Effective alpha theory certification using interval arithmetic: alpha theory over regions

TL;DR

This paper enhances Smale's alpha theory by integrating interval arithmetic to efficiently certify numerical solutions of analytic systems, ensuring quadratic convergence without costly exact computations.

Contribution

It introduces an interval arithmetic-based method for alpha theory certification, improving computational efficiency over traditional approaches.

Findings

Interval arithmetic enables efficient alpha theory certification.

The method guarantees quadratic convergence of Newton's method.

Computational performance surpasses existing software implementations.

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.