Based On The New Quasi-newton Equation Of Nonlinear Least Squares, A New Class Of Algorithms

With the comprehensive application of nonlinear least squares problems, moreand more attentions have been given to the study of the algorithm, and many newmethods are advanced in recent years. The first part of this paper goes to a thoroughreview of studies on methods for nonlinear least squares problems, which are then,from the perspective of arithmetic design, classified into five major types: methodsbased on quasi-Newton equation, hybrid methods, factorized quasi-Newton method,trust-region methods, and self-scaling method. The second part of this paper turns thefocus toward a type of new algorithms for nonlinear least squares problems based onnew quasi-Newton equation.In this paper, the new quasi-Newton equation [36], proposed by Zhang andothers in 2001 by using the tensor method and belong to huang's quasi-Newton, isemployed to replace the quasi-Newton equation in Chen's paper [39]. Then Chen'salgorithm is generalized to Broyden's by applying the dual principle as well as thestructure principle of the secant method. Under such a design, the algorithm isinvariant under an orthogonal matrix transformation on variables and be invulnerableto the good or bad quality of problems in the process of actual calculation. After that,three corresponding algorithms are presented, followed by a testimony to show thatall the three algorithms own local and super-linear convergence. Finally, numericalexperiments are conducted to demonstrate that all the three algorithms shown in thispaper are better than that in Chen's paper [39]. In addition, improvements of thealgorithms are given to enable them to be immune of the influence of the initial pointand each iteration point, a sound effect of which has been approved by the numericalresults, thus expanding the application scope of such algorithms.