Convergence and Stability of a New Parametric Class of Iterative Processes for Nonlinear Systems

In this manuscript, we carry out a study on the generalization of a known family of multipoint scalar iterative processes for approximating the solutions of nonlinear systems. The convergence analysis of the proposed class under various smooth conditions is provided. We also study the stability of this family, analyzing the fixed and critical points of the rational operator resulting from applying the family on low-degree polynomials, as well as the basins of attraction and the orbits (periodic or not) that these points produce. This dynamical study also allows us to observe which members of the family are more stable and which have chaotic behavior. Graphical analyses of dynamical planes, parameter line and bifurcation planes are also studied. Numerical tests are performed on different nonlinear systems for checking the theoretical results and to compare the proposed schemes with other known ones.