FullPivHouseholderQR.h

00001 // This file is part of Eigen, a lightweight C++ template library
00002 // for linear algebra.
00003 //
00004 // Copyright (C) 2008-2009 Gael Guennebaud <gael.guennebaud@inria.fr>
00005 // Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
00006 //
00007 // This Source Code Form is subject to the terms of the Mozilla
00008 // Public License v. 2.0. If a copy of the MPL was not distributed
00009 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
00010 
00011 #ifndef EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H
00012 #define EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H
00013 
00014 namespace Eigen { 
00015 
00016 namespace internal {
00017 
00018 template<typename MatrixType> struct FullPivHouseholderQRMatrixQReturnType;
00019 
00020 template<typename MatrixType>
00021 struct traits<FullPivHouseholderQRMatrixQReturnType<MatrixType> >
00022 {
00023   typedef typename MatrixType::PlainObject ReturnType;
00024 };
00025 
00026 }
00027 
00028 /** \ingroup QR_Module
00029   *
00030   * \class FullPivHouseholderQR
00031   *
00032   * \brief Householder rank-revealing QR decomposition of a matrix with full pivoting
00033   *
00034   * \param MatrixType the type of the matrix of which we are computing the QR decomposition
00035   *
00036   * This class performs a rank-revealing QR decomposition of a matrix \b A into matrices \b P, \b Q and \b R
00037   * such that 
00038   * \f[
00039   *  \mathbf{A} \, \mathbf{P} = \mathbf{Q} \, \mathbf{R}
00040   * \f]
00041   * by using Householder transformations. Here, \b P is a permutation matrix, \b Q a unitary matrix and \b R an 
00042   * upper triangular matrix.
00043   *
00044   * This decomposition performs a very prudent full pivoting in order to be rank-revealing and achieve optimal
00045   * numerical stability. The trade-off is that it is slower than HouseholderQR and ColPivHouseholderQR.
00046   *
00047   * \sa MatrixBase::fullPivHouseholderQr()
00048   */
00049 template<typename _MatrixType> class FullPivHouseholderQR
00050 {
00051   public:
00052 
00053     typedef _MatrixType MatrixType;
00054     enum {
00055       RowsAtCompileTime = MatrixType::RowsAtCompileTime,
00056       ColsAtCompileTime = MatrixType::ColsAtCompileTime,
00057       Options = MatrixType::Options,
00058       MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
00059       MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
00060     };
00061     typedef typename MatrixType::Scalar Scalar;
00062     typedef typename MatrixType::RealScalar RealScalar;
00063     typedef typename MatrixType::Index Index;
00064     typedef internal::FullPivHouseholderQRMatrixQReturnType<MatrixType> MatrixQReturnType;
00065     typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
00066     typedef Matrix<Index, 1,
00067                    EIGEN_SIZE_MIN_PREFER_DYNAMIC(ColsAtCompileTime,RowsAtCompileTime), RowMajor, 1,
00068                    EIGEN_SIZE_MIN_PREFER_FIXED(MaxColsAtCompileTime,MaxRowsAtCompileTime)> IntDiagSizeVectorType ;
00069     typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime>  PermutationType ;
00070     typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
00071     typedef typename internal::plain_col_type<MatrixType>::type ColVectorType;
00072 
00073     /** \brief Default Constructor.
00074       *
00075       * The default constructor is useful in cases in which the user intends to
00076       * perform decompositions via FullPivHouseholderQR::compute(const MatrixType&).
00077       */
00078     FullPivHouseholderQR()
00079       : m_qr(),
00080         m_hCoeffs(),
00081         m_rows_transpositions(),
00082         m_cols_transpositions(),
00083         m_cols_permutation(),
00084         m_temp(),
00085         m_isInitialized(false),
00086         m_usePrescribedThreshold(false) {}
00087 
00088     /** \brief Default Constructor with memory preallocation
00089       *
00090       * Like the default constructor but with preallocation of the internal data
00091       * according to the specified problem \a size.
00092       * \sa FullPivHouseholderQR()
00093       */
00094     FullPivHouseholderQR(Index rows, Index cols)
00095       : m_qr(rows, cols),
00096         m_hCoeffs((std::min)(rows,cols)),
00097         m_rows_transpositions((std::min)(rows,cols)),
00098         m_cols_transpositions((std::min)(rows,cols)),
00099         m_cols_permutation(cols),
00100         m_temp(cols),
00101         m_isInitialized(false),
00102         m_usePrescribedThreshold(false) {}
00103 
00104     /** \brief Constructs a QR factorization from a given matrix
00105       *
00106       * This constructor computes the QR factorization of the matrix \a matrix by calling
00107       * the method compute(). It is a short cut for:
00108       * 
00109       * \code
00110       * FullPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
00111       * qr.compute(matrix);
00112       * \endcode
00113       * 
00114       * \sa compute()
00115       */
00116     FullPivHouseholderQR(const MatrixType& matrix)
00117       : m_qr(matrix.rows(), matrix.cols()),
00118         m_hCoeffs((std::min)(matrix.rows(), matrix.cols())),
00119         m_rows_transpositions((std::min)(matrix.rows(), matrix.cols())),
00120         m_cols_transpositions((std::min)(matrix.rows(), matrix.cols())),
00121         m_cols_permutation(matrix.cols()),
00122         m_temp(matrix.cols()),
00123         m_isInitialized(false),
00124         m_usePrescribedThreshold(false)
00125     {
00126       compute(matrix);
00127     }
00128 
00129     /** This method finds a solution x to the equation Ax=b, where A is the matrix of which
00130       * \c *this is the QR decomposition.
00131       *
00132       * \param b the right-hand-side of the equation to solve.
00133       *
00134       * \returns the exact or least-square solution if the rank is greater or equal to the number of columns of A,
00135       * and an arbitrary solution otherwise.
00136       *
00137       * \note The case where b is a matrix is not yet implemented. Also, this
00138       *       code is space inefficient.
00139       *
00140       * \note_about_checking_solutions
00141       *
00142       * \note_about_arbitrary_choice_of_solution
00143       *
00144       * Example: \include FullPivHouseholderQR_solve.cpp
00145       * Output: \verbinclude FullPivHouseholderQR_solve.out
00146       */
00147     template<typename Rhs>
00148     inline const internal::solve_retval<FullPivHouseholderQR, Rhs>
00149     solve(const MatrixBase<Rhs>& b) const
00150     {
00151       eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
00152       return internal::solve_retval<FullPivHouseholderQR, Rhs>(*this, b.derived());
00153     }
00154 
00155     /** \returns Expression object representing the matrix Q
00156       */
00157     MatrixQReturnType matrixQ (void) const;
00158 
00159     /** \returns a reference to the matrix where the Householder QR decomposition is stored
00160       */
00161     const MatrixType& matrixQR () const
00162     {
00163       eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
00164       return m_qr;
00165     }
00166 
00167     FullPivHouseholderQR& compute(const MatrixType& matrix);
00168 
00169     /** \returns a const reference to the column permutation matrix */
00170     const PermutationType & colsPermutation () const
00171     {
00172       eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
00173       return m_cols_permutation;
00174     }
00175 
00176     /** \returns a const reference to the vector of indices representing the rows transpositions */
00177     const IntDiagSizeVectorType & rowsTranspositions () const
00178     {
00179       eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
00180       return m_rows_transpositions;
00181     }
00182 
00183     /** \returns the absolute value of the determinant of the matrix of which
00184       * *this is the QR decomposition. It has only linear complexity
00185       * (that is, O(n) where n is the dimension of the square matrix)
00186       * as the QR decomposition has already been computed.
00187       *
00188       * \note This is only for square matrices.
00189       *
00190       * \warning a determinant can be very big or small, so for matrices
00191       * of large enough dimension, there is a risk of overflow/underflow.
00192       * One way to work around that is to use logAbsDeterminant() instead.
00193       *
00194       * \sa logAbsDeterminant(), MatrixBase::determinant()
00195       */
00196     typename MatrixType::RealScalar absDeterminant () const;
00197 
00198     /** \returns the natural log of the absolute value of the determinant of the matrix of which
00199       * *this is the QR decomposition. It has only linear complexity
00200       * (that is, O(n) where n is the dimension of the square matrix)
00201       * as the QR decomposition has already been computed.
00202       *
00203       * \note This is only for square matrices.
00204       *
00205       * \note This method is useful to work around the risk of overflow/underflow that's inherent
00206       * to determinant computation.
00207       *
00208       * \sa absDeterminant(), MatrixBase::determinant()
00209       */
00210     typename MatrixType::RealScalar logAbsDeterminant () const;
00211 
00212     /** \returns the rank of the matrix of which *this is the QR decomposition.
00213       *
00214       * \note This method has to determine which pivots should be considered nonzero.
00215       *       For that, it uses the threshold value that you can control by calling
00216       *       setThreshold(const RealScalar&).
00217       */
00218     inline Index rank () const
00219     {
00220       using std::abs;
00221       eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
00222       RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold();
00223       Index result = 0;
00224       for(Index i = 0; i < m_nonzero_pivots; ++i)
00225         result += (abs(m_qr.coeff(i,i)) > premultiplied_threshold);
00226       return result;
00227     }
00228 
00229     /** \returns the dimension of the kernel of the matrix of which *this is the QR decomposition.
00230       *
00231       * \note This method has to determine which pivots should be considered nonzero.
00232       *       For that, it uses the threshold value that you can control by calling
00233       *       setThreshold(const RealScalar&).
00234       */
00235     inline Index dimensionOfKernel () const
00236     {
00237       eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
00238       return cols() - rank ();
00239     }
00240 
00241     /** \returns true if the matrix of which *this is the QR decomposition represents an injective
00242       *          linear map, i.e. has trivial kernel; false otherwise.
00243       *
00244       * \note This method has to determine which pivots should be considered nonzero.
00245       *       For that, it uses the threshold value that you can control by calling
00246       *       setThreshold(const RealScalar&).
00247       */
00248     inline bool isInjective () const
00249     {
00250       eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
00251       return rank () == cols();
00252     }
00253 
00254     /** \returns true if the matrix of which *this is the QR decomposition represents a surjective
00255       *          linear map; false otherwise.
00256       *
00257       * \note This method has to determine which pivots should be considered nonzero.
00258       *       For that, it uses the threshold value that you can control by calling
00259       *       setThreshold(const RealScalar&).
00260       */
00261     inline bool isSurjective () const
00262     {
00263       eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
00264       return rank () == rows();
00265     }
00266 
00267     /** \returns true if the matrix of which *this is the QR decomposition is invertible.
00268       *
00269       * \note This method has to determine which pivots should be considered nonzero.
00270       *       For that, it uses the threshold value that you can control by calling
00271       *       setThreshold(const RealScalar&).
00272       */
00273     inline bool isInvertible () const
00274     {
00275       eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
00276       return isInjective () && isSurjective ();
00277     }
00278 
00279     /** \returns the inverse of the matrix of which *this is the QR decomposition.
00280       *
00281       * \note If this matrix is not invertible, the returned matrix has undefined coefficients.
00282       *       Use isInvertible() to first determine whether this matrix is invertible.
00283       */    inline const
00284     internal::solve_retval<FullPivHouseholderQR, typename MatrixType::IdentityReturnType>
00285     inverse () const
00286     {
00287       eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
00288       return internal::solve_retval<FullPivHouseholderQR,typename MatrixType::IdentityReturnType>
00289                (*this, MatrixType::Identity(m_qr.rows(), m_qr.cols()));
00290     }
00291 
00292     inline Index rows() const { return m_qr.rows(); }
00293     inline Index cols() const { return m_qr.cols(); }
00294     
00295     /** \returns a const reference to the vector of Householder coefficients used to represent the factor \c Q.
00296       * 
00297       * For advanced uses only.
00298       */
00299     const HCoeffsType& hCoeffs () const { return m_hCoeffs; }
00300 
00301     /** Allows to prescribe a threshold to be used by certain methods, such as rank(),
00302       * who need to determine when pivots are to be considered nonzero. This is not used for the
00303       * QR decomposition itself.
00304       *
00305       * When it needs to get the threshold value, Eigen calls threshold(). By default, this
00306       * uses a formula to automatically determine a reasonable threshold.
00307       * Once you have called the present method setThreshold(const RealScalar&),
00308       * your value is used instead.
00309       *
00310       * \param threshold The new value to use as the threshold.
00311       *
00312       * A pivot will be considered nonzero if its absolute value is strictly greater than
00313       *  \f$ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert \f$
00314       * where maxpivot is the biggest pivot.
00315       *
00316       * If you want to come back to the default behavior, call setThreshold(Default_t)
00317       */
00318     FullPivHouseholderQR& setThreshold(const RealScalar& threshold)
00319     {
00320       m_usePrescribedThreshold = true;
00321       m_prescribedThreshold = threshold;
00322       return *this;
00323     }
00324 
00325     /** Allows to come back to the default behavior, letting Eigen use its default formula for
00326       * determining the threshold.
00327       *
00328       * You should pass the special object Eigen::Default as parameter here.
00329       * \code qr.setThreshold(Eigen::Default); \endcode
00330       *
00331       * See the documentation of setThreshold(const RealScalar&).
00332       */
00333     FullPivHouseholderQR& setThreshold(Default_t)
00334     {
00335       m_usePrescribedThreshold = false;
00336       return *this;
00337     }
00338 
00339     /** Returns the threshold that will be used by certain methods such as rank().
00340       *
00341       * See the documentation of setThreshold(const RealScalar&).
00342       */
00343     RealScalar threshold() const
00344     {
00345       eigen_assert(m_isInitialized || m_usePrescribedThreshold);
00346       return m_usePrescribedThreshold ? m_prescribedThreshold
00347       // this formula comes from experimenting (see "LU precision tuning" thread on the list)
00348       // and turns out to be identical to Higham's formula used already in LDLt.
00349                                       : NumTraits<Scalar>::epsilon() * RealScalar(m_qr.diagonalSize());
00350     }
00351 
00352     /** \returns the number of nonzero pivots in the QR decomposition.
00353       * Here nonzero is meant in the exact sense, not in a fuzzy sense.
00354       * So that notion isn't really intrinsically interesting, but it is
00355       * still useful when implementing algorithms.
00356       *
00357       * \sa rank()
00358       */
00359     inline Index nonzeroPivots () const
00360     {
00361       eigen_assert(m_isInitialized && "LU is not initialized.");
00362       return m_nonzero_pivots;
00363     }
00364 
00365     /** \returns the absolute value of the biggest pivot, i.e. the biggest
00366       *          diagonal coefficient of U.
00367       */
00368     RealScalar maxPivot () const { return m_maxpivot; }
00369 
00370   protected:
00371     
00372     static void check_template_parameters()
00373     {
00374       EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
00375     }
00376     
00377     MatrixType m_qr;
00378     HCoeffsType m_hCoeffs;
00379     IntDiagSizeVectorType m_rows_transpositions;
00380     IntDiagSizeVectorType m_cols_transpositions;
00381     PermutationType m_cols_permutation;
00382     RowVectorType m_temp;
00383     bool m_isInitialized, m_usePrescribedThreshold;
00384     RealScalar m_prescribedThreshold, m_maxpivot;
00385     Index m_nonzero_pivots;
00386     RealScalar m_precision;
00387     Index m_det_pq;
00388 };
00389 
00390 template<typename MatrixType>
00391 typename MatrixType::RealScalar FullPivHouseholderQR<MatrixType>::absDeterminant () const
00392 {
00393   using std::abs;
00394   eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
00395   eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
00396   return abs(m_qr.diagonal().prod());
00397 }
00398 
00399 template<typename MatrixType>
00400 typename MatrixType::RealScalar FullPivHouseholderQR<MatrixType>::logAbsDeterminant() const
00401 {
00402   eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
00403   eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
00404   return m_qr.diagonal().cwiseAbs().array().log().sum();
00405 }
00406 
00407 /** Performs the QR factorization of the given matrix \a matrix. The result of
00408   * the factorization is stored into \c *this, and a reference to \c *this
00409   * is returned.
00410   *
00411   * \sa class FullPivHouseholderQR, FullPivHouseholderQR(const MatrixType&)
00412   */
00413 template<typename MatrixType>
00414 FullPivHouseholderQR<MatrixType> & FullPivHouseholderQR<MatrixType>::compute(const MatrixType& matrix)
00415 {
00416   check_template_parameters();
00417   
00418   using std::abs;
00419   Index rows = matrix.rows();
00420   Index cols = matrix.cols();
00421   Index size = (std::min)(rows,cols);
00422 
00423   m_qr = matrix;
00424   m_hCoeffs.resize(size);
00425 
00426   m_temp.resize(cols);
00427 
00428   m_precision = NumTraits<Scalar>::epsilon() * RealScalar(size);
00429 
00430   m_rows_transpositions.resize(size);
00431   m_cols_transpositions.resize(size);
00432   Index number_of_transpositions = 0;
00433 
00434   RealScalar biggest(0);
00435 
00436   m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case)
00437   m_maxpivot = RealScalar(0);
00438 
00439   for (Index k = 0; k < size; ++k)
00440   {
00441     Index row_of_biggest_in_corner, col_of_biggest_in_corner;
00442     RealScalar biggest_in_corner;
00443 
00444     biggest_in_corner = m_qr.bottomRightCorner(rows-k, cols-k)
00445                             .cwiseAbs()
00446                             .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
00447     row_of_biggest_in_corner += k;
00448     col_of_biggest_in_corner += k;
00449     if(k==0) biggest = biggest_in_corner;
00450 
00451     // if the corner is negligible, then we have less than full rank, and we can finish early
00452     if(internal::isMuchSmallerThan(biggest_in_corner, biggest, m_precision))
00453     {
00454       m_nonzero_pivots = k;
00455       for(Index i = k; i < size; i++)
00456       {
00457         m_rows_transpositions.coeffRef(i) = i;
00458         m_cols_transpositions.coeffRef(i) = i;
00459         m_hCoeffs.coeffRef(i) = Scalar(0);
00460       }
00461       break;
00462     }
00463 
00464     m_rows_transpositions.coeffRef(k) = row_of_biggest_in_corner;
00465     m_cols_transpositions.coeffRef(k) = col_of_biggest_in_corner;
00466     if(k != row_of_biggest_in_corner) {
00467       m_qr.row(k).tail(cols-k).swap(m_qr.row(row_of_biggest_in_corner).tail(cols-k));
00468       ++number_of_transpositions;
00469     }
00470     if(k != col_of_biggest_in_corner) {
00471       m_qr.col(k).swap(m_qr.col(col_of_biggest_in_corner));
00472       ++number_of_transpositions;
00473     }
00474 
00475     RealScalar beta;
00476     m_qr.col(k).tail(rows-k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);
00477     m_qr.coeffRef(k,k) = beta;
00478 
00479     // remember the maximum absolute value of diagonal coefficients
00480     if(abs(beta) > m_maxpivot) m_maxpivot = abs(beta);
00481 
00482     m_qr.bottomRightCorner(rows-k, cols-k-1)
00483         .applyHouseholderOnTheLeft(m_qr.col(k).tail(rows-k-1), m_hCoeffs.coeffRef(k), &m_temp.coeffRef(k+1));
00484   }
00485 
00486   m_cols_permutation.setIdentity(cols);
00487   for(Index k = 0; k < size; ++k)
00488     m_cols_permutation.applyTranspositionOnTheRight(k, m_cols_transpositions.coeff(k));
00489 
00490   m_det_pq = (number_of_transpositions%2) ? -1 : 1;
00491   m_isInitialized = true;
00492 
00493   return *this;
00494 }
00495 
00496 namespace internal {
00497 
00498 template<typename _MatrixType, typename Rhs>
00499 struct solve_retval<FullPivHouseholderQR<_MatrixType>, Rhs>
00500   : solve_retval_base<FullPivHouseholderQR<_MatrixType>, Rhs>
00501 {
00502   EIGEN_MAKE_SOLVE_HELPERS(FullPivHouseholderQR<_MatrixType>,Rhs)
00503 
00504   template<typename Dest> void evalTo(Dest& dst) const
00505   {
00506     const Index rows = dec().rows(), cols = dec().cols();
00507     eigen_assert(rhs().rows() == rows);
00508 
00509     // FIXME introduce nonzeroPivots() and use it here. and more generally,
00510     // make the same improvements in this dec as in FullPivLU.
00511     if(dec().rank()==0)
00512     {
00513       dst.setZero();
00514       return;
00515     }
00516 
00517     typename Rhs::PlainObject c(rhs());
00518 
00519     Matrix<Scalar,1,Rhs::ColsAtCompileTime> temp(rhs().cols());
00520     for (Index k = 0; k < dec().rank(); ++k)
00521     {
00522       Index remainingSize = rows-k;
00523       c.row(k).swap(c.row(dec().rowsTranspositions().coeff(k)));
00524       c.bottomRightCorner(remainingSize, rhs().cols())
00525        .applyHouseholderOnTheLeft(dec().matrixQR().col(k).tail(remainingSize-1),
00526                                   dec().hCoeffs().coeff(k), &temp.coeffRef(0));
00527     }
00528 
00529     dec().matrixQR()
00530        .topLeftCorner(dec().rank(), dec().rank())
00531        .template triangularView<Upper>()
00532        .solveInPlace(c.topRows(dec().rank()));
00533 
00534     for(Index i = 0; i < dec().rank(); ++i) dst.row(dec().colsPermutation().indices().coeff(i)) = c.row(i);
00535     for(Index i = dec().rank(); i < cols; ++i) dst.row(dec().colsPermutation().indices().coeff(i)).setZero();
00536   }
00537 };
00538 
00539 /** \ingroup QR_Module
00540   *
00541   * \brief Expression type for return value of FullPivHouseholderQR::matrixQ()
00542   *
00543   * \tparam MatrixType type of underlying dense matrix
00544   */
00545 template<typename MatrixType> struct FullPivHouseholderQRMatrixQReturnType
00546   : public ReturnByValue<FullPivHouseholderQRMatrixQReturnType<MatrixType> >
00547 {
00548 public:
00549   typedef typename MatrixType::Index Index;
00550   typedef typename FullPivHouseholderQR<MatrixType>::IntDiagSizeVectorType IntDiagSizeVectorType;
00551   typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
00552   typedef Matrix<typename MatrixType::Scalar, 1, MatrixType::RowsAtCompileTime, RowMajor, 1,
00553                  MatrixType::MaxRowsAtCompileTime> WorkVectorType;
00554 
00555   FullPivHouseholderQRMatrixQReturnType(const MatrixType&       qr,
00556                                         const HCoeffsType&      hCoeffs,
00557                                         const IntDiagSizeVectorType& rowsTranspositions)
00558     : m_qr(qr),
00559       m_hCoeffs(hCoeffs),
00560       m_rowsTranspositions(rowsTranspositions)
00561       {}
00562 
00563   template <typename ResultType>
00564   void evalTo(ResultType& result) const
00565   {
00566     const Index rows = m_qr.rows();
00567     WorkVectorType workspace(rows);
00568     evalTo(result, workspace);
00569   }
00570 
00571   template <typename ResultType>
00572   void evalTo(ResultType& result, WorkVectorType& workspace) const
00573   {
00574     using numext::conj;
00575     // compute the product H'_0 H'_1 ... H'_n-1,
00576     // where H_k is the k-th Householder transformation I - h_k v_k v_k'
00577     // and v_k is the k-th Householder vector [1,m_qr(k+1,k), m_qr(k+2,k), ...]
00578     const Index rows = m_qr.rows();
00579     const Index cols = m_qr.cols();
00580     const Index size = (std::min)(rows, cols);
00581     workspace.resize(rows);
00582     result.setIdentity(rows, rows);
00583     for (Index k = size-1; k >= 0; k--)
00584     {
00585       result.block(k, k, rows-k, rows-k)
00586             .applyHouseholderOnTheLeft(m_qr.col(k).tail(rows-k-1), conj(m_hCoeffs.coeff(k)), &workspace.coeffRef(k));
00587       result.row(k).swap(result.row(m_rowsTranspositions.coeff(k)));
00588     }
00589   }
00590 
00591     Index rows() const { return m_qr.rows(); }
00592     Index cols() const { return m_qr.rows(); }
00593 
00594 protected:
00595   typename MatrixType::Nested m_qr;
00596   typename HCoeffsType::Nested m_hCoeffs;
00597   typename IntDiagSizeVectorType::Nested m_rowsTranspositions;
00598 };
00599 
00600 } // end namespace internal
00601 
00602 template<typename MatrixType>
00603 inline typename FullPivHouseholderQR<MatrixType>::MatrixQReturnType FullPivHouseholderQR<MatrixType>::matrixQ() const
00604 {
00605   eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
00606   return MatrixQReturnType(m_qr, m_hCoeffs, m_rows_transpositions);
00607 }
00608 
00609 /** \return the full-pivoting Householder QR decomposition of \c *this.
00610   *
00611   * \sa class FullPivHouseholderQR
00612   */
00613 template<typename Derived>
00614 const FullPivHouseholderQR<typename MatrixBase<Derived>::PlainObject>
00615 MatrixBase<Derived>::fullPivHouseholderQr() const
00616 {
00617   return FullPivHouseholderQR<PlainObject>(eval());
00618 }
00619 
00620 } // end namespace Eigen
00621 
00622 #endif // EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H