The eigen-structures of real (skew) circulant matrices with some applications

Detalhes bibliográficos
Autor(a) principal: Liu, Zhongyun
Data de Publicação: 2019
Outros Autores: Chen, Siheng, Xu, Weijin, Zhang, Yulin
Tipo de documento: Artigo
Idioma: eng
Título da fonte: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
Texto Completo: https://hdl.handle.net/1822/62618
Resumo: The circulant matrices and skew-circulant matrices are two special classes of Toeplitz matrices and play vital roles in the computation of Toeplitz matrices. In this paper, we focus on real circulant and skew-circulant matrices. We first investigate their real Schur forms, which are closely related to the family of discrete cosine transform (DCT) and discrete sine transform (DST). Using those real Schur forms, we then develop some fast algorithms for computing real circulant, skew-circulant and Toeplitz matrix-real vector multiplications. Also, we develop a DCT-DST version of circulant and skew-circulant splitting (CSCS) iteration for real positive definite Toeplitz systems. Compared with the fast Fourier transform (FFT) version of CSCS iteration, the DCT-DST version is more efficient and saves a half storage. Numerical experiments are presented to illustrate the effectiveness of our method.
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spelling The eigen-structures of real (skew) circulant matrices with some applicationsReal Schur formReal circulant matricesReal skew-circulant matricesCSCS iterationReal Toeplitz matricesCiências Naturais::MatemáticasScience & TechnologyThe circulant matrices and skew-circulant matrices are two special classes of Toeplitz matrices and play vital roles in the computation of Toeplitz matrices. In this paper, we focus on real circulant and skew-circulant matrices. We first investigate their real Schur forms, which are closely related to the family of discrete cosine transform (DCT) and discrete sine transform (DST). Using those real Schur forms, we then develop some fast algorithms for computing real circulant, skew-circulant and Toeplitz matrix-real vector multiplications. Also, we develop a DCT-DST version of circulant and skew-circulant splitting (CSCS) iteration for real positive definite Toeplitz systems. Compared with the fast Fourier transform (FFT) version of CSCS iteration, the DCT-DST version is more efficient and saves a half storage. Numerical experiments are presented to illustrate the effectiveness of our method.The authors would like to thank the supports of the National Natural Science Foundationof China under Grant No. 11371075, the Hunan Key Laboratory of Mathematical Modeling and Analysis inEngineering, and the Portuguese Funds through FCT-Fundação para a Ciência, within the Project UID/ MAT/00013/2013.Springer NatureUniversidade do MinhoLiu, ZhongyunChen, SihengXu, WeijinZhang, Yulin20192019-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/1822/62618eng2238-36031807-030210.1007/s40314-019-0971-9https://link.springer.com/article/10.1007/s40314-019-0971-9info:eu-repo/semantics/openAccessreponame:Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)instname:FCCN, serviços digitais da FCT – Fundação para a Ciência e a Tecnologiainstacron:RCAAP2024-05-11T05:02:21Zoai:repositorium.sdum.uminho.pt:1822/62618Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireinfo@rcaap.ptopendoar:https://opendoar.ac.uk/repository/71602025-05-28T15:06:20.352288Repositórios Científicos de Acesso Aberto de Portugal (RCAAP) - FCCN, serviços digitais da FCT – Fundação para a Ciência e a Tecnologiafalse
dc.title.none.fl_str_mv The eigen-structures of real (skew) circulant matrices with some applications
title The eigen-structures of real (skew) circulant matrices with some applications
spellingShingle The eigen-structures of real (skew) circulant matrices with some applications
Liu, Zhongyun
Real Schur form
Real circulant matrices
Real skew-circulant matrices
CSCS iteration
Real Toeplitz matrices
Ciências Naturais::Matemáticas
Science & Technology
title_short The eigen-structures of real (skew) circulant matrices with some applications
title_full The eigen-structures of real (skew) circulant matrices with some applications
title_fullStr The eigen-structures of real (skew) circulant matrices with some applications
title_full_unstemmed The eigen-structures of real (skew) circulant matrices with some applications
title_sort The eigen-structures of real (skew) circulant matrices with some applications
author Liu, Zhongyun
author_facet Liu, Zhongyun
Chen, Siheng
Xu, Weijin
Zhang, Yulin
author_role author
author2 Chen, Siheng
Xu, Weijin
Zhang, Yulin
author2_role author
author
author
dc.contributor.none.fl_str_mv Universidade do Minho
dc.contributor.author.fl_str_mv Liu, Zhongyun
Chen, Siheng
Xu, Weijin
Zhang, Yulin
dc.subject.por.fl_str_mv Real Schur form
Real circulant matrices
Real skew-circulant matrices
CSCS iteration
Real Toeplitz matrices
Ciências Naturais::Matemáticas
Science & Technology
topic Real Schur form
Real circulant matrices
Real skew-circulant matrices
CSCS iteration
Real Toeplitz matrices
Ciências Naturais::Matemáticas
Science & Technology
description The circulant matrices and skew-circulant matrices are two special classes of Toeplitz matrices and play vital roles in the computation of Toeplitz matrices. In this paper, we focus on real circulant and skew-circulant matrices. We first investigate their real Schur forms, which are closely related to the family of discrete cosine transform (DCT) and discrete sine transform (DST). Using those real Schur forms, we then develop some fast algorithms for computing real circulant, skew-circulant and Toeplitz matrix-real vector multiplications. Also, we develop a DCT-DST version of circulant and skew-circulant splitting (CSCS) iteration for real positive definite Toeplitz systems. Compared with the fast Fourier transform (FFT) version of CSCS iteration, the DCT-DST version is more efficient and saves a half storage. Numerical experiments are presented to illustrate the effectiveness of our method.
publishDate 2019
dc.date.none.fl_str_mv 2019
2019-01-01T00:00:00Z
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 https://hdl.handle.net/1822/62618
url https://hdl.handle.net/1822/62618
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv 2238-3603
1807-0302
10.1007/s40314-019-0971-9
https://link.springer.com/article/10.1007/s40314-019-0971-9
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 Springer Nature
publisher.none.fl_str_mv Springer Nature
dc.source.none.fl_str_mv reponame:Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
instname:FCCN, serviços digitais da FCT – Fundação para a Ciência e a Tecnologia
instacron:RCAAP
instname_str FCCN, serviços digitais da FCT – Fundação para a Ciência e a Tecnologia
instacron_str RCAAP
institution RCAAP
reponame_str Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
collection Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
repository.name.fl_str_mv Repositórios Científicos de Acesso Aberto de Portugal (RCAAP) - FCCN, serviços digitais da FCT – Fundação para a Ciência e a Tecnologia
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