Efficient Newton-Type Solvers Based on for Finding the Solution of Nonlinear Algebraic Problems

The purpose of this study is to improve the computational efficiency of solvers for nonlinear algebraic problems with simple roots. To this end, a multi-step solver based on Newton’s method is utilized. Divided difference operators are applied at two substeps in various forms to enhance the convergence speed and, consequently, the solver’s efficiency index. Attraction basins for the proposed solvers and their competitors are presented, demonstrating that the proposed solvers exhibit large attraction basins in the scalar case while maintaining high convergence rates. Theoretical findings are supported by numerical experiments.