Certified surface approximations using the interval Krawczyk test

TL;DR

This paper introduces a new algorithm that extends the Krawczyk test using interval arithmetic to certify surface approximations, applicable to complex systems and higher-dimensional varieties, with practical implementation and examples.

Contribution

It generalizes the Krawczyk test for non-square systems and higher-dimensional varieties, enabling certified surface approximations.

Findings

Successfully extended the Krawczyk test to broader classes of systems.

Provided a prototype implementation demonstrating the method.

Validated the approach with multiple illustrative examples.

Abstract

We propose an algorithm to construct a certified approximation of a surface by generalizing the Krawczyk test. The Krawczyk test is based on interval arithmetic, and confirms the existence and uniqueness of a solution to a square system of analytic equations in a region. By generalizing this test, we extend the reach of this technique to non-square systems and higher-dimensional varieties. We provide a prototype implementation and illustrate its use on several examples.