HouseholderQR.h

00001 // This file is part of Eigen, a lightweight C++ template library
00002 // for linear algebra.
00003 //
00004 // Copyright (C) 2008-2010 Gael Guennebaud <gael.guennebaud@inria.fr>
00005 // Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
00006 // Copyright (C) 2010 Vincent Lejeune
00007 //
00008 // This Source Code Form is subject to the terms of the Mozilla
00009 // Public License v. 2.0. If a copy of the MPL was not distributed
00010 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
00011 
00012 #ifndef EIGEN_QR_H
00013 #define EIGEN_QR_H
00014 
00015 namespace Eigen { 
00016 
00017 /** \ingroup QR_Module
00018   *
00019   *
00020   * \class HouseholderQR
00021   *
00022   * \brief Householder QR decomposition of a matrix
00023   *
00024   * \param MatrixType the type of the matrix of which we are computing the QR decomposition
00025   *
00026   * This class performs a QR decomposition of a matrix \b A into matrices \b Q and \b R
00027   * such that 
00028   * \f[
00029   *  \mathbf{A} = \mathbf{Q} \, \mathbf{R}
00030   * \f]
00031   * by using Householder transformations. Here, \b Q a unitary matrix and \b R an upper triangular matrix.
00032   * The result is stored in a compact way compatible with LAPACK.
00033   *
00034   * Note that no pivoting is performed. This is \b not a rank-revealing decomposition.
00035   * If you want that feature, use FullPivHouseholderQR or ColPivHouseholderQR instead.
00036   *
00037   * This Householder QR decomposition is faster, but less numerically stable and less feature-full than
00038   * FullPivHouseholderQR or ColPivHouseholderQR.
00039   *
00040   * \sa MatrixBase::householderQr()
00041   */
00042 template<typename _MatrixType> class HouseholderQR
00043 {
00044   public:
00045 
00046     typedef _MatrixType MatrixType;
00047     enum {
00048       RowsAtCompileTime = MatrixType::RowsAtCompileTime,
00049       ColsAtCompileTime = MatrixType::ColsAtCompileTime,
00050       Options = MatrixType::Options,
00051       MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
00052       MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
00053     };
00054     typedef typename MatrixType::Scalar Scalar;
00055     typedef typename MatrixType::RealScalar RealScalar;
00056     typedef typename MatrixType::Index Index;
00057     typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, (MatrixType::Flags&RowMajorBit) ? RowMajor : ColMajor, MaxRowsAtCompileTime, MaxRowsAtCompileTime> MatrixQType;
00058     typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
00059     typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
00060     typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename HCoeffsType::ConjugateReturnType>::type > HouseholderSequenceType ;
00061 
00062     /**
00063       * \brief Default Constructor.
00064       *
00065       * The default constructor is useful in cases in which the user intends to
00066       * perform decompositions via HouseholderQR::compute(const MatrixType&).
00067       */
00068     HouseholderQR() : m_qr(), m_hCoeffs(), m_temp(), m_isInitialized(false) {}
00069 
00070     /** \brief Default Constructor with memory preallocation
00071       *
00072       * Like the default constructor but with preallocation of the internal data
00073       * according to the specified problem \a size.
00074       * \sa HouseholderQR()
00075       */
00076     HouseholderQR(Index rows, Index cols)
00077       : m_qr(rows, cols),
00078         m_hCoeffs((std::min)(rows,cols)),
00079         m_temp(cols),
00080         m_isInitialized(false) {}
00081 
00082     /** \brief Constructs a QR factorization from a given matrix
00083       *
00084       * This constructor computes the QR factorization of the matrix \a matrix by calling
00085       * the method compute(). It is a short cut for:
00086       * 
00087       * \code
00088       * HouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
00089       * qr.compute(matrix);
00090       * \endcode
00091       * 
00092       * \sa compute()
00093       */
00094     HouseholderQR(const MatrixType& matrix)
00095       : m_qr(matrix.rows(), matrix.cols()),
00096         m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
00097         m_temp(matrix.cols()),
00098         m_isInitialized(false)
00099     {
00100       compute(matrix);
00101     }
00102 
00103     /** This method finds a solution x to the equation Ax=b, where A is the matrix of which
00104       * *this is the QR decomposition, if any exists.
00105       *
00106       * \param b the right-hand-side of the equation to solve.
00107       *
00108       * \returns a solution.
00109       *
00110       * \note The case where b is a matrix is not yet implemented. Also, this
00111       *       code is space inefficient.
00112       *
00113       * \note_about_checking_solutions
00114       *
00115       * \note_about_arbitrary_choice_of_solution
00116       *
00117       * Example: \include HouseholderQR_solve.cpp
00118       * Output: \verbinclude HouseholderQR_solve.out
00119       */
00120     template<typename Rhs>
00121     inline const internal::solve_retval<HouseholderQR, Rhs>
00122     solve(const MatrixBase<Rhs>& b) const
00123     {
00124       eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
00125       return internal::solve_retval<HouseholderQR, Rhs>(*this, b.derived());
00126     }
00127 
00128     /** This method returns an expression of the unitary matrix Q as a sequence of Householder transformations.
00129       *
00130       * The returned expression can directly be used to perform matrix products. It can also be assigned to a dense Matrix object.
00131       * Here is an example showing how to recover the full or thin matrix Q, as well as how to perform matrix products using operator*:
00132       *
00133       * Example: \include HouseholderQR_householderQ.cpp
00134       * Output: \verbinclude HouseholderQR_householderQ.out
00135       */
00136     HouseholderSequenceType  householderQ() const
00137     {
00138       eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
00139       return HouseholderSequenceType (m_qr, m_hCoeffs.conjugate());
00140     }
00141 
00142     /** \returns a reference to the matrix where the Householder QR decomposition is stored
00143       * in a LAPACK-compatible way.
00144       */
00145     const MatrixType& matrixQR () const
00146     {
00147         eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
00148         return m_qr;
00149     }
00150 
00151     HouseholderQR& compute(const MatrixType& matrix);
00152 
00153     /** \returns the absolute value of the determinant of the matrix of which
00154       * *this is the QR decomposition. It has only linear complexity
00155       * (that is, O(n) where n is the dimension of the square matrix)
00156       * as the QR decomposition has already been computed.
00157       *
00158       * \note This is only for square matrices.
00159       *
00160       * \warning a determinant can be very big or small, so for matrices
00161       * of large enough dimension, there is a risk of overflow/underflow.
00162       * One way to work around that is to use logAbsDeterminant() instead.
00163       *
00164       * \sa logAbsDeterminant(), MatrixBase::determinant()
00165       */
00166     typename MatrixType::RealScalar absDeterminant () const;
00167 
00168     /** \returns the natural log of the absolute value of the determinant of the matrix of which
00169       * *this is the QR decomposition. It has only linear complexity
00170       * (that is, O(n) where n is the dimension of the square matrix)
00171       * as the QR decomposition has already been computed.
00172       *
00173       * \note This is only for square matrices.
00174       *
00175       * \note This method is useful to work around the risk of overflow/underflow that's inherent
00176       * to determinant computation.
00177       *
00178       * \sa absDeterminant(), MatrixBase::determinant()
00179       */
00180     typename MatrixType::RealScalar logAbsDeterminant () const;
00181 
00182     inline Index rows() const { return m_qr.rows(); }
00183     inline Index cols() const { return m_qr.cols(); }
00184     
00185     /** \returns a const reference to the vector of Householder coefficients used to represent the factor \c Q.
00186       * 
00187       * For advanced uses only.
00188       */
00189     const HCoeffsType& hCoeffs () const { return m_hCoeffs; }
00190 
00191   protected:
00192     
00193     static void check_template_parameters()
00194     {
00195       EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
00196     }
00197     
00198     MatrixType m_qr;
00199     HCoeffsType m_hCoeffs;
00200     RowVectorType m_temp;
00201     bool m_isInitialized;
00202 };
00203 
00204 template<typename MatrixType>
00205 typename MatrixType::RealScalar HouseholderQR<MatrixType>::absDeterminant () const
00206 {
00207   using std::abs;
00208   eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
00209   eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
00210   return abs(m_qr.diagonal().prod());
00211 }
00212 
00213 template<typename MatrixType>
00214 typename MatrixType::RealScalar HouseholderQR<MatrixType>::logAbsDeterminant() const
00215 {
00216   eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
00217   eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
00218   return m_qr.diagonal().cwiseAbs().array().log().sum();
00219 }
00220 
00221 namespace internal {
00222 
00223 /** \internal */
00224 template<typename MatrixQR, typename HCoeffs>
00225 void householder_qr_inplace_unblocked(MatrixQR& mat, HCoeffs& hCoeffs, typename MatrixQR::Scalar* tempData = 0)
00226 {
00227   typedef typename MatrixQR::Index Index;
00228   typedef typename MatrixQR::Scalar Scalar;
00229   typedef typename MatrixQR::RealScalar RealScalar;
00230   Index rows = mat.rows();
00231   Index cols = mat.cols();
00232   Index size = (std::min)(rows,cols);
00233 
00234   eigen_assert(hCoeffs.size() == size);
00235 
00236   typedef Matrix<Scalar,MatrixQR::ColsAtCompileTime,1> TempType;
00237   TempType tempVector;
00238   if(tempData==0)
00239   {
00240     tempVector.resize(cols);
00241     tempData = tempVector.data();
00242   }
00243 
00244   for(Index k = 0; k < size; ++k)
00245   {
00246     Index remainingRows = rows - k;
00247     Index remainingCols = cols - k - 1;
00248 
00249     RealScalar beta;
00250     mat.col(k).tail(remainingRows).makeHouseholderInPlace(hCoeffs.coeffRef(k), beta);
00251     mat.coeffRef(k,k) = beta;
00252 
00253     // apply H to remaining part of m_qr from the left
00254     mat.bottomRightCorner(remainingRows, remainingCols)
00255         .applyHouseholderOnTheLeft(mat.col(k).tail(remainingRows-1), hCoeffs.coeffRef(k), tempData+k+1);
00256   }
00257 }
00258 
00259 /** \internal */
00260 template<typename MatrixQR, typename HCoeffs,
00261   typename MatrixQRScalar = typename MatrixQR::Scalar,
00262   bool InnerStrideIsOne = (MatrixQR::InnerStrideAtCompileTime == 1 && HCoeffs::InnerStrideAtCompileTime == 1)>
00263 struct householder_qr_inplace_blocked
00264 {
00265   // This is specialized for MKL-supported Scalar types in HouseholderQR_MKL.h
00266   static void run(MatrixQR& mat, HCoeffs& hCoeffs,
00267       typename MatrixQR::Index maxBlockSize=32,
00268       typename MatrixQR::Scalar* tempData = 0)
00269   {
00270     typedef typename MatrixQR::Index Index;
00271     typedef typename MatrixQR::Scalar Scalar;
00272     typedef Block<MatrixQR,Dynamic,Dynamic> BlockType;
00273 
00274     Index rows = mat.rows();
00275     Index cols = mat.cols();
00276     Index size = (std::min)(rows, cols);
00277 
00278     typedef Matrix<Scalar,Dynamic,1,ColMajor,MatrixQR::MaxColsAtCompileTime,1> TempType;
00279     TempType tempVector;
00280     if(tempData==0)
00281     {
00282       tempVector.resize(cols);
00283       tempData = tempVector.data();
00284     }
00285 
00286     Index blockSize = (std::min)(maxBlockSize,size);
00287 
00288     Index k = 0;
00289     for (k = 0; k < size; k += blockSize)
00290     {
00291       Index bs = (std::min)(size-k,blockSize);  // actual size of the block
00292       Index tcols = cols - k - bs;            // trailing columns
00293       Index brows = rows-k;                   // rows of the block
00294 
00295       // partition the matrix:
00296       //        A00 | A01 | A02
00297       // mat  = A10 | A11 | A12
00298       //        A20 | A21 | A22
00299       // and performs the qr dec of [A11^T A12^T]^T
00300       // and update [A21^T A22^T]^T using level 3 operations.
00301       // Finally, the algorithm continue on A22
00302 
00303       BlockType A11_21 = mat.block(k,k,brows,bs);
00304       Block<HCoeffs,Dynamic,1> hCoeffsSegment = hCoeffs.segment(k,bs);
00305 
00306       householder_qr_inplace_unblocked(A11_21, hCoeffsSegment, tempData);
00307 
00308       if(tcols)
00309       {
00310         BlockType A21_22 = mat.block(k,k+bs,brows,tcols);
00311         apply_block_householder_on_the_left(A21_22,A11_21,hCoeffsSegment.adjoint());
00312       }
00313     }
00314   }
00315 };
00316 
00317 template<typename _MatrixType, typename Rhs>
00318 struct solve_retval<HouseholderQR<_MatrixType>, Rhs>
00319   : solve_retval_base<HouseholderQR<_MatrixType>, Rhs>
00320 {
00321   EIGEN_MAKE_SOLVE_HELPERS(HouseholderQR<_MatrixType>,Rhs)
00322 
00323   template<typename Dest> void evalTo(Dest& dst) const
00324   {
00325     const Index rows = dec().rows(), cols = dec().cols();
00326     const Index rank = (std::min)(rows, cols);
00327     eigen_assert(rhs().rows() == rows);
00328 
00329     typename Rhs::PlainObject c(rhs());
00330 
00331     // Note that the matrix Q = H_0^* H_1^*... so its inverse is Q^* = (H_0 H_1 ...)^T
00332     c.applyOnTheLeft(householderSequence(
00333       dec().matrixQR().leftCols(rank),
00334       dec().hCoeffs().head(rank)).transpose()
00335     );
00336 
00337     dec().matrixQR()
00338        .topLeftCorner(rank, rank)
00339        .template triangularView<Upper>()
00340        .solveInPlace(c.topRows(rank));
00341 
00342     dst.topRows(rank) = c.topRows(rank);
00343     dst.bottomRows(cols-rank).setZero();
00344   }
00345 };
00346 
00347 } // end namespace internal
00348 
00349 /** Performs the QR factorization of the given matrix \a matrix. The result of
00350   * the factorization is stored into \c *this, and a reference to \c *this
00351   * is returned.
00352   *
00353   * \sa class HouseholderQR, HouseholderQR(const MatrixType&)
00354   */
00355 template<typename MatrixType>
00356 HouseholderQR<MatrixType> & HouseholderQR<MatrixType>::compute(const MatrixType& matrix)
00357 {
00358   check_template_parameters();
00359   
00360   Index rows = matrix.rows();
00361   Index cols = matrix.cols();
00362   Index size = (std::min)(rows,cols);
00363 
00364   m_qr = matrix;
00365   m_hCoeffs.resize(size);
00366 
00367   m_temp.resize(cols);
00368 
00369   internal::householder_qr_inplace_blocked<MatrixType, HCoeffsType>::run(m_qr, m_hCoeffs, 48, m_temp.data());
00370 
00371   m_isInitialized = true;
00372   return *this;
00373 }
00374 
00375 /** \return the Householder QR decomposition of \c *this.
00376   *
00377   * \sa class HouseholderQR
00378   */
00379 template<typename Derived>
00380 const HouseholderQR<typename MatrixBase<Derived>::PlainObject>
00381 MatrixBase<Derived>::householderQr() const
00382 {
00383   return HouseholderQR<PlainObject>(eval());
00384 }
00385 
00386 } // end namespace Eigen
00387 
00388 #endif // EIGEN_QR_H