371 lines
15 KiB
C
371 lines
15 KiB
C
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// This file is part of Eigen, a lightweight C++ template library
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// for linear algebra.
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//
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// Copyright (C) 2012 Gael Guennebaud <gael.guennebaud@inria.fr>
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// Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk>
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//
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// This Source Code Form is subject to the terms of the Mozilla
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// Public License v. 2.0. If a copy of the MPL was not distributed
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// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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#ifndef EIGEN_GENERALIZEDEIGENSOLVER_H
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#define EIGEN_GENERALIZEDEIGENSOLVER_H
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#include "./RealQZ.h"
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namespace Eigen {
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/** \eigenvalues_module \ingroup Eigenvalues_Module
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*
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*
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* \class GeneralizedEigenSolver
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*
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* \brief Computes the generalized eigenvalues and eigenvectors of a pair of general matrices
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*
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* \tparam _MatrixType the type of the matrices of which we are computing the
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* eigen-decomposition; this is expected to be an instantiation of the Matrix
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* class template. Currently, only real matrices are supported.
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*
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* The generalized eigenvalues and eigenvectors of a matrix pair \f$ A \f$ and \f$ B \f$ are scalars
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* \f$ \lambda \f$ and vectors \f$ v \f$ such that \f$ Av = \lambda Bv \f$. If
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* \f$ D \f$ is a diagonal matrix with the eigenvalues on the diagonal, and
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* \f$ V \f$ is a matrix with the eigenvectors as its columns, then \f$ A V =
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* B V D \f$. The matrix \f$ V \f$ is almost always invertible, in which case we
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* have \f$ A = B V D V^{-1} \f$. This is called the generalized eigen-decomposition.
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*
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* The generalized eigenvalues and eigenvectors of a matrix pair may be complex, even when the
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* matrices are real. Moreover, the generalized eigenvalue might be infinite if the matrix B is
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* singular. To workaround this difficulty, the eigenvalues are provided as a pair of complex \f$ \alpha \f$
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* and real \f$ \beta \f$ such that: \f$ \lambda_i = \alpha_i / \beta_i \f$. If \f$ \beta_i \f$ is (nearly) zero,
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* then one can consider the well defined left eigenvalue \f$ \mu = \beta_i / \alpha_i\f$ such that:
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* \f$ \mu_i A v_i = B v_i \f$, or even \f$ \mu_i u_i^T A = u_i^T B \f$ where \f$ u_i \f$ is
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* called the left eigenvector.
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*
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* Call the function compute() to compute the generalized eigenvalues and eigenvectors of
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* a given matrix pair. Alternatively, you can use the
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* GeneralizedEigenSolver(const MatrixType&, const MatrixType&, bool) constructor which computes the
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* eigenvalues and eigenvectors at construction time. Once the eigenvalue and
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* eigenvectors are computed, they can be retrieved with the eigenvalues() and
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* eigenvectors() functions.
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*
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* Here is an usage example of this class:
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* Example: \include GeneralizedEigenSolver.cpp
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* Output: \verbinclude GeneralizedEigenSolver.out
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*
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* \sa MatrixBase::eigenvalues(), class ComplexEigenSolver, class SelfAdjointEigenSolver
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*/
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template<typename _MatrixType> class GeneralizedEigenSolver
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{
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public:
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/** \brief Synonym for the template parameter \p _MatrixType. */
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typedef _MatrixType MatrixType;
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enum {
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RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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Options = MatrixType::Options,
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MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
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};
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/** \brief Scalar type for matrices of type #MatrixType. */
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typedef typename MatrixType::Scalar Scalar;
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typedef typename NumTraits<Scalar>::Real RealScalar;
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typedef typename MatrixType::Index Index;
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/** \brief Complex scalar type for #MatrixType.
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*
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* This is \c std::complex<Scalar> if #Scalar is real (e.g.,
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* \c float or \c double) and just \c Scalar if #Scalar is
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* complex.
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*/
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typedef std::complex<RealScalar> ComplexScalar;
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/** \brief Type for vector of real scalar values eigenvalues as returned by betas().
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*
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* This is a column vector with entries of type #Scalar.
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* The length of the vector is the size of #MatrixType.
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*/
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typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> VectorType;
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/** \brief Type for vector of complex scalar values eigenvalues as returned by betas().
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*
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* This is a column vector with entries of type #ComplexScalar.
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* The length of the vector is the size of #MatrixType.
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*/
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typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ComplexVectorType;
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/** \brief Expression type for the eigenvalues as returned by eigenvalues().
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*/
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typedef CwiseBinaryOp<internal::scalar_quotient_op<ComplexScalar,Scalar>,ComplexVectorType,VectorType> EigenvalueType;
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/** \brief Type for matrix of eigenvectors as returned by eigenvectors().
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*
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* This is a square matrix with entries of type #ComplexScalar.
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* The size is the same as the size of #MatrixType.
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*/
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typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> EigenvectorsType;
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/** \brief Default constructor.
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*
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* The default constructor is useful in cases in which the user intends to
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* perform decompositions via EigenSolver::compute(const MatrixType&, bool).
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*
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* \sa compute() for an example.
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*/
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GeneralizedEigenSolver() : m_eivec(), m_alphas(), m_betas(), m_isInitialized(false), m_realQZ(), m_matS(), m_tmp() {}
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/** \brief Default constructor with memory preallocation
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*
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* Like the default constructor but with preallocation of the internal data
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* according to the specified problem \a size.
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* \sa GeneralizedEigenSolver()
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*/
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GeneralizedEigenSolver(Index size)
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: m_eivec(size, size),
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m_alphas(size),
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m_betas(size),
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m_isInitialized(false),
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m_eigenvectorsOk(false),
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m_realQZ(size),
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m_matS(size, size),
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m_tmp(size)
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{}
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/** \brief Constructor; computes the generalized eigendecomposition of given matrix pair.
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*
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* \param[in] A Square matrix whose eigendecomposition is to be computed.
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* \param[in] B Square matrix whose eigendecomposition is to be computed.
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* \param[in] computeEigenvectors If true, both the eigenvectors and the
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* eigenvalues are computed; if false, only the eigenvalues are computed.
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*
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* This constructor calls compute() to compute the generalized eigenvalues
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* and eigenvectors.
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*
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* \sa compute()
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*/
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GeneralizedEigenSolver(const MatrixType& A, const MatrixType& B, bool computeEigenvectors = true)
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: m_eivec(A.rows(), A.cols()),
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m_alphas(A.cols()),
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m_betas(A.cols()),
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m_isInitialized(false),
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m_eigenvectorsOk(false),
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m_realQZ(A.cols()),
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m_matS(A.rows(), A.cols()),
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m_tmp(A.cols())
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{
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compute(A, B, computeEigenvectors);
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}
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/* \brief Returns the computed generalized eigenvectors.
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*
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* \returns %Matrix whose columns are the (possibly complex) eigenvectors.
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*
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* \pre Either the constructor
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* GeneralizedEigenSolver(const MatrixType&,const MatrixType&, bool) or the member function
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* compute(const MatrixType&, const MatrixType& bool) has been called before, and
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* \p computeEigenvectors was set to true (the default).
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*
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* Column \f$ k \f$ of the returned matrix is an eigenvector corresponding
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* to eigenvalue number \f$ k \f$ as returned by eigenvalues(). The
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* eigenvectors are normalized to have (Euclidean) norm equal to one. The
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* matrix returned by this function is the matrix \f$ V \f$ in the
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* generalized eigendecomposition \f$ A = B V D V^{-1} \f$, if it exists.
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*
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* \sa eigenvalues()
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*/
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// EigenvectorsType eigenvectors() const;
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/** \brief Returns an expression of the computed generalized eigenvalues.
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*
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* \returns An expression of the column vector containing the eigenvalues.
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*
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* It is a shortcut for \code this->alphas().cwiseQuotient(this->betas()); \endcode
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* Not that betas might contain zeros. It is therefore not recommended to use this function,
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* but rather directly deal with the alphas and betas vectors.
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*
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* \pre Either the constructor
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* GeneralizedEigenSolver(const MatrixType&,const MatrixType&,bool) or the member function
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* compute(const MatrixType&,const MatrixType&,bool) has been called before.
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*
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* The eigenvalues are repeated according to their algebraic multiplicity,
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* so there are as many eigenvalues as rows in the matrix. The eigenvalues
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* are not sorted in any particular order.
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*
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* \sa alphas(), betas(), eigenvectors()
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*/
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EigenvalueType eigenvalues() const
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{
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eigen_assert(m_isInitialized && "GeneralizedEigenSolver is not initialized.");
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return EigenvalueType(m_alphas,m_betas);
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}
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/** \returns A const reference to the vectors containing the alpha values
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*
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* This vector permits to reconstruct the j-th eigenvalues as alphas(i)/betas(j).
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*
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* \sa betas(), eigenvalues() */
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ComplexVectorType alphas() const
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{
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eigen_assert(m_isInitialized && "GeneralizedEigenSolver is not initialized.");
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return m_alphas;
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}
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/** \returns A const reference to the vectors containing the beta values
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*
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* This vector permits to reconstruct the j-th eigenvalues as alphas(i)/betas(j).
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*
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* \sa alphas(), eigenvalues() */
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VectorType betas() const
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{
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eigen_assert(m_isInitialized && "GeneralizedEigenSolver is not initialized.");
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return m_betas;
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}
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/** \brief Computes generalized eigendecomposition of given matrix.
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*
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* \param[in] A Square matrix whose eigendecomposition is to be computed.
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* \param[in] B Square matrix whose eigendecomposition is to be computed.
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* \param[in] computeEigenvectors If true, both the eigenvectors and the
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* eigenvalues are computed; if false, only the eigenvalues are
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* computed.
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* \returns Reference to \c *this
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*
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* This function computes the eigenvalues of the real matrix \p matrix.
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* The eigenvalues() function can be used to retrieve them. If
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* \p computeEigenvectors is true, then the eigenvectors are also computed
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* and can be retrieved by calling eigenvectors().
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*
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* The matrix is first reduced to real generalized Schur form using the RealQZ
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* class. The generalized Schur decomposition is then used to compute the eigenvalues
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* and eigenvectors.
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*
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* The cost of the computation is dominated by the cost of the
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* generalized Schur decomposition.
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*
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* This method reuses of the allocated data in the GeneralizedEigenSolver object.
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*/
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GeneralizedEigenSolver& compute(const MatrixType& A, const MatrixType& B, bool computeEigenvectors = true);
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ComputationInfo info() const
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{
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eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
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return m_realQZ.info();
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}
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/** Sets the maximal number of iterations allowed.
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*/
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GeneralizedEigenSolver& setMaxIterations(Index maxIters)
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{
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m_realQZ.setMaxIterations(maxIters);
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return *this;
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}
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protected:
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static void check_template_parameters()
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{
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EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
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EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsComplex, NUMERIC_TYPE_MUST_BE_REAL);
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}
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MatrixType m_eivec;
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ComplexVectorType m_alphas;
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VectorType m_betas;
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bool m_isInitialized;
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bool m_eigenvectorsOk;
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RealQZ<MatrixType> m_realQZ;
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MatrixType m_matS;
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typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;
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ColumnVectorType m_tmp;
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};
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//template<typename MatrixType>
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//typename GeneralizedEigenSolver<MatrixType>::EigenvectorsType GeneralizedEigenSolver<MatrixType>::eigenvectors() const
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//{
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// eigen_assert(m_isInitialized && "EigenSolver is not initialized.");
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// eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
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// Index n = m_eivec.cols();
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// EigenvectorsType matV(n,n);
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// // TODO
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// return matV;
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//}
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template<typename MatrixType>
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GeneralizedEigenSolver<MatrixType>&
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GeneralizedEigenSolver<MatrixType>::compute(const MatrixType& A, const MatrixType& B, bool computeEigenvectors)
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{
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check_template_parameters();
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using std::sqrt;
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using std::abs;
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eigen_assert(A.cols() == A.rows() && B.cols() == A.rows() && B.cols() == B.rows());
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// Reduce to generalized real Schur form:
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// A = Q S Z and B = Q T Z
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m_realQZ.compute(A, B, computeEigenvectors);
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if (m_realQZ.info() == Success)
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{
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m_matS = m_realQZ.matrixS();
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if (computeEigenvectors)
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m_eivec = m_realQZ.matrixZ().transpose();
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// Compute eigenvalues from matS
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m_alphas.resize(A.cols());
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m_betas.resize(A.cols());
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Index i = 0;
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while (i < A.cols())
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{
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if (i == A.cols() - 1 || m_matS.coeff(i+1, i) == Scalar(0))
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{
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m_alphas.coeffRef(i) = m_matS.coeff(i, i);
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m_betas.coeffRef(i) = m_realQZ.matrixT().coeff(i,i);
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++i;
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}
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else
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{
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// We need to extract the generalized eigenvalues of the pair of a general 2x2 block S and a triangular 2x2 block T
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// From the eigen decomposition of T = U * E * U^-1,
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// we can extract the eigenvalues of (U^-1 * S * U) / E
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// Here, we can take advantage that E = diag(T), and U = [ 1 T_01 ; 0 T_11-T_00], and U^-1 = [1 -T_11/(T_11-T_00) ; 0 1/(T_11-T_00)].
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// Then taking beta=T_00*T_11*(T_11-T_00), we can avoid any division, and alpha is the eigenvalues of A = (U^-1 * S * U) * diag(T_11,T_00) * (T_11-T_00):
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// T = [a b ; 0 c]
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// S = [e f ; g h]
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RealScalar a = m_realQZ.matrixT().coeff(i, i), b = m_realQZ.matrixT().coeff(i, i+1), c = m_realQZ.matrixT().coeff(i+1, i+1);
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RealScalar e = m_matS.coeff(i, i), f = m_matS.coeff(i, i+1), g = m_matS.coeff(i+1, i), h = m_matS.coeff(i+1, i+1);
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RealScalar d = c-a;
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RealScalar gb = g*b;
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Matrix<RealScalar,2,2> A;
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A << (e*d-gb)*c, ((e*b+f*d-h*b)*d-gb*b)*a,
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g*c , (gb+h*d)*a;
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// NOTE, we could also compute the SVD of T's block during the QZ factorization so that the respective T block is guaranteed to be diagonal,
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// and then we could directly apply the formula below (while taking care of scaling S columns by T11,T00):
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Scalar p = Scalar(0.5) * (A.coeff(i, i) - A.coeff(i+1, i+1));
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Scalar z = sqrt(abs(p * p + A.coeff(i+1, i) * A.coeff(i, i+1)));
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m_alphas.coeffRef(i) = ComplexScalar(A.coeff(i+1, i+1) + p, z);
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m_alphas.coeffRef(i+1) = ComplexScalar(A.coeff(i+1, i+1) + p, -z);
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m_betas.coeffRef(i) =
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m_betas.coeffRef(i+1) = a*c*d;
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i += 2;
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||
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}
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}
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}
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m_isInitialized = true;
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m_eigenvectorsOk = false;//computeEigenvectors;
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return *this;
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||
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|
}
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} // end namespace Eigen
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#endif // EIGEN_GENERALIZEDEIGENSOLVER_H
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