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- W2025473789 abstract "Recently, Lu and Hurvich [Y. Lu, C. Hurvich, On the complexity of the preconditioned conjugate gradient algorithm for solving toeplitz systems with a Fisher–Hartwig singularity, SIAM J. Matrix Anal. Appl. 27 (2005) 638–653] used the preconditioned conjugate gradient method with the optimal circulant preconditioner proposed in Chan [T. Chan, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Statist. Comput. 9 (1988) 766–771] for solving the Toeplitz system Tn(f)x=b where the generating function f is given byf(ω)=|1-e-iω|-2dh(ω)with d∈-12,12⧹{0}. The function h(ω) is positive continuous on [-π,π] and differentiable on [-π,π]⧹{0}. In this paper, we will use the superoptimal circulant preconditioner proposed by Tyrtyshnikov [E. Tyrtyshnikov, Optimal and superoptimal circulant preconditioners, SIAM J. Matrix Anal. Appl. 13 (1992) 459–473] to solve the same problem when 0<d<1/2. Our convergence analysis shows that the number of iterations is bounded by O(log3n) and therefore the complexity of our algorithm is O(nlog4n). We notice that the numerical performance of superoptimal preconditioner is almost the same as that of the optimal preconditioner." @default.
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- W2025473789 date "2008-01-01" @default.
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- W2025473789 title "Convergence analysis of superoptimal PCG algorithm for Toeplitz systems with a Fisher–Hartwig singularity" @default.
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