Boundary value problems for second order linear difference equations: application to the computation of the inverse of generalized Jacobi matrices

We have named generalized Jacobi matrices to those that are practically tridiagonal, except for the two final entries and the two first entries of its first andits last row respectively. This class of matrices encompasses both standard Jacobiand periodic Jacobi matrices that appear in many contexts...

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Autores: Encinas Bachiller, Andrés Marcos|||0000-0001-5588-0373, Jiménez Jiménez, María José|||0000-0003-3502-462X
Tipo de recurso: artículo
Fecha de publicación:2019
País:España
Institución:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/168258
Acceso en línea:https://hdl.handle.net/2117/168258
https://dx.doi.org/10.1007/s13398-019-00723-3
Access Level:acceso abierto
Palabra clave:Second order difference equation
Boundary value problem
Generalized Jacobi matrix
Anàlisi matricial
Classificació AMS::34 Ordinary differential equations::34B Boundary value problems
Classificació AMS::39 Difference and functional equations::39A Difference equations
Classificació AMS::15 Linear and multilinear algebra
matrix theory
Classificació AMS::11 Number theory::11B Sequences and sets
Àrees temàtiques de la UPC::Matemàtiques i estadística
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repository_id_str
spelling Boundary value problems for second order linear difference equations: application to the computation of the inverse of generalized Jacobi matricesEncinas Bachiller, Andrés Marcos|||0000-0001-5588-0373Jiménez Jiménez, María José|||0000-0003-3502-462XSecond order difference equationBoundary value problemGeneralized Jacobi matrixAnàlisi matricialClassificació AMS::34 Ordinary differential equations::34B Boundary value problemsClassificació AMS::39 Difference and functional equations::39A Difference equationsClassificació AMS::15 Linear and multilinear algebramatrix theoryClassificació AMS::11 Number theory::11B Sequences and setsÀrees temàtiques de la UPC::Matemàtiques i estadísticaWe have named generalized Jacobi matrices to those that are practically tridiagonal, except for the two final entries and the two first entries of its first andits last row respectively. This class of matrices encompasses both standard Jacobiand periodic Jacobi matrices that appear in many contexts in pure and appliedmathematics. Therefore, the study of the inverse of these matrices becomes ofspecific interest. However, explicit formulas for inverses are known only in a fewcases, in particular when the coefficients of the diagonal entries are subjected tosome restrictions.We will show that the inverse of generalized Jacobi matrices can be raisedin terms of the resolution of a boundary value problem associated with a secondorder linear difference equation. In fact, recent advances in the study of lineardifference equations, allow us to compute the solution of this kind of boundaryvalue problems. So, the conditions that ensure the uniqueness of the solution ofthe boundary value problem leads to the invertibility conditions for the matrix,whereas that solutions for suitable problems provide explicitly the entries of theinverse matrix.Peer ReviewedSpringer20192019-07-2220192019-09-16journal articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/168258https://dx.doi.org/10.1007/s13398-019-00723-3reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NonCommercial-NoDerivs 3.0 Spainhttp://creativecommons.org/licenses/by-nc-nd/3.0/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/1682582026-05-27T15:37:01Z
dc.title.none.fl_str_mv Boundary value problems for second order linear difference equations: application to the computation of the inverse of generalized Jacobi matrices
title Boundary value problems for second order linear difference equations: application to the computation of the inverse of generalized Jacobi matrices
spellingShingle Boundary value problems for second order linear difference equations: application to the computation of the inverse of generalized Jacobi matrices
Encinas Bachiller, Andrés Marcos|||0000-0001-5588-0373
Second order difference equation
Boundary value problem
Generalized Jacobi matrix
Anàlisi matricial
Classificació AMS::34 Ordinary differential equations::34B Boundary value problems
Classificació AMS::39 Difference and functional equations::39A Difference equations
Classificació AMS::15 Linear and multilinear algebra
matrix theory
Classificació AMS::11 Number theory::11B Sequences and sets
Àrees temàtiques de la UPC::Matemàtiques i estadística
title_short Boundary value problems for second order linear difference equations: application to the computation of the inverse of generalized Jacobi matrices
title_full Boundary value problems for second order linear difference equations: application to the computation of the inverse of generalized Jacobi matrices
title_fullStr Boundary value problems for second order linear difference equations: application to the computation of the inverse of generalized Jacobi matrices
title_full_unstemmed Boundary value problems for second order linear difference equations: application to the computation of the inverse of generalized Jacobi matrices
title_sort Boundary value problems for second order linear difference equations: application to the computation of the inverse of generalized Jacobi matrices
dc.creator.none.fl_str_mv Encinas Bachiller, Andrés Marcos|||0000-0001-5588-0373
Jiménez Jiménez, María José|||0000-0003-3502-462X
author Encinas Bachiller, Andrés Marcos|||0000-0001-5588-0373
author_facet Encinas Bachiller, Andrés Marcos|||0000-0001-5588-0373
Jiménez Jiménez, María José|||0000-0003-3502-462X
author_role author
author2 Jiménez Jiménez, María José|||0000-0003-3502-462X
author2_role author
dc.subject.none.fl_str_mv Second order difference equation
Boundary value problem
Generalized Jacobi matrix
Anàlisi matricial
Classificació AMS::34 Ordinary differential equations::34B Boundary value problems
Classificació AMS::39 Difference and functional equations::39A Difference equations
Classificació AMS::15 Linear and multilinear algebra
matrix theory
Classificació AMS::11 Number theory::11B Sequences and sets
Àrees temàtiques de la UPC::Matemàtiques i estadística
topic Second order difference equation
Boundary value problem
Generalized Jacobi matrix
Anàlisi matricial
Classificació AMS::34 Ordinary differential equations::34B Boundary value problems
Classificació AMS::39 Difference and functional equations::39A Difference equations
Classificació AMS::15 Linear and multilinear algebra
matrix theory
Classificació AMS::11 Number theory::11B Sequences and sets
Àrees temàtiques de la UPC::Matemàtiques i estadística
description We have named generalized Jacobi matrices to those that are practically tridiagonal, except for the two final entries and the two first entries of its first andits last row respectively. This class of matrices encompasses both standard Jacobiand periodic Jacobi matrices that appear in many contexts in pure and appliedmathematics. Therefore, the study of the inverse of these matrices becomes ofspecific interest. However, explicit formulas for inverses are known only in a fewcases, in particular when the coefficients of the diagonal entries are subjected tosome restrictions.We will show that the inverse of generalized Jacobi matrices can be raisedin terms of the resolution of a boundary value problem associated with a secondorder linear difference equation. In fact, recent advances in the study of lineardifference equations, allow us to compute the solution of this kind of boundaryvalue problems. So, the conditions that ensure the uniqueness of the solution ofthe boundary value problem leads to the invertibility conditions for the matrix,whereas that solutions for suitable problems provide explicitly the entries of theinverse matrix.
publishDate 2019
dc.date.none.fl_str_mv 2019
2019-07-22
2019
2019-09-16
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
AM
http://purl.org/coar/version/c_ab4af688f83e57aa
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/168258
https://dx.doi.org/10.1007/s13398-019-00723-3
url https://hdl.handle.net/2117/168258
https://dx.doi.org/10.1007/s13398-019-00723-3
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-NonCommercial-NoDerivs 3.0 Spain
http://creativecommons.org/licenses/by-nc-nd/3.0/es/
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-NonCommercial-NoDerivs 3.0 Spain
http://creativecommons.org/licenses/by-nc-nd/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.300719

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