About the stability of self-accelerating parameters in vectorial iterative methods without memory

Abstract

This study is motivated by the need to improve the efficiency of iterative methods for solving nonlinear systems, overcoming some limitations of Newton’s method and other classical schemes. Building on the recent work of Singh, Sharma, and Kumar—which introduced a self-accelerating parameter dependent on the last iterate—an in-depth stability analysis of these without memory methods in a vectorial framework is proposed. The main objective is to evaluate how parameters’ selection and variation affect the process’s convergence and stability using tools from discrete dynamical systems, bifurcation diagrams, and dynamical planes. The results demonstrate that certain members of the family, particularly those with fourth-order convergence, exhibit superior stability properties compared to the initially proposed fifth-order scheme. Additionally, numerical tests on various nonlinear systems confirm the theoretical stability results. The stable methods of these families can be successfully applied on realistic problems such as preliminary orbit determination and on the nonlinear systems resulting from the discretization of Fisher’s equation used in population genetics and ecology, convection-diffusion equations appearing in engineering problems, etc.