Some Special Nonlinear Least Squares Problems

Abstract

A least squares problem is called separable if the fitting function can be written as a linear combination of functions involving further parameters in a nonlinear manner. Here it is shown that a separable problem can be transformed to a minimization problem involving the nonlinear parameters only. This result can be interpreted as a generalization of the classical technique of Prony, and it also shows how the numerical difficulties associated with Prony’s method can be overcome. The transformation is then worked out in detail for two examples (exponential fitting to equispaced data and rational fitting). In both cases the condition for a stationary value of the transformed problem leads to a nonlinear eigenvalue problem. Two algorithms are then suggested and illustrated by numerical examples.