[136] | 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);
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| 475 | diag[2] = mat(2,2);
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| 476 | subdiag[0] = mat(1,0);
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| 477 | subdiag[1] = mat(2,1);
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| 478 | if (extractQ)
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| 479 | mat.setIdentity();
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| 480 | }
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| 481 | else
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| 482 | {
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| 483 | RealScalar beta = sqrt(numext::abs2(mat(1,0)) + v1norm2);
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| 484 | RealScalar invBeta = RealScalar(1)/beta;
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| 485 | Scalar m01 = mat(1,0) * invBeta;
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| 486 | Scalar m02 = mat(2,0) * invBeta;
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| 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;
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| 489 | diag[2] = mat(2,2) - m02*q;
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| 490 | subdiag[0] = beta;
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| 491 | subdiag[1] = mat(2,1) - m01 * q;
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| 492 | if (extractQ)
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| 493 | {
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| 494 | mat << 1, 0, 0,
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| 495 | 0, m01, m02,
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| 496 | 0, m02, -m01;
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| 497 | }
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| 498 | }
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| 499 | }
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| 500 | };
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| 501 |
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| 502 | /** \internal
|
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| 503 | * Trivial specialization for 1x1 matrices
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| 504 | */
|
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| 505 | template<typename MatrixType, bool IsComplex>
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| 506 | struct tridiagonalization_inplace_selector<MatrixType,1,IsComplex>
|
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| 507 | {
|
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| 508 | typedef typename MatrixType::Scalar Scalar;
|
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| 509 |
|
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| 510 | template<typename DiagonalType, typename SubDiagonalType>
|
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| 511 | static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType&, bool extractQ)
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| 512 | {
|
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| 513 | diag(0,0) = numext::real(mat(0,0));
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| 514 | if(extractQ)
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| 515 | mat(0,0) = Scalar(1);
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| 516 | }
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| 517 | };
|
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| 518 |
|
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| 519 | /** \internal
|
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| 520 | * \eigenvalues_module \ingroup Eigenvalues_Module
|
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| 521 | *
|
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| 522 | * \brief Expression type for return value of Tridiagonalization::matrixT()
|
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| 523 | *
|
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| 524 | * \tparam MatrixType type of underlying dense matrix
|
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| 525 | */
|
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| 526 | template<typename MatrixType> struct TridiagonalizationMatrixTReturnType
|
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| 527 | : public ReturnByValue<TridiagonalizationMatrixTReturnType<MatrixType> >
|
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| 528 | {
|
---|
| 529 | typedef typename MatrixType::Index Index;
|
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| 530 | public:
|
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| 531 | /** \brief Constructor.
|
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| 532 | *
|
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| 533 | * \param[in] mat The underlying dense matrix
|
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| 534 | */
|
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| 535 | TridiagonalizationMatrixTReturnType(const MatrixType& mat) : m_matrix(mat) { }
|
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| 536 |
|
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| 537 | template <typename ResultType>
|
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| 538 | inline void evalTo(ResultType& result) const
|
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| 539 | {
|
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| 540 | result.setZero();
|
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| 541 | result.template diagonal<1>() = m_matrix.template diagonal<-1>().conjugate();
|
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| 542 | result.diagonal() = m_matrix.diagonal();
|
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| 543 | result.template diagonal<-1>() = m_matrix.template diagonal<-1>();
|
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| 544 | }
|
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| 545 |
|
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| 546 | Index rows() const { return m_matrix.rows(); }
|
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| 547 | Index cols() const { return m_matrix.cols(); }
|
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| 548 |
|
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| 549 | protected:
|
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| 550 | typename MatrixType::Nested m_matrix;
|
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| 551 | };
|
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| 552 |
|
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| 553 | } // end namespace internal
|
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| 554 |
|
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| 555 | } // end namespace Eigen
|
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| 556 |
|
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| 557 | #endif // EIGEN_TRIDIAGONALIZATION_H
|
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