Fast band-Toeplitz preconditioners for Hermitian Toeplitz systems

Journal Article · Sat Jan 01 04:00:00 EST 1994 · SIAM Journal on Scientific and Statistical Computing (Society for Industrial and Applied Mathematics); (United States)

DOI:https://doi.org/10.1137/0915011· OSTI ID:5072141

Chan, R H ^[1]; Tang, P T.P. ^[2]

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1. Univ. of Hong Kong (Hong Kong). Dept. of Mathematics
2. Argonne National Lab., IL (United States). Mathematics and Computer Science Div.

This paper considers the solutions of Hermitian Toeplitz systems where the Toeplitz matrices are generated by nonnegative functions [line integral]. The preconditioned conjugate gradient method with well-known circulant preconditioners fails in the case when [line integral] has zeros. This paper employs Toeplitz matrices of fixed bandwidth as preconditioners. Their generating functions g are trigonometric polynomials of fixed degree and are determined by minimizing the maximum relative error [parallel](f [minus] g)/[line integral][parallel][infinity]. It is shown that the conditions number of systems preconditioned by the band-Toeplitz matrices are O(1) for [line integral], with or without zeros. When [line integral] is positive, the preconditions systems converge at the same rate as other well-known circulant preconditioned systems. An a priori bound of the number of iterations required for convergence is also given.

• DOE Contract Number: FG03-87ER25037
• OSTI ID: 5072141
• Journal Information: SIAM Journal on Scientific and Statistical Computing (Society for Industrial and Applied Mathematics); (United States), Journal Name: SIAM Journal on Scientific and Statistical Computing (Society for Industrial and Applied Mathematics); (United States) Vol. 15:1; ISSN SIJCD4; ISSN 0196-5204
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
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MATRICES
PERFORMANCE