Effective alpha theory certification using interval arithmetic: alpha theory over regions (Q6637824)

Smale's alpha theory provides a robust framework for certifying numerical solutions to analytic systems. In this work, the author introduces a novel approach to Smale's alpha theory computation based on interval arithmetic. This variation enhances computational efficiency compared to traditional implementations of alpha theory.\N\NIn the experimental section, the author investigates the behavior of alpha values with respect to the radius of the input intervals and the precision of computation. To demonstrate the efficiency of the proposed method, the elapsed times for certifying solutions to Fano problems are compared between the interval arithmetic implementation of alpha theory over regions and \textit{alphaCertified} which is an implementation of alpha theory with exact arithmetic.\N\NOverall, this paper makes a contribution to improving the computational aspects of Smale's alpha theory and highlights the usefulness of applying interval arithmetic to the alpha theory.\N\NFor the entire collection see [Zbl 1548.68018].