On the accuracy of solving confluent Prony systems

TL;DR

This paper analyzes the stability and maximal accuracy of solving Prony-type systems, revealing that common algorithms are suboptimal compared to theoretical local limits, with implications for frequency estimation and Fourier inversion.

Contribution

It provides geometric-based accuracy estimates for Prony systems and compares practical algorithms to these theoretical bounds, highlighting their suboptimality.

Findings

Prony's algorithm and ESPRIT are less accurate than theoretical local bounds.

Accuracy estimates are expressed in simple geometric terms.

Numerical tests confirm the suboptimality of common methods.

Abstract

In this paper we consider several nonlinear systems of algebraic equations which can be called "Prony-type". These systems arise in various reconstruction problems in several branches of theoretical and applied mathematics, such as frequency estimation and nonlinear Fourier inversion. Consequently, the question of stability of solution with respect to errors in the right-hand side becomes critical for the success of any particular application. We investigate the question of "maximal possible accuracy" of solving Prony-type systems, putting stress on the "local" behavior which approximates situations with low absolute measurement error. The accuracy estimates are formulated in very simple geometric terms, shedding some light on the structure of the problem. Numerical tests suggest that "global" solution techniques such as Prony's algorithm and ESPRIT method are suboptimal when compared to this theoretical "best local" behavior.