Different approximations of the parameter for low-order iterative methods with memory

Abstract

A technique for generating iterative methods for solving nonlinear equations with memory can be constructed from a method without memory that includes a parameter, provided the parameter is present in the error equation. Generally, the parameter depends on the evaluation of the function and its derivatives in the solution. However, this information is not available. So this parameter is approximated using interpolation techniques, taking the current iterate 𝑥𝑘 and the previous iterates 𝑥𝑘−1, 𝑥𝑘−2, . . . In this paper we explore different interpolation techniques to obtain both the convergence order of the new methods and their stability characteristics.