Tools for Rigorous Computing using Chebyshev Series Approximations

Performing numerical computations, yet being able to provide rigorous mathematical statements about the obtained result, is required in many domains like global optimization, ODE solving or integration. Taylor models are a widely used rigorous computation tool: they associate to a function a pair made of a Taylor approximation polynomial and a rigorous remainder bound. This approach benefits from the advantages of numerical methods, but also gives the ability to make reliable statements about the approximated function. A natural idea is to try to replace Taylor polynomials with better approximations such as minimax approximation, Chebyshev truncated series or interpolation polynomials. Despite their features, an analogous to Taylor models, based on such polynomials, has not been yet well-established in the field of validated numerics. In this talk we propose two approaches for computing such models : one is based on interpolation polynomials at Chebyshev nodes; the other on using Chebyshev truncated series. We compare the quality of the obtained remainders and the performance of the approaches to the ones provided by Taylor models. We also present two practical examples where this tool can be used: supremum norm computation of approximation errors and rigorous quadrature. This talk is based on a joint work with Nicolas Brisebarre.