[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) 2009-2010 Benoit Jacob <jacob.benoit.1@gmail.com>
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| 5 | //
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| 6 | // This Source Code Form is subject to the terms of the Mozilla
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| 7 | // Public License v. 2.0. If a copy of the MPL was not distributed
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| 8 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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| 9 |
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| 10 | #ifndef EIGEN_JACOBISVD_H
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| 11 | #define EIGEN_JACOBISVD_H
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| 12 |
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| 13 | namespace Eigen {
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| 14 |
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| 15 | namespace internal {
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| 16 | // forward declaration (needed by ICC)
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| 17 | // the empty body is required by MSVC
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| 18 | template<typename MatrixType, int QRPreconditioner,
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| 19 | bool IsComplex = NumTraits<typename MatrixType::Scalar>::IsComplex>
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| 20 | struct svd_precondition_2x2_block_to_be_real {};
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| 21 |
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| 22 | /*** QR preconditioners (R-SVD)
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| 23 | ***
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| 24 | *** Their role is to reduce the problem of computing the SVD to the case of a square matrix.
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| 25 | *** This approach, known as R-SVD, is an optimization for rectangular-enough matrices, and is a requirement for
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| 26 | *** JacobiSVD which by itself is only able to work on square matrices.
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| 27 | ***/
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| 28 |
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| 29 | enum { PreconditionIfMoreColsThanRows, PreconditionIfMoreRowsThanCols };
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| 30 |
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| 31 | template<typename MatrixType, int QRPreconditioner, int Case>
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| 32 | struct qr_preconditioner_should_do_anything
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| 33 | {
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| 34 | enum { a = MatrixType::RowsAtCompileTime != Dynamic &&
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| 35 | MatrixType::ColsAtCompileTime != Dynamic &&
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| 36 | MatrixType::ColsAtCompileTime <= MatrixType::RowsAtCompileTime,
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| 37 | b = MatrixType::RowsAtCompileTime != Dynamic &&
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| 38 | MatrixType::ColsAtCompileTime != Dynamic &&
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| 39 | MatrixType::RowsAtCompileTime <= MatrixType::ColsAtCompileTime,
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| 40 | ret = !( (QRPreconditioner == NoQRPreconditioner) ||
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| 41 | (Case == PreconditionIfMoreColsThanRows && bool(a)) ||
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| 42 | (Case == PreconditionIfMoreRowsThanCols && bool(b)) )
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| 43 | };
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| 44 | };
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| 45 |
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| 46 | template<typename MatrixType, int QRPreconditioner, int Case,
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| 47 | bool DoAnything = qr_preconditioner_should_do_anything<MatrixType, QRPreconditioner, Case>::ret
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| 48 | > struct qr_preconditioner_impl {};
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| 49 |
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| 50 | template<typename MatrixType, int QRPreconditioner, int Case>
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| 51 | class qr_preconditioner_impl<MatrixType, QRPreconditioner, Case, false>
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| 52 | {
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| 53 | public:
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| 54 | typedef typename MatrixType::Index Index;
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| 55 | void allocate(const JacobiSVD<MatrixType, QRPreconditioner>&) {}
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| 56 | bool run(JacobiSVD<MatrixType, QRPreconditioner>&, const MatrixType&)
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| 57 | {
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| 58 | return false;
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| 59 | }
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| 60 | };
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| 61 |
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| 62 | /*** preconditioner using FullPivHouseholderQR ***/
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| 63 |
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| 64 | template<typename MatrixType>
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| 65 | class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
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| 66 | {
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| 67 | public:
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| 68 | typedef typename MatrixType::Index Index;
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| 69 | typedef typename MatrixType::Scalar Scalar;
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| 70 | enum
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| 71 | {
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| 72 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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| 73 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime
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| 74 | };
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| 75 | typedef Matrix<Scalar, 1, RowsAtCompileTime, RowMajor, 1, MaxRowsAtCompileTime> WorkspaceType;
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| 76 |
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| 77 | void allocate(const JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd)
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| 78 | {
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| 79 | if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
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| 80 | {
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| 81 | m_qr.~QRType();
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| 82 | ::new (&m_qr) QRType(svd.rows(), svd.cols());
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| 83 | }
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| 84 | if (svd.m_computeFullU) m_workspace.resize(svd.rows());
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| 85 | }
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| 86 |
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| 87 | bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
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| 88 | {
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| 89 | if(matrix.rows() > matrix.cols())
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| 90 | {
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| 91 | m_qr.compute(matrix);
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| 92 | svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
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| 93 | if(svd.m_computeFullU) m_qr.matrixQ().evalTo(svd.m_matrixU, m_workspace);
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| 94 | if(svd.computeV()) svd.m_matrixV = m_qr.colsPermutation();
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| 95 | return true;
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| 96 | }
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| 97 | return false;
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| 98 | }
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| 99 | private:
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| 100 | typedef FullPivHouseholderQR<MatrixType> QRType;
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| 101 | QRType m_qr;
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| 102 | WorkspaceType m_workspace;
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| 103 | };
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| 104 |
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| 105 | template<typename MatrixType>
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| 106 | class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
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| 107 | {
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| 108 | public:
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| 109 | typedef typename MatrixType::Index Index;
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| 110 | typedef typename MatrixType::Scalar Scalar;
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| 111 | enum
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| 112 | {
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| 113 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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| 114 | ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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| 115 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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| 116 | MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
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| 117 | Options = MatrixType::Options
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| 118 | };
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| 119 | typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>
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| 120 | TransposeTypeWithSameStorageOrder;
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| 121 |
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| 122 | void allocate(const JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd)
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| 123 | {
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| 124 | if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
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| 125 | {
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| 126 | m_qr.~QRType();
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| 127 | ::new (&m_qr) QRType(svd.cols(), svd.rows());
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| 128 | }
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| 129 | m_adjoint.resize(svd.cols(), svd.rows());
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| 130 | if (svd.m_computeFullV) m_workspace.resize(svd.cols());
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| 131 | }
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| 132 |
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| 133 | bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
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| 134 | {
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| 135 | if(matrix.cols() > matrix.rows())
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| 136 | {
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| 137 | m_adjoint = matrix.adjoint();
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| 138 | m_qr.compute(m_adjoint);
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| 139 | svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
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| 140 | if(svd.m_computeFullV) m_qr.matrixQ().evalTo(svd.m_matrixV, m_workspace);
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| 141 | if(svd.computeU()) svd.m_matrixU = m_qr.colsPermutation();
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| 142 | return true;
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| 143 | }
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| 144 | else return false;
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| 145 | }
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| 146 | private:
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| 147 | typedef FullPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
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| 148 | QRType m_qr;
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| 149 | TransposeTypeWithSameStorageOrder m_adjoint;
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| 150 | typename internal::plain_row_type<MatrixType>::type m_workspace;
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| 151 | };
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| 152 |
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| 153 | /*** preconditioner using ColPivHouseholderQR ***/
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| 154 |
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| 155 | template<typename MatrixType>
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| 156 | class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
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| 157 | {
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| 158 | public:
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| 159 | typedef typename MatrixType::Index Index;
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| 160 |
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| 161 | void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd)
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| 162 | {
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| 163 | if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
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| 164 | {
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| 165 | m_qr.~QRType();
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| 166 | ::new (&m_qr) QRType(svd.rows(), svd.cols());
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| 167 | }
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| 168 | if (svd.m_computeFullU) m_workspace.resize(svd.rows());
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| 169 | else if (svd.m_computeThinU) m_workspace.resize(svd.cols());
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| 170 | }
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| 171 |
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| 172 | bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
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| 173 | {
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| 174 | if(matrix.rows() > matrix.cols())
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| 175 | {
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| 176 | m_qr.compute(matrix);
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| 177 | svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
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| 178 | if(svd.m_computeFullU) m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace);
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| 179 | else if(svd.m_computeThinU)
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| 180 | {
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| 181 | svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
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| 182 | m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace);
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| 183 | }
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| 184 | if(svd.computeV()) svd.m_matrixV = m_qr.colsPermutation();
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| 185 | return true;
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| 186 | }
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| 187 | return false;
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| 188 | }
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| 189 |
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| 190 | private:
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| 191 | typedef ColPivHouseholderQR<MatrixType> QRType;
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| 192 | QRType m_qr;
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| 193 | typename internal::plain_col_type<MatrixType>::type m_workspace;
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| 194 | };
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| 195 |
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| 196 | template<typename MatrixType>
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| 197 | class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
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| 198 | {
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| 199 | public:
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| 200 | typedef typename MatrixType::Index Index;
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| 201 | typedef typename MatrixType::Scalar Scalar;
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| 202 | enum
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| 203 | {
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| 204 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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| 205 | ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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| 206 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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| 207 | MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
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| 208 | Options = MatrixType::Options
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| 209 | };
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| 210 |
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| 211 | typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>
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| 212 | TransposeTypeWithSameStorageOrder;
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| 213 |
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| 214 | void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd)
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| 215 | {
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| 216 | if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
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| 217 | {
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| 218 | m_qr.~QRType();
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| 219 | ::new (&m_qr) QRType(svd.cols(), svd.rows());
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| 220 | }
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| 221 | if (svd.m_computeFullV) m_workspace.resize(svd.cols());
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| 222 | else if (svd.m_computeThinV) m_workspace.resize(svd.rows());
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| 223 | m_adjoint.resize(svd.cols(), svd.rows());
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| 224 | }
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| 225 |
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| 226 | bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
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| 227 | {
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| 228 | if(matrix.cols() > matrix.rows())
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| 229 | {
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| 230 | m_adjoint = matrix.adjoint();
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| 231 | m_qr.compute(m_adjoint);
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| 232 |
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| 233 | svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
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| 234 | if(svd.m_computeFullV) m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace);
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| 235 | else if(svd.m_computeThinV)
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| 236 | {
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| 237 | svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
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| 238 | m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace);
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| 239 | }
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| 240 | if(svd.computeU()) svd.m_matrixU = m_qr.colsPermutation();
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| 241 | return true;
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| 242 | }
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| 243 | else return false;
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| 244 | }
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| 245 |
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| 246 | private:
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| 247 | typedef ColPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
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| 248 | QRType m_qr;
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| 249 | TransposeTypeWithSameStorageOrder m_adjoint;
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| 250 | typename internal::plain_row_type<MatrixType>::type m_workspace;
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| 251 | };
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| 252 |
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| 253 | /*** preconditioner using HouseholderQR ***/
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| 254 |
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| 255 | template<typename MatrixType>
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| 256 | class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
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| 257 | {
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| 258 | public:
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| 259 | typedef typename MatrixType::Index Index;
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| 260 |
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| 261 | void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd)
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| 262 | {
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| 263 | if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
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| 264 | {
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| 265 | m_qr.~QRType();
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| 266 | ::new (&m_qr) QRType(svd.rows(), svd.cols());
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| 267 | }
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| 268 | if (svd.m_computeFullU) m_workspace.resize(svd.rows());
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| 269 | else if (svd.m_computeThinU) m_workspace.resize(svd.cols());
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| 270 | }
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| 271 |
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| 272 | bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix)
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| 273 | {
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| 274 | if(matrix.rows() > matrix.cols())
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| 275 | {
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| 276 | m_qr.compute(matrix);
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| 277 | svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
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| 278 | if(svd.m_computeFullU) m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace);
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| 279 | else if(svd.m_computeThinU)
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| 280 | {
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| 281 | svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
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| 282 | m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace);
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| 283 | }
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| 284 | if(svd.computeV()) svd.m_matrixV.setIdentity(matrix.cols(), matrix.cols());
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| 285 | return true;
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| 286 | }
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| 287 | return false;
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| 288 | }
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| 289 | private:
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| 290 | typedef HouseholderQR<MatrixType> QRType;
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| 291 | QRType m_qr;
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| 292 | typename internal::plain_col_type<MatrixType>::type m_workspace;
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| 293 | };
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| 294 |
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| 295 | template<typename MatrixType>
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| 296 | class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
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| 297 | {
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| 298 | public:
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| 299 | typedef typename MatrixType::Index Index;
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| 300 | typedef typename MatrixType::Scalar Scalar;
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| 301 | enum
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| 302 | {
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| 303 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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| 304 | ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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| 305 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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| 306 | MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
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| 307 | Options = MatrixType::Options
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| 308 | };
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| 309 |
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| 310 | typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>
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| 311 | TransposeTypeWithSameStorageOrder;
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| 312 |
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| 313 | void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd)
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| 314 | {
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| 315 | if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
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| 316 | {
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| 317 | m_qr.~QRType();
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| 318 | ::new (&m_qr) QRType(svd.cols(), svd.rows());
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| 319 | }
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| 320 | if (svd.m_computeFullV) m_workspace.resize(svd.cols());
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| 321 | else if (svd.m_computeThinV) m_workspace.resize(svd.rows());
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| 322 | m_adjoint.resize(svd.cols(), svd.rows());
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| 323 | }
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| 324 |
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| 325 | bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix)
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| 326 | {
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| 327 | if(matrix.cols() > matrix.rows())
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| 328 | {
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| 329 | m_adjoint = matrix.adjoint();
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| 330 | m_qr.compute(m_adjoint);
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| 331 |
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| 332 | svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
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| 333 | if(svd.m_computeFullV) m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace);
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| 334 | else if(svd.m_computeThinV)
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| 335 | {
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| 336 | svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
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| 337 | m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace);
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| 338 | }
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| 339 | if(svd.computeU()) svd.m_matrixU.setIdentity(matrix.rows(), matrix.rows());
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| 340 | return true;
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| 341 | }
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| 342 | else return false;
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| 343 | }
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| 344 |
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| 345 | private:
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| 346 | typedef HouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
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| 347 | QRType m_qr;
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| 348 | TransposeTypeWithSameStorageOrder m_adjoint;
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| 349 | typename internal::plain_row_type<MatrixType>::type m_workspace;
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| 350 | };
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| 351 |
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| 352 | /*** 2x2 SVD implementation
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| 353 | ***
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| 354 | *** JacobiSVD consists in performing a series of 2x2 SVD subproblems
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| 355 | ***/
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| 356 |
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| 357 | template<typename MatrixType, int QRPreconditioner>
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| 358 | struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, false>
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| 359 | {
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| 360 | typedef JacobiSVD<MatrixType, QRPreconditioner> SVD;
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| 361 | typedef typename SVD::Index Index;
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| 362 | static void run(typename SVD::WorkMatrixType&, SVD&, Index, Index) {}
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| 363 | };
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| 364 |
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| 365 | template<typename MatrixType, int QRPreconditioner>
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| 366 | struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, true>
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| 367 | {
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| 368 | typedef JacobiSVD<MatrixType, QRPreconditioner> SVD;
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| 369 | typedef typename MatrixType::Scalar Scalar;
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| 370 | typedef typename MatrixType::RealScalar RealScalar;
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| 371 | typedef typename SVD::Index Index;
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| 372 | static void run(typename SVD::WorkMatrixType& work_matrix, SVD& svd, Index p, Index q)
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| 373 | {
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| 374 | using std::sqrt;
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| 375 | Scalar z;
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| 376 | JacobiRotation<Scalar> rot;
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| 377 | RealScalar n = sqrt(numext::abs2(work_matrix.coeff(p,p)) + numext::abs2(work_matrix.coeff(q,p)));
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| 378 | if(n==0)
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| 379 | {
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| 380 | z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q);
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| 381 | work_matrix.row(p) *= z;
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| 382 | if(svd.computeU()) svd.m_matrixU.col(p) *= conj(z);
|
---|
| 383 | z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q);
|
---|
| 384 | work_matrix.row(q) *= z;
|
---|
| 385 | if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z);
|
---|
| 386 | }
|
---|
| 387 | else
|
---|
| 388 | {
|
---|
| 389 | rot.c() = conj(work_matrix.coeff(p,p)) / n;
|
---|
| 390 | rot.s() = work_matrix.coeff(q,p) / n;
|
---|
| 391 | work_matrix.applyOnTheLeft(p,q,rot);
|
---|
| 392 | if(svd.computeU()) svd.m_matrixU.applyOnTheRight(p,q,rot.adjoint());
|
---|
| 393 | if(work_matrix.coeff(p,q) != Scalar(0))
|
---|
| 394 | {
|
---|
| 395 | Scalar z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q);
|
---|
| 396 | work_matrix.col(q) *= z;
|
---|
| 397 | if(svd.computeV()) svd.m_matrixV.col(q) *= z;
|
---|
| 398 | }
|
---|
| 399 | if(work_matrix.coeff(q,q) != Scalar(0))
|
---|
| 400 | {
|
---|
| 401 | z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q);
|
---|
| 402 | work_matrix.row(q) *= z;
|
---|
| 403 | if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z);
|
---|
| 404 | }
|
---|
| 405 | }
|
---|
| 406 | }
|
---|
| 407 | };
|
---|
| 408 |
|
---|
| 409 | template<typename MatrixType, typename RealScalar, typename Index>
|
---|
| 410 | void real_2x2_jacobi_svd(const MatrixType& matrix, Index p, Index q,
|
---|
| 411 | JacobiRotation<RealScalar> *j_left,
|
---|
| 412 | JacobiRotation<RealScalar> *j_right)
|
---|
| 413 | {
|
---|
| 414 | using std::sqrt;
|
---|
| 415 | Matrix<RealScalar,2,2> m;
|
---|
| 416 | m << numext::real(matrix.coeff(p,p)), numext::real(matrix.coeff(p,q)),
|
---|
| 417 | numext::real(matrix.coeff(q,p)), numext::real(matrix.coeff(q,q));
|
---|
| 418 | JacobiRotation<RealScalar> rot1;
|
---|
| 419 | RealScalar t = m.coeff(0,0) + m.coeff(1,1);
|
---|
| 420 | RealScalar d = m.coeff(1,0) - m.coeff(0,1);
|
---|
| 421 | if(t == RealScalar(0))
|
---|
| 422 | {
|
---|
| 423 | rot1.c() = RealScalar(0);
|
---|
| 424 | rot1.s() = d > RealScalar(0) ? RealScalar(1) : RealScalar(-1);
|
---|
| 425 | }
|
---|
| 426 | else
|
---|
| 427 | {
|
---|
| 428 | RealScalar u = d / t;
|
---|
| 429 | rot1.c() = RealScalar(1) / sqrt(RealScalar(1) + numext::abs2(u));
|
---|
| 430 | rot1.s() = rot1.c() * u;
|
---|
| 431 | }
|
---|
| 432 | m.applyOnTheLeft(0,1,rot1);
|
---|
| 433 | j_right->makeJacobi(m,0,1);
|
---|
| 434 | *j_left = rot1 * j_right->transpose();
|
---|
| 435 | }
|
---|
| 436 |
|
---|
| 437 | } // end namespace internal
|
---|
| 438 |
|
---|
| 439 | /** \ingroup SVD_Module
|
---|
| 440 | *
|
---|
| 441 | *
|
---|
| 442 | * \class JacobiSVD
|
---|
| 443 | *
|
---|
| 444 | * \brief Two-sided Jacobi SVD decomposition of a rectangular matrix
|
---|
| 445 | *
|
---|
| 446 | * \param MatrixType the type of the matrix of which we are computing the SVD decomposition
|
---|
| 447 | * \param QRPreconditioner this optional parameter allows to specify the type of QR decomposition that will be used internally
|
---|
| 448 | * for the R-SVD step for non-square matrices. See discussion of possible values below.
|
---|
| 449 | *
|
---|
| 450 | * SVD decomposition consists in decomposing any n-by-p matrix \a A as a product
|
---|
| 451 | * \f[ A = U S V^* \f]
|
---|
| 452 | * where \a U is a n-by-n unitary, \a V is a p-by-p unitary, and \a S is a n-by-p real positive matrix which is zero outside of its main diagonal;
|
---|
| 453 | * the diagonal entries of S are known as the \em singular \em values of \a A and the columns of \a U and \a V are known as the left
|
---|
| 454 | * and right \em singular \em vectors of \a A respectively.
|
---|
| 455 | *
|
---|
| 456 | * Singular values are always sorted in decreasing order.
|
---|
| 457 | *
|
---|
| 458 | * This JacobiSVD decomposition computes only the singular values by default. If you want \a U or \a V, you need to ask for them explicitly.
|
---|
| 459 | *
|
---|
| 460 | * You can ask for only \em thin \a U or \a V to be computed, meaning the following. In case of a rectangular n-by-p matrix, letting \a m be the
|
---|
| 461 | * smaller value among \a n and \a p, there are only \a m singular vectors; the remaining columns of \a U and \a V do not correspond to actual
|
---|
| 462 | * singular vectors. Asking for \em thin \a U or \a V means asking for only their \a m first columns to be formed. So \a U is then a n-by-m matrix,
|
---|
| 463 | * and \a V is then a p-by-m matrix. Notice that thin \a U and \a V are all you need for (least squares) solving.
|
---|
| 464 | *
|
---|
| 465 | * Here's an example demonstrating basic usage:
|
---|
| 466 | * \include JacobiSVD_basic.cpp
|
---|
| 467 | * Output: \verbinclude JacobiSVD_basic.out
|
---|
| 468 | *
|
---|
| 469 | * This JacobiSVD class is a two-sided Jacobi R-SVD decomposition, ensuring optimal reliability and accuracy. The downside is that it's slower than
|
---|
| 470 | * bidiagonalizing SVD algorithms for large square matrices; however its complexity is still \f$ O(n^2p) \f$ where \a n is the smaller dimension and
|
---|
| 471 | * \a p is the greater dimension, meaning that it is still of the same order of complexity as the faster bidiagonalizing R-SVD algorithms.
|
---|
| 472 | * In particular, like any R-SVD, it takes advantage of non-squareness in that its complexity is only linear in the greater dimension.
|
---|
| 473 | *
|
---|
| 474 | * If the input matrix has inf or nan coefficients, the result of the computation is undefined, but the computation is guaranteed to
|
---|
| 475 | * terminate in finite (and reasonable) time.
|
---|
| 476 | *
|
---|
| 477 | * The possible values for QRPreconditioner are:
|
---|
| 478 | * \li ColPivHouseholderQRPreconditioner is the default. In practice it's very safe. It uses column-pivoting QR.
|
---|
| 479 | * \li FullPivHouseholderQRPreconditioner, is the safest and slowest. It uses full-pivoting QR.
|
---|
| 480 | * Contrary to other QRs, it doesn't allow computing thin unitaries.
|
---|
| 481 | * \li HouseholderQRPreconditioner is the fastest, and less safe and accurate than the pivoting variants. It uses non-pivoting QR.
|
---|
| 482 | * This is very similar in safety and accuracy to the bidiagonalization process used by bidiagonalizing SVD algorithms (since bidiagonalization
|
---|
| 483 | * is inherently non-pivoting). However the resulting SVD is still more reliable than bidiagonalizing SVDs because the Jacobi-based iterarive
|
---|
| 484 | * process is more reliable than the optimized bidiagonal SVD iterations.
|
---|
| 485 | * \li NoQRPreconditioner allows not to use a QR preconditioner at all. This is useful if you know that you will only be computing
|
---|
| 486 | * JacobiSVD decompositions of square matrices. Non-square matrices require a QR preconditioner. Using this option will result in
|
---|
| 487 | * faster compilation and smaller executable code. It won't significantly speed up computation, since JacobiSVD is always checking
|
---|
| 488 | * if QR preconditioning is needed before applying it anyway.
|
---|
| 489 | *
|
---|
| 490 | * \sa MatrixBase::jacobiSvd()
|
---|
| 491 | */
|
---|
| 492 | template<typename _MatrixType, int QRPreconditioner>
|
---|
| 493 | class JacobiSVD : public SVDBase<_MatrixType>
|
---|
| 494 | {
|
---|
| 495 | public:
|
---|
| 496 |
|
---|
| 497 | typedef _MatrixType MatrixType;
|
---|
| 498 | typedef typename MatrixType::Scalar Scalar;
|
---|
| 499 | typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
|
---|
| 500 | typedef typename MatrixType::Index Index;
|
---|
| 501 | enum {
|
---|
| 502 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
|
---|
| 503 | ColsAtCompileTime = MatrixType::ColsAtCompileTime,
|
---|
| 504 | DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime,ColsAtCompileTime),
|
---|
| 505 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
|
---|
| 506 | MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
|
---|
| 507 | MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime,MaxColsAtCompileTime),
|
---|
| 508 | MatrixOptions = MatrixType::Options
|
---|
| 509 | };
|
---|
| 510 |
|
---|
| 511 | typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime,
|
---|
| 512 | MatrixOptions, MaxRowsAtCompileTime, MaxRowsAtCompileTime>
|
---|
| 513 | MatrixUType;
|
---|
| 514 | typedef Matrix<Scalar, ColsAtCompileTime, ColsAtCompileTime,
|
---|
| 515 | MatrixOptions, MaxColsAtCompileTime, MaxColsAtCompileTime>
|
---|
| 516 | MatrixVType;
|
---|
| 517 | typedef typename internal::plain_diag_type<MatrixType, RealScalar>::type SingularValuesType;
|
---|
| 518 | typedef typename internal::plain_row_type<MatrixType>::type RowType;
|
---|
| 519 | typedef typename internal::plain_col_type<MatrixType>::type ColType;
|
---|
| 520 | typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime,
|
---|
| 521 | MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime>
|
---|
| 522 | WorkMatrixType;
|
---|
| 523 |
|
---|
| 524 | /** \brief Default Constructor.
|
---|
| 525 | *
|
---|
| 526 | * The default constructor is useful in cases in which the user intends to
|
---|
| 527 | * perform decompositions via JacobiSVD::compute(const MatrixType&).
|
---|
| 528 | */
|
---|
| 529 | JacobiSVD()
|
---|
| 530 | : SVDBase<_MatrixType>::SVDBase()
|
---|
| 531 | {}
|
---|
| 532 |
|
---|
| 533 |
|
---|
| 534 | /** \brief Default Constructor with memory preallocation
|
---|
| 535 | *
|
---|
| 536 | * Like the default constructor but with preallocation of the internal data
|
---|
| 537 | * according to the specified problem size.
|
---|
| 538 | * \sa JacobiSVD()
|
---|
| 539 | */
|
---|
| 540 | JacobiSVD(Index rows, Index cols, unsigned int computationOptions = 0)
|
---|
| 541 | : SVDBase<_MatrixType>::SVDBase()
|
---|
| 542 | {
|
---|
| 543 | allocate(rows, cols, computationOptions);
|
---|
| 544 | }
|
---|
| 545 |
|
---|
| 546 | /** \brief Constructor performing the decomposition of given matrix.
|
---|
| 547 | *
|
---|
| 548 | * \param matrix the matrix to decompose
|
---|
| 549 | * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
|
---|
| 550 | * By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU,
|
---|
| 551 | * #ComputeFullV, #ComputeThinV.
|
---|
| 552 | *
|
---|
| 553 | * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
|
---|
| 554 | * available with the (non-default) FullPivHouseholderQR preconditioner.
|
---|
| 555 | */
|
---|
| 556 | JacobiSVD(const MatrixType& matrix, unsigned int computationOptions = 0)
|
---|
| 557 | : SVDBase<_MatrixType>::SVDBase()
|
---|
| 558 | {
|
---|
| 559 | compute(matrix, computationOptions);
|
---|
| 560 | }
|
---|
| 561 |
|
---|
| 562 | /** \brief Method performing the decomposition of given matrix using custom options.
|
---|
| 563 | *
|
---|
| 564 | * \param matrix the matrix to decompose
|
---|
| 565 | * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
|
---|
| 566 | * By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU,
|
---|
| 567 | * #ComputeFullV, #ComputeThinV.
|
---|
| 568 | *
|
---|
| 569 | * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
|
---|
| 570 | * available with the (non-default) FullPivHouseholderQR preconditioner.
|
---|
| 571 | */
|
---|
| 572 | SVDBase<MatrixType>& compute(const MatrixType& matrix, unsigned int computationOptions);
|
---|
| 573 |
|
---|
| 574 | /** \brief Method performing the decomposition of given matrix using current options.
|
---|
| 575 | *
|
---|
| 576 | * \param matrix the matrix to decompose
|
---|
| 577 | *
|
---|
| 578 | * This method uses the current \a computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int).
|
---|
| 579 | */
|
---|
| 580 | SVDBase<MatrixType>& compute(const MatrixType& matrix)
|
---|
| 581 | {
|
---|
| 582 | return compute(matrix, this->m_computationOptions);
|
---|
| 583 | }
|
---|
| 584 |
|
---|
| 585 | /** \returns a (least squares) solution of \f$ A x = b \f$ using the current SVD decomposition of A.
|
---|
| 586 | *
|
---|
| 587 | * \param b the right-hand-side of the equation to solve.
|
---|
| 588 | *
|
---|
| 589 | * \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.
|
---|
| 590 | *
|
---|
| 591 | * \note SVD solving is implicitly least-squares. Thus, this method serves both purposes of exact solving and least-squares solving.
|
---|
| 592 | * In other words, the returned solution is guaranteed to minimize the Euclidean norm \f$ \Vert A x - b \Vert \f$.
|
---|
| 593 | */
|
---|
| 594 | template<typename Rhs>
|
---|
| 595 | inline const internal::solve_retval<JacobiSVD, Rhs>
|
---|
| 596 | solve(const MatrixBase<Rhs>& b) const
|
---|
| 597 | {
|
---|
| 598 | eigen_assert(this->m_isInitialized && "JacobiSVD is not initialized.");
|
---|
| 599 | eigen_assert(SVDBase<MatrixType>::computeU() && SVDBase<MatrixType>::computeV() && "JacobiSVD::solve() requires both unitaries U and V to be computed (thin unitaries suffice).");
|
---|
| 600 | return internal::solve_retval<JacobiSVD, Rhs>(*this, b.derived());
|
---|
| 601 | }
|
---|
| 602 |
|
---|
| 603 |
|
---|
| 604 |
|
---|
| 605 | private:
|
---|
| 606 | void allocate(Index rows, Index cols, unsigned int computationOptions);
|
---|
| 607 |
|
---|
| 608 | protected:
|
---|
| 609 | WorkMatrixType m_workMatrix;
|
---|
| 610 |
|
---|
| 611 | template<typename __MatrixType, int _QRPreconditioner, bool _IsComplex>
|
---|
| 612 | friend struct internal::svd_precondition_2x2_block_to_be_real;
|
---|
| 613 | template<typename __MatrixType, int _QRPreconditioner, int _Case, bool _DoAnything>
|
---|
| 614 | friend struct internal::qr_preconditioner_impl;
|
---|
| 615 |
|
---|
| 616 | internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows> m_qr_precond_morecols;
|
---|
| 617 | internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols> m_qr_precond_morerows;
|
---|
| 618 | };
|
---|
| 619 |
|
---|
| 620 | template<typename MatrixType, int QRPreconditioner>
|
---|
| 621 | void JacobiSVD<MatrixType, QRPreconditioner>::allocate(Index rows, Index cols, unsigned int computationOptions)
|
---|
| 622 | {
|
---|
| 623 | if (SVDBase<MatrixType>::allocate(rows, cols, computationOptions)) return;
|
---|
| 624 |
|
---|
| 625 | if (QRPreconditioner == FullPivHouseholderQRPreconditioner)
|
---|
| 626 | {
|
---|
| 627 | eigen_assert(!(this->m_computeThinU || this->m_computeThinV) &&
|
---|
| 628 | "JacobiSVD: can't compute thin U or thin V with the FullPivHouseholderQR preconditioner. "
|
---|
| 629 | "Use the ColPivHouseholderQR preconditioner instead.");
|
---|
| 630 | }
|
---|
| 631 |
|
---|
| 632 | m_workMatrix.resize(this->m_diagSize, this->m_diagSize);
|
---|
| 633 |
|
---|
| 634 | if(this->m_cols>this->m_rows) m_qr_precond_morecols.allocate(*this);
|
---|
| 635 | if(this->m_rows>this->m_cols) m_qr_precond_morerows.allocate(*this);
|
---|
| 636 | }
|
---|
| 637 |
|
---|
| 638 | template<typename MatrixType, int QRPreconditioner>
|
---|
| 639 | SVDBase<MatrixType>&
|
---|
| 640 | JacobiSVD<MatrixType, QRPreconditioner>::compute(const MatrixType& matrix, unsigned int computationOptions)
|
---|
| 641 | {
|
---|
| 642 | using std::abs;
|
---|
| 643 | allocate(matrix.rows(), matrix.cols(), computationOptions);
|
---|
| 644 |
|
---|
| 645 | // currently we stop when we reach precision 2*epsilon as the last bit of precision can require an unreasonable number of iterations,
|
---|
| 646 | // only worsening the precision of U and V as we accumulate more rotations
|
---|
| 647 | const RealScalar precision = RealScalar(2) * NumTraits<Scalar>::epsilon();
|
---|
| 648 |
|
---|
| 649 | // limit for very small denormal numbers to be considered zero in order to avoid infinite loops (see bug 286)
|
---|
| 650 | const RealScalar considerAsZero = RealScalar(2) * std::numeric_limits<RealScalar>::denorm_min();
|
---|
| 651 |
|
---|
| 652 | /*** step 1. The R-SVD step: we use a QR decomposition to reduce to the case of a square matrix */
|
---|
| 653 |
|
---|
| 654 | if(!m_qr_precond_morecols.run(*this, matrix) && !m_qr_precond_morerows.run(*this, matrix))
|
---|
| 655 | {
|
---|
| 656 | m_workMatrix = matrix.block(0,0,this->m_diagSize,this->m_diagSize);
|
---|
| 657 | if(this->m_computeFullU) this->m_matrixU.setIdentity(this->m_rows,this->m_rows);
|
---|
| 658 | if(this->m_computeThinU) this->m_matrixU.setIdentity(this->m_rows,this->m_diagSize);
|
---|
| 659 | if(this->m_computeFullV) this->m_matrixV.setIdentity(this->m_cols,this->m_cols);
|
---|
| 660 | if(this->m_computeThinV) this->m_matrixV.setIdentity(this->m_cols, this->m_diagSize);
|
---|
| 661 | }
|
---|
| 662 |
|
---|
| 663 | /*** step 2. The main Jacobi SVD iteration. ***/
|
---|
| 664 |
|
---|
| 665 | bool finished = false;
|
---|
| 666 | while(!finished)
|
---|
| 667 | {
|
---|
| 668 | finished = true;
|
---|
| 669 |
|
---|
| 670 | // do a sweep: for all index pairs (p,q), perform SVD of the corresponding 2x2 sub-matrix
|
---|
| 671 |
|
---|
| 672 | for(Index p = 1; p < this->m_diagSize; ++p)
|
---|
| 673 | {
|
---|
| 674 | for(Index q = 0; q < p; ++q)
|
---|
| 675 | {
|
---|
| 676 | // if this 2x2 sub-matrix is not diagonal already...
|
---|
| 677 | // notice that this comparison will evaluate to false if any NaN is involved, ensuring that NaN's don't
|
---|
| 678 | // keep us iterating forever. Similarly, small denormal numbers are considered zero.
|
---|
| 679 | using std::max;
|
---|
| 680 | RealScalar threshold = (max)(considerAsZero, precision * (max)(abs(m_workMatrix.coeff(p,p)),
|
---|
| 681 | abs(m_workMatrix.coeff(q,q))));
|
---|
| 682 | if((max)(abs(m_workMatrix.coeff(p,q)),abs(m_workMatrix.coeff(q,p))) > threshold)
|
---|
| 683 | {
|
---|
| 684 | finished = false;
|
---|
| 685 |
|
---|
| 686 | // perform SVD decomposition of 2x2 sub-matrix corresponding to indices p,q to make it diagonal
|
---|
| 687 | internal::svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner>::run(m_workMatrix, *this, p, q);
|
---|
| 688 | JacobiRotation<RealScalar> j_left, j_right;
|
---|
| 689 | internal::real_2x2_jacobi_svd(m_workMatrix, p, q, &j_left, &j_right);
|
---|
| 690 |
|
---|
| 691 | // accumulate resulting Jacobi rotations
|
---|
| 692 | m_workMatrix.applyOnTheLeft(p,q,j_left);
|
---|
| 693 | if(SVDBase<MatrixType>::computeU()) this->m_matrixU.applyOnTheRight(p,q,j_left.transpose());
|
---|
| 694 |
|
---|
| 695 | m_workMatrix.applyOnTheRight(p,q,j_right);
|
---|
| 696 | if(SVDBase<MatrixType>::computeV()) this->m_matrixV.applyOnTheRight(p,q,j_right);
|
---|
| 697 | }
|
---|
| 698 | }
|
---|
| 699 | }
|
---|
| 700 | }
|
---|
| 701 |
|
---|
| 702 | /*** step 3. The work matrix is now diagonal, so ensure it's positive so its diagonal entries are the singular values ***/
|
---|
| 703 |
|
---|
| 704 | for(Index i = 0; i < this->m_diagSize; ++i)
|
---|
| 705 | {
|
---|
| 706 | RealScalar a = abs(m_workMatrix.coeff(i,i));
|
---|
| 707 | this->m_singularValues.coeffRef(i) = a;
|
---|
| 708 | if(SVDBase<MatrixType>::computeU() && (a!=RealScalar(0))) this->m_matrixU.col(i) *= this->m_workMatrix.coeff(i,i)/a;
|
---|
| 709 | }
|
---|
| 710 |
|
---|
| 711 | /*** step 4. Sort singular values in descending order and compute the number of nonzero singular values ***/
|
---|
| 712 |
|
---|
| 713 | this->m_nonzeroSingularValues = this->m_diagSize;
|
---|
| 714 | for(Index i = 0; i < this->m_diagSize; i++)
|
---|
| 715 | {
|
---|
| 716 | Index pos;
|
---|
| 717 | RealScalar maxRemainingSingularValue = this->m_singularValues.tail(this->m_diagSize-i).maxCoeff(&pos);
|
---|
| 718 | if(maxRemainingSingularValue == RealScalar(0))
|
---|
| 719 | {
|
---|
| 720 | this->m_nonzeroSingularValues = i;
|
---|
| 721 | break;
|
---|
| 722 | }
|
---|
| 723 | if(pos)
|
---|
| 724 | {
|
---|
| 725 | pos += i;
|
---|
| 726 | std::swap(this->m_singularValues.coeffRef(i), this->m_singularValues.coeffRef(pos));
|
---|
| 727 | if(SVDBase<MatrixType>::computeU()) this->m_matrixU.col(pos).swap(this->m_matrixU.col(i));
|
---|
| 728 | if(SVDBase<MatrixType>::computeV()) this->m_matrixV.col(pos).swap(this->m_matrixV.col(i));
|
---|
| 729 | }
|
---|
| 730 | }
|
---|
| 731 |
|
---|
| 732 | this->m_isInitialized = true;
|
---|
| 733 | return *this;
|
---|
| 734 | }
|
---|
| 735 |
|
---|
| 736 | namespace internal {
|
---|
| 737 | template<typename _MatrixType, int QRPreconditioner, typename Rhs>
|
---|
| 738 | struct solve_retval<JacobiSVD<_MatrixType, QRPreconditioner>, Rhs>
|
---|
| 739 | : solve_retval_base<JacobiSVD<_MatrixType, QRPreconditioner>, Rhs>
|
---|
| 740 | {
|
---|
| 741 | typedef JacobiSVD<_MatrixType, QRPreconditioner> JacobiSVDType;
|
---|
| 742 | EIGEN_MAKE_SOLVE_HELPERS(JacobiSVDType,Rhs)
|
---|
| 743 |
|
---|
| 744 | template<typename Dest> void evalTo(Dest& dst) const
|
---|
| 745 | {
|
---|
| 746 | eigen_assert(rhs().rows() == dec().rows());
|
---|
| 747 |
|
---|
| 748 | // A = U S V^*
|
---|
| 749 | // So A^{-1} = V S^{-1} U^*
|
---|
| 750 |
|
---|
| 751 | Index diagSize = (std::min)(dec().rows(), dec().cols());
|
---|
| 752 | typename JacobiSVDType::SingularValuesType invertedSingVals(diagSize);
|
---|
| 753 |
|
---|
| 754 | Index nonzeroSingVals = dec().nonzeroSingularValues();
|
---|
| 755 | invertedSingVals.head(nonzeroSingVals) = dec().singularValues().head(nonzeroSingVals).array().inverse();
|
---|
| 756 | invertedSingVals.tail(diagSize - nonzeroSingVals).setZero();
|
---|
| 757 |
|
---|
| 758 | dst = dec().matrixV().leftCols(diagSize)
|
---|
| 759 | * invertedSingVals.asDiagonal()
|
---|
| 760 | * dec().matrixU().leftCols(diagSize).adjoint()
|
---|
| 761 | * rhs();
|
---|
| 762 | }
|
---|
| 763 | };
|
---|
| 764 | } // end namespace internal
|
---|
| 765 |
|
---|
| 766 | /** \svd_module
|
---|
| 767 | *
|
---|
| 768 | * \return the singular value decomposition of \c *this computed by two-sided
|
---|
| 769 | * Jacobi transformations.
|
---|
| 770 | *
|
---|
| 771 | * \sa class JacobiSVD
|
---|
| 772 | */
|
---|
| 773 | template<typename Derived>
|
---|
| 774 | JacobiSVD<typename MatrixBase<Derived>::PlainObject>
|
---|
| 775 | MatrixBase<Derived>::jacobiSvd(unsigned int computationOptions) const
|
---|
| 776 | {
|
---|
| 777 | return JacobiSVD<PlainObject>(*this, computationOptions);
|
---|
| 778 | }
|
---|
| 779 |
|
---|
| 780 | } // end namespace Eigen
|
---|
| 781 |
|
---|
| 782 | #endif // EIGEN_JACOBISVD_H
|
---|