Extended Parametric Family of Two-Step Methods with Applications for Solving Nonlinear Equations or Systems

The parametric family of two-step methods, with its special cases, has been introduced in various papers. However, in most cases, the local convergence analysis relies on the existence of derivatives of orders that the method does not require. Moreover, the more challenging semi-local convergence analysis was not introduced for this class of methods. These drawbacks are considered in this paper. We determine the radius of convergence and the uniqueness of the solution based on generalized continuity conditions. We also present the semi-local convergence analysis for this family of methods, which has not been studied before, using majorizing sequences. Numerical experiments and basins of attraction are included to validate the theoretical conditions and demonstrate the stability of the methods.