A derivative-free scaling memoryless Broyden-Fletcher-Goldfarb-Shanno method for solving a system of monotone nonlinear equations

This paper presents the two-parameter scaling memoryless Broyden-Fletcher-Goldfarb-Shanno (BFGS) method for solving a system of monotone nonlinear equations. The optimal values of the scaling parameters are obtained by minimizing the measure function involving all the eigenvalues of the memoryless BFGS matrix. The optimal values can be used in the analysis of the quasi-Newton method for ill-conditioned matrices. This algorithm can also be described as a combination of the projection technique and memoryless BGFS method. Global convergence of the method is provided. For validation and efficiency of the scheme, some test problems are computed and compared with existing results.