Improving the accessibility of Steffensen’s method by decomposition of operators

摘要

Solving equations of the form H(x)=0 is usually done by applying iterative methods. The main interest of this paper is to improve the domain of starting points for Steffensen’s method. In general, the accessibility of iterative methods that use divided differences in their algorithms is reduced, since there are difficulties in the choice of starting points to guarantee the convergence of the methods. In particular, by using a decomposition of the operator H and applying a special type of iterative methods, which combine two iterative schemes in the algorithms, we can improve the accessibility of Steffensen’s method. Moreover, we analyze the local convergence of the new iterative method proposed in two cases: when H is differentiable and H is non-differentiable. The dynamical properties show that the method also improves the region of accessibility of Steffensen’s method for non-differentiable operators. So, we present an alternative for the non-applicability of Newton’s method to non-differentiable operators that improves the accessibility of Steffensen’s method. The theoretical results are illustrated with numerical experiments.