Eigen/src/QR/ColPivHouseholderQR.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/.

#ifndef EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
#define EIGEN_COLPIVOTINGHOUSEHOLDERQR_H

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

namespace Eigen {

namespace internal {
template <typename MatrixType_, typename PermutationIndex_>
struct traits<ColPivHouseholderQR<MatrixType_, PermutationIndex_>> : traits<MatrixType_> {
  typedef MatrixXpr XprKind;
  typedef SolverStorage StorageKind;
  typedef PermutationIndex_ PermutationIndex;
  enum { Flags = 0 };
};

}  // end namespace internal

/** \ingroup QR_Module
 *
 * \class ColPivHouseholderQR
 *
 * \brief Householder rank-revealing QR decomposition of a matrix with column-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 P, \b Q and \b R
 * such that
 * \f[
 *  \mathbf{A} \, \mathbf{P} = \mathbf{Q} \, \mathbf{R}
 * \f]
 * by using Householder transformations. Here, \b P is a permutation matrix, \b Q a unitary matrix and \b R an
 * upper triangular matrix.
 *
 * This decomposition performs column pivoting in order to be rank-revealing and improve
 * numerical stability. It is slower than HouseholderQR, and faster than FullPivHouseholderQR.
 *
 * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
 *
 * \sa MatrixBase::colPivHouseholderQr()
 */
template <typename MatrixType_, typename PermutationIndex_>
class ColPivHouseholderQR : public SolverBase<ColPivHouseholderQR<MatrixType_, PermutationIndex_>> {
 public:
  typedef MatrixType_ MatrixType;
  typedef SolverBase<ColPivHouseholderQR> Base;
  friend class SolverBase<ColPivHouseholderQR>;
  typedef PermutationIndex_ PermutationIndex;
  EIGEN_GENERIC_PUBLIC_INTERFACE(ColPivHouseholderQR)

  enum {
    MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
    MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
  };
  typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
  typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime, PermutationIndex> PermutationType;
  typedef typename internal::plain_row_type<MatrixType, PermutationIndex>::type IntRowVectorType;
  typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
  typedef typename internal::plain_row_type<MatrixType, RealScalar>::type RealRowVectorType;
  typedef HouseholderSequence<MatrixType, internal::remove_all_t<typename HCoeffsType::ConjugateReturnType>>
      HouseholderSequenceType;
  typedef typename MatrixType::PlainObject PlainObject;

 private:
  void init(Index rows, Index cols) {
    Index diag = numext::mini(rows, cols);
    m_hCoeffs.resize(diag);
    m_colsPermutation.resize(cols);
    m_colsTranspositions.resize(cols);
    m_temp.resize(cols);
    m_colNormsUpdated.resize(cols);
    m_colNormsDirect.resize(cols);
    m_isInitialized = false;
    m_usePrescribedThreshold = false;
  }

 public:
  /**
   * \brief Default Constructor.
   *
   * The default constructor is useful in cases in which the user intends to
   * perform decompositions via ColPivHouseholderQR::compute(const MatrixType&).
   */
  ColPivHouseholderQR()
      : m_qr(),
        m_hCoeffs(),
        m_colsPermutation(),
        m_colsTranspositions(),
        m_temp(),
        m_colNormsUpdated(),
        m_colNormsDirect(),
        m_isInitialized(false),
        m_usePrescribedThreshold(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 ColPivHouseholderQR()
   */
  ColPivHouseholderQR(Index rows, Index cols) : m_qr(rows, cols) { init(rows, cols); }

  /** \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
   * ColPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
   * qr.compute(matrix);
   * \endcode
   *
   * \sa compute()
   */
  template <typename InputType>
  explicit ColPivHouseholderQR(const EigenBase<InputType>& matrix) : m_qr(matrix.rows(), matrix.cols()) {
    init(matrix.rows(), matrix.cols());
    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 ColPivHouseholderQR(const EigenBase&)
   */
  template <typename InputType>
  explicit ColPivHouseholderQR(EigenBase<InputType>& matrix) : m_qr(matrix.derived()) {
    init(matrix.rows(), matrix.cols());
    computeInPlace();
  }

#ifdef EIGEN_PARSED_BY_DOXYGEN
  /** This method finds a solution x to the equation Ax=b, where A is the matrix of which
   * *this is the QR decomposition, if any exists.
   *
   * \param b the right-hand-side of the equation to solve.
   *
   * \returns a solution.
   *
   * \note_about_checking_solutions
   *
   * \note_about_arbitrary_choice_of_solution
   *
   * Example: \include ColPivHouseholderQR_solve.cpp
   * Output: \verbinclude ColPivHouseholderQR_solve.out
   */
  template <typename Rhs>
  inline const Solve<ColPivHouseholderQR, Rhs> solve(const MatrixBase<Rhs>& b) const;
#endif

  HouseholderSequenceType householderQ() const;
  HouseholderSequenceType matrixQ() const { return householderQ(); }

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

  /** \returns a reference to the matrix where the result Householder QR is stored
   * \warning The strict lower part of this matrix contains internal values.
   * Only the upper triangular part should be referenced. To get it, use
   * \code matrixR().template triangularView<Upper>() \endcode
   * For rank-deficient matrices, use
   * \code
   * matrixR().topLeftCorner(rank(), rank()).template triangularView<Upper>()
   * \endcode
   */
  const MatrixType& matrixR() const {
    eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
    return m_qr;
  }

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

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

  /** \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 rank of the matrix of which *this is the QR decomposition.
   *
   * \note This method has to determine which pivots should be considered nonzero.
   *       For that, it uses the threshold value that you can control by calling
   *       setThreshold(const RealScalar&).
   */
  inline Index rank() const {
    using std::abs;
    eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
    RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold();
    Index result = 0;
    for (Index i = 0; i < m_nonzero_pivots; ++i) result += (abs(m_qr.coeff(i, i)) > premultiplied_threshold);
    return result;
  }

  /** \returns the dimension of the kernel of the matrix of which *this is the QR decomposition.
   *
   * \note This method has to determine which pivots should be considered nonzero.
   *       For that, it uses the threshold value that you can control by calling
   *       setThreshold(const RealScalar&).
   */
  inline Index dimensionOfKernel() const {
    eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
    return cols() - rank();
  }

  /** \returns true if the matrix of which *this is the QR decomposition represents an injective
   *          linear map, i.e. has trivial kernel; false otherwise.
   *
   * \note This method has to determine which pivots should be considered nonzero.
   *       For that, it uses the threshold value that you can control by calling
   *       setThreshold(const RealScalar&).
   */
  inline bool isInjective() const {
    eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
    return rank() == cols();
  }

  /** \returns true if the matrix of which *this is the QR decomposition represents a surjective
   *          linear map; false otherwise.
   *
   * \note This method has to determine which pivots should be considered nonzero.
   *       For that, it uses the threshold value that you can control by calling
   *       setThreshold(const RealScalar&).
   */
  inline bool isSurjective() const {
    eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
    return rank() == rows();
  }

  /** \returns true if the matrix of which *this is the QR decomposition is invertible.
   *
   * \note This method has to determine which pivots should be considered nonzero.
   *       For that, it uses the threshold value that you can control by calling
   *       setThreshold(const RealScalar&).
   */
  inline bool isInvertible() const {
    eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
    return isInjective() && isSurjective();
  }

  /** \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 const Inverse<ColPivHouseholderQR> inverse() const {
    eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
    return Inverse<ColPivHouseholderQR>(*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; }

  /** Allows to prescribe a threshold to be used by certain methods, such as rank(),
   * who need to determine when pivots are to be considered nonzero. This is not used for the
   * QR decomposition itself.
   *
   * When it needs to get the threshold value, Eigen calls threshold(). By default, this
   * uses a formula to automatically determine a reasonable threshold.
   * Once you have called the present method setThreshold(const RealScalar&),
   * your value is used instead.
   *
   * \param threshold The new value to use as the threshold.
   *
   * A pivot will be considered nonzero if its absolute value is strictly greater than
   *  \f$ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert \f$
   * where maxpivot is the biggest pivot.
   *
   * If you want to come back to the default behavior, call setThreshold(Default_t)
   */
  ColPivHouseholderQR& setThreshold(const RealScalar& threshold) {
    m_usePrescribedThreshold = true;
    m_prescribedThreshold = threshold;
    return *this;
  }

  /** Allows to come back to the default behavior, letting Eigen use its default formula for
   * determining the threshold.
   *
   * You should pass the special object Eigen::Default as parameter here.
   * \code qr.setThreshold(Eigen::Default); \endcode
   *
   * See the documentation of setThreshold(const RealScalar&).
   */
  ColPivHouseholderQR& setThreshold(Default_t) {
    m_usePrescribedThreshold = false;
    return *this;
  }

  /** Returns the threshold that will be used by certain methods such as rank().
   *
   * See the documentation of setThreshold(const RealScalar&).
   */
  RealScalar threshold() const {
    eigen_assert(m_isInitialized || m_usePrescribedThreshold);
    return m_usePrescribedThreshold ? m_prescribedThreshold
                                    // this formula comes from experimenting (see "LU precision tuning" thread on the
                                    // list) and turns out to be identical to Higham's formula used already in LDLt.
                                    : NumTraits<Scalar>::epsilon() * RealScalar(m_qr.diagonalSize());
  }

  /** \returns the number of nonzero pivots in the QR decomposition.
   * Here nonzero is meant in the exact sense, not in a fuzzy sense.
   * So that notion isn't really intrinsically interesting, but it is
   * still useful when implementing algorithms.
   *
   * \sa rank()
   */
  inline Index nonzeroPivots() const {
    eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
    return m_nonzero_pivots;
  }

  /** \returns the absolute value of the biggest pivot, i.e. the biggest
   *          diagonal coefficient of R.
   */
  RealScalar maxPivot() const { return m_maxpivot; }

  /** \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 && "Decomposition is not initialized.");
    return Success;
  }

#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:
  friend class CompleteOrthogonalDecomposition<MatrixType, PermutationIndex>;

  EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)

  void computeInPlace();

  MatrixType m_qr;
  HCoeffsType m_hCoeffs;
  PermutationType m_colsPermutation;
  IntRowVectorType m_colsTranspositions;
  RowVectorType m_temp;
  RealRowVectorType m_colNormsUpdated;
  RealRowVectorType m_colNormsDirect;
  bool m_isInitialized, m_usePrescribedThreshold;
  RealScalar m_prescribedThreshold, m_maxpivot;
  Index m_nonzero_pivots;
  Index m_det_p;
};

template <typename MatrixType, typename PermutationIndex>
typename MatrixType::Scalar ColPivHouseholderQR<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 ColPivHouseholderQR<MatrixType, PermutationIndex>::absDeterminant() const {
  using std::abs;
  eigen_assert(m_isInitialized && "ColPivHouseholderQR 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 ColPivHouseholderQR<MatrixType, PermutationIndex>::logAbsDeterminant() const {
  eigen_assert(m_isInitialized && "ColPivHouseholderQR 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 ColPivHouseholderQR<MatrixType, PermutationIndex>::signDeterminant() const {
  eigen_assert(m_isInitialized && "ColPivHouseholderQR 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 ColPivHouseholderQR, ColPivHouseholderQR(const MatrixType&)
 */
template <typename MatrixType, typename PermutationIndex>
template <typename InputType>
ColPivHouseholderQR<MatrixType, PermutationIndex>& ColPivHouseholderQR<MatrixType, PermutationIndex>::compute(
    const EigenBase<InputType>& matrix) {
  m_qr = matrix.derived();
  computeInPlace();
  return *this;
}

template <typename MatrixType, typename PermutationIndex>
void ColPivHouseholderQR<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 = m_qr.diagonalSize();

  m_hCoeffs.resize(size);

  m_temp.resize(cols);

  m_colsTranspositions.resize(m_qr.cols());
  Index number_of_transpositions = 0;

  m_colNormsUpdated.resize(cols);
  m_colNormsDirect.resize(cols);
  for (Index k = 0; k < cols; ++k) {
    // colNormsDirect(k) caches the most recent directly computed norm of
    // column k.
    m_colNormsDirect.coeffRef(k) = m_qr.col(k).norm();
    m_colNormsUpdated.coeffRef(k) = m_colNormsDirect.coeffRef(k);
  }

  RealScalar threshold_helper =
      numext::abs2<RealScalar>(m_colNormsUpdated.maxCoeff() * NumTraits<RealScalar>::epsilon()) / RealScalar(rows);
  RealScalar norm_downdate_threshold = numext::sqrt(NumTraits<RealScalar>::epsilon());

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

  for (Index k = 0; k < size; ++k) {
    // first, we look up in our table m_colNormsUpdated which column has the biggest norm
    Index biggest_col_index;
    RealScalar biggest_col_sq_norm = numext::abs2(m_colNormsUpdated.tail(cols - k).maxCoeff(&biggest_col_index));
    biggest_col_index += k;

    // Track the number of meaningful pivots but do not stop the decomposition to make
    // sure that the initial matrix is properly reproduced. See bug 941.
    if (m_nonzero_pivots == size && biggest_col_sq_norm < threshold_helper * RealScalar(rows - k)) m_nonzero_pivots = k;

    // apply the transposition to the columns
    m_colsTranspositions.coeffRef(k) = static_cast<PermutationIndex>(biggest_col_index);
    if (k != biggest_col_index) {
      m_qr.col(k).swap(m_qr.col(biggest_col_index));
      std::swap(m_colNormsUpdated.coeffRef(k), m_colNormsUpdated.coeffRef(biggest_col_index));
      std::swap(m_colNormsDirect.coeffRef(k), m_colNormsDirect.coeffRef(biggest_col_index));
      ++number_of_transpositions;
    }

    // generate the householder vector, store it below the diagonal
    RealScalar beta;
    m_qr.col(k).tail(rows - k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);

    // apply the householder transformation to the diagonal coefficient
    m_qr.coeffRef(k, k) = beta;

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

    // apply the householder transformation
    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));

    // update our table of norms of the columns
    for (Index j = k + 1; j < cols; ++j) {
      // The following implements the stable norm downgrade step discussed in
      // http://www.netlib.org/lapack/lawnspdf/lawn176.pdf
      // and used in LAPACK routines xGEQPF and xGEQP3.
      // See lines 278-297 in http://www.netlib.org/lapack/explore-html/dc/df4/sgeqpf_8f_source.html
      if (!numext::is_exactly_zero(m_colNormsUpdated.coeffRef(j))) {
        RealScalar temp = abs(m_qr.coeffRef(k, j)) / m_colNormsUpdated.coeffRef(j);
        temp = (RealScalar(1) + temp) * (RealScalar(1) - temp);
        temp = temp < RealScalar(0) ? RealScalar(0) : temp;
        RealScalar temp2 =
            temp * numext::abs2<RealScalar>(m_colNormsUpdated.coeffRef(j) / m_colNormsDirect.coeffRef(j));
        if (temp2 <= norm_downdate_threshold) {
          // The updated norm has become too inaccurate so re-compute the column
          // norm directly.
          m_colNormsDirect.coeffRef(j) = m_qr.col(j).tail(rows - k - 1).norm();
          m_colNormsUpdated.coeffRef(j) = m_colNormsDirect.coeffRef(j);
        } else {
          m_colNormsUpdated.coeffRef(j) *= numext::sqrt(temp);
        }
      }
    }
  }

  m_colsPermutation.setIdentity(cols);
  for (Index k = 0; k < size /*m_nonzero_pivots*/; ++k)
    m_colsPermutation.applyTranspositionOnTheRight(k, static_cast<Index>(m_colsTranspositions.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 ColPivHouseholderQR<MatrixType_, PermutationIndex_>::_solve_impl(const RhsType& rhs, DstType& dst) const {
  const Index nonzero_pivots = nonzeroPivots();

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

  typename RhsType::PlainObject c(rhs);

  c.applyOnTheLeft(householderQ().setLength(nonzero_pivots).adjoint());

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

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

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

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

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

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

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

  dst.applyOnTheLeft(householderQ().setLength(nonzero_pivots).template conjugateIf<!Conjugate>());
}
#endif

namespace internal {

template <typename DstXprType, typename MatrixType, typename PermutationIndex>
struct Assignment<DstXprType, Inverse<ColPivHouseholderQR<MatrixType, PermutationIndex>>,
                  internal::assign_op<typename DstXprType::Scalar,
                                      typename ColPivHouseholderQR<MatrixType, PermutationIndex>::Scalar>,
                  Dense2Dense> {
  typedef ColPivHouseholderQR<MatrixType, PermutationIndex> QrType;
  typedef Inverse<QrType> SrcXprType;
  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()));
  }
};

}  // end namespace internal

/** \returns the matrix Q as a sequence of householder transformations.
 * You can extract the meaningful part only by using:
 * \code qr.householderQ().setLength(qr.nonzeroPivots()) \endcode*/
template <typename MatrixType, typename PermutationIndex>
typename ColPivHouseholderQR<MatrixType, PermutationIndex>::HouseholderSequenceType
ColPivHouseholderQR<MatrixType, PermutationIndex>::householderQ() const {
  eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
  return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate());
}

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

}  // end namespace Eigen

#endif  // EIGEN_COLPIVOTINGHOUSEHOLDERQR_H