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_TRIDIAGONALIZATION_H
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12 | #define EIGEN_TRIDIAGONALIZATION_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 MatrixType> struct TridiagonalizationMatrixTReturnType;
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19 | template<typename MatrixType>
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20 | struct traits<TridiagonalizationMatrixTReturnType<MatrixType> >
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21 | {
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22 | typedef typename MatrixType::PlainObject ReturnType;
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23 | };
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24 |
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25 | template<typename MatrixType, typename CoeffVectorType>
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26 | void tridiagonalization_inplace(MatrixType& matA, CoeffVectorType& hCoeffs);
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27 | }
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28 |
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29 | /** \eigenvalues_module \ingroup Eigenvalues_Module
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30 | *
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31 | *
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32 | * \class Tridiagonalization
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33 | *
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34 | * \brief Tridiagonal decomposition of a selfadjoint matrix
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35 | *
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36 | * \tparam _MatrixType the type of the matrix of which we are computing the
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37 | * tridiagonal decomposition; this is expected to be an instantiation of the
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38 | * Matrix class template.
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39 | *
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40 | * This class performs a tridiagonal decomposition of a selfadjoint matrix \f$ A \f$ such that:
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41 | * \f$ A = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real symmetric tridiagonal matrix.
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42 | *
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43 | * A tridiagonal matrix is a matrix which has nonzero elements only on the
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44 | * main diagonal and the first diagonal below and above it. The Hessenberg
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45 | * decomposition of a selfadjoint matrix is in fact a tridiagonal
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46 | * decomposition. This class is used in SelfAdjointEigenSolver to compute the
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47 | * eigenvalues and eigenvectors of a selfadjoint matrix.
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48 | *
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49 | * Call the function compute() to compute the tridiagonal decomposition of a
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50 | * given matrix. Alternatively, you can use the Tridiagonalization(const MatrixType&)
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51 | * constructor which computes the tridiagonal Schur decomposition at
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52 | * construction time. Once the decomposition is computed, you can use the
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53 | * matrixQ() and matrixT() functions to retrieve the matrices Q and T in the
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54 | * decomposition.
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55 | *
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56 | * The documentation of Tridiagonalization(const MatrixType&) contains an
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57 | * example of the typical use of this class.
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58 | *
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59 | * \sa class HessenbergDecomposition, class SelfAdjointEigenSolver
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60 | */
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61 | template<typename _MatrixType> class Tridiagonalization
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62 | {
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63 | public:
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64 |
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65 | /** \brief Synonym for the template parameter \p _MatrixType. */
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66 | typedef _MatrixType MatrixType;
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67 |
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68 | typedef typename MatrixType::Scalar Scalar;
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69 | typedef typename NumTraits<Scalar>::Real RealScalar;
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70 | typedef typename MatrixType::Index Index;
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71 |
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72 | enum {
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73 | Size = MatrixType::RowsAtCompileTime,
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74 | SizeMinusOne = Size == Dynamic ? Dynamic : (Size > 1 ? Size - 1 : 1),
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75 | Options = MatrixType::Options,
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76 | MaxSize = MatrixType::MaxRowsAtCompileTime,
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77 | MaxSizeMinusOne = MaxSize == Dynamic ? Dynamic : (MaxSize > 1 ? MaxSize - 1 : 1)
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78 | };
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79 |
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80 | typedef Matrix<Scalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> CoeffVectorType;
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81 | typedef typename internal::plain_col_type<MatrixType, RealScalar>::type DiagonalType;
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82 | typedef Matrix<RealScalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> SubDiagonalType;
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83 | typedef typename internal::remove_all<typename MatrixType::RealReturnType>::type MatrixTypeRealView;
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84 | typedef internal::TridiagonalizationMatrixTReturnType<MatrixTypeRealView> MatrixTReturnType;
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85 |
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86 | typedef typename internal::conditional<NumTraits<Scalar>::IsComplex,
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87 | typename internal::add_const_on_value_type<typename Diagonal<const MatrixType>::RealReturnType>::type,
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88 | const Diagonal<const MatrixType>
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89 | >::type DiagonalReturnType;
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90 |
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91 | typedef typename internal::conditional<NumTraits<Scalar>::IsComplex,
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92 | typename internal::add_const_on_value_type<typename Diagonal<
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93 | Block<const MatrixType,SizeMinusOne,SizeMinusOne> >::RealReturnType>::type,
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94 | const Diagonal<
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95 | Block<const MatrixType,SizeMinusOne,SizeMinusOne> >
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96 | >::type SubDiagonalReturnType;
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97 |
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98 | /** \brief Return type of matrixQ() */
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99 | typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename CoeffVectorType::ConjugateReturnType>::type> HouseholderSequenceType;
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100 |
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101 | /** \brief Default constructor.
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102 | *
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103 | * \param [in] size Positive integer, size of the matrix whose tridiagonal
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104 | * decomposition will be computed.
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105 | *
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106 | * The default constructor is useful in cases in which the user intends to
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107 | * perform decompositions via compute(). The \p size parameter is only
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108 | * used as a hint. It is not an error to give a wrong \p size, but it may
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109 | * impair performance.
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110 | *
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111 | * \sa compute() for an example.
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112 | */
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113 | Tridiagonalization(Index size = Size==Dynamic ? 2 : Size)
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114 | : m_matrix(size,size),
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115 | m_hCoeffs(size > 1 ? size-1 : 1),
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116 | m_isInitialized(false)
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117 | {}
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118 |
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119 | /** \brief Constructor; computes tridiagonal decomposition of given matrix.
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120 | *
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121 | * \param[in] matrix Selfadjoint matrix whose tridiagonal decomposition
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122 | * is to be computed.
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123 | *
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124 | * This constructor calls compute() to compute the tridiagonal decomposition.
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125 | *
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126 | * Example: \include Tridiagonalization_Tridiagonalization_MatrixType.cpp
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127 | * Output: \verbinclude Tridiagonalization_Tridiagonalization_MatrixType.out
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128 | */
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129 | Tridiagonalization(const MatrixType& matrix)
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130 | : m_matrix(matrix),
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131 | m_hCoeffs(matrix.cols() > 1 ? matrix.cols()-1 : 1),
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132 | m_isInitialized(false)
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133 | {
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134 | internal::tridiagonalization_inplace(m_matrix, m_hCoeffs);
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135 | m_isInitialized = true;
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136 | }
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137 |
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138 | /** \brief Computes tridiagonal decomposition of given matrix.
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139 | *
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140 | * \param[in] matrix Selfadjoint matrix whose tridiagonal decomposition
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141 | * is to be computed.
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142 | * \returns Reference to \c *this
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143 | *
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144 | * The tridiagonal decomposition is computed by bringing the columns of
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145 | * the matrix successively in the required form using Householder
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146 | * reflections. The cost is \f$ 4n^3/3 \f$ flops, where \f$ n \f$ denotes
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147 | * the size of the given matrix.
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148 | *
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149 | * This method reuses of the allocated data in the Tridiagonalization
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150 | * object, if the size of the matrix does not change.
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151 | *
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152 | * Example: \include Tridiagonalization_compute.cpp
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153 | * Output: \verbinclude Tridiagonalization_compute.out
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154 | */
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155 | Tridiagonalization& compute(const MatrixType& matrix)
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156 | {
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157 | m_matrix = matrix;
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158 | m_hCoeffs.resize(matrix.rows()-1, 1);
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159 | internal::tridiagonalization_inplace(m_matrix, m_hCoeffs);
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160 | m_isInitialized = true;
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161 | return *this;
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162 | }
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163 |
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164 | /** \brief Returns the Householder coefficients.
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165 | *
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166 | * \returns a const reference to the vector of Householder coefficients
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167 | *
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168 | * \pre Either the constructor Tridiagonalization(const MatrixType&) or
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169 | * the member function compute(const MatrixType&) has been called before
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170 | * to compute the tridiagonal decomposition of a matrix.
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171 | *
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172 | * The Householder coefficients allow the reconstruction of the matrix
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173 | * \f$ Q \f$ in the tridiagonal decomposition from the packed data.
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174 | *
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175 | * Example: \include Tridiagonalization_householderCoefficients.cpp
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176 | * Output: \verbinclude Tridiagonalization_householderCoefficients.out
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177 | *
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178 | * \sa packedMatrix(), \ref Householder_Module "Householder module"
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179 | */
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180 | inline CoeffVectorType householderCoefficients() const
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181 | {
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182 | eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
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183 | return m_hCoeffs;
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184 | }
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185 |
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186 | /** \brief Returns the internal representation of the decomposition
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187 | *
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188 | * \returns a const reference to a matrix with the internal representation
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189 | * of the decomposition.
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190 | *
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191 | * \pre Either the constructor Tridiagonalization(const MatrixType&) or
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192 | * the member function compute(const MatrixType&) has been called before
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193 | * to compute the tridiagonal decomposition of a matrix.
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194 | *
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195 | * The returned matrix contains the following information:
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196 | * - the strict upper triangular part is equal to the input matrix A.
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197 | * - the diagonal and lower sub-diagonal represent the real tridiagonal
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198 | * symmetric matrix T.
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199 | * - the rest of the lower part contains the Householder vectors that,
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200 | * combined with Householder coefficients returned by
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201 | * householderCoefficients(), allows to reconstruct the matrix Q as
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202 | * \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
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203 | * Here, the matrices \f$ H_i \f$ are the Householder transformations
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204 | * \f$ H_i = (I - h_i v_i v_i^T) \f$
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205 | * where \f$ h_i \f$ is the \f$ i \f$th Householder coefficient and
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206 | * \f$ v_i \f$ is the Householder vector defined by
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207 | * \f$ v_i = [ 0, \ldots, 0, 1, M(i+2,i), \ldots, M(N-1,i) ]^T \f$
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208 | * with M the matrix returned by this function.
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209 | *
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210 | * See LAPACK for further details on this packed storage.
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211 | *
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212 | * Example: \include Tridiagonalization_packedMatrix.cpp
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213 | * Output: \verbinclude Tridiagonalization_packedMatrix.out
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214 | *
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215 | * \sa householderCoefficients()
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216 | */
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217 | inline const MatrixType& packedMatrix() const
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218 | {
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219 | eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
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220 | return m_matrix;
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221 | }
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222 |
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223 | /** \brief Returns the unitary matrix Q in the decomposition
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224 | *
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225 | * \returns object representing the matrix Q
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226 | *
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227 | * \pre Either the constructor Tridiagonalization(const MatrixType&) or
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228 | * the member function compute(const MatrixType&) has been called before
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229 | * to compute the tridiagonal decomposition of a matrix.
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230 | *
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231 | * This function returns a light-weight object of template class
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232 | * HouseholderSequence. You can either apply it directly to a matrix or
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233 | * you can convert it to a matrix of type #MatrixType.
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234 | *
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235 | * \sa Tridiagonalization(const MatrixType&) for an example,
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236 | * matrixT(), class HouseholderSequence
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237 | */
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238 | HouseholderSequenceType matrixQ() const
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239 | {
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240 | eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
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241 | return HouseholderSequenceType(m_matrix, m_hCoeffs.conjugate())
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242 | .setLength(m_matrix.rows() - 1)
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243 | .setShift(1);
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244 | }
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245 |
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246 | /** \brief Returns an expression of the tridiagonal matrix T in the decomposition
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247 | *
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248 | * \returns expression object representing the matrix T
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249 | *
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250 | * \pre Either the constructor Tridiagonalization(const MatrixType&) or
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251 | * the member function compute(const MatrixType&) has been called before
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252 | * to compute the tridiagonal decomposition of a matrix.
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253 | *
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254 | * Currently, this function can be used to extract the matrix T from internal
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255 | * data and copy it to a dense matrix object. In most cases, it may be
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256 | * sufficient to directly use the packed matrix or the vector expressions
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257 | * returned by diagonal() and subDiagonal() instead of creating a new
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258 | * dense copy matrix with this function.
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259 | *
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260 | * \sa Tridiagonalization(const MatrixType&) for an example,
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261 | * matrixQ(), packedMatrix(), diagonal(), subDiagonal()
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262 | */
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263 | MatrixTReturnType matrixT() const
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264 | {
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265 | eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
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266 | return MatrixTReturnType(m_matrix.real());
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267 | }
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268 |
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269 | /** \brief Returns the diagonal of the tridiagonal matrix T in the decomposition.
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270 | *
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271 | * \returns expression representing the diagonal of T
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272 | *
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273 | * \pre Either the constructor Tridiagonalization(const MatrixType&) or
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274 | * the member function compute(const MatrixType&) has been called before
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275 | * to compute the tridiagonal decomposition of a matrix.
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276 | *
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277 | * Example: \include Tridiagonalization_diagonal.cpp
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278 | * Output: \verbinclude Tridiagonalization_diagonal.out
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279 | *
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280 | * \sa matrixT(), subDiagonal()
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281 | */
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282 | DiagonalReturnType diagonal() const;
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283 |
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284 | /** \brief Returns the subdiagonal of the tridiagonal matrix T in the decomposition.
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285 | *
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286 | * \returns expression representing the subdiagonal of T
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287 | *
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288 | * \pre Either the constructor Tridiagonalization(const MatrixType&) or
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289 | * the member function compute(const MatrixType&) has been called before
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290 | * to compute the tridiagonal decomposition of a matrix.
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291 | *
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292 | * \sa diagonal() for an example, matrixT()
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293 | */
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294 | SubDiagonalReturnType subDiagonal() const;
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295 |
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296 | protected:
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297 |
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298 | MatrixType m_matrix;
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299 | CoeffVectorType m_hCoeffs;
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300 | bool m_isInitialized;
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301 | };
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302 |
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303 | template<typename MatrixType>
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304 | typename Tridiagonalization<MatrixType>::DiagonalReturnType
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305 | Tridiagonalization<MatrixType>::diagonal() const
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306 | {
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307 | eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
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308 | return m_matrix.diagonal();
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309 | }
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310 |
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311 | template<typename MatrixType>
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312 | typename Tridiagonalization<MatrixType>::SubDiagonalReturnType
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313 | Tridiagonalization<MatrixType>::subDiagonal() const
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314 | {
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315 | eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
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316 | Index n = m_matrix.rows();
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317 | return Block<const MatrixType,SizeMinusOne,SizeMinusOne>(m_matrix, 1, 0, n-1,n-1).diagonal();
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318 | }
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319 |
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320 | namespace internal {
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321 |
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322 | /** \internal
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323 | * Performs a tridiagonal decomposition of the selfadjoint matrix \a matA in-place.
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324 | *
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325 | * \param[in,out] matA On input the selfadjoint matrix. Only the \b lower triangular part is referenced.
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326 | * On output, the strict upper part is left unchanged, and the lower triangular part
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327 | * represents the T and Q matrices in packed format has detailed below.
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328 | * \param[out] hCoeffs returned Householder coefficients (see below)
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329 | *
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330 | * On output, the tridiagonal selfadjoint matrix T is stored in the diagonal
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331 | * and lower sub-diagonal of the matrix \a matA.
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332 | * The unitary matrix Q is represented in a compact way as a product of
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333 | * Householder reflectors \f$ H_i \f$ such that:
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334 | * \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
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335 | * The Householder reflectors are defined as
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336 | * \f$ H_i = (I - h_i v_i v_i^T) \f$
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337 | * where \f$ h_i = hCoeffs[i]\f$ is the \f$ i \f$th Householder coefficient and
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338 | * \f$ v_i \f$ is the Householder vector defined by
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339 | * \f$ v_i = [ 0, \ldots, 0, 1, matA(i+2,i), \ldots, matA(N-1,i) ]^T \f$.
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340 | *
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341 | * Implemented from Golub's "Matrix Computations", algorithm 8.3.1.
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342 | *
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343 | * \sa Tridiagonalization::packedMatrix()
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344 | */
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345 | template<typename MatrixType, typename CoeffVectorType>
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346 | void tridiagonalization_inplace(MatrixType& matA, CoeffVectorType& hCoeffs)
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347 | {
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348 | using numext::conj;
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349 | typedef typename MatrixType::Index Index;
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350 | typedef typename MatrixType::Scalar Scalar;
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351 | typedef typename MatrixType::RealScalar RealScalar;
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352 | Index n = matA.rows();
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353 | eigen_assert(n==matA.cols());
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354 | eigen_assert(n==hCoeffs.size()+1 || n==1);
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355 |
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356 | for (Index i = 0; i<n-1; ++i)
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357 | {
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358 | Index remainingSize = n-i-1;
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359 | RealScalar beta;
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360 | Scalar h;
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361 | matA.col(i).tail(remainingSize).makeHouseholderInPlace(h, beta);
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362 |
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363 | // Apply similarity transformation to remaining columns,
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364 | // i.e., A = H A H' where H = I - h v v' and v = matA.col(i).tail(n-i-1)
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365 | matA.col(i).coeffRef(i+1) = 1;
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366 |
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367 | hCoeffs.tail(n-i-1).noalias() = (matA.bottomRightCorner(remainingSize,remainingSize).template selfadjointView<Lower>()
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368 | * (conj(h) * matA.col(i).tail(remainingSize)));
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369 |
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370 | hCoeffs.tail(n-i-1) += (conj(h)*RealScalar(-0.5)*(hCoeffs.tail(remainingSize).dot(matA.col(i).tail(remainingSize)))) * matA.col(i).tail(n-i-1);
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371 |
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372 | matA.bottomRightCorner(remainingSize, remainingSize).template selfadjointView<Lower>()
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373 | .rankUpdate(matA.col(i).tail(remainingSize), hCoeffs.tail(remainingSize), Scalar(-1));
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374 |
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375 | matA.col(i).coeffRef(i+1) = beta;
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376 | hCoeffs.coeffRef(i) = h;
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377 | }
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378 | }
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379 |
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380 | // forward declaration, implementation at the end of this file
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381 | template<typename MatrixType,
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382 | int Size=MatrixType::ColsAtCompileTime,
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383 | bool IsComplex=NumTraits<typename MatrixType::Scalar>::IsComplex>
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384 | struct tridiagonalization_inplace_selector;
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385 |
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386 | /** \brief Performs a full tridiagonalization in place
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387 | *
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388 | * \param[in,out] mat On input, the selfadjoint matrix whose tridiagonal
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389 | * decomposition is to be computed. Only the lower triangular part referenced.
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390 | * The rest is left unchanged. On output, the orthogonal matrix Q
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391 | * in the decomposition if \p extractQ is true.
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392 | * \param[out] diag The diagonal of the tridiagonal matrix T in the
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393 | * decomposition.
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394 | * \param[out] subdiag The subdiagonal of the tridiagonal matrix T in
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395 | * the decomposition.
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396 | * \param[in] extractQ If true, the orthogonal matrix Q in the
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397 | * decomposition is computed and stored in \p mat.
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398 | *
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399 | * Computes the tridiagonal decomposition of the selfadjoint matrix \p mat in place
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400 | * such that \f$ mat = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real
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401 | * symmetric tridiagonal matrix.
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402 | *
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403 | * The tridiagonal matrix T is passed to the output parameters \p diag and \p subdiag. If
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404 | * \p extractQ is true, then the orthogonal matrix Q is passed to \p mat. Otherwise the lower
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405 | * part of the matrix \p mat is destroyed.
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406 | *
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407 | * The vectors \p diag and \p subdiag are not resized. The function
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408 | * assumes that they are already of the correct size. The length of the
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409 | * vector \p diag should equal the number of rows in \p mat, and the
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410 | * length of the vector \p subdiag should be one left.
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411 | *
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412 | * This implementation contains an optimized path for 3-by-3 matrices
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413 | * which is especially useful for plane fitting.
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414 | *
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415 | * \note Currently, it requires two temporary vectors to hold the intermediate
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416 | * Householder coefficients, and to reconstruct the matrix Q from the Householder
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417 | * reflectors.
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418 | *
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419 | * Example (this uses the same matrix as the example in
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420 | * Tridiagonalization::Tridiagonalization(const MatrixType&)):
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421 | * \include Tridiagonalization_decomposeInPlace.cpp
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422 | * Output: \verbinclude Tridiagonalization_decomposeInPlace.out
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423 | *
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424 | * \sa class Tridiagonalization
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425 | */
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426 | template<typename MatrixType, typename DiagonalType, typename SubDiagonalType>
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427 | void tridiagonalization_inplace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
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428 | {
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429 | eigen_assert(mat.cols()==mat.rows() && diag.size()==mat.rows() && subdiag.size()==mat.rows()-1);
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430 | tridiagonalization_inplace_selector<MatrixType>::run(mat, diag, subdiag, extractQ);
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431 | }
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432 |
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433 | /** \internal
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434 | * General full tridiagonalization
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435 | */
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436 | template<typename MatrixType, int Size, bool IsComplex>
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437 | struct tridiagonalization_inplace_selector
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438 | {
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439 | typedef typename Tridiagonalization<MatrixType>::CoeffVectorType CoeffVectorType;
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440 | typedef typename Tridiagonalization<MatrixType>::HouseholderSequenceType HouseholderSequenceType;
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441 | typedef typename MatrixType::Index Index;
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442 | template<typename DiagonalType, typename SubDiagonalType>
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443 | static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
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444 | {
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445 | CoeffVectorType hCoeffs(mat.cols()-1);
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446 | tridiagonalization_inplace(mat,hCoeffs);
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447 | diag = mat.diagonal().real();
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448 | subdiag = mat.template diagonal<-1>().real();
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449 | if(extractQ)
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450 | mat = HouseholderSequenceType(mat, hCoeffs.conjugate())
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451 | .setLength(mat.rows() - 1)
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452 | .setShift(1);
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453 | }
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454 | };
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455 |
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456 | /** \internal
|
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457 | * Specialization for 3x3 real matrices.
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458 | * Especially useful for plane fitting.
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459 | */
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460 | template<typename MatrixType>
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461 | struct tridiagonalization_inplace_selector<MatrixType,3,false>
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462 | {
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463 | typedef typename MatrixType::Scalar Scalar;
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464 | typedef typename MatrixType::RealScalar RealScalar;
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465 |
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466 | template<typename DiagonalType, typename SubDiagonalType>
|
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467 | static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
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468 | {
|
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469 | using std::sqrt;
|
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470 | diag[0] = mat(0,0);
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471 | RealScalar v1norm2 = numext::abs2(mat(2,0));
|
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472 | if(v1norm2 == RealScalar(0))
|
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473 | {
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474 | diag[1] = mat(1,1);
|
---|
475 | diag[2] = mat(2,2);
|
---|
476 | subdiag[0] = mat(1,0);
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477 | subdiag[1] = mat(2,1);
|
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478 | if (extractQ)
|
---|
479 | mat.setIdentity();
|
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480 | }
|
---|
481 | else
|
---|
482 | {
|
---|
483 | RealScalar beta = sqrt(numext::abs2(mat(1,0)) + v1norm2);
|
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484 | RealScalar invBeta = RealScalar(1)/beta;
|
---|
485 | Scalar m01 = mat(1,0) * invBeta;
|
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486 | Scalar m02 = mat(2,0) * invBeta;
|
---|
487 | Scalar q = RealScalar(2)*m01*mat(2,1) + m02*(mat(2,2) - mat(1,1));
|
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488 | diag[1] = mat(1,1) + m02*q;
|
---|
489 | diag[2] = mat(2,2) - m02*q;
|
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490 | subdiag[0] = beta;
|
---|
491 | subdiag[1] = mat(2,1) - m01 * q;
|
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492 | if (extractQ)
|
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493 | {
|
---|
494 | mat << 1, 0, 0,
|
---|
495 | 0, m01, m02,
|
---|
496 | 0, m02, -m01;
|
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497 | }
|
---|
498 | }
|
---|
499 | }
|
---|
500 | };
|
---|
501 |
|
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502 | /** \internal
|
---|
503 | * Trivial specialization for 1x1 matrices
|
---|
504 | */
|
---|
505 | template<typename MatrixType, bool IsComplex>
|
---|
506 | struct tridiagonalization_inplace_selector<MatrixType,1,IsComplex>
|
---|
507 | {
|
---|
508 | typedef typename MatrixType::Scalar Scalar;
|
---|
509 |
|
---|
510 | template<typename DiagonalType, typename SubDiagonalType>
|
---|
511 | static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType&, bool extractQ)
|
---|
512 | {
|
---|
513 | diag(0,0) = numext::real(mat(0,0));
|
---|
514 | if(extractQ)
|
---|
515 | mat(0,0) = Scalar(1);
|
---|
516 | }
|
---|
517 | };
|
---|
518 |
|
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519 | /** \internal
|
---|
520 | * \eigenvalues_module \ingroup Eigenvalues_Module
|
---|
521 | *
|
---|
522 | * \brief Expression type for return value of Tridiagonalization::matrixT()
|
---|
523 | *
|
---|
524 | * \tparam MatrixType type of underlying dense matrix
|
---|
525 | */
|
---|
526 | template<typename MatrixType> struct TridiagonalizationMatrixTReturnType
|
---|
527 | : public ReturnByValue<TridiagonalizationMatrixTReturnType<MatrixType> >
|
---|
528 | {
|
---|
529 | typedef typename MatrixType::Index Index;
|
---|
530 | public:
|
---|
531 | /** \brief Constructor.
|
---|
532 | *
|
---|
533 | * \param[in] mat The underlying dense matrix
|
---|
534 | */
|
---|
535 | TridiagonalizationMatrixTReturnType(const MatrixType& mat) : m_matrix(mat) { }
|
---|
536 |
|
---|
537 | template <typename ResultType>
|
---|
538 | inline void evalTo(ResultType& result) const
|
---|
539 | {
|
---|
540 | result.setZero();
|
---|
541 | result.template diagonal<1>() = m_matrix.template diagonal<-1>().conjugate();
|
---|
542 | result.diagonal() = m_matrix.diagonal();
|
---|
543 | result.template diagonal<-1>() = m_matrix.template diagonal<-1>();
|
---|
544 | }
|
---|
545 |
|
---|
546 | Index rows() const { return m_matrix.rows(); }
|
---|
547 | Index cols() const { return m_matrix.cols(); }
|
---|
548 |
|
---|
549 | protected:
|
---|
550 | typename MatrixType::Nested m_matrix;
|
---|
551 | };
|
---|
552 |
|
---|
553 | } // end namespace internal
|
---|
554 |
|
---|
555 | } // end namespace Eigen
|
---|
556 |
|
---|
557 | #endif // EIGEN_TRIDIAGONALIZATION_H
|
---|