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Convergence analysis of superoptimal PCG algorithm for Toeplitz systems with a Fisher-Hartwig singularity
Vong S.-W.; Wang W.; Jin X.-Q.
2008-01-15
Source PublicationLinear Algebra and Its Applications
ISSN00243795
Volume428Issue:2-3Pages:535-549
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 T (f) x = b where the generating function f is given byf (ω) = | 1 - e | h (ω)with d ∈ fenced(- frac(1, 2), frac(1, 2)) {minus 45 degree rule} {0}. The function h (ω) is positive continuous on [- π, π] and differentiable on [- π, π] {minus 45 degree rule} {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 (log n) and therefore the complexity of our algorithm is O (n log n). We notice that the numerical performance of superoptimal preconditioner is almost the same as that of the optimal preconditioner. © 2007 Elsevier Inc. All rights reserved.

KeywordFisher-hartwig Singularity Optimal Circulant Preconditioner Pcg Method Superoptimal Circulant Preconditioner Toeplitz System
DOIhttps://doi.org/10.1016/j.laa.2007.05.032
URLView the original
WOS Research AreaMathematics
WOS SubjectMathematics, Applied ; Mathematics
WOS IDWOS:000252172800011
PublisherELSEVIER SCIENCE INC
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Document TypeJournal article
CollectionDEPARTMENT OF MATHEMATICS
Corresponding AuthorJin X.-Q.
AffiliationDepartment of Mathematics, University of Macau, Macao, China
First Author AffilicationUniversity of Macau
Corresponding Author AffilicationUniversity of Macau
Recommended Citation
GB/T 7714
Vong S.-W.,Wang W.,Jin X.-Q.. Convergence analysis of superoptimal PCG algorithm for Toeplitz systems with a Fisher-Hartwig singularity[J]. Linear Algebra and Its Applications,2008,428(2-3):535-549.
APA Vong S.-W.,Wang W.,&Jin X.-Q..(2008).Convergence analysis of superoptimal PCG algorithm for Toeplitz systems with a Fisher-Hartwig singularity.Linear Algebra and Its Applications,428(2-3),535-549.
MLA Vong S.-W.,et al."Convergence analysis of superoptimal PCG algorithm for Toeplitz systems with a Fisher-Hartwig singularity".Linear Algebra and Its Applications 428.2-3(2008):535-549.
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