A neural network-based approach for approximating arbitrary roots of polynomials
Autor(a) principal: | |
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Data de Publicação: | 2021 |
Outros Autores: | , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10400.13/3739 |
Resumo: | Finding arbitrary roots of polynomials is a fundamental problem in various areas of science and engineering. A myriad of methods was suggested to address this problem, such as the sequential Newton’s method and the Durand–Kerner (D–K) simultaneous iterative method. The sequential iterative methods, on the one hand, need to use a deflation procedure in order to compute approximations to all the roots of a given polynomial, which can produce inaccurate results due to the accumulation of rounding errors. On the other hand, the simultaneous iterative methods require good initial guesses to converge. However, Artificial Neural Networks (ANNs) are widely known by their capacity to find complex mappings between the dependent and independent variables. In view of this, this paper aims to determine, based on comparative results, whether ANNs can be used to compute approximations to the real and complex roots of a given polynomial, as an alternative to simultaneous iterative algorithms like the D–K method. Although the results are very encouraging and demonstrate the viability and potentiality of the suggested approach, the ANNs were not able to surpass the accuracy of the D–K method. The results indicated, however, that the use of the approximations computed by the ANNs as the initial guesses for the D–K method can be beneficial to the accuracy of this method |
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A neural network-based approach for approximating arbitrary roots of polynomialsPolynomial rootsArtificial neural networksParticle swarm optimizationDurand– Kerner methodPerformance analysis.Faculdade de Ciências Exatas e da EngenhariaFinding arbitrary roots of polynomials is a fundamental problem in various areas of science and engineering. A myriad of methods was suggested to address this problem, such as the sequential Newton’s method and the Durand–Kerner (D–K) simultaneous iterative method. The sequential iterative methods, on the one hand, need to use a deflation procedure in order to compute approximations to all the roots of a given polynomial, which can produce inaccurate results due to the accumulation of rounding errors. On the other hand, the simultaneous iterative methods require good initial guesses to converge. However, Artificial Neural Networks (ANNs) are widely known by their capacity to find complex mappings between the dependent and independent variables. In view of this, this paper aims to determine, based on comparative results, whether ANNs can be used to compute approximations to the real and complex roots of a given polynomial, as an alternative to simultaneous iterative algorithms like the D–K method. Although the results are very encouraging and demonstrate the viability and potentiality of the suggested approach, the ANNs were not able to surpass the accuracy of the D–K method. The results indicated, however, that the use of the approximations computed by the ANNs as the initial guesses for the D–K method can be beneficial to the accuracy of this methodMDPIDigitUMaFreitas, DiogoLopes, LuizMorgado-Dias, Fernando2021-10-20T12:39:05Z20212021-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10400.13/3739engFreitas, D., Lopes Guerreiro, L., & Morgado-Dias, F. (2021). A neural network-based approach for approximating arbitrary roots of polynomials. Mathematics, 9(4), 317. https://doi.org/10.3390/math904031710.3390/math9040317info: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:RCAAP2023-03-19T05:36:04Zoai:digituma.uma.pt:10400.13/3739Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T15:07:08.082355Repositório 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 |
A neural network-based approach for approximating arbitrary roots of polynomials |
title |
A neural network-based approach for approximating arbitrary roots of polynomials |
spellingShingle |
A neural network-based approach for approximating arbitrary roots of polynomials Freitas, Diogo Polynomial roots Artificial neural networks Particle swarm optimization Durand– Kerner method Performance analysis . Faculdade de Ciências Exatas e da Engenharia |
title_short |
A neural network-based approach for approximating arbitrary roots of polynomials |
title_full |
A neural network-based approach for approximating arbitrary roots of polynomials |
title_fullStr |
A neural network-based approach for approximating arbitrary roots of polynomials |
title_full_unstemmed |
A neural network-based approach for approximating arbitrary roots of polynomials |
title_sort |
A neural network-based approach for approximating arbitrary roots of polynomials |
author |
Freitas, Diogo |
author_facet |
Freitas, Diogo Lopes, Luiz Morgado-Dias, Fernando |
author_role |
author |
author2 |
Lopes, Luiz Morgado-Dias, Fernando |
author2_role |
author author |
dc.contributor.none.fl_str_mv |
DigitUMa |
dc.contributor.author.fl_str_mv |
Freitas, Diogo Lopes, Luiz Morgado-Dias, Fernando |
dc.subject.por.fl_str_mv |
Polynomial roots Artificial neural networks Particle swarm optimization Durand– Kerner method Performance analysis . Faculdade de Ciências Exatas e da Engenharia |
topic |
Polynomial roots Artificial neural networks Particle swarm optimization Durand– Kerner method Performance analysis . Faculdade de Ciências Exatas e da Engenharia |
description |
Finding arbitrary roots of polynomials is a fundamental problem in various areas of science and engineering. A myriad of methods was suggested to address this problem, such as the sequential Newton’s method and the Durand–Kerner (D–K) simultaneous iterative method. The sequential iterative methods, on the one hand, need to use a deflation procedure in order to compute approximations to all the roots of a given polynomial, which can produce inaccurate results due to the accumulation of rounding errors. On the other hand, the simultaneous iterative methods require good initial guesses to converge. However, Artificial Neural Networks (ANNs) are widely known by their capacity to find complex mappings between the dependent and independent variables. In view of this, this paper aims to determine, based on comparative results, whether ANNs can be used to compute approximations to the real and complex roots of a given polynomial, as an alternative to simultaneous iterative algorithms like the D–K method. Although the results are very encouraging and demonstrate the viability and potentiality of the suggested approach, the ANNs were not able to surpass the accuracy of the D–K method. The results indicated, however, that the use of the approximations computed by the ANNs as the initial guesses for the D–K method can be beneficial to the accuracy of this method |
publishDate |
2021 |
dc.date.none.fl_str_mv |
2021-10-20T12:39:05Z 2021 2021-01-01T00:00:00Z |
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/10400.13/3739 |
url |
http://hdl.handle.net/10400.13/3739 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Freitas, D., Lopes Guerreiro, L., & Morgado-Dias, F. (2021). A neural network-based approach for approximating arbitrary roots of polynomials. Mathematics, 9(4), 317. https://doi.org/10.3390/math9040317 10.3390/math9040317 |
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 |
MDPI |
publisher.none.fl_str_mv |
MDPI |
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 |
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Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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RCAAP |
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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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1799129941052227584 |