A numerical algorithm for zero counting, I: Complexity and accuracy

Autores
Cucker, Felipe; Krick, Teresa Elena Genoveva; Malajovich, Gregorio; Wschebor, Mario
Año de publicación
2008
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f. The algorithm performs O(log(nDκ(f)))iterations (grid refinements) where n is the number of polynomials (as well as the dimension of the ambient space), D is a bound on the polynomials’ degree, and κ(f) is a condition number for the system. Each iteration uses an exponential number of operations. The algorithm uses finite-precision arithmetic and a major feature of our results is a bound for the precision required to ensure that the returned output is correct which is polynomial in n and D and logarithmic in κ(f). The algorithm parallelizes well in the sense that each iteration can be computed in parallel polynomial time in n, log D and log(κ(f)).
Fil: Cucker, Felipe. City University of Hong Kong; Hong Kong
Fil: Krick, Teresa Elena Genoveva. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
Fil: Malajovich, Gregorio. Universidade Federal do Rio de Janeiro; Brasil
Fil: Wschebor, Mario. Universidad de la República; Uruguay
Materia
Polynomial system
Condition
Number of roots
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/275406
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/275406
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling A numerical algorithm for zero counting, I: Complexity and accuracyCucker, FelipeKrick, Teresa Elena GenovevaMalajovich, GregorioWschebor, MarioPolynomial systemConditionNumber of rootshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f. The algorithm performs O(log(nDκ(f)))iterations (grid refinements) where n is the number of polynomials (as well as the dimension of the ambient space), D is a bound on the polynomials’ degree, and κ(f) is a condition number for the system. Each iteration uses an exponential number of operations. The algorithm uses finite-precision arithmetic and a major feature of our results is a bound for the precision required to ensure that the returned output is correct which is polynomial in n and D and logarithmic in κ(f). The algorithm parallelizes well in the sense that each iteration can be computed in parallel polynomial time in n, log D and log(κ(f)).Fil: Cucker, Felipe. City University of Hong Kong; Hong KongFil: Krick, Teresa Elena Genoveva. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Malajovich, Gregorio. Universidade Federal do Rio de Janeiro; BrasilFil: Wschebor, Mario. Universidad de la República; UruguayAcademic Press Inc Elsevier Science2008-10info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/275406Cucker, Felipe; Krick, Teresa Elena Genoveva; Malajovich, Gregorio; Wschebor, Mario; A numerical algorithm for zero counting, I: Complexity and accuracy; Academic Press Inc Elsevier Science; Journal Of Complexity; 24; 5-6; 10-2008; 582-6050885-064XCONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0885064X08000162info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jco.2008.03.001info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-12-23T14:54:40Zoai:ri.conicet.gov.ar:11336/275406instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-12-23 14:54:40.994CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv A numerical algorithm for zero counting, I: Complexity and accuracy
title A numerical algorithm for zero counting, I: Complexity and accuracy
spellingShingle A numerical algorithm for zero counting, I: Complexity and accuracy
Cucker, Felipe
Polynomial system
Condition
Number of roots
title_short A numerical algorithm for zero counting, I: Complexity and accuracy
title_full A numerical algorithm for zero counting, I: Complexity and accuracy
title_fullStr A numerical algorithm for zero counting, I: Complexity and accuracy
title_full_unstemmed A numerical algorithm for zero counting, I: Complexity and accuracy
title_sort A numerical algorithm for zero counting, I: Complexity and accuracy
dc.creator.none.fl_str_mv Cucker, Felipe
Krick, Teresa Elena Genoveva
Malajovich, Gregorio
Wschebor, Mario
author Cucker, Felipe
author_facet Cucker, Felipe
Krick, Teresa Elena Genoveva
Malajovich, Gregorio
Wschebor, Mario
author_role author
author2 Krick, Teresa Elena Genoveva
Malajovich, Gregorio
Wschebor, Mario
author2_role author
author
author
dc.subject.none.fl_str_mv Polynomial system
Condition
Number of roots
topic Polynomial system
Condition
Number of roots
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv We describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f. The algorithm performs O(log(nDκ(f)))iterations (grid refinements) where n is the number of polynomials (as well as the dimension of the ambient space), D is a bound on the polynomials’ degree, and κ(f) is a condition number for the system. Each iteration uses an exponential number of operations. The algorithm uses finite-precision arithmetic and a major feature of our results is a bound for the precision required to ensure that the returned output is correct which is polynomial in n and D and logarithmic in κ(f). The algorithm parallelizes well in the sense that each iteration can be computed in parallel polynomial time in n, log D and log(κ(f)).
Fil: Cucker, Felipe. City University of Hong Kong; Hong Kong
Fil: Krick, Teresa Elena Genoveva. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
Fil: Malajovich, Gregorio. Universidade Federal do Rio de Janeiro; Brasil
Fil: Wschebor, Mario. Universidad de la República; Uruguay
description We describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f. The algorithm performs O(log(nDκ(f)))iterations (grid refinements) where n is the number of polynomials (as well as the dimension of the ambient space), D is a bound on the polynomials’ degree, and κ(f) is a condition number for the system. Each iteration uses an exponential number of operations. The algorithm uses finite-precision arithmetic and a major feature of our results is a bound for the precision required to ensure that the returned output is correct which is polynomial in n and D and logarithmic in κ(f). The algorithm parallelizes well in the sense that each iteration can be computed in parallel polynomial time in n, log D and log(κ(f)).
publishDate 2008
dc.date.none.fl_str_mv 2008-10
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/275406
Cucker, Felipe; Krick, Teresa Elena Genoveva; Malajovich, Gregorio; Wschebor, Mario; A numerical algorithm for zero counting, I: Complexity and accuracy; Academic Press Inc Elsevier Science; Journal Of Complexity; 24; 5-6; 10-2008; 582-605
0885-064X
CONICET Digital
CONICET
url http://hdl.handle.net/11336/275406
identifier_str_mv Cucker, Felipe; Krick, Teresa Elena Genoveva; Malajovich, Gregorio; Wschebor, Mario; A numerical algorithm for zero counting, I: Complexity and accuracy; Academic Press Inc Elsevier Science; Journal Of Complexity; 24; 5-6; 10-2008; 582-605
0885-064X
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0885064X08000162
info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jco.2008.03.001
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Academic Press Inc Elsevier Science
publisher.none.fl_str_mv Academic Press Inc Elsevier Science
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 12.952241

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