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

International audienceWe 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