Diff: src/QR/FullPivHouseholderQR.h

Revision:

0:13a5d365ba16

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/QR/FullPivHouseholderQR.h	Thu Oct 13 04:07:23 2016 +0000
@@ -0,0 +1,622 @@
+// 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_FULLPIVOTINGHOUSEHOLDERQR_H
+#define EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H
+
+namespace Eigen { 
+
+namespace internal {
+
+template<typename MatrixType> struct FullPivHouseholderQRMatrixQReturnType;
+
+template<typename MatrixType>
+struct traits<FullPivHouseholderQRMatrixQReturnType<MatrixType> >
+{
+  typedef typename MatrixType::PlainObject ReturnType;
+};
+
+}
+
+/** \ingroup QR_Module
+  *
+  * \class FullPivHouseholderQR
+  *
+  * \brief Householder rank-revealing QR decomposition of a matrix with full pivoting
+  *
+  * \param 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 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.
+  *
+  * \sa MatrixBase::fullPivHouseholderQr()
+  */
+template<typename _MatrixType> class FullPivHouseholderQR
+{
+  public:
+
+    typedef _MatrixType MatrixType;
+    enum {
+      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
+      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
+      Options = MatrixType::Options,
+      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
+      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
+    };
+    typedef typename MatrixType::Scalar Scalar;
+    typedef typename MatrixType::RealScalar RealScalar;
+    typedef typename MatrixType::Index Index;
+    typedef internal::FullPivHouseholderQRMatrixQReturnType<MatrixType> MatrixQReturnType;
+    typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
+    typedef Matrix<Index, 1,
+                   EIGEN_SIZE_MIN_PREFER_DYNAMIC(ColsAtCompileTime,RowsAtCompileTime), RowMajor, 1,
+                   EIGEN_SIZE_MIN_PREFER_FIXED(MaxColsAtCompileTime,MaxRowsAtCompileTime)> IntDiagSizeVectorType;
+    typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationType;
+    typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
+    typedef typename internal::plain_col_type<MatrixType>::type ColVectorType;
+
+    /** \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),
+        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 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),
+        m_usePrescribedThreshold(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()
+      */
+    FullPivHouseholderQR(const MatrixType& 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),
+        m_usePrescribedThreshold(false)
+    {
+      compute(matrix);
+    }
+
+    /** 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.
+      *
+      * \note The case where b is a matrix is not yet implemented. Also, this
+      *       code is space inefficient.
+      *
+      * \note_about_checking_solutions
+      *
+      * \note_about_arbitrary_choice_of_solution
+      *
+      * Example: \include FullPivHouseholderQR_solve.cpp
+      * Output: \verbinclude FullPivHouseholderQR_solve.out
+      */
+    template<typename Rhs>
+    inline const internal::solve_retval<FullPivHouseholderQR, Rhs>
+    solve(const MatrixBase<Rhs>& b) const
+    {
+      eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
+      return internal::solve_retval<FullPivHouseholderQR, Rhs>(*this, b.derived());
+    }
+
+    /** \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;
+    }
+
+    FullPivHouseholderQR& compute(const MatrixType& 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 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 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 absDeterminant(), MatrixBase::determinant()
+      */
+    typename MatrixType::RealScalar logAbsDeterminant() 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 && "FullPivHouseholderQR 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 && "FullPivHouseholderQR 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 && "FullPivHouseholderQR 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 && "FullPivHouseholderQR 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 && "FullPivHouseholderQR 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
+    internal::solve_retval<FullPivHouseholderQR, typename MatrixType::IdentityReturnType>
+    inverse() const
+    {
+      eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
+      return internal::solve_retval<FullPivHouseholderQR,typename MatrixType::IdentityReturnType>
+               (*this, MatrixType::Identity(m_qr.rows(), m_qr.cols()));
+    }
+
+    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)
+      */
+    FullPivHouseholderQR& 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&).
+      */
+    FullPivHouseholderQR& 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 && "LU is not initialized.");
+      return m_nonzero_pivots;
+    }
+
+    /** \returns the absolute value of the biggest pivot, i.e. the biggest
+      *          diagonal coefficient of U.
+      */
+    RealScalar maxPivot() const { return m_maxpivot; }
+
+  protected:
+    
+    static void check_template_parameters()
+    {
+      EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
+    }
+    
+    MatrixType m_qr;
+    HCoeffsType m_hCoeffs;
+    IntDiagSizeVectorType m_rows_transpositions;
+    IntDiagSizeVectorType m_cols_transpositions;
+    PermutationType m_cols_permutation;
+    RowVectorType m_temp;
+    bool m_isInitialized, m_usePrescribedThreshold;
+    RealScalar m_prescribedThreshold, m_maxpivot;
+    Index m_nonzero_pivots;
+    RealScalar m_precision;
+    Index m_det_pq;
+};
+
+template<typename MatrixType>
+typename MatrixType::RealScalar FullPivHouseholderQR<MatrixType>::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 abs(m_qr.diagonal().prod());
+}
+
+template<typename MatrixType>
+typename MatrixType::RealScalar FullPivHouseholderQR<MatrixType>::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 m_qr.diagonal().cwiseAbs().array().log().sum();
+}
+
+/** 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>
+FullPivHouseholderQR<MatrixType>& FullPivHouseholderQR<MatrixType>::compute(const MatrixType& matrix)
+{
+  check_template_parameters();
+  
+  using std::abs;
+  Index rows = matrix.rows();
+  Index cols = matrix.cols();
+  Index size = (std::min)(rows,cols);
+
+  m_qr = matrix;
+  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);
+
+  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)
+  {
+    Index row_of_biggest_in_corner, col_of_biggest_in_corner;
+    RealScalar biggest_in_corner;
+
+    biggest_in_corner = m_qr.bottomRightCorner(rows-k, cols-k)
+                            .cwiseAbs()
+                            .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
+    row_of_biggest_in_corner += k;
+    col_of_biggest_in_corner += k;
+    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))
+    {
+      m_nonzero_pivots = k;
+      for(Index i = k; i < size; i++)
+      {
+        m_rows_transpositions.coeffRef(i) = i;
+        m_cols_transpositions.coeffRef(i) = i;
+        m_hCoeffs.coeffRef(i) = Scalar(0);
+      }
+      break;
+    }
+
+    m_rows_transpositions.coeffRef(k) = row_of_biggest_in_corner;
+    m_cols_transpositions.coeffRef(k) = 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) > m_maxpivot) 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_pq = (number_of_transpositions%2) ? -1 : 1;
+  m_isInitialized = true;
+
+  return *this;
+}
+
+namespace internal {
+
+template<typename _MatrixType, typename Rhs>
+struct solve_retval<FullPivHouseholderQR<_MatrixType>, Rhs>
+  : solve_retval_base<FullPivHouseholderQR<_MatrixType>, Rhs>
+{
+  EIGEN_MAKE_SOLVE_HELPERS(FullPivHouseholderQR<_MatrixType>,Rhs)
+
+  template<typename Dest> void evalTo(Dest& dst) const
+  {
+    const Index rows = dec().rows(), cols = dec().cols();
+    eigen_assert(rhs().rows() == rows);
+
+    // FIXME introduce nonzeroPivots() and use it here. and more generally,
+    // make the same improvements in this dec as in FullPivLU.
+    if(dec().rank()==0)
+    {
+      dst.setZero();
+      return;
+    }
+
+    typename Rhs::PlainObject c(rhs());
+
+    Matrix<Scalar,1,Rhs::ColsAtCompileTime> temp(rhs().cols());
+    for (Index k = 0; k < dec().rank(); ++k)
+    {
+      Index remainingSize = rows-k;
+      c.row(k).swap(c.row(dec().rowsTranspositions().coeff(k)));
+      c.bottomRightCorner(remainingSize, rhs().cols())
+       .applyHouseholderOnTheLeft(dec().matrixQR().col(k).tail(remainingSize-1),
+                                  dec().hCoeffs().coeff(k), &temp.coeffRef(0));
+    }
+
+    dec().matrixQR()
+       .topLeftCorner(dec().rank(), dec().rank())
+       .template triangularView<Upper>()
+       .solveInPlace(c.topRows(dec().rank()));
+
+    for(Index i = 0; i < dec().rank(); ++i) dst.row(dec().colsPermutation().indices().coeff(i)) = c.row(i);
+    for(Index i = dec().rank(); i < cols; ++i) dst.row(dec().colsPermutation().indices().coeff(i)).setZero();
+  }
+};
+
+/** \ingroup QR_Module
+  *
+  * \brief Expression type for return value of FullPivHouseholderQR::matrixQ()
+  *
+  * \tparam MatrixType type of underlying dense matrix
+  */
+template<typename MatrixType> struct FullPivHouseholderQRMatrixQReturnType
+  : public ReturnByValue<FullPivHouseholderQRMatrixQReturnType<MatrixType> >
+{
+public:
+  typedef typename MatrixType::Index Index;
+  typedef typename FullPivHouseholderQR<MatrixType>::IntDiagSizeVectorType IntDiagSizeVectorType;
+  typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
+  typedef Matrix<typename MatrixType::Scalar, 1, MatrixType::RowsAtCompileTime, RowMajor, 1,
+                 MatrixType::MaxRowsAtCompileTime> WorkVectorType;
+
+  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>
+inline typename FullPivHouseholderQR<MatrixType>::MatrixQReturnType FullPivHouseholderQR<MatrixType>::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>
+const FullPivHouseholderQR<typename MatrixBase<Derived>::PlainObject>
+MatrixBase<Derived>::fullPivHouseholderQr() const
+{
+  return FullPivHouseholderQR<PlainObject>(eval());
+}
+
+} // end namespace Eigen
+
+#endif // EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H
\ No newline at end of file