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diff --git a/Eigen/src/SVD/JacobiSVD.h b/Eigen/src/SVD/JacobiSVD.h new file mode 100644 index 000000000..a7dbf0737 --- /dev/null +++ b/Eigen/src/SVD/JacobiSVD.h @@ -0,0 +1,867 @@ +// This file is part of Eigen, a lightweight C++ template library +// for linear algebra. +// +// Copyright (C) 2009-2010 Benoit Jacob <jacob.benoit.1@gmail.com> +// +// This Source Code Form is subject to the terms of the Mozilla +// Public License v. 2.0. If a copy of the MPL was not distributed +// with this file, You can obtain one at http://mozilla.org/MPL/2.0/. + +#ifndef EIGEN_JACOBISVD_H +#define EIGEN_JACOBISVD_H + +namespace Eigen { + +namespace internal { +// forward declaration (needed by ICC) +// the empty body is required by MSVC +template<typename MatrixType, int QRPreconditioner, + bool IsComplex = NumTraits<typename MatrixType::Scalar>::IsComplex> +struct svd_precondition_2x2_block_to_be_real {}; + +/*** QR preconditioners (R-SVD) + *** + *** Their role is to reduce the problem of computing the SVD to the case of a square matrix. + *** This approach, known as R-SVD, is an optimization for rectangular-enough matrices, and is a requirement for + *** JacobiSVD which by itself is only able to work on square matrices. + ***/ + +enum { PreconditionIfMoreColsThanRows, PreconditionIfMoreRowsThanCols }; + +template<typename MatrixType, int QRPreconditioner, int Case> +struct qr_preconditioner_should_do_anything +{ + enum { a = MatrixType::RowsAtCompileTime != Dynamic && + MatrixType::ColsAtCompileTime != Dynamic && + MatrixType::ColsAtCompileTime <= MatrixType::RowsAtCompileTime, + b = MatrixType::RowsAtCompileTime != Dynamic && + MatrixType::ColsAtCompileTime != Dynamic && + MatrixType::RowsAtCompileTime <= MatrixType::ColsAtCompileTime, + ret = !( (QRPreconditioner == NoQRPreconditioner) || + (Case == PreconditionIfMoreColsThanRows && bool(a)) || + (Case == PreconditionIfMoreRowsThanCols && bool(b)) ) + }; +}; + +template<typename MatrixType, int QRPreconditioner, int Case, + bool DoAnything = qr_preconditioner_should_do_anything<MatrixType, QRPreconditioner, Case>::ret +> struct qr_preconditioner_impl {}; + +template<typename MatrixType, int QRPreconditioner, int Case> +class qr_preconditioner_impl<MatrixType, QRPreconditioner, Case, false> +{ +public: + typedef typename MatrixType::Index Index; + void allocate(const JacobiSVD<MatrixType, QRPreconditioner>&) {} + bool run(JacobiSVD<MatrixType, QRPreconditioner>&, const MatrixType&) + { + return false; + } +}; + +/*** preconditioner using FullPivHouseholderQR ***/ + +template<typename MatrixType> +class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true> +{ +public: + typedef typename MatrixType::Index Index; + typedef typename MatrixType::Scalar Scalar; + enum + { + RowsAtCompileTime = MatrixType::RowsAtCompileTime, + MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime + }; + typedef Matrix<Scalar, 1, RowsAtCompileTime, RowMajor, 1, MaxRowsAtCompileTime> WorkspaceType; + + void allocate(const JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd) + { + if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols()) + { + m_qr = FullPivHouseholderQR<MatrixType>(svd.rows(), svd.cols()); + } + if (svd.m_computeFullU) m_workspace.resize(svd.rows()); + } + + bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix) + { + if(matrix.rows() > matrix.cols()) + { + m_qr.compute(matrix); + svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>(); + if(svd.m_computeFullU) m_qr.matrixQ().evalTo(svd.m_matrixU, m_workspace); + if(svd.computeV()) svd.m_matrixV = m_qr.colsPermutation(); + return true; + } + return false; + } +private: + FullPivHouseholderQR<MatrixType> m_qr; + WorkspaceType m_workspace; +}; + +template<typename MatrixType> +class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true> +{ +public: + typedef typename MatrixType::Index Index; + typedef typename MatrixType::Scalar Scalar; + enum + { + RowsAtCompileTime = MatrixType::RowsAtCompileTime, + ColsAtCompileTime = MatrixType::ColsAtCompileTime, + MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, + MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime, + Options = MatrixType::Options + }; + typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime> + TransposeTypeWithSameStorageOrder; + + void allocate(const JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd) + { + if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols()) + { + m_qr = FullPivHouseholderQR<TransposeTypeWithSameStorageOrder>(svd.cols(), svd.rows()); + } + m_adjoint.resize(svd.cols(), svd.rows()); + if (svd.m_computeFullV) m_workspace.resize(svd.cols()); + } + + bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix) + { + if(matrix.cols() > matrix.rows()) + { + m_adjoint = matrix.adjoint(); + m_qr.compute(m_adjoint); + svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint(); + if(svd.m_computeFullV) m_qr.matrixQ().evalTo(svd.m_matrixV, m_workspace); + if(svd.computeU()) svd.m_matrixU = m_qr.colsPermutation(); + return true; + } + else return false; + } +private: + FullPivHouseholderQR<TransposeTypeWithSameStorageOrder> m_qr; + TransposeTypeWithSameStorageOrder m_adjoint; + typename internal::plain_row_type<MatrixType>::type m_workspace; +}; + +/*** preconditioner using ColPivHouseholderQR ***/ + +template<typename MatrixType> +class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true> +{ +public: + typedef typename MatrixType::Index Index; + + void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd) + { + if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols()) + { + m_qr = ColPivHouseholderQR<MatrixType>(svd.rows(), svd.cols()); + } + if (svd.m_computeFullU) m_workspace.resize(svd.rows()); + else if (svd.m_computeThinU) m_workspace.resize(svd.cols()); + } + + bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix) + { + if(matrix.rows() > matrix.cols()) + { + m_qr.compute(matrix); + svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>(); + if(svd.m_computeFullU) m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace); + else if(svd.m_computeThinU) + { + svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols()); + m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace); + } + if(svd.computeV()) svd.m_matrixV = m_qr.colsPermutation(); + return true; + } + return false; + } + +private: + ColPivHouseholderQR<MatrixType> m_qr; + typename internal::plain_col_type<MatrixType>::type m_workspace; +}; + +template<typename MatrixType> +class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true> +{ +public: + typedef typename MatrixType::Index Index; + typedef typename MatrixType::Scalar Scalar; + enum + { + RowsAtCompileTime = MatrixType::RowsAtCompileTime, + ColsAtCompileTime = MatrixType::ColsAtCompileTime, + MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, + MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime, + Options = MatrixType::Options + }; + + typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime> + TransposeTypeWithSameStorageOrder; + + void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd) + { + if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols()) + { + m_qr = ColPivHouseholderQR<TransposeTypeWithSameStorageOrder>(svd.cols(), svd.rows()); + } + if (svd.m_computeFullV) m_workspace.resize(svd.cols()); + else if (svd.m_computeThinV) m_workspace.resize(svd.rows()); + m_adjoint.resize(svd.cols(), svd.rows()); + } + + bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix) + { + if(matrix.cols() > matrix.rows()) + { + m_adjoint = matrix.adjoint(); + m_qr.compute(m_adjoint); + + svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint(); + if(svd.m_computeFullV) m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace); + else if(svd.m_computeThinV) + { + svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows()); + m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace); + } + if(svd.computeU()) svd.m_matrixU = m_qr.colsPermutation(); + return true; + } + else return false; + } + +private: + ColPivHouseholderQR<TransposeTypeWithSameStorageOrder> m_qr; + TransposeTypeWithSameStorageOrder m_adjoint; + typename internal::plain_row_type<MatrixType>::type m_workspace; +}; + +/*** preconditioner using HouseholderQR ***/ + +template<typename MatrixType> +class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true> +{ +public: + typedef typename MatrixType::Index Index; + + void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd) + { + if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols()) + { + m_qr = HouseholderQR<MatrixType>(svd.rows(), svd.cols()); + } + if (svd.m_computeFullU) m_workspace.resize(svd.rows()); + else if (svd.m_computeThinU) m_workspace.resize(svd.cols()); + } + + bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix) + { + if(matrix.rows() > matrix.cols()) + { + m_qr.compute(matrix); + svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>(); + if(svd.m_computeFullU) m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace); + else if(svd.m_computeThinU) + { + svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols()); + m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace); + } + if(svd.computeV()) svd.m_matrixV.setIdentity(matrix.cols(), matrix.cols()); + return true; + } + return false; + } +private: + HouseholderQR<MatrixType> m_qr; + typename internal::plain_col_type<MatrixType>::type m_workspace; +}; + +template<typename MatrixType> +class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true> +{ +public: + typedef typename MatrixType::Index Index; + typedef typename MatrixType::Scalar Scalar; + enum + { + RowsAtCompileTime = MatrixType::RowsAtCompileTime, + ColsAtCompileTime = MatrixType::ColsAtCompileTime, + MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, + MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime, + Options = MatrixType::Options + }; + + typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime> + TransposeTypeWithSameStorageOrder; + + void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd) + { + if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols()) + { + m_qr = HouseholderQR<TransposeTypeWithSameStorageOrder>(svd.cols(), svd.rows()); + } + if (svd.m_computeFullV) m_workspace.resize(svd.cols()); + else if (svd.m_computeThinV) m_workspace.resize(svd.rows()); + m_adjoint.resize(svd.cols(), svd.rows()); + } + + bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix) + { + if(matrix.cols() > matrix.rows()) + { + m_adjoint = matrix.adjoint(); + m_qr.compute(m_adjoint); + + svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint(); + if(svd.m_computeFullV) m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace); + else if(svd.m_computeThinV) + { + svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows()); + m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace); + } + if(svd.computeU()) svd.m_matrixU.setIdentity(matrix.rows(), matrix.rows()); + return true; + } + else return false; + } + +private: + HouseholderQR<TransposeTypeWithSameStorageOrder> m_qr; + TransposeTypeWithSameStorageOrder m_adjoint; + typename internal::plain_row_type<MatrixType>::type m_workspace; +}; + +/*** 2x2 SVD implementation + *** + *** JacobiSVD consists in performing a series of 2x2 SVD subproblems + ***/ + +template<typename MatrixType, int QRPreconditioner> +struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, false> +{ + typedef JacobiSVD<MatrixType, QRPreconditioner> SVD; + typedef typename SVD::Index Index; + static void run(typename SVD::WorkMatrixType&, SVD&, Index, Index) {} +}; + +template<typename MatrixType, int QRPreconditioner> +struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, true> +{ + typedef JacobiSVD<MatrixType, QRPreconditioner> SVD; + typedef typename MatrixType::Scalar Scalar; + typedef typename MatrixType::RealScalar RealScalar; + typedef typename SVD::Index Index; + static void run(typename SVD::WorkMatrixType& work_matrix, SVD& svd, Index p, Index q) + { + Scalar z; + JacobiRotation<Scalar> rot; + RealScalar n = sqrt(abs2(work_matrix.coeff(p,p)) + abs2(work_matrix.coeff(q,p))); + if(n==0) + { + z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q); + work_matrix.row(p) *= z; + if(svd.computeU()) svd.m_matrixU.col(p) *= conj(z); + z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q); + work_matrix.row(q) *= z; + if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z); + } + else + { + rot.c() = conj(work_matrix.coeff(p,p)) / n; + rot.s() = work_matrix.coeff(q,p) / n; + work_matrix.applyOnTheLeft(p,q,rot); + if(svd.computeU()) svd.m_matrixU.applyOnTheRight(p,q,rot.adjoint()); + if(work_matrix.coeff(p,q) != Scalar(0)) + { + Scalar z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q); + work_matrix.col(q) *= z; + if(svd.computeV()) svd.m_matrixV.col(q) *= z; + } + if(work_matrix.coeff(q,q) != Scalar(0)) + { + z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q); + work_matrix.row(q) *= z; + if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z); + } + } + } +}; + +template<typename MatrixType, typename RealScalar, typename Index> +void real_2x2_jacobi_svd(const MatrixType& matrix, Index p, Index q, + JacobiRotation<RealScalar> *j_left, + JacobiRotation<RealScalar> *j_right) +{ + Matrix<RealScalar,2,2> m; + m << real(matrix.coeff(p,p)), real(matrix.coeff(p,q)), + real(matrix.coeff(q,p)), real(matrix.coeff(q,q)); + JacobiRotation<RealScalar> rot1; + RealScalar t = m.coeff(0,0) + m.coeff(1,1); + RealScalar d = m.coeff(1,0) - m.coeff(0,1); + if(t == RealScalar(0)) + { + rot1.c() = RealScalar(0); + rot1.s() = d > RealScalar(0) ? RealScalar(1) : RealScalar(-1); + } + else + { + RealScalar u = d / t; + rot1.c() = RealScalar(1) / sqrt(RealScalar(1) + abs2(u)); + rot1.s() = rot1.c() * u; + } + m.applyOnTheLeft(0,1,rot1); + j_right->makeJacobi(m,0,1); + *j_left = rot1 * j_right->transpose(); +} + +} // end namespace internal + +/** \ingroup SVD_Module + * + * + * \class JacobiSVD + * + * \brief Two-sided Jacobi SVD decomposition of a rectangular matrix + * + * \param MatrixType the type of the matrix of which we are computing the SVD decomposition + * \param QRPreconditioner this optional parameter allows to specify the type of QR decomposition that will be used internally + * for the R-SVD step for non-square matrices. See discussion of possible values below. + * + * SVD decomposition consists in decomposing any n-by-p matrix \a A as a product + * \f[ A = U S V^* \f] + * 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; + * 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 + * and right \em singular \em vectors of \a A respectively. + * + * Singular values are always sorted in decreasing order. + * + * 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. + * + * 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 + * 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 + * 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, + * 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. + * + * Here's an example demonstrating basic usage: + * \include JacobiSVD_basic.cpp + * Output: \verbinclude JacobiSVD_basic.out + * + * This JacobiSVD class is a two-sided Jacobi R-SVD decomposition, ensuring optimal reliability and accuracy. The downside is that it's slower than + * 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 + * \a p is the greater dimension, meaning that it is still of the same order of complexity as the faster bidiagonalizing R-SVD algorithms. + * In particular, like any R-SVD, it takes advantage of non-squareness in that its complexity is only linear in the greater dimension. + * + * If the input matrix has inf or nan coefficients, the result of the computation is undefined, but the computation is guaranteed to + * terminate in finite (and reasonable) time. + * + * The possible values for QRPreconditioner are: + * \li ColPivHouseholderQRPreconditioner is the default. In practice it's very safe. It uses column-pivoting QR. + * \li FullPivHouseholderQRPreconditioner, is the safest and slowest. It uses full-pivoting QR. + * Contrary to other QRs, it doesn't allow computing thin unitaries. + * \li HouseholderQRPreconditioner is the fastest, and less safe and accurate than the pivoting variants. It uses non-pivoting QR. + * This is very similar in safety and accuracy to the bidiagonalization process used by bidiagonalizing SVD algorithms (since bidiagonalization + * is inherently non-pivoting). However the resulting SVD is still more reliable than bidiagonalizing SVDs because the Jacobi-based iterarive + * process is more reliable than the optimized bidiagonal SVD iterations. + * \li NoQRPreconditioner allows not to use a QR preconditioner at all. This is useful if you know that you will only be computing + * JacobiSVD decompositions of square matrices. Non-square matrices require a QR preconditioner. Using this option will result in + * faster compilation and smaller executable code. It won't significantly speed up computation, since JacobiSVD is always checking + * if QR preconditioning is needed before applying it anyway. + * + * \sa MatrixBase::jacobiSvd() + */ +template<typename _MatrixType, int QRPreconditioner> class JacobiSVD +{ + public: + + typedef _MatrixType MatrixType; + typedef typename MatrixType::Scalar Scalar; + typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar; + typedef typename MatrixType::Index Index; + enum { + RowsAtCompileTime = MatrixType::RowsAtCompileTime, + ColsAtCompileTime = MatrixType::ColsAtCompileTime, + DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime,ColsAtCompileTime), + MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, + MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime, + MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime,MaxColsAtCompileTime), + MatrixOptions = MatrixType::Options + }; + + typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, + MatrixOptions, MaxRowsAtCompileTime, MaxRowsAtCompileTime> + MatrixUType; + typedef Matrix<Scalar, ColsAtCompileTime, ColsAtCompileTime, + MatrixOptions, MaxColsAtCompileTime, MaxColsAtCompileTime> + MatrixVType; + typedef typename internal::plain_diag_type<MatrixType, RealScalar>::type SingularValuesType; + typedef typename internal::plain_row_type<MatrixType>::type RowType; + typedef typename internal::plain_col_type<MatrixType>::type ColType; + typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime, + MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime> + WorkMatrixType; + + /** \brief Default Constructor. + * + * The default constructor is useful in cases in which the user intends to + * perform decompositions via JacobiSVD::compute(const MatrixType&). + */ + JacobiSVD() + : m_isInitialized(false), + m_isAllocated(false), + m_computationOptions(0), + m_rows(-1), m_cols(-1) + {} + + + /** \brief Default Constructor with memory preallocation + * + * Like the default constructor but with preallocation of the internal data + * according to the specified problem size. + * \sa JacobiSVD() + */ + JacobiSVD(Index rows, Index cols, unsigned int computationOptions = 0) + : m_isInitialized(false), + m_isAllocated(false), + m_computationOptions(0), + m_rows(-1), m_cols(-1) + { + allocate(rows, cols, computationOptions); + } + + /** \brief Constructor performing the decomposition of given matrix. + * + * \param matrix the matrix to decompose + * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed. + * By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU, + * #ComputeFullV, #ComputeThinV. + * + * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not + * available with the (non-default) FullPivHouseholderQR preconditioner. + */ + JacobiSVD(const MatrixType& matrix, unsigned int computationOptions = 0) + : m_isInitialized(false), + m_isAllocated(false), + m_computationOptions(0), + m_rows(-1), m_cols(-1) + { + compute(matrix, computationOptions); + } + + /** \brief Method performing the decomposition of given matrix using custom options. + * + * \param matrix the matrix to decompose + * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed. + * By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU, + * #ComputeFullV, #ComputeThinV. + * + * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not + * available with the (non-default) FullPivHouseholderQR preconditioner. + */ + JacobiSVD& compute(const MatrixType& matrix, unsigned int computationOptions); + + /** \brief Method performing the decomposition of given matrix using current options. + * + * \param matrix the matrix to decompose + * + * This method uses the current \a computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int). + */ + JacobiSVD& compute(const MatrixType& matrix) + { + return compute(matrix, m_computationOptions); + } + + /** \returns the \a U matrix. + * + * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p, + * the U matrix is n-by-n if you asked for #ComputeFullU, and is n-by-m if you asked for #ComputeThinU. + * + * The \a m first columns of \a U are the left singular vectors of the matrix being decomposed. + * + * This method asserts that you asked for \a U to be computed. + */ + const MatrixUType& matrixU() const + { + eigen_assert(m_isInitialized && "JacobiSVD is not initialized."); + eigen_assert(computeU() && "This JacobiSVD decomposition didn't compute U. Did you ask for it?"); + return m_matrixU; + } + + /** \returns the \a V matrix. + * + * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p, + * the V matrix is p-by-p if you asked for #ComputeFullV, and is p-by-m if you asked for ComputeThinV. + * + * The \a m first columns of \a V are the right singular vectors of the matrix being decomposed. + * + * This method asserts that you asked for \a V to be computed. + */ + const MatrixVType& matrixV() const + { + eigen_assert(m_isInitialized && "JacobiSVD is not initialized."); + eigen_assert(computeV() && "This JacobiSVD decomposition didn't compute V. Did you ask for it?"); + return m_matrixV; + } + + /** \returns the vector of singular values. + * + * For the SVD decomposition of a n-by-p matrix, letting \a m be the minimum of \a n and \a p, the + * returned vector has size \a m. Singular values are always sorted in decreasing order. + */ + const SingularValuesType& singularValues() const + { + eigen_assert(m_isInitialized && "JacobiSVD is not initialized."); + return m_singularValues; + } + + /** \returns true if \a U (full or thin) is asked for in this SVD decomposition */ + inline bool computeU() const { return m_computeFullU || m_computeThinU; } + /** \returns true if \a V (full or thin) is asked for in this SVD decomposition */ + inline bool computeV() const { return m_computeFullV || m_computeThinV; } + + /** \returns a (least squares) solution of \f$ A x = b \f$ using the current SVD decomposition of A. + * + * \param b the right-hand-side of the equation to solve. + * + * \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. + * + * \note SVD solving is implicitly least-squares. Thus, this method serves both purposes of exact solving and least-squares solving. + * In other words, the returned solution is guaranteed to minimize the Euclidean norm \f$ \Vert A x - b \Vert \f$. + */ + template<typename Rhs> + inline const internal::solve_retval<JacobiSVD, Rhs> + solve(const MatrixBase<Rhs>& b) const + { + eigen_assert(m_isInitialized && "JacobiSVD is not initialized."); + eigen_assert(computeU() && computeV() && "JacobiSVD::solve() requires both unitaries U and V to be computed (thin unitaries suffice)."); + return internal::solve_retval<JacobiSVD, Rhs>(*this, b.derived()); + } + + /** \returns the number of singular values that are not exactly 0 */ + Index nonzeroSingularValues() const + { + eigen_assert(m_isInitialized && "JacobiSVD is not initialized."); + return m_nonzeroSingularValues; + } + + inline Index rows() const { return m_rows; } + inline Index cols() const { return m_cols; } + + private: + void allocate(Index rows, Index cols, unsigned int computationOptions); + + protected: + MatrixUType m_matrixU; + MatrixVType m_matrixV; + SingularValuesType m_singularValues; + WorkMatrixType m_workMatrix; + bool m_isInitialized, m_isAllocated; + bool m_computeFullU, m_computeThinU; + bool m_computeFullV, m_computeThinV; + unsigned int m_computationOptions; + Index m_nonzeroSingularValues, m_rows, m_cols, m_diagSize; + + template<typename __MatrixType, int _QRPreconditioner, bool _IsComplex> + friend struct internal::svd_precondition_2x2_block_to_be_real; + template<typename __MatrixType, int _QRPreconditioner, int _Case, bool _DoAnything> + friend struct internal::qr_preconditioner_impl; + + internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows> m_qr_precond_morecols; + internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols> m_qr_precond_morerows; +}; + +template<typename MatrixType, int QRPreconditioner> +void JacobiSVD<MatrixType, QRPreconditioner>::allocate(Index rows, Index cols, unsigned int computationOptions) +{ + eigen_assert(rows >= 0 && cols >= 0); + + if (m_isAllocated && + rows == m_rows && + cols == m_cols && + computationOptions == m_computationOptions) + { + return; + } + + m_rows = rows; + m_cols = cols; + m_isInitialized = false; + m_isAllocated = true; + m_computationOptions = computationOptions; + m_computeFullU = (computationOptions & ComputeFullU) != 0; + m_computeThinU = (computationOptions & ComputeThinU) != 0; + m_computeFullV = (computationOptions & ComputeFullV) != 0; + m_computeThinV = (computationOptions & ComputeThinV) != 0; + eigen_assert(!(m_computeFullU && m_computeThinU) && "JacobiSVD: you can't ask for both full and thin U"); + eigen_assert(!(m_computeFullV && m_computeThinV) && "JacobiSVD: you can't ask for both full and thin V"); + eigen_assert(EIGEN_IMPLIES(m_computeThinU || m_computeThinV, MatrixType::ColsAtCompileTime==Dynamic) && + "JacobiSVD: thin U and V are only available when your matrix has a dynamic number of columns."); + if (QRPreconditioner == FullPivHouseholderQRPreconditioner) + { + eigen_assert(!(m_computeThinU || m_computeThinV) && + "JacobiSVD: can't compute thin U or thin V with the FullPivHouseholderQR preconditioner. " + "Use the ColPivHouseholderQR preconditioner instead."); + } + m_diagSize = (std::min)(m_rows, m_cols); + m_singularValues.resize(m_diagSize); + m_matrixU.resize(m_rows, m_computeFullU ? m_rows + : m_computeThinU ? m_diagSize + : 0); + m_matrixV.resize(m_cols, m_computeFullV ? m_cols + : m_computeThinV ? m_diagSize + : 0); + m_workMatrix.resize(m_diagSize, m_diagSize); + + if(m_cols>m_rows) m_qr_precond_morecols.allocate(*this); + if(m_rows>m_cols) m_qr_precond_morerows.allocate(*this); +} + +template<typename MatrixType, int QRPreconditioner> +JacobiSVD<MatrixType, QRPreconditioner>& +JacobiSVD<MatrixType, QRPreconditioner>::compute(const MatrixType& matrix, unsigned int computationOptions) +{ + allocate(matrix.rows(), matrix.cols(), computationOptions); + + // currently we stop when we reach precision 2*epsilon as the last bit of precision can require an unreasonable number of iterations, + // only worsening the precision of U and V as we accumulate more rotations + const RealScalar precision = RealScalar(2) * NumTraits<Scalar>::epsilon(); + + // limit for very small denormal numbers to be considered zero in order to avoid infinite loops (see bug 286) + const RealScalar considerAsZero = RealScalar(2) * std::numeric_limits<RealScalar>::denorm_min(); + + /*** step 1. The R-SVD step: we use a QR decomposition to reduce to the case of a square matrix */ + + if(!m_qr_precond_morecols.run(*this, matrix) && !m_qr_precond_morerows.run(*this, matrix)) + { + m_workMatrix = matrix.block(0,0,m_diagSize,m_diagSize); + if(m_computeFullU) m_matrixU.setIdentity(m_rows,m_rows); + if(m_computeThinU) m_matrixU.setIdentity(m_rows,m_diagSize); + if(m_computeFullV) m_matrixV.setIdentity(m_cols,m_cols); + if(m_computeThinV) m_matrixV.setIdentity(m_cols, m_diagSize); + } + + /*** step 2. The main Jacobi SVD iteration. ***/ + + bool finished = false; + while(!finished) + { + finished = true; + + // do a sweep: for all index pairs (p,q), perform SVD of the corresponding 2x2 sub-matrix + + for(Index p = 1; p < m_diagSize; ++p) + { + for(Index q = 0; q < p; ++q) + { + // if this 2x2 sub-matrix is not diagonal already... + // notice that this comparison will evaluate to false if any NaN is involved, ensuring that NaN's don't + // keep us iterating forever. Similarly, small denormal numbers are considered zero. + using std::max; + RealScalar threshold = (max)(considerAsZero, precision * (max)(internal::abs(m_workMatrix.coeff(p,p)), + internal::abs(m_workMatrix.coeff(q,q)))); + if((max)(internal::abs(m_workMatrix.coeff(p,q)),internal::abs(m_workMatrix.coeff(q,p))) > threshold) + { + finished = false; + + // perform SVD decomposition of 2x2 sub-matrix corresponding to indices p,q to make it diagonal + internal::svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner>::run(m_workMatrix, *this, p, q); + JacobiRotation<RealScalar> j_left, j_right; + internal::real_2x2_jacobi_svd(m_workMatrix, p, q, &j_left, &j_right); + + // accumulate resulting Jacobi rotations + m_workMatrix.applyOnTheLeft(p,q,j_left); + if(computeU()) m_matrixU.applyOnTheRight(p,q,j_left.transpose()); + + m_workMatrix.applyOnTheRight(p,q,j_right); + if(computeV()) m_matrixV.applyOnTheRight(p,q,j_right); + } + } + } + } + + /*** step 3. The work matrix is now diagonal, so ensure it's positive so its diagonal entries are the singular values ***/ + + for(Index i = 0; i < m_diagSize; ++i) + { + RealScalar a = internal::abs(m_workMatrix.coeff(i,i)); + m_singularValues.coeffRef(i) = a; + if(computeU() && (a!=RealScalar(0))) m_matrixU.col(i) *= m_workMatrix.coeff(i,i)/a; + } + + /*** step 4. Sort singular values in descending order and compute the number of nonzero singular values ***/ + + m_nonzeroSingularValues = m_diagSize; + for(Index i = 0; i < m_diagSize; i++) + { + Index pos; + RealScalar maxRemainingSingularValue = m_singularValues.tail(m_diagSize-i).maxCoeff(&pos); + if(maxRemainingSingularValue == RealScalar(0)) + { + m_nonzeroSingularValues = i; + break; + } + if(pos) + { + pos += i; + std::swap(m_singularValues.coeffRef(i), m_singularValues.coeffRef(pos)); + if(computeU()) m_matrixU.col(pos).swap(m_matrixU.col(i)); + if(computeV()) m_matrixV.col(pos).swap(m_matrixV.col(i)); + } + } + + m_isInitialized = true; + return *this; +} + +namespace internal { +template<typename _MatrixType, int QRPreconditioner, typename Rhs> +struct solve_retval<JacobiSVD<_MatrixType, QRPreconditioner>, Rhs> + : solve_retval_base<JacobiSVD<_MatrixType, QRPreconditioner>, Rhs> +{ + typedef JacobiSVD<_MatrixType, QRPreconditioner> JacobiSVDType; + EIGEN_MAKE_SOLVE_HELPERS(JacobiSVDType,Rhs) + + template<typename Dest> void evalTo(Dest& dst) const + { + eigen_assert(rhs().rows() == dec().rows()); + + // A = U S V^* + // So A^{-1} = V S^{-1} U^* + + Index diagSize = (std::min)(dec().rows(), dec().cols()); + typename JacobiSVDType::SingularValuesType invertedSingVals(diagSize); + + Index nonzeroSingVals = dec().nonzeroSingularValues(); + invertedSingVals.head(nonzeroSingVals) = dec().singularValues().head(nonzeroSingVals).array().inverse(); + invertedSingVals.tail(diagSize - nonzeroSingVals).setZero(); + + dst = dec().matrixV().leftCols(diagSize) + * invertedSingVals.asDiagonal() + * dec().matrixU().leftCols(diagSize).adjoint() + * rhs(); + } +}; +} // end namespace internal + +/** \svd_module + * + * \return the singular value decomposition of \c *this computed by two-sided + * Jacobi transformations. + * + * \sa class JacobiSVD + */ +template<typename Derived> +JacobiSVD<typename MatrixBase<Derived>::PlainObject> +MatrixBase<Derived>::jacobiSvd(unsigned int computationOptions) const +{ + return JacobiSVD<PlainObject>(*this, computationOptions); +} + +} // end namespace Eigen + +#endif // EIGEN_JACOBISVD_H |