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;
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432 | shiftInfo.setConstant(Scalar(0.964));
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433 | }
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434 | }
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435 | }
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436 |
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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)
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440 | {
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441 | using std::abs;
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442 | Vector3s& v = firstHouseholderVector; // alias to save typing
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443 |
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444 | for (im = iu-2; im >= il; --im)
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445 | {
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446 | const Scalar Tmm = m_matT.coeff(im,im);
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447 | const Scalar r = shiftInfo.coeff(0) - Tmm;
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448 | const Scalar s = shiftInfo.coeff(1) - Tmm;
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449 | v.coeffRef(0) = (r * s - shiftInfo.coeff(2)) / m_matT.coeff(im+1,im) + m_matT.coeff(im,im+1);
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450 | v.coeffRef(1) = m_matT.coeff(im+1,im+1) - Tmm - r - s;
|
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451 | v.coeffRef(2) = m_matT.coeff(im+2,im+1);
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452 | if (im == il) {
|
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453 | break;
|
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454 | }
|
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455 | const Scalar lhs = m_matT.coeff(im,im-1) * (abs(v.coeff(1)) + abs(v.coeff(2)));
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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)));
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457 | if (abs(lhs) < NumTraits<Scalar>::epsilon() * rhs)
|
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458 | break;
|
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459 | }
|
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460 | }
|
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461 |
|
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462 | /** \internal Perform a Francis QR step involving rows il:iu and columns im:iu. */
|
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463 | template<typename MatrixType>
|
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464 | inline void RealSchur<MatrixType>::performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace)
|
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465 | {
|
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466 | eigen_assert(im >= il);
|
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467 | eigen_assert(im <= iu-2);
|
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468 |
|
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469 | const Index size = m_matT.cols();
|
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470 |
|
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471 | for (Index k = im; k <= iu-2; ++k)
|
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472 | {
|
---|
473 | bool firstIteration = (k == im);
|
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474 |
|
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475 | Vector3s v;
|
---|
476 | if (firstIteration)
|
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477 | v = firstHouseholderVector;
|
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478 | else
|
---|
479 | v = m_matT.template block<3,1>(k,k-1);
|
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480 |
|
---|
481 | Scalar tau, beta;
|
---|
482 | Matrix<Scalar, 2, 1> ess;
|
---|
483 | v.makeHouseholder(ess, tau, beta);
|
---|
484 |
|
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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
|
---|