| // 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) 2010 Jitse Niesen <jitse@maths.leeds.ac.uk> |
| // |
| // 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_HESSENBERGDECOMPOSITION_H |
| #define EIGEN_HESSENBERGDECOMPOSITION_H |
| |
| namespace internal { |
| |
| template<typename MatrixType> struct HessenbergDecompositionMatrixHReturnType; |
| template<typename MatrixType> |
| struct traits<HessenbergDecompositionMatrixHReturnType<MatrixType> > |
| { |
| typedef MatrixType ReturnType; |
| }; |
| |
| } |
| |
| /** \eigenvalues_module \ingroup Eigenvalues_Module |
| * |
| * |
| * \class HessenbergDecomposition |
| * |
| * \brief Reduces a square matrix to Hessenberg form by an orthogonal similarity transformation |
| * |
| * \tparam _MatrixType the type of the matrix of which we are computing the Hessenberg decomposition |
| * |
| * This class performs an Hessenberg decomposition of a matrix \f$ A \f$. In |
| * the real case, the Hessenberg decomposition consists of an orthogonal |
| * matrix \f$ Q \f$ and a Hessenberg matrix \f$ H \f$ such that \f$ A = Q H |
| * Q^T \f$. An orthogonal matrix is a matrix whose inverse equals its |
| * transpose (\f$ Q^{-1} = Q^T \f$). A Hessenberg matrix has zeros below the |
| * subdiagonal, so it is almost upper triangular. The Hessenberg decomposition |
| * of a complex matrix is \f$ A = Q H Q^* \f$ with \f$ Q \f$ unitary (that is, |
| * \f$ Q^{-1} = Q^* \f$). |
| * |
| * Call the function compute() to compute the Hessenberg decomposition of a |
| * given matrix. Alternatively, you can use the |
| * HessenbergDecomposition(const MatrixType&) constructor which computes the |
| * Hessenberg decomposition at construction time. Once the decomposition is |
| * computed, you can use the matrixH() and matrixQ() functions to construct |
| * the matrices H and Q in the decomposition. |
| * |
| * The documentation for matrixH() contains an example of the typical use of |
| * this class. |
| * |
| * \sa class ComplexSchur, class Tridiagonalization, \ref QR_Module "QR Module" |
| */ |
| template<typename _MatrixType> class HessenbergDecomposition |
| { |
| public: |
| |
| /** \brief Synonym for the template parameter \p _MatrixType. */ |
| typedef _MatrixType MatrixType; |
| |
| enum { |
| Size = MatrixType::RowsAtCompileTime, |
| SizeMinusOne = Size == Dynamic ? Dynamic : Size - 1, |
| Options = MatrixType::Options, |
| MaxSize = MatrixType::MaxRowsAtCompileTime, |
| MaxSizeMinusOne = MaxSize == Dynamic ? Dynamic : MaxSize - 1 |
| }; |
| |
| /** \brief Scalar type for matrices of type #MatrixType. */ |
| typedef typename MatrixType::Scalar Scalar; |
| typedef typename MatrixType::Index Index; |
| |
| /** \brief Type for vector of Householder coefficients. |
| * |
| * This is column vector with entries of type #Scalar. The length of the |
| * vector is one less than the size of #MatrixType, if it is a fixed-side |
| * type. |
| */ |
| typedef Matrix<Scalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> CoeffVectorType; |
| |
| /** \brief Return type of matrixQ() */ |
| typedef typename HouseholderSequence<MatrixType,CoeffVectorType>::ConjugateReturnType HouseholderSequenceType; |
| |
| typedef internal::HessenbergDecompositionMatrixHReturnType<MatrixType> MatrixHReturnType; |
| |
| /** \brief Default constructor; the decomposition will be computed later. |
| * |
| * \param [in] size The size of the matrix whose Hessenberg decomposition will be computed. |
| * |
| * The default constructor is useful in cases in which the user intends to |
| * perform decompositions via compute(). The \p size parameter is only |
| * used as a hint. It is not an error to give a wrong \p size, but it may |
| * impair performance. |
| * |
| * \sa compute() for an example. |
| */ |
| HessenbergDecomposition(Index size = Size==Dynamic ? 2 : Size) |
| : m_matrix(size,size), |
| m_temp(size), |
| m_isInitialized(false) |
| { |
| if(size>1) |
| m_hCoeffs.resize(size-1); |
| } |
| |
| /** \brief Constructor; computes Hessenberg decomposition of given matrix. |
| * |
| * \param[in] matrix Square matrix whose Hessenberg decomposition is to be computed. |
| * |
| * This constructor calls compute() to compute the Hessenberg |
| * decomposition. |
| * |
| * \sa matrixH() for an example. |
| */ |
| HessenbergDecomposition(const MatrixType& matrix) |
| : m_matrix(matrix), |
| m_temp(matrix.rows()), |
| m_isInitialized(false) |
| { |
| if(matrix.rows()<2) |
| { |
| m_isInitialized = true; |
| return; |
| } |
| m_hCoeffs.resize(matrix.rows()-1,1); |
| _compute(m_matrix, m_hCoeffs, m_temp); |
| m_isInitialized = true; |
| } |
| |
| /** \brief Computes Hessenberg decomposition of given matrix. |
| * |
| * \param[in] matrix Square matrix whose Hessenberg decomposition is to be computed. |
| * \returns Reference to \c *this |
| * |
| * The Hessenberg decomposition is computed by bringing the columns of the |
| * matrix successively in the required form using Householder reflections |
| * (see, e.g., Algorithm 7.4.2 in Golub \& Van Loan, <i>%Matrix |
| * Computations</i>). The cost is \f$ 10n^3/3 \f$ flops, where \f$ n \f$ |
| * denotes the size of the given matrix. |
| * |
| * This method reuses of the allocated data in the HessenbergDecomposition |
| * object. |
| * |
| * Example: \include HessenbergDecomposition_compute.cpp |
| * Output: \verbinclude HessenbergDecomposition_compute.out |
| */ |
| HessenbergDecomposition& compute(const MatrixType& matrix) |
| { |
| m_matrix = matrix; |
| if(matrix.rows()<2) |
| { |
| m_isInitialized = true; |
| return *this; |
| } |
| m_hCoeffs.resize(matrix.rows()-1,1); |
| _compute(m_matrix, m_hCoeffs, m_temp); |
| m_isInitialized = true; |
| return *this; |
| } |
| |
| /** \brief Returns the Householder coefficients. |
| * |
| * \returns a const reference to the vector of Householder coefficients |
| * |
| * \pre Either the constructor HessenbergDecomposition(const MatrixType&) |
| * or the member function compute(const MatrixType&) has been called |
| * before to compute the Hessenberg decomposition of a matrix. |
| * |
| * The Householder coefficients allow the reconstruction of the matrix |
| * \f$ Q \f$ in the Hessenberg decomposition from the packed data. |
| * |
| * \sa packedMatrix(), \ref Householder_Module "Householder module" |
| */ |
| const CoeffVectorType& householderCoefficients() const |
| { |
| eigen_assert(m_isInitialized && "HessenbergDecomposition is not initialized."); |
| return m_hCoeffs; |
| } |
| |
| /** \brief Returns the internal representation of the decomposition |
| * |
| * \returns a const reference to a matrix with the internal representation |
| * of the decomposition. |
| * |
| * \pre Either the constructor HessenbergDecomposition(const MatrixType&) |
| * or the member function compute(const MatrixType&) has been called |
| * before to compute the Hessenberg decomposition of a matrix. |
| * |
| * The returned matrix contains the following information: |
| * - the upper part and lower sub-diagonal represent the Hessenberg matrix H |
| * - the rest of the lower part contains the Householder vectors that, combined with |
| * Householder coefficients returned by householderCoefficients(), |
| * allows to reconstruct the matrix Q as |
| * \f$ Q = H_{N-1} \ldots H_1 H_0 \f$. |
| * Here, the matrices \f$ H_i \f$ are the Householder transformations |
| * \f$ H_i = (I - h_i v_i v_i^T) \f$ |
| * where \f$ h_i \f$ is the \f$ i \f$th Householder coefficient and |
| * \f$ v_i \f$ is the Householder vector defined by |
| * \f$ v_i = [ 0, \ldots, 0, 1, M(i+2,i), \ldots, M(N-1,i) ]^T \f$ |
| * with M the matrix returned by this function. |
| * |
| * See LAPACK for further details on this packed storage. |
| * |
| * Example: \include HessenbergDecomposition_packedMatrix.cpp |
| * Output: \verbinclude HessenbergDecomposition_packedMatrix.out |
| * |
| * \sa householderCoefficients() |
| */ |
| const MatrixType& packedMatrix() const |
| { |
| eigen_assert(m_isInitialized && "HessenbergDecomposition is not initialized."); |
| return m_matrix; |
| } |
| |
| /** \brief Reconstructs the orthogonal matrix Q in the decomposition |
| * |
| * \returns object representing the matrix Q |
| * |
| * \pre Either the constructor HessenbergDecomposition(const MatrixType&) |
| * or the member function compute(const MatrixType&) has been called |
| * before to compute the Hessenberg decomposition of a matrix. |
| * |
| * This function returns a light-weight object of template class |
| * HouseholderSequence. You can either apply it directly to a matrix or |
| * you can convert it to a matrix of type #MatrixType. |
| * |
| * \sa matrixH() for an example, class HouseholderSequence |
| */ |
| HouseholderSequenceType matrixQ() const |
| { |
| eigen_assert(m_isInitialized && "HessenbergDecomposition is not initialized."); |
| return HouseholderSequenceType(m_matrix, m_hCoeffs.conjugate()) |
| .setLength(m_matrix.rows() - 1) |
| .setShift(1); |
| } |
| |
| /** \brief Constructs the Hessenberg matrix H in the decomposition |
| * |
| * \returns expression object representing the matrix H |
| * |
| * \pre Either the constructor HessenbergDecomposition(const MatrixType&) |
| * or the member function compute(const MatrixType&) has been called |
| * before to compute the Hessenberg decomposition of a matrix. |
| * |
| * The object returned by this function constructs the Hessenberg matrix H |
| * when it is assigned to a matrix or otherwise evaluated. The matrix H is |
| * constructed from the packed matrix as returned by packedMatrix(): The |
| * upper part (including the subdiagonal) of the packed matrix contains |
| * the matrix H. It may sometimes be better to directly use the packed |
| * matrix instead of constructing the matrix H. |
| * |
| * Example: \include HessenbergDecomposition_matrixH.cpp |
| * Output: \verbinclude HessenbergDecomposition_matrixH.out |
| * |
| * \sa matrixQ(), packedMatrix() |
| */ |
| MatrixHReturnType matrixH() const |
| { |
| eigen_assert(m_isInitialized && "HessenbergDecomposition is not initialized."); |
| return MatrixHReturnType(*this); |
| } |
| |
| private: |
| |
| typedef Matrix<Scalar, 1, Size, Options | RowMajor, 1, MaxSize> VectorType; |
| typedef typename NumTraits<Scalar>::Real RealScalar; |
| static void _compute(MatrixType& matA, CoeffVectorType& hCoeffs, VectorType& temp); |
| |
| protected: |
| MatrixType m_matrix; |
| CoeffVectorType m_hCoeffs; |
| VectorType m_temp; |
| bool m_isInitialized; |
| }; |
| |
| /** \internal |
| * Performs a tridiagonal decomposition of \a matA in place. |
| * |
| * \param matA the input selfadjoint matrix |
| * \param hCoeffs returned Householder coefficients |
| * |
| * The result is written in the lower triangular part of \a matA. |
| * |
| * Implemented from Golub's "%Matrix Computations", algorithm 8.3.1. |
| * |
| * \sa packedMatrix() |
| */ |
| template<typename MatrixType> |
| void HessenbergDecomposition<MatrixType>::_compute(MatrixType& matA, CoeffVectorType& hCoeffs, VectorType& temp) |
| { |
| assert(matA.rows()==matA.cols()); |
| Index n = matA.rows(); |
| temp.resize(n); |
| for (Index i = 0; i<n-1; ++i) |
| { |
| // let's consider the vector v = i-th column starting at position i+1 |
| Index remainingSize = n-i-1; |
| RealScalar beta; |
| Scalar h; |
| matA.col(i).tail(remainingSize).makeHouseholderInPlace(h, beta); |
| matA.col(i).coeffRef(i+1) = beta; |
| hCoeffs.coeffRef(i) = h; |
| |
| // Apply similarity transformation to remaining columns, |
| // i.e., compute A = H A H' |
| |
| // A = H A |
| matA.bottomRightCorner(remainingSize, remainingSize) |
| .applyHouseholderOnTheLeft(matA.col(i).tail(remainingSize-1), h, &temp.coeffRef(0)); |
| |
| // A = A H' |
| matA.rightCols(remainingSize) |
| .applyHouseholderOnTheRight(matA.col(i).tail(remainingSize-1).conjugate(), internal::conj(h), &temp.coeffRef(0)); |
| } |
| } |
| |
| namespace internal { |
| |
| /** \eigenvalues_module \ingroup Eigenvalues_Module |
| * |
| * |
| * \brief Expression type for return value of HessenbergDecomposition::matrixH() |
| * |
| * \tparam MatrixType type of matrix in the Hessenberg decomposition |
| * |
| * Objects of this type represent the Hessenberg matrix in the Hessenberg |
| * decomposition of some matrix. The object holds a reference to the |
| * HessenbergDecomposition class until the it is assigned or evaluated for |
| * some other reason (the reference should remain valid during the life time |
| * of this object). This class is the return type of |
| * HessenbergDecomposition::matrixH(); there is probably no other use for this |
| * class. |
| */ |
| template<typename MatrixType> struct HessenbergDecompositionMatrixHReturnType |
| : public ReturnByValue<HessenbergDecompositionMatrixHReturnType<MatrixType> > |
| { |
| typedef typename MatrixType::Index Index; |
| public: |
| /** \brief Constructor. |
| * |
| * \param[in] hess Hessenberg decomposition |
| */ |
| HessenbergDecompositionMatrixHReturnType(const HessenbergDecomposition<MatrixType>& hess) : m_hess(hess) { } |
| |
| /** \brief Hessenberg matrix in decomposition. |
| * |
| * \param[out] result Hessenberg matrix in decomposition \p hess which |
| * was passed to the constructor |
| */ |
| template <typename ResultType> |
| inline void evalTo(ResultType& result) const |
| { |
| result = m_hess.packedMatrix(); |
| Index n = result.rows(); |
| if (n>2) |
| result.bottomLeftCorner(n-2, n-2).template triangularView<Lower>().setZero(); |
| } |
| |
| Index rows() const { return m_hess.packedMatrix().rows(); } |
| Index cols() const { return m_hess.packedMatrix().cols(); } |
| |
| protected: |
| const HessenbergDecomposition<MatrixType>& m_hess; |
| }; |
| |
| } |
| |
| #endif // EIGEN_HESSENBERGDECOMPOSITION_H |