Lower complexity bounds for interpolation algorithms

Autores
Gimenez, Nardo Ariel; Heintz, Joos Ulrich; Matera, Guillermo; Solernó, Pablo Luis
Año de publicación
2011
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We introduce and discuss a new computational model for the HermiteLagrange interpolation with nonlinear classes of polynomial interpolants. We distinguish between an interpolation problem and an algorithm that solves it. Our model includes also coalescence phenomena and captures a large variety of known HermiteLagrange interpolation problems and algorithms. Like in traditional HermiteLagrange interpolation, our model is based on the execution of arithmetic operations (including divisions) in the field where the data (nodes and values) are interpreted and arithmetic operations are counted at unit cost. This leads us to a new view of rational functions and maps defined on arbitrary constructible subsets of complex affine spaces. For this purpose we have to develop new tools in algebraic geometry which themselves are mainly based on Zariski's Main Theorem and the theory of places (or equivalently: valuations). We finish this paper by exhibiting two examples of Lagrange interpolation problems with nonlinear classes of interpolants, which do not admit efficient interpolation algorithms (one of these interpolation problems requires even an exponential quantity of arithmetic operations in terms of the number of the given nodes in order to represent some of the interpolants). In other words, classic Lagrange interpolation algorithms are asymptotically optimal for the solution of these selected interpolation problems and nothing is gained by allowing interpolation algorithms and classes of interpolants to be nonlinear. We show also that classic Lagrange interpolation algorithms are almost optimal for generic nodes and values. This generic data cannot be substantially compressed by using nonlinear techniques. We finish this paper highlighting the close connection of our complexity results in HermiteLagrange interpolation with a modern trend in software engineering: architecture tradeoff analysis methods (ATAM).
Fil: Gimenez, Nardo Ariel. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina
Fil: Heintz, Joos Ulrich. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria; Argentina
Fil: Matera, Guillermo. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Solernó, Pablo Luis. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria; Argentina
Materia
HERMITE--LAGRANGE INTERPOLATION
INTERPOLATION PROBLEM
INTERPOLATION ALGORITHM
COMPUTATIONAL COMPLEXITY
LOWER COMPLEXITY BOUNDS
CONSTRUCTIBLE MAP
RATIONAL MAP
TOPOLOGICALLY ROBUST MAP
GEOMETRICALLY ROBUST MAP
HERMITE--LAGRANGE INTERPOLATION
INTERPOLATION PROBLEM
INTERPOLATION ALGORITHM
COMPUTATIONAL COMPLEXITY
LOWER COMPLEXITY BOUNDS
CONSTRUCTIBLE MAP
RATIONAL MAP
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/113310
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/113310
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Lower complexity bounds for interpolation algorithmsGimenez, Nardo ArielHeintz, Joos UlrichMatera, GuillermoSolernó, Pablo LuisHERMITE--LAGRANGE INTERPOLATIONINTERPOLATION PROBLEMINTERPOLATION ALGORITHMCOMPUTATIONAL COMPLEXITYLOWER COMPLEXITY BOUNDSCONSTRUCTIBLE MAPRATIONAL MAPTOPOLOGICALLY ROBUST MAPGEOMETRICALLY ROBUST MAPHERMITE--LAGRANGE INTERPOLATIONINTERPOLATION PROBLEMINTERPOLATION ALGORITHMCOMPUTATIONAL COMPLEXITYLOWER COMPLEXITY BOUNDSCONSTRUCTIBLE MAPRATIONAL MAPhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We introduce and discuss a new computational model for the HermiteLagrange interpolation with nonlinear classes of polynomial interpolants. We distinguish between an interpolation problem and an algorithm that solves it. Our model includes also coalescence phenomena and captures a large variety of known HermiteLagrange interpolation problems and algorithms. Like in traditional HermiteLagrange interpolation, our model is based on the execution of arithmetic operations (including divisions) in the field where the data (nodes and values) are interpreted and arithmetic operations are counted at unit cost. This leads us to a new view of rational functions and maps defined on arbitrary constructible subsets of complex affine spaces. For this purpose we have to develop new tools in algebraic geometry which themselves are mainly based on Zariski's Main Theorem and the theory of places (or equivalently: valuations). We finish this paper by exhibiting two examples of Lagrange interpolation problems with nonlinear classes of interpolants, which do not admit efficient interpolation algorithms (one of these interpolation problems requires even an exponential quantity of arithmetic operations in terms of the number of the given nodes in order to represent some of the interpolants). In other words, classic Lagrange interpolation algorithms are asymptotically optimal for the solution of these selected interpolation problems and nothing is gained by allowing interpolation algorithms and classes of interpolants to be nonlinear. We show also that classic Lagrange interpolation algorithms are almost optimal for generic nodes and values. This generic data cannot be substantially compressed by using nonlinear techniques. We finish this paper highlighting the close connection of our complexity results in HermiteLagrange interpolation with a modern trend in software engineering: architecture tradeoff analysis methods (ATAM).Fil: Gimenez, Nardo Ariel. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; ArgentinaFil: Heintz, Joos Ulrich. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria; ArgentinaFil: Matera, Guillermo. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Solernó, Pablo Luis. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria; ArgentinaAcademic Press Inc Elsevier Science2011-04info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/113310Gimenez, Nardo Ariel; Heintz, Joos Ulrich; Matera, Guillermo; Solernó, Pablo Luis; Lower complexity bounds for interpolation algorithms; Academic Press Inc Elsevier Science; Journal Of Complexity; 27; 2; 4-2011; 151-1870885-064XCONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0885064X10000956info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jco.2010.10.003info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-03T09:44:02Zoai:ri.conicet.gov.ar:11336/113310instacron: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:34982025-09-03 09:44:02.35CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Lower complexity bounds for interpolation algorithms
title Lower complexity bounds for interpolation algorithms
spellingShingle Lower complexity bounds for interpolation algorithms
Gimenez, Nardo Ariel
HERMITE--LAGRANGE INTERPOLATION
INTERPOLATION PROBLEM
INTERPOLATION ALGORITHM
COMPUTATIONAL COMPLEXITY
LOWER COMPLEXITY BOUNDS
CONSTRUCTIBLE MAP
RATIONAL MAP
TOPOLOGICALLY ROBUST MAP
GEOMETRICALLY ROBUST MAP
HERMITE--LAGRANGE INTERPOLATION
INTERPOLATION PROBLEM
INTERPOLATION ALGORITHM
COMPUTATIONAL COMPLEXITY
LOWER COMPLEXITY BOUNDS
CONSTRUCTIBLE MAP
RATIONAL MAP
title_short Lower complexity bounds for interpolation algorithms
title_full Lower complexity bounds for interpolation algorithms
title_fullStr Lower complexity bounds for interpolation algorithms
title_full_unstemmed Lower complexity bounds for interpolation algorithms
title_sort Lower complexity bounds for interpolation algorithms
dc.creator.none.fl_str_mv Gimenez, Nardo Ariel
Heintz, Joos Ulrich
Matera, Guillermo
Solernó, Pablo Luis
author Gimenez, Nardo Ariel
author_facet Gimenez, Nardo Ariel
Heintz, Joos Ulrich
Matera, Guillermo
Solernó, Pablo Luis
author_role author
author2 Heintz, Joos Ulrich
Matera, Guillermo
Solernó, Pablo Luis
author2_role author
author
author
dc.subject.none.fl_str_mv HERMITE--LAGRANGE INTERPOLATION
INTERPOLATION PROBLEM
INTERPOLATION ALGORITHM
COMPUTATIONAL COMPLEXITY
LOWER COMPLEXITY BOUNDS
CONSTRUCTIBLE MAP
RATIONAL MAP
TOPOLOGICALLY ROBUST MAP
GEOMETRICALLY ROBUST MAP
HERMITE--LAGRANGE INTERPOLATION
INTERPOLATION PROBLEM
INTERPOLATION ALGORITHM
COMPUTATIONAL COMPLEXITY
LOWER COMPLEXITY BOUNDS
CONSTRUCTIBLE MAP
RATIONAL MAP
topic HERMITE--LAGRANGE INTERPOLATION
INTERPOLATION PROBLEM
INTERPOLATION ALGORITHM
COMPUTATIONAL COMPLEXITY
LOWER COMPLEXITY BOUNDS
CONSTRUCTIBLE MAP
RATIONAL MAP
TOPOLOGICALLY ROBUST MAP
GEOMETRICALLY ROBUST MAP
HERMITE--LAGRANGE INTERPOLATION
INTERPOLATION PROBLEM
INTERPOLATION ALGORITHM
COMPUTATIONAL COMPLEXITY
LOWER COMPLEXITY BOUNDS
CONSTRUCTIBLE MAP
RATIONAL MAP
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 introduce and discuss a new computational model for the HermiteLagrange interpolation with nonlinear classes of polynomial interpolants. We distinguish between an interpolation problem and an algorithm that solves it. Our model includes also coalescence phenomena and captures a large variety of known HermiteLagrange interpolation problems and algorithms. Like in traditional HermiteLagrange interpolation, our model is based on the execution of arithmetic operations (including divisions) in the field where the data (nodes and values) are interpreted and arithmetic operations are counted at unit cost. This leads us to a new view of rational functions and maps defined on arbitrary constructible subsets of complex affine spaces. For this purpose we have to develop new tools in algebraic geometry which themselves are mainly based on Zariski's Main Theorem and the theory of places (or equivalently: valuations). We finish this paper by exhibiting two examples of Lagrange interpolation problems with nonlinear classes of interpolants, which do not admit efficient interpolation algorithms (one of these interpolation problems requires even an exponential quantity of arithmetic operations in terms of the number of the given nodes in order to represent some of the interpolants). In other words, classic Lagrange interpolation algorithms are asymptotically optimal for the solution of these selected interpolation problems and nothing is gained by allowing interpolation algorithms and classes of interpolants to be nonlinear. We show also that classic Lagrange interpolation algorithms are almost optimal for generic nodes and values. This generic data cannot be substantially compressed by using nonlinear techniques. We finish this paper highlighting the close connection of our complexity results in HermiteLagrange interpolation with a modern trend in software engineering: architecture tradeoff analysis methods (ATAM).
Fil: Gimenez, Nardo Ariel. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina
Fil: Heintz, Joos Ulrich. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria; Argentina
Fil: Matera, Guillermo. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Solernó, Pablo Luis. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria; Argentina
description We introduce and discuss a new computational model for the HermiteLagrange interpolation with nonlinear classes of polynomial interpolants. We distinguish between an interpolation problem and an algorithm that solves it. Our model includes also coalescence phenomena and captures a large variety of known HermiteLagrange interpolation problems and algorithms. Like in traditional HermiteLagrange interpolation, our model is based on the execution of arithmetic operations (including divisions) in the field where the data (nodes and values) are interpreted and arithmetic operations are counted at unit cost. This leads us to a new view of rational functions and maps defined on arbitrary constructible subsets of complex affine spaces. For this purpose we have to develop new tools in algebraic geometry which themselves are mainly based on Zariski's Main Theorem and the theory of places (or equivalently: valuations). We finish this paper by exhibiting two examples of Lagrange interpolation problems with nonlinear classes of interpolants, which do not admit efficient interpolation algorithms (one of these interpolation problems requires even an exponential quantity of arithmetic operations in terms of the number of the given nodes in order to represent some of the interpolants). In other words, classic Lagrange interpolation algorithms are asymptotically optimal for the solution of these selected interpolation problems and nothing is gained by allowing interpolation algorithms and classes of interpolants to be nonlinear. We show also that classic Lagrange interpolation algorithms are almost optimal for generic nodes and values. This generic data cannot be substantially compressed by using nonlinear techniques. We finish this paper highlighting the close connection of our complexity results in HermiteLagrange interpolation with a modern trend in software engineering: architecture tradeoff analysis methods (ATAM).
publishDate 2011
dc.date.none.fl_str_mv 2011-04
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/11336/113310
Gimenez, Nardo Ariel; Heintz, Joos Ulrich; Matera, Guillermo; Solernó, Pablo Luis; Lower complexity bounds for interpolation algorithms; Academic Press Inc Elsevier Science; Journal Of Complexity; 27; 2; 4-2011; 151-187
0885-064X
CONICET Digital
CONICET
url http://hdl.handle.net/11336/113310
identifier_str_mv Gimenez, Nardo Ariel; Heintz, Joos Ulrich; Matera, Guillermo; Solernó, Pablo Luis; Lower complexity bounds for interpolation algorithms; Academic Press Inc Elsevier Science; Journal Of Complexity; 27; 2; 4-2011; 151-187
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/S0885064X10000956
info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jco.2010.10.003
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
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 13.13397

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