Trigonometric transform splitting methods for real symmetric Toeplitz systems

Detalhes bibliográficos
Autor(a) principal: Zhongyun Liu
Data de Publicação: 2018
Outros Autores: Nianci Wu, Xiaorong Qin, Zhang, Yulin
Tipo de documento: Artigo
Idioma: eng
Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
Texto Completo: http://hdl.handle.net/1822/54634
Resumo: In this paper, we study efficient iterative methods for real symmetric Toeplitz systems based on the trigonometric transformation splitting (TTS) of the real symmetric Toeplitz matrix A. Theoretical analyses show that if the generating function f of the n × n Toeplitz matrix A is a real positive even function, then the TTS iterative methods converge to the unique solution of the linear system of equations for sufficient large n. Moreover, we derive an upper bound of the contraction factor of the TTS iteration which is dependent solely on the spectra of the two TTS matrices involved. Different from the CSCS iterative method in Ng (2003) in which all operations counts concern complex operations when the DFTs are employed, even if the Toeplitz matrix A is real and symmetric, our method only involves real arithmetics when the DCTs and DSTs are used. The numerical experiments show that our method works better than CSCS iterative method and much better than the positive definite and skew-symmetric splitting (PSS) iterative method in Bai et al. (2005) and the symmetric Gauss–Seidel (SGS) iterative method.
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spelling Trigonometric transform splitting methods for real symmetric Toeplitz systemsSine transformCosine transformMatrix splitting Iterative methodsReal Toeplitz matricesIterative methodsMatrix splittingCiências Naturais::MatemáticasScience & TechnologyIn this paper, we study efficient iterative methods for real symmetric Toeplitz systems based on the trigonometric transformation splitting (TTS) of the real symmetric Toeplitz matrix A. Theoretical analyses show that if the generating function f of the n × n Toeplitz matrix A is a real positive even function, then the TTS iterative methods converge to the unique solution of the linear system of equations for sufficient large n. Moreover, we derive an upper bound of the contraction factor of the TTS iteration which is dependent solely on the spectra of the two TTS matrices involved. Different from the CSCS iterative method in Ng (2003) in which all operations counts concern complex operations when the DFTs are employed, even if the Toeplitz matrix A is real and symmetric, our method only involves real arithmetics when the DCTs and DSTs are used. The numerical experiments show that our method works better than CSCS iterative method and much better than the positive definite and skew-symmetric splitting (PSS) iterative method in Bai et al. (2005) and the symmetric Gauss–Seidel (SGS) iterative method.National Natural Science Foundation of China under Grant No. 11371075info:eu-repo/semantics/publishedVersionElsevierUniversidade do MinhoZhongyun LiuNianci WuXiaorong QinZhang, Yulin20182018-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/1822/54634eng0898-122110.1016/j.camwa.2018.01.008www.elsevier.com/locate/camwainfo:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2023-07-21T12:37:10Zoai:repositorium.sdum.uminho.pt:1822/54634Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T19:33:25.538729Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse
dc.title.none.fl_str_mv Trigonometric transform splitting methods for real symmetric Toeplitz systems
title Trigonometric transform splitting methods for real symmetric Toeplitz systems
spellingShingle Trigonometric transform splitting methods for real symmetric Toeplitz systems
Zhongyun Liu
Sine transform
Cosine transform
Matrix splitting Iterative methods
Real Toeplitz matrices
Iterative methods
Matrix splitting
Ciências Naturais::Matemáticas
Science & Technology
title_short Trigonometric transform splitting methods for real symmetric Toeplitz systems
title_full Trigonometric transform splitting methods for real symmetric Toeplitz systems
title_fullStr Trigonometric transform splitting methods for real symmetric Toeplitz systems
title_full_unstemmed Trigonometric transform splitting methods for real symmetric Toeplitz systems
title_sort Trigonometric transform splitting methods for real symmetric Toeplitz systems
author Zhongyun Liu
author_facet Zhongyun Liu
Nianci Wu
Xiaorong Qin
Zhang, Yulin
author_role author
author2 Nianci Wu
Xiaorong Qin
Zhang, Yulin
author2_role author
author
author
dc.contributor.none.fl_str_mv Universidade do Minho
dc.contributor.author.fl_str_mv Zhongyun Liu
Nianci Wu
Xiaorong Qin
Zhang, Yulin
dc.subject.por.fl_str_mv Sine transform
Cosine transform
Matrix splitting Iterative methods
Real Toeplitz matrices
Iterative methods
Matrix splitting
Ciências Naturais::Matemáticas
Science & Technology
topic Sine transform
Cosine transform
Matrix splitting Iterative methods
Real Toeplitz matrices
Iterative methods
Matrix splitting
Ciências Naturais::Matemáticas
Science & Technology
description In this paper, we study efficient iterative methods for real symmetric Toeplitz systems based on the trigonometric transformation splitting (TTS) of the real symmetric Toeplitz matrix A. Theoretical analyses show that if the generating function f of the n × n Toeplitz matrix A is a real positive even function, then the TTS iterative methods converge to the unique solution of the linear system of equations for sufficient large n. Moreover, we derive an upper bound of the contraction factor of the TTS iteration which is dependent solely on the spectra of the two TTS matrices involved. Different from the CSCS iterative method in Ng (2003) in which all operations counts concern complex operations when the DFTs are employed, even if the Toeplitz matrix A is real and symmetric, our method only involves real arithmetics when the DCTs and DSTs are used. The numerical experiments show that our method works better than CSCS iterative method and much better than the positive definite and skew-symmetric splitting (PSS) iterative method in Bai et al. (2005) and the symmetric Gauss–Seidel (SGS) iterative method.
publishDate 2018
dc.date.none.fl_str_mv 2018
2018-01-01T00:00:00Z
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/article
format article
status_str publishedVersion
dc.identifier.uri.fl_str_mv http://hdl.handle.net/1822/54634
url http://hdl.handle.net/1822/54634
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv 0898-1221
10.1016/j.camwa.2018.01.008
www.elsevier.com/locate/camwa
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
instacron:RCAAP
instname_str Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
instacron_str RCAAP
institution RCAAP
reponame_str Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
collection Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
repository.name.fl_str_mv Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
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