| 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) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
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| 5 | // Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk>
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| 6 | //
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| 7 | // This Source Code Form is subject to the terms of the Mozilla
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| 8 | // Public License v. 2.0. If a copy of the MPL was not distributed
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| 9 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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| 10 |
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| 11 | #ifndef EIGEN_REAL_SCHUR_H
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| 12 | #define EIGEN_REAL_SCHUR_H
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| 13 |
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| 14 | #include "./HessenbergDecomposition.h"
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| 15 |
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| 16 | namespace Eigen {
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| 17 |
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| 18 | /** \eigenvalues_module \ingroup Eigenvalues_Module
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| 19 | *
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| 20 | *
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| 21 | * \class RealSchur
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| 22 | *
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| 23 | * \brief Performs a real Schur decomposition of a square matrix
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| 24 | *
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| 25 | * \tparam _MatrixType the type of the matrix of which we are computing the
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| 26 | * real Schur decomposition; this is expected to be an instantiation of the
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| 27 | * Matrix class template.
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| 28 | *
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| 29 | * Given a real square matrix A, this class computes the real Schur
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| 30 | * decomposition: \f$ A = U T U^T \f$ where U is a real orthogonal matrix and
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| 31 | * T is a real quasi-triangular matrix. An orthogonal matrix is a matrix whose
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| 32 | * inverse is equal to its transpose, \f$ U^{-1} = U^T \f$. A quasi-triangular
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| 33 | * matrix is a block-triangular matrix whose diagonal consists of 1-by-1
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| 34 | * blocks and 2-by-2 blocks with complex eigenvalues. The eigenvalues of the
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| 35 | * blocks on the diagonal of T are the same as the eigenvalues of the matrix
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| 36 | * A, and thus the real Schur decomposition is used in EigenSolver to compute
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| 37 | * the eigendecomposition of a matrix.
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| 38 | *
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| 39 | * Call the function compute() to compute the real Schur decomposition of a
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| 40 | * given matrix. Alternatively, you can use the RealSchur(const MatrixType&, bool)
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| 41 | * constructor which computes the real Schur decomposition at construction
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| 42 | * time. Once the decomposition is computed, you can use the matrixU() and
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| 43 | * matrixT() functions to retrieve the matrices U and T in the decomposition.
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| 44 | *
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| 45 | * The documentation of RealSchur(const MatrixType&, bool) contains an example
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| 46 | * of the typical use of this class.
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| 47 | *
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| 48 | * \note The implementation is adapted from
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| 49 | * <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> (public domain).
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| 50 | * Their code is based on EISPACK.
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| 51 | *
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| 52 | * \sa class ComplexSchur, class EigenSolver, class ComplexEigenSolver
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| 53 | */
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| 54 | template<typename _MatrixType> class RealSchur
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| 55 | {
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| 56 | public:
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| 57 | typedef _MatrixType MatrixType;
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| 58 | enum {
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| 59 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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| 60 | ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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| 61 | Options = MatrixType::Options,
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| 62 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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| 63 | MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
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| 64 | };
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| 65 | typedef typename MatrixType::Scalar Scalar;
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| 66 | typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
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| 67 | typedef typename MatrixType::Index Index;
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| 68 |
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| 69 | typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType;
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| 70 | typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;
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| 71 |
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| 72 | /** \brief Default constructor.
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| 73 | *
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| 74 | * \param [in] size Positive integer, size of the matrix whose Schur decomposition will be computed.
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| 75 | *
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| 76 | * The default constructor is useful in cases in which the user intends to
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| 77 | * perform decompositions via compute(). The \p size parameter is only
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| 78 | * used as a hint. It is not an error to give a wrong \p size, but it may
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| 79 | * impair performance.
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| 80 | *
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| 81 | * \sa compute() for an example.
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| 82 | */
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| 83 | RealSchur(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime)
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| 84 | : m_matT(size, size),
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| 85 | m_matU(size, size),
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| 86 | m_workspaceVector(size),
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| 87 | m_hess(size),
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| 88 | m_isInitialized(false),
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| 89 | m_matUisUptodate(false),
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| 90 | m_maxIters(-1)
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| 91 | { }
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| 92 |
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| 93 | /** \brief Constructor; computes real Schur decomposition of given matrix.
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| 94 | *
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| 95 | * \param[in] matrix Square matrix whose Schur decomposition is to be computed.
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| 96 | * \param[in] computeU If true, both T and U are computed; if false, only T is computed.
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| 97 | *
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| 98 | * This constructor calls compute() to compute the Schur decomposition.
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| 99 | *
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| 100 | * Example: \include RealSchur_RealSchur_MatrixType.cpp
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| 101 | * Output: \verbinclude RealSchur_RealSchur_MatrixType.out
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| 102 | */
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| 103 | RealSchur(const MatrixType& matrix, bool computeU = true)
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| 104 | : m_matT(matrix.rows(),matrix.cols()),
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| 105 | m_matU(matrix.rows(),matrix.cols()),
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| 106 | m_workspaceVector(matrix.rows()),
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| 107 | m_hess(matrix.rows()),
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| 108 | m_isInitialized(false),
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| 109 | m_matUisUptodate(false),
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| 110 | m_maxIters(-1)
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| 111 | {
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| 112 | compute(matrix, computeU);
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| 113 | }
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| 114 |
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| 115 | /** \brief Returns the orthogonal matrix in the Schur decomposition.
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| 116 | *
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| 117 | * \returns A const reference to the matrix U.
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| 118 | *
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| 119 | * \pre Either the constructor RealSchur(const MatrixType&, bool) or the
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| 120 | * member function compute(const MatrixType&, bool) has been called before
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| 121 | * to compute the Schur decomposition of a matrix, and \p computeU was set
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| 122 | * to true (the default value).
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| 123 | *
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| 124 | * \sa RealSchur(const MatrixType&, bool) for an example
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| 125 | */
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| 126 | const MatrixType& matrixU() const
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| 127 | {
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| 128 | eigen_assert(m_isInitialized && "RealSchur is not initialized.");
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| 129 | eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the RealSchur decomposition.");
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| 130 | return m_matU;
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| 131 | }
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| 132 |
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| 133 | /** \brief Returns the quasi-triangular matrix in the Schur decomposition.
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| 134 | *
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| 135 | * \returns A const reference to the matrix T.
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| 136 | *
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| 137 | * \pre Either the constructor RealSchur(const MatrixType&, bool) or the
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| 138 | * member function compute(const MatrixType&, bool) has been called before
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| 139 | * to compute the Schur decomposition of a matrix.
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| 140 | *
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| 141 | * \sa RealSchur(const MatrixType&, bool) for an example
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| 142 | */
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| 143 | const MatrixType& matrixT() const
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| 144 | {
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| 145 | eigen_assert(m_isInitialized && "RealSchur is not initialized.");
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| 146 | return m_matT;
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| 147 | }
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| 148 |
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| 149 | /** \brief Computes Schur decomposition of given matrix.
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| 150 | *
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| 151 | * \param[in] matrix Square matrix whose Schur decomposition is to be computed.
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| 152 | * \param[in] computeU If true, both T and U are computed; if false, only T is computed.
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| 153 | * \returns Reference to \c *this
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| 154 | *
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| 155 | * The Schur decomposition is computed by first reducing the matrix to
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| 156 | * Hessenberg form using the class HessenbergDecomposition. The Hessenberg
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| 157 | * matrix is then reduced to triangular form by performing Francis QR
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| 158 | * iterations with implicit double shift. The cost of computing the Schur
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| 159 | * decomposition depends on the number of iterations; as a rough guide, it
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| 160 | * may be taken to be \f$25n^3\f$ flops if \a computeU is true and
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| 161 | * \f$10n^3\f$ flops if \a computeU is false.
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| 162 | *
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| 163 | * Example: \include RealSchur_compute.cpp
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| 164 | * Output: \verbinclude RealSchur_compute.out
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| 165 | *
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| 166 | * \sa compute(const MatrixType&, bool, Index)
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| 167 | */
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| 168 | RealSchur& compute(const MatrixType& matrix, bool computeU = true);
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| 169 |
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| 170 | /** \brief Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T
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| 171 | * \param[in] matrixH Matrix in Hessenberg form H
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| 172 | * \param[in] matrixQ orthogonal matrix Q that transform a matrix A to H : A = Q H Q^T
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| 173 | * \param computeU Computes the matriX U of the Schur vectors
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| 174 | * \return Reference to \c *this
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| 175 | *
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| 176 | * This routine assumes that the matrix is already reduced in Hessenberg form matrixH
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| 177 | * using either the class HessenbergDecomposition or another mean.
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| 178 | * It computes the upper quasi-triangular matrix T of the Schur decomposition of H
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| 179 | * When computeU is true, this routine computes the matrix U such that
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| 180 | * A = U T U^T = (QZ) T (QZ)^T = Q H Q^T where A is the initial matrix
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| 181 | *
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| 182 | * NOTE Q is referenced if computeU is true; so, if the initial orthogonal matrix
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| 183 | * is not available, the user should give an identity matrix (Q.setIdentity())
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| 184 | *
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| 185 | * \sa compute(const MatrixType&, bool)
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| 186 | */
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| 187 | template<typename HessMatrixType, typename OrthMatrixType>
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| 188 | RealSchur& computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU);
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| 189 | /** \brief Reports whether previous computation was successful.
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| 190 | *
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| 191 | * \returns \c Success if computation was succesful, \c NoConvergence otherwise.
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| 192 | */
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| 193 | ComputationInfo info() const
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| 194 | {
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| 195 | eigen_assert(m_isInitialized && "RealSchur is not initialized.");
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| 196 | return m_info;
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| 197 | }
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| 198 |
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| 199 | /** \brief Sets the maximum number of iterations allowed.
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| 200 | *
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| 201 | * If not specified by the user, the maximum number of iterations is m_maxIterationsPerRow times the size
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| 202 | * of the matrix.
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| 203 | */
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| 204 | RealSchur& setMaxIterations(Index maxIters)
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| 205 | {
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| 206 | m_maxIters = maxIters;
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| 207 | return *this;
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| 208 | }
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| 209 |
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| 210 | /** \brief Returns the maximum number of iterations. */
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| 211 | Index getMaxIterations()
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| 212 | {
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| 213 | return m_maxIters;
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| 214 | }
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| 215 |
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| 216 | /** \brief Maximum number of iterations per row.
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| 217 | *
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| 218 | * If not otherwise specified, the maximum number of iterations is this number times the size of the
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| 219 | * matrix. It is currently set to 40.
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| 220 | */
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| 221 | static const int m_maxIterationsPerRow = 40;
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| 222 |
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| 223 | private:
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| 224 |
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| 225 | MatrixType m_matT;
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| 226 | MatrixType m_matU;
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| 227 | ColumnVectorType m_workspaceVector;
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| 228 | HessenbergDecomposition<MatrixType> m_hess;
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| 229 | ComputationInfo m_info;
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| 230 | bool m_isInitialized;
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| 231 | bool m_matUisUptodate;
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| 232 | Index m_maxIters;
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| 233 |
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| 234 | typedef Matrix<Scalar,3,1> Vector3s;
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| 235 |
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| 236 | Scalar computeNormOfT();
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| 237 | Index findSmallSubdiagEntry(Index iu);
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| 238 | void splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift);
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| 239 | void computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo);
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| 240 | void initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector);
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| 241 | void performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace);
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| 242 | };
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| 243 |
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| 244 |
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| 245 | template<typename MatrixType>
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| 246 | RealSchur<MatrixType>& RealSchur<MatrixType>::compute(const MatrixType& matrix, bool computeU)
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| 247 | {
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| 248 | eigen_assert(matrix.cols() == matrix.rows());
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| 249 | Index maxIters = m_maxIters;
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| 250 | if (maxIters == -1)
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| 251 | maxIters = m_maxIterationsPerRow * matrix.rows();
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| 252 |
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| 253 | // Step 1. Reduce to Hessenberg form
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| 254 | m_hess.compute(matrix);
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| 255 |
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| 256 | // Step 2. Reduce to real Schur form
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| 257 | computeFromHessenberg(m_hess.matrixH(), m_hess.matrixQ(), computeU);
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| 258 |
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| 259 | return *this;
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| 260 | }
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| 261 | template<typename MatrixType>
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| 262 | template<typename HessMatrixType, typename OrthMatrixType>
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| 263 | RealSchur<MatrixType>& RealSchur<MatrixType>::computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU)
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| 264 | {
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| 265 | m_matT = matrixH;
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| 266 | if(computeU)
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| 267 | m_matU = matrixQ;
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| 268 |
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| 269 | Index maxIters = m_maxIters;
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| 270 | if (maxIters == -1)
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| 271 | maxIters = m_maxIterationsPerRow * matrixH.rows();
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| 272 | m_workspaceVector.resize(m_matT.cols());
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| 273 | Scalar* workspace = &m_workspaceVector.coeffRef(0);
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| 274 |
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| 275 | // The matrix m_matT is divided in three parts.
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| 276 | // Rows 0,...,il-1 are decoupled from the rest because m_matT(il,il-1) is zero.
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| 277 | // Rows il,...,iu is the part we are working on (the active window).
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| 278 | // Rows iu+1,...,end are already brought in triangular form.
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| 279 | Index iu = m_matT.cols() - 1;
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| 280 | Index iter = 0; // iteration count for current eigenvalue
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| 281 | Index totalIter = 0; // iteration count for whole matrix
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| 282 | Scalar exshift(0); // sum of exceptional shifts
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| 283 | Scalar norm = computeNormOfT();
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| 284 |
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| 285 | if(norm!=0)
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| 286 | {
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| 287 | while (iu >= 0)
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| 288 | {
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| 289 | Index il = findSmallSubdiagEntry(iu);
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| 290 |
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| 291 | // Check for convergence
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| 292 | if (il == iu) // One root found
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| 293 | {
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| 294 | m_matT.coeffRef(iu,iu) = m_matT.coeff(iu,iu) + exshift;
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| 295 | if (iu > 0)
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| 296 | m_matT.coeffRef(iu, iu-1) = Scalar(0);
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| 297 | iu--;
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| 298 | iter = 0;
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| 299 | }
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| 300 | else if (il == iu-1) // Two roots found
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| 301 | {
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| 302 | splitOffTwoRows(iu, computeU, exshift);
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| 303 | iu -= 2;
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| 304 | iter = 0;
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| 305 | }
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| 306 | else // No convergence yet
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| 307 | {
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| 308 | // The firstHouseholderVector vector has to be initialized to something to get rid of a silly GCC warning (-O1 -Wall -DNDEBUG )
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| 309 | Vector3s firstHouseholderVector(0,0,0), shiftInfo;
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| 310 | computeShift(iu, iter, exshift, shiftInfo);
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| 311 | iter = iter + 1;
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| 312 | totalIter = totalIter + 1;
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| 313 | if (totalIter > maxIters) break;
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| 314 | Index im;
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| 315 | initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector);
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| 316 | performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace);
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| 317 | }
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| 318 | }
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| 319 | }
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| 320 | if(totalIter <= maxIters)
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| 321 | m_info = Success;
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| 322 | else
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| 323 | m_info = NoConvergence;
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| 324 |
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| 325 | m_isInitialized = true;
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| 326 | m_matUisUptodate = computeU;
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| 327 | return *this;
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| 328 | }
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| 329 |
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| 330 | /** \internal Computes and returns vector L1 norm of T */
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| 331 | template<typename MatrixType>
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| 332 | inline typename MatrixType::Scalar RealSchur<MatrixType>::computeNormOfT()
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| 333 | {
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| 334 | const Index size = m_matT.cols();
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| 335 | // FIXME to be efficient the following would requires a triangular reduxion code
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| 336 | // Scalar norm = m_matT.upper().cwiseAbs().sum()
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| 337 | // + m_matT.bottomLeftCorner(size-1,size-1).diagonal().cwiseAbs().sum();
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| 338 | Scalar norm(0);
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| 339 | for (Index j = 0; j < size; ++j)
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| 340 | norm += m_matT.col(j).segment(0, (std::min)(size,j+2)).cwiseAbs().sum();
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| 341 | return norm;
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| 342 | }
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| 343 |
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| 344 | /** \internal Look for single small sub-diagonal element and returns its index */
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| 345 | template<typename MatrixType>
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| 346 | inline typename MatrixType::Index RealSchur<MatrixType>::findSmallSubdiagEntry(Index iu)
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| 347 | {
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| 348 | using std::abs;
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| 349 | Index res = iu;
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| 350 | while (res > 0)
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| 351 | {
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| 352 | Scalar s = abs(m_matT.coeff(res-1,res-1)) + abs(m_matT.coeff(res,res));
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| 353 | if (abs(m_matT.coeff(res,res-1)) <= NumTraits<Scalar>::epsilon() * s)
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| 354 | break;
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| 355 | res--;
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| 356 | }
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| 357 | return res;
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| 358 | }
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| 359 |
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| 360 | /** \internal Update T given that rows iu-1 and iu decouple from the rest. */
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| 361 | template<typename MatrixType>
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| 362 | inline void RealSchur<MatrixType>::splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift)
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| 363 | {
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| 364 | using std::sqrt;
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| 365 | using std::abs;
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| 366 | const Index size = m_matT.cols();
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| 367 |
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| 368 | // The eigenvalues of the 2x2 matrix [a b; c d] are
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| 369 | // trace +/- sqrt(discr/4) where discr = tr^2 - 4*det, tr = a + d, det = ad - bc
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| 370 | Scalar p = Scalar(0.5) * (m_matT.coeff(iu-1,iu-1) - m_matT.coeff(iu,iu));
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| 371 | Scalar q = p * p + m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu); // q = tr^2 / 4 - det = discr/4
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| 372 | m_matT.coeffRef(iu,iu) += exshift;
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| 373 | m_matT.coeffRef(iu-1,iu-1) += exshift;
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| 374 |
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| 375 | if (q >= Scalar(0)) // Two real eigenvalues
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| 376 | {
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| 377 | Scalar z = sqrt(abs(q));
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| 378 | JacobiRotation<Scalar> rot;
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| 379 | if (p >= Scalar(0))
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| 380 | rot.makeGivens(p + z, m_matT.coeff(iu, iu-1));
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| 381 | else
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| 382 | rot.makeGivens(p - z, m_matT.coeff(iu, iu-1));
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| 383 |
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| 384 | m_matT.rightCols(size-iu+1).applyOnTheLeft(iu-1, iu, rot.adjoint());
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| 385 | m_matT.topRows(iu+1).applyOnTheRight(iu-1, iu, rot);
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| 386 | m_matT.coeffRef(iu, iu-1) = Scalar(0);
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| 387 | if (computeU)
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| 388 | m_matU.applyOnTheRight(iu-1, iu, rot);
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| 389 | }
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| 390 |
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| 391 | if (iu > 1)
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| 392 | m_matT.coeffRef(iu-1, iu-2) = Scalar(0);
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| 393 | }
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| 394 |
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| 395 | /** \internal Form shift in shiftInfo, and update exshift if an exceptional shift is performed. */
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| 396 | template<typename MatrixType>
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| 397 | inline void RealSchur<MatrixType>::computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo)
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| 398 | {
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| 399 | using std::sqrt;
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| 400 | using std::abs;
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| 401 | shiftInfo.coeffRef(0) = m_matT.coeff(iu,iu);
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| 402 | shiftInfo.coeffRef(1) = m_matT.coeff(iu-1,iu-1);
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| 403 | shiftInfo.coeffRef(2) = m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);
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| 404 |
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| 405 | // Wilkinson's original ad hoc shift
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| 406 | if (iter == 10)
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| 407 | {
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| 408 | exshift += shiftInfo.coeff(0);
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| 409 | for (Index i = 0; i <= iu; ++i)
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| 410 | m_matT.coeffRef(i,i) -= shiftInfo.coeff(0);
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| 411 | Scalar s = abs(m_matT.coeff(iu,iu-1)) + abs(m_matT.coeff(iu-1,iu-2));
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| 412 | shiftInfo.coeffRef(0) = Scalar(0.75) * s;
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| 413 | shiftInfo.coeffRef(1) = Scalar(0.75) * s;
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| 414 | shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s;
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| 415 | }
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| 416 |
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| 417 | // MATLAB's new ad hoc shift
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| 418 | if (iter == 30)
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| 419 | {
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| 420 | Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
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| 421 | s = s * s + shiftInfo.coeff(2);
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| 422 | if (s > Scalar(0))
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| 423 | {
|
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| 424 | s = sqrt(s);
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| 425 | if (shiftInfo.coeff(1) < shiftInfo.coeff(0))
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| 426 | s = -s;
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| 427 | s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
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|---|
| 428 | s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s;
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|---|
| 429 | exshift += s;
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|---|
| 430 | for (Index i = 0; i <= iu; ++i)
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|---|
| 431 | m_matT.coeffRef(i,i) -= s;
|
|---|
| 432 | shiftInfo.setConstant(Scalar(0.964));
|
|---|
| 433 | }
|
|---|
| 434 | }
|
|---|
| 435 | }
|
|---|
| 436 |
|
|---|
| 437 | /** \internal Compute index im at which Francis QR step starts and the first Householder vector. */
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|---|
| 438 | template<typename MatrixType>
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|---|
| 439 | inline void RealSchur<MatrixType>::initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector)
|
|---|
| 440 | {
|
|---|
| 441 | using std::abs;
|
|---|
| 442 | Vector3s& v = firstHouseholderVector; // alias to save typing
|
|---|
| 443 |
|
|---|
| 444 | for (im = iu-2; im >= il; --im)
|
|---|
| 445 | {
|
|---|
| 446 | const Scalar Tmm = m_matT.coeff(im,im);
|
|---|
| 447 | const Scalar r = shiftInfo.coeff(0) - Tmm;
|
|---|
| 448 | const Scalar s = shiftInfo.coeff(1) - Tmm;
|
|---|
| 449 | v.coeffRef(0) = (r * s - shiftInfo.coeff(2)) / m_matT.coeff(im+1,im) + m_matT.coeff(im,im+1);
|
|---|
| 450 | v.coeffRef(1) = m_matT.coeff(im+1,im+1) - Tmm - r - s;
|
|---|
| 451 | v.coeffRef(2) = m_matT.coeff(im+2,im+1);
|
|---|
| 452 | if (im == il) {
|
|---|
| 453 | break;
|
|---|
| 454 | }
|
|---|
| 455 | const Scalar lhs = m_matT.coeff(im,im-1) * (abs(v.coeff(1)) + abs(v.coeff(2)));
|
|---|
| 456 | const Scalar rhs = v.coeff(0) * (abs(m_matT.coeff(im-1,im-1)) + abs(Tmm) + abs(m_matT.coeff(im+1,im+1)));
|
|---|
| 457 | if (abs(lhs) < NumTraits<Scalar>::epsilon() * rhs)
|
|---|
| 458 | break;
|
|---|
| 459 | }
|
|---|
| 460 | }
|
|---|
| 461 |
|
|---|
| 462 | /** \internal Perform a Francis QR step involving rows il:iu and columns im:iu. */
|
|---|
| 463 | template<typename MatrixType>
|
|---|
| 464 | inline void RealSchur<MatrixType>::performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace)
|
|---|
| 465 | {
|
|---|
| 466 | eigen_assert(im >= il);
|
|---|
| 467 | eigen_assert(im <= iu-2);
|
|---|
| 468 |
|
|---|
| 469 | const Index size = m_matT.cols();
|
|---|
| 470 |
|
|---|
| 471 | for (Index k = im; k <= iu-2; ++k)
|
|---|
| 472 | {
|
|---|
| 473 | bool firstIteration = (k == im);
|
|---|
| 474 |
|
|---|
| 475 | Vector3s v;
|
|---|
| 476 | if (firstIteration)
|
|---|
| 477 | v = firstHouseholderVector;
|
|---|
| 478 | else
|
|---|
| 479 | v = m_matT.template block<3,1>(k,k-1);
|
|---|
| 480 |
|
|---|
| 481 | Scalar tau, beta;
|
|---|
| 482 | Matrix<Scalar, 2, 1> ess;
|
|---|
| 483 | v.makeHouseholder(ess, tau, beta);
|
|---|
| 484 |
|
|---|
| 485 | if (beta != Scalar(0)) // if v is not zero
|
|---|
| 486 | {
|
|---|
| 487 | if (firstIteration && k > il)
|
|---|
| 488 | m_matT.coeffRef(k,k-1) = -m_matT.coeff(k,k-1);
|
|---|
| 489 | else if (!firstIteration)
|
|---|
| 490 | m_matT.coeffRef(k,k-1) = beta;
|
|---|
| 491 |
|
|---|
| 492 | // These Householder transformations form the O(n^3) part of the algorithm
|
|---|
| 493 | m_matT.block(k, k, 3, size-k).applyHouseholderOnTheLeft(ess, tau, workspace);
|
|---|
| 494 | m_matT.block(0, k, (std::min)(iu,k+3) + 1, 3).applyHouseholderOnTheRight(ess, tau, workspace);
|
|---|
| 495 | if (computeU)
|
|---|
| 496 | m_matU.block(0, k, size, 3).applyHouseholderOnTheRight(ess, tau, workspace);
|
|---|
| 497 | }
|
|---|
| 498 | }
|
|---|
| 499 |
|
|---|
| 500 | Matrix<Scalar, 2, 1> v = m_matT.template block<2,1>(iu-1, iu-2);
|
|---|
| 501 | Scalar tau, beta;
|
|---|
| 502 | Matrix<Scalar, 1, 1> ess;
|
|---|
| 503 | v.makeHouseholder(ess, tau, beta);
|
|---|
| 504 |
|
|---|
| 505 | if (beta != Scalar(0)) // if v is not zero
|
|---|
| 506 | {
|
|---|
| 507 | m_matT.coeffRef(iu-1, iu-2) = beta;
|
|---|
| 508 | m_matT.block(iu-1, iu-1, 2, size-iu+1).applyHouseholderOnTheLeft(ess, tau, workspace);
|
|---|
| 509 | m_matT.block(0, iu-1, iu+1, 2).applyHouseholderOnTheRight(ess, tau, workspace);
|
|---|
| 510 | if (computeU)
|
|---|
| 511 | m_matU.block(0, iu-1, size, 2).applyHouseholderOnTheRight(ess, tau, workspace);
|
|---|
| 512 | }
|
|---|
| 513 |
|
|---|
| 514 | // clean up pollution due to round-off errors
|
|---|
| 515 | for (Index i = im+2; i <= iu; ++i)
|
|---|
| 516 | {
|
|---|
| 517 | m_matT.coeffRef(i,i-2) = Scalar(0);
|
|---|
| 518 | if (i > im+2)
|
|---|
| 519 | m_matT.coeffRef(i,i-3) = Scalar(0);
|
|---|
| 520 | }
|
|---|
| 521 | }
|
|---|
| 522 |
|
|---|
| 523 | } // end namespace Eigen
|
|---|
| 524 |
|
|---|
| 525 | #endif // EIGEN_REAL_SCHUR_H
|
|---|
