Effective certification of approximate solutions to systems of equations involving analytic functions

TL;DR

This paper presents algorithms to certify approximate solutions to systems of equations with analytic functions, utilizing alpha-theory and Krawczyk methods, with implementations for D-finite functions.

Contribution

It introduces new certification algorithms based on alpha-theory and Krawczyk iteration, applicable to systems involving D-finite analytic functions.

Findings

Algorithms successfully certify solutions for D-finite functions.

Comparison of alpha-theory and Krawczyk approaches implemented in SageMath.

Effective certification methods demonstrated on systems of analytic equations.

Abstract

We develop algorithms for certifying an approximation to a nonsingular solution of a square system of equations built from univariate analytic functions. These algorithms are based on the existence of oracles for evaluating basic data about the input analytic functions. One approach for certification is based on alpha-theory while the other is based on the Krawczyk generalization of Newton's iteration. We show that the necessary oracles exist for D-finite functions and compare the two algorithmic approaches for this case using our software implementation in SageMath.