SSL HessenbergDecomposition.h
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// 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> // // This Source Code Form is subject to the terms of the Mozilla // Public License v. 2.0. If a copy of the MPL was not distributed // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
/** \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;
/** \brief Scalar type for matrices of type #MatrixType. */ typedeftypename MatrixType::Scalar Scalar; typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3
/** \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 HouseholderSequence<MatrixType,typename internal::remove_all<typename CoeffVectorType::ConjugateReturnType>::type> HouseholderSequenceType;
/** \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.
*/ explicit 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.
*/ template<typename InputType> explicit HessenbergDecomposition(const EigenBase<InputType>& matrix)
: m_matrix(matrix.derived()),
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
*/ template<typename InputType>
HessenbergDecomposition& compute(const EigenBase<InputType>& matrix)
{
m_matrix = matrix.derived(); 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);
}
/** \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)
{
eigen_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), numext::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> >
{ 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> inlinevoid 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(); }
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