qr_colpivoting.cpp File Reference

#include "main.h"
#include <Eigen/QR>
#include <Eigen/SVD>
#include "solverbase.h"

Functions
template<typename MatrixType >
void	cod ()
template<typename MatrixType , int Cols2>
void	cod_fixedsize ()
template<typename MatrixType >
void	qr ()
template<typename MatrixType , int Cols2>
void	qr_fixedsize ()
template<typename MatrixType >
void	qr_kahan_matrix ()
template<typename MatrixType >
void	qr_invertible ()
template<typename MatrixType >
void	qr_verify_assert ()
template<typename MatrixType >
void	cod_verify_assert ()
	EIGEN_DECLARE_TEST (qr_colpivoting)

Function Documentation

cod()

template<typename MatrixType >

void cod

(

)

           {
  Index rows = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
  Index cols = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
  Index cols2 = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
  Index rank = internal::random<Index>(1, (std::min)(rows, cols) - 1);
 
  typedef typename MatrixType::Scalar Scalar;
  typedef typename MatrixType::RealScalar RealScalar;
  typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, MatrixType::RowsAtCompileTime> MatrixQType;
  MatrixType matrix;
  createRandomPIMatrixOfRank(rank, rows, cols, matrix);
  CompleteOrthogonalDecomposition<MatrixType> cod(matrix);
  VERIFY(rank == cod.rank());
  VERIFY(cols - cod.rank() == cod.dimensionOfKernel());
  VERIFY(!cod.isInjective());
  VERIFY(!cod.isInvertible());
  VERIFY(!cod.isSurjective());
 
  MatrixQType q = cod.householderQ();
  VERIFY_IS_UNITARY(q);
 
  MatrixType z = cod.matrixZ();
  VERIFY_IS_UNITARY(z);
 
  MatrixType t;
  t.setZero(rows, cols);
  t.topLeftCorner(rank, rank) = cod.matrixT().topLeftCorner(rank, rank).template triangularView<Upper>();
 
  MatrixType c = q * t * z * cod.colsPermutation().inverse();
  VERIFY_IS_APPROX(matrix, c);
 
  check_solverbase<MatrixType, MatrixType>(matrix, cod, rows, cols, cols2);
 
  // Verify that we get the same minimum-norm solution as the SVD.
  MatrixType exact_solution = MatrixType::Random(cols, cols2);
  MatrixType rhs = matrix * exact_solution;
  MatrixType cod_solution = cod.solve(rhs);
  JacobiSVD<MatrixType, ComputeThinU | ComputeThinV> svd(matrix);
  MatrixType svd_solution = svd.solve(rhs);
  VERIFY_IS_APPROX(cod_solution, svd_solution);
 
  MatrixType pinv = cod.pseudoInverse();
  VERIFY_IS_APPROX(cod_solution, pinv * rhs);
 
  // now construct a (square) matrix with prescribed determinant
  Index size = internal::random<Index>(2, 20);
  matrix.setZero(size, size);
  for (int i = 0; i < size; i++) {
    matrix(i, i) = internal::random<Scalar>();
  }
  Scalar det = matrix.diagonal().prod();
  RealScalar absdet = numext::abs(det);
  CompleteOrthogonalDecomposition<MatrixType> cod2(matrix);
  cod2.compute(matrix);
  q = cod2.householderQ();
  matrix = q * matrix * q.adjoint();
  VERIFY_IS_APPROX(det, cod2.determinant());
  VERIFY_IS_APPROX(absdet, cod2.absDeterminant());
  VERIFY_IS_APPROX(numext::log(absdet), cod2.logAbsDeterminant());
  VERIFY_IS_APPROX(numext::sign(det), cod2.signDeterminant());
}

References abs(), Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::absDeterminant(), calibrate::c, cols, Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::compute(), Eigen::createRandomPIMatrixOfRank(), Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::determinant(), EIGEN_TEST_MAX_SIZE, Global_Parameters::exact_solution(), Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::householderQ(), i, Eigen::bfloat16_impl::log(), Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::logAbsDeterminant(), matrix(), min, Global_Physical_Variables::pinv, Eigen::numext::q, rows, SYCL::sign(), Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::signDeterminant(), size, svd(), plotPSD::t, VERIFY, VERIFY_IS_APPROX, and VERIFY_IS_UNITARY.

Referenced by bench(), cod_fixedsize(), and cod_verify_assert().

cod_fixedsize()

template<typename MatrixType , int Cols2>

void cod_fixedsize

(

)

                     {
  enum { Rows = MatrixType::RowsAtCompileTime, Cols = MatrixType::ColsAtCompileTime };
  typedef typename MatrixType::Scalar Scalar;
  typedef CompleteOrthogonalDecomposition<Matrix<Scalar, Rows, Cols> > COD;
  int rank = internal::random<int>(1, (std::min)(int(Rows), int(Cols)) - 1);
  Matrix<Scalar, Rows, Cols> matrix;
  createRandomPIMatrixOfRank(rank, Rows, Cols, matrix);
  COD cod(matrix);
  VERIFY(rank == cod.rank());
  VERIFY(Cols - cod.rank() == cod.dimensionOfKernel());
  VERIFY(cod.isInjective() == (rank == Rows));
  VERIFY(cod.isSurjective() == (rank == Cols));
  VERIFY(cod.isInvertible() == (cod.isInjective() && cod.isSurjective()));
 
  check_solverbase<Matrix<Scalar, Cols, Cols2>, Matrix<Scalar, Rows, Cols2> >(matrix, cod, Rows, Cols, Cols2);
 
  // Verify that we get the same minimum-norm solution as the SVD.
  Matrix<Scalar, Cols, Cols2> exact_solution;
  exact_solution.setRandom(Cols, Cols2);
  Matrix<Scalar, Rows, Cols2> rhs = matrix * exact_solution;
  Matrix<Scalar, Cols, Cols2> cod_solution = cod.solve(rhs);
  JacobiSVD<MatrixType, ComputeFullU | ComputeFullV> svd(matrix);
  Matrix<Scalar, Cols, Cols2> svd_solution = svd.solve(rhs);
  VERIFY_IS_APPROX(cod_solution, svd_solution);
 
  typename Inverse<COD>::PlainObject pinv = cod.pseudoInverse();
  VERIFY_IS_APPROX(cod_solution, pinv * rhs);
}

References cod(), Eigen::createRandomPIMatrixOfRank(), Global_Parameters::exact_solution(), matrix(), min, Global_Physical_Variables::pinv, svd(), VERIFY, and VERIFY_IS_APPROX.

Referenced by EIGEN_DECLARE_TEST().

cod_verify_assert()

template<typename MatrixType >

void cod_verify_assert

(

)

                         {
  MatrixType tmp;
 
  CompleteOrthogonalDecomposition<MatrixType> cod;
  VERIFY_RAISES_ASSERT(cod.matrixQTZ())
  VERIFY_RAISES_ASSERT(cod.solve(tmp))
  VERIFY_RAISES_ASSERT(cod.transpose().solve(tmp))
  VERIFY_RAISES_ASSERT(cod.adjoint().solve(tmp))
  VERIFY_RAISES_ASSERT(cod.householderQ())
  VERIFY_RAISES_ASSERT(cod.dimensionOfKernel())
  VERIFY_RAISES_ASSERT(cod.isInjective())
  VERIFY_RAISES_ASSERT(cod.isSurjective())
  VERIFY_RAISES_ASSERT(cod.isInvertible())
  VERIFY_RAISES_ASSERT(cod.pseudoInverse())
  VERIFY_RAISES_ASSERT(cod.determinant())
  VERIFY_RAISES_ASSERT(cod.absDeterminant())
  VERIFY_RAISES_ASSERT(cod.logAbsDeterminant())
  VERIFY_RAISES_ASSERT(cod.signDeterminant())
}

References cod(), tmp, and VERIFY_RAISES_ASSERT.

EIGEN_DECLARE_TEST()

EIGEN_DECLARE_TEST

(

qr_colpivoting

)

                                   {
  for (int i = 0; i < g_repeat; i++) {
    CALL_SUBTEST_1(qr<MatrixXf>());
    CALL_SUBTEST_2(qr<MatrixXd>());
    CALL_SUBTEST_3(qr<MatrixXcd>());
    CALL_SUBTEST_4((qr_fixedsize<Matrix<float, 3, 5>, 4>()));
    CALL_SUBTEST_5((qr_fixedsize<Matrix<double, 6, 2>, 3>()));
    CALL_SUBTEST_5((qr_fixedsize<Matrix<double, 1, 1>, 1>()));
  }
 
  for (int i = 0; i < g_repeat; i++) {
    CALL_SUBTEST_1(cod<MatrixXf>());
    CALL_SUBTEST_2(cod<MatrixXd>());
    CALL_SUBTEST_3(cod<MatrixXcd>());
    CALL_SUBTEST_4((cod_fixedsize<Matrix<float, 3, 5>, 4>()));
    CALL_SUBTEST_5((cod_fixedsize<Matrix<double, 6, 2>, 3>()));
    CALL_SUBTEST_5((cod_fixedsize<Matrix<double, 1, 1>, 1>()));
  }
 
  for (int i = 0; i < g_repeat; i++) {
    CALL_SUBTEST_1(qr_invertible<MatrixXf>());
    CALL_SUBTEST_2(qr_invertible<MatrixXd>());
    CALL_SUBTEST_6(qr_invertible<MatrixXcf>());
    CALL_SUBTEST_3(qr_invertible<MatrixXcd>());
  }
 
  CALL_SUBTEST_7(qr_verify_assert<Matrix3f>());
  CALL_SUBTEST_8(qr_verify_assert<Matrix3d>());
  CALL_SUBTEST_1(qr_verify_assert<MatrixXf>());
  CALL_SUBTEST_2(qr_verify_assert<MatrixXd>());
  CALL_SUBTEST_6(qr_verify_assert<MatrixXcf>());
  CALL_SUBTEST_3(qr_verify_assert<MatrixXcd>());
 
  CALL_SUBTEST_7(cod_verify_assert<Matrix3f>());
  CALL_SUBTEST_8(cod_verify_assert<Matrix3d>());
  CALL_SUBTEST_1(cod_verify_assert<MatrixXf>());
  CALL_SUBTEST_2(cod_verify_assert<MatrixXd>());
  CALL_SUBTEST_6(cod_verify_assert<MatrixXcf>());
  CALL_SUBTEST_3(cod_verify_assert<MatrixXcd>());
 
  // Test problem size constructors
  CALL_SUBTEST_9(ColPivHouseholderQR<MatrixXf>(10, 20));
 
  CALL_SUBTEST_1(qr_kahan_matrix<MatrixXf>());
  CALL_SUBTEST_2(qr_kahan_matrix<MatrixXd>());
}

References CALL_SUBTEST_1, CALL_SUBTEST_2, CALL_SUBTEST_3, CALL_SUBTEST_4, CALL_SUBTEST_5, CALL_SUBTEST_6, CALL_SUBTEST_7, CALL_SUBTEST_8, CALL_SUBTEST_9, cod_fixedsize(), Eigen::g_repeat, i, and qr_fixedsize().

qr()

template<typename MatrixType >

void qr

(

)

          {
  using std::sqrt;
 
  Index rows = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE), cols = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE),
        cols2 = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
  Index rank = internal::random<Index>(1, (std::min)(rows, cols) - 1);
 
  typedef typename MatrixType::Scalar Scalar;
  typedef typename MatrixType::RealScalar RealScalar;
  typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, MatrixType::RowsAtCompileTime> MatrixQType;
  MatrixType m1;
  createRandomPIMatrixOfRank(rank, rows, cols, m1);
  ColPivHouseholderQR<MatrixType> qr(m1);
  VERIFY_IS_EQUAL(rank, qr.rank());
  VERIFY_IS_EQUAL(cols - qr.rank(), qr.dimensionOfKernel());
  VERIFY(!qr.isInjective());
  VERIFY(!qr.isInvertible());
  VERIFY(!qr.isSurjective());
 
  MatrixQType q = qr.householderQ();
  VERIFY_IS_UNITARY(q);
 
  MatrixType r = qr.matrixQR().template triangularView<Upper>();
  MatrixType c = q * r * qr.colsPermutation().inverse();
  VERIFY_IS_APPROX(m1, c);
 
  // Verify that the absolute value of the diagonal elements in R are
  // non-increasing until they reach the singularity threshold.
  RealScalar threshold = sqrt(RealScalar(rows)) * numext::abs(r(0, 0)) * NumTraits<Scalar>::epsilon();
  for (Index i = 0; i < (std::min)(rows, cols) - 1; ++i) {
    RealScalar x = numext::abs(r(i, i));
    RealScalar y = numext::abs(r(i + 1, i + 1));
    if (x < threshold && y < threshold) continue;
    if (!test_isApproxOrLessThan(y, x)) {
      for (Index j = 0; j < (std::min)(rows, cols); ++j) {
        std::cout << "i = " << j << ", |r_ii| = " << numext::abs(r(j, j)) << std::endl;
      }
      std::cout << "Failure at i=" << i << ", rank=" << rank << ", threshold=" << threshold << std::endl;
    }
    VERIFY_IS_APPROX_OR_LESS_THAN(y, x);
  }
 
  check_solverbase<MatrixType, MatrixType>(m1, qr, rows, cols, cols2);
 
  {
    MatrixType m2, m3;
    Index size = rows;
    do {
      m1 = MatrixType::Random(size, size);
      qr.compute(m1);
    } while (!qr.isInvertible());
    MatrixType m1_inv = qr.inverse();
    m3 = m1 * MatrixType::Random(size, cols2);
    m2 = qr.solve(m3);
    VERIFY_IS_APPROX(m2, m1_inv * m3);
  }
}

References abs(), calibrate::c, cols, Eigen::createRandomPIMatrixOfRank(), EIGEN_TEST_MAX_SIZE, oomph::SarahBL::epsilon, i, j, m1, m2(), min, Eigen::numext::q, UniformPSDSelfTest::r, rows, size, sqrt(), boost::multiprecision::test_isApproxOrLessThan(), VERIFY, VERIFY_IS_APPROX, VERIFY_IS_APPROX_OR_LESS_THAN, VERIFY_IS_EQUAL, VERIFY_IS_UNITARY, plotDoE::x, and y.

Referenced by qr_fixedsize(), qr_invertible(), qr_kahan_matrix(), and qr_verify_assert().

qr_fixedsize()

template<typename MatrixType , int Cols2>

void qr_fixedsize

(

)

                    {
  using std::abs;
  using std::sqrt;
  enum { Rows = MatrixType::RowsAtCompileTime, Cols = MatrixType::ColsAtCompileTime };
  typedef typename MatrixType::Scalar Scalar;
  typedef typename MatrixType::RealScalar RealScalar;
  int rank = internal::random<int>(1, (std::min)(int(Rows), int(Cols)) - 1);
  Matrix<Scalar, Rows, Cols> m1;
  createRandomPIMatrixOfRank(rank, Rows, Cols, m1);
  ColPivHouseholderQR<Matrix<Scalar, Rows, Cols> > qr(m1);
  VERIFY_IS_EQUAL(rank, qr.rank());
  VERIFY_IS_EQUAL(Cols - qr.rank(), qr.dimensionOfKernel());
  VERIFY_IS_EQUAL(qr.isInjective(), (rank == Rows));
  VERIFY_IS_EQUAL(qr.isSurjective(), (rank == Cols));
  VERIFY_IS_EQUAL(qr.isInvertible(), (qr.isInjective() && qr.isSurjective()));
 
  Matrix<Scalar, Rows, Cols> r = qr.matrixQR().template triangularView<Upper>();
  Matrix<Scalar, Rows, Cols> c = qr.householderQ() * r * qr.colsPermutation().inverse();
  VERIFY_IS_APPROX(m1, c);
 
  check_solverbase<Matrix<Scalar, Cols, Cols2>, Matrix<Scalar, Rows, Cols2> >(m1, qr, Rows, Cols, Cols2);
 
  // Verify that the absolute value of the diagonal elements in R are
  // non-increasing until they reache the singularity threshold.
  RealScalar threshold = sqrt(RealScalar(Rows)) * (std::abs)(r(0, 0)) * NumTraits<Scalar>::epsilon();
  for (Index i = 0; i < (std::min)(int(Rows), int(Cols)) - 1; ++i) {
    RealScalar x = numext::abs(r(i, i));
    RealScalar y = numext::abs(r(i + 1, i + 1));
    if (x < threshold && y < threshold) continue;
    if (!test_isApproxOrLessThan(y, x)) {
      for (Index j = 0; j < (std::min)(int(Rows), int(Cols)); ++j) {
        std::cout << "i = " << j << ", |r_ii| = " << numext::abs(r(j, j)) << std::endl;
      }
      std::cout << "Failure at i=" << i << ", rank=" << rank << ", threshold=" << threshold << std::endl;
    }
    VERIFY_IS_APPROX_OR_LESS_THAN(y, x);
  }
}

References abs(), calibrate::c, Eigen::createRandomPIMatrixOfRank(), i, int(), j, m1, min, qr(), UniformPSDSelfTest::r, sqrt(), boost::multiprecision::test_isApproxOrLessThan(), VERIFY_IS_APPROX, VERIFY_IS_APPROX_OR_LESS_THAN, VERIFY_IS_EQUAL, plotDoE::x, and y.

Referenced by EIGEN_DECLARE_TEST().

qr_invertible()

template<typename MatrixType >

void qr_invertible

(

)

                     {
  using std::abs;
  using std::log;
  typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
  typedef typename MatrixType::Scalar Scalar;
 
  int size = internal::random<int>(10, 50);
 
  MatrixType m1(size, size), m2(size, size), m3(size, size);
  m1 = MatrixType::Random(size, size);
 
  if (internal::is_same<RealScalar, float>::value) {
    // let's build a matrix more stable to inverse
    MatrixType a = MatrixType::Random(size, size * 2);
    m1 += a * a.adjoint();
  }
 
  ColPivHouseholderQR<MatrixType> qr(m1);
 
  check_solverbase<MatrixType, MatrixType>(m1, qr, size, size, size);
 
  // now construct a matrix with prescribed determinant
  m1.setZero();
  for (int i = 0; i < size; i++) m1(i, i) = internal::random<Scalar>();
  Scalar det = m1.diagonal().prod();
  RealScalar absdet = abs(det);
  m3 = qr.householderQ();  // get a unitary
  m1 = m3 * m1 * m3.adjoint();
  qr.compute(m1);
  VERIFY_IS_APPROX(det, qr.determinant());
  VERIFY_IS_APPROX(absdet, qr.absDeterminant());
  VERIFY_IS_APPROX(log(absdet), qr.logAbsDeterminant());
  VERIFY_IS_APPROX(numext::sign(det), qr.signDeterminant());
}

References a, abs(), i, Eigen::bfloat16_impl::log(), m1, m2(), qr(), SYCL::sign(), size, Eigen::value, and VERIFY_IS_APPROX.

qr_kahan_matrix()

template<typename MatrixType >

void qr_kahan_matrix

(

)

                       {
  using std::abs;
  using std::sqrt;
  typedef typename MatrixType::Scalar Scalar;
  typedef typename MatrixType::RealScalar RealScalar;
 
  Index rows = 300, cols = rows;
 
  MatrixType m1;
  m1.setZero(rows, cols);
  RealScalar s = std::pow(NumTraits<RealScalar>::epsilon(), 1.0 / rows);
  RealScalar c = std::sqrt(1 - s * s);
  RealScalar pow_s_i(1.0);  // pow(s,i)
  for (Index i = 0; i < rows; ++i) {
    m1(i, i) = pow_s_i;
    m1.row(i).tail(rows - i - 1) = -pow_s_i * c * MatrixType::Ones(1, rows - i - 1);
    pow_s_i *= s;
  }
  m1 = (m1 + m1.transpose()).eval();
  ColPivHouseholderQR<MatrixType> qr(m1);
  MatrixType r = qr.matrixQR().template triangularView<Upper>();
 
  RealScalar threshold = std::sqrt(RealScalar(rows)) * numext::abs(r(0, 0)) * NumTraits<Scalar>::epsilon();
  for (Index i = 0; i < (std::min)(rows, cols) - 1; ++i) {
    RealScalar x = numext::abs(r(i, i));
    RealScalar y = numext::abs(r(i + 1, i + 1));
    if (x < threshold && y < threshold) continue;
    if (!test_isApproxOrLessThan(y, x)) {
      for (Index j = 0; j < (std::min)(rows, cols); ++j) {
        std::cout << "i = " << j << ", |r_ii| = " << numext::abs(r(j, j)) << std::endl;
      }
      std::cout << "Failure at i=" << i << ", rank=" << qr.rank() << ", threshold=" << threshold << std::endl;
    }
    VERIFY_IS_APPROX_OR_LESS_THAN(y, x);
  }
}

References abs(), calibrate::c, cols, oomph::SarahBL::epsilon, eval(), i, j, m1, min, Eigen::bfloat16_impl::pow(), qr(), UniformPSDSelfTest::r, rows, s, sqrt(), boost::multiprecision::test_isApproxOrLessThan(), VERIFY_IS_APPROX_OR_LESS_THAN, plotDoE::x, and y.

qr_verify_assert()

template<typename MatrixType >

void qr_verify_assert

(

)

                        {
  MatrixType tmp;
 
  ColPivHouseholderQR<MatrixType> qr;
  VERIFY_RAISES_ASSERT(qr.matrixQR())
  VERIFY_RAISES_ASSERT(qr.solve(tmp))
  VERIFY_RAISES_ASSERT(qr.transpose().solve(tmp))
  VERIFY_RAISES_ASSERT(qr.adjoint().solve(tmp))
  VERIFY_RAISES_ASSERT(qr.householderQ())
  VERIFY_RAISES_ASSERT(qr.dimensionOfKernel())
  VERIFY_RAISES_ASSERT(qr.isInjective())
  VERIFY_RAISES_ASSERT(qr.isSurjective())
  VERIFY_RAISES_ASSERT(qr.isInvertible())
  VERIFY_RAISES_ASSERT(qr.inverse())
  VERIFY_RAISES_ASSERT(qr.determinant())
  VERIFY_RAISES_ASSERT(qr.absDeterminant())
  VERIFY_RAISES_ASSERT(qr.logAbsDeterminant())
  VERIFY_RAISES_ASSERT(qr.signDeterminant())
}

References qr(), tmp, and VERIFY_RAISES_ASSERT.