SML000104

SML000104 - SIAM J. MATRIX ANAL. APPL. Vol. 18, No. 1, pp....

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STABILITY ISSUES IN THE FACTORIZATION OF STRUCTURED MATRICES * MICHAEL STEWART AND PAUL VAN DOOREN SIAM J. M ATRIX A NAL. A PPL. c ± 1997 Society for Industrial and Applied Mathematics Vol. 18, No. 1, pp. 104–118, January 1997 009 Abstract. This paper provides an error analysis of the generalized Schur algorithm of Kailath and Chun [ SIAM J. Matrix Anal. Appl. , 15 (1994), pp. 114–128]—a class of algorithms which can be used to factorize Toeplitz-like matrices, including block-Toeplitz matrices, and matrices of the form T T T , where T is Toeplitz. The conclusion drawn is that if this algorithm is implemented with hyperbolic transformations in the factored form which is well known to provide numerical stability in the context of Cholesky downdating, then the generalized Schur algorithm will be stable. If a more direct implementation of the hyperbolic transformations is used, then it will be unstable. In this respect, the algorithm is analogous to Cholesky downdating; the details of implementation of the hyperbolic transformations are essential for stability. An example which illustrates this instability is given. This result is in contrast to the ordinary Schur algorithm for which an analysis by Bojanczyk, Brent, De Hoog, and Sweet [ SIAM J. Matrix Anal. Appl. , 16 (1995), pp. 40–57] shows that the sta- bility of the algorithm is not dependent on the implementation of the hyperbolic transformations. Key words. Schur algorithm, structured matrices, Toeplitz matrices, stability AMS subject classifications. 65F05, 65F30, 15A06, 15A23 PII. S089547989528692X 1. Introduction. The Schur algorithm is a popular and fast method for the Cholesky factorization of a square, positive-definite Toeplitz matrix T . It performs reliably, and in [3] it was shown to be stable in the sense that if the algorithm runs to completion and ˆ C is the computed Cholesky factor, k T - ˆ C T ˆ C k is guaranteed to be small. This paper will perform a similar stability analysis which applies to several special cases of the generalized Schur algorithm [10]. In its full generality, the generalized Schur algorithm can be adapted to the factorization of a wide variety of structured matrices. The analysis given here is primarily of interest for block- Toeplitz and Toeplitz-block matrices, as well as for matrices of the form T T T , where T is rectangular and Toeplitz. The key notion behind the general algorithm is the concept of displacement rank [10]. One of the most significant examples is the Cholesky factorization of T T T . This factor is also the factor R in the QR factorization of the rectangular Toeplitz matrix T , and the obvious application of this fact to the solution of Toeplitz least squares problems is explored in [1]. However, the analysis given there assumes the use of the algorithm presented in [2] rather than the generalized Schur algorithm. The basic idea is to obtain R without bothering about finding Q , thus avoiding any problems associated with the loss of orthogonality which are common to all fast Toeplitz QR algorithms. The method of seminormal equations, possibly with iterative refinement,
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This note was uploaded on 07/21/2011 for the course MAD 5932 taught by Professor Gallivan during the Fall '06 term at FSU.

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SML000104 - SIAM J. MATRIX ANAL. APPL. Vol. 18, No. 1, pp....

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