Computing the singular value decomposition with high relative accuracy
type: Item
instance of: scholarly article
title: Computing the singular value decomposition with high relative accuracy
zbMATH Open document ID: 0952.65032
DOI: 10.1016/S0024-3795(99)00134-2
author: James Demmel
author: Ivan Slapničar
author: Krešimir Veselić
author: Zlatko Drmač
author: Ming Gu
author: Stanley C. Eisenstat
published in: Linear Algebra and its Applications
publication date: 2001-01-02T00:00:00Z
review text: Although classical methods can compute singular values of matrices with very small absolute errors, the estimates they give for very small singular values often contain no correct significant figures, even when these small singular values are well determined by the physical problem being modeled. Many recent papers have identified special classes of matrices for which the singular value decomposition may be computed with high relative accuracy, but the perturbation theory and algorithms for the different cases have been very different. This paper gives a common perturbation theory and a common strategy which makes possible the computation of the singular value decomposition with high relative accuracy not only for many classes of matrices previously dealt with but also for many new cases as well. The first step is to compute a ``rank revealing'' decomposition \(XDY^T\) of the given matrix, where \(D\) is diagonal and \(X\) and \(Y\) have at least as many rows as columns and are well-conditioned in a certain sense. Often the \(LDU\) decomposition is a suitable choice. Several algorithms for computing the singular value decomposition from \(XDY^T\) are compared and a complete analysis is given for the main recommended method. The paper includes a discussion of several open problems, and 80 references to the related literature.
reviewed by: Alan L. Andrew
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 65G50
Mathematics Subject Classification ID: 65F20
zbMATH DE Number: 1399444
zbMATH Keywords: singular value decomposition
zbMATH Keywords: structured matrices
zbMATH Keywords: high relative accuracy
zbMATH Keywords: rank revealing decomposition
zbMATH Keywords: algorithms
describes a project that uses: BLAS
describes a project that uses: LAPACK
describes a project that uses: mctoolbox
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1399444