A Newton-like Validation Method for Chebyshev Approximate Solutions of Linear Ordinary Differential Systems

Florent Bréhard 1, 2, 3
1 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
2 PLUME - Preuves et Langages
LIP - Laboratoire de l'Informatique du Parallélisme
3 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
LAAS - Laboratoire d'analyse et d'architecture des systèmes [Toulouse]
Abstract : We provide a new framework for a posteriori validation of vector-valued problems with componentwise tight error enclosures, and use it to design a symbolic-numeric Newton-like validation algorithm for Chebyshev approximate solutions of coupled systems of linear ordinary differential equations. More precisely, given a coupled differential system of dimension p with polynomial coefficients over a compact interval (or continuous coefficients rigorously approximated by poly-nomials) and polynomial approximate solutions Φ • i in Chebyshev basis (1 i p), the algorithm outputs rigorous upper bounds ε i for the approximation error of Φ • i to the exact solution Φ i , with respect to the uniform norm over the interval under consideration. A complexity analysis shows that the number of arithmetic operations needed by this algorithm (in floating-point or interval arith-metics) is proportional to the approximation degree when the differential equation is considered fixed. Finally, we illustrate the efficiency of this fully automated validation method on an example of a coupled Airy-like system.
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Florent Bréhard. A Newton-like Validation Method for Chebyshev Approximate Solutions of Linear Ordinary Differential Systems. Rapport LAAS n° 17444. 2018. 〈hal-01654396〉

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