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#### Article: Recursive-Based PCG Methods for Toeplitz Systems with Nonnegative Generating Functions

Title Recursive-Based PCG Methods for Toeplitz Systems with Nonnegative Generating Functions Ng, KPSun, HWJin, XQ Toeplitz matricesGohberg--semencul formulaRecursive-based methodPreconditioned conjugate gradient methodPreconditioners 2003 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, 2003, v. 24 n. 5, p. 1507-1529 How to Cite? In this paper, we consider the solutions of symmetric positive definite, but ill-conditioned, Toeplitz systems An x = b. Here we propose to solve the system by the recursive-based preconditioned conjugate gradient method. The idea is to use the inverse of Am (the principal submatrix of An with the Gohberg--Semencul formula as a preconditioner for An. The inverse of Am can be generated recursively by using the formula until m is small enough. The construction of the preconditioners requires only the entries of An and does not require the explicit knowledge of the generating function f of An. We show that if f is a nonnegative, bounded, and piecewise continuous even function with a finite number of zeros of even order, the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers. Hence the conjugate gradient method, when applied to solving the preconditioned system, converges very quickly. Numerical results are included to illustrate the effectiveness of our approach. http://hdl.handle.net/10722/43003 1064-82752015 Impact Factor: 1.7922015 SCImago Journal Rankings: 2.166 WOS:000183166600003

DC FieldValueLanguage
dc.contributor.authorNg, KPen_HK
dc.contributor.authorSun, HWen_HK
dc.contributor.authorJin, XQen_HK
dc.date.accessioned2007-03-23T04:36:39Z-
dc.date.available2007-03-23T04:36:39Z-
dc.date.issued2003en_HK
dc.identifier.citationSIAM Journal on Scientific Computing, 2003, v. 24 n. 5, p. 1507-1529en_HK
dc.identifier.issn1064-8275en_HK
dc.identifier.urihttp://hdl.handle.net/10722/43003-
dc.description.abstractIn this paper, we consider the solutions of symmetric positive definite, but ill-conditioned, Toeplitz systems An x = b. Here we propose to solve the system by the recursive-based preconditioned conjugate gradient method. The idea is to use the inverse of Am (the principal submatrix of An with the Gohberg--Semencul formula as a preconditioner for An. The inverse of Am can be generated recursively by using the formula until m is small enough. The construction of the preconditioners requires only the entries of An and does not require the explicit knowledge of the generating function f of An. We show that if f is a nonnegative, bounded, and piecewise continuous even function with a finite number of zeros of even order, the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers. Hence the conjugate gradient method, when applied to solving the preconditioned system, converges very quickly. Numerical results are included to illustrate the effectiveness of our approach.en_HK
dc.format.extent247568 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 matricesen_HK
dc.subjectGohberg--semencul formulaen_HK
dc.subjectRecursive-based methoden_HK
dc.subjectPreconditionersen_HK
dc.titleRecursive-Based PCG Methods for Toeplitz Systems with Nonnegative Generating Functionsen_HK
dc.typeArticleen_HK
dc.identifier.openurlhttp://library.hku.hk:4550/resserv?sid=HKU:IR&issn=1064-8275&volume=24&issue=5&spage=1507&epage=1529&date=2003&atitle=Recursive-Based+PCG+Methods+for+Toeplitz+Systems+with+Nonnegative+Generating+Functionsen_HK
dc.description.naturepublished_or_final_versionen_HK
dc.identifier.doi10.1137/S1064827500378155en_HK
dc.identifier.scopuseid_2-s2.0-0141940605-
dc.identifier.hkuros89443-
dc.identifier.isiWOS:000183166600003-