On the Accuracy of Solving Confluent Prony Systems

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 the ESPRIT method are suboptimal when compared to this theoretical “best local” behavior.