Eigen/src/QR/ColPivHouseholderQR.h - mirror - Git at Google

 // 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>
 //
 // Eigen is free software; you can redistribute it and/or
 // modify it under the terms of the GNU Lesser General Public
 // License as published by the Free Software Foundation; either
 // version 3 of the License, or (at your option) any later version.
 //
 // Alternatively, you can redistribute it and/or
 // modify it under the terms of the GNU General Public License as
 // published by the Free Software Foundation; either version 2 of
 // the License, or (at your option) any later version.
 //
 // Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
 // WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
 // FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
 // GNU General Public License for more details.
 //
 // You should have received a copy of the GNU Lesser General Public
 // License and a copy of the GNU General Public License along with
 // Eigen. If not, see <http://www.gnu.org/licenses/>.

 #ifndef EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
 #define EIGEN_COLPIVOTINGHOUSEHOLDERQR_H

 namespace Eigen {

 /** \ingroup QR_Module
   *
   * \class ColPivHouseholderQR
   *
   * \brief Householder rank-revealing QR decomposition of a matrix with column-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 column pivoting in order to be rank-revealing and improve
   * numerical stability. It is slower than HouseholderQR, and faster than FullPivHouseholderQR.
   *
   * \sa MatrixBase::colPivHouseholderQr()
   */
 template<typename _MatrixType> class ColPivHouseholderQR
 {
   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 Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, Options, MaxRowsAtCompileTime, MaxRowsAtCompileTime> MatrixQType;
     typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
     typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationType;
     typedef typename internal::plain_row_type<MatrixType, Index>::type IntRowVectorType;
     typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
     typedef typename internal::plain_row_type<MatrixType, RealScalar>::type RealRowVectorType;
     typedef typename HouseholderSequence<MatrixType,HCoeffsType>::ConjugateReturnType HouseholderSequenceType;

     /**
     * \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_colSqNorms(),
         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 ColPivHouseholderQR()
       */
     ColPivHouseholderQR(Index rows, Index cols)
       : m_qr(rows, cols),
         m_hCoeffs((std::min)(rows,cols)),
         m_colsPermutation(cols),
         m_colsTranspositions(cols),
         m_temp(cols),
         m_colSqNorms(cols),
         m_isInitialized(false),
         m_usePrescribedThreshold(false) {}

     ColPivHouseholderQR(const MatrixType& matrix)
       : m_qr(matrix.rows(), matrix.cols()),
         m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
         m_colsPermutation(matrix.cols()),
         m_colsTranspositions(matrix.cols()),
         m_temp(matrix.cols()),
         m_colSqNorms(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
       * *this is the QR decomposition, if any exists.
       *
       * \param b the right-hand-side of the equation to solve.
       *
       * \returns a solution.
       *
       * \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 ColPivHouseholderQR_solve.cpp
       * Output: \verbinclude ColPivHouseholderQR_solve.out
       */
     template<typename Rhs>
     inline const internal::solve_retval<ColPivHouseholderQR, Rhs>
     solve(const MatrixBase<Rhs>& b) const
     {
       eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
       return internal::solve_retval<ColPivHouseholderQR, Rhs>(*this, b.derived());
     }

     HouseholderSequenceType householderQ(void) const;

     /** \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;
     }

     ColPivHouseholderQR& compute(const MatrixType& matrix);

     const PermutationType& colsPermutation() const
     {
       eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
       return m_colsPermutation;
     }

     /** \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
     {
       eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
       RealScalar premultiplied_threshold = internal::abs(m_maxpivot) * threshold();
       Index result = 0;
       for(Index i = 0; i < m_nonzero_pivots; ++i)
         result += (internal::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
     internal::solve_retval<ColPivHouseholderQR, typename MatrixType::IdentityReturnType>
     inverse() const
     {
       eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
       return internal::solve_retval<ColPivHouseholderQR,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(); }
     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() * 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; }

   protected:
     MatrixType m_qr;
     HCoeffsType m_hCoeffs;
     PermutationType m_colsPermutation;
     IntRowVectorType m_colsTranspositions;
     RowVectorType m_temp;
     RealRowVectorType m_colSqNorms;
     bool m_isInitialized, m_usePrescribedThreshold;
     RealScalar m_prescribedThreshold, m_maxpivot;
     Index m_nonzero_pivots;
     Index m_det_pq;
 };

 template<typename MatrixType>
 typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::absDeterminant() 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 internal::abs(m_qr.diagonal().prod());
 }

 template<typename MatrixType>
 typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::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 m_qr.diagonal().cwiseAbs().array().log().sum();
 }

 template<typename MatrixType>
 ColPivHouseholderQR<MatrixType>& ColPivHouseholderQR<MatrixType>::compute(const MatrixType& matrix)
 {
   Index rows = matrix.rows();
   Index cols = matrix.cols();
   Index size = matrix.diagonalSize();

   m_qr = matrix;
   m_hCoeffs.resize(size);

   m_temp.resize(cols);

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

   m_colSqNorms.resize(cols);
   for(Index k = 0; k < cols; ++k)
     m_colSqNorms.coeffRef(k) = m_qr.col(k).squaredNorm();

   RealScalar threshold_helper = m_colSqNorms.maxCoeff() * internal::abs2(NumTraits<Scalar>::epsilon()) / RealScalar(rows);

   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_colSqNorms which column has the biggest squared norm
     Index biggest_col_index;
     RealScalar biggest_col_sq_norm = m_colSqNorms.tail(cols-k).maxCoeff(&biggest_col_index);
     biggest_col_index += k;

     // since our table m_colSqNorms accumulates imprecision at every step, we must now recompute
     // the actual squared norm of the selected column.
     // Note that not doing so does result in solve() sometimes returning inf/nan values
     // when running the unit test with 1000 repetitions.
     biggest_col_sq_norm = m_qr.col(biggest_col_index).tail(rows-k).squaredNorm();

     // we store that back into our table: it can't hurt to correct our table.
     m_colSqNorms.coeffRef(biggest_col_index) = biggest_col_sq_norm;

     // if the current biggest column is smaller than epsilon times the initial biggest column,
     // terminate to avoid generating nan/inf values.
     // Note that here, if we test instead for "biggest == 0", we get a failure every 1000 (or so)
     // repetitions of the unit test, with the result of solve() filled with large values of the order
     // of 1/(size*epsilon).
     if(biggest_col_sq_norm < threshold_helper * RealScalar(rows-k))
     {
       m_nonzero_pivots = k;
       m_hCoeffs.tail(size-k).setZero();
       m_qr.bottomRightCorner(rows-k,cols-k)
           .template triangularView<StrictlyLower>()
           .setZero();
       break;
     }

     // apply the transposition to the columns
     m_colsTranspositions.coeffRef(k) = biggest_col_index;
     if(k != biggest_col_index) {
       m_qr.col(k).swap(m_qr.col(biggest_col_index));
       std::swap(m_colSqNorms.coeffRef(k), m_colSqNorms.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(internal::abs(beta) > m_maxpivot) m_maxpivot = internal::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 squared norms of the columns
     m_colSqNorms.tail(cols-k-1) -= m_qr.row(k).tail(cols-k-1).cwiseAbs2();
   }

   m_colsPermutation.setIdentity(cols);
   for(Index k = 0; k < m_nonzero_pivots; ++k)
     m_colsPermutation.applyTranspositionOnTheRight(k, m_colsTranspositions.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<ColPivHouseholderQR<_MatrixType>, Rhs>
   : solve_retval_base<ColPivHouseholderQR<_MatrixType>, Rhs>
 {
   EIGEN_MAKE_SOLVE_HELPERS(ColPivHouseholderQR<_MatrixType>,Rhs)

   template<typename Dest> void evalTo(Dest& dst) const
   {
     eigen_assert(rhs().rows() == dec().rows());

     const int cols = dec().cols(),
     nonzero_pivots = dec().nonzeroPivots();

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

     typename Rhs::PlainObject c(rhs());

     // Note that the matrix Q = H_0^* H_1^*... so its inverse is Q^* = (H_0 H_1 ...)^T
     c.applyOnTheLeft(householderSequence(dec().matrixQR(), dec().hCoeffs())
                      .setLength(dec().nonzeroPivots())
 		     .transpose()
       );

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


     typename Rhs::PlainObject d(c);
     d.topRows(nonzero_pivots)
       = dec().matrixQR()
        .topLeftCorner(nonzero_pivots, nonzero_pivots)
        .template triangularView<Upper>()
        * c.topRows(nonzero_pivots);

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

 } // end namespace internal

 /** \returns the matrix Q as a sequence of householder transformations */
 template<typename MatrixType>
 typename ColPivHouseholderQR<MatrixType>::HouseholderSequenceType ColPivHouseholderQR<MatrixType>
   ::householderQ() const
 {
   eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
   return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate()).setLength(m_nonzero_pivots);
 }

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

 } // end namespace Eigen

 #endif // EIGEN_COLPIVOTINGHOUSEHOLDERQR_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>
	//
	// Eigen is free software; you can redistribute it and/or
	// modify it under the terms of the GNU Lesser General Public
	// License as published by the Free Software Foundation; either
	// version 3 of the License, or (at your option) any later version.
	//
	// Alternatively, you can redistribute it and/or
	// modify it under the terms of the GNU General Public License as
	// published by the Free Software Foundation; either version 2 of
	// the License, or (at your option) any later version.
	//
	// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
	// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
	// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
	// GNU General Public License for more details.
	//
	// You should have received a copy of the GNU Lesser General Public
	// License and a copy of the GNU General Public License along with
	// Eigen. If not, see <http://www.gnu.org/licenses/>.

	#ifndef EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
	#define EIGEN_COLPIVOTINGHOUSEHOLDERQR_H

	namespace Eigen {

	/** \ingroup QR_Module
	*
	* \class ColPivHouseholderQR
	*
	* \brief Householder rank-revealing QR decomposition of a matrix with column-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 column pivoting in order to be rank-revealing and improve
	* numerical stability. It is slower than HouseholderQR, and faster than FullPivHouseholderQR.
	*
	* \sa MatrixBase::colPivHouseholderQr()
	*/
	template<typename _MatrixType> class ColPivHouseholderQR
	{
	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 Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, Options, MaxRowsAtCompileTime, MaxRowsAtCompileTime> MatrixQType;
	typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
	typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationType;
	typedef typename internal::plain_row_type<MatrixType, Index>::type IntRowVectorType;
	typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
	typedef typename internal::plain_row_type<MatrixType, RealScalar>::type RealRowVectorType;
	typedef typename HouseholderSequence<MatrixType,HCoeffsType>::ConjugateReturnType HouseholderSequenceType;

	/**
	* \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_colSqNorms(),
	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 ColPivHouseholderQR()
	*/
	ColPivHouseholderQR(Index rows, Index cols)
	: m_qr(rows, cols),
	m_hCoeffs((std::min)(rows,cols)),
	m_colsPermutation(cols),
	m_colsTranspositions(cols),
	m_temp(cols),
	m_colSqNorms(cols),
	m_isInitialized(false),
	m_usePrescribedThreshold(false) {}

	ColPivHouseholderQR(const MatrixType& matrix)
	: m_qr(matrix.rows(), matrix.cols()),
	m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
	m_colsPermutation(matrix.cols()),
	m_colsTranspositions(matrix.cols()),
	m_temp(matrix.cols()),
	m_colSqNorms(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
	* *this is the QR decomposition, if any exists.
	*
	* \param b the right-hand-side of the equation to solve.
	*
	* \returns a solution.
	*
	* \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 ColPivHouseholderQR_solve.cpp
	* Output: \verbinclude ColPivHouseholderQR_solve.out
	*/
	template<typename Rhs>
	inline const internal::solve_retval<ColPivHouseholderQR, Rhs>
	solve(const MatrixBase<Rhs>& b) const
	{
	eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
	return internal::solve_retval<ColPivHouseholderQR, Rhs>(*this, b.derived());
	}

	HouseholderSequenceType householderQ(void) const;

	/** \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;
	}

	ColPivHouseholderQR& compute(const MatrixType& matrix);

	const PermutationType& colsPermutation() const
	{
	eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
	return m_colsPermutation;
	}

	/** \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
	{
	eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
	RealScalar premultiplied_threshold = internal::abs(m_maxpivot) * threshold();
	Index result = 0;
	for(Index i = 0; i < m_nonzero_pivots; ++i)
	result += (internal::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
	internal::solve_retval<ColPivHouseholderQR, typename MatrixType::IdentityReturnType>
	inverse() const
	{
	eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
	return internal::solve_retval<ColPivHouseholderQR,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(); }
	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() * 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; }

	protected:
	MatrixType m_qr;
	HCoeffsType m_hCoeffs;
	PermutationType m_colsPermutation;
	IntRowVectorType m_colsTranspositions;
	RowVectorType m_temp;
	RealRowVectorType m_colSqNorms;
	bool m_isInitialized, m_usePrescribedThreshold;
	RealScalar m_prescribedThreshold, m_maxpivot;
	Index m_nonzero_pivots;
	Index m_det_pq;
	};

	template<typename MatrixType>
	typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::absDeterminant() 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 internal::abs(m_qr.diagonal().prod());
	}

	template<typename MatrixType>
	typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::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 m_qr.diagonal().cwiseAbs().array().log().sum();
	}

	template<typename MatrixType>
	ColPivHouseholderQR<MatrixType>& ColPivHouseholderQR<MatrixType>::compute(const MatrixType& matrix)
	{
	Index rows = matrix.rows();
	Index cols = matrix.cols();
	Index size = matrix.diagonalSize();

	m_qr = matrix;
	m_hCoeffs.resize(size);

	m_temp.resize(cols);

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

	m_colSqNorms.resize(cols);
	for(Index k = 0; k < cols; ++k)
	m_colSqNorms.coeffRef(k) = m_qr.col(k).squaredNorm();

	RealScalar threshold_helper = m_colSqNorms.maxCoeff() * internal::abs2(NumTraits<Scalar>::epsilon()) / RealScalar(rows);

	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_colSqNorms which column has the biggest squared norm
	Index biggest_col_index;
	RealScalar biggest_col_sq_norm = m_colSqNorms.tail(cols-k).maxCoeff(&biggest_col_index);
	biggest_col_index += k;

	// since our table m_colSqNorms accumulates imprecision at every step, we must now recompute
	// the actual squared norm of the selected column.
	// Note that not doing so does result in solve() sometimes returning inf/nan values
	// when running the unit test with 1000 repetitions.
	biggest_col_sq_norm = m_qr.col(biggest_col_index).tail(rows-k).squaredNorm();

	// we store that back into our table: it can't hurt to correct our table.
	m_colSqNorms.coeffRef(biggest_col_index) = biggest_col_sq_norm;

	// if the current biggest column is smaller than epsilon times the initial biggest column,
	// terminate to avoid generating nan/inf values.
	// Note that here, if we test instead for "biggest == 0", we get a failure every 1000 (or so)
	// repetitions of the unit test, with the result of solve() filled with large values of the order
	// of 1/(size*epsilon).
	if(biggest_col_sq_norm < threshold_helper * RealScalar(rows-k))
	{
	m_nonzero_pivots = k;
	m_hCoeffs.tail(size-k).setZero();
	m_qr.bottomRightCorner(rows-k,cols-k)
	.template triangularView<StrictlyLower>()
	.setZero();
	break;
	}

	// apply the transposition to the columns
	m_colsTranspositions.coeffRef(k) = biggest_col_index;
	if(k != biggest_col_index) {
	m_qr.col(k).swap(m_qr.col(biggest_col_index));
	std::swap(m_colSqNorms.coeffRef(k), m_colSqNorms.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(internal::abs(beta) > m_maxpivot) m_maxpivot = internal::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 squared norms of the columns
	m_colSqNorms.tail(cols-k-1) -= m_qr.row(k).tail(cols-k-1).cwiseAbs2();
	}

	m_colsPermutation.setIdentity(cols);
	for(Index k = 0; k < m_nonzero_pivots; ++k)
	m_colsPermutation.applyTranspositionOnTheRight(k, m_colsTranspositions.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<ColPivHouseholderQR<_MatrixType>, Rhs>
	: solve_retval_base<ColPivHouseholderQR<_MatrixType>, Rhs>
	{
	EIGEN_MAKE_SOLVE_HELPERS(ColPivHouseholderQR<_MatrixType>,Rhs)

	template<typename Dest> void evalTo(Dest& dst) const
	{
	eigen_assert(rhs().rows() == dec().rows());

	const int cols = dec().cols(),
	nonzero_pivots = dec().nonzeroPivots();

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

	typename Rhs::PlainObject c(rhs());

	// Note that the matrix Q = H_0^* H_1^... so its inverse is Q^ = (H_0 H_1 ...)^T
	c.applyOnTheLeft(householderSequence(dec().matrixQR(), dec().hCoeffs())
	.setLength(dec().nonzeroPivots())
	.transpose()
	);

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


	typename Rhs::PlainObject d(c);
	d.topRows(nonzero_pivots)
	= dec().matrixQR()
	.topLeftCorner(nonzero_pivots, nonzero_pivots)
	.template triangularView<Upper>()
	* c.topRows(nonzero_pivots);

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

	} // end namespace internal

	/** \returns the matrix Q as a sequence of householder transformations */
	template<typename MatrixType>
	typename ColPivHouseholderQR<MatrixType>::HouseholderSequenceType ColPivHouseholderQR<MatrixType>
	::householderQ() const
	{
	eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
	return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate()).setLength(m_nonzero_pivots);
	}

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

	} // end namespace Eigen

	#endif // EIGEN_COLPIVOTINGHOUSEHOLDERQR_H