Multi-step root solvers of Traub’s type in real interval arithmetic

A multi-step self-validated iterative method for solving nonlinear equations is constructed. The main advantages of this method are the feasibility to provide global convergence and to produce automatic computation of rigorous error bound of approximations, given by the radius of the resulting inclusion interval. The convergence analysis and numerical examples are included to demonstrate convergence properties of the presented method. A special attention is devoted to two-step and three-step methods for their high computational efficiency. In particular, extended interval arithmetic is used for the construction of never-failing variant of the proposed method.