A singular value decomposition updating algorithm for subspace tracking

The noise averaged Jacobi-type SVD updating algorithm presented in this paper is capable of simultaneously tracking the signal subspace and its dimension, while preserving both the low computational cost of and the parallel structure of the method, as demonstrated in a systolic implementation.Furthermore, the algorithm tracks all signal singular values.The first two were developed independently of each other and have dist ..." In this paper we review the state of affairs in the area of approximation of large-scale systems. The first two were developed independently of each other and have distinct sets of attributes and drawbacks. 7 3.1.2 Proper Orthogonal Decomposition (POD) methods . The upper triangular structure is then used to ze...Their squares are estimates of the powers in the orthogonal modes of the signal.

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. Recently, Stewart [52] gave an excellent survey on the early history of the SVD back to the contributions of E. With the development of the Kogbetliantz algorithm for computing the SVD of a product of two matrices by Heath, Laub, Paige and Ward =-=[31]-=- and Hari and Veseli'c [30], Paige proposed a generalization of the Kogbetliantz algorithm to compute the GSVD directly. In this paper, a Jacobi-type correction scheme is described, that continuously annihilates accumulated errors and thus stabilizes the overall scheme. A number of problems arising from dynamical systems and other areas leads to problems of computing eigenvalues/vectors and singular value decomposisitions of products of matrices. However, when finite precision arithmetic is used, round-off errors apparently accumulate unboundedly, so that after a number of updates the computed least squares solutions turn out to be useless. These generalizations can be obtained for any number of matrices of compatible dimensions. Typical examples are adaptive beamforming, direction finding, spectral analysis, pattern recognition, etc. An adaptive algorithm can be constructed by interlacing a Jacobi-type SVD procedure (Kogbetliantz's algorithm [9], modified for triangular matrices =-=[8, 10]-=-) with repeated QR updates. Initialization V (0) ( I n2n R (0) ( O n2n Loop for k = 1; : : : ; 1 input new measurement vector a (k) a (k) T ? These generalizations can be obtained for any number of matrices of compatible dimensions ..." In this paper, we discuss multi-matrix generalizations of two well-known orthogonal rank factorizations of a matrix: the generalized singular value decomposition and the generalized QR-(or URV-) decomposition.

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