Reliable eigenvalues of symmetric tridiagonals
Autor(a) principal: | |
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Data de Publicação: | 2011 |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/1822/16203 |
Resumo: | For the eigenvalues of a symmetric tridiagonal matrix T, the most accurate algorithms deliver approximations which are the exact eigenvalues of a matrix whose entries differ from the corresponding entries of T by small relative perturbations. However, for matrices with eigenvalues of different magnitudes, the number of correct digits in the computed approximations for eigenvalues of size smaller than ‖T‖₂ depends on how well such eigenvalues are defined by the data. Some classes of matrices are known to define their eigenvalues to high relative accuracy but, in general, there is no simple way to estimate well the number of correct digits in the approximations. To remedy this, we propose a method that provides sharp bounds for the eigenvalues of T. We present some numerical examples to illustrate the usefulness of our method. |
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Reliable eigenvalues of symmetric tridiagonalsSymmetric tridiagonalsBisection methodBounds for eigenvaluesScience & TechnologyFor the eigenvalues of a symmetric tridiagonal matrix T, the most accurate algorithms deliver approximations which are the exact eigenvalues of a matrix whose entries differ from the corresponding entries of T by small relative perturbations. However, for matrices with eigenvalues of different magnitudes, the number of correct digits in the computed approximations for eigenvalues of size smaller than ‖T‖₂ depends on how well such eigenvalues are defined by the data. Some classes of matrices are known to define their eigenvalues to high relative accuracy but, in general, there is no simple way to estimate well the number of correct digits in the approximations. To remedy this, we propose a method that provides sharp bounds for the eigenvalues of T. We present some numerical examples to illustrate the usefulness of our method.FEDER (Programa Operacional Factores de Competitividade)FCT (Projecto PEst-C/MAT/UI0013/2011Society for Industrial and Applied MathematicsUniversidade do MinhoRalha, Rui2011-122011-12-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/1822/16203eng0895-479810.1137/100817413http://epubs.siam.org/sima/resource/1/sjmael/v32/i4/p1524_s1?isAuthorized=noinfo: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:RCAAP2023-07-21T12:12:28Zoai:repositorium.sdum.uminho.pt:1822/16203Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T19:04:23.684083Repositório 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 |
Reliable eigenvalues of symmetric tridiagonals |
title |
Reliable eigenvalues of symmetric tridiagonals |
spellingShingle |
Reliable eigenvalues of symmetric tridiagonals Ralha, Rui Symmetric tridiagonals Bisection method Bounds for eigenvalues Science & Technology |
title_short |
Reliable eigenvalues of symmetric tridiagonals |
title_full |
Reliable eigenvalues of symmetric tridiagonals |
title_fullStr |
Reliable eigenvalues of symmetric tridiagonals |
title_full_unstemmed |
Reliable eigenvalues of symmetric tridiagonals |
title_sort |
Reliable eigenvalues of symmetric tridiagonals |
author |
Ralha, Rui |
author_facet |
Ralha, Rui |
author_role |
author |
dc.contributor.none.fl_str_mv |
Universidade do Minho |
dc.contributor.author.fl_str_mv |
Ralha, Rui |
dc.subject.por.fl_str_mv |
Symmetric tridiagonals Bisection method Bounds for eigenvalues Science & Technology |
topic |
Symmetric tridiagonals Bisection method Bounds for eigenvalues Science & Technology |
description |
For the eigenvalues of a symmetric tridiagonal matrix T, the most accurate algorithms deliver approximations which are the exact eigenvalues of a matrix whose entries differ from the corresponding entries of T by small relative perturbations. However, for matrices with eigenvalues of different magnitudes, the number of correct digits in the computed approximations for eigenvalues of size smaller than ‖T‖₂ depends on how well such eigenvalues are defined by the data. Some classes of matrices are known to define their eigenvalues to high relative accuracy but, in general, there is no simple way to estimate well the number of correct digits in the approximations. To remedy this, we propose a method that provides sharp bounds for the eigenvalues of T. We present some numerical examples to illustrate the usefulness of our method. |
publishDate |
2011 |
dc.date.none.fl_str_mv |
2011-12 2011-12-01T00:00:00Z |
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/1822/16203 |
url |
http://hdl.handle.net/1822/16203 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
0895-4798 10.1137/100817413 http://epubs.siam.org/sima/resource/1/sjmael/v32/i4/p1524_s1?isAuthorized=no |
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 |
Society for Industrial and Applied Mathematics |
publisher.none.fl_str_mv |
Society for Industrial and Applied Mathematics |
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 |
repository.mail.fl_str_mv |
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1799132453593415680 |