Effective alpha theory certification using interval arithmetic: alpha theory over regions
Kisun Lee
TL;DR
The paper extends Smale's alpha theory to certify numerical solutions of analytic systems by introducing alpha theory over regions, which uses interval arithmetic to form bounds $\alpha(F,I)$, $\beta(F,I)$, and $\gamma(F,I)$ over input intervals. It proves that small enough $\alpha(F,I)$ guarantees that all points in the region converge to the same associated solution and provides interval-based bounds for $\gamma$ via $\mu(F,I)$ and system degree; it also discusses corollaries for certifying distinct solutions. The implementation leverages MPFI for arbitrary precision and a dual-interval LU approach to compute tight inverse interval matrices, improving computational efficiency over exact arithmetic. Experiments on cyclic and Fano-type polynomial systems show that region-based certification scales with interval radius and precision, and generally outperforms exact-alphaCertified in runtime while maintaining reliability. Overall, alpha theory over regions offers a faster, interval-arithmetic-based route to rigorously certify Newton-based solutions to analytic systems.
Abstract
We reexamine Smale's alpha theory as a way to certify a numerical solution to an analytic system. For a given point and a system, Smale's alpha theory determines whether Newton's method applied to this point shows the quadratic convergence to an exact solution. We introduce the alpha theory computation using interval arithmetic to avoid costly exact arithmetic. As a straightforward variation of the alpha theory, our work improves computational efficiency compared to software employing the traditional alpha theory.