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

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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. LOTTI , 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 classifications. 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 defi- 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 ) operations, N being the size of the system. Unfortunately, when a simple operation like multiplication or inversion or low rank modification 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. For 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 definite matrices has been proved in [2, 5]. Look-ahead modifications 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,
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern