test/lu.cpp

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2009 Benoit Jacob <jacob.benoit.1@gmail.com>
//
// 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/LU>
#include "solverbase.h"
using namespace std;

template <typename MatrixType>
typename MatrixType::RealScalar matrix_l1_norm(const MatrixType& m) {
  return m.cwiseAbs().colwise().sum().maxCoeff();
}

template <typename MatrixType>
void lu_non_invertible() {
  typedef typename MatrixType::RealScalar RealScalar;
  /* this test covers the following files:
     LU.h
  */
  Index rows, cols, cols2;
  if (MatrixType::RowsAtCompileTime == Dynamic) {
    rows = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
  } else {
    rows = MatrixType::RowsAtCompileTime;
  }
  if (MatrixType::ColsAtCompileTime == Dynamic) {
    cols = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
    cols2 = internal::random<int>(2, EIGEN_TEST_MAX_SIZE);
  } else {
    cols2 = cols = MatrixType::ColsAtCompileTime;
  }

  enum { RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime };
  typedef typename internal::kernel_retval_base<FullPivLU<MatrixType> >::ReturnType KernelMatrixType;
  typedef typename internal::image_retval_base<FullPivLU<MatrixType> >::ReturnType ImageMatrixType;
  typedef Matrix<typename MatrixType::Scalar, ColsAtCompileTime, ColsAtCompileTime> CMatrixType;
  typedef Matrix<typename MatrixType::Scalar, RowsAtCompileTime, RowsAtCompileTime> RMatrixType;

  Index rank = internal::random<Index>(1, (std::min)(rows, cols) - 1);

  // The image of the zero matrix should consist of a single (zero) column vector
  VERIFY((MatrixType::Zero(rows, cols).fullPivLu().image(MatrixType::Zero(rows, cols)).cols() == 1));

  // The kernel of the zero matrix is the entire space, and thus is an invertible matrix of dimensions cols.
  KernelMatrixType kernel = MatrixType::Zero(rows, cols).fullPivLu().kernel();
  VERIFY((kernel.fullPivLu().isInvertible()));

  MatrixType m1(rows, cols), m3(rows, cols2);
  CMatrixType m2(cols, cols2);
  createRandomPIMatrixOfRank(rank, rows, cols, m1);

  FullPivLU<MatrixType> lu;

  // The special value 0.01 below works well in tests. Keep in mind that we're only computing the rank
  // of singular values are either 0 or 1.
  // So it's not clear at all that the epsilon should play any role there.
  lu.setThreshold(RealScalar(0.01));
  lu.compute(m1);

  MatrixType u(rows, cols);
  u = lu.matrixLU().template triangularView<Upper>();
  RMatrixType l = RMatrixType::Identity(rows, rows);
  l.block(0, 0, rows, (std::min)(rows, cols)).template triangularView<StrictlyLower>() =
      lu.matrixLU().block(0, 0, rows, (std::min)(rows, cols));

  VERIFY_IS_APPROX(lu.permutationP() * m1 * lu.permutationQ(), l * u);

  KernelMatrixType m1kernel = lu.kernel();
  ImageMatrixType m1image = lu.image(m1);

  VERIFY_IS_APPROX(m1, lu.reconstructedMatrix());
  VERIFY(rank == lu.rank());
  VERIFY(cols - lu.rank() == lu.dimensionOfKernel());
  VERIFY(!lu.isInjective());
  VERIFY(!lu.isInvertible());
  VERIFY(!lu.isSurjective());
  VERIFY_IS_MUCH_SMALLER_THAN((m1 * m1kernel), m1);
  VERIFY(m1image.fullPivLu().rank() == rank);
  VERIFY_IS_APPROX(m1 * m1.adjoint() * m1image, m1image);

  check_solverbase<CMatrixType, MatrixType>(m1, lu, rows, cols, cols2);

  m2 = CMatrixType::Random(cols, cols2);
  m3 = m1 * m2;
  m2 = CMatrixType::Random(cols, cols2);
  // test that the code, which does resize(), may be applied to an xpr
  m2.block(0, 0, m2.rows(), m2.cols()) = lu.solve(m3);
  VERIFY_IS_APPROX(m3, m1 * m2);
}

template <typename MatrixType>
void lu_invertible() {
  /* this test covers the following files:
     FullPivLU.h
  */
  typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
  Index size = MatrixType::RowsAtCompileTime;
  if (size == Dynamic) size = internal::random<Index>(1, EIGEN_TEST_MAX_SIZE);

  MatrixType m1(size, size), m2(size, size), m3(size, size);
  FullPivLU<MatrixType> lu;
  lu.setThreshold(RealScalar(0.01));
  // Create a random diagonally dominant (thus invertible) matrix.
  m1 = MatrixType::Random(size, size);
  m1.diagonal().array() += RealScalar(2 * size);
  lu.compute(m1);

  VERIFY_IS_APPROX(m1, lu.reconstructedMatrix());
  VERIFY(0 == lu.dimensionOfKernel());
  VERIFY(lu.kernel().cols() == 1);  // the kernel() should consist of a single (zero) column vector
  VERIFY(size == lu.rank());
  VERIFY(lu.isInjective());
  VERIFY(lu.isSurjective());
  VERIFY(lu.isInvertible());
  VERIFY(lu.image(m1).fullPivLu().isInvertible());

  check_solverbase<MatrixType, MatrixType>(m1, lu, size, size, size);

  MatrixType m1_inverse = lu.inverse();
  m3 = MatrixType::Random(size, size);
  m2 = lu.solve(m3);
  VERIFY_IS_APPROX(m2, m1_inverse * m3);

  RealScalar rcond = (RealScalar(1) / matrix_l1_norm(m1)) / matrix_l1_norm(m1_inverse);
  const RealScalar rcond_est = lu.rcond();
  // Verify that the estimated condition number is within a factor of 10 of the
  // truth.
  VERIFY(rcond_est > rcond / 10 && rcond_est < rcond * 10);

  // Regression test for Bug 302
  MatrixType m4 = MatrixType::Random(size, size);
  VERIFY_IS_APPROX(lu.solve(m3 * m4), lu.solve(m3) * m4);
}

template <typename MatrixType>
void lu_partial_piv(Index size = MatrixType::ColsAtCompileTime) {
  /* this test covers the following files:
     PartialPivLU.h
  */
  typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;

  MatrixType m1(size, size), m2(size, size), m3(size, size);
  m1.setRandom();
  PartialPivLU<MatrixType> plu(m1);

  VERIFY_IS_APPROX(m1, plu.reconstructedMatrix());

  check_solverbase<MatrixType, MatrixType>(m1, plu, size, size, size);

  MatrixType m1_inverse = plu.inverse();
  m3 = MatrixType::Random(size, size);
  m2 = plu.solve(m3);
  VERIFY_IS_APPROX(m2, m1_inverse * m3);

  RealScalar rcond = (RealScalar(1) / matrix_l1_norm(m1)) / matrix_l1_norm(m1_inverse);
  const RealScalar rcond_est = plu.rcond();
  // Verify that the estimate is within a factor of 10 of the truth.
  VERIFY(rcond_est > rcond / 10 && rcond_est < rcond * 10);
}

template <typename MatrixType>
void lu_verify_assert() {
  MatrixType tmp;

  FullPivLU<MatrixType> lu;
  VERIFY_RAISES_ASSERT(lu.matrixLU())
  VERIFY_RAISES_ASSERT(lu.permutationP())
  VERIFY_RAISES_ASSERT(lu.permutationQ())
  VERIFY_RAISES_ASSERT(lu.kernel())
  VERIFY_RAISES_ASSERT(lu.image(tmp))
  VERIFY_RAISES_ASSERT(lu.solve(tmp))
  VERIFY_RAISES_ASSERT(lu.transpose().solve(tmp))
  VERIFY_RAISES_ASSERT(lu.adjoint().solve(tmp))
  VERIFY_RAISES_ASSERT(lu.determinant())
  VERIFY_RAISES_ASSERT(lu.rank())
  VERIFY_RAISES_ASSERT(lu.dimensionOfKernel())
  VERIFY_RAISES_ASSERT(lu.isInjective())
  VERIFY_RAISES_ASSERT(lu.isSurjective())
  VERIFY_RAISES_ASSERT(lu.isInvertible())
  VERIFY_RAISES_ASSERT(lu.inverse())

  PartialPivLU<MatrixType> plu;
  VERIFY_RAISES_ASSERT(plu.matrixLU())
  VERIFY_RAISES_ASSERT(plu.permutationP())
  VERIFY_RAISES_ASSERT(plu.solve(tmp))
  VERIFY_RAISES_ASSERT(plu.transpose().solve(tmp))
  VERIFY_RAISES_ASSERT(plu.adjoint().solve(tmp))
  VERIFY_RAISES_ASSERT(plu.determinant())
  VERIFY_RAISES_ASSERT(plu.inverse())
}

// Rank-deficient matrix returns 0.
// https://gitlab.com/libeigen/eigen/-/issues/2889
void test_2889() {
  Eigen::MatrixXd A =
      Eigen::MatrixXd({{0.0000000000000000, 0.0000000000000000, 1.0000000000000000, 0.0000000000000000,
                        0.34149999916553497, 0.0000000000000000, 0.79877008515664061},
                       {0.0000000000000000, 1.0000000000000000, 0.0000000000000000, 0.29200000315904617,
                        0.0000000000000000, -0.37149999849498272, -0.16425902650844920},
                       {0.0000000000000000, 0.0000000000000000, 1.0000000000000000, 0.0000000000000000,
                        0.34149999916553497, 0.0000000000000000, 0.79877008515664061},
                       {0.0000000000000000, 1.0000000000000000, 0.0000000000000000, 0.040500000119209290,
                        0.0000000000000000, -0.30099999904632568, -0.081170580429391403},
                       {1.0000000000000000, 0.0000000000000000, 0.0000000000000000, 0.0000000000000000,
                        0.0000000000000000, 0.0000000000000000, -0.0000000000000000},
                       {0.0000000000000000, 0.70710689672598170, 0.70710666564709435, 0.027000000700354562,
                        0.025455838867477515, -0.025455847186317101, -0.0068972271572272821},
                       {1.0000000000000000, 0.0000000000000000, 0.0000000000000000, 0.0000000000000000,
                        0.0000000000000000, 0.0000000000000000, -0.0000000000000000}});
  Eigen::FullPivLU<Eigen::MatrixXd> lu_factorization(A);
  double rcond = lu_factorization.rcond();
  VERIFY_IS_EQUAL(rcond, 0.0);
}

EIGEN_DECLARE_TEST(lu) {
  for (int i = 0; i < g_repeat; i++) {
    CALL_SUBTEST_1(lu_non_invertible<Matrix3f>());
    CALL_SUBTEST_1(lu_invertible<Matrix3f>());
    CALL_SUBTEST_1(lu_verify_assert<Matrix3f>());
    CALL_SUBTEST_1(lu_partial_piv<Matrix3f>());

    CALL_SUBTEST_2((lu_non_invertible<Matrix<double, 4, 6> >()));
    CALL_SUBTEST_2((lu_verify_assert<Matrix<double, 4, 6> >()));
    CALL_SUBTEST_2(lu_partial_piv<Matrix2d>());
    CALL_SUBTEST_2(lu_partial_piv<Matrix4d>());
    CALL_SUBTEST_2((lu_partial_piv<Matrix<double, 6, 6> >()));

    CALL_SUBTEST_3(lu_non_invertible<MatrixXf>());
    CALL_SUBTEST_3(lu_invertible<MatrixXf>());
    CALL_SUBTEST_3(lu_verify_assert<MatrixXf>());

    CALL_SUBTEST_4(lu_non_invertible<MatrixXd>());
    CALL_SUBTEST_4(lu_invertible<MatrixXd>());
    CALL_SUBTEST_4(lu_partial_piv<MatrixXd>(internal::random<int>(1, EIGEN_TEST_MAX_SIZE)));
    CALL_SUBTEST_4(lu_verify_assert<MatrixXd>());

    CALL_SUBTEST_5(lu_non_invertible<MatrixXcf>());
    CALL_SUBTEST_5(lu_invertible<MatrixXcf>());
    CALL_SUBTEST_5(lu_verify_assert<MatrixXcf>());

    CALL_SUBTEST_6(lu_non_invertible<MatrixXcd>());
    CALL_SUBTEST_6(lu_invertible<MatrixXcd>());
    CALL_SUBTEST_6(lu_partial_piv<MatrixXcd>(internal::random<int>(1, EIGEN_TEST_MAX_SIZE)));
    CALL_SUBTEST_6(lu_verify_assert<MatrixXcd>());

    CALL_SUBTEST_7((lu_non_invertible<Matrix<float, Dynamic, 16> >()));

    // Test problem size constructors
    CALL_SUBTEST_8(PartialPivLU<MatrixXf>(10));
    CALL_SUBTEST_8(FullPivLU<MatrixXf>(10, 20););

    CALL_SUBTEST_9(test_2889());
  }
}