Certifying solutions to square systems of polynomial-exponential equations

TL;DR

This paper develops a certification algorithm for solutions to polynomial-exponential equations using bounds on higher order derivatives, extending Smale's alpha-theory and implementing it in alphaCertified.

Contribution

It introduces a new bound for higher derivatives of polynomial-exponential systems and provides a complete certification algorithm based on this bound.

Findings

The certification algorithm successfully certifies solutions to polynomial-exponential systems.

Implementation in alphaCertified demonstrates practical applicability.

Examples validate the effectiveness of the certification method.

Abstract

Smale's alpha-theory certifies that Newton iterations will converge quadratically to a solution of a square system of analytic functions based on the Newton residual and all higher order derivatives at the given point. Shub and Smale presented a bound for the higher order derivatives of a system of polynomial equations based in part on the degrees of the equations. For a given system of polynomial-exponential equations, we consider a related system of polynomial-exponential equations and provide a bound on the higher order derivatives of this related system. This bound yields a complete algorithm for certifying solutions to polynomial-exponential systems, which is implemented in alphaCertified. Examples are presented to demonstrate this certification algorithm.