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Conditioning and solution of Hermitian (block) Toeplitz systems by means of preconditioned conjugate gradient methods
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Paper Abstract

Let {An(f)} be a sequence of nested n X n Toeplitz matrices generated by a Lebesgue integrable real-function f defined on (-(pi) , (pi) ). In this paper, we first present some results about the spectral properties of An(f) (density, range, behavior of the extreme eigenvalues etc.), then we apply these results to the preconditioning problem. We analyze in detail the preconditioned conjugate gradient method, where the proposed poreconditioners An1(g)An(f): we obtain new results about the range, the density and the extremal properties of their spectra. In particular we deal with the critical case where the matrices An(f) are asymptotically ill-conditioned, i.e., zero belongs to the convex hull of the essential range of f. We consider positive definite Toeplitz linear systems (f >= 0), nondefinite Toeplitz linear systems (f with nondefinite sign), with zeros of generic orders. Moreover, these analyses and techniques are partially extended to the block case, too.

Paper Details

Date Published: 7 June 1995
PDF: 12 pages
Proc. SPIE 2563, Advanced Signal Processing Algorithms, (7 June 1995); doi: 10.1117/12.211409
Show Author Affiliations
Stefano Serra-Capizzano, Univ. di Pisa (Italy)

Published in SPIE Proceedings Vol. 2563:
Advanced Signal Processing Algorithms
Franklin T. Luk, Editor(s)

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