[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) 2008-2011 Gael Guennebaud <gael.guennebaud@inria.fr>
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| 5 | // Copyright (C) 2009 Keir Mierle <mierle@gmail.com>
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| 6 | // Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
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| 7 | // Copyright (C) 2011 Timothy E. Holy <tim.holy@gmail.com >
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| 8 | //
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| 9 | // This Source Code Form is subject to the terms of the Mozilla
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| 10 | // Public License v. 2.0. If a copy of the MPL was not distributed
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| 11 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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| 12 |
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| 13 | #ifndef EIGEN_LDLT_H
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| 14 | #define EIGEN_LDLT_H
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| 15 |
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| 16 | namespace Eigen {
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| 17 |
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| 18 | namespace internal {
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| 19 | template<typename MatrixType, int UpLo> struct LDLT_Traits;
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| 20 |
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| 21 | // PositiveSemiDef means positive semi-definite and non-zero; same for NegativeSemiDef
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| 22 | enum SignMatrix { PositiveSemiDef, NegativeSemiDef, ZeroSign, Indefinite };
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| 23 | }
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| 24 |
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| 25 | /** \ingroup Cholesky_Module
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| 26 | *
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| 27 | * \class LDLT
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| 28 | *
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| 29 | * \brief Robust Cholesky decomposition of a matrix with pivoting
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| 30 | *
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| 31 | * \param MatrixType the type of the matrix of which to compute the LDL^T Cholesky decomposition
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| 32 | * \param UpLo the triangular part that will be used for the decompositon: Lower (default) or Upper.
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| 33 | * The other triangular part won't be read.
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| 34 | *
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| 35 | * Perform a robust Cholesky decomposition of a positive semidefinite or negative semidefinite
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| 36 | * matrix \f$ A \f$ such that \f$ A = P^TLDL^*P \f$, where P is a permutation matrix, L
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| 37 | * is lower triangular with a unit diagonal and D is a diagonal matrix.
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| 38 | *
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| 39 | * The decomposition uses pivoting to ensure stability, so that L will have
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| 40 | * zeros in the bottom right rank(A) - n submatrix. Avoiding the square root
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| 41 | * on D also stabilizes the computation.
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| 42 | *
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| 43 | * Remember that Cholesky decompositions are not rank-revealing. Also, do not use a Cholesky
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| 44 | * decomposition to determine whether a system of equations has a solution.
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| 45 | *
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| 46 | * \sa MatrixBase::ldlt(), class LLT
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| 47 | */
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| 48 | template<typename _MatrixType, int _UpLo> class LDLT
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| 49 | {
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| 50 | public:
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| 51 | typedef _MatrixType MatrixType;
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| 52 | enum {
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| 53 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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| 54 | ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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| 55 | Options = MatrixType::Options & ~RowMajorBit, // these are the options for the TmpMatrixType, we need a ColMajor matrix here!
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| 56 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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| 57 | MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
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| 58 | UpLo = _UpLo
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| 59 | };
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| 60 | typedef typename MatrixType::Scalar Scalar;
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| 61 | typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
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| 62 | typedef typename MatrixType::Index Index;
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| 63 | typedef Matrix<Scalar, RowsAtCompileTime, 1, Options, MaxRowsAtCompileTime, 1> TmpMatrixType;
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| 64 |
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| 65 | typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> TranspositionType;
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| 66 | typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationType;
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| 67 |
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| 68 | typedef internal::LDLT_Traits<MatrixType,UpLo> Traits;
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| 69 |
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| 70 | /** \brief Default Constructor.
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| 71 | *
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| 72 | * The default constructor is useful in cases in which the user intends to
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| 73 | * perform decompositions via LDLT::compute(const MatrixType&).
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| 74 | */
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| 75 | LDLT()
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| 76 | : m_matrix(),
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| 77 | m_transpositions(),
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| 78 | m_sign(internal::ZeroSign),
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| 79 | m_isInitialized(false)
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| 80 | {}
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| 81 |
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| 82 | /** \brief Default Constructor with memory preallocation
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| 83 | *
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| 84 | * Like the default constructor but with preallocation of the internal data
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| 85 | * according to the specified problem \a size.
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| 86 | * \sa LDLT()
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| 87 | */
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| 88 | LDLT(Index size)
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| 89 | : m_matrix(size, size),
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| 90 | m_transpositions(size),
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| 91 | m_temporary(size),
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| 92 | m_sign(internal::ZeroSign),
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| 93 | m_isInitialized(false)
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| 94 | {}
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| 95 |
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| 96 | /** \brief Constructor with decomposition
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| 97 | *
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| 98 | * This calculates the decomposition for the input \a matrix.
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| 99 | * \sa LDLT(Index size)
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| 100 | */
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| 101 | LDLT(const MatrixType& matrix)
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| 102 | : m_matrix(matrix.rows(), matrix.cols()),
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| 103 | m_transpositions(matrix.rows()),
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| 104 | m_temporary(matrix.rows()),
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| 105 | m_sign(internal::ZeroSign),
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| 106 | m_isInitialized(false)
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| 107 | {
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| 108 | compute(matrix);
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| 109 | }
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| 110 |
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| 111 | /** Clear any existing decomposition
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| 112 | * \sa rankUpdate(w,sigma)
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| 113 | */
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| 114 | void setZero()
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| 115 | {
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| 116 | m_isInitialized = false;
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| 117 | }
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| 118 |
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| 119 | /** \returns a view of the upper triangular matrix U */
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| 120 | inline typename Traits::MatrixU matrixU() const
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| 121 | {
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| 122 | eigen_assert(m_isInitialized && "LDLT is not initialized.");
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| 123 | return Traits::getU(m_matrix);
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| 124 | }
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| 125 |
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| 126 | /** \returns a view of the lower triangular matrix L */
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| 127 | inline typename Traits::MatrixL matrixL() const
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| 128 | {
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| 129 | eigen_assert(m_isInitialized && "LDLT is not initialized.");
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| 130 | return Traits::getL(m_matrix);
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| 131 | }
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| 132 |
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| 133 | /** \returns the permutation matrix P as a transposition sequence.
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| 134 | */
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| 135 | inline const TranspositionType& transpositionsP() const
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| 136 | {
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| 137 | eigen_assert(m_isInitialized && "LDLT is not initialized.");
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| 138 | return m_transpositions;
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| 139 | }
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| 140 |
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| 141 | /** \returns the coefficients of the diagonal matrix D */
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| 142 | inline Diagonal<const MatrixType> vectorD() const
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| 143 | {
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| 144 | eigen_assert(m_isInitialized && "LDLT is not initialized.");
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| 145 | return m_matrix.diagonal();
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| 146 | }
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| 147 |
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| 148 | /** \returns true if the matrix is positive (semidefinite) */
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| 149 | inline bool isPositive() const
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| 150 | {
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| 151 | eigen_assert(m_isInitialized && "LDLT is not initialized.");
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| 152 | return m_sign == internal::PositiveSemiDef || m_sign == internal::ZeroSign;
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| 153 | }
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| 154 |
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| 155 | #ifdef EIGEN2_SUPPORT
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| 156 | inline bool isPositiveDefinite() const
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| 157 | {
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| 158 | return isPositive();
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| 159 | }
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| 160 | #endif
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| 161 |
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| 162 | /** \returns true if the matrix is negative (semidefinite) */
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| 163 | inline bool isNegative(void) const
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| 164 | {
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| 165 | eigen_assert(m_isInitialized && "LDLT is not initialized.");
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| 166 | return m_sign == internal::NegativeSemiDef || m_sign == internal::ZeroSign;
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| 167 | }
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| 168 |
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| 169 | /** \returns a solution x of \f$ A x = b \f$ using the current decomposition of A.
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| 170 | *
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| 171 | * This function also supports in-place solves using the syntax <tt>x = decompositionObject.solve(x)</tt> .
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| 172 | *
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| 173 | * \note_about_checking_solutions
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| 174 | *
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| 175 | * More precisely, this method solves \f$ A x = b \f$ using the decomposition \f$ A = P^T L D L^* P \f$
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| 176 | * by solving the systems \f$ P^T y_1 = b \f$, \f$ L y_2 = y_1 \f$, \f$ D y_3 = y_2 \f$,
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| 177 | * \f$ L^* y_4 = y_3 \f$ and \f$ P x = y_4 \f$ in succession. If the matrix \f$ A \f$ is singular, then
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| 178 | * \f$ D \f$ will also be singular (all the other matrices are invertible). In that case, the
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| 179 | * least-square solution of \f$ D y_3 = y_2 \f$ is computed. This does not mean that this function
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| 180 | * computes the least-square solution of \f$ A x = b \f$ is \f$ A \f$ is singular.
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| 181 | *
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| 182 | * \sa MatrixBase::ldlt()
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| 183 | */
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| 184 | template<typename Rhs>
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| 185 | inline const internal::solve_retval<LDLT, Rhs>
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| 186 | solve(const MatrixBase<Rhs>& b) const
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| 187 | {
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| 188 | eigen_assert(m_isInitialized && "LDLT is not initialized.");
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| 189 | eigen_assert(m_matrix.rows()==b.rows()
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| 190 | && "LDLT::solve(): invalid number of rows of the right hand side matrix b");
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| 191 | return internal::solve_retval<LDLT, Rhs>(*this, b.derived());
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| 192 | }
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| 193 |
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| 194 | #ifdef EIGEN2_SUPPORT
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| 195 | template<typename OtherDerived, typename ResultType>
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| 196 | bool solve(const MatrixBase<OtherDerived>& b, ResultType *result) const
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| 197 | {
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| 198 | *result = this->solve(b);
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| 199 | return true;
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| 200 | }
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| 201 | #endif
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| 202 |
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| 203 | template<typename Derived>
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| 204 | bool solveInPlace(MatrixBase<Derived> &bAndX) const;
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| 205 |
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| 206 | LDLT& compute(const MatrixType& matrix);
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| 207 |
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| 208 | template <typename Derived>
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| 209 | LDLT& rankUpdate(const MatrixBase<Derived>& w, const RealScalar& alpha=1);
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| 210 |
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| 211 | /** \returns the internal LDLT decomposition matrix
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| 212 | *
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| 213 | * TODO: document the storage layout
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| 214 | */
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| 215 | inline const MatrixType& matrixLDLT() const
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| 216 | {
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| 217 | eigen_assert(m_isInitialized && "LDLT is not initialized.");
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| 218 | return m_matrix;
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| 219 | }
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| 220 |
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| 221 | MatrixType reconstructedMatrix() const;
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| 222 |
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| 223 | inline Index rows() const { return m_matrix.rows(); }
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| 224 | inline Index cols() const { return m_matrix.cols(); }
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| 225 |
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| 226 | /** \brief Reports whether previous computation was successful.
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| 227 | *
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| 228 | * \returns \c Success if computation was succesful,
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| 229 | * \c NumericalIssue if the matrix.appears to be negative.
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| 230 | */
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| 231 | ComputationInfo info() const
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| 232 | {
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| 233 | eigen_assert(m_isInitialized && "LDLT is not initialized.");
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| 234 | return Success;
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| 235 | }
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| 236 |
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| 237 | protected:
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| 238 |
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| 239 | static void check_template_parameters()
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| 240 | {
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| 241 | EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
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| 242 | }
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| 243 |
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| 244 | /** \internal
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| 245 | * Used to compute and store the Cholesky decomposition A = L D L^* = U^* D U.
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| 246 | * The strict upper part is used during the decomposition, the strict lower
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| 247 | * part correspond to the coefficients of L (its diagonal is equal to 1 and
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| 248 | * is not stored), and the diagonal entries correspond to D.
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| 249 | */
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| 250 | MatrixType m_matrix;
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| 251 | TranspositionType m_transpositions;
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| 252 | TmpMatrixType m_temporary;
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| 253 | internal::SignMatrix m_sign;
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| 254 | bool m_isInitialized;
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| 255 | };
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| 256 |
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| 257 | namespace internal {
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| 258 |
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| 259 | template<int UpLo> struct ldlt_inplace;
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| 260 |
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| 261 | template<> struct ldlt_inplace<Lower>
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| 262 | {
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| 263 | template<typename MatrixType, typename TranspositionType, typename Workspace>
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| 264 | static bool unblocked(MatrixType& mat, TranspositionType& transpositions, Workspace& temp, SignMatrix& sign)
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| 265 | {
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| 266 | using std::abs;
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| 267 | typedef typename MatrixType::Scalar Scalar;
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| 268 | typedef typename MatrixType::RealScalar RealScalar;
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| 269 | typedef typename MatrixType::Index Index;
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| 270 | eigen_assert(mat.rows()==mat.cols());
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| 271 | const Index size = mat.rows();
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| 272 |
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| 273 | if (size <= 1)
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| 274 | {
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| 275 | transpositions.setIdentity();
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| 276 | if (numext::real(mat.coeff(0,0)) > 0) sign = PositiveSemiDef;
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| 277 | else if (numext::real(mat.coeff(0,0)) < 0) sign = NegativeSemiDef;
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| 278 | else sign = ZeroSign;
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| 279 | return true;
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| 280 | }
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| 281 |
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| 282 | for (Index k = 0; k < size; ++k)
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| 283 | {
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| 284 | // Find largest diagonal element
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| 285 | Index index_of_biggest_in_corner;
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| 286 | mat.diagonal().tail(size-k).cwiseAbs().maxCoeff(&index_of_biggest_in_corner);
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| 287 | index_of_biggest_in_corner += k;
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| 288 |
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| 289 | transpositions.coeffRef(k) = index_of_biggest_in_corner;
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| 290 | if(k != index_of_biggest_in_corner)
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| 291 | {
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| 292 | // apply the transposition while taking care to consider only
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| 293 | // the lower triangular part
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| 294 | Index s = size-index_of_biggest_in_corner-1; // trailing size after the biggest element
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| 295 | mat.row(k).head(k).swap(mat.row(index_of_biggest_in_corner).head(k));
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| 296 | mat.col(k).tail(s).swap(mat.col(index_of_biggest_in_corner).tail(s));
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| 297 | std::swap(mat.coeffRef(k,k),mat.coeffRef(index_of_biggest_in_corner,index_of_biggest_in_corner));
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| 298 | for(int i=k+1;i<index_of_biggest_in_corner;++i)
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| 299 | {
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| 300 | Scalar tmp = mat.coeffRef(i,k);
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| 301 | mat.coeffRef(i,k) = numext::conj(mat.coeffRef(index_of_biggest_in_corner,i));
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| 302 | mat.coeffRef(index_of_biggest_in_corner,i) = numext::conj(tmp);
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| 303 | }
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| 304 | if(NumTraits<Scalar>::IsComplex)
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| 305 | mat.coeffRef(index_of_biggest_in_corner,k) = numext::conj(mat.coeff(index_of_biggest_in_corner,k));
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| 306 | }
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| 307 |
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| 308 | // partition the matrix:
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| 309 | // A00 | - | -
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| 310 | // lu = A10 | A11 | -
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| 311 | // A20 | A21 | A22
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| 312 | Index rs = size - k - 1;
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| 313 | Block<MatrixType,Dynamic,1> A21(mat,k+1,k,rs,1);
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| 314 | Block<MatrixType,1,Dynamic> A10(mat,k,0,1,k);
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| 315 | Block<MatrixType,Dynamic,Dynamic> A20(mat,k+1,0,rs,k);
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| 316 |
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| 317 | if(k>0)
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| 318 | {
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| 319 | temp.head(k) = mat.diagonal().real().head(k).asDiagonal() * A10.adjoint();
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| 320 | mat.coeffRef(k,k) -= (A10 * temp.head(k)).value();
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| 321 | if(rs>0)
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| 322 | A21.noalias() -= A20 * temp.head(k);
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| 323 | }
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| 324 |
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| 325 | // In some previous versions of Eigen (e.g., 3.2.1), the scaling was omitted if the pivot
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| 326 | // was smaller than the cutoff value. However, soince LDLT is not rank-revealing
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| 327 | // we should only make sure we do not introduce INF or NaN values.
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| 328 | // LAPACK also uses 0 as the cutoff value.
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| 329 | RealScalar realAkk = numext::real(mat.coeffRef(k,k));
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| 330 | if((rs>0) && (abs(realAkk) > RealScalar(0)))
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| 331 | A21 /= realAkk;
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| 332 |
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| 333 | if (sign == PositiveSemiDef) {
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| 334 | if (realAkk < 0) sign = Indefinite;
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| 335 | } else if (sign == NegativeSemiDef) {
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| 336 | if (realAkk > 0) sign = Indefinite;
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| 337 | } else if (sign == ZeroSign) {
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| 338 | if (realAkk > 0) sign = PositiveSemiDef;
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| 339 | else if (realAkk < 0) sign = NegativeSemiDef;
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| 340 | }
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| 341 | }
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| 342 |
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| 343 | return true;
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| 344 | }
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| 345 |
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| 346 | // Reference for the algorithm: Davis and Hager, "Multiple Rank
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| 347 | // Modifications of a Sparse Cholesky Factorization" (Algorithm 1)
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| 348 | // Trivial rearrangements of their computations (Timothy E. Holy)
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| 349 | // allow their algorithm to work for rank-1 updates even if the
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| 350 | // original matrix is not of full rank.
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| 351 | // Here only rank-1 updates are implemented, to reduce the
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| 352 | // requirement for intermediate storage and improve accuracy
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| 353 | template<typename MatrixType, typename WDerived>
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| 354 | static bool updateInPlace(MatrixType& mat, MatrixBase<WDerived>& w, const typename MatrixType::RealScalar& sigma=1)
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| 355 | {
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| 356 | using numext::isfinite;
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| 357 | typedef typename MatrixType::Scalar Scalar;
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| 358 | typedef typename MatrixType::RealScalar RealScalar;
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| 359 | typedef typename MatrixType::Index Index;
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| 360 |
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| 361 | const Index size = mat.rows();
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| 362 | eigen_assert(mat.cols() == size && w.size()==size);
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| 363 |
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| 364 | RealScalar alpha = 1;
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| 365 |
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| 366 | // Apply the update
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| 367 | for (Index j = 0; j < size; j++)
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| 368 | {
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| 369 | // Check for termination due to an original decomposition of low-rank
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| 370 | if (!(isfinite)(alpha))
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| 371 | break;
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| 372 |
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| 373 | // Update the diagonal terms
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| 374 | RealScalar dj = numext::real(mat.coeff(j,j));
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| 375 | Scalar wj = w.coeff(j);
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| 376 | RealScalar swj2 = sigma*numext::abs2(wj);
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| 377 | RealScalar gamma = dj*alpha + swj2;
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| 378 |
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| 379 | mat.coeffRef(j,j) += swj2/alpha;
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| 380 | alpha += swj2/dj;
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| 381 |
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| 382 |
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| 383 | // Update the terms of L
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| 384 | Index rs = size-j-1;
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| 385 | w.tail(rs) -= wj * mat.col(j).tail(rs);
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| 386 | if(gamma != 0)
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| 387 | mat.col(j).tail(rs) += (sigma*numext::conj(wj)/gamma)*w.tail(rs);
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| 388 | }
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| 389 | return true;
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| 390 | }
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| 391 |
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| 392 | template<typename MatrixType, typename TranspositionType, typename Workspace, typename WType>
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| 393 | static bool update(MatrixType& mat, const TranspositionType& transpositions, Workspace& tmp, const WType& w, const typename MatrixType::RealScalar& sigma=1)
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| 394 | {
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| 395 | // Apply the permutation to the input w
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| 396 | tmp = transpositions * w;
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| 397 |
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| 398 | return ldlt_inplace<Lower>::updateInPlace(mat,tmp,sigma);
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| 399 | }
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| 400 | };
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| 401 |
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| 402 | template<> struct ldlt_inplace<Upper>
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| 403 | {
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| 404 | template<typename MatrixType, typename TranspositionType, typename Workspace>
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| 405 | static EIGEN_STRONG_INLINE bool unblocked(MatrixType& mat, TranspositionType& transpositions, Workspace& temp, SignMatrix& sign)
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| 406 | {
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| 407 | Transpose<MatrixType> matt(mat);
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| 408 | return ldlt_inplace<Lower>::unblocked(matt, transpositions, temp, sign);
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| 409 | }
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| 410 |
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| 411 | template<typename MatrixType, typename TranspositionType, typename Workspace, typename WType>
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| 412 | static EIGEN_STRONG_INLINE bool update(MatrixType& mat, TranspositionType& transpositions, Workspace& tmp, WType& w, const typename MatrixType::RealScalar& sigma=1)
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| 413 | {
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| 414 | Transpose<MatrixType> matt(mat);
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| 415 | return ldlt_inplace<Lower>::update(matt, transpositions, tmp, w.conjugate(), sigma);
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| 416 | }
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| 417 | };
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| 418 |
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| 419 | template<typename MatrixType> struct LDLT_Traits<MatrixType,Lower>
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| 420 | {
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| 421 | typedef const TriangularView<const MatrixType, UnitLower> MatrixL;
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| 422 | typedef const TriangularView<const typename MatrixType::AdjointReturnType, UnitUpper> MatrixU;
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| 423 | static inline MatrixL getL(const MatrixType& m) { return m; }
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| 424 | static inline MatrixU getU(const MatrixType& m) { return m.adjoint(); }
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| 425 | };
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| 426 |
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| 427 | template<typename MatrixType> struct LDLT_Traits<MatrixType,Upper>
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| 428 | {
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| 429 | typedef const TriangularView<const typename MatrixType::AdjointReturnType, UnitLower> MatrixL;
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| 430 | typedef const TriangularView<const MatrixType, UnitUpper> MatrixU;
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| 431 | static inline MatrixL getL(const MatrixType& m) { return m.adjoint(); }
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| 432 | static inline MatrixU getU(const MatrixType& m) { return m; }
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| 433 | };
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| 434 |
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| 435 | } // end namespace internal
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| 436 |
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| 437 | /** Compute / recompute the LDLT decomposition A = L D L^* = U^* D U of \a matrix
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| 438 | */
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| 439 | template<typename MatrixType, int _UpLo>
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| 440 | LDLT<MatrixType,_UpLo>& LDLT<MatrixType,_UpLo>::compute(const MatrixType& a)
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| 441 | {
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| 442 | check_template_parameters();
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| 443 |
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| 444 | eigen_assert(a.rows()==a.cols());
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| 445 | const Index size = a.rows();
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| 446 |
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| 447 | m_matrix = a;
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| 448 |
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| 449 | m_transpositions.resize(size);
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| 450 | m_isInitialized = false;
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| 451 | m_temporary.resize(size);
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| 452 | m_sign = internal::ZeroSign;
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| 453 |
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| 454 | internal::ldlt_inplace<UpLo>::unblocked(m_matrix, m_transpositions, m_temporary, m_sign);
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| 455 |
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| 456 | m_isInitialized = true;
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| 457 | return *this;
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| 458 | }
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| 459 |
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| 460 | /** Update the LDLT decomposition: given A = L D L^T, efficiently compute the decomposition of A + sigma w w^T.
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| 461 | * \param w a vector to be incorporated into the decomposition.
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| 462 | * \param sigma a scalar, +1 for updates and -1 for "downdates," which correspond to removing previously-added column vectors. Optional; default value is +1.
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| 463 | * \sa setZero()
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| 464 | */
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| 465 | template<typename MatrixType, int _UpLo>
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| 466 | template<typename Derived>
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| 467 | LDLT<MatrixType,_UpLo>& LDLT<MatrixType,_UpLo>::rankUpdate(const MatrixBase<Derived>& w, const typename LDLT<MatrixType,_UpLo>::RealScalar& sigma)
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| 468 | {
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| 469 | const Index size = w.rows();
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| 470 | if (m_isInitialized)
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| 471 | {
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| 472 | eigen_assert(m_matrix.rows()==size);
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| 473 | }
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| 474 | else
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| 475 | {
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| 476 | m_matrix.resize(size,size);
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| 477 | m_matrix.setZero();
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| 478 | m_transpositions.resize(size);
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| 479 | for (Index i = 0; i < size; i++)
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| 480 | m_transpositions.coeffRef(i) = i;
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| 481 | m_temporary.resize(size);
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| 482 | m_sign = sigma>=0 ? internal::PositiveSemiDef : internal::NegativeSemiDef;
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| 483 | m_isInitialized = true;
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| 484 | }
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| 485 |
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| 486 | internal::ldlt_inplace<UpLo>::update(m_matrix, m_transpositions, m_temporary, w, sigma);
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| 487 |
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| 488 | return *this;
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| 489 | }
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| 490 |
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| 491 | namespace internal {
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| 492 | template<typename _MatrixType, int _UpLo, typename Rhs>
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| 493 | struct solve_retval<LDLT<_MatrixType,_UpLo>, Rhs>
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| 494 | : solve_retval_base<LDLT<_MatrixType,_UpLo>, Rhs>
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| 495 | {
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| 496 | typedef LDLT<_MatrixType,_UpLo> LDLTType;
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| 497 | EIGEN_MAKE_SOLVE_HELPERS(LDLTType,Rhs)
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| 498 |
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| 499 | template<typename Dest> void evalTo(Dest& dst) const
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| 500 | {
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| 501 | eigen_assert(rhs().rows() == dec().matrixLDLT().rows());
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| 502 | // dst = P b
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| 503 | dst = dec().transpositionsP() * rhs();
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| 504 |
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| 505 | // dst = L^-1 (P b)
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| 506 | dec().matrixL().solveInPlace(dst);
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| 507 |
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| 508 | // dst = D^-1 (L^-1 P b)
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| 509 | // more precisely, use pseudo-inverse of D (see bug 241)
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| 510 | using std::abs;
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| 511 | using std::max;
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| 512 | typedef typename LDLTType::MatrixType MatrixType;
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| 513 | typedef typename LDLTType::RealScalar RealScalar;
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| 514 | const typename Diagonal<const MatrixType>::RealReturnType vectorD(dec().vectorD());
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| 515 | // In some previous versions, tolerance was set to the max of 1/highest and the maximal diagonal entry * epsilon
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| 516 | // as motivated by LAPACK's xGELSS:
|
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| 517 | // RealScalar tolerance = (max)(vectorD.array().abs().maxCoeff() *NumTraits<RealScalar>::epsilon(),RealScalar(1) / NumTraits<RealScalar>::highest());
|
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| 518 | // However, LDLT is not rank revealing, and so adjusting the tolerance wrt to the highest
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| 519 | // diagonal element is not well justified and to numerical issues in some cases.
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| 520 | // Moreover, Lapack's xSYTRS routines use 0 for the tolerance.
|
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| 521 | RealScalar tolerance = RealScalar(1) / NumTraits<RealScalar>::highest();
|
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| 522 |
|
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| 523 | for (Index i = 0; i < vectorD.size(); ++i) {
|
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| 524 | if(abs(vectorD(i)) > tolerance)
|
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| 525 | dst.row(i) /= vectorD(i);
|
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| 526 | else
|
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| 527 | dst.row(i).setZero();
|
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| 528 | }
|
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| 529 |
|
---|
| 530 | // dst = L^-T (D^-1 L^-1 P b)
|
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| 531 | dec().matrixU().solveInPlace(dst);
|
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| 532 |
|
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| 533 | // dst = P^-1 (L^-T D^-1 L^-1 P b) = A^-1 b
|
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| 534 | dst = dec().transpositionsP().transpose() * dst;
|
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| 535 | }
|
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| 536 | };
|
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| 537 | }
|
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| 538 |
|
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| 539 | /** \internal use x = ldlt_object.solve(x);
|
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| 540 | *
|
---|
| 541 | * This is the \em in-place version of solve().
|
---|
| 542 | *
|
---|
| 543 | * \param bAndX represents both the right-hand side matrix b and result x.
|
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| 544 | *
|
---|
| 545 | * \returns true always! If you need to check for existence of solutions, use another decomposition like LU, QR, or SVD.
|
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| 546 | *
|
---|
| 547 | * This version avoids a copy when the right hand side matrix b is not
|
---|
| 548 | * needed anymore.
|
---|
| 549 | *
|
---|
| 550 | * \sa LDLT::solve(), MatrixBase::ldlt()
|
---|
| 551 | */
|
---|
| 552 | template<typename MatrixType,int _UpLo>
|
---|
| 553 | template<typename Derived>
|
---|
| 554 | bool LDLT<MatrixType,_UpLo>::solveInPlace(MatrixBase<Derived> &bAndX) const
|
---|
| 555 | {
|
---|
| 556 | eigen_assert(m_isInitialized && "LDLT is not initialized.");
|
---|
| 557 | eigen_assert(m_matrix.rows() == bAndX.rows());
|
---|
| 558 |
|
---|
| 559 | bAndX = this->solve(bAndX);
|
---|
| 560 |
|
---|
| 561 | return true;
|
---|
| 562 | }
|
---|
| 563 |
|
---|
| 564 | /** \returns the matrix represented by the decomposition,
|
---|
| 565 | * i.e., it returns the product: P^T L D L^* P.
|
---|
| 566 | * This function is provided for debug purpose. */
|
---|
| 567 | template<typename MatrixType, int _UpLo>
|
---|
| 568 | MatrixType LDLT<MatrixType,_UpLo>::reconstructedMatrix() const
|
---|
| 569 | {
|
---|
| 570 | eigen_assert(m_isInitialized && "LDLT is not initialized.");
|
---|
| 571 | const Index size = m_matrix.rows();
|
---|
| 572 | MatrixType res(size,size);
|
---|
| 573 |
|
---|
| 574 | // P
|
---|
| 575 | res.setIdentity();
|
---|
| 576 | res = transpositionsP() * res;
|
---|
| 577 | // L^* P
|
---|
| 578 | res = matrixU() * res;
|
---|
| 579 | // D(L^*P)
|
---|
| 580 | res = vectorD().real().asDiagonal() * res;
|
---|
| 581 | // L(DL^*P)
|
---|
| 582 | res = matrixL() * res;
|
---|
| 583 | // P^T (LDL^*P)
|
---|
| 584 | res = transpositionsP().transpose() * res;
|
---|
| 585 |
|
---|
| 586 | return res;
|
---|
| 587 | }
|
---|
| 588 |
|
---|
| 589 | /** \cholesky_module
|
---|
| 590 | * \returns the Cholesky decomposition with full pivoting without square root of \c *this
|
---|
| 591 | */
|
---|
| 592 | template<typename MatrixType, unsigned int UpLo>
|
---|
| 593 | inline const LDLT<typename SelfAdjointView<MatrixType, UpLo>::PlainObject, UpLo>
|
---|
| 594 | SelfAdjointView<MatrixType, UpLo>::ldlt() const
|
---|
| 595 | {
|
---|
| 596 | return LDLT<PlainObject,UpLo>(m_matrix);
|
---|
| 597 | }
|
---|
| 598 |
|
---|
| 599 | /** \cholesky_module
|
---|
| 600 | * \returns the Cholesky decomposition with full pivoting without square root of \c *this
|
---|
| 601 | */
|
---|
| 602 | template<typename Derived>
|
---|
| 603 | inline const LDLT<typename MatrixBase<Derived>::PlainObject>
|
---|
| 604 | MatrixBase<Derived>::ldlt() const
|
---|
| 605 | {
|
---|
| 606 | return LDLT<PlainObject>(derived());
|
---|
| 607 | }
|
---|
| 608 |
|
---|
| 609 | } // end namespace Eigen
|
---|
| 610 |
|
---|
| 611 | #endif // EIGEN_LDLT_H
|
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