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 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_MATRIXBASEEIGENVALUES_H
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12 | #define EIGEN_MATRIXBASEEIGENVALUES_H
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13 |
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14 | namespace Eigen {
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15 |
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16 | namespace internal {
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17 |
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18 | template<typename Derived, bool IsComplex>
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19 | struct eigenvalues_selector
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20 | {
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21 | // this is the implementation for the case IsComplex = true
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22 | static inline typename MatrixBase<Derived>::EigenvaluesReturnType const
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23 | run(const MatrixBase<Derived>& m)
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24 | {
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25 | typedef typename Derived::PlainObject PlainObject;
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26 | PlainObject m_eval(m);
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27 | return ComplexEigenSolver<PlainObject>(m_eval, false).eigenvalues();
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28 | }
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29 | };
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30 |
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31 | template<typename Derived>
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32 | struct eigenvalues_selector<Derived, false>
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33 | {
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34 | static inline typename MatrixBase<Derived>::EigenvaluesReturnType const
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35 | run(const MatrixBase<Derived>& m)
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36 | {
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37 | typedef typename Derived::PlainObject PlainObject;
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38 | PlainObject m_eval(m);
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39 | return EigenSolver<PlainObject>(m_eval, false).eigenvalues();
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40 | }
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41 | };
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42 |
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43 | } // end namespace internal
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44 |
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45 | /** \brief Computes the eigenvalues of a matrix
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46 | * \returns Column vector containing the eigenvalues.
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47 | *
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48 | * \eigenvalues_module
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49 | * This function computes the eigenvalues with the help of the EigenSolver
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50 | * class (for real matrices) or the ComplexEigenSolver class (for complex
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51 | * matrices).
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52 | *
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53 | * The eigenvalues are repeated according to their algebraic multiplicity,
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54 | * so there are as many eigenvalues as rows in the matrix.
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55 | *
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56 | * The SelfAdjointView class provides a better algorithm for selfadjoint
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57 | * matrices.
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58 | *
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59 | * Example: \include MatrixBase_eigenvalues.cpp
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60 | * Output: \verbinclude MatrixBase_eigenvalues.out
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61 | *
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62 | * \sa EigenSolver::eigenvalues(), ComplexEigenSolver::eigenvalues(),
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63 | * SelfAdjointView::eigenvalues()
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64 | */
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65 | template<typename Derived>
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66 | inline typename MatrixBase<Derived>::EigenvaluesReturnType
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67 | MatrixBase<Derived>::eigenvalues() const
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68 | {
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69 | typedef typename internal::traits<Derived>::Scalar Scalar;
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70 | return internal::eigenvalues_selector<Derived, NumTraits<Scalar>::IsComplex>::run(derived());
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71 | }
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72 |
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73 | /** \brief Computes the eigenvalues of a matrix
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74 | * \returns Column vector containing the eigenvalues.
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75 | *
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76 | * \eigenvalues_module
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77 | * This function computes the eigenvalues with the help of the
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78 | * SelfAdjointEigenSolver class. The eigenvalues are repeated according to
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79 | * their algebraic multiplicity, so there are as many eigenvalues as rows in
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80 | * the matrix.
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81 | *
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82 | * Example: \include SelfAdjointView_eigenvalues.cpp
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83 | * Output: \verbinclude SelfAdjointView_eigenvalues.out
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84 | *
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85 | * \sa SelfAdjointEigenSolver::eigenvalues(), MatrixBase::eigenvalues()
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86 | */
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87 | template<typename MatrixType, unsigned int UpLo>
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88 | inline typename SelfAdjointView<MatrixType, UpLo>::EigenvaluesReturnType
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89 | SelfAdjointView<MatrixType, UpLo>::eigenvalues() const
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90 | {
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91 | typedef typename SelfAdjointView<MatrixType, UpLo>::PlainObject PlainObject;
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92 | PlainObject thisAsMatrix(*this);
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93 | return SelfAdjointEigenSolver<PlainObject>(thisAsMatrix, false).eigenvalues();
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94 | }
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95 |
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96 |
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97 |
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98 | /** \brief Computes the L2 operator norm
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99 | * \returns Operator norm of the matrix.
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100 | *
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101 | * \eigenvalues_module
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102 | * This function computes the L2 operator norm of a matrix, which is also
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103 | * known as the spectral norm. The norm of a matrix \f$ A \f$ is defined to be
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104 | * \f[ \|A\|_2 = \max_x \frac{\|Ax\|_2}{\|x\|_2} \f]
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105 | * where the maximum is over all vectors and the norm on the right is the
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106 | * Euclidean vector norm. The norm equals the largest singular value, which is
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107 | * the square root of the largest eigenvalue of the positive semi-definite
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108 | * matrix \f$ A^*A \f$.
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109 | *
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110 | * The current implementation uses the eigenvalues of \f$ A^*A \f$, as computed
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111 | * by SelfAdjointView::eigenvalues(), to compute the operator norm of a
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112 | * matrix. The SelfAdjointView class provides a better algorithm for
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113 | * selfadjoint matrices.
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114 | *
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115 | * Example: \include MatrixBase_operatorNorm.cpp
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116 | * Output: \verbinclude MatrixBase_operatorNorm.out
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117 | *
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118 | * \sa SelfAdjointView::eigenvalues(), SelfAdjointView::operatorNorm()
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119 | */
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120 | template<typename Derived>
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121 | inline typename MatrixBase<Derived>::RealScalar
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122 | MatrixBase<Derived>::operatorNorm() const
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123 | {
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124 | using std::sqrt;
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125 | typename Derived::PlainObject m_eval(derived());
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126 | // FIXME if it is really guaranteed that the eigenvalues are already sorted,
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127 | // then we don't need to compute a maxCoeff() here, comparing the 1st and last ones is enough.
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128 | return sqrt((m_eval*m_eval.adjoint())
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129 | .eval()
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130 | .template selfadjointView<Lower>()
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131 | .eigenvalues()
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132 | .maxCoeff()
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133 | );
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134 | }
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135 |
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136 | /** \brief Computes the L2 operator norm
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137 | * \returns Operator norm of the matrix.
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138 | *
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139 | * \eigenvalues_module
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140 | * This function computes the L2 operator norm of a self-adjoint matrix. For a
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141 | * self-adjoint matrix, the operator norm is the largest eigenvalue.
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142 | *
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143 | * The current implementation uses the eigenvalues of the matrix, as computed
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144 | * by eigenvalues(), to compute the operator norm of the matrix.
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145 | *
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146 | * Example: \include SelfAdjointView_operatorNorm.cpp
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147 | * Output: \verbinclude SelfAdjointView_operatorNorm.out
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148 | *
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149 | * \sa eigenvalues(), MatrixBase::operatorNorm()
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150 | */
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151 | template<typename MatrixType, unsigned int UpLo>
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152 | inline typename SelfAdjointView<MatrixType, UpLo>::RealScalar
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153 | SelfAdjointView<MatrixType, UpLo>::operatorNorm() const
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154 | {
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155 | return eigenvalues().cwiseAbs().maxCoeff();
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156 | }
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157 |
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158 | } // end namespace Eigen
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159 |
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160 | #endif
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