Abstract
This paper is concerned with the solution of a system of linear equations T M, N X=b, where T M, N denotes a positive definite doubly symmetric block-Toeplitz matrix with Toeplitz blocks arising from a generating function f of the Wiener class. We derive optimal and Strang-type trigonometric preconditioners P M, N of T M, N from the Fejér and Fourier sum of f, respectively. Using relations between trigonometric transforms and Toeplitz matrices, we prove that for all ε > 0 and sufficiently large M, N, at most O(M) + O(N) eigenvalues of lie outside the interval (1 — ε, l + ε) such that the preconditioned conjugate gradient method converges in at most O(M) + O(N) steps.
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© 1997 Springer Basel AG
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Potts, D., Steidl, G., Tasche, M. (1997). Trigonometric Preconditioners for Block Toeplitz Systems. In: Nürnberger, G., Schmidt, J.W., Walz, G. (eds) Multivariate Approximation and Splines. ISNM International Series of Numerical Mathematics, vol 125. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8871-4_18
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DOI: https://doi.org/10.1007/978-3-0348-8871-4_18
Publisher Name: Birkhäuser, Basel
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