Approximation of inverse operators by a new family of high-order iterative methods

The main goal of this paper is to approximate inverse operators by high-order Newton-type methods with the important feature of not using inverse operators. We analyse the semilocal convergence, the speed of convergence, and the efficiency of these methods. We determine that Chebyshev's method is the most efficient method and test it on two problems: one associated to the heat equation and the other one to a boundary value problem. We consider examples with matrices that are close to be singular and/or are badly conditioned. We check the robustness and the stability of the methods by considering situations with many steps and noised data. Copyright (C) 2013 John Wiley & Sons, Ltd.