A Symbolic-Numeric Validation Algorithm ...
Type de document :
Compte-rendu et recension critique d'ouvrage
Titre :
A Symbolic-Numeric Validation Algorithm for Linear ODEs with Newton-Picard Method
Auteur(s) :
Bréhard, Florent [Auteur]
Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 [CRIStAL]
Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 [CRIStAL]
Titre de la revue :
Mathematics in Computer Science
Éditeur :
Springer
Date de publication :
2021
ISSN :
1661-8270
Mot(s)-clé(s) en anglais :
validated numerics
a posteriori validation
Chebyshev spectral methods
linear differential equations
D-finite functions
a posteriori validation
Chebyshev spectral methods
linear differential equations
D-finite functions
Discipline(s) HAL :
Informatique [cs]/Calcul formel [cs.SC]
Informatique [cs]/Analyse numérique [cs.NA]
Informatique [cs]/Analyse numérique [cs.NA]
Résumé en anglais : [en]
A symbolic-numeric validation algorithm is developed to compute rigorous and tight uniform error bounds for polynomial approximate solutions to linear ordinary differential equations, and in particular D-finite functions. ...
Lire la suite >A symbolic-numeric validation algorithm is developed to compute rigorous and tight uniform error bounds for polynomial approximate solutions to linear ordinary differential equations, and in particular D-finite functions. It relies on an a posteriori validation scheme, where such an error bound is computed afterwards, independently from how the approximation was built. Contrary to Newton-Galerkin validation methods, widely used in the mathematical community of computer-assisted proofs, our algorithm does not rely on finite-dimensional truncations of differential or integral operators, but on an efficient approximation of the resolvent kernel using a Chebyshev spectral method. The result is a much better complexity of the validation process, carefully investigated throughout this article. Indeed, the approximation degree for the resolvent kernel depends linearly on the magnitude of the input equation, while the truncation order used in Newton-Galerkin may be exponential in the same quantity. Numerical experiments based on an implementation in C corroborate this complexity advantage over other a posteriori validation methods, including Newton-Galerkin.Lire moins >
Lire la suite >A symbolic-numeric validation algorithm is developed to compute rigorous and tight uniform error bounds for polynomial approximate solutions to linear ordinary differential equations, and in particular D-finite functions. It relies on an a posteriori validation scheme, where such an error bound is computed afterwards, independently from how the approximation was built. Contrary to Newton-Galerkin validation methods, widely used in the mathematical community of computer-assisted proofs, our algorithm does not rely on finite-dimensional truncations of differential or integral operators, but on an efficient approximation of the resolvent kernel using a Chebyshev spectral method. The result is a much better complexity of the validation process, carefully investigated throughout this article. Indeed, the approximation degree for the resolvent kernel depends linearly on the magnitude of the input equation, while the truncation order used in Newton-Galerkin may be exponential in the same quantity. Numerical experiments based on an implementation in C corroborate this complexity advantage over other a posteriori validation methods, including Newton-Galerkin.Lire moins >
Langue :
Anglais
Vulgarisation :
Non
Collections :
Source :
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