SML002531 - c 2010 Society for Industrial and Applied...

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Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. M ATRIX A NAL. A PPL . c ± 2010 Society for Industrial and Applied Mathematics Vol. 31, No. 5, pp. 2531–2552 STABILITY OF THE LEVINSON ALGORITHM FOR TOEPLITZ-LIKE SYSTEMS P. FAVATI ,G .LOTT I , AND O. MENCHI § Abstract. Numerical stability of the Levinson algorithm, generalized for Toeplitz-like systems, is studied. Arguments based on the analytic results of an error analysis for floating point arithmetic produce an upper bound on the norm of the residual vector, which grows exponentially with respect to the size of the problem. The base of such an exponential function can be small for diagonally dominant Toeplitz-like matrices. Numerical experiments show that, for these matrices, Gaussian elimination by row and the Levinson algorithm have residuals of the same order of magnitude. As expected, the empirical results point out that the theoretical bound is too pessimistic. Key words. Levinson algorithm, Toeplitz-like matrices, stability AMS subject classifcations. 15A06, 15B05 DOI. 10.1137/090753619 1. Introduction. Toeplitz systems arise frequently in linear algebra (see [2] for a list of possible sources), and special fast and superfast algorithms have been devised to solve them. Starting from the original Durbin algorithm to solve the Yule–Walker equations, the Levinson algorithm has been proposed for symmetric positive deF- nite Toeplitz matrices and extended to the case of general Toeplitz matrices [8]. The Levinson algorithm is a fast method, i.e., it has a cost of O ( N 2 )operat ions , N being the size of the system. Unfortunately, when a simple operation like multiplication or inversion or low rank modiFcation is applied to a Toeplitz matrix, the Toeplitz struc- ture is lost and more general structures must be considered. The class of Toeplitz-like matrices, which is closed for the most common operations applied in numerical algo- rithms, seems ideal from this point of view. It is based on the concept of displacement rank introduced in [15] and has been studied by many authors (see, for example, [11, 13, 14]). The displacement operator allows a compact representation of the ma- trices of this class by means of a set of generators. The standard situation, when dealing with a structured matrix, assumes that its entries are not exactly known but can be computed from the generators when they are needed. ±or the Toeplitz-like matrices, fast and superfast algorithms have been devised as well (see the extensive bibliography in [14]), but the question of their stability is still a matter of discussion. The Levinson algorithm, too, has been generalized for this class of matrices, maintaining its computational cost [6, 13]. The numerical stability of the original Levinson algorithm for symmetric positive deFnite matrices has been proved in [2, 5]. Look-ahead modiFcations have been pro- posed in [3, 4] to obviate instability in the general Toeplitz case. We are interested in studying the stability of the Levinson algorithm generalized for Toeplitz-like systems,
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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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SML002531 - c 2010 Society for Industrial and Applied...

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