Derivative free analogues of the Levenberg-Marquardt and Gauss algorithms for nonlinear least squares approximation

In this paper we give two derivative-free computational algorithms for nonlinear least squares approximation. The algorithms are finite difference analogues of the Levenberg-Marquardt and Gauss methods. Local convergence theorems for the algorithms are proven. In the special case when the residuals are zero at the minimum, we show that certain computationally simple choices of the parameters lead to quadratic convergence. Numerical examples are included.On leave 1970–71 at Yale UniversityThe work of this author was supported in part by the National Science Foundation under Grant GJ-844.