[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 | // We used the "A Divide-And-Conquer Algorithm for the Bidiagonal SVD"
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| 5 | // research report written by Ming Gu and Stanley C.Eisenstat
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| 6 | // The code variable names correspond to the names they used in their
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| 7 | // report
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| 8 | //
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| 9 | // Copyright (C) 2013 Gauthier Brun <brun.gauthier@gmail.com>
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| 10 | // Copyright (C) 2013 Nicolas Carre <nicolas.carre@ensimag.fr>
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| 11 | // Copyright (C) 2013 Jean Ceccato <jean.ceccato@ensimag.fr>
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| 12 | // Copyright (C) 2013 Pierre Zoppitelli <pierre.zoppitelli@ensimag.fr>
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| 13 | //
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| 14 | // Source Code Form is subject to the terms of the Mozilla
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| 15 | // Public License v. 2.0. If a copy of the MPL was not distributed
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| 16 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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| 17 |
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| 18 | #ifndef EIGEN_BDCSVD_H
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| 19 | #define EIGEN_BDCSVD_H
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| 20 |
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| 21 | #define EPSILON 0.0000000000000001
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| 22 |
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| 23 | #define ALGOSWAP 32
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| 24 |
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| 25 | namespace Eigen {
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| 26 | /** \ingroup SVD_Module
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| 27 | *
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| 28 | *
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| 29 | * \class BDCSVD
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| 30 | *
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| 31 | * \brief class Bidiagonal Divide and Conquer SVD
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| 32 | *
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| 33 | * \param MatrixType the type of the matrix of which we are computing the SVD decomposition
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| 34 | * We plan to have a very similar interface to JacobiSVD on this class.
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| 35 | * It should be used to speed up the calcul of SVD for big matrices.
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| 36 | */
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| 37 | template<typename _MatrixType>
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| 38 | class BDCSVD : public SVDBase<_MatrixType>
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| 39 | {
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| 40 | typedef SVDBase<_MatrixType> Base;
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| 41 |
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| 42 | public:
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| 43 | using Base::rows;
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| 44 | using Base::cols;
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| 45 |
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| 46 | typedef _MatrixType MatrixType;
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| 47 | typedef typename MatrixType::Scalar Scalar;
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| 48 | typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
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| 49 | typedef typename MatrixType::Index Index;
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| 50 | enum {
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| 51 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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| 52 | ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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| 53 | DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime, ColsAtCompileTime),
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| 54 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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| 55 | MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
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| 56 | MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime, MaxColsAtCompileTime),
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| 57 | MatrixOptions = MatrixType::Options
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| 58 | };
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| 59 |
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| 60 | typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime,
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| 61 | MatrixOptions, MaxRowsAtCompileTime, MaxRowsAtCompileTime>
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| 62 | MatrixUType;
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| 63 | typedef Matrix<Scalar, ColsAtCompileTime, ColsAtCompileTime,
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| 64 | MatrixOptions, MaxColsAtCompileTime, MaxColsAtCompileTime>
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| 65 | MatrixVType;
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| 66 | typedef typename internal::plain_diag_type<MatrixType, RealScalar>::type SingularValuesType;
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| 67 | typedef typename internal::plain_row_type<MatrixType>::type RowType;
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| 68 | typedef typename internal::plain_col_type<MatrixType>::type ColType;
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| 69 | typedef Matrix<Scalar, Dynamic, Dynamic> MatrixX;
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| 70 | typedef Matrix<RealScalar, Dynamic, Dynamic> MatrixXr;
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| 71 | typedef Matrix<RealScalar, Dynamic, 1> VectorType;
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| 72 |
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| 73 | /** \brief Default Constructor.
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| 74 | *
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| 75 | * The default constructor is useful in cases in which the user intends to
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| 76 | * perform decompositions via BDCSVD::compute(const MatrixType&).
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| 77 | */
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| 78 | BDCSVD()
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| 79 | : SVDBase<_MatrixType>::SVDBase(),
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| 80 | algoswap(ALGOSWAP)
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| 81 | {}
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| 82 |
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| 83 |
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| 84 | /** \brief Default Constructor with memory preallocation
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| 85 | *
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| 86 | * Like the default constructor but with preallocation of the internal data
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| 87 | * according to the specified problem size.
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| 88 | * \sa BDCSVD()
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| 89 | */
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| 90 | BDCSVD(Index rows, Index cols, unsigned int computationOptions = 0)
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| 91 | : SVDBase<_MatrixType>::SVDBase(),
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| 92 | algoswap(ALGOSWAP)
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| 93 | {
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| 94 | allocate(rows, cols, computationOptions);
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| 95 | }
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| 96 |
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| 97 | /** \brief Constructor performing the decomposition of given matrix.
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| 98 | *
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| 99 | * \param matrix the matrix to decompose
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| 100 | * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
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| 101 | * By default, none is computed. This is a bit - field, the possible bits are #ComputeFullU, #ComputeThinU,
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| 102 | * #ComputeFullV, #ComputeThinV.
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| 103 | *
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| 104 | * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
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| 105 | * available with the (non - default) FullPivHouseholderQR preconditioner.
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| 106 | */
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| 107 | BDCSVD(const MatrixType& matrix, unsigned int computationOptions = 0)
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| 108 | : SVDBase<_MatrixType>::SVDBase(),
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| 109 | algoswap(ALGOSWAP)
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| 110 | {
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| 111 | compute(matrix, computationOptions);
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| 112 | }
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| 113 |
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| 114 | ~BDCSVD()
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| 115 | {
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| 116 | }
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| 117 | /** \brief Method performing the decomposition of given matrix using custom options.
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| 118 | *
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| 119 | * \param matrix the matrix to decompose
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| 120 | * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
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| 121 | * By default, none is computed. This is a bit - field, the possible bits are #ComputeFullU, #ComputeThinU,
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| 122 | * #ComputeFullV, #ComputeThinV.
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| 123 | *
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| 124 | * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
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| 125 | * available with the (non - default) FullPivHouseholderQR preconditioner.
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| 126 | */
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| 127 | SVDBase<MatrixType>& compute(const MatrixType& matrix, unsigned int computationOptions);
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| 128 |
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| 129 | /** \brief Method performing the decomposition of given matrix using current options.
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| 130 | *
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| 131 | * \param matrix the matrix to decompose
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| 132 | *
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| 133 | * This method uses the current \a computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int).
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| 134 | */
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| 135 | SVDBase<MatrixType>& compute(const MatrixType& matrix)
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| 136 | {
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| 137 | return compute(matrix, this->m_computationOptions);
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| 138 | }
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| 139 |
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| 140 | void setSwitchSize(int s)
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| 141 | {
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| 142 | eigen_assert(s>3 && "BDCSVD the size of the algo switch has to be greater than 4");
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| 143 | algoswap = s;
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| 144 | }
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| 145 |
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| 146 |
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| 147 | /** \returns a (least squares) solution of \f$ A x = b \f$ using the current SVD decomposition of A.
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| 148 | *
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| 149 | * \param b the right - hand - side of the equation to solve.
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| 150 | *
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| 151 | * \note Solving requires both U and V to be computed. Thin U and V are enough, there is no need for full U or V.
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| 152 | *
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| 153 | * \note SVD solving is implicitly least - squares. Thus, this method serves both purposes of exact solving and least - squares solving.
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| 154 | * In other words, the returned solution is guaranteed to minimize the Euclidean norm \f$ \Vert A x - b \Vert \f$.
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| 155 | */
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| 156 | template<typename Rhs>
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| 157 | inline const internal::solve_retval<BDCSVD, Rhs>
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| 158 | solve(const MatrixBase<Rhs>& b) const
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| 159 | {
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| 160 | eigen_assert(this->m_isInitialized && "BDCSVD is not initialized.");
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| 161 | eigen_assert(SVDBase<_MatrixType>::computeU() && SVDBase<_MatrixType>::computeV() &&
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| 162 | "BDCSVD::solve() requires both unitaries U and V to be computed (thin unitaries suffice).");
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| 163 | return internal::solve_retval<BDCSVD, Rhs>(*this, b.derived());
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| 164 | }
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| 165 |
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| 166 |
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| 167 | const MatrixUType& matrixU() const
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| 168 | {
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| 169 | eigen_assert(this->m_isInitialized && "SVD is not initialized.");
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| 170 | if (isTranspose){
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| 171 | eigen_assert(this->computeV() && "This SVD decomposition didn't compute U. Did you ask for it?");
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| 172 | return this->m_matrixV;
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| 173 | }
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| 174 | else
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| 175 | {
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| 176 | eigen_assert(this->computeU() && "This SVD decomposition didn't compute U. Did you ask for it?");
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| 177 | return this->m_matrixU;
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| 178 | }
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| 179 |
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| 180 | }
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| 181 |
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| 182 |
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| 183 | const MatrixVType& matrixV() const
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| 184 | {
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| 185 | eigen_assert(this->m_isInitialized && "SVD is not initialized.");
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| 186 | if (isTranspose){
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| 187 | eigen_assert(this->computeU() && "This SVD decomposition didn't compute V. Did you ask for it?");
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| 188 | return this->m_matrixU;
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| 189 | }
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| 190 | else
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| 191 | {
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| 192 | eigen_assert(this->computeV() && "This SVD decomposition didn't compute V. Did you ask for it?");
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| 193 | return this->m_matrixV;
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| 194 | }
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| 195 | }
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| 196 |
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| 197 | private:
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| 198 | void allocate(Index rows, Index cols, unsigned int computationOptions);
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| 199 | void divide (Index firstCol, Index lastCol, Index firstRowW,
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| 200 | Index firstColW, Index shift);
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| 201 | void deflation43(Index firstCol, Index shift, Index i, Index size);
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| 202 | void deflation44(Index firstColu , Index firstColm, Index firstRowW, Index firstColW, Index i, Index j, Index size);
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| 203 | void deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW, Index shift);
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| 204 | void copyUV(MatrixXr naiveU, MatrixXr naiveV, MatrixX householderU, MatrixX houseHolderV);
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| 205 |
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| 206 | protected:
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| 207 | MatrixXr m_naiveU, m_naiveV;
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| 208 | MatrixXr m_computed;
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| 209 | Index nRec;
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| 210 | int algoswap;
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| 211 | bool isTranspose, compU, compV;
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| 212 |
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| 213 | }; //end class BDCSVD
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| 214 |
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| 215 |
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| 216 | // Methode to allocate ans initialize matrix and attributs
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| 217 | template<typename MatrixType>
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| 218 | void BDCSVD<MatrixType>::allocate(Index rows, Index cols, unsigned int computationOptions)
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| 219 | {
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| 220 | isTranspose = (cols > rows);
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| 221 | if (SVDBase<MatrixType>::allocate(rows, cols, computationOptions)) return;
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| 222 | m_computed = MatrixXr::Zero(this->m_diagSize + 1, this->m_diagSize );
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| 223 | if (isTranspose){
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| 224 | compU = this->computeU();
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| 225 | compV = this->computeV();
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| 226 | }
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| 227 | else
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| 228 | {
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| 229 | compV = this->computeU();
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| 230 | compU = this->computeV();
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| 231 | }
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| 232 | if (compU) m_naiveU = MatrixXr::Zero(this->m_diagSize + 1, this->m_diagSize + 1 );
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| 233 | else m_naiveU = MatrixXr::Zero(2, this->m_diagSize + 1 );
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| 234 |
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| 235 | if (compV) m_naiveV = MatrixXr::Zero(this->m_diagSize, this->m_diagSize);
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| 236 |
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| 237 |
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| 238 | //should be changed for a cleaner implementation
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| 239 | if (isTranspose){
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| 240 | bool aux;
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| 241 | if (this->computeU()||this->computeV()){
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| 242 | aux = this->m_computeFullU;
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| 243 | this->m_computeFullU = this->m_computeFullV;
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| 244 | this->m_computeFullV = aux;
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| 245 | aux = this->m_computeThinU;
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| 246 | this->m_computeThinU = this->m_computeThinV;
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| 247 | this->m_computeThinV = aux;
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| 248 | }
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| 249 | }
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| 250 | }// end allocate
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| 251 |
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| 252 | // Methode which compute the BDCSVD for the int
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| 253 | template<>
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| 254 | SVDBase<Matrix<int, Dynamic, Dynamic> >&
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| 255 | BDCSVD<Matrix<int, Dynamic, Dynamic> >::compute(const MatrixType& matrix, unsigned int computationOptions) {
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| 256 | allocate(matrix.rows(), matrix.cols(), computationOptions);
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| 257 | this->m_nonzeroSingularValues = 0;
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| 258 | m_computed = Matrix<int, Dynamic, Dynamic>::Zero(rows(), cols());
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| 259 | for (int i=0; i<this->m_diagSize; i++) {
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| 260 | this->m_singularValues.coeffRef(i) = 0;
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| 261 | }
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| 262 | if (this->m_computeFullU) this->m_matrixU = Matrix<int, Dynamic, Dynamic>::Zero(rows(), rows());
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| 263 | if (this->m_computeFullV) this->m_matrixV = Matrix<int, Dynamic, Dynamic>::Zero(cols(), cols());
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| 264 | this->m_isInitialized = true;
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| 265 | return *this;
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| 266 | }
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| 267 |
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| 268 |
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| 269 | // Methode which compute the BDCSVD
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| 270 | template<typename MatrixType>
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| 271 | SVDBase<MatrixType>&
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| 272 | BDCSVD<MatrixType>::compute(const MatrixType& matrix, unsigned int computationOptions)
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| 273 | {
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| 274 | allocate(matrix.rows(), matrix.cols(), computationOptions);
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| 275 | using std::abs;
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| 276 |
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| 277 | //**** step 1 Bidiagonalization isTranspose = (matrix.cols()>matrix.rows()) ;
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| 278 | MatrixType copy;
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| 279 | if (isTranspose) copy = matrix.adjoint();
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| 280 | else copy = matrix;
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| 281 |
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| 282 | internal::UpperBidiagonalization<MatrixX > bid(copy);
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| 283 |
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| 284 | //**** step 2 Divide
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| 285 | // this is ugly and has to be redone (care of complex cast)
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| 286 | MatrixXr temp;
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| 287 | temp = bid.bidiagonal().toDenseMatrix().transpose();
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| 288 | m_computed.setZero();
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| 289 | for (int i=0; i<this->m_diagSize - 1; i++) {
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| 290 | m_computed(i, i) = temp(i, i);
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| 291 | m_computed(i + 1, i) = temp(i + 1, i);
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| 292 | }
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| 293 | m_computed(this->m_diagSize - 1, this->m_diagSize - 1) = temp(this->m_diagSize - 1, this->m_diagSize - 1);
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| 294 | divide(0, this->m_diagSize - 1, 0, 0, 0);
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| 295 |
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| 296 | //**** step 3 copy
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| 297 | for (int i=0; i<this->m_diagSize; i++) {
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| 298 | RealScalar a = abs(m_computed.coeff(i, i));
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| 299 | this->m_singularValues.coeffRef(i) = a;
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| 300 | if (a == 0){
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| 301 | this->m_nonzeroSingularValues = i;
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| 302 | break;
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| 303 | }
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| 304 | else if (i == this->m_diagSize - 1)
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| 305 | {
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| 306 | this->m_nonzeroSingularValues = i + 1;
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| 307 | break;
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| 308 | }
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| 309 | }
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| 310 | copyUV(m_naiveV, m_naiveU, bid.householderU(), bid.householderV());
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| 311 | this->m_isInitialized = true;
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| 312 | return *this;
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| 313 | }// end compute
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| 314 |
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| 315 |
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| 316 | template<typename MatrixType>
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| 317 | void BDCSVD<MatrixType>::copyUV(MatrixXr naiveU, MatrixXr naiveV, MatrixX householderU, MatrixX householderV){
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| 318 | if (this->computeU()){
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| 319 | MatrixX temp = MatrixX::Zero(naiveU.rows(), naiveU.cols());
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| 320 | temp.real() = naiveU;
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| 321 | if (this->m_computeThinU){
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| 322 | this->m_matrixU = MatrixX::Identity(householderU.cols(), this->m_nonzeroSingularValues );
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| 323 | this->m_matrixU.block(0, 0, this->m_diagSize, this->m_nonzeroSingularValues) =
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| 324 | temp.block(0, 0, this->m_diagSize, this->m_nonzeroSingularValues);
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| 325 | this->m_matrixU = householderU * this->m_matrixU ;
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| 326 | }
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| 327 | else
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| 328 | {
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| 329 | this->m_matrixU = MatrixX::Identity(householderU.cols(), householderU.cols());
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| 330 | this->m_matrixU.block(0, 0, this->m_diagSize, this->m_diagSize) = temp.block(0, 0, this->m_diagSize, this->m_diagSize);
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| 331 | this->m_matrixU = householderU * this->m_matrixU ;
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| 332 | }
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| 333 | }
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| 334 | if (this->computeV()){
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| 335 | MatrixX temp = MatrixX::Zero(naiveV.rows(), naiveV.cols());
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| 336 | temp.real() = naiveV;
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| 337 | if (this->m_computeThinV){
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| 338 | this->m_matrixV = MatrixX::Identity(householderV.cols(),this->m_nonzeroSingularValues );
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| 339 | this->m_matrixV.block(0, 0, this->m_nonzeroSingularValues, this->m_nonzeroSingularValues) =
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| 340 | temp.block(0, 0, this->m_nonzeroSingularValues, this->m_nonzeroSingularValues);
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| 341 | this->m_matrixV = householderV * this->m_matrixV ;
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| 342 | }
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| 343 | else
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| 344 | {
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| 345 | this->m_matrixV = MatrixX::Identity(householderV.cols(), householderV.cols());
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| 346 | this->m_matrixV.block(0, 0, this->m_diagSize, this->m_diagSize) = temp.block(0, 0, this->m_diagSize, this->m_diagSize);
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| 347 | this->m_matrixV = householderV * this->m_matrixV;
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| 348 | }
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| 349 | }
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| 350 | }
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| 351 |
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| 352 | // The divide algorithm is done "in place", we are always working on subsets of the same matrix. The divide methods takes as argument the
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| 353 | // place of the submatrix we are currently working on.
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| 354 |
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| 355 | //@param firstCol : The Index of the first column of the submatrix of m_computed and for m_naiveU;
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| 356 | //@param lastCol : The Index of the last column of the submatrix of m_computed and for m_naiveU;
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| 357 | // lastCol + 1 - firstCol is the size of the submatrix.
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| 358 | //@param firstRowW : The Index of the first row of the matrix W that we are to change. (see the reference paper section 1 for more information on W)
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| 359 | //@param firstRowW : Same as firstRowW with the column.
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| 360 | //@param shift : Each time one takes the left submatrix, one must add 1 to the shift. Why? Because! We actually want the last column of the U submatrix
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| 361 | // to become the first column (*coeff) and to shift all the other columns to the right. There are more details on the reference paper.
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| 362 | template<typename MatrixType>
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| 363 | void BDCSVD<MatrixType>::divide (Index firstCol, Index lastCol, Index firstRowW,
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| 364 | Index firstColW, Index shift)
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| 365 | {
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| 366 | // requires nbRows = nbCols + 1;
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| 367 | using std::pow;
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| 368 | using std::sqrt;
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| 369 | using std::abs;
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| 370 | const Index n = lastCol - firstCol + 1;
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| 371 | const Index k = n/2;
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| 372 | RealScalar alphaK;
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| 373 | RealScalar betaK;
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| 374 | RealScalar r0;
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| 375 | RealScalar lambda, phi, c0, s0;
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| 376 | MatrixXr l, f;
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| 377 | // We use the other algorithm which is more efficient for small
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| 378 | // matrices.
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| 379 | if (n < algoswap){
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| 380 | JacobiSVD<MatrixXr> b(m_computed.block(firstCol, firstCol, n + 1, n),
|
---|
| 381 | ComputeFullU | (ComputeFullV * compV)) ;
|
---|
| 382 | if (compU) m_naiveU.block(firstCol, firstCol, n + 1, n + 1).real() << b.matrixU();
|
---|
| 383 | else
|
---|
| 384 | {
|
---|
| 385 | m_naiveU.row(0).segment(firstCol, n + 1).real() << b.matrixU().row(0);
|
---|
| 386 | m_naiveU.row(1).segment(firstCol, n + 1).real() << b.matrixU().row(n);
|
---|
| 387 | }
|
---|
| 388 | if (compV) m_naiveV.block(firstRowW, firstColW, n, n).real() << b.matrixV();
|
---|
| 389 | m_computed.block(firstCol + shift, firstCol + shift, n + 1, n).setZero();
|
---|
| 390 | for (int i=0; i<n; i++)
|
---|
| 391 | {
|
---|
| 392 | m_computed(firstCol + shift + i, firstCol + shift +i) = b.singularValues().coeffRef(i);
|
---|
| 393 | }
|
---|
| 394 | return;
|
---|
| 395 | }
|
---|
| 396 | // We use the divide and conquer algorithm
|
---|
| 397 | alphaK = m_computed(firstCol + k, firstCol + k);
|
---|
| 398 | betaK = m_computed(firstCol + k + 1, firstCol + k);
|
---|
| 399 | // The divide must be done in that order in order to have good results. Divide change the data inside the submatrices
|
---|
| 400 | // and the divide of the right submatrice reads one column of the left submatrice. That's why we need to treat the
|
---|
| 401 | // right submatrix before the left one.
|
---|
| 402 | divide(k + 1 + firstCol, lastCol, k + 1 + firstRowW, k + 1 + firstColW, shift);
|
---|
| 403 | divide(firstCol, k - 1 + firstCol, firstRowW, firstColW + 1, shift + 1);
|
---|
| 404 | if (compU)
|
---|
| 405 | {
|
---|
| 406 | lambda = m_naiveU(firstCol + k, firstCol + k);
|
---|
| 407 | phi = m_naiveU(firstCol + k + 1, lastCol + 1);
|
---|
| 408 | }
|
---|
| 409 | else
|
---|
| 410 | {
|
---|
| 411 | lambda = m_naiveU(1, firstCol + k);
|
---|
| 412 | phi = m_naiveU(0, lastCol + 1);
|
---|
| 413 | }
|
---|
| 414 | r0 = sqrt((abs(alphaK * lambda) * abs(alphaK * lambda))
|
---|
| 415 | + abs(betaK * phi) * abs(betaK * phi));
|
---|
| 416 | if (compU)
|
---|
| 417 | {
|
---|
| 418 | l = m_naiveU.row(firstCol + k).segment(firstCol, k);
|
---|
| 419 | f = m_naiveU.row(firstCol + k + 1).segment(firstCol + k + 1, n - k - 1);
|
---|
| 420 | }
|
---|
| 421 | else
|
---|
| 422 | {
|
---|
| 423 | l = m_naiveU.row(1).segment(firstCol, k);
|
---|
| 424 | f = m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1);
|
---|
| 425 | }
|
---|
| 426 | if (compV) m_naiveV(firstRowW+k, firstColW) = 1;
|
---|
| 427 | if (r0 == 0)
|
---|
| 428 | {
|
---|
| 429 | c0 = 1;
|
---|
| 430 | s0 = 0;
|
---|
| 431 | }
|
---|
| 432 | else
|
---|
| 433 | {
|
---|
| 434 | c0 = alphaK * lambda / r0;
|
---|
| 435 | s0 = betaK * phi / r0;
|
---|
| 436 | }
|
---|
| 437 | if (compU)
|
---|
| 438 | {
|
---|
| 439 | MatrixXr q1 (m_naiveU.col(firstCol + k).segment(firstCol, k + 1));
|
---|
| 440 | // we shiftW Q1 to the right
|
---|
| 441 | for (Index i = firstCol + k - 1; i >= firstCol; i--)
|
---|
| 442 | {
|
---|
| 443 | m_naiveU.col(i + 1).segment(firstCol, k + 1) << m_naiveU.col(i).segment(firstCol, k + 1);
|
---|
| 444 | }
|
---|
| 445 | // we shift q1 at the left with a factor c0
|
---|
| 446 | m_naiveU.col(firstCol).segment( firstCol, k + 1) << (q1 * c0);
|
---|
| 447 | // last column = q1 * - s0
|
---|
| 448 | m_naiveU.col(lastCol + 1).segment(firstCol, k + 1) << (q1 * ( - s0));
|
---|
| 449 | // first column = q2 * s0
|
---|
| 450 | m_naiveU.col(firstCol).segment(firstCol + k + 1, n - k) <<
|
---|
| 451 | m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) *s0;
|
---|
| 452 | // q2 *= c0
|
---|
| 453 | m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) *= c0;
|
---|
| 454 | }
|
---|
| 455 | else
|
---|
| 456 | {
|
---|
| 457 | RealScalar q1 = (m_naiveU(0, firstCol + k));
|
---|
| 458 | // we shift Q1 to the right
|
---|
| 459 | for (Index i = firstCol + k - 1; i >= firstCol; i--)
|
---|
| 460 | {
|
---|
| 461 | m_naiveU(0, i + 1) = m_naiveU(0, i);
|
---|
| 462 | }
|
---|
| 463 | // we shift q1 at the left with a factor c0
|
---|
| 464 | m_naiveU(0, firstCol) = (q1 * c0);
|
---|
| 465 | // last column = q1 * - s0
|
---|
| 466 | m_naiveU(0, lastCol + 1) = (q1 * ( - s0));
|
---|
| 467 | // first column = q2 * s0
|
---|
| 468 | m_naiveU(1, firstCol) = m_naiveU(1, lastCol + 1) *s0;
|
---|
| 469 | // q2 *= c0
|
---|
| 470 | m_naiveU(1, lastCol + 1) *= c0;
|
---|
| 471 | m_naiveU.row(1).segment(firstCol + 1, k).setZero();
|
---|
| 472 | m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1).setZero();
|
---|
| 473 | }
|
---|
| 474 | m_computed(firstCol + shift, firstCol + shift) = r0;
|
---|
| 475 | m_computed.col(firstCol + shift).segment(firstCol + shift + 1, k) << alphaK * l.transpose().real();
|
---|
| 476 | m_computed.col(firstCol + shift).segment(firstCol + shift + k + 1, n - k - 1) << betaK * f.transpose().real();
|
---|
| 477 |
|
---|
| 478 |
|
---|
| 479 | // the line below do the deflation of the matrix for the third part of the algorithm
|
---|
| 480 | // Here the deflation is commented because the third part of the algorithm is not implemented
|
---|
| 481 | // the third part of the algorithm is a fast SVD on the matrix m_computed which works thanks to the deflation
|
---|
| 482 |
|
---|
| 483 | deflation(firstCol, lastCol, k, firstRowW, firstColW, shift);
|
---|
| 484 |
|
---|
| 485 | // Third part of the algorithm, since the real third part of the algorithm is not implemeted we use a JacobiSVD
|
---|
| 486 | JacobiSVD<MatrixXr> res= JacobiSVD<MatrixXr>(m_computed.block(firstCol + shift, firstCol +shift, n + 1, n),
|
---|
| 487 | ComputeFullU | (ComputeFullV * compV)) ;
|
---|
| 488 | if (compU) m_naiveU.block(firstCol, firstCol, n + 1, n + 1) *= res.matrixU();
|
---|
| 489 | else m_naiveU.block(0, firstCol, 2, n + 1) *= res.matrixU();
|
---|
| 490 |
|
---|
| 491 | if (compV) m_naiveV.block(firstRowW, firstColW, n, n) *= res.matrixV();
|
---|
| 492 | m_computed.block(firstCol + shift, firstCol + shift, n, n) << MatrixXr::Zero(n, n);
|
---|
| 493 | for (int i=0; i<n; i++)
|
---|
| 494 | m_computed(firstCol + shift + i, firstCol + shift +i) = res.singularValues().coeffRef(i);
|
---|
| 495 | // end of the third part
|
---|
| 496 |
|
---|
| 497 |
|
---|
| 498 | }// end divide
|
---|
| 499 |
|
---|
| 500 |
|
---|
| 501 | // page 12_13
|
---|
| 502 | // i >= 1, di almost null and zi non null.
|
---|
| 503 | // We use a rotation to zero out zi applied to the left of M
|
---|
| 504 | template <typename MatrixType>
|
---|
| 505 | void BDCSVD<MatrixType>::deflation43(Index firstCol, Index shift, Index i, Index size){
|
---|
| 506 | using std::abs;
|
---|
| 507 | using std::sqrt;
|
---|
| 508 | using std::pow;
|
---|
| 509 | RealScalar c = m_computed(firstCol + shift, firstCol + shift);
|
---|
| 510 | RealScalar s = m_computed(i, firstCol + shift);
|
---|
| 511 | RealScalar r = sqrt(pow(abs(c), 2) + pow(abs(s), 2));
|
---|
| 512 | if (r == 0){
|
---|
| 513 | m_computed(i, i)=0;
|
---|
| 514 | return;
|
---|
| 515 | }
|
---|
| 516 | c/=r;
|
---|
| 517 | s/=r;
|
---|
| 518 | m_computed(firstCol + shift, firstCol + shift) = r;
|
---|
| 519 | m_computed(i, firstCol + shift) = 0;
|
---|
| 520 | m_computed(i, i) = 0;
|
---|
| 521 | if (compU){
|
---|
| 522 | m_naiveU.col(firstCol).segment(firstCol,size) =
|
---|
| 523 | c * m_naiveU.col(firstCol).segment(firstCol, size) -
|
---|
| 524 | s * m_naiveU.col(i).segment(firstCol, size) ;
|
---|
| 525 |
|
---|
| 526 | m_naiveU.col(i).segment(firstCol, size) =
|
---|
| 527 | (c + s*s/c) * m_naiveU.col(i).segment(firstCol, size) +
|
---|
| 528 | (s/c) * m_naiveU.col(firstCol).segment(firstCol,size);
|
---|
| 529 | }
|
---|
| 530 | }// end deflation 43
|
---|
| 531 |
|
---|
| 532 |
|
---|
| 533 | // page 13
|
---|
| 534 | // i,j >= 1, i != j and |di - dj| < epsilon * norm2(M)
|
---|
| 535 | // We apply two rotations to have zj = 0;
|
---|
| 536 | template <typename MatrixType>
|
---|
| 537 | void BDCSVD<MatrixType>::deflation44(Index firstColu , Index firstColm, Index firstRowW, Index firstColW, Index i, Index j, Index size){
|
---|
| 538 | using std::abs;
|
---|
| 539 | using std::sqrt;
|
---|
| 540 | using std::conj;
|
---|
| 541 | using std::pow;
|
---|
| 542 | RealScalar c = m_computed(firstColm, firstColm + j - 1);
|
---|
| 543 | RealScalar s = m_computed(firstColm, firstColm + i - 1);
|
---|
| 544 | RealScalar r = sqrt(pow(abs(c), 2) + pow(abs(s), 2));
|
---|
| 545 | if (r==0){
|
---|
| 546 | m_computed(firstColm + i, firstColm + i) = m_computed(firstColm + j, firstColm + j);
|
---|
| 547 | return;
|
---|
| 548 | }
|
---|
| 549 | c/=r;
|
---|
| 550 | s/=r;
|
---|
| 551 | m_computed(firstColm + i, firstColm) = r;
|
---|
| 552 | m_computed(firstColm + i, firstColm + i) = m_computed(firstColm + j, firstColm + j);
|
---|
| 553 | m_computed(firstColm + j, firstColm) = 0;
|
---|
| 554 | if (compU){
|
---|
| 555 | m_naiveU.col(firstColu + i).segment(firstColu, size) =
|
---|
| 556 | c * m_naiveU.col(firstColu + i).segment(firstColu, size) -
|
---|
| 557 | s * m_naiveU.col(firstColu + j).segment(firstColu, size) ;
|
---|
| 558 |
|
---|
| 559 | m_naiveU.col(firstColu + j).segment(firstColu, size) =
|
---|
| 560 | (c + s*s/c) * m_naiveU.col(firstColu + j).segment(firstColu, size) +
|
---|
| 561 | (s/c) * m_naiveU.col(firstColu + i).segment(firstColu, size);
|
---|
| 562 | }
|
---|
| 563 | if (compV){
|
---|
| 564 | m_naiveV.col(firstColW + i).segment(firstRowW, size - 1) =
|
---|
| 565 | c * m_naiveV.col(firstColW + i).segment(firstRowW, size - 1) +
|
---|
| 566 | s * m_naiveV.col(firstColW + j).segment(firstRowW, size - 1) ;
|
---|
| 567 |
|
---|
| 568 | m_naiveV.col(firstColW + j).segment(firstRowW, size - 1) =
|
---|
| 569 | (c + s*s/c) * m_naiveV.col(firstColW + j).segment(firstRowW, size - 1) -
|
---|
| 570 | (s/c) * m_naiveV.col(firstColW + i).segment(firstRowW, size - 1);
|
---|
| 571 | }
|
---|
| 572 | }// end deflation 44
|
---|
| 573 |
|
---|
| 574 |
|
---|
| 575 |
|
---|
| 576 | template <typename MatrixType>
|
---|
| 577 | void BDCSVD<MatrixType>::deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW, Index shift){
|
---|
| 578 | //condition 4.1
|
---|
| 579 | RealScalar EPS = EPSILON * (std::max<RealScalar>(m_computed(firstCol + shift + 1, firstCol + shift + 1), m_computed(firstCol + k, firstCol + k)));
|
---|
| 580 | const Index length = lastCol + 1 - firstCol;
|
---|
| 581 | if (m_computed(firstCol + shift, firstCol + shift) < EPS){
|
---|
| 582 | m_computed(firstCol + shift, firstCol + shift) = EPS;
|
---|
| 583 | }
|
---|
| 584 | //condition 4.2
|
---|
| 585 | for (Index i=firstCol + shift + 1;i<=lastCol + shift;i++){
|
---|
| 586 | if (std::abs(m_computed(i, firstCol + shift)) < EPS){
|
---|
| 587 | m_computed(i, firstCol + shift) = 0;
|
---|
| 588 | }
|
---|
| 589 | }
|
---|
| 590 |
|
---|
| 591 | //condition 4.3
|
---|
| 592 | for (Index i=firstCol + shift + 1;i<=lastCol + shift; i++){
|
---|
| 593 | if (m_computed(i, i) < EPS){
|
---|
| 594 | deflation43(firstCol, shift, i, length);
|
---|
| 595 | }
|
---|
| 596 | }
|
---|
| 597 |
|
---|
| 598 | //condition 4.4
|
---|
| 599 |
|
---|
| 600 | Index i=firstCol + shift + 1, j=firstCol + shift + k + 1;
|
---|
| 601 | //we stock the final place of each line
|
---|
| 602 | Index *permutation = new Index[length];
|
---|
| 603 |
|
---|
| 604 | for (Index p =1; p < length; p++) {
|
---|
| 605 | if (i> firstCol + shift + k){
|
---|
| 606 | permutation[p] = j;
|
---|
| 607 | j++;
|
---|
| 608 | } else if (j> lastCol + shift)
|
---|
| 609 | {
|
---|
| 610 | permutation[p] = i;
|
---|
| 611 | i++;
|
---|
| 612 | }
|
---|
| 613 | else
|
---|
| 614 | {
|
---|
| 615 | if (m_computed(i, i) < m_computed(j, j)){
|
---|
| 616 | permutation[p] = j;
|
---|
| 617 | j++;
|
---|
| 618 | }
|
---|
| 619 | else
|
---|
| 620 | {
|
---|
| 621 | permutation[p] = i;
|
---|
| 622 | i++;
|
---|
| 623 | }
|
---|
| 624 | }
|
---|
| 625 | }
|
---|
| 626 | //we do the permutation
|
---|
| 627 | RealScalar aux;
|
---|
| 628 | //we stock the current index of each col
|
---|
| 629 | //and the column of each index
|
---|
| 630 | Index *realInd = new Index[length];
|
---|
| 631 | Index *realCol = new Index[length];
|
---|
| 632 | for (int pos = 0; pos< length; pos++){
|
---|
| 633 | realCol[pos] = pos + firstCol + shift;
|
---|
| 634 | realInd[pos] = pos;
|
---|
| 635 | }
|
---|
| 636 | const Index Zero = firstCol + shift;
|
---|
| 637 | VectorType temp;
|
---|
| 638 | for (int i = 1; i < length - 1; i++){
|
---|
| 639 | const Index I = i + Zero;
|
---|
| 640 | const Index realI = realInd[i];
|
---|
| 641 | const Index j = permutation[length - i] - Zero;
|
---|
| 642 | const Index J = realCol[j];
|
---|
| 643 |
|
---|
| 644 | //diag displace
|
---|
| 645 | aux = m_computed(I, I);
|
---|
| 646 | m_computed(I, I) = m_computed(J, J);
|
---|
| 647 | m_computed(J, J) = aux;
|
---|
| 648 |
|
---|
| 649 | //firstrow displace
|
---|
| 650 | aux = m_computed(I, Zero);
|
---|
| 651 | m_computed(I, Zero) = m_computed(J, Zero);
|
---|
| 652 | m_computed(J, Zero) = aux;
|
---|
| 653 |
|
---|
| 654 | // change columns
|
---|
| 655 | if (compU) {
|
---|
| 656 | temp = m_naiveU.col(I - shift).segment(firstCol, length + 1);
|
---|
| 657 | m_naiveU.col(I - shift).segment(firstCol, length + 1) <<
|
---|
| 658 | m_naiveU.col(J - shift).segment(firstCol, length + 1);
|
---|
| 659 | m_naiveU.col(J - shift).segment(firstCol, length + 1) << temp;
|
---|
| 660 | }
|
---|
| 661 | else
|
---|
| 662 | {
|
---|
| 663 | temp = m_naiveU.col(I - shift).segment(0, 2);
|
---|
| 664 | m_naiveU.col(I - shift).segment(0, 2) <<
|
---|
| 665 | m_naiveU.col(J - shift).segment(0, 2);
|
---|
| 666 | m_naiveU.col(J - shift).segment(0, 2) << temp;
|
---|
| 667 | }
|
---|
| 668 | if (compV) {
|
---|
| 669 | const Index CWI = I + firstColW - Zero;
|
---|
| 670 | const Index CWJ = J + firstColW - Zero;
|
---|
| 671 | temp = m_naiveV.col(CWI).segment(firstRowW, length);
|
---|
| 672 | m_naiveV.col(CWI).segment(firstRowW, length) << m_naiveV.col(CWJ).segment(firstRowW, length);
|
---|
| 673 | m_naiveV.col(CWJ).segment(firstRowW, length) << temp;
|
---|
| 674 | }
|
---|
| 675 |
|
---|
| 676 | //update real pos
|
---|
| 677 | realCol[realI] = J;
|
---|
| 678 | realCol[j] = I;
|
---|
| 679 | realInd[J - Zero] = realI;
|
---|
| 680 | realInd[I - Zero] = j;
|
---|
| 681 | }
|
---|
| 682 | for (Index i = firstCol + shift + 1; i<lastCol + shift;i++){
|
---|
| 683 | if ((m_computed(i + 1, i + 1) - m_computed(i, i)) < EPS){
|
---|
| 684 | deflation44(firstCol ,
|
---|
| 685 | firstCol + shift,
|
---|
| 686 | firstRowW,
|
---|
| 687 | firstColW,
|
---|
| 688 | i - Zero,
|
---|
| 689 | i + 1 - Zero,
|
---|
| 690 | length);
|
---|
| 691 | }
|
---|
| 692 | }
|
---|
| 693 | delete [] permutation;
|
---|
| 694 | delete [] realInd;
|
---|
| 695 | delete [] realCol;
|
---|
| 696 |
|
---|
| 697 | }//end deflation
|
---|
| 698 |
|
---|
| 699 |
|
---|
| 700 | namespace internal{
|
---|
| 701 |
|
---|
| 702 | template<typename _MatrixType, typename Rhs>
|
---|
| 703 | struct solve_retval<BDCSVD<_MatrixType>, Rhs>
|
---|
| 704 | : solve_retval_base<BDCSVD<_MatrixType>, Rhs>
|
---|
| 705 | {
|
---|
| 706 | typedef BDCSVD<_MatrixType> BDCSVDType;
|
---|
| 707 | EIGEN_MAKE_SOLVE_HELPERS(BDCSVDType, Rhs)
|
---|
| 708 |
|
---|
| 709 | template<typename Dest> void evalTo(Dest& dst) const
|
---|
| 710 | {
|
---|
| 711 | eigen_assert(rhs().rows() == dec().rows());
|
---|
| 712 | // A = U S V^*
|
---|
| 713 | // So A^{ - 1} = V S^{ - 1} U^*
|
---|
| 714 | Index diagSize = (std::min)(dec().rows(), dec().cols());
|
---|
| 715 | typename BDCSVDType::SingularValuesType invertedSingVals(diagSize);
|
---|
| 716 | Index nonzeroSingVals = dec().nonzeroSingularValues();
|
---|
| 717 | invertedSingVals.head(nonzeroSingVals) = dec().singularValues().head(nonzeroSingVals).array().inverse();
|
---|
| 718 | invertedSingVals.tail(diagSize - nonzeroSingVals).setZero();
|
---|
| 719 |
|
---|
| 720 | dst = dec().matrixV().leftCols(diagSize)
|
---|
| 721 | * invertedSingVals.asDiagonal()
|
---|
| 722 | * dec().matrixU().leftCols(diagSize).adjoint()
|
---|
| 723 | * rhs();
|
---|
| 724 | return;
|
---|
| 725 | }
|
---|
| 726 | };
|
---|
| 727 |
|
---|
| 728 | } //end namespace internal
|
---|
| 729 |
|
---|
| 730 | /** \svd_module
|
---|
| 731 | *
|
---|
| 732 | * \return the singular value decomposition of \c *this computed by
|
---|
| 733 | * BDC Algorithm
|
---|
| 734 | *
|
---|
| 735 | * \sa class BDCSVD
|
---|
| 736 | */
|
---|
| 737 | /*
|
---|
| 738 | template<typename Derived>
|
---|
| 739 | BDCSVD<typename MatrixBase<Derived>::PlainObject>
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| 740 | MatrixBase<Derived>::bdcSvd(unsigned int computationOptions) const
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| 741 | {
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| 742 | return BDCSVD<PlainObject>(*this, computationOptions);
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| 743 | }
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| 744 | */
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| 745 |
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| 746 | } // end namespace Eigen
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| 747 |
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| 748 | #endif
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