An algorithm derivative-free to improve the steffensen-type methods

[EN] Solving equations of the form H(x)=0 is one of the most faced problem in mathematics and in other science fields such as chemistry or physics. This kind of equations cannot be solved without the use of iterative methods. The Steffensen-type methods, defined using divided differences are derivat...

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Autores: Hernández-Verón, Miguel A., Yadav, Sonia, Magreñán, Ángel Alberto, Singh, Sukhjit, Martínez Molada, Eulalia|||0000-0003-2869-4334
Tipo de documento: artigo
Data de publicação:2022
País:España
Recursos:Universitat Politècnica de València (UPV)
Repositório:RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia
Idioma:inglês
OAI Identifier:oai:riunet.upv.es:10251/192218
Acesso em linha:https://riunet.upv.es/handle/10251/192218
Access Level:Acceso aberto
Palavra-chave:Iterative method
Local convergence
Non-differentiable operator
Dynamics
Steffensen&apos
s method
MATEMATICA APLICADA
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repository_id_str
spelling An algorithm derivative-free to improve the steffensen-type methodsHernández-Verón, Miguel A.Yadav, SoniaMagreñán, Ángel AlbertoSingh, SukhjitMartínez Molada, Eulalia|||0000-0003-2869-4334Iterative methodLocal convergenceNon-differentiable operatorDynamicsSteffensen&aposs methodMATEMATICA APLICADA[EN] Solving equations of the form H(x)=0 is one of the most faced problem in mathematics and in other science fields such as chemistry or physics. This kind of equations cannot be solved without the use of iterative methods. The Steffensen-type methods, defined using divided differences are derivative free, are usually considered to solve these problems when H is a non-differentiable operator due to its accuracy and efficiency. However, in general, the accessibility of these iterative methods is small. The main interest of this paper is to improve the accessibility of Steffensen-type methods, this is the set of starting points that converge to the roots applying those methods. So, by means of using a predictor-corrector iterative process we can improve this accessibility. For this, we use a predictor iterative process, using symmetric divided differences, with good accessibility and then, as corrector method, we consider the Center-Steffensen method with quadratic convergence. In addition, the dynamical studies presented show, in an experimental way, that this iterative process also improves the region of accessibility of Steffensen-type methods. Moreover, we analyze the semilocal convergence of the predictor-corrector iterative process proposed in two cases: when H is differentiable and H is non-differentiable. Summing up, we present an effective alternative for Newton's method to non-differentiable operators, where this method cannot be applied. The theoretical results are illustrated with numerical experiments.This research was partially supported by the project PGC2018-095896-B-C21-C22 of Spanish Ministry of Economy and Competitiveness and by the project of Generalitat Valenciana Prometeo/2016/089.MDPI AGEscuela Técnica Superior de Ingeniería de TelecomunicaciónDepartamento de Matemática AplicadaInstituto Universitario de Matemática MultidisciplinarGeneralitat ValencianaAGENCIA ESTATAL DE INVESTIGACIONAgencia Estatal de InvestigaciónRepositorio Institucional de la Universitat Politècnica de València Riunet20222022-01-01journal articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://riunet.upv.es/handle/10251/192218reponame:RiuNet. Repositorio Institucional de la Universitat Politécnica de Valénciainstname:Universitat Politècnica de València (UPV)InglésengAgencia Estatal de Investigación http://dx.doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020 PGC2018-095896-B-C21 DISEÑO, ANALISIS Y ESTABILIDAD DE PROCESOS ITERATIVOS APLICADOS A LAS ECUACIONES INTEGRALES Y MATRICIALES Y A LA COMUNICACION AEROESPACIALGeneralitat Valenciana https://doi.org/10.13039/501100003359 PROMETEO%2F2016%2F089 Resolución de ecuaciones y sistemas no lineales mediante técnicas iterativas: análisis dinámico y aplicacionesAgencia Estatal de Investigación http://dx.doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020 PGC2018-095896-B-C22 DISEÑO, ANALISIS Y ESTABILIDAD DE PROCESOS ITERATIVOS APLICADOS A LAS ECUACIONES INTEGRALES Y MATRICIALES Y A LA COMUNICACION AEROESPACIALopen accesshttp://purl.org/coar/access_right/c_abf2Reconocimiento (by)http://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:riunet.upv.es:10251/1922182026-06-13T07:49:27Z
dc.title.none.fl_str_mv An algorithm derivative-free to improve the steffensen-type methods
title An algorithm derivative-free to improve the steffensen-type methods
spellingShingle An algorithm derivative-free to improve the steffensen-type methods
Hernández-Verón, Miguel A.
Iterative method
Local convergence
Non-differentiable operator
Dynamics
Steffensen&apos
s method
MATEMATICA APLICADA
title_short An algorithm derivative-free to improve the steffensen-type methods
title_full An algorithm derivative-free to improve the steffensen-type methods
title_fullStr An algorithm derivative-free to improve the steffensen-type methods
title_full_unstemmed An algorithm derivative-free to improve the steffensen-type methods
title_sort An algorithm derivative-free to improve the steffensen-type methods
dc.creator.none.fl_str_mv Hernández-Verón, Miguel A.
Yadav, Sonia
Magreñán, Ángel Alberto
Singh, Sukhjit
Martínez Molada, Eulalia|||0000-0003-2869-4334
author Hernández-Verón, Miguel A.
author_facet Hernández-Verón, Miguel A.
Yadav, Sonia
Magreñán, Ángel Alberto
Singh, Sukhjit
Martínez Molada, Eulalia|||0000-0003-2869-4334
author_role author
author2 Yadav, Sonia
Magreñán, Ángel Alberto
Singh, Sukhjit
Martínez Molada, Eulalia|||0000-0003-2869-4334
author2_role author
author
author
author
dc.contributor.none.fl_str_mv Escuela Técnica Superior de Ingeniería de Telecomunicación
Departamento de Matemática Aplicada
Instituto Universitario de Matemática Multidisciplinar
Generalitat Valenciana
AGENCIA ESTATAL DE INVESTIGACION
Agencia Estatal de Investigación
Repositorio Institucional de la Universitat Politècnica de València Riunet
dc.subject.none.fl_str_mv Iterative method
Local convergence
Non-differentiable operator
Dynamics
Steffensen&apos
s method
MATEMATICA APLICADA
topic Iterative method
Local convergence
Non-differentiable operator
Dynamics
Steffensen&apos
s method
MATEMATICA APLICADA
description [EN] Solving equations of the form H(x)=0 is one of the most faced problem in mathematics and in other science fields such as chemistry or physics. This kind of equations cannot be solved without the use of iterative methods. The Steffensen-type methods, defined using divided differences are derivative free, are usually considered to solve these problems when H is a non-differentiable operator due to its accuracy and efficiency. However, in general, the accessibility of these iterative methods is small. The main interest of this paper is to improve the accessibility of Steffensen-type methods, this is the set of starting points that converge to the roots applying those methods. So, by means of using a predictor-corrector iterative process we can improve this accessibility. For this, we use a predictor iterative process, using symmetric divided differences, with good accessibility and then, as corrector method, we consider the Center-Steffensen method with quadratic convergence. In addition, the dynamical studies presented show, in an experimental way, that this iterative process also improves the region of accessibility of Steffensen-type methods. Moreover, we analyze the semilocal convergence of the predictor-corrector iterative process proposed in two cases: when H is differentiable and H is non-differentiable. Summing up, we present an effective alternative for Newton's method to non-differentiable operators, where this method cannot be applied. The theoretical results are illustrated with numerical experiments.
publishDate 2022
dc.date.none.fl_str_mv 2022
2022-01-01
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://riunet.upv.es/handle/10251/192218
url https://riunet.upv.es/handle/10251/192218
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Agencia Estatal de Investigación http://dx.doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020 PGC2018-095896-B-C21 DISEÑO, ANALISIS Y ESTABILIDAD DE PROCESOS ITERATIVOS APLICADOS A LAS ECUACIONES INTEGRALES Y MATRICIALES Y A LA COMUNICACION AEROESPACIAL
Generalitat Valenciana https://doi.org/10.13039/501100003359 PROMETEO%2F2016%2F089 Resolución de ecuaciones y sistemas no lineales mediante técnicas iterativas: análisis dinámico y aplicaciones
Agencia Estatal de Investigación http://dx.doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020 PGC2018-095896-B-C22 DISEÑO, ANALISIS Y ESTABILIDAD DE PROCESOS ITERATIVOS APLICADOS A LAS ECUACIONES INTEGRALES Y MATRICIALES Y A LA COMUNICACION AEROESPACIAL
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Reconocimiento (by)
http://creativecommons.org/licenses/by/4.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Reconocimiento (by)
http://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv MDPI AG
publisher.none.fl_str_mv MDPI AG
dc.source.none.fl_str_mv reponame:RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia
instname:Universitat Politècnica de València (UPV)
instname_str Universitat Politècnica de València (UPV)
reponame_str RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia
collection RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,301603

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