Accurate solution of near-colliding Prony systems via decimation and homotopy continuation

TL;DR

This paper introduces a novel algorithm combining decimation and homotopy continuation to accurately solve near-colliding Prony systems, which are challenging due to their nonlinear structure and data perturbations.

Contribution

The authors develop a new method transforming Prony systems into Hankel-type polynomial systems and apply homotopy continuation, improving robustness in near-colliding cases.

Findings

Achieves high-accuracy solutions in perturbed data scenarios.

Effectively handles near-colliding configurations.

Provides a practical algorithm for complex polynomial systems.

Abstract

We consider polynomial systems of Prony type, appearing in many areas of mathematics. Their robust numerical solution is considered to be difficult, especially in "near-colliding" situations. We consider a case when the structure of the system is a-priori fixed. We transform the nonlinear part of the Prony system into a Hankel-type polynomial system. Combining this representation with a recently discovered "decimation" technique, we present an algorithm which applies homotopy continuation to an appropriately chosen Hankel-type system as above. In this way, we are able to solve for the nonlinear variables of the original system with high accuracy when the data is perturbed.