A Jacobi-type updating algorithm for the SVD or URV decomposition is developed, which is related to the QR algorithm for the symmetric eigenvalue problem. The algorithm employs one-sided transformations, and therefore provides a cheap alternative to earlier developed updating algorithms based on two-sided transformations. The present algorithm as well as the corresponding systolic implementation is therefore roughly twice as fast, compared to the former method, while the tracking properties are preserved. The algorithm is also extended to the 2-matrix QSVD or QURV case. Finally, the differences are discussed with a number of closely related algorithms that have recently been proposed. I. Introduction In an earlier report [16], an adaptive algorithm has been developed for tracking the singular value decomposition of a data matrix, when new observations (rows) are added continuously. The algorithm may be organized such that it provides at each time a certain approximation for the exact ...

Skew-Hamiltonian and Hamiltonian eigenvalue problems arise from a number of applications, particularly in systems and control theory. The preservation of the underlying matrix structures often plays an important role in these applications and may lead to more accurate and more efficient computational methods. We will discuss the relation of structured and unstructured condition numbers for these problems as well as algorithms exploiting the given matrix structures. Applications of Hamiltonian and skew-Hamiltonian eigenproblems are briefly described.

. In this note, we propose an implicit method for applying orthogonal transformations on both sides of a product of upper triangular 2 \Theta 2 matrices that preserve upper triangularity of the factors. Such problems arise in Jacobi type methods for computing the PSVD of a product of several matrices, and in ordering eigenvalues in the periodic Schur decomposition. Key Words. Orthogonal transformations, SVD, PSVD, Schur decompostion. AMS(MOS) Subject Classifications. 15A23, 65F25. 1 Introduction The problem of computing the singular value decomposition (SVD) of a product of matrices (PSVD) has been considered in [1],[2], [3], [10]. The computation proceeds in two stages. In the first stage the matrices are transformed into the upper triangular forms. In the second iterative stage an implicit Jacobi-type method is applied to the triangular matrices. It is important that after each iteration the matrices stay triangular [8]. A crucial aspect in such implicit Jacobi iterations is the ...

This paper gives an overview of frequency domain total least squares (TLS) estimators for rational transfer function models of linear time-invariant multivariable systems. The statistical performance of the different approaches are analyzed through their equivalent cost functions. Both generalized and bootstrapped total least squares (GTLS and BTLS) methods require the exact knowledge of the noise covariance matrix. The paper also studies the asymptotic (the number of data points going to infinity) behavior of the GTLS and BTLS estimators when the exact noise covariance matrix is replaced by the sample noise covariance matrix obtained from a (small) number of independent data sets. Even if only two independent repeated observations are available, it is shown that the estimates are still strongly consistent without any increase in the asymptotic uncertainty.

Since the CS decomposition (CSD) and the generalized singular value decomposition (GSVD) emerged as the generalization of the singular value decomposition about fifteen years ago, they have been proved to be very useful tools in numerical linear algebra. In this paper, we review the theoretical and numerical development of the decompositions, discuss some of their applications and present some new results and observations. We also point out some open problems. A Fortran 77 code has been written that computes the CSD and the GSVD. Keywords: singular value decomposition, CS decomposition, generalized singular value decomposition. Subject Classifications: AMS(MOS): 65F30; CR:G1.3 1 Introduction The singular value decomposition (SVD) of a matrix is one of the most important tools in numerical linear algebra. It has been widely used in scientific computing. Recently, Stewart [52] gave an excellent survey on the early history of the SVD back to the contributions of E. Beltrami and C. Jord...

Abstract. Avariational formulation for the generalized singular value decomposition (GSVD) of a pair of matrices A 2 R m n and B 2 R p n is presented. In particular, a duality theory analogous to that of the SVD provides new understanding of left and right generalized singular vectors. It is shown that the intersection of row spaces of A and B playsakey role in the GSVD duality theory. The main result that characterizes left GSVD vectors involves a generalized singular value de ation process.

Many algorithms exist for computing the symmetric eigendecomposition, the singular value decomposition and the generalized singular value decomposition. In this thesis, we present several new algorithms and improvements on old algorithms, analyzing them with respect to their speed, accuracy, and storage requirements. We rst discuss the variations on the bisection algorithm for nding eigenvalues of symmetric tridiagonal matrices. We show the challenges in implementing a correct al-gorithm with oating point arithmetic. We show how reasonable looking but incorrect implementations can fail. We carefully de ne correctness, and present several implementa-tions that we rigorously prove correct. We then discuss a fast implementation of bisection using parallel pre x. We show many numerical examples of the instability of this algorithm, and then discuss its forward error and backward error analysis. We also discuss possible ways to stabilize it by using iterative re nement. Finally, we discuss how to use a divide-and-conquer algorithm to compute the sin-gular value decomposition and solve the linear least squares problem, and how to implement

We survey recent developments in dense numerical linear algebra, covering linear systems, least squares problems and eigenproblems. Topics considered include the design and analysis of block, partitioned and parallel algorithms, condition number estimation, componentwise error analysis, and the computation of practical error bounds. Frequent reference is made to LAPACK, the state of the art package of Fortran software designed to solve linear algebra problems efficiently and accurately on high-performance computers.

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Recently, Chu, Funderlic and Golub [ SIAM J. Matrix Anal. Appl., 18:1082--1092, 1997] presented a variational formulation for the quotient singular value decomposition (QSVD) of two matrices A 2 R n\Thetam ; C 2 R p\Thetam which is a generalization of that one for the ordinary singular value decomposition (OSVD) and characterizes the role of two orthogonal matrices in QSVD. In this paper, we give an alternative derivation of this variational formulation and extend it to establish an analogous variational formulation for the Restricted Singular Value Decomposition (RSVD) of Matrix Triplets A 2 R n\Thetam ; B 2 R n\Thetal ; C 2 R p\Thetam which provides new understanding of the orthogonal matrices appearing in this decomposition. Keywords: OSVD, QSVD, RSVD, Generalized Singular Value, Variational Formulation, Stationary Value, Stationary Point. AMS subject classification: 65F15, 65H15. 1 Introduction The ordinary singular value decomposition (OSVD) of a given matrix A 2 R ...