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// This file is part of Eigen, a lightweight C++ template library
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// Copyright (C) 2009-2010 Benoit Jacob <jacob.benoit.1@gmail.com>
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// Eigen is free software; you can redistribute it and/or
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// modify it under the terms of the GNU Lesser General Public
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// License as published by the Free Software Foundation; either
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// version 3 of the License, or (at your option) any later version.
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// Alternatively, you can redistribute it and/or
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// modify it under the terms of the GNU General Public License as
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// published by the Free Software Foundation; either version 2 of
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// the License, or (at your option) any later version.
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// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
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// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
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// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
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// GNU General Public License for more details.
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// You should have received a copy of the GNU Lesser General Public
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// License and a copy of the GNU General Public License along with
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// Eigen. If not, see <http://www.gnu.org/licenses/>.
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#ifndef EIGEN_JACOBISVD_H
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#define EIGEN_JACOBISVD_H
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// forward declaration (needed by ICC)
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// the empty body is required by MSVC
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template<typename MatrixType, int QRPreconditioner,
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bool IsComplex = NumTraits<typename MatrixType::Scalar>::IsComplex>
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struct svd_precondition_2x2_block_to_be_real {};
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/*** QR preconditioners (R-SVD)
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*** Their role is to reduce the problem of computing the SVD to the case of a square matrix.
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*** This approach, known as R-SVD, is an optimization for rectangular-enough matrices, and is a requirement for
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*** JacobiSVD which by itself is only able to work on square matrices.
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enum { PreconditionIfMoreColsThanRows, PreconditionIfMoreRowsThanCols };
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template<typename MatrixType, int QRPreconditioner, int Case>
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struct qr_preconditioner_should_do_anything
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enum { a = MatrixType::RowsAtCompileTime != Dynamic &&
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MatrixType::ColsAtCompileTime != Dynamic &&
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MatrixType::ColsAtCompileTime <= MatrixType::RowsAtCompileTime,
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b = MatrixType::RowsAtCompileTime != Dynamic &&
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MatrixType::ColsAtCompileTime != Dynamic &&
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MatrixType::RowsAtCompileTime <= MatrixType::ColsAtCompileTime,
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ret = !( (QRPreconditioner == NoQRPreconditioner) ||
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(Case == PreconditionIfMoreColsThanRows && bool(a)) ||
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(Case == PreconditionIfMoreRowsThanCols && bool(b)) )
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template<typename MatrixType, int QRPreconditioner, int Case,
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bool DoAnything = qr_preconditioner_should_do_anything<MatrixType, QRPreconditioner, Case>::ret
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> struct qr_preconditioner_impl {};
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template<typename MatrixType, int QRPreconditioner, int Case>
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struct qr_preconditioner_impl<MatrixType, QRPreconditioner, Case, false>
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static bool run(JacobiSVD<MatrixType, QRPreconditioner>&, const MatrixType&)
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/*** preconditioner using FullPivHouseholderQR ***/
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template<typename MatrixType>
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struct qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
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static bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
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if(matrix.rows() > matrix.cols())
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FullPivHouseholderQR<MatrixType> qr(matrix);
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svd.m_workMatrix = qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
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if(svd.m_computeFullU) svd.m_matrixU = qr.matrixQ();
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if(svd.computeV()) svd.m_matrixV = qr.colsPermutation();
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template<typename MatrixType>
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struct qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
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static bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
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if(matrix.cols() > matrix.rows())
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typedef Matrix<typename MatrixType::Scalar, MatrixType::ColsAtCompileTime, MatrixType::RowsAtCompileTime,
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MatrixType::Options, MatrixType::MaxColsAtCompileTime, MatrixType::MaxRowsAtCompileTime>
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TransposeTypeWithSameStorageOrder;
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FullPivHouseholderQR<TransposeTypeWithSameStorageOrder> qr(matrix.adjoint());
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svd.m_workMatrix = qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
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if(svd.m_computeFullV) svd.m_matrixV = qr.matrixQ();
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if(svd.computeU()) svd.m_matrixU = qr.colsPermutation();
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/*** preconditioner using ColPivHouseholderQR ***/
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template<typename MatrixType>
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struct qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
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static bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
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if(matrix.rows() > matrix.cols())
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ColPivHouseholderQR<MatrixType> qr(matrix);
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svd.m_workMatrix = qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
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if(svd.m_computeFullU) svd.m_matrixU = qr.householderQ();
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else if(svd.m_computeThinU) {
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svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
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qr.householderQ().applyThisOnTheLeft(svd.m_matrixU);
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if(svd.computeV()) svd.m_matrixV = qr.colsPermutation();
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template<typename MatrixType>
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struct qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
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static bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
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if(matrix.cols() > matrix.rows())
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typedef Matrix<typename MatrixType::Scalar, MatrixType::ColsAtCompileTime, MatrixType::RowsAtCompileTime,
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MatrixType::Options, MatrixType::MaxColsAtCompileTime, MatrixType::MaxRowsAtCompileTime>
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TransposeTypeWithSameStorageOrder;
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ColPivHouseholderQR<TransposeTypeWithSameStorageOrder> qr(matrix.adjoint());
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svd.m_workMatrix = qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
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if(svd.m_computeFullV) svd.m_matrixV = qr.householderQ();
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else if(svd.m_computeThinV) {
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svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
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qr.householderQ().applyThisOnTheLeft(svd.m_matrixV);
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if(svd.computeU()) svd.m_matrixU = qr.colsPermutation();
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/*** preconditioner using HouseholderQR ***/
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template<typename MatrixType>
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struct qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
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static bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix)
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if(matrix.rows() > matrix.cols())
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HouseholderQR<MatrixType> qr(matrix);
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svd.m_workMatrix = qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
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if(svd.m_computeFullU) svd.m_matrixU = qr.householderQ();
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else if(svd.m_computeThinU) {
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svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
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qr.householderQ().applyThisOnTheLeft(svd.m_matrixU);
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if(svd.computeV()) svd.m_matrixV.setIdentity(matrix.cols(), matrix.cols());
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template<typename MatrixType>
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struct qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
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static bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix)
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if(matrix.cols() > matrix.rows())
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typedef Matrix<typename MatrixType::Scalar, MatrixType::ColsAtCompileTime, MatrixType::RowsAtCompileTime,
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MatrixType::Options, MatrixType::MaxColsAtCompileTime, MatrixType::MaxRowsAtCompileTime>
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TransposeTypeWithSameStorageOrder;
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HouseholderQR<TransposeTypeWithSameStorageOrder> qr(matrix.adjoint());
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svd.m_workMatrix = qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
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if(svd.m_computeFullV) svd.m_matrixV = qr.householderQ();
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else if(svd.m_computeThinV) {
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svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
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qr.householderQ().applyThisOnTheLeft(svd.m_matrixV);
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if(svd.computeU()) svd.m_matrixU.setIdentity(matrix.rows(), matrix.rows());
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/*** 2x2 SVD implementation
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*** JacobiSVD consists in performing a series of 2x2 SVD subproblems
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template<typename MatrixType, int QRPreconditioner>
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struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, false>
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typedef JacobiSVD<MatrixType, QRPreconditioner> SVD;
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typedef typename SVD::Index Index;
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static void run(typename SVD::WorkMatrixType&, SVD&, Index, Index) {}
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template<typename MatrixType, int QRPreconditioner>
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struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, true>
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typedef JacobiSVD<MatrixType, QRPreconditioner> SVD;
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typedef typename MatrixType::Scalar Scalar;
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typedef typename MatrixType::RealScalar RealScalar;
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typedef typename SVD::Index Index;
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static void run(typename SVD::WorkMatrixType& work_matrix, SVD& svd, Index p, Index q)
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JacobiRotation<Scalar> rot;
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RealScalar n = sqrt(abs2(work_matrix.coeff(p,p)) + abs2(work_matrix.coeff(q,p)));
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z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q);
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work_matrix.row(p) *= z;
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if(svd.computeU()) svd.m_matrixU.col(p) *= conj(z);
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z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q);
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work_matrix.row(q) *= z;
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if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z);
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rot.c() = conj(work_matrix.coeff(p,p)) / n;
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rot.s() = work_matrix.coeff(q,p) / n;
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work_matrix.applyOnTheLeft(p,q,rot);
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if(svd.computeU()) svd.m_matrixU.applyOnTheRight(p,q,rot.adjoint());
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if(work_matrix.coeff(p,q) != Scalar(0))
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Scalar z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q);
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work_matrix.col(q) *= z;
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if(svd.computeV()) svd.m_matrixV.col(q) *= z;
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if(work_matrix.coeff(q,q) != Scalar(0))
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z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q);
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work_matrix.row(q) *= z;
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if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z);
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template<typename MatrixType, typename RealScalar, typename Index>
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void real_2x2_jacobi_svd(const MatrixType& matrix, Index p, Index q,
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JacobiRotation<RealScalar> *j_left,
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JacobiRotation<RealScalar> *j_right)
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Matrix<RealScalar,2,2> m;
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m << real(matrix.coeff(p,p)), real(matrix.coeff(p,q)),
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real(matrix.coeff(q,p)), real(matrix.coeff(q,q));
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JacobiRotation<RealScalar> rot1;
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RealScalar t = m.coeff(0,0) + m.coeff(1,1);
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RealScalar d = m.coeff(1,0) - m.coeff(0,1);
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if(t == RealScalar(0))
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rot1.c() = RealScalar(0);
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rot1.s() = d > RealScalar(0) ? RealScalar(1) : RealScalar(-1);
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RealScalar u = d / t;
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rot1.c() = RealScalar(1) / sqrt(RealScalar(1) + abs2(u));
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rot1.s() = rot1.c() * u;
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m.applyOnTheLeft(0,1,rot1);
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j_right->makeJacobi(m,0,1);
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*j_left = rot1 * j_right->transpose();
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} // end namespace internal
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/** \ingroup SVD_Module
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* \brief Two-sided Jacobi SVD decomposition of a rectangular matrix
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* \param MatrixType the type of the matrix of which we are computing the SVD decomposition
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* \param QRPreconditioner this optional parameter allows to specify the type of QR decomposition that will be used internally
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* for the R-SVD step for non-square matrices. See discussion of possible values below.
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* SVD decomposition consists in decomposing any n-by-p matrix \a A as a product
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* \f[ A = U S V^* \f]
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* 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;
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* 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
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* and right \em singular \em vectors of \a A respectively.
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* Singular values are always sorted in decreasing order.
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* 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.
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* 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
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* 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
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* 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,
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* 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.
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* Here's an example demonstrating basic usage:
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* \include JacobiSVD_basic.cpp
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* Output: \verbinclude JacobiSVD_basic.out
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* This JacobiSVD class is a two-sided Jacobi R-SVD decomposition, ensuring optimal reliability and accuracy. The downside is that it's slower than
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* 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
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* \a p is the greater dimension, meaning that it is still of the same order of complexity as the faster bidiagonalizing R-SVD algorithms.
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* In particular, like any R-SVD, it takes advantage of non-squareness in that its complexity is only linear in the greater dimension.
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* If the input matrix has inf or nan coefficients, the result of the computation is undefined, but the computation is guaranteed to
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* terminate in finite (and reasonable) time.
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* The possible values for QRPreconditioner are:
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* \li ColPivHouseholderQRPreconditioner is the default. In practice it's very safe. It uses column-pivoting QR.
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* \li FullPivHouseholderQRPreconditioner, is the safest and slowest. It uses full-pivoting QR.
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* Contrary to other QRs, it doesn't allow computing thin unitaries.
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* \li HouseholderQRPreconditioner is the fastest, and less safe and accurate than the pivoting variants. It uses non-pivoting QR.
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* This is very similar in safety and accuracy to the bidiagonalization process used by bidiagonalizing SVD algorithms (since bidiagonalization
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* is inherently non-pivoting). However the resulting SVD is still more reliable than bidiagonalizing SVDs because the Jacobi-based iterarive
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* process is more reliable than the optimized bidiagonal SVD iterations.
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* \li NoQRPreconditioner allows not to use a QR preconditioner at all. This is useful if you know that you will only be computing
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* JacobiSVD decompositions of square matrices. Non-square matrices require a QR preconditioner. Using this option will result in
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* faster compilation and smaller executable code. It won't significantly speed up computation, since JacobiSVD is always checking
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* if QR preconditioning is needed before applying it anyway.
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* \sa MatrixBase::jacobiSvd()
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template<typename _MatrixType, int QRPreconditioner> class JacobiSVD
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typedef _MatrixType MatrixType;
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typedef typename MatrixType::Scalar Scalar;
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typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
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typedef typename MatrixType::Index Index;
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RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime,ColsAtCompileTime),
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MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
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MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime,MaxColsAtCompileTime),
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MatrixOptions = MatrixType::Options
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typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime,
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MatrixOptions, MaxRowsAtCompileTime, MaxRowsAtCompileTime>
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typedef Matrix<Scalar, ColsAtCompileTime, ColsAtCompileTime,
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MatrixOptions, MaxColsAtCompileTime, MaxColsAtCompileTime>
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typedef typename internal::plain_diag_type<MatrixType, RealScalar>::type SingularValuesType;
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typedef typename internal::plain_row_type<MatrixType>::type RowType;
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typedef typename internal::plain_col_type<MatrixType>::type ColType;
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typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime,
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MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime>
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/** \brief Default Constructor.
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* The default constructor is useful in cases in which the user intends to
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* perform decompositions via JacobiSVD::compute(const MatrixType&).
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: m_isInitialized(false),
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m_isAllocated(false),
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m_computationOptions(0),
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m_rows(-1), m_cols(-1)
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/** \brief Default Constructor with memory preallocation
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* Like the default constructor but with preallocation of the internal data
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* according to the specified problem size.
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JacobiSVD(Index rows, Index cols, unsigned int computationOptions = 0)
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: m_isInitialized(false),
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m_isAllocated(false),
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m_computationOptions(0),
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m_rows(-1), m_cols(-1)
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allocate(rows, cols, computationOptions);
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/** \brief Constructor performing the decomposition of given matrix.
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* \param matrix the matrix to decompose
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* \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
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* By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU,
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* #ComputeFullV, #ComputeThinV.
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* 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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* available with the (non-default) FullPivHouseholderQR preconditioner.
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JacobiSVD(const MatrixType& matrix, unsigned int computationOptions = 0)
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: m_isInitialized(false),
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m_isAllocated(false),
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m_computationOptions(0),
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m_rows(-1), m_cols(-1)
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compute(matrix, computationOptions);
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/** \brief Method performing the decomposition of given matrix using custom options.
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* \param matrix the matrix to decompose
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* \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
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* By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU,
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* #ComputeFullV, #ComputeThinV.
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* 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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* available with the (non-default) FullPivHouseholderQR preconditioner.
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JacobiSVD& compute(const MatrixType& matrix, unsigned int computationOptions);
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/** \brief Method performing the decomposition of given matrix using current options.
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* \param matrix the matrix to decompose
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* This method uses the current \a computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int).
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JacobiSVD& compute(const MatrixType& matrix)
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return compute(matrix, m_computationOptions);
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/** \returns the \a U matrix.
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* For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p,
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* the U matrix is n-by-n if you asked for #ComputeFullU, and is n-by-m if you asked for #ComputeThinU.
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* The \a m first columns of \a U are the left singular vectors of the matrix being decomposed.
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* This method asserts that you asked for \a U to be computed.
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const MatrixUType& matrixU() const
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eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
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eigen_assert(computeU() && "This JacobiSVD decomposition didn't compute U. Did you ask for it?");
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/** \returns the \a V matrix.
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* For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p,
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* the V matrix is p-by-p if you asked for #ComputeFullV, and is p-by-m if you asked for ComputeThinV.
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* The \a m first columns of \a V are the right singular vectors of the matrix being decomposed.
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* This method asserts that you asked for \a V to be computed.
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const MatrixVType& matrixV() const
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eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
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eigen_assert(computeV() && "This JacobiSVD decomposition didn't compute V. Did you ask for it?");
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/** \returns the vector of singular values.
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* For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p, the
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* returned vector has size \a m. Singular values are always sorted in decreasing order.
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const SingularValuesType& singularValues() const
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eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
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return m_singularValues;
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/** \returns true if \a U (full or thin) is asked for in this SVD decomposition */
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inline bool computeU() const { return m_computeFullU || m_computeThinU; }
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/** \returns true if \a V (full or thin) is asked for in this SVD decomposition */
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inline bool computeV() const { return m_computeFullV || m_computeThinV; }
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/** \returns a (least squares) solution of \f$ A x = b \f$ using the current SVD decomposition of A.
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* \param b the right-hand-side of the equation to solve.
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* \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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* \note SVD solving is implicitly least-squares. Thus, this method serves both purposes of exact solving and least-squares solving.
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* In other words, the returned solution is guaranteed to minimize the Euclidean norm \f$ \Vert A x - b \Vert \f$.
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template<typename Rhs>
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inline const internal::solve_retval<JacobiSVD, Rhs>
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solve(const MatrixBase<Rhs>& b) const
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eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
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eigen_assert(computeU() && computeV() && "JacobiSVD::solve() requires both unitaries U and V to be computed (thin unitaries suffice).");
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return internal::solve_retval<JacobiSVD, Rhs>(*this, b.derived());
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/** \returns the number of singular values that are not exactly 0 */
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Index nonzeroSingularValues() const
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eigen_assert(m_isInitialized && "JacobiSVD is not initialized.");
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return m_nonzeroSingularValues;
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inline Index rows() const { return m_rows; }
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inline Index cols() const { return m_cols; }
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void allocate(Index rows, Index cols, unsigned int computationOptions);
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MatrixUType m_matrixU;
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MatrixVType m_matrixV;
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SingularValuesType m_singularValues;
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WorkMatrixType m_workMatrix;
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bool m_isInitialized, m_isAllocated;
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bool m_computeFullU, m_computeThinU;
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bool m_computeFullV, m_computeThinV;
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unsigned int m_computationOptions;
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Index m_nonzeroSingularValues, m_rows, m_cols, m_diagSize;
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template<typename __MatrixType, int _QRPreconditioner, bool _IsComplex>
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friend struct internal::svd_precondition_2x2_block_to_be_real;
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template<typename __MatrixType, int _QRPreconditioner, int _Case, bool _DoAnything>
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friend struct internal::qr_preconditioner_impl;
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template<typename MatrixType, int QRPreconditioner>
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void JacobiSVD<MatrixType, QRPreconditioner>::allocate(Index rows, Index cols, unsigned int computationOptions)
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eigen_assert(rows >= 0 && cols >= 0);
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computationOptions == m_computationOptions)
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m_isInitialized = false;
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m_isAllocated = true;
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m_computationOptions = computationOptions;
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m_computeFullU = (computationOptions & ComputeFullU) != 0;
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m_computeThinU = (computationOptions & ComputeThinU) != 0;
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m_computeFullV = (computationOptions & ComputeFullV) != 0;
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m_computeThinV = (computationOptions & ComputeThinV) != 0;
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eigen_assert(!(m_computeFullU && m_computeThinU) && "JacobiSVD: you can't ask for both full and thin U");
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eigen_assert(!(m_computeFullV && m_computeThinV) && "JacobiSVD: you can't ask for both full and thin V");
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eigen_assert(EIGEN_IMPLIES(m_computeThinU || m_computeThinV, MatrixType::ColsAtCompileTime==Dynamic) &&
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"JacobiSVD: thin U and V are only available when your matrix has a dynamic number of columns.");
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if (QRPreconditioner == FullPivHouseholderQRPreconditioner)
568
eigen_assert(!(m_computeThinU || m_computeThinV) &&
569
"JacobiSVD: can't compute thin U or thin V with the FullPivHouseholderQR preconditioner. "
570
"Use the ColPivHouseholderQR preconditioner instead.");
572
m_diagSize = std::min(m_rows, m_cols);
573
m_singularValues.resize(m_diagSize);
574
m_matrixU.resize(m_rows, m_computeFullU ? m_rows
575
: m_computeThinU ? m_diagSize
577
m_matrixV.resize(m_cols, m_computeFullV ? m_cols
578
: m_computeThinV ? m_diagSize
580
m_workMatrix.resize(m_diagSize, m_diagSize);
583
template<typename MatrixType, int QRPreconditioner>
584
JacobiSVD<MatrixType, QRPreconditioner>&
585
JacobiSVD<MatrixType, QRPreconditioner>::compute(const MatrixType& matrix, unsigned int computationOptions)
587
allocate(matrix.rows(), matrix.cols(), computationOptions);
589
// currently we stop when we reach precision 2*epsilon as the last bit of precision can require an unreasonable number of iterations,
590
// only worsening the precision of U and V as we accumulate more rotations
591
const RealScalar precision = RealScalar(2) * NumTraits<Scalar>::epsilon();
593
/*** step 1. The R-SVD step: we use a QR decomposition to reduce to the case of a square matrix */
595
if(!internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows>::run(*this, matrix)
596
&& !internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols>::run(*this, matrix))
598
m_workMatrix = matrix.block(0,0,m_diagSize,m_diagSize);
599
if(m_computeFullU) m_matrixU.setIdentity(m_rows,m_rows);
600
if(m_computeThinU) m_matrixU.setIdentity(m_rows,m_diagSize);
601
if(m_computeFullV) m_matrixV.setIdentity(m_cols,m_cols);
602
if(m_computeThinV) m_matrixV.setIdentity(m_cols, m_diagSize);
605
/*** step 2. The main Jacobi SVD iteration. ***/
607
bool finished = false;
612
// do a sweep: for all index pairs (p,q), perform SVD of the corresponding 2x2 sub-matrix
614
for(Index p = 1; p < m_diagSize; ++p)
616
for(Index q = 0; q < p; ++q)
618
// if this 2x2 sub-matrix is not diagonal already...
619
// notice that this comparison will evaluate to false if any NaN is involved, ensuring that NaN's don't
620
// keep us iterating forever.
622
if(max(internal::abs(m_workMatrix.coeff(p,q)),internal::abs(m_workMatrix.coeff(q,p)))
623
> max(internal::abs(m_workMatrix.coeff(p,p)),internal::abs(m_workMatrix.coeff(q,q)))*precision)
627
// perform SVD decomposition of 2x2 sub-matrix corresponding to indices p,q to make it diagonal
628
internal::svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner>::run(m_workMatrix, *this, p, q);
629
JacobiRotation<RealScalar> j_left, j_right;
630
internal::real_2x2_jacobi_svd(m_workMatrix, p, q, &j_left, &j_right);
632
// accumulate resulting Jacobi rotations
633
m_workMatrix.applyOnTheLeft(p,q,j_left);
634
if(computeU()) m_matrixU.applyOnTheRight(p,q,j_left.transpose());
636
m_workMatrix.applyOnTheRight(p,q,j_right);
637
if(computeV()) m_matrixV.applyOnTheRight(p,q,j_right);
643
/*** step 3. The work matrix is now diagonal, so ensure it's positive so its diagonal entries are the singular values ***/
645
for(Index i = 0; i < m_diagSize; ++i)
647
RealScalar a = internal::abs(m_workMatrix.coeff(i,i));
648
m_singularValues.coeffRef(i) = a;
649
if(computeU() && (a!=RealScalar(0))) m_matrixU.col(i) *= m_workMatrix.coeff(i,i)/a;
652
/*** step 4. Sort singular values in descending order and compute the number of nonzero singular values ***/
654
m_nonzeroSingularValues = m_diagSize;
655
for(Index i = 0; i < m_diagSize; i++)
658
RealScalar maxRemainingSingularValue = m_singularValues.tail(m_diagSize-i).maxCoeff(&pos);
659
if(maxRemainingSingularValue == RealScalar(0))
661
m_nonzeroSingularValues = i;
667
std::swap(m_singularValues.coeffRef(i), m_singularValues.coeffRef(pos));
668
if(computeU()) m_matrixU.col(pos).swap(m_matrixU.col(i));
669
if(computeV()) m_matrixV.col(pos).swap(m_matrixV.col(i));
673
m_isInitialized = true;
678
template<typename _MatrixType, int QRPreconditioner, typename Rhs>
679
struct solve_retval<JacobiSVD<_MatrixType, QRPreconditioner>, Rhs>
680
: solve_retval_base<JacobiSVD<_MatrixType, QRPreconditioner>, Rhs>
682
typedef JacobiSVD<_MatrixType, QRPreconditioner> JacobiSVDType;
683
EIGEN_MAKE_SOLVE_HELPERS(JacobiSVDType,Rhs)
685
template<typename Dest> void evalTo(Dest& dst) const
687
eigen_assert(rhs().rows() == dec().rows());
690
// So A^{-1} = V S^{-1} U^*
692
Index diagSize = std::min(dec().rows(), dec().cols());
693
typename JacobiSVDType::SingularValuesType invertedSingVals(diagSize);
695
Index nonzeroSingVals = dec().nonzeroSingularValues();
696
invertedSingVals.head(nonzeroSingVals) = dec().singularValues().head(nonzeroSingVals).array().inverse();
697
invertedSingVals.tail(diagSize - nonzeroSingVals).setZero();
699
dst = dec().matrixV().leftCols(diagSize)
700
* invertedSingVals.asDiagonal()
701
* dec().matrixU().leftCols(diagSize).adjoint()
705
} // end namespace internal
707
template<typename Derived>
708
JacobiSVD<typename MatrixBase<Derived>::PlainObject>
709
MatrixBase<Derived>::jacobiSvd(unsigned int computationOptions) const
711
return JacobiSVD<PlainObject>(*this, computationOptions);
716
#endif // EIGEN_JACOBISVD_H
added
removed
thirdparty/eigen3/Eigen/src/SVD/JacobiSVD.h
