On the Stability of the Generalized Schur Algorithm

Abstract

The generalized Schur algorithm (GSA) is a fast method to compute the Cholesky factorization of a wide variety of structured matrices. The stability property of the GSA depends on the way it is implemented. In [15] GSA was shown to be as stable as the Schur algorithm, provided one hyperbolic rotation in factored form [3] is performed at each iteration. Fast and efficient algorithms for solving Structured Total Least Squares problems [14],[15] are based on a particular implementation of GSA requiring two hyperbolic transformations at each iteration. In this paper the authors prove the stability property of such implementation provided the hyperbolic transformations are performed in factored form [3].

S. Van Huffel is a Senior Research Associate with the F.W.O. (Fund for Scientific Research - Flanders). This paper presents research results of the Belgian Programme on Interuniversity Poles of Attraction (IUAP P4-02 and P4-24), initiated by the Belgian State, Prime Minister’s Office - Federal Office for Scientific, Technical and Cultural Affairs, of the European Community TMR Programme, Networks, project CHRXCT97-0160, of the Brite Euram Programme, Thematic Network BRRT-CT97-5040 ‘Niconet’, of the Concerted Research Action (GOA) projects of the Flemish Government MEFISTO-666 (Mathematical Engineering for Information and Communication Systems Technology), of the IDO/99/03 project (K.U.Leuven) “Predictive computer models for medical classification problems using patient data and expert knowledge”, of the FWO “Krediet aan navorsers” G.0326.98 and the FWO project G.0200.00.