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Methods for large scale total least squares problems
Linköping University, Department of Mathematics, Scientific Computing. Linköping University, The Institute of Technology.
Department of Informatics, University of Bergen, NO-5020 Bergen, Norway.
2001 (English)In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 22, no 2, 413-429 p.Article in journal (Refereed) Published
Abstract [en]

The solution of the total least squares (TLS) problems, minE,f ?(E,f)?F subject to (A + E)x = b + f, can in the generic case be obtained from the right singular vector corresponding to the smallest singular value sn+1 of (A, b). When A is large and sparse (or structured) a method based on Rayleigh quotient iteration (RQI) has been suggested by Björck. In this method the problem is reduced to the solution of a sequence of symmetric, positive definite linear systems of the form (ATA - s¯2I)z = g, where s¯ is an approximation to sn+1. These linear systems are then solved by a preconditioned conjugate gradient method (PCGTLS). For TLS problems where A is large and sparse a (possibly incomplete) Cholesky factor of AT A can usually be computed, and this provides a very efficient preconditioner. The resulting method can be used to solve a much wider range of problems than it is possible to solve by using Lanczos-type algorithms directly for the singular value problem. In this paper the RQI-PCGTLS method is further developed, and the choice of initial approximation and termination criteria are discussed. Numerical results confirm that the given algorithm achieves rapid convergence and good accuracy.

Place, publisher, year, edition, pages
2001. Vol. 22, no 2, 413-429 p.
Keyword [en]
Conjugate gradient method, Rayleigh quotient iteration, Singular values, Total least squares
National Category
Natural Sciences
Identifiers
URN: urn:nbn:se:liu:diva-47174DOI: 10.1137/S0895479899355414OAI: oai:DiVA.org:liu-47174DiVA: diva2:268070
Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2017-12-13

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Björck, Åke

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