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Article: Inverse Toeplitz preconditioners for Hermitian Toeplitz systems

TitleInverse Toeplitz preconditioners for Hermitian Toeplitz systems
Authors
KeywordsGenerating function
Preconditioned conjugate gradient method
Preconditioner
Toeplitz system
Issue Date2005
PublisherJohn Wiley & Sons Ltd.
Citation
Numerical Linear Algebra With Applications, 2005, v. 12 n. 2-3, p. 221-229 How to Cite?
AbstractIn this paper we consider solving Hermitian Toeplitz systems T nx = b by using the preconditioned conjugate gradient (PCG) method. Here the Toeplitz matrices Tn are assumed to be generated by a non-negative continuous 2π-periodic function f, i.e. Tn = script J sign[f]. It was proved in (Linear Algebra Appl. 1993; 190:181) that if f is positive then the spectrum of script J signn[1/f]script J sign n[f] is clustered around 1. We prove that the trigonometric polynomial qn (s) (s ≥ 2, cf. (2) and (3)) converges to 1/f uniformly as n → ∞ under the condition that 1/f is in Wiener class. It follows that the computational cost of the PCG method can be reduced by replacing 1/f with qN (2) where N < n. We also extend our method to construct efficient preconditioners for Tn when f has finite zeros of even orders. We prove that with our preconditioners, the preconditioned matrix has spectrum clustered around 1. It follows that the PCG methods converge very fast when applied to solve the preconditioned systems. Numerical results are given to demonstrate the efficiency of our preconditioners. Copyright © 2004 John Wiley & Sons, Ltd.
Persistent Identifierhttp://hdl.handle.net/10722/75215
ISSN
2015 Impact Factor: 1.431
2015 SCImago Journal Rankings: 1.250
ISI Accession Number ID
References

 

DC FieldValueLanguage
dc.contributor.authorLin, FRen_HK
dc.contributor.authorChing, WKen_HK
dc.date.accessioned2010-09-06T07:09:02Z-
dc.date.available2010-09-06T07:09:02Z-
dc.date.issued2005en_HK
dc.identifier.citationNumerical Linear Algebra With Applications, 2005, v. 12 n. 2-3, p. 221-229en_HK
dc.identifier.issn1070-5325en_HK
dc.identifier.urihttp://hdl.handle.net/10722/75215-
dc.description.abstractIn this paper we consider solving Hermitian Toeplitz systems T nx = b by using the preconditioned conjugate gradient (PCG) method. Here the Toeplitz matrices Tn are assumed to be generated by a non-negative continuous 2π-periodic function f, i.e. Tn = script J sign[f]. It was proved in (Linear Algebra Appl. 1993; 190:181) that if f is positive then the spectrum of script J signn[1/f]script J sign n[f] is clustered around 1. We prove that the trigonometric polynomial qn (s) (s ≥ 2, cf. (2) and (3)) converges to 1/f uniformly as n → ∞ under the condition that 1/f is in Wiener class. It follows that the computational cost of the PCG method can be reduced by replacing 1/f with qN (2) where N < n. We also extend our method to construct efficient preconditioners for Tn when f has finite zeros of even orders. We prove that with our preconditioners, the preconditioned matrix has spectrum clustered around 1. It follows that the PCG methods converge very fast when applied to solve the preconditioned systems. Numerical results are given to demonstrate the efficiency of our preconditioners. Copyright © 2004 John Wiley & Sons, Ltd.en_HK
dc.languageengen_HK
dc.publisherJohn Wiley & Sons Ltd.en_HK
dc.relation.ispartofNumerical Linear Algebra with Applicationsen_HK
dc.rightsNumerical Linear Algebra with Applications. Copyright © John Wiley & Sons Ltd.en_HK
dc.subjectGenerating functionen_HK
dc.subjectPreconditioned conjugate gradient methoden_HK
dc.subjectPreconditioneren_HK
dc.subjectToeplitz systemen_HK
dc.titleInverse Toeplitz preconditioners for Hermitian Toeplitz systemsen_HK
dc.typeArticleen_HK
dc.identifier.openurlhttp://library.hku.hk:4550/resserv?sid=HKU:IR&issn=1070-5325&volume=12 no2-3&spage=221&epage=229&date=2005&atitle=Inverse+Toeplitz+Preconditioners+for+Hermitian+Toeplitz+Systemsen_HK
dc.identifier.emailChing, WK:wching@hku.hken_HK
dc.identifier.authorityChing, WK=rp00679en_HK
dc.description.naturelink_to_subscribed_fulltext-
dc.identifier.doi10.1002/nla.397en_HK
dc.identifier.scopuseid_2-s2.0-20744457219en_HK
dc.identifier.hkuros97994en_HK
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-20744457219&selection=ref&src=s&origin=recordpageen_HK
dc.identifier.volume12en_HK
dc.identifier.issue2-3en_HK
dc.identifier.spage221en_HK
dc.identifier.epage229en_HK
dc.identifier.isiWOS:000228112200016-
dc.publisher.placeUnited Kingdomen_HK
dc.identifier.scopusauthoridLin, FR=7402777425en_HK
dc.identifier.scopusauthoridChing, WK=13310265500en_HK

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