Eigen/src/QR/HouseholderQR.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2010 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
// Copyright (C) 2010 Vincent Lejeune
//
// 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_QR_H
#define EIGEN_QR_H

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

namespace Eigen {

namespace internal {
template <typename MatrixType_>
struct traits<HouseholderQR<MatrixType_>> : traits<MatrixType_> {
  using XprKind = MatrixXpr;
  using StorageKind = SolverStorage;
  using StorageIndex = int;
  enum { Flags = 0 };
};

}  // end namespace internal

/** \ingroup QR_Module
 *
 *
 * \class HouseholderQR
 *
 * \brief Householder QR decomposition of a matrix
 *
 * \tparam MatrixType_ the type of the matrix of which we are computing the QR decomposition
 *
 * This class performs a QR decomposition of a matrix \b A into matrices \b Q and \b R
 * such that
 * \f[
 *  \mathbf{A} = \mathbf{Q} \, \mathbf{R}
 * \f]
 * by using Householder transformations. Here, \b Q a unitary matrix and \b R an upper triangular matrix.
 * The result is stored in a compact way compatible with LAPACK.
 *
 * Note that no pivoting is performed. This is \b not a rank-revealing decomposition.
 * If you want that feature, use FullPivHouseholderQR or ColPivHouseholderQR instead.
 *
 * This Householder QR decomposition is faster, but less numerically stable and less feature-full than
 * FullPivHouseholderQR or ColPivHouseholderQR.
 *
 * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
 *
 * \warning For a wide m-by-n matrix (m < n), transpose().solve() and adjoint().solve() use only
 * the first m equations, through the leading m-by-m triangular block of R. This block can be
 * ill-conditioned or singular even when A is well-conditioned and has full row rank. The remaining
 * equations are satisfied only insofar as the right-hand side is consistent and rounding errors permit;
 * these calls do not compute a least-squares solution for an inconsistent right-hand side.
 * To solve using all equations when A has full row rank, factor the transpose directly:
 * \code
 * x = A.transpose().householderQr().solve(b);
 * \endcode
 * For rank-deficient A, or to reuse a decomposition of A, use CompleteOrthogonalDecomposition
 * to obtain the minimum-norm least-squares solution:
 * \code
 * auto cod = A.completeOrthogonalDecomposition();
 * x = cod.transpose().solve(b);
 * \endcode
 * Replace transpose() with adjoint() in these examples for the adjoint system.
 *
 * \sa MatrixBase::householderQr()
 */
template <typename MatrixType_>
class HouseholderQR : public SolverBase<HouseholderQR<MatrixType_>> {
 public:
  using MatrixType = MatrixType_;
  using Base = SolverBase<HouseholderQR>;
  friend class SolverBase<HouseholderQR>;

  EIGEN_GENERIC_PUBLIC_INTERFACE(HouseholderQR)
  enum {
    MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
    MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
  };
  using MatrixQType =
      Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, (MatrixType::Flags & RowMajorBit) ? RowMajor : ColMajor,
             MaxRowsAtCompileTime, MaxRowsAtCompileTime>;
  using HCoeffsType = typename internal::plain_diag_type<MatrixType>::type;
  using RowVectorType = typename internal::plain_row_type<MatrixType>::type;
  using HouseholderSequenceType =
      HouseholderSequence<MatrixType, internal::remove_all_t<typename HCoeffsType::ConjugateReturnType>>;

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

  /**
   * \brief Default Constructor.
   *
   * The default constructor is useful in cases in which the user intends to
   * perform decompositions via HouseholderQR::compute(const MatrixType&).
   */
  HouseholderQR() : m_qr(), m_hCoeffs(), 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 HouseholderQR()
   */
  HouseholderQR(Index rows, Index cols)
      : m_qr(rows, cols), m_hCoeffs((std::min)(rows, 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
   * HouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
   * qr.compute(matrix);
   * \endcode
   *
   * \sa compute()
   */
  template <typename InputType>
  explicit HouseholderQR(const EigenBase<InputType>& matrix)
      : m_qr(matrix.rows(), matrix.cols()),
        m_hCoeffs((std::min)(matrix.rows(), 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 HouseholderQR(const EigenBase&)
   */
  template <typename InputType>
  explicit HouseholderQR(EigenBase<InputType>& matrix)
      : m_qr(matrix.derived()),
        m_hCoeffs((std::min)(matrix.rows(), 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
   * *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 HouseholderQR_solve.cpp
   * Output: \verbinclude HouseholderQR_solve.out
   */
  template <typename Rhs>
  inline Solve<HouseholderQR, Rhs> solve(const MatrixBase<Rhs>& b) const;
#endif

  /** This method returns an expression of the unitary matrix Q as a sequence of Householder transformations.
   *
   * The returned expression can directly be used to perform matrix products. It can also be assigned to a dense Matrix
   * object. Here is an example showing how to recover the full or thin matrix Q, as well as how to perform matrix
   * products using operator*:
   *
   * Example: \include HouseholderQR_householderQ.cpp
   * Output: \verbinclude HouseholderQR_householderQ.out
   */
  HouseholderSequenceType householderQ() const {
    eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
    return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate());
  }

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

  template <typename InputType>
  HouseholderQR& compute(const EigenBase<InputType>& matrix) {
    m_qr = matrix.derived();
    computeInPlace();
    return *this;
  }

  /** \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.
   * Also, do not rely on the determinant being exactly zero for testing
   * singularity or rank-deficiency.
   *
   * \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.
   * Also, do not rely on the determinant being exactly zero for testing
   * singularity or rank-deficiency.
   *
   * \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.
   *
   * \warning Do not rely on the determinant being exactly zero for testing
   * singularity or rank-deficiency.
   *
   * \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.
   *
   * \warning Do not rely on the determinant being exactly zero for testing
   * singularity or rank-deficiency.
   *
   * \sa determinant(), absDeterminant(), MatrixBase::determinant()
   */
  typename MatrixType::Scalar signDeterminant() const;

  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;
  RowVectorType m_temp;
  bool m_isInitialized;
};

namespace internal {

/** \internal */
template <typename HCoeffs, typename Scalar, bool IsComplex>
struct householder_determinant {
  static void run(const HCoeffs& hCoeffs, Scalar& out_det) {
    out_det = Scalar(1);
    Index size = hCoeffs.rows();
    for (Index i = 0; i < size; i++) {
      // For each valid reflection Q_n,
      // det(Q_n) = - conj(h_n) / h_n
      // where h_n is the Householder coefficient.
      if (hCoeffs(i) != Scalar(0)) out_det *= -numext::conj(hCoeffs(i)) / hCoeffs(i);
    }
  }
};

/** \internal */
template <typename HCoeffs, typename Scalar>
struct householder_determinant<HCoeffs, Scalar, false> {
  static void run(const HCoeffs& hCoeffs, Scalar& out_det) {
    bool negated = false;
    Index size = hCoeffs.rows();
    for (Index i = 0; i < size; i++) {
      // Each valid reflection negates the determinant.
      if (hCoeffs(i) != Scalar(0)) negated ^= true;
    }
    out_det = negated ? Scalar(-1) : Scalar(1);
  }
};

}  // end namespace internal

template <typename MatrixType>
typename MatrixType::Scalar HouseholderQR<MatrixType>::determinant() const {
  eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
  EIGEN_INTERNAL_ASSERT_SQUARE_MSG(m_qr, "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 m_qr.diagonal().prod() * detQ;
}

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

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

template <typename MatrixType>
typename MatrixType::Scalar HouseholderQR<MatrixType>::signDeterminant() const {
  eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
  EIGEN_INTERNAL_ASSERT_SQUARE_MSG(m_qr, "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 detQ * m_qr.diagonal().array().sign().prod();
}

namespace internal {

/** \internal */
template <typename MatrixQR, typename HCoeffs>
void householder_qr_inplace_unblocked(MatrixQR& mat, HCoeffs& hCoeffs, typename MatrixQR::Scalar* tempData = 0) {
  using Scalar = typename MatrixQR::Scalar;
  using RealScalar = typename MatrixQR::RealScalar;
  Index rows = mat.rows();
  Index cols = mat.cols();
  Index size = (std::min)(rows, cols);

  eigen_assert(hCoeffs.size() == size);

  using TempType = Matrix<Scalar, MatrixQR::ColsAtCompileTime, 1>;
  TempType tempVector;
  if (tempData == 0) {
    tempVector.resize(cols);
    tempData = tempVector.data();
  }

  for (Index k = 0; k < size; ++k) {
    Index remainingRows = rows - k;
    Index remainingCols = cols - k - 1;

    RealScalar beta;
    mat.col(k).tail(remainingRows).makeHouseholderInPlace(hCoeffs.coeffRef(k), beta);
    mat.coeffRef(k, k) = beta;

    // apply H to remaining part of mat from the left
    mat.bottomRightCorner(remainingRows, remainingCols)
        .applyHouseholderOnTheLeft(mat.col(k).tail(remainingRows - 1), hCoeffs.coeffRef(k), tempData + k + 1);
  }
}

/** \internal
 * Column-insert / column-replace helper for a compact-storage Householder QR.
 * Given a matrix @c mat and @c hCoeffs holding the QR factorization of the first @c k columns of
 * some matrix A, this function replaces column @c k of that factorization with @c newColumn: it
 * applies the existing k Householder reflectors (stored in columns 0..k-1 of @c mat and in
 * @c hCoeffs.head(k)) to @c newColumn, then computes the k-th reflector in place. On exit
 * @c mat.leftCols(k+1) and @c hCoeffs.head(k+1) hold the QR factorization of A.leftCols(k) with
 * @c newColumn inserted at position @c k. @c tempData must point to at least @c mat.cols() scalars.
 *
 * Despite the historical "rank-1 update" label, this is not a full QR update in the
 * Gill-Golub-Murray-Saunders sense: there is no public API for @c QR(A + u vT) or for column/row
 * delete. See libeigen/eigen#3072 for a tracker of that feature gap. Currently only NNLS relies on
 * this helper; the signature is tuned to its active-set bookkeeping. */
template <typename MatrixQR, typename HCoeffs, typename VectorQR>
void householder_qr_inplace_update(MatrixQR& mat, HCoeffs& hCoeffs, const VectorQR& newColumn,
                                   typename MatrixQR::Index k, typename MatrixQR::Scalar* tempData) {
  using Index = typename MatrixQR::Index;
  using RealScalar = typename MatrixQR::RealScalar;
  Index rows = mat.rows();

  eigen_assert(k < mat.cols());
  eigen_assert(k < rows);
  EIGEN_INTERNAL_ASSERT_SIZE_MATCHES_COLS(hCoeffs, mat);
  EIGEN_INTERNAL_ASSERT_SIZE_MATCHES_ROWS(newColumn, mat);
  eigen_assert(tempData);

  // Store new column in mat at column k
  mat.col(k) = newColumn;
  // Apply H = H_1...H_{k-1} on newColumn (skip if k=0)
  for (Index i = 0; i < k; ++i) {
    Index remainingRows = rows - i;
    mat.col(k)
        .tail(remainingRows)
        .applyHouseholderOnTheLeft(mat.col(i).tail(remainingRows - 1), hCoeffs.coeffRef(i), tempData + i + 1);
  }
  // Construct Householder projector in-place in column k
  RealScalar beta;
  mat.col(k).tail(rows - k).makeHouseholderInPlace(hCoeffs.coeffRef(k), beta);
  mat.coeffRef(k, k) = beta;
}

/** \internal */
template <typename MatrixQR, typename HCoeffs, typename MatrixQRScalar = typename MatrixQR::Scalar,
          bool InnerStrideIsOne = (MatrixQR::InnerStrideAtCompileTime == 1 && HCoeffs::InnerStrideAtCompileTime == 1)>
struct householder_qr_inplace_blocked {
  // This is specialized for LAPACK-supported Scalar types in HouseholderQR_LAPACKE.h
  static void run(MatrixQR& mat, HCoeffs& hCoeffs, Index maxBlockSize = 32, typename MatrixQR::Scalar* tempData = 0) {
    using Scalar = typename MatrixQR::Scalar;
    using BlockType = Block<MatrixQR, Dynamic, Dynamic>;

    Index rows = mat.rows();
    Index cols = mat.cols();
    Index size = (std::min)(rows, cols);

    using TempType = Matrix<Scalar, Dynamic, 1, ColMajor, MatrixQR::MaxColsAtCompileTime, 1>;
    TempType tempVector;
    if (tempData == 0) {
      tempVector.resize(cols);
      tempData = tempVector.data();
    }

    Index blockSize = (std::min)(maxBlockSize, size);

    Index k = 0;
    for (k = 0; k < size; k += blockSize) {
      Index bs = (std::min)(size - k, blockSize);  // actual size of the block
      Index tcols = cols - k - bs;                 // trailing columns
      Index brows = rows - k;                      // rows of the block

      // partition the matrix:
      //        A00 | A01 | A02
      // mat  = A10 | A11 | A12
      //        A20 | A21 | A22
      // and performs the qr dec of [A11^T A12^T]^T
      // and update [A21^T A22^T]^T using level 3 operations.
      // Finally, the algorithm continues on A22

      BlockType A11_21 = mat.block(k, k, brows, bs);
      Block<HCoeffs, Dynamic, 1> hCoeffsSegment = hCoeffs.segment(k, bs);

      householder_qr_inplace_unblocked(A11_21, hCoeffsSegment, tempData);

      if (tcols) {
        BlockType A21_22 = mat.block(k, k + bs, brows, tcols);
        apply_block_householder_on_the_left(A21_22, A11_21, hCoeffsSegment, false);  // false == backward
      }
    }
  }
};

/** \internal
 * Panel width b for householder_qr_inplace_blocked on a rows x cols matrix, with s = min(rows, cols).
 * Factoring a panel streams it O(b) times at level 2 and each trailing update streams the trailing block O(1) times,
 * so the factorization moves
 *   ~ c_p * b * P + c_t * Q / b,   P = sum_{k<s} (rows - k),   Q = sum_{k<s} (rows - k) * (cols - k)
 * coefficients (P and Q to leading order below), least at b = sqrt(kappa * Q / P); peak flop rate cancels, and
 * kappa = 3 fits the measured optimum on AVX-512, NEON and SME. The cost is flat in log(b) near the minimum, so b is
 * the nearest of 8, 16, 24, 32, 48 that is a whole number of packets, or one packet if none is: b is the row count of
 * the product V^* * C, which slows sharply on a partial packet. Panels wider than 48 measured no faster. Up to
 * rows * cols * s = 64^3 the per-panel overhead (forming T, product setup) outweighs the traffic saved, and a single
 * panel of width s is returned.
 */
template <typename Scalar>
Index householder_qr_panel_width(Index rows, Index cols) {
  const Index size = numext::mini(rows, cols);
  const double m = double(rows), n = double(cols), s = double(size);
  if (m * n * s <= 64.0 * 64.0 * 64.0) return size;
  const double P = m * s - s * s / 2;
  const double Q = m * n * s - (m + n) * s * s / 2 + s * s * s / 3;
  const double target = numext::sqrt(3 * Q / P);
  constexpr Index kPacketSize = packet_traits<Scalar>::size;
  const Index kWidths[] = {8, 16, 24, 32, 48};
  Index width = kPacketSize;
  double distance = NumTraits<double>::highest();
  for (Index w : kWidths) {
    const double d = numext::maxi(double(w) / target, target / double(w));
    if (w % kPacketSize == 0 && d < distance) {
      width = w;
      distance = d;
    }
  }
  return width;
}

}  // end namespace internal

#ifndef EIGEN_PARSED_BY_DOXYGEN
template <typename MatrixType_>
template <typename RhsType, typename DstType>
void HouseholderQR<MatrixType_>::_solve_impl(const RhsType& rhs, DstType& dst) const {
  const Index rank = (std::min)(rows(), cols());

  typename RhsType::PlainObject c(rhs);

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

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

  dst.topRows(rank) = c.topRows(rank);
  dst.bottomRows(cols() - rank).setZero();
}

template <typename MatrixType_>
template <bool Conjugate, typename RhsType, typename DstType>
void HouseholderQR<MatrixType_>::_solve_impl_transposed(const RhsType& rhs, DstType& dst) const {
  const Index rank = (std::min)(rows(), cols());

  typename RhsType::PlainObject c(rhs);

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

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

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

/** 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 HouseholderQR, HouseholderQR(const MatrixType&)
 */
template <typename MatrixType>
void HouseholderQR<MatrixType>::computeInPlace() {
  Index rows = m_qr.rows();
  Index cols = m_qr.cols();
  Index size = (std::min)(rows, cols);

  m_hCoeffs.resize(size);

  m_temp.resize(cols);

  const Index maxBlockSize = internal::householder_qr_panel_width<Scalar>(rows, cols);
  internal::householder_qr_inplace_blocked<MatrixType, HCoeffsType>::run(m_qr, m_hCoeffs, maxBlockSize, m_temp.data());

  m_isInitialized = true;
}

/** \return the Householder QR decomposition of \c *this.
 *
 * \sa class HouseholderQR
 */
template <typename Derived>
HouseholderQR<typename MatrixBase<Derived>::PlainObject> MatrixBase<Derived>::householderQr() const {
  return HouseholderQR<PlainObject>(eval());
}

}  // end namespace Eigen

#endif  // EIGEN_QR_H