A predictor-corrector iterative scheme for improving the accessibility of the Steffensen-type methods

Abstract

Solving equations of the form 𝐻(𝑥) = 0 is usually done by applying iterative methods. The Steffensen-type methods, defined by means divided differences and derivative free, are usually considered to solve these problems when 𝐻 is a non-differentiable operator due its accuracy and efficiency. But, in general, the accessibility of iterative methods that use divided differences in their algorithms is reduced. The main interest of this paper is to improve the accessibility, domain of starting points, for Setffensen-type methods. So, by using a predictor-corrector iterative process we can improve this accessibility. For this, we use a predictor iterative process with a good accessibility and after we consider a Steffensen-type iterative method for a good accuracy, since this type of iterative process has quadratic convergence. Thus we will obtain a predictor-corrector iterative process with good accessibility, given by the predictor iterative process, and an accuracy like the Steffensen-type methods. Moreover, we analyze the semilocal convergence of the predictor-corrector iterative process proposed in two cases: when 𝐻 is differentiable and 𝐻 is non-differentiable. So, we present a good alternative for the non-applicability of Newton’s method to non-differentiable operators. The theoretical results are illustrated with numerical experiments. CEDYA/CMA 2020.