A Modified Prony Algorithm for Fitting Functions Defined by Difference Equations

This paper reformulates, generalizes and investigates the stability of the modified Prony algorithm introduced by Osborne (1975), with special reference to rational and exponential fitting. The algorithm, originally for exponential functions, is generalized to the least squares fitting of any function which satisfies a linear homogeneous difference equation. Using the difference equation formulation, the problem is expressed as a separable regression, and hence as a nonlinear eigenproblem in terms of the coefficients of the difference equation. The eigenproblem involves finding the null space of a matrix of data differences B, and is solved using a variant of inverse iteration. Stability of the algorithm is shown to depend on the fact that B closely approximates the Hessian of the sum of squares. The expectations of B and the Hessian are evaluated. In the case of rational fitting, the relative difference between B and the Hessian is shown to converge to zero almost surely. Some details of the implementation of the algorithm are given. A simulation study compares the modified Prony algorithm with the Levenberg algorithm on a rational fitting problem, and supports the theoretical results.