Suitable approximations for the self-accelerating parameters in iterative methods with memory

Abstract

Solving the nonlinear equation f(x)=0 is a common problem in several areas of Science and Engineering. Since exact solutions of the nonlinear equation are hardly available, scientists best rely on numerical solutions, such as those given by iterative methods. Iterative procedures can be classified according to different criteria: the order of convergence, the presence of derivatives or the number of previous iterations to obtain the current one, among others. Focusing on the last type of schemes, they are known as iterative methods with memory. The major advantage of this sort of methods is to enhance the order of convergence of the original method without introducing new evaluations of f. One technique to obtain iterative schemes with memory starts with an iterative method without memory that includes a parameter. Depending on the error equation of the method, the parameter can be replaced by an expression that includes the previous iterations. The strategy to obtain these expressions is essential. In this work we propose suitable approximations that are present in the literature for different iterative methods with memory. On the one hand, the polynomials approximation, such as Newton's interpolation polynomial. On the other hand, the non-polynomials approximation, such as Padé's approximants. Finally, we decide which is the most suitable choice in terms of convergence and stability.