSymmetric().toDenseMatrix() * A;
    const SquareMatrix A_times_skB = A * v1.asSkewSymmetric().toDenseMatrix();

    // c1: dense - skew
    const SquareMatrix c1 = A - SkewSymmetricMatrix3<Scalar>(A * b);
    VERIFY_IS_APPROX(c1, A - skA);

    // c2: Product<dense,skew> - skew
    const SquareMatrix c2 = A * SkewSymmetricMatrix3<Scalar>(v1) - SkewSymmetricMatrix3<Scalar>(A * b);
    VERIFY_IS_APPROX(c2, A_times_skB - skA);

    // c3: Product<skew,dense> - skew
    const SquareMatrix c3 = SkewSymmetricMatrix3<Scalar>(v1) * A - SkewSymmetricMatrix3<Scalar>(A * b);
    VERIFY_IS_APPROX(c3, skB_times_A - skA);

    // M + S and S + M (commutativity check).
    VERIFY_IS_APPROX(SquareMatrix(A + sk1), A + skB);
    VERIFY_IS_APPROX(SquareMatrix(sk1 + A), skB + A);
    VERIFY_IS_APPROX(SquareMatrix(A + sk1), SquareMatrix(sk1 + A));

    // M - S and S - M (anti-commutativity sanity).
    VERIFY_IS_APPROX(SquareMatrix(A - sk1), A - skB);
    VERIFY_IS_APPROX(SquareMatrix(sk1 - A), skB - A);
    VERIFY_IS_APPROX(SquareMatrix(A - sk1), -SquareMatrix(sk1 - A));

    // Product expression on the LHS should still fuse (no explicit
    // .toDenseMatrix() needed on the dense side).
    VERIFY_IS_APPROX(SquareMatrix(A * A - sk1), (A * A).eval() - skB);
  }
}

template <typename Scalar>
void multiplyScale() {
  typedef Matrix<Scalar, 3, 1> Vector;
  typedef Matrix<Scalar, 3, 3> SquareMatrix;

  const Vector v1 = Vector::Random();
  SquareMatrix sq1;
  sq1 = v1.asSkewSymmetric();
  SkewSymmetricMatrix3<Scalar> sk1;
  sk1 = v1.asSkewSymmetric();

  const Scalar s1 = internal::random<Scalar>();
  VERIFY_IS_APPROX(SkewSymmetricMatrix3<Scalar>(sk1 * s1).vector(), sk1.vector() * s1);
  VERIFY_IS_APPROX(SkewSymmetricMatrix3<Scalar>(s1 * sk1).vector(), s1 * sk1.vector());
  VERIFY_IS_APPROX(sq1 * (sk1 * s1), (sq1 * sk1) * s1);

  const Vector v2 = Vector::Random();
  SquareMatrix sq2;
  sq2 = v2.asSkewSymmetric();
  SkewSymmetricMatrix3<Scalar> sk2;
  sk2 = v2.asSkewSymmetric();
  VERIFY_IS_APPROX(sk1 * sk2, sq1 * sq2);

  // null space
  VERIFY((sk1 * v1).isZero());
  VERIFY((sk2 * v2).isZero());
}

template <typename Scalar>
void skewSymmetricDiagonalProducts() {
  using Vector = Matrix<Scalar, 3, 1>;
  using SquareMatrix = Matrix<Scalar, 3, 3>;

  const Vector v = Vector::Random();
  const Vector d = Vector::Random();
  const SkewSymmetricMatrix3<Scalar> sk = v.asSkewSymmetric();
  const DiagonalMatrix<Scalar, 3> dm(d);
  const SquareMatrix skDense = sk.toDenseMatrix();
  const SquareMatrix dmDense = dm.toDenseMatrix();

  SquareMatrix result = sk * dm;
  VERIFY_IS_APPROX(result, skDense * dmDense);
  result = dm * sk;
  VERIFY_IS_APPROX(result, dmDense * skDense);
  result = v.asSkewSymmetric() * d.asDiagonal();
  VERIFY_IS_APPROX(result, skDense * dmDense);
  result = d.asDiagonal() * v.asSkewSymmetric();
  VERIFY_IS_APPROX(result, dmDense * skDense);

  result.setZero();
  result += sk * dm;
  result -= dm * sk;
  VERIFY_IS_APPROX(result, skDense * dmDense - dmDense * skDense);
  VERIFY_IS_APPROX((sk * dm) * v, skDense * (dmDense * v));
}

template <typename Matrix>
void skewSymmetricMultiplication(const Matrix& m) {
  typedef Eigen::Matrix<typename Matrix::Scalar, 3, 1> Vector;
  const Vector v = Vector::Random();
  const Matrix m1 = Matrix::Random(m.rows(), m.cols());
  const SkewSymmetricMatrix3<typename Matrix::Scalar> sk = v.asSkewSymmetric();
  VERIFY_IS_APPROX(m1.transpose() * (sk * m1), (m1.transpose() * sk) * m1);
  VERIFY((m1.transpose() * (sk * m1)).isSkewSymmetric());
}

template <typename Scalar>
void traceAndDet() {
  typedef Matrix<Scalar, 3, 1> Vector;
  const Vector v = Vector::Random();
  // this does not work, values larger than 1.e-08 can be seen
  // VERIFY_IS_APPROX(sq.determinant(), static_cast<Scalar>(0));
  VERIFY_IS_APPROX(v.asSkewSymmetric().determinant(), static_cast<Scalar>(0));
  VERIFY_IS_APPROX(v.asSkewSymmetric().toDenseMatrix().trace(), static_cast<Scalar>(0));
}

template <typename Scalar>
void transpose() {
  typedef Matrix<Scalar, 3, 1> Vector;
  const Vector v = Vector::Random();
  // By definition of a skew symmetric matrix: A^T = -A
  VERIFY_IS_APPROX(v.asSkewSymmetric().toDenseMatrix().transpose(), v.asSkewSymmetric().transpose().toDenseMatrix());
  VERIFY_IS_APPROX(v.asSkewSymmetric().transpose().vector(), (-v).asSkewSymmetric().vector());
}

template <typename Scalar>
void exponentialIdentity() {
  typedef Matrix<Scalar, 3, 1> Vector;
  const Vector v1 = Vector::Zero();
  VERIFY(v1.asSkewSymmetric().exponential().isIdentity());

  Vector v2 = Vector::Random();
  v2.normalize();
  VERIFY((2 * EIGEN_PI * v2).asSkewSymmetric().exponential().isIdentity());

  Vector v3;
  const auto precision = static_cast<Scalar>(1.1) * NumTraits<Scalar>::dummy_precision();
  v3 << 0, 0, precision;
  VERIFY(v3.asSkewSymmetric().exponential().isIdentity(precision));

  // a small rotation is I + K(v) + O(|v|^2), not the identity
  typedef Matrix<Scalar, 3, 3> SquareMatrix;
  const Scalar angle = Scalar(0.5) * NumTraits<Scalar>::dummy_precision();
  const Vector v4(0, 0, angle);
  const SquareMatrix r4 = v4.asSkewSymmetric().exponential();
  VERIFY_IS_APPROX(r4(1, 0), angle);
  VERIFY_IS_APPROX(r4(0, 1), -angle);
}

template <typename Scalar>
void exponentialOrthogonality() {
  typedef Matrix<Scalar, 3, 1> Vector;
  typedef Matrix<Scalar, 3, 3> SquareMatrix;
  const Vector v = Vector::Random();
  SquareMatrix sq = v.asSkewSymmetric().exponential();
  VERIFY(sq.isUnitary());
}

template <typename Scalar>
void exponentialRotation() {
  typedef Matrix<Scalar, 3, 1> Vector;
  typedef Matrix<Scalar, 3, 3> SquareMatrix;

  // rotation axis is invariant
  const Vector v1 = Vector::Random();
  const SquareMatrix r1 = v1.asSkewSymmetric().exponential();
  VERIFY_IS_APPROX(r1 * v1, v1);

  // rotate around z-axis
  Vector v2;
  v2 << 0, 0, Scalar(EIGEN_PI);
  const SquareMatrix r2 = v2.asSkewSymmetric().exponential();
  VERIFY_IS_APPROX(r2 * (Vector() << 1, 0, 0).finished(), (Vector() << -1, 0, 0).finished());
  VERIFY_IS_APPROX(r2 * (Vector() << 0, 1, 0).finished(), (Vector() << 0, -1, 0).finished());
}

}  // namespace

EIGEN_DECLARE_TEST(skew_symmetric_matrix3) {
  for (int i = 0; i < g_repeat; i++) {
    CALL_SUBTEST_1(constructors<float>());
    CALL_SUBTEST_1(constructors<double>());
    CALL_SUBTEST_1(assignments<float>());
    CALL_SUBTEST_1(assignments<double>());

    CALL_SUBTEST_2(plusMinus<float>());
    CALL_SUBTEST_2(plusMinus<double>());
    CALL_SUBTEST_2(multiplyScale<float>());
    CALL_SUBTEST_2(multiplyScale<double>());
    CALL_SUBTEST_2(skewSymmetricMultiplication(MatrixXf(3, internal::random<int>(3, EIGEN_TEST_MAX_SIZE))));
    CALL_SUBTEST_2(skewSymmetricMultiplication(MatrixXd(3, internal::random<int>(3, EIGEN_TEST_MAX_SIZE))));
    CALL_SUBTEST_2(skewSymmetricDiagonalProducts<float>());
    CALL_SUBTEST_2(skewSymmetricDiagonalProducts<double>());
    CALL_SUBTEST_2(traceAndDet<float>());
    CALL_SUBTEST_2(traceAndDet<double>());
    CALL_SUBTEST_2(transpose<float>());
    CALL_SUBTEST_2(transpose<double>());

    CALL_SUBTEST_3(exponentialIdentity<float>());
    CALL_SUBTEST_3(exponentialIdentity<double>());
    CALL_SUBTEST_3(exponentialOrthogonality<float>());
    CALL_SUBTEST_3(exponentialOrthogonality<double>());
    CALL_SUBTEST_3(exponentialRotation<float>());
    CALL_SUBTEST_3(exponentialRotation<double>());
  }
}