Title :

A Symbolic-Numeric Validation Algorithm for Linear ODEs with Newton-Picard Method

Author(s) :

Bréhard, Florent [Auteur]
Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 [CRIStAL]

Journal title :

Mathematics in Computer Science

Publisher :

Springer

Publication date :

2021

ISSN :

1661-8270

English keyword(s) :

validated numerics
a posteriori validation
Chebyshev spectral methods
linear differential equations
D-finite functions

HAL domain(s) :

Informatique [cs]/Calcul formel [cs.SC]
Informatique [cs]/Analyse numérique [cs.NA]

English abstract : [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. ...
Show more >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.Show less >

Language :

Anglais

Peer reviewed article :

Oui

Audience :

Internationale

Popular science :

Non

Collections :

• Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) - UMR 9189

Source :

Harvested from HAL