625 lines
22 KiB
C++
625 lines
22 KiB
C++
// 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 Alexey Korepanov <kaikaikai@yandex.ru>
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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_REAL_QZ_H
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#define EIGEN_REAL_QZ_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 RealQZ
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*
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* \brief Performs a real QZ decomposition of a pair of square matrices
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*
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* \tparam _MatrixType the type of the matrix of which we are computing the
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* real QZ decomposition; this is expected to be an instantiation of the
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* Matrix class template.
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*
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* Given a real square matrices A and B, this class computes the real QZ
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* decomposition: \f$ A = Q S Z \f$, \f$ B = Q T Z \f$ where Q and Z are
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* real orthogonal matrixes, T is upper-triangular matrix, and S is upper
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* quasi-triangular matrix. An orthogonal matrix is a matrix whose
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* inverse is equal to its transpose, \f$ U^{-1} = U^T \f$. A quasi-triangular
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* matrix is a block-triangular matrix whose diagonal consists of 1-by-1
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* blocks and 2-by-2 blocks where further reduction is impossible due to
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* complex eigenvalues.
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*
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* The eigenvalues of the pencil \f$ A - z B \f$ can be obtained from
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* 1x1 and 2x2 blocks on the diagonals of S and T.
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*
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* Call the function compute() to compute the real QZ decomposition of a
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* given pair of matrices. Alternatively, you can use the
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* RealQZ(const MatrixType& B, const MatrixType& B, bool computeQZ)
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* constructor which computes the real QZ decomposition at construction
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* time. Once the decomposition is computed, you can use the matrixS(),
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* matrixT(), matrixQ() and matrixZ() functions to retrieve the matrices
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* S, T, Q and Z in the decomposition. If computeQZ==false, some time
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* is saved by not computing matrices Q and Z.
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*
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* Example: \include RealQZ_compute.cpp
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* Output: \include RealQZ_compute.out
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*
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* \note The implementation is based on the algorithm in "Matrix Computations"
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* by Gene H. Golub and Charles F. Van Loan, and a paper "An algorithm for
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* generalized eigenvalue problems" by C.B.Moler and G.W.Stewart.
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*
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* \sa class RealSchur, class ComplexSchur, class EigenSolver, class ComplexEigenSolver
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*/
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template<typename _MatrixType> class RealQZ
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{
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public:
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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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typedef typename MatrixType::Scalar Scalar;
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typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
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typedef typename MatrixType::Index Index;
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typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType;
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typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;
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/** \brief Default constructor.
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*
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* \param [in] size Positive integer, size of the matrix whose QZ decomposition will be computed.
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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 compute(). The \p size parameter is only
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* used as a hint. It is not an error to give a wrong \p size, but it may
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* impair performance.
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*
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* \sa compute() for an example.
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*/
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RealQZ(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime) :
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m_S(size, size),
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m_T(size, size),
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m_Q(size, size),
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m_Z(size, size),
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m_workspace(size*2),
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m_maxIters(400),
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m_isInitialized(false)
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{ }
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/** \brief Constructor; computes real QZ decomposition of given matrices
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*
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* \param[in] A Matrix A.
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* \param[in] B Matrix B.
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* \param[in] computeQZ If false, A and Z are not computed.
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*
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* This constructor calls compute() to compute the QZ decomposition.
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*/
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RealQZ(const MatrixType& A, const MatrixType& B, bool computeQZ = true) :
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m_S(A.rows(),A.cols()),
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m_T(A.rows(),A.cols()),
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m_Q(A.rows(),A.cols()),
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m_Z(A.rows(),A.cols()),
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m_workspace(A.rows()*2),
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m_maxIters(400),
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m_isInitialized(false) {
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compute(A, B, computeQZ);
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}
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/** \brief Returns matrix Q in the QZ decomposition.
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*
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* \returns A const reference to the matrix Q.
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*/
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const MatrixType& matrixQ() const {
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eigen_assert(m_isInitialized && "RealQZ is not initialized.");
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eigen_assert(m_computeQZ && "The matrices Q and Z have not been computed during the QZ decomposition.");
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return m_Q;
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}
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/** \brief Returns matrix Z in the QZ decomposition.
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*
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* \returns A const reference to the matrix Z.
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*/
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const MatrixType& matrixZ() const {
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eigen_assert(m_isInitialized && "RealQZ is not initialized.");
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eigen_assert(m_computeQZ && "The matrices Q and Z have not been computed during the QZ decomposition.");
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return m_Z;
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}
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/** \brief Returns matrix S in the QZ decomposition.
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*
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* \returns A const reference to the matrix S.
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*/
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const MatrixType& matrixS() const {
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eigen_assert(m_isInitialized && "RealQZ is not initialized.");
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return m_S;
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}
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/** \brief Returns matrix S in the QZ decomposition.
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*
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* \returns A const reference to the matrix S.
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*/
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const MatrixType& matrixT() const {
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eigen_assert(m_isInitialized && "RealQZ is not initialized.");
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return m_T;
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}
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/** \brief Computes QZ decomposition of given matrix.
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*
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* \param[in] A Matrix A.
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* \param[in] B Matrix B.
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* \param[in] computeQZ If false, A and Z are not computed.
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* \returns Reference to \c *this
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*/
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RealQZ& compute(const MatrixType& A, const MatrixType& B, bool computeQZ = true);
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/** \brief Reports whether previous computation was successful.
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*
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* \returns \c Success if computation was succesful, \c NoConvergence otherwise.
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*/
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ComputationInfo info() const
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{
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eigen_assert(m_isInitialized && "RealQZ is not initialized.");
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return m_info;
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}
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/** \brief Returns number of performed QR-like iterations.
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*/
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Index iterations() const
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{
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eigen_assert(m_isInitialized && "RealQZ is not initialized.");
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return m_global_iter;
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}
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/** Sets the maximal number of iterations allowed to converge to one eigenvalue
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* or decouple the problem.
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*/
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RealQZ& setMaxIterations(Index maxIters)
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{
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m_maxIters = maxIters;
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return *this;
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}
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private:
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MatrixType m_S, m_T, m_Q, m_Z;
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Matrix<Scalar,Dynamic,1> m_workspace;
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ComputationInfo m_info;
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Index m_maxIters;
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bool m_isInitialized;
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bool m_computeQZ;
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Scalar m_normOfT, m_normOfS;
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Index m_global_iter;
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typedef Matrix<Scalar,3,1> Vector3s;
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typedef Matrix<Scalar,2,1> Vector2s;
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typedef Matrix<Scalar,2,2> Matrix2s;
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typedef JacobiRotation<Scalar> JRs;
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void hessenbergTriangular();
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void computeNorms();
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Index findSmallSubdiagEntry(Index iu);
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Index findSmallDiagEntry(Index f, Index l);
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void splitOffTwoRows(Index i);
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void pushDownZero(Index z, Index f, Index l);
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void step(Index f, Index l, Index iter);
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}; // RealQZ
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/** \internal Reduces S and T to upper Hessenberg - triangular form */
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template<typename MatrixType>
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void RealQZ<MatrixType>::hessenbergTriangular()
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{
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const Index dim = m_S.cols();
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// perform QR decomposition of T, overwrite T with R, save Q
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HouseholderQR<MatrixType> qrT(m_T);
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m_T = qrT.matrixQR();
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m_T.template triangularView<StrictlyLower>().setZero();
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m_Q = qrT.householderQ();
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// overwrite S with Q* S
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m_S.applyOnTheLeft(m_Q.adjoint());
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// init Z as Identity
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if (m_computeQZ)
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m_Z = MatrixType::Identity(dim,dim);
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// reduce S to upper Hessenberg with Givens rotations
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for (Index j=0; j<=dim-3; j++) {
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for (Index i=dim-1; i>=j+2; i--) {
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JRs G;
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// kill S(i,j)
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if(m_S.coeff(i,j) != 0)
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{
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G.makeGivens(m_S.coeff(i-1,j), m_S.coeff(i,j), &m_S.coeffRef(i-1, j));
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m_S.coeffRef(i,j) = Scalar(0.0);
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m_S.rightCols(dim-j-1).applyOnTheLeft(i-1,i,G.adjoint());
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m_T.rightCols(dim-i+1).applyOnTheLeft(i-1,i,G.adjoint());
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// update Q
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if (m_computeQZ)
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m_Q.applyOnTheRight(i-1,i,G);
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}
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// kill T(i,i-1)
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if(m_T.coeff(i,i-1)!=Scalar(0))
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{
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G.makeGivens(m_T.coeff(i,i), m_T.coeff(i,i-1), &m_T.coeffRef(i,i));
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m_T.coeffRef(i,i-1) = Scalar(0.0);
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m_S.applyOnTheRight(i,i-1,G);
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m_T.topRows(i).applyOnTheRight(i,i-1,G);
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// update Z
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if (m_computeQZ)
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m_Z.applyOnTheLeft(i,i-1,G.adjoint());
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}
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}
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}
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}
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/** \internal Computes vector L1 norms of S and T when in Hessenberg-Triangular form already */
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template<typename MatrixType>
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inline void RealQZ<MatrixType>::computeNorms()
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{
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const Index size = m_S.cols();
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m_normOfS = Scalar(0.0);
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m_normOfT = Scalar(0.0);
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for (Index j = 0; j < size; ++j)
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{
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m_normOfS += m_S.col(j).segment(0, (std::min)(size,j+2)).cwiseAbs().sum();
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m_normOfT += m_T.row(j).segment(j, size - j).cwiseAbs().sum();
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}
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}
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/** \internal Look for single small sub-diagonal element S(res, res-1) and return res (or 0) */
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template<typename MatrixType>
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inline typename MatrixType::Index RealQZ<MatrixType>::findSmallSubdiagEntry(Index iu)
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{
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using std::abs;
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Index res = iu;
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while (res > 0)
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{
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Scalar s = abs(m_S.coeff(res-1,res-1)) + abs(m_S.coeff(res,res));
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if (s == Scalar(0.0))
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s = m_normOfS;
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if (abs(m_S.coeff(res,res-1)) < NumTraits<Scalar>::epsilon() * s)
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break;
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res--;
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}
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return res;
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}
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/** \internal Look for single small diagonal element T(res, res) for res between f and l, and return res (or f-1) */
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template<typename MatrixType>
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inline typename MatrixType::Index RealQZ<MatrixType>::findSmallDiagEntry(Index f, Index l)
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{
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using std::abs;
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Index res = l;
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while (res >= f) {
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if (abs(m_T.coeff(res,res)) <= NumTraits<Scalar>::epsilon() * m_normOfT)
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break;
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res--;
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}
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return res;
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}
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/** \internal decouple 2x2 diagonal block in rows i, i+1 if eigenvalues are real */
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template<typename MatrixType>
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inline void RealQZ<MatrixType>::splitOffTwoRows(Index i)
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{
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using std::abs;
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using std::sqrt;
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const Index dim=m_S.cols();
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if (abs(m_S.coeff(i+1,i))==Scalar(0))
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return;
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Index z = findSmallDiagEntry(i,i+1);
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if (z==i-1)
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{
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// block of (S T^{-1})
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Matrix2s STi = m_T.template block<2,2>(i,i).template triangularView<Upper>().
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template solve<OnTheRight>(m_S.template block<2,2>(i,i));
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Scalar p = Scalar(0.5)*(STi(0,0)-STi(1,1));
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Scalar q = p*p + STi(1,0)*STi(0,1);
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if (q>=0) {
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Scalar z = sqrt(q);
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// one QR-like iteration for ABi - lambda I
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// is enough - when we know exact eigenvalue in advance,
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// convergence is immediate
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JRs G;
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if (p>=0)
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G.makeGivens(p + z, STi(1,0));
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else
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G.makeGivens(p - z, STi(1,0));
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m_S.rightCols(dim-i).applyOnTheLeft(i,i+1,G.adjoint());
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m_T.rightCols(dim-i).applyOnTheLeft(i,i+1,G.adjoint());
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// update Q
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if (m_computeQZ)
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m_Q.applyOnTheRight(i,i+1,G);
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G.makeGivens(m_T.coeff(i+1,i+1), m_T.coeff(i+1,i));
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m_S.topRows(i+2).applyOnTheRight(i+1,i,G);
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m_T.topRows(i+2).applyOnTheRight(i+1,i,G);
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// update Z
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if (m_computeQZ)
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m_Z.applyOnTheLeft(i+1,i,G.adjoint());
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m_S.coeffRef(i+1,i) = Scalar(0.0);
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m_T.coeffRef(i+1,i) = Scalar(0.0);
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}
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}
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else
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{
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pushDownZero(z,i,i+1);
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}
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}
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/** \internal use zero in T(z,z) to zero S(l,l-1), working in block f..l */
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template<typename MatrixType>
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inline void RealQZ<MatrixType>::pushDownZero(Index z, Index f, Index l)
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{
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JRs G;
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const Index dim = m_S.cols();
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for (Index zz=z; zz<l; zz++)
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{
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// push 0 down
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Index firstColS = zz>f ? (zz-1) : zz;
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G.makeGivens(m_T.coeff(zz, zz+1), m_T.coeff(zz+1, zz+1));
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m_S.rightCols(dim-firstColS).applyOnTheLeft(zz,zz+1,G.adjoint());
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m_T.rightCols(dim-zz).applyOnTheLeft(zz,zz+1,G.adjoint());
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m_T.coeffRef(zz+1,zz+1) = Scalar(0.0);
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// update Q
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if (m_computeQZ)
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m_Q.applyOnTheRight(zz,zz+1,G);
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// kill S(zz+1, zz-1)
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if (zz>f)
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{
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G.makeGivens(m_S.coeff(zz+1, zz), m_S.coeff(zz+1,zz-1));
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m_S.topRows(zz+2).applyOnTheRight(zz, zz-1,G);
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m_T.topRows(zz+1).applyOnTheRight(zz, zz-1,G);
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m_S.coeffRef(zz+1,zz-1) = Scalar(0.0);
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// update Z
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if (m_computeQZ)
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m_Z.applyOnTheLeft(zz,zz-1,G.adjoint());
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}
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}
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// finally kill S(l,l-1)
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G.makeGivens(m_S.coeff(l,l), m_S.coeff(l,l-1));
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m_S.applyOnTheRight(l,l-1,G);
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m_T.applyOnTheRight(l,l-1,G);
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m_S.coeffRef(l,l-1)=Scalar(0.0);
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// update Z
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if (m_computeQZ)
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m_Z.applyOnTheLeft(l,l-1,G.adjoint());
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}
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/** \internal QR-like iterative step for block f..l */
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template<typename MatrixType>
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inline void RealQZ<MatrixType>::step(Index f, Index l, Index iter)
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{
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using std::abs;
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const Index dim = m_S.cols();
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// x, y, z
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Scalar x, y, z;
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if (iter==10)
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{
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// Wilkinson ad hoc shift
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const Scalar
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a11=m_S.coeff(f+0,f+0), a12=m_S.coeff(f+0,f+1),
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a21=m_S.coeff(f+1,f+0), a22=m_S.coeff(f+1,f+1), a32=m_S.coeff(f+2,f+1),
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b12=m_T.coeff(f+0,f+1),
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b11i=Scalar(1.0)/m_T.coeff(f+0,f+0),
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b22i=Scalar(1.0)/m_T.coeff(f+1,f+1),
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a87=m_S.coeff(l-1,l-2),
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a98=m_S.coeff(l-0,l-1),
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b77i=Scalar(1.0)/m_T.coeff(l-2,l-2),
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b88i=Scalar(1.0)/m_T.coeff(l-1,l-1);
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Scalar ss = abs(a87*b77i) + abs(a98*b88i),
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lpl = Scalar(1.5)*ss,
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ll = ss*ss;
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x = ll + a11*a11*b11i*b11i - lpl*a11*b11i + a12*a21*b11i*b22i
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- a11*a21*b12*b11i*b11i*b22i;
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y = a11*a21*b11i*b11i - lpl*a21*b11i + a21*a22*b11i*b22i
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- a21*a21*b12*b11i*b11i*b22i;
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z = a21*a32*b11i*b22i;
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}
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else if (iter==16)
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{
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// another exceptional shift
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x = m_S.coeff(f,f)/m_T.coeff(f,f)-m_S.coeff(l,l)/m_T.coeff(l,l) + m_S.coeff(l,l-1)*m_T.coeff(l-1,l) /
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(m_T.coeff(l-1,l-1)*m_T.coeff(l,l));
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y = m_S.coeff(f+1,f)/m_T.coeff(f,f);
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z = 0;
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}
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else if (iter>23 && !(iter%8))
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{
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// extremely exceptional shift
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x = internal::random<Scalar>(-1.0,1.0);
|
|
y = internal::random<Scalar>(-1.0,1.0);
|
|
z = internal::random<Scalar>(-1.0,1.0);
|
|
}
|
|
else
|
|
{
|
|
// Compute the shifts: (x,y,z,0...) = (AB^-1 - l1 I) (AB^-1 - l2 I) e1
|
|
// where l1 and l2 are the eigenvalues of the 2x2 matrix C = U V^-1 where
|
|
// U and V are 2x2 bottom right sub matrices of A and B. Thus:
|
|
// = AB^-1AB^-1 + l1 l2 I - (l1+l2)(AB^-1)
|
|
// = AB^-1AB^-1 + det(M) - tr(M)(AB^-1)
|
|
// Since we are only interested in having x, y, z with a correct ratio, we have:
|
|
const Scalar
|
|
a11 = m_S.coeff(f,f), a12 = m_S.coeff(f,f+1),
|
|
a21 = m_S.coeff(f+1,f), a22 = m_S.coeff(f+1,f+1),
|
|
a32 = m_S.coeff(f+2,f+1),
|
|
|
|
a88 = m_S.coeff(l-1,l-1), a89 = m_S.coeff(l-1,l),
|
|
a98 = m_S.coeff(l,l-1), a99 = m_S.coeff(l,l),
|
|
|
|
b11 = m_T.coeff(f,f), b12 = m_T.coeff(f,f+1),
|
|
b22 = m_T.coeff(f+1,f+1),
|
|
|
|
b88 = m_T.coeff(l-1,l-1), b89 = m_T.coeff(l-1,l),
|
|
b99 = m_T.coeff(l,l);
|
|
|
|
x = ( (a88/b88 - a11/b11)*(a99/b99 - a11/b11) - (a89/b99)*(a98/b88) + (a98/b88)*(b89/b99)*(a11/b11) ) * (b11/a21)
|
|
+ a12/b22 - (a11/b11)*(b12/b22);
|
|
y = (a22/b22-a11/b11) - (a21/b11)*(b12/b22) - (a88/b88-a11/b11) - (a99/b99-a11/b11) + (a98/b88)*(b89/b99);
|
|
z = a32/b22;
|
|
}
|
|
|
|
JRs G;
|
|
|
|
for (Index k=f; k<=l-2; k++)
|
|
{
|
|
// variables for Householder reflections
|
|
Vector2s essential2;
|
|
Scalar tau, beta;
|
|
|
|
Vector3s hr(x,y,z);
|
|
|
|
// Q_k to annihilate S(k+1,k-1) and S(k+2,k-1)
|
|
hr.makeHouseholderInPlace(tau, beta);
|
|
essential2 = hr.template bottomRows<2>();
|
|
Index fc=(std::max)(k-1,Index(0)); // first col to update
|
|
m_S.template middleRows<3>(k).rightCols(dim-fc).applyHouseholderOnTheLeft(essential2, tau, m_workspace.data());
|
|
m_T.template middleRows<3>(k).rightCols(dim-fc).applyHouseholderOnTheLeft(essential2, tau, m_workspace.data());
|
|
if (m_computeQZ)
|
|
m_Q.template middleCols<3>(k).applyHouseholderOnTheRight(essential2, tau, m_workspace.data());
|
|
if (k>f)
|
|
m_S.coeffRef(k+2,k-1) = m_S.coeffRef(k+1,k-1) = Scalar(0.0);
|
|
|
|
// Z_{k1} to annihilate T(k+2,k+1) and T(k+2,k)
|
|
hr << m_T.coeff(k+2,k+2),m_T.coeff(k+2,k),m_T.coeff(k+2,k+1);
|
|
hr.makeHouseholderInPlace(tau, beta);
|
|
essential2 = hr.template bottomRows<2>();
|
|
{
|
|
Index lr = (std::min)(k+4,dim); // last row to update
|
|
Map<Matrix<Scalar,Dynamic,1> > tmp(m_workspace.data(),lr);
|
|
// S
|
|
tmp = m_S.template middleCols<2>(k).topRows(lr) * essential2;
|
|
tmp += m_S.col(k+2).head(lr);
|
|
m_S.col(k+2).head(lr) -= tau*tmp;
|
|
m_S.template middleCols<2>(k).topRows(lr) -= (tau*tmp) * essential2.adjoint();
|
|
// T
|
|
tmp = m_T.template middleCols<2>(k).topRows(lr) * essential2;
|
|
tmp += m_T.col(k+2).head(lr);
|
|
m_T.col(k+2).head(lr) -= tau*tmp;
|
|
m_T.template middleCols<2>(k).topRows(lr) -= (tau*tmp) * essential2.adjoint();
|
|
}
|
|
if (m_computeQZ)
|
|
{
|
|
// Z
|
|
Map<Matrix<Scalar,1,Dynamic> > tmp(m_workspace.data(),dim);
|
|
tmp = essential2.adjoint()*(m_Z.template middleRows<2>(k));
|
|
tmp += m_Z.row(k+2);
|
|
m_Z.row(k+2) -= tau*tmp;
|
|
m_Z.template middleRows<2>(k) -= essential2 * (tau*tmp);
|
|
}
|
|
m_T.coeffRef(k+2,k) = m_T.coeffRef(k+2,k+1) = Scalar(0.0);
|
|
|
|
// Z_{k2} to annihilate T(k+1,k)
|
|
G.makeGivens(m_T.coeff(k+1,k+1), m_T.coeff(k+1,k));
|
|
m_S.applyOnTheRight(k+1,k,G);
|
|
m_T.applyOnTheRight(k+1,k,G);
|
|
// update Z
|
|
if (m_computeQZ)
|
|
m_Z.applyOnTheLeft(k+1,k,G.adjoint());
|
|
m_T.coeffRef(k+1,k) = Scalar(0.0);
|
|
|
|
// update x,y,z
|
|
x = m_S.coeff(k+1,k);
|
|
y = m_S.coeff(k+2,k);
|
|
if (k < l-2)
|
|
z = m_S.coeff(k+3,k);
|
|
} // loop over k
|
|
|
|
// Q_{n-1} to annihilate y = S(l,l-2)
|
|
G.makeGivens(x,y);
|
|
m_S.applyOnTheLeft(l-1,l,G.adjoint());
|
|
m_T.applyOnTheLeft(l-1,l,G.adjoint());
|
|
if (m_computeQZ)
|
|
m_Q.applyOnTheRight(l-1,l,G);
|
|
m_S.coeffRef(l,l-2) = Scalar(0.0);
|
|
|
|
// Z_{n-1} to annihilate T(l,l-1)
|
|
G.makeGivens(m_T.coeff(l,l),m_T.coeff(l,l-1));
|
|
m_S.applyOnTheRight(l,l-1,G);
|
|
m_T.applyOnTheRight(l,l-1,G);
|
|
if (m_computeQZ)
|
|
m_Z.applyOnTheLeft(l,l-1,G.adjoint());
|
|
m_T.coeffRef(l,l-1) = Scalar(0.0);
|
|
}
|
|
|
|
|
|
template<typename MatrixType>
|
|
RealQZ<MatrixType>& RealQZ<MatrixType>::compute(const MatrixType& A_in, const MatrixType& B_in, bool computeQZ)
|
|
{
|
|
|
|
const Index dim = A_in.cols();
|
|
|
|
eigen_assert (A_in.rows()==dim && A_in.cols()==dim
|
|
&& B_in.rows()==dim && B_in.cols()==dim
|
|
&& "Need square matrices of the same dimension");
|
|
|
|
m_isInitialized = true;
|
|
m_computeQZ = computeQZ;
|
|
m_S = A_in; m_T = B_in;
|
|
m_workspace.resize(dim*2);
|
|
m_global_iter = 0;
|
|
|
|
// entrance point: hessenberg triangular decomposition
|
|
hessenbergTriangular();
|
|
// compute L1 vector norms of T, S into m_normOfS, m_normOfT
|
|
computeNorms();
|
|
|
|
Index l = dim-1,
|
|
f,
|
|
local_iter = 0;
|
|
|
|
while (l>0 && local_iter<m_maxIters)
|
|
{
|
|
f = findSmallSubdiagEntry(l);
|
|
// now rows and columns f..l (including) decouple from the rest of the problem
|
|
if (f>0) m_S.coeffRef(f,f-1) = Scalar(0.0);
|
|
if (f == l) // One root found
|
|
{
|
|
l--;
|
|
local_iter = 0;
|
|
}
|
|
else if (f == l-1) // Two roots found
|
|
{
|
|
splitOffTwoRows(f);
|
|
l -= 2;
|
|
local_iter = 0;
|
|
}
|
|
else // No convergence yet
|
|
{
|
|
// if there's zero on diagonal of T, we can isolate an eigenvalue with Givens rotations
|
|
Index z = findSmallDiagEntry(f,l);
|
|
if (z>=f)
|
|
{
|
|
// zero found
|
|
pushDownZero(z,f,l);
|
|
}
|
|
else
|
|
{
|
|
// We are sure now that S.block(f,f, l-f+1,l-f+1) is underuced upper-Hessenberg
|
|
// and T.block(f,f, l-f+1,l-f+1) is invertible uper-triangular, which allows to
|
|
// apply a QR-like iteration to rows and columns f..l.
|
|
step(f,l, local_iter);
|
|
local_iter++;
|
|
m_global_iter++;
|
|
}
|
|
}
|
|
}
|
|
// check if we converged before reaching iterations limit
|
|
m_info = (local_iter<m_maxIters) ? Success : NoConvergence;
|
|
return *this;
|
|
} // end compute
|
|
|
|
} // end namespace Eigen
|
|
|
|
#endif //EIGEN_REAL_QZ
|