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#### Article: Preconditioned Lanczos Methods for the Minimum Eigenvalue of a Symmetric Positive Definite Toeplitz Matrix

Title Preconditioned Lanczos Methods for the Minimum Eigenvalue of a Symmetric Positive Definite Toeplitz Matrix Ng, KP Toeplitz matrixSine transform matrixFast sine transformPreconditioningLanczos method 2000 Society for Industrial and Applied Mathematics. The Journal's web site is located at http://epubs.siam.org/sam-bin/dbq/toclist/SISC SIAM Journal on Scientific Computing, 2000, v. 21 n. 6, p. 1973-1986 How to Cite? In this paper, we apply the preconditioned Lanczos (PL) method to compute the minimum eigenvalue of a symmetric positive definite Toeplitz matrix. The sine transform-based preconditioner is used to speed up the convergence rate of the PL method. The resulting method involves only Toeplitz and sine transform matrix-vector multiplications and hence can be computed efficiently by fast transform algorithms. We show that if the symmetric Toeplitz matrix is generated by a positive $2 \pi$-periodic even continuous function, then the PL method will converge sufficiently fast. Numerical results including Toeplitz and non-Toeplitz matrices are reported to illustrate the effectiveness of the method. http://hdl.handle.net/10722/42998 1064-82752015 Impact Factor: 1.7922015 SCImago Journal Rankings: 2.166 WOS:000087640400002

DC FieldValueLanguage
dc.contributor.authorNg, KPen_HK
dc.date.accessioned2007-03-23T04:36:32Z-
dc.date.available2007-03-23T04:36:32Z-
dc.date.issued2000en_HK
dc.identifier.citationSIAM Journal on Scientific Computing, 2000, v. 21 n. 6, p. 1973-1986en_HK
dc.identifier.issn1064-8275en_HK
dc.identifier.urihttp://hdl.handle.net/10722/42998-
dc.description.abstractIn this paper, we apply the preconditioned Lanczos (PL) method to compute the minimum eigenvalue of a symmetric positive definite Toeplitz matrix. The sine transform-based preconditioner is used to speed up the convergence rate of the PL method. The resulting method involves only Toeplitz and sine transform matrix-vector multiplications and hence can be computed efficiently by fast transform algorithms. We show that if the symmetric Toeplitz matrix is generated by a positive $2 \pi$-periodic even continuous function, then the PL method will converge sufficiently fast. Numerical results including Toeplitz and non-Toeplitz matrices are reported to illustrate the effectiveness of the method.en_HK
dc.format.extent193210 bytes-
dc.format.extent26112 bytes-
dc.format.mimetypeapplication/pdf-
dc.format.mimetypeapplication/msword-
dc.languageengen_HK
dc.publisherSociety for Industrial and Applied Mathematics. The Journal's web site is located at http://epubs.siam.org/sam-bin/dbq/toclist/SISCen_HK
dc.relation.ispartofSIAM Journal on Scientific Computing-
dc.subjectToeplitz matrixen_HK
dc.subjectSine transform matrixen_HK
dc.subjectFast sine transformen_HK
dc.subjectPreconditioningen_HK
dc.subjectLanczos methoden_HK
dc.titlePreconditioned Lanczos Methods for the Minimum Eigenvalue of a Symmetric Positive Definite Toeplitz Matrixen_HK
dc.typeArticleen_HK
dc.identifier.openurlhttp://library.hku.hk:4550/resserv?sid=HKU:IR&issn=1064-8275&volume=21&issue=6&spage=1973&epage=1986&date=2000&atitle=Preconditioned+Lanczos+Methods+for+the+Minimum+Eigenvalue+of+a+Symmetric+Positive+Definite+Toeplitz+Matrixen_HK
dc.description.naturepublished_or_final_versionen_HK
dc.identifier.doi10.1137/S1064827597330169en_HK
dc.identifier.scopuseid_2-s2.0-0034513884-
dc.identifier.hkuros63220-
dc.identifier.isiWOS:000087640400002-