[136] | 1 | // This file is part of Eigen, a lightweight C++ template library
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| 2 | // for linear algebra.
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| 3 | //
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| 4 | // Copyright (C) 2009 Claire Maurice
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| 5 | // Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
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| 6 | // Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk>
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| 7 | //
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| 8 | // This Source Code Form is subject to the terms of the Mozilla
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| 9 | // Public License v. 2.0. If a copy of the MPL was not distributed
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| 10 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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| 11 |
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| 12 | #ifndef EIGEN_COMPLEX_SCHUR_H
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| 13 | #define EIGEN_COMPLEX_SCHUR_H
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| 14 |
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| 15 | #include "./HessenbergDecomposition.h"
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| 16 |
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| 17 | namespace Eigen {
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| 18 |
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| 19 | namespace internal {
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| 20 | template<typename MatrixType, bool IsComplex> struct complex_schur_reduce_to_hessenberg;
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| 21 | }
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| 22 |
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| 23 | /** \eigenvalues_module \ingroup Eigenvalues_Module
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| 24 | *
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| 25 | *
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| 26 | * \class ComplexSchur
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| 27 | *
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| 28 | * \brief Performs a complex Schur decomposition of a real or complex square matrix
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| 29 | *
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| 30 | * \tparam _MatrixType the type of the matrix of which we are
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| 31 | * computing the Schur decomposition; this is expected to be an
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| 32 | * instantiation of the Matrix class template.
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| 33 | *
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| 34 | * Given a real or complex square matrix A, this class computes the
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| 35 | * Schur decomposition: \f$ A = U T U^*\f$ where U is a unitary
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| 36 | * complex matrix, and T is a complex upper triangular matrix. The
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| 37 | * diagonal of the matrix T corresponds to the eigenvalues of the
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| 38 | * matrix A.
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| 39 | *
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| 40 | * Call the function compute() to compute the Schur decomposition of
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| 41 | * a given matrix. Alternatively, you can use the
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| 42 | * ComplexSchur(const MatrixType&, bool) constructor which computes
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| 43 | * the Schur decomposition at construction time. Once the
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| 44 | * decomposition is computed, you can use the matrixU() and matrixT()
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| 45 | * functions to retrieve the matrices U and V in the decomposition.
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| 46 | *
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| 47 | * \note This code is inspired from Jampack
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| 48 | *
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| 49 | * \sa class RealSchur, class EigenSolver, class ComplexEigenSolver
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| 50 | */
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| 51 | template<typename _MatrixType> class ComplexSchur
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| 52 | {
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| 53 | public:
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| 54 | typedef _MatrixType MatrixType;
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| 55 | enum {
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| 56 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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| 57 | ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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| 58 | Options = MatrixType::Options,
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| 59 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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| 60 | MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
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| 61 | };
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| 62 |
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| 63 | /** \brief Scalar type for matrices of type \p _MatrixType. */
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| 64 | typedef typename MatrixType::Scalar Scalar;
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| 65 | typedef typename NumTraits<Scalar>::Real RealScalar;
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| 66 | typedef typename MatrixType::Index Index;
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| 67 |
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| 68 | /** \brief Complex scalar type for \p _MatrixType.
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| 69 | *
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| 70 | * This is \c std::complex<Scalar> if #Scalar is real (e.g.,
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| 71 | * \c float or \c double) and just \c Scalar if #Scalar is
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| 72 | * complex.
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| 73 | */
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| 74 | typedef std::complex<RealScalar> ComplexScalar;
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| 75 |
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| 76 | /** \brief Type for the matrices in the Schur decomposition.
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| 77 | *
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| 78 | * This is a square matrix with entries of type #ComplexScalar.
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| 79 | * The size is the same as the size of \p _MatrixType.
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| 80 | */
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| 81 | typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> ComplexMatrixType;
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| 82 |
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| 83 | /** \brief Default constructor.
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| 84 | *
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| 85 | * \param [in] size Positive integer, size of the matrix whose Schur decomposition will be computed.
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| 86 | *
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| 87 | * The default constructor is useful in cases in which the user
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| 88 | * intends to perform decompositions via compute(). The \p size
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| 89 | * parameter is only used as a hint. It is not an error to give a
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| 90 | * wrong \p size, but it may impair performance.
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| 91 | *
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| 92 | * \sa compute() for an example.
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| 93 | */
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| 94 | ComplexSchur(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime)
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| 95 | : m_matT(size,size),
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| 96 | m_matU(size,size),
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| 97 | m_hess(size),
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| 98 | m_isInitialized(false),
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| 99 | m_matUisUptodate(false),
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| 100 | m_maxIters(-1)
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| 101 | {}
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| 102 |
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| 103 | /** \brief Constructor; computes Schur decomposition of given matrix.
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| 104 | *
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| 105 | * \param[in] matrix Square matrix whose Schur decomposition is to be computed.
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| 106 | * \param[in] computeU If true, both T and U are computed; if false, only T is computed.
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| 107 | *
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| 108 | * This constructor calls compute() to compute the Schur decomposition.
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| 109 | *
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| 110 | * \sa matrixT() and matrixU() for examples.
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| 111 | */
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| 112 | ComplexSchur(const MatrixType& matrix, bool computeU = true)
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| 113 | : m_matT(matrix.rows(),matrix.cols()),
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| 114 | m_matU(matrix.rows(),matrix.cols()),
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| 115 | m_hess(matrix.rows()),
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| 116 | m_isInitialized(false),
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| 117 | m_matUisUptodate(false),
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| 118 | m_maxIters(-1)
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| 119 | {
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| 120 | compute(matrix, computeU);
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| 121 | }
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| 122 |
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| 123 | /** \brief Returns the unitary matrix in the Schur decomposition.
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| 124 | *
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| 125 | * \returns A const reference to the matrix U.
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| 126 | *
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| 127 | * It is assumed that either the constructor
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| 128 | * ComplexSchur(const MatrixType& matrix, bool computeU) or the
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| 129 | * member function compute(const MatrixType& matrix, bool computeU)
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| 130 | * has been called before to compute the Schur decomposition of a
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| 131 | * matrix, and that \p computeU was set to true (the default
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| 132 | * value).
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| 133 | *
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| 134 | * Example: \include ComplexSchur_matrixU.cpp
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| 135 | * Output: \verbinclude ComplexSchur_matrixU.out
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| 136 | */
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| 137 | const ComplexMatrixType& matrixU() const
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| 138 | {
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| 139 | eigen_assert(m_isInitialized && "ComplexSchur is not initialized.");
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| 140 | eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the ComplexSchur decomposition.");
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| 141 | return m_matU;
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| 142 | }
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| 143 |
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| 144 | /** \brief Returns the triangular matrix in the Schur decomposition.
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| 145 | *
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| 146 | * \returns A const reference to the matrix T.
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| 147 | *
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| 148 | * It is assumed that either the constructor
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| 149 | * ComplexSchur(const MatrixType& matrix, bool computeU) or the
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| 150 | * member function compute(const MatrixType& matrix, bool computeU)
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| 151 | * has been called before to compute the Schur decomposition of a
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| 152 | * matrix.
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| 153 | *
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| 154 | * Note that this function returns a plain square matrix. If you want to reference
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| 155 | * only the upper triangular part, use:
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| 156 | * \code schur.matrixT().triangularView<Upper>() \endcode
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| 157 | *
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| 158 | * Example: \include ComplexSchur_matrixT.cpp
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| 159 | * Output: \verbinclude ComplexSchur_matrixT.out
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| 160 | */
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| 161 | const ComplexMatrixType& matrixT() const
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| 162 | {
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| 163 | eigen_assert(m_isInitialized && "ComplexSchur is not initialized.");
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| 164 | return m_matT;
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| 165 | }
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| 166 |
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| 167 | /** \brief Computes Schur decomposition of given matrix.
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| 168 | *
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| 169 | * \param[in] matrix Square matrix whose Schur decomposition is to be computed.
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| 170 | * \param[in] computeU If true, both T and U are computed; if false, only T is computed.
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| 171 |
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| 172 | * \returns Reference to \c *this
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| 173 | *
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| 174 | * The Schur decomposition is computed by first reducing the
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| 175 | * matrix to Hessenberg form using the class
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| 176 | * HessenbergDecomposition. The Hessenberg matrix is then reduced
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| 177 | * to triangular form by performing QR iterations with a single
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| 178 | * shift. The cost of computing the Schur decomposition depends
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| 179 | * on the number of iterations; as a rough guide, it may be taken
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| 180 | * on the number of iterations; as a rough guide, it may be taken
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| 181 | * to be \f$25n^3\f$ complex flops, or \f$10n^3\f$ complex flops
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| 182 | * if \a computeU is false.
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| 183 | *
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| 184 | * Example: \include ComplexSchur_compute.cpp
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| 185 | * Output: \verbinclude ComplexSchur_compute.out
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| 186 | *
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| 187 | * \sa compute(const MatrixType&, bool, Index)
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| 188 | */
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| 189 | ComplexSchur& compute(const MatrixType& matrix, bool computeU = true);
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| 190 |
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| 191 | /** \brief Compute Schur decomposition from a given Hessenberg matrix
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| 192 | * \param[in] matrixH Matrix in Hessenberg form H
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| 193 | * \param[in] matrixQ orthogonal matrix Q that transform a matrix A to H : A = Q H Q^T
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| 194 | * \param computeU Computes the matriX U of the Schur vectors
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| 195 | * \return Reference to \c *this
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| 196 | *
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| 197 | * This routine assumes that the matrix is already reduced in Hessenberg form matrixH
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| 198 | * using either the class HessenbergDecomposition or another mean.
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| 199 | * It computes the upper quasi-triangular matrix T of the Schur decomposition of H
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| 200 | * When computeU is true, this routine computes the matrix U such that
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| 201 | * A = U T U^T = (QZ) T (QZ)^T = Q H Q^T where A is the initial matrix
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| 202 | *
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| 203 | * NOTE Q is referenced if computeU is true; so, if the initial orthogonal matrix
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| 204 | * is not available, the user should give an identity matrix (Q.setIdentity())
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| 205 | *
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| 206 | * \sa compute(const MatrixType&, bool)
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| 207 | */
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| 208 | template<typename HessMatrixType, typename OrthMatrixType>
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| 209 | ComplexSchur& computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU=true);
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| 210 |
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| 211 | /** \brief Reports whether previous computation was successful.
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| 212 | *
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| 213 | * \returns \c Success if computation was succesful, \c NoConvergence otherwise.
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| 214 | */
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| 215 | ComputationInfo info() const
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| 216 | {
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| 217 | eigen_assert(m_isInitialized && "ComplexSchur is not initialized.");
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| 218 | return m_info;
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| 219 | }
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| 220 |
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| 221 | /** \brief Sets the maximum number of iterations allowed.
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| 222 | *
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| 223 | * If not specified by the user, the maximum number of iterations is m_maxIterationsPerRow times the size
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| 224 | * of the matrix.
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| 225 | */
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| 226 | ComplexSchur& setMaxIterations(Index maxIters)
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| 227 | {
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| 228 | m_maxIters = maxIters;
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| 229 | return *this;
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| 230 | }
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| 231 |
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| 232 | /** \brief Returns the maximum number of iterations. */
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| 233 | Index getMaxIterations()
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| 234 | {
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| 235 | return m_maxIters;
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| 236 | }
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| 237 |
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| 238 | /** \brief Maximum number of iterations per row.
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| 239 | *
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| 240 | * If not otherwise specified, the maximum number of iterations is this number times the size of the
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| 241 | * matrix. It is currently set to 30.
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| 242 | */
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| 243 | static const int m_maxIterationsPerRow = 30;
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| 244 |
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| 245 | protected:
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| 246 | ComplexMatrixType m_matT, m_matU;
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| 247 | HessenbergDecomposition<MatrixType> m_hess;
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| 248 | ComputationInfo m_info;
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| 249 | bool m_isInitialized;
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| 250 | bool m_matUisUptodate;
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| 251 | Index m_maxIters;
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| 252 |
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| 253 | private:
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| 254 | bool subdiagonalEntryIsNeglegible(Index i);
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| 255 | ComplexScalar computeShift(Index iu, Index iter);
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| 256 | void reduceToTriangularForm(bool computeU);
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| 257 | friend struct internal::complex_schur_reduce_to_hessenberg<MatrixType, NumTraits<Scalar>::IsComplex>;
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| 258 | };
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| 259 |
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| 260 | /** If m_matT(i+1,i) is neglegible in floating point arithmetic
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| 261 | * compared to m_matT(i,i) and m_matT(j,j), then set it to zero and
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| 262 | * return true, else return false. */
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| 263 | template<typename MatrixType>
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| 264 | inline bool ComplexSchur<MatrixType>::subdiagonalEntryIsNeglegible(Index i)
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| 265 | {
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| 266 | RealScalar d = numext::norm1(m_matT.coeff(i,i)) + numext::norm1(m_matT.coeff(i+1,i+1));
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| 267 | RealScalar sd = numext::norm1(m_matT.coeff(i+1,i));
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| 268 | if (internal::isMuchSmallerThan(sd, d, NumTraits<RealScalar>::epsilon()))
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| 269 | {
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| 270 | m_matT.coeffRef(i+1,i) = ComplexScalar(0);
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| 271 | return true;
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| 272 | }
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| 273 | return false;
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| 274 | }
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| 275 |
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| 276 |
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| 277 | /** Compute the shift in the current QR iteration. */
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| 278 | template<typename MatrixType>
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| 279 | typename ComplexSchur<MatrixType>::ComplexScalar ComplexSchur<MatrixType>::computeShift(Index iu, Index iter)
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| 280 | {
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| 281 | using std::abs;
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| 282 | if (iter == 10 || iter == 20)
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| 283 | {
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| 284 | // exceptional shift, taken from http://www.netlib.org/eispack/comqr.f
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| 285 | return abs(numext::real(m_matT.coeff(iu,iu-1))) + abs(numext::real(m_matT.coeff(iu-1,iu-2)));
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| 286 | }
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| 287 |
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| 288 | // compute the shift as one of the eigenvalues of t, the 2x2
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| 289 | // diagonal block on the bottom of the active submatrix
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| 290 | Matrix<ComplexScalar,2,2> t = m_matT.template block<2,2>(iu-1,iu-1);
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| 291 | RealScalar normt = t.cwiseAbs().sum();
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| 292 | t /= normt; // the normalization by sf is to avoid under/overflow
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| 293 |
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| 294 | ComplexScalar b = t.coeff(0,1) * t.coeff(1,0);
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| 295 | ComplexScalar c = t.coeff(0,0) - t.coeff(1,1);
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| 296 | ComplexScalar disc = sqrt(c*c + RealScalar(4)*b);
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| 297 | ComplexScalar det = t.coeff(0,0) * t.coeff(1,1) - b;
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| 298 | ComplexScalar trace = t.coeff(0,0) + t.coeff(1,1);
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| 299 | ComplexScalar eival1 = (trace + disc) / RealScalar(2);
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| 300 | ComplexScalar eival2 = (trace - disc) / RealScalar(2);
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| 301 |
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| 302 | if(numext::norm1(eival1) > numext::norm1(eival2))
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| 303 | eival2 = det / eival1;
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| 304 | else
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| 305 | eival1 = det / eival2;
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| 306 |
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| 307 | // choose the eigenvalue closest to the bottom entry of the diagonal
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| 308 | if(numext::norm1(eival1-t.coeff(1,1)) < numext::norm1(eival2-t.coeff(1,1)))
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| 309 | return normt * eival1;
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| 310 | else
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| 311 | return normt * eival2;
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| 312 | }
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| 313 |
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| 314 |
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| 315 | template<typename MatrixType>
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| 316 | ComplexSchur<MatrixType>& ComplexSchur<MatrixType>::compute(const MatrixType& matrix, bool computeU)
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| 317 | {
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| 318 | m_matUisUptodate = false;
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| 319 | eigen_assert(matrix.cols() == matrix.rows());
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| 320 |
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| 321 | if(matrix.cols() == 1)
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| 322 | {
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| 323 | m_matT = matrix.template cast<ComplexScalar>();
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| 324 | if(computeU) m_matU = ComplexMatrixType::Identity(1,1);
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| 325 | m_info = Success;
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| 326 | m_isInitialized = true;
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| 327 | m_matUisUptodate = computeU;
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| 328 | return *this;
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| 329 | }
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| 330 |
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| 331 | internal::complex_schur_reduce_to_hessenberg<MatrixType, NumTraits<Scalar>::IsComplex>::run(*this, matrix, computeU);
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| 332 | computeFromHessenberg(m_matT, m_matU, computeU);
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| 333 | return *this;
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| 334 | }
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| 335 |
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| 336 | template<typename MatrixType>
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| 337 | template<typename HessMatrixType, typename OrthMatrixType>
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| 338 | ComplexSchur<MatrixType>& ComplexSchur<MatrixType>::computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU)
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| 339 | {
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| 340 | m_matT = matrixH;
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| 341 | if(computeU)
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| 342 | m_matU = matrixQ;
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| 343 | reduceToTriangularForm(computeU);
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| 344 | return *this;
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| 345 | }
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| 346 | namespace internal {
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| 347 |
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| 348 | /* Reduce given matrix to Hessenberg form */
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| 349 | template<typename MatrixType, bool IsComplex>
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| 350 | struct complex_schur_reduce_to_hessenberg
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| 351 | {
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| 352 | // this is the implementation for the case IsComplex = true
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| 353 | static void run(ComplexSchur<MatrixType>& _this, const MatrixType& matrix, bool computeU)
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| 354 | {
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| 355 | _this.m_hess.compute(matrix);
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| 356 | _this.m_matT = _this.m_hess.matrixH();
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| 357 | if(computeU) _this.m_matU = _this.m_hess.matrixQ();
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| 358 | }
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| 359 | };
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| 360 |
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| 361 | template<typename MatrixType>
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| 362 | struct complex_schur_reduce_to_hessenberg<MatrixType, false>
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| 363 | {
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| 364 | static void run(ComplexSchur<MatrixType>& _this, const MatrixType& matrix, bool computeU)
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| 365 | {
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| 366 | typedef typename ComplexSchur<MatrixType>::ComplexScalar ComplexScalar;
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| 367 |
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| 368 | // Note: m_hess is over RealScalar; m_matT and m_matU is over ComplexScalar
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| 369 | _this.m_hess.compute(matrix);
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| 370 | _this.m_matT = _this.m_hess.matrixH().template cast<ComplexScalar>();
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| 371 | if(computeU)
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| 372 | {
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| 373 | // This may cause an allocation which seems to be avoidable
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| 374 | MatrixType Q = _this.m_hess.matrixQ();
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| 375 | _this.m_matU = Q.template cast<ComplexScalar>();
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| 376 | }
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| 377 | }
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| 378 | };
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| 379 |
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| 380 | } // end namespace internal
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| 381 |
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| 382 | // Reduce the Hessenberg matrix m_matT to triangular form by QR iteration.
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| 383 | template<typename MatrixType>
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| 384 | void ComplexSchur<MatrixType>::reduceToTriangularForm(bool computeU)
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| 385 | {
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| 386 | Index maxIters = m_maxIters;
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| 387 | if (maxIters == -1)
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| 388 | maxIters = m_maxIterationsPerRow * m_matT.rows();
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| 389 |
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| 390 | // The matrix m_matT is divided in three parts.
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| 391 | // Rows 0,...,il-1 are decoupled from the rest because m_matT(il,il-1) is zero.
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| 392 | // Rows il,...,iu is the part we are working on (the active submatrix).
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| 393 | // Rows iu+1,...,end are already brought in triangular form.
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| 394 | Index iu = m_matT.cols() - 1;
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| 395 | Index il;
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| 396 | Index iter = 0; // number of iterations we are working on the (iu,iu) element
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| 397 | Index totalIter = 0; // number of iterations for whole matrix
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| 398 |
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| 399 | while(true)
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| 400 | {
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| 401 | // find iu, the bottom row of the active submatrix
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| 402 | while(iu > 0)
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| 403 | {
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| 404 | if(!subdiagonalEntryIsNeglegible(iu-1)) break;
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| 405 | iter = 0;
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| 406 | --iu;
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| 407 | }
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| 408 |
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| 409 | // if iu is zero then we are done; the whole matrix is triangularized
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| 410 | if(iu==0) break;
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| 411 |
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| 412 | // if we spent too many iterations, we give up
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| 413 | iter++;
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| 414 | totalIter++;
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| 415 | if(totalIter > maxIters) break;
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| 416 |
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| 417 | // find il, the top row of the active submatrix
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| 418 | il = iu-1;
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| 419 | while(il > 0 && !subdiagonalEntryIsNeglegible(il-1))
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| 420 | {
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| 421 | --il;
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| 422 | }
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| 423 |
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| 424 | /* perform the QR step using Givens rotations. The first rotation
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| 425 | creates a bulge; the (il+2,il) element becomes nonzero. This
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| 426 | bulge is chased down to the bottom of the active submatrix. */
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| 427 |
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| 428 | ComplexScalar shift = computeShift(iu, iter);
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| 429 | JacobiRotation<ComplexScalar> rot;
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| 430 | rot.makeGivens(m_matT.coeff(il,il) - shift, m_matT.coeff(il+1,il));
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| 431 | m_matT.rightCols(m_matT.cols()-il).applyOnTheLeft(il, il+1, rot.adjoint());
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| 432 | m_matT.topRows((std::min)(il+2,iu)+1).applyOnTheRight(il, il+1, rot);
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| 433 | if(computeU) m_matU.applyOnTheRight(il, il+1, rot);
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| 434 |
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| 435 | for(Index i=il+1 ; i<iu ; i++)
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| 436 | {
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| 437 | rot.makeGivens(m_matT.coeffRef(i,i-1), m_matT.coeffRef(i+1,i-1), &m_matT.coeffRef(i,i-1));
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| 438 | m_matT.coeffRef(i+1,i-1) = ComplexScalar(0);
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| 439 | m_matT.rightCols(m_matT.cols()-i).applyOnTheLeft(i, i+1, rot.adjoint());
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| 440 | m_matT.topRows((std::min)(i+2,iu)+1).applyOnTheRight(i, i+1, rot);
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| 441 | if(computeU) m_matU.applyOnTheRight(i, i+1, rot);
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| 442 | }
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| 443 | }
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| 444 |
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| 445 | if(totalIter <= maxIters)
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| 446 | m_info = Success;
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| 447 | else
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| 448 | m_info = NoConvergence;
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| 449 |
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| 450 | m_isInitialized = true;
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| 451 | m_matUisUptodate = computeU;
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| 452 | }
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| 453 |
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| 454 | } // end namespace Eigen
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| 455 |
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| 456 | #endif // EIGEN_COMPLEX_SCHUR_H
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