Some Numerical Methods for Nonlinear Least Squares Problems

Abstract

Nonlinear least-square problems appear in estimating parameters and ehecking the hypotheses of mathematical statisticsj in estimating parameter of physical process from measurement, and in managing of different objects, processes, etc. For solving such problems the Gauss-Newton method and Levenberg-Marquardt method (Dennis and Schnabel 1983, Ortega and Rheinboldt 1970, Schwetlick 1991) are usually applied. Iterative-difference analogue to GaussNewton method is investigated by Shakhno and Gnatyshyn (1999). Some updatings Gauss-Newton method are proposed by Bartish and Shakhno (1993). In this work we propose methods for solving nonlinear least-squares problems. These methods are constructed by using a combination of known iterative methods with the aim of obtaining greater efficiency in regards to the number of iterations and the number of calculations. The theorems about convergence conditions of the methods as weIl as speed of iteration convergence are formulated and proved. A comparison is made between these methods and the GaussNewton method. The results of extensive numerical experiments are demonstrated on the basis of tested problems, whieh are widely known though rather complex. Conclusions have been made on the basis ofthese experimental results.