A Heuristic Method for Solving Polynomial Matrix Equations

We propose a heuristic method to solve polynomial matrix equations of the type ∑=1= , where are scalar coefficients and X and B are square matrices of order n. The method is based on the decomposition of the B matrix as a linear combination of the identity matrix and an idempotent, involutive, or ni...

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Detalles Bibliográficos
Autores: González-Santander Martínez, Juan Luis|||0000-0001-5348-4967, Sánchez Lasheras, Fernando|||0000-0002-7052-2811
Tipo de recurso: artículo
Fecha de publicación:2024
País:España
Institución:Universidad de Oviedo (UNIOVI)
Repositorio:RUO. Repositorio Institucional de la Universidad de Oviedo
Idioma:inglés
OAI Identifier:oai:digibuo.uniovi.es:10651/72148
Acceso en línea:https://hdl.handle.net/10651/72148
https://dx.doi.org/10.3390/axioms13040239
Access Level:acceso abierto
Palabra clave:polynomial matrix equations
idempotent matrix
involutive matrix
nilpotent matrix
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spelling A Heuristic Method for Solving Polynomial Matrix EquationsGonzález-Santander Martínez, Juan Luis|||0000-0001-5348-4967Sánchez Lasheras, Fernando|||0000-0002-7052-2811polynomial matrix equationsidempotent matrixinvolutive matrixnilpotent matrixWe propose a heuristic method to solve polynomial matrix equations of the type ∑=1= , where are scalar coefficients and X and B are square matrices of order n. The method is based on the decomposition of the B matrix as a linear combination of the identity matrix and an idempotent, involutive, or nilpotent matrix. We prove that this decomposition is always possible when =2. Moreover, in some cases we can compute solutions when we have an infinite number of them (singular solutions). This method has been coded in MATLAB and has been compared to other methods found in the existing literature, such as the diagonalization and the interpolation methods. It turns out that the proposed method is considerably faster than the latter methods. Furthermore, the proposed method can calculate solutions when diagonalization and interpolation methods fail or calculate singular solutions when these methods are not capable of doing so.20242024-04-04journal articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articlehttps://hdl.handle.net/10651/72148https://dx.doi.org/10.3390/axioms13040239reponame:RUO. Repositorio Institucional de la Universidad de Oviedoinstname:Universidad de Oviedo (UNIOVI)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:digibuo.uniovi.es:10651/721482026-06-07T06:38:51Z
dc.title.none.fl_str_mv A Heuristic Method for Solving Polynomial Matrix Equations
title A Heuristic Method for Solving Polynomial Matrix Equations
spellingShingle A Heuristic Method for Solving Polynomial Matrix Equations
González-Santander Martínez, Juan Luis|||0000-0001-5348-4967
polynomial matrix equations
idempotent matrix
involutive matrix
nilpotent matrix
title_short A Heuristic Method for Solving Polynomial Matrix Equations
title_full A Heuristic Method for Solving Polynomial Matrix Equations
title_fullStr A Heuristic Method for Solving Polynomial Matrix Equations
title_full_unstemmed A Heuristic Method for Solving Polynomial Matrix Equations
title_sort A Heuristic Method for Solving Polynomial Matrix Equations
dc.creator.none.fl_str_mv González-Santander Martínez, Juan Luis|||0000-0001-5348-4967
Sánchez Lasheras, Fernando|||0000-0002-7052-2811
author González-Santander Martínez, Juan Luis|||0000-0001-5348-4967
author_facet González-Santander Martínez, Juan Luis|||0000-0001-5348-4967
Sánchez Lasheras, Fernando|||0000-0002-7052-2811
author_role author
author2 Sánchez Lasheras, Fernando|||0000-0002-7052-2811
author2_role author
dc.subject.none.fl_str_mv polynomial matrix equations
idempotent matrix
involutive matrix
nilpotent matrix
topic polynomial matrix equations
idempotent matrix
involutive matrix
nilpotent matrix
description We propose a heuristic method to solve polynomial matrix equations of the type ∑=1= , where are scalar coefficients and X and B are square matrices of order n. The method is based on the decomposition of the B matrix as a linear combination of the identity matrix and an idempotent, involutive, or nilpotent matrix. We prove that this decomposition is always possible when =2. Moreover, in some cases we can compute solutions when we have an infinite number of them (singular solutions). This method has been coded in MATLAB and has been compared to other methods found in the existing literature, such as the diagonalization and the interpolation methods. It turns out that the proposed method is considerably faster than the latter methods. Furthermore, the proposed method can calculate solutions when diagonalization and interpolation methods fail or calculate singular solutions when these methods are not capable of doing so.
publishDate 2024
dc.date.none.fl_str_mv 2024
2024-04-04
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://hdl.handle.net/10651/72148
https://dx.doi.org/10.3390/axioms13040239
url https://hdl.handle.net/10651/72148
https://dx.doi.org/10.3390/axioms13040239
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution 4.0 International
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
Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
dc.source.none.fl_str_mv reponame:RUO. Repositorio Institucional de la Universidad de Oviedo
instname:Universidad de Oviedo (UNIOVI)
instname_str Universidad de Oviedo (UNIOVI)
reponame_str RUO. Repositorio Institucional de la Universidad de Oviedo
collection RUO. Repositorio Institucional de la Universidad de Oviedo
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,301603

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