Eigen/src/QR/FullPivHouseholderQR.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2009 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 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/.
// SPDX-License-Identifier: MPL-2.0

#ifndef EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H
#define EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H

// IWYU pragma: private
#include "./InternalHeaderCheck.h"

namespace Eigen {

namespace internal {

template <typename MatrixType_, typename PermutationIndex_>
struct traits<FullPivHouseholderQR<MatrixType_, PermutationIndex_> > : traits<MatrixType_> {
  using XprKind = MatrixXpr;
  using StorageKind = SolverStorage;
  using PermutationIndex = PermutationIndex_;
  enum { Flags = 0 };
};

template <typename MatrixType, typename PermutationIndex>
struct FullPivHouseholderQRMatrixQReturnType;

template <typename MatrixType, typename PermutationIndex>
struct traits<FullPivHouseholderQRMatrixQReturnType<MatrixType, PermutationIndex> > {
  using ReturnType = typename MatrixType::PlainObject;
};

}  // end namespace internal

/** \ingroup QR_Module
 *
 * \class FullPivHouseholderQR
 *
 * \brief Householder rank-revealing QR decomposition of a matrix with full pivoting
 *
 * \tparam MatrixType_ the type of the matrix of which we are computing the QR decomposition
 *
 * This class performs a rank-revealing QR decomposition of a matrix \b A into matrices \b Q, \b R and a column
 * permutation \b P such that
 * \f[
 *  \mathbf{A} \, \mathbf{P} = \mathbf{Q} \, \mathbf{R}
 * \f]
 * by using Householder transformations. Equivalently, \f$ \mathbf{A} = \mathbf{Q} \, \mathbf{R} \, \mathbf{P}^{-1} \f$.
 * Here \b P is the column permutation returned by colsPermutation(), \b Q is the unitary matrix returned by matrixQ(),
 * and \b R is an upper triangular matrix. Row transpositions used during pivoting are folded into \b Q.
 *
 * This decomposition performs a very prudent full pivoting in order to be rank-revealing and achieve optimal
 * numerical stability. The trade-off is that it is slower than HouseholderQR and ColPivHouseholderQR.
 *
 * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
 *
 * \sa MatrixBase::fullPivHouseholderQr()
 */
template <typename MatrixType_, typename PermutationIndex_>
class FullPivHouseholderQR : public SolverBase<FullPivHouseholderQR<MatrixType_, PermutationIndex_> >,
                             public RankRevealingBase<FullPivHouseholderQR<MatrixType_, PermutationIndex_> > {
 public:
  using MatrixType = MatrixType_;
  using Base = SolverBase<FullPivHouseholderQR>;
  using RankRevealingBase_ = RankRevealingBase<FullPivHouseholderQR>;
  friend class SolverBase<FullPivHouseholderQR>;
  friend class RankRevealingBase<FullPivHouseholderQR>;
  using RankRevealingBase_::dimensionOfKernel;
  using RankRevealingBase_::isInjective;
  using RankRevealingBase_::isInvertible;
  using RankRevealingBase_::isSurjective;
  using RankRevealingBase_::maxPivot;
  using RankRevealingBase_::nonzeroPivots;
  using RankRevealingBase_::rank;
  using RankRevealingBase_::setThreshold;
  using RankRevealingBase_::threshold;
  using PermutationIndex = PermutationIndex_;
  EIGEN_GENERIC_PUBLIC_INTERFACE(FullPivHouseholderQR)

  enum {
    MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
    MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
  };
  using MatrixQReturnType = internal::FullPivHouseholderQRMatrixQReturnType<MatrixType, PermutationIndex>;
  using HCoeffsType = typename internal::plain_diag_type<MatrixType>::type;
  using IntDiagSizeVectorType =
      Matrix<PermutationIndex, 1, internal::min_size_prefer_dynamic(ColsAtCompileTime, RowsAtCompileTime), RowMajor, 1,
             internal::min_size_prefer_fixed(MaxColsAtCompileTime, MaxRowsAtCompileTime)>;
  using PermutationType = PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime, PermutationIndex>;
  using RowVectorType = typename internal::plain_row_type<MatrixType>::type;
  using ColVectorType = typename internal::plain_col_type<MatrixType>::type;
  using PlainObject = typename MatrixType::PlainObject;

  /** \brief Reports whether the QR factorization was successful.
   *
   * \note This function always returns \c Success. It is provided for compatibility
   * with other factorization routines.
   * \returns \c Success
   */
  ComputationInfo info() const {
    eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
    return Success;
  }

  /** \brief Default Constructor.
   *
   * The default constructor is useful in cases in which the user intends to
   * perform decompositions via FullPivHouseholderQR::compute(const MatrixType&).
   */
  FullPivHouseholderQR()
      : m_qr(),
        m_hCoeffs(),
        m_rows_transpositions(),
        m_cols_transpositions(),
        m_cols_permutation(),
        m_temp(),
        m_isInitialized(false) {}

  /** \brief Default Constructor with memory preallocation
   *
   * Like the default constructor but with preallocation of the internal data
   * according to the specified problem \a size.
   * \sa FullPivHouseholderQR()
   */
  FullPivHouseholderQR(Index rows, Index cols)
      : m_qr(rows, cols),
        m_hCoeffs((std::min)(rows, cols)),
        m_rows_transpositions((std::min)(rows, cols)),
        m_cols_transpositions((std::min)(rows, cols)),
        m_cols_permutation(cols),
        m_temp(cols),
        m_isInitialized(false) {}

  /** \brief Constructs a QR factorization from a given matrix
   *
   * This constructor computes the QR factorization of the matrix \a matrix by calling
   * the method compute(). It is a short cut for:
   *
   * \code
   * FullPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
   * qr.compute(matrix);
   * \endcode
   *
   * \sa compute()
   */
  template <typename InputType>
  explicit FullPivHouseholderQR(const EigenBase<InputType>& matrix)
      : m_qr(matrix.rows(), matrix.cols()),
        m_hCoeffs((std::min)(matrix.rows(), matrix.cols())),
        m_rows_transpositions((std::min)(matrix.rows(), matrix.cols())),
        m_cols_transpositions((std::min)(matrix.rows(), matrix.cols())),
        m_cols_permutation(matrix.cols()),
        m_temp(matrix.cols()),
        m_isInitialized(false) {
    compute(matrix.derived());
  }

  /** \brief Constructs a QR factorization from a given matrix
   *
   * This overloaded constructor is provided for \link InplaceDecomposition inplace decomposition \endlink when \c
   * MatrixType is a Eigen::Ref.
   *
   * \sa FullPivHouseholderQR(const EigenBase&)
   */
  template <typename InputType>
  explicit FullPivHouseholderQR(EigenBase<InputType>& matrix)
      : m_qr(matrix.derived()),
        m_hCoeffs((std::min)(matrix.rows(), matrix.cols())),
        m_rows_transpositions((std::min)(matrix.rows(), matrix.cols())),
        m_cols_transpositions((std::min)(matrix.rows(), matrix.cols())),
        m_cols_permutation(matrix.cols()),
        m_temp(matrix.cols()),
        m_isInitialized(false) {
    computeInPlace();
  }

#ifdef EIGEN_PARSED_BY_DOXYGEN
  /** This method finds a solution x to the equation Ax=b, where A is the matrix of which
   * \c *this is the QR decomposition.
   *
   * \param b the right-hand-side of the equation to solve.
   *
   * \returns the exact or least-square solution if the rank is greater or equal to the number of columns of A,
   * and an arbitrary solution otherwise.
   * For overdetermined systems, the least-square solution minimizes the Euclidean norm \f$ \Vert A x - b \Vert \f$.
   * For rank-deficient matrices, this method generally does not compute the minimum-norm solution; use
   * CompleteOrthogonalDecomposition or an SVD if that is required.
   *
   * \note_about_checking_solutions
   *
   * \note_about_arbitrary_choice_of_solution
   *
   * Example: \include FullPivHouseholderQR_solve.cpp
   * Output: \verbinclude FullPivHouseholderQR_solve.out
   */
  template <typename Rhs>
  inline Solve<FullPivHouseholderQR, Rhs> solve(const MatrixBase<Rhs>& b) const;
#endif

  /** \returns Expression object representing the matrix Q
   */
  MatrixQReturnType matrixQ(void) const;

  /** \returns a reference to the matrix where the Householder QR decomposition is stored
   */
  const MatrixType& matrixQR() const {
    eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
    return m_qr;
  }

  template <typename InputType>
  FullPivHouseholderQR& compute(const EigenBase<InputType>& matrix);

  /** \returns a const reference to the column permutation matrix */
  const PermutationType& colsPermutation() const {
    eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
    return m_cols_permutation;
  }

  /** \returns a const reference to the vector of indices representing the rows transpositions */
  const IntDiagSizeVectorType& rowsTranspositions() const {
    eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
    return m_rows_transpositions;
  }

  /** \returns the determinant of the matrix of which
   * *this is the QR decomposition. It has only linear complexity
   * (that is, O(n) where n is the dimension of the square matrix)
   * as the QR decomposition has already been computed.
   *
   * \note This is only for square matrices.
   *
   * \warning a determinant can be very big or small, so for matrices
   * of large enough dimension, there is a risk of overflow/underflow.
   * One way to work around that is to use logAbsDeterminant() instead.
   *
   * \sa absDeterminant(), logAbsDeterminant(), MatrixBase::determinant()
   */
  typename MatrixType::Scalar determinant() const;

  /** \returns the absolute value of the determinant of the matrix of which
   * *this is the QR decomposition. It has only linear complexity
   * (that is, O(n) where n is the dimension of the square matrix)
   * as the QR decomposition has already been computed.
   *
   * \note This is only for square matrices.
   *
   * \warning a determinant can be very big or small, so for matrices
   * of large enough dimension, there is a risk of overflow/underflow.
   * One way to work around that is to use logAbsDeterminant() instead.
   *
   * \sa determinant(), logAbsDeterminant(), MatrixBase::determinant()
   */
  typename MatrixType::RealScalar absDeterminant() const;

  /** \returns the natural log of the absolute value of the determinant of the matrix of which
   * *this is the QR decomposition. It has only linear complexity
   * (that is, O(n) where n is the dimension of the square matrix)
   * as the QR decomposition has already been computed.
   *
   * \note This is only for square matrices.
   *
   * \note This method is useful to work around the risk of overflow/underflow that's inherent
   * to determinant computation.
   *
   * \sa determinant(), absDeterminant(), MatrixBase::determinant()
   */
  typename MatrixType::RealScalar logAbsDeterminant() const;

  /** \returns the sign of the determinant of the matrix of which
   * *this is the QR decomposition. It has only linear complexity
   * (that is, O(n) where n is the dimension of the square matrix)
   * as the QR decomposition has already been computed.
   *
   * \note This is only for square matrices.
   *
   * \note This method is useful to work around the risk of overflow/underflow that's inherent
   * to determinant computation.
   *
   * \sa determinant(), absDeterminant(), logAbsDeterminant(), MatrixBase::determinant()
   */
  typename MatrixType::Scalar signDeterminant() const;

  /** \returns the absolute value of the i-th pivot coefficient (for RankRevealingBase). */
  RealScalar pivotCoeff(Index i) const {
    using std::abs;
    return abs(m_qr.coeff(i, i));
  }

  /** \returns the inverse of the matrix of which *this is the QR decomposition.
   *
   * \note If this matrix is not invertible, the returned matrix has undefined coefficients.
   *       Use isInvertible() to first determine whether this matrix is invertible.
   */
  inline Inverse<FullPivHouseholderQR> inverse() const {
    eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
    return Inverse<FullPivHouseholderQR>(*this);
  }

  inline Index rows() const { return m_qr.rows(); }
  inline Index cols() const { return m_qr.cols(); }

  /** \returns a const reference to the vector of Householder coefficients used to represent the factor \c Q.
   *
   * For advanced uses only.
   */
  const HCoeffsType& hCoeffs() const { return m_hCoeffs; }

#ifndef EIGEN_PARSED_BY_DOXYGEN
  template <typename RhsType, typename DstType>
  void _solve_impl(const RhsType& rhs, DstType& dst) const;

  template <bool Conjugate, typename RhsType, typename DstType>
  void _solve_impl_transposed(const RhsType& rhs, DstType& dst) const;
#endif

 protected:
  EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)

  void computeInPlace();

  MatrixType m_qr;
  HCoeffsType m_hCoeffs;
  IntDiagSizeVectorType m_rows_transpositions;
  IntDiagSizeVectorType m_cols_transpositions;
  PermutationType m_cols_permutation;
  RowVectorType m_temp;
  bool m_isInitialized;
  RealScalar m_precision;
  Index m_det_p;
};

template <typename MatrixType, typename PermutationIndex>
typename MatrixType::Scalar FullPivHouseholderQR<MatrixType, PermutationIndex>::determinant() const {
  eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
  eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
  Scalar detQ;
  internal::householder_determinant<HCoeffsType, Scalar, NumTraits<Scalar>::IsComplex>::run(m_hCoeffs, detQ);
  return isInjective() ? (detQ * Scalar(m_det_p)) * m_qr.diagonal().prod() : Scalar(0);
}

template <typename MatrixType, typename PermutationIndex>
typename MatrixType::RealScalar FullPivHouseholderQR<MatrixType, PermutationIndex>::absDeterminant() const {
  using std::abs;
  eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
  eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
  return isInjective() ? abs(m_qr.diagonal().prod()) : RealScalar(0);
}

template <typename MatrixType, typename PermutationIndex>
typename MatrixType::RealScalar FullPivHouseholderQR<MatrixType, PermutationIndex>::logAbsDeterminant() const {
  eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
  eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
  return isInjective() ? m_qr.diagonal().cwiseAbs().array().log().sum() : -NumTraits<RealScalar>::infinity();
}

template <typename MatrixType, typename PermutationIndex>
typename MatrixType::Scalar FullPivHouseholderQR<MatrixType, PermutationIndex>::signDeterminant() const {
  eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
  eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
  Scalar detQ;
  internal::householder_determinant<HCoeffsType, Scalar, NumTraits<Scalar>::IsComplex>::run(m_hCoeffs, detQ);
  return isInjective() ? (detQ * Scalar(m_det_p)) * m_qr.diagonal().array().sign().prod() : Scalar(0);
}

/** Performs the QR factorization of the given matrix \a matrix. The result of
 * the factorization is stored into \c *this, and a reference to \c *this
 * is returned.
 *
 * \sa class FullPivHouseholderQR, FullPivHouseholderQR(const MatrixType&)
 */
template <typename MatrixType, typename PermutationIndex>
template <typename InputType>
FullPivHouseholderQR<MatrixType, PermutationIndex>& FullPivHouseholderQR<MatrixType, PermutationIndex>::compute(
    const EigenBase<InputType>& matrix) {
  m_qr = matrix.derived();
  computeInPlace();
  return *this;
}

template <typename MatrixType, typename PermutationIndex>
void FullPivHouseholderQR<MatrixType, PermutationIndex>::computeInPlace() {
  eigen_assert(m_qr.cols() <= NumTraits<PermutationIndex>::highest());
  using std::abs;
  Index rows = m_qr.rows();
  Index cols = m_qr.cols();
  Index size = (std::min)(rows, cols);

  m_hCoeffs.resize(size);

  m_temp.resize(cols);

  m_precision = NumTraits<Scalar>::epsilon() * RealScalar(size);

  m_rows_transpositions.resize(size);
  m_cols_transpositions.resize(size);
  Index number_of_transpositions = 0;

  RealScalar biggest(0);

  this->m_nonzero_pivots = size;  // the generic case is that in which all pivots are nonzero (invertible case)
  this->m_maxpivot = RealScalar(0);

  for (Index k = 0; k < size; ++k) {
    Index row_of_biggest_in_corner, col_of_biggest_in_corner;
    using Scoring = internal::scalar_score_coeff_op<Scalar>;
    using Score = typename Scoring::result_type;

    Score score = m_qr.bottomRightCorner(rows - k, cols - k)
                      .unaryExpr(Scoring())
                      .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
    row_of_biggest_in_corner += k;
    col_of_biggest_in_corner += k;
    RealScalar biggest_in_corner =
        internal::abs_knowing_score<Scalar>()(m_qr(row_of_biggest_in_corner, col_of_biggest_in_corner), score);
    if (k == 0) biggest = biggest_in_corner;

    // if the corner is negligible, then we have less than full rank, and we can finish early
    if (internal::isMuchSmallerThan(biggest_in_corner, biggest, m_precision)) {
      this->m_nonzero_pivots = k;
      for (Index i = k; i < size; i++) {
        m_rows_transpositions.coeffRef(i) = internal::convert_index<PermutationIndex>(i);
        m_cols_transpositions.coeffRef(i) = internal::convert_index<PermutationIndex>(i);
        m_hCoeffs.coeffRef(i) = Scalar(0);
      }
      break;
    }

    m_rows_transpositions.coeffRef(k) = internal::convert_index<PermutationIndex>(row_of_biggest_in_corner);
    m_cols_transpositions.coeffRef(k) = internal::convert_index<PermutationIndex>(col_of_biggest_in_corner);
    if (k != row_of_biggest_in_corner) {
      m_qr.row(k).tail(cols - k).swap(m_qr.row(row_of_biggest_in_corner).tail(cols - k));
      ++number_of_transpositions;
    }
    if (k != col_of_biggest_in_corner) {
      m_qr.col(k).swap(m_qr.col(col_of_biggest_in_corner));
      ++number_of_transpositions;
    }

    RealScalar beta;
    m_qr.col(k).tail(rows - k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);
    m_qr.coeffRef(k, k) = beta;

    // remember the maximum absolute value of diagonal coefficients
    if (abs(beta) > this->m_maxpivot) this->m_maxpivot = abs(beta);

    m_qr.bottomRightCorner(rows - k, cols - k - 1)
        .applyHouseholderOnTheLeft(m_qr.col(k).tail(rows - k - 1), m_hCoeffs.coeffRef(k), &m_temp.coeffRef(k + 1));
  }

  m_cols_permutation.setIdentity(cols);
  for (Index k = 0; k < size; ++k) m_cols_permutation.applyTranspositionOnTheRight(k, m_cols_transpositions.coeff(k));

  m_det_p = (number_of_transpositions % 2) ? -1 : 1;
  m_isInitialized = true;
}

#ifndef EIGEN_PARSED_BY_DOXYGEN
template <typename MatrixType_, typename PermutationIndex_>
template <typename RhsType, typename DstType>
void FullPivHouseholderQR<MatrixType_, PermutationIndex_>::_solve_impl(const RhsType& rhs, DstType& dst) const {
  const Index l_rank = rank();

  // FIXME: introduce nonzeroPivots() and apply the same improvements as in FullPivLU.
  if (l_rank == 0) {
    dst.setZero();
    return;
  }

  typename RhsType::PlainObject c(rhs);

  Matrix<typename RhsType::Scalar, 1, RhsType::ColsAtCompileTime> temp(rhs.cols());
  for (Index k = 0; k < l_rank; ++k) {
    Index remainingSize = rows() - k;
    c.row(k).swap(c.row(m_rows_transpositions.coeff(k)));
    c.bottomRightCorner(remainingSize, rhs.cols())
        .applyHouseholderOnTheLeft(m_qr.col(k).tail(remainingSize - 1), m_hCoeffs.coeff(k), &temp.coeffRef(0));
  }

  m_qr.topLeftCorner(l_rank, l_rank).template triangularView<Upper>().solveInPlace(c.topRows(l_rank));

  for (Index i = 0; i < l_rank; ++i) dst.row(m_cols_permutation.indices().coeff(i)) = c.row(i);
  for (Index i = l_rank; i < cols(); ++i) dst.row(m_cols_permutation.indices().coeff(i)).setZero();
}

template <typename MatrixType_, typename PermutationIndex_>
template <bool Conjugate, typename RhsType, typename DstType>
void FullPivHouseholderQR<MatrixType_, PermutationIndex_>::_solve_impl_transposed(const RhsType& rhs,
                                                                                  DstType& dst) const {
  const Index l_rank = rank();

  if (l_rank == 0) {
    dst.setZero();
    return;
  }

  typename RhsType::PlainObject c(m_cols_permutation.transpose() * rhs);

  m_qr.topLeftCorner(l_rank, l_rank)
      .template triangularView<Upper>()
      .transpose()
      .template conjugateIf<Conjugate>()
      .solveInPlace(c.topRows(l_rank));

  dst.topRows(l_rank) = c.topRows(l_rank);
  dst.bottomRows(rows() - l_rank).setZero();

  Matrix<Scalar, 1, DstType::ColsAtCompileTime> temp(dst.cols());
  const Index size = (std::min)(rows(), cols());
  for (Index k = size - 1; k >= 0; --k) {
    Index remainingSize = rows() - k;

    dst.bottomRightCorner(remainingSize, dst.cols())
        .applyHouseholderOnTheLeft(m_qr.col(k).tail(remainingSize - 1).template conjugateIf<!Conjugate>(),
                                   m_hCoeffs.template conjugateIf<Conjugate>().coeff(k), &temp.coeffRef(0));

    dst.row(k).swap(dst.row(m_rows_transpositions.coeff(k)));
  }
}
#endif

namespace internal {

template <typename DstXprType, typename MatrixType, typename PermutationIndex>
struct Assignment<DstXprType, Inverse<FullPivHouseholderQR<MatrixType, PermutationIndex> >,
                  internal::assign_op<typename DstXprType::Scalar,
                                      typename FullPivHouseholderQR<MatrixType, PermutationIndex>::Scalar>,
                  Dense2Dense> {
  using QrType = FullPivHouseholderQR<MatrixType, PermutationIndex>;
  using SrcXprType = Inverse<QrType>;
  static void run(DstXprType& dst, const SrcXprType& src,
                  const internal::assign_op<typename DstXprType::Scalar, typename QrType::Scalar>&) {
    dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols()));
  }
};

/** \ingroup QR_Module
 *
 * \brief Expression type for return value of FullPivHouseholderQR::matrixQ()
 *
 * \tparam MatrixType type of underlying dense matrix
 */
template <typename MatrixType, typename PermutationIndex>
struct FullPivHouseholderQRMatrixQReturnType
    : public ReturnByValue<FullPivHouseholderQRMatrixQReturnType<MatrixType, PermutationIndex> > {
 public:
  using IntDiagSizeVectorType = typename FullPivHouseholderQR<MatrixType, PermutationIndex>::IntDiagSizeVectorType;
  using HCoeffsType = typename internal::plain_diag_type<MatrixType>::type;
  using WorkVectorType = Matrix<typename MatrixType::Scalar, 1, MatrixType::RowsAtCompileTime, RowMajor, 1,
                                MatrixType::MaxRowsAtCompileTime>;

  FullPivHouseholderQRMatrixQReturnType(const MatrixType& qr, const HCoeffsType& hCoeffs,
                                        const IntDiagSizeVectorType& rowsTranspositions)
      : m_qr(qr), m_hCoeffs(hCoeffs), m_rowsTranspositions(rowsTranspositions) {}

  template <typename ResultType>
  void evalTo(ResultType& result) const {
    const Index rows = m_qr.rows();
    WorkVectorType workspace(rows);
    evalTo(result, workspace);
  }

  template <typename ResultType>
  void evalTo(ResultType& result, WorkVectorType& workspace) const {
    using numext::conj;
    // compute the product H'_0 H'_1 ... H'_n-1,
    // where H_k is the k-th Householder transformation I - h_k v_k v_k'
    // and v_k is the k-th Householder vector [1,m_qr(k+1,k), m_qr(k+2,k), ...]
    const Index rows = m_qr.rows();
    const Index cols = m_qr.cols();
    const Index size = (std::min)(rows, cols);
    workspace.resize(rows);
    result.setIdentity(rows, rows);
    for (Index k = size - 1; k >= 0; k--) {
      result.block(k, k, rows - k, rows - k)
          .applyHouseholderOnTheLeft(m_qr.col(k).tail(rows - k - 1), conj(m_hCoeffs.coeff(k)), &workspace.coeffRef(k));
      result.row(k).swap(result.row(m_rowsTranspositions.coeff(k)));
    }
  }

  Index rows() const { return m_qr.rows(); }
  Index cols() const { return m_qr.rows(); }

 protected:
  typename MatrixType::Nested m_qr;
  typename HCoeffsType::Nested m_hCoeffs;
  typename IntDiagSizeVectorType::Nested m_rowsTranspositions;
};

}  // end namespace internal

template <typename MatrixType, typename PermutationIndex>
inline typename FullPivHouseholderQR<MatrixType, PermutationIndex>::MatrixQReturnType
FullPivHouseholderQR<MatrixType, PermutationIndex>::matrixQ() const {
  eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
  return MatrixQReturnType(m_qr, m_hCoeffs, m_rows_transpositions);
}

/** \return the full-pivoting Householder QR decomposition of \c *this.
 *
 * \sa class FullPivHouseholderQR
 */
template <typename Derived>
template <typename PermutationIndex>
FullPivHouseholderQR<typename MatrixBase<Derived>::PlainObject, PermutationIndex>
MatrixBase<Derived>::fullPivHouseholderQr() const {
  return FullPivHouseholderQR<PlainObject, PermutationIndex>(eval());
}

}  // end namespace Eigen

#endif  // EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H