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_EIGEN_SOLVER_H
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13 | #define EIGEN_COMPLEX_EIGEN_SOLVER_H
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14 |
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15 | #include "./ComplexSchur.h"
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16 |
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17 | namespace Eigen {
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18 |
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19 | /** \eigenvalues_module \ingroup Eigenvalues_Module
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20 | *
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21 | *
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22 | * \class ComplexEigenSolver
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23 | *
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24 | * \brief Computes eigenvalues and eigenvectors of general complex matrices
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25 | *
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26 | * \tparam _MatrixType the type of the matrix of which we are
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27 | * computing the eigendecomposition; this is expected to be an
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28 | * instantiation of the Matrix class template.
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29 | *
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30 | * The eigenvalues and eigenvectors of a matrix \f$ A \f$ are scalars
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31 | * \f$ \lambda \f$ and vectors \f$ v \f$ such that \f$ Av = \lambda v
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32 | * \f$. If \f$ D \f$ is a diagonal matrix with the eigenvalues on
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33 | * the diagonal, and \f$ V \f$ is a matrix with the eigenvectors as
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34 | * its columns, then \f$ A V = V D \f$. The matrix \f$ V \f$ is
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35 | * almost always invertible, in which case we have \f$ A = V D V^{-1}
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36 | * \f$. This is called the eigendecomposition.
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37 | *
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38 | * The main function in this class is compute(), which computes the
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39 | * eigenvalues and eigenvectors of a given function. The
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40 | * documentation for that function contains an example showing the
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41 | * main features of the class.
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42 | *
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43 | * \sa class EigenSolver, class SelfAdjointEigenSolver
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44 | */
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45 | template<typename _MatrixType> class ComplexEigenSolver
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46 | {
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47 | public:
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48 |
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49 | /** \brief Synonym for the template parameter \p _MatrixType. */
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50 | typedef _MatrixType MatrixType;
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51 |
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52 | enum {
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53 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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54 | ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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55 | Options = MatrixType::Options,
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56 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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57 | MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
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58 | };
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59 |
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60 | /** \brief Scalar type for matrices of type #MatrixType. */
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61 | typedef typename MatrixType::Scalar Scalar;
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62 | typedef typename NumTraits<Scalar>::Real RealScalar;
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63 | typedef typename MatrixType::Index Index;
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64 |
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65 | /** \brief Complex scalar type for #MatrixType.
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66 | *
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67 | * This is \c std::complex<Scalar> if #Scalar is real (e.g.,
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68 | * \c float or \c double) and just \c Scalar if #Scalar is
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69 | * complex.
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70 | */
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71 | typedef std::complex<RealScalar> ComplexScalar;
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72 |
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73 | /** \brief Type for vector of eigenvalues as returned by eigenvalues().
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74 | *
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75 | * This is a column vector with entries of type #ComplexScalar.
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76 | * The length of the vector is the size of #MatrixType.
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77 | */
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78 | typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options&(~RowMajor), MaxColsAtCompileTime, 1> EigenvalueType;
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79 |
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80 | /** \brief Type for matrix of eigenvectors as returned by eigenvectors().
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81 | *
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82 | * This is a square matrix with entries of type #ComplexScalar.
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83 | * The size is the same as the size of #MatrixType.
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84 | */
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85 | typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> EigenvectorType;
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86 |
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87 | /** \brief Default constructor.
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88 | *
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89 | * The default constructor is useful in cases in which the user intends to
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90 | * perform decompositions via compute().
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91 | */
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92 | ComplexEigenSolver()
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93 | : m_eivec(),
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94 | m_eivalues(),
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95 | m_schur(),
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96 | m_isInitialized(false),
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97 | m_eigenvectorsOk(false),
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98 | m_matX()
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99 | {}
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100 |
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101 | /** \brief Default Constructor with memory preallocation
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102 | *
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103 | * Like the default constructor but with preallocation of the internal data
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104 | * according to the specified problem \a size.
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105 | * \sa ComplexEigenSolver()
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106 | */
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107 | ComplexEigenSolver(Index size)
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108 | : m_eivec(size, size),
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109 | m_eivalues(size),
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110 | m_schur(size),
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111 | m_isInitialized(false),
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112 | m_eigenvectorsOk(false),
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113 | m_matX(size, size)
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114 | {}
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115 |
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116 | /** \brief Constructor; computes eigendecomposition of given matrix.
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117 | *
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118 | * \param[in] matrix Square matrix whose eigendecomposition is to be computed.
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119 | * \param[in] computeEigenvectors If true, both the eigenvectors and the
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120 | * eigenvalues are computed; if false, only the eigenvalues are
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121 | * computed.
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122 | *
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123 | * This constructor calls compute() to compute the eigendecomposition.
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124 | */
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125 | ComplexEigenSolver(const MatrixType& matrix, bool computeEigenvectors = true)
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126 | : m_eivec(matrix.rows(),matrix.cols()),
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127 | m_eivalues(matrix.cols()),
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128 | m_schur(matrix.rows()),
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129 | m_isInitialized(false),
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130 | m_eigenvectorsOk(false),
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131 | m_matX(matrix.rows(),matrix.cols())
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132 | {
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133 | compute(matrix, computeEigenvectors);
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134 | }
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135 |
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136 | /** \brief Returns the eigenvectors of given matrix.
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137 | *
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138 | * \returns A const reference to the matrix whose columns are the eigenvectors.
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139 | *
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140 | * \pre Either the constructor
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141 | * ComplexEigenSolver(const MatrixType& matrix, bool) or the member
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142 | * function compute(const MatrixType& matrix, bool) has been called before
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143 | * to compute the eigendecomposition of a matrix, and
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144 | * \p computeEigenvectors was set to true (the default).
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145 | *
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146 | * This function returns a matrix whose columns are the eigenvectors. Column
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147 | * \f$ k \f$ is an eigenvector corresponding to eigenvalue number \f$ k
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148 | * \f$ as returned by eigenvalues(). The eigenvectors are normalized to
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149 | * have (Euclidean) norm equal to one. The matrix returned by this
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150 | * function is the matrix \f$ V \f$ in the eigendecomposition \f$ A = V D
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151 | * V^{-1} \f$, if it exists.
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152 | *
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153 | * Example: \include ComplexEigenSolver_eigenvectors.cpp
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154 | * Output: \verbinclude ComplexEigenSolver_eigenvectors.out
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155 | */
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156 | const EigenvectorType& eigenvectors() const
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157 | {
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158 | eigen_assert(m_isInitialized && "ComplexEigenSolver is not initialized.");
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159 | eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
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160 | return m_eivec;
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161 | }
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162 |
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163 | /** \brief Returns the eigenvalues of given matrix.
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164 | *
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165 | * \returns A const reference to the column vector containing the eigenvalues.
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166 | *
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167 | * \pre Either the constructor
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168 | * ComplexEigenSolver(const MatrixType& matrix, bool) or the member
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169 | * function compute(const MatrixType& matrix, bool) has been called before
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170 | * to compute the eigendecomposition of a matrix.
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171 | *
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172 | * This function returns a column vector containing the
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173 | * eigenvalues. Eigenvalues are repeated according to their
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174 | * algebraic multiplicity, so there are as many eigenvalues as
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175 | * rows in the matrix. The eigenvalues are not sorted in any particular
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176 | * order.
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177 | *
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178 | * Example: \include ComplexEigenSolver_eigenvalues.cpp
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179 | * Output: \verbinclude ComplexEigenSolver_eigenvalues.out
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180 | */
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181 | const EigenvalueType& eigenvalues() const
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182 | {
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183 | eigen_assert(m_isInitialized && "ComplexEigenSolver is not initialized.");
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184 | return m_eivalues;
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185 | }
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186 |
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187 | /** \brief Computes eigendecomposition of given matrix.
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188 | *
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189 | * \param[in] matrix Square matrix whose eigendecomposition is to be computed.
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190 | * \param[in] computeEigenvectors If true, both the eigenvectors and the
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191 | * eigenvalues are computed; if false, only the eigenvalues are
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192 | * computed.
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193 | * \returns Reference to \c *this
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194 | *
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195 | * This function computes the eigenvalues of the complex matrix \p matrix.
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196 | * The eigenvalues() function can be used to retrieve them. If
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197 | * \p computeEigenvectors is true, then the eigenvectors are also computed
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198 | * and can be retrieved by calling eigenvectors().
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199 | *
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200 | * The matrix is first reduced to Schur form using the
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201 | * ComplexSchur class. The Schur decomposition is then used to
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202 | * compute the eigenvalues and eigenvectors.
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203 | *
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204 | * The cost of the computation is dominated by the cost of the
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205 | * Schur decomposition, which is \f$ O(n^3) \f$ where \f$ n \f$
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206 | * is the size of the matrix.
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207 | *
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208 | * Example: \include ComplexEigenSolver_compute.cpp
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209 | * Output: \verbinclude ComplexEigenSolver_compute.out
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210 | */
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211 | ComplexEigenSolver& compute(const MatrixType& matrix, bool computeEigenvectors = true);
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212 |
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213 | /** \brief Reports whether previous computation was successful.
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214 | *
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215 | * \returns \c Success if computation was succesful, \c NoConvergence otherwise.
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216 | */
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217 | ComputationInfo info() const
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218 | {
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219 | eigen_assert(m_isInitialized && "ComplexEigenSolver is not initialized.");
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220 | return m_schur.info();
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221 | }
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222 |
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223 | /** \brief Sets the maximum number of iterations allowed. */
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224 | ComplexEigenSolver& setMaxIterations(Index maxIters)
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225 | {
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226 | m_schur.setMaxIterations(maxIters);
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227 | return *this;
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228 | }
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229 |
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230 | /** \brief Returns the maximum number of iterations. */
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231 | Index getMaxIterations()
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232 | {
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233 | return m_schur.getMaxIterations();
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234 | }
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235 |
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236 | protected:
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237 |
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238 | static void check_template_parameters()
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239 | {
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240 | EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
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241 | }
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242 |
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243 | EigenvectorType m_eivec;
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244 | EigenvalueType m_eivalues;
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245 | ComplexSchur<MatrixType> m_schur;
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246 | bool m_isInitialized;
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247 | bool m_eigenvectorsOk;
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248 | EigenvectorType m_matX;
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249 |
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250 | private:
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251 | void doComputeEigenvectors(const RealScalar& matrixnorm);
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252 | void sortEigenvalues(bool computeEigenvectors);
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253 | };
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254 |
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255 |
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256 | template<typename MatrixType>
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257 | ComplexEigenSolver<MatrixType>&
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258 | ComplexEigenSolver<MatrixType>::compute(const MatrixType& matrix, bool computeEigenvectors)
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259 | {
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260 | check_template_parameters();
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261 |
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262 | // this code is inspired from Jampack
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263 | eigen_assert(matrix.cols() == matrix.rows());
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264 |
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265 | // Do a complex Schur decomposition, A = U T U^*
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266 | // The eigenvalues are on the diagonal of T.
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267 | m_schur.compute(matrix, computeEigenvectors);
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268 |
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269 | if(m_schur.info() == Success)
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270 | {
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271 | m_eivalues = m_schur.matrixT().diagonal();
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272 | if(computeEigenvectors)
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273 | doComputeEigenvectors(matrix.norm());
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274 | sortEigenvalues(computeEigenvectors);
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275 | }
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276 |
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277 | m_isInitialized = true;
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278 | m_eigenvectorsOk = computeEigenvectors;
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279 | return *this;
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280 | }
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281 |
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282 |
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283 | template<typename MatrixType>
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284 | void ComplexEigenSolver<MatrixType>::doComputeEigenvectors(const RealScalar& matrixnorm)
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285 | {
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286 | const Index n = m_eivalues.size();
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287 |
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288 | // Compute X such that T = X D X^(-1), where D is the diagonal of T.
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289 | // The matrix X is unit triangular.
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290 | m_matX = EigenvectorType::Zero(n, n);
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291 | for(Index k=n-1 ; k>=0 ; k--)
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292 | {
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293 | m_matX.coeffRef(k,k) = ComplexScalar(1.0,0.0);
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294 | // Compute X(i,k) using the (i,k) entry of the equation X T = D X
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295 | for(Index i=k-1 ; i>=0 ; i--)
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296 | {
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297 | m_matX.coeffRef(i,k) = -m_schur.matrixT().coeff(i,k);
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298 | if(k-i-1>0)
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299 | m_matX.coeffRef(i,k) -= (m_schur.matrixT().row(i).segment(i+1,k-i-1) * m_matX.col(k).segment(i+1,k-i-1)).value();
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300 | ComplexScalar z = m_schur.matrixT().coeff(i,i) - m_schur.matrixT().coeff(k,k);
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301 | if(z==ComplexScalar(0))
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302 | {
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303 | // If the i-th and k-th eigenvalue are equal, then z equals 0.
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304 | // Use a small value instead, to prevent division by zero.
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305 | numext::real_ref(z) = NumTraits<RealScalar>::epsilon() * matrixnorm;
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306 | }
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307 | m_matX.coeffRef(i,k) = m_matX.coeff(i,k) / z;
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308 | }
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309 | }
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310 |
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311 | // Compute V as V = U X; now A = U T U^* = U X D X^(-1) U^* = V D V^(-1)
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312 | m_eivec.noalias() = m_schur.matrixU() * m_matX;
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313 | // .. and normalize the eigenvectors
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314 | for(Index k=0 ; k<n ; k++)
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315 | {
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316 | m_eivec.col(k).normalize();
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317 | }
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318 | }
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319 |
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320 |
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321 | template<typename MatrixType>
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322 | void ComplexEigenSolver<MatrixType>::sortEigenvalues(bool computeEigenvectors)
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323 | {
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324 | const Index n = m_eivalues.size();
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325 | for (Index i=0; i<n; i++)
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326 | {
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327 | Index k;
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328 | m_eivalues.cwiseAbs().tail(n-i).minCoeff(&k);
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329 | if (k != 0)
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330 | {
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331 | k += i;
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332 | std::swap(m_eivalues[k],m_eivalues[i]);
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333 | if(computeEigenvectors)
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334 | m_eivec.col(i).swap(m_eivec.col(k));
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335 | }
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336 | }
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337 | }
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338 |
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339 | } // end namespace Eigen
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340 |
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341 | #endif // EIGEN_COMPLEX_EIGEN_SOLVER_H
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