SML000669 - c 2003 Society for Industrial and Applied...

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A SUPERFAST TOEPLITZ SOLVER WITH IMPROVED NUMERICAL STABILITY MICHAEL STEWART SIAM J. M ATRIX A NAL. A PPL . c ° 2003 Society for Industrial and Applied Mathematics Vol. 25, No. 3, pp. 669–693 Abstract. This paper describes a new O ( n log 3 ( n )) solver for the positive deFnite Toeplitz system Tx = b . Instead of computing generators for the inverse of T , the new algorithm adjoins b to T and applies a superfast Schur algorithm to the resulting augmented matrix. The genera- tors of this augmented matrix and its Schur complements are used by a divide-and-conquer block back-substitution routine to complete the solution of the system. The goal is to avoid the well- known numerical instability inherent in explicit inversion. Experiments suggest that the algorithm is backward stable in most cases. Key words. Toeplitz matrix, Schur algorithm AMS subject classifcation. 65±05 DOI. 10.1137/S089547980241791X 1. Background. We start with the positive defnite Toeplitz matrix T = t 0 t 1 ··· ··· t m 1 t 1 t 0 t 1 . . . . . . t 1 . . . . . . . . . . . . . . . . . . t 1 t m 1 t 1 t 0 C m × m and the system oF equations = b . There are several classes oF algorithms For solving such systems: these include slow algorithms requiring O ( n 3 ) unstructured matrix computations, Fast O ( n 2 ) algorithms that exploit the Toeplitz structure, and superFast algorithms that achieve a complexity strictly less than O ( n 2 ). Examples oF Fast algorithms include the Schur and Levinson algorithms. SuperFast algorithms have been developed in [3, 5, 10, 1, 7, 2]. One way to view the related approaches oF [10, 1, 7, 2] is as a divide-and-conquer variant oF the O ( n 2 ) Schur algorithm with Fast polynomial multiplication via the ±±T used to extend computations From submatrices and Schur complements to the Full matrix T . The underlying Schur algorithm, along with several generalizations, is numerically stable [4, 15, 6], but it has not been shown that this stability extends to the superFast Schur algorithm. In Fact, the proposed application oF the algorithm to linear systems involves computing generators oF T 1 and then Forming T 1 b using the ±±T. Numerical methods based on explicit inversion are usually unstable [9]. Experiments presented in section 5 show that the superFast Schur algorithm is no exception; it is not a backward stable algorithm. We will propose an alternative method that parallels the conventional and stable method oF triangular Factorization and back-substitution. Instead oF inverting T we Received by the editors November 13, 2002; accepted for publication (in revised form) by L. Reichel May 6, 2003; published electronically December 17, 2003. Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303 ([email protected]).
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SML000669 - c 2003 Society for Industrial and Applied...

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