On approximating the solutions of equations by the chord method and a method of Aitken-Steffensen type

Abstract

In [13] we have studied the existence and the convergence of iterative methods that use generalized abstract divided differences (this notion being defined there). We have indicated a construction model for these differences as well. A special place has been given to the iterative method of the chord for which we have established a convergence theorem which in the same time ensures the existence of the solution of the considered equation. We have obtained the convergence order with the value \(\tfrac{1+\sqrt{5}}{2}.\) This value is inferior to \(2,\) this last value representing the convergence order of the method of Newton-Kantorovich. This diminuation of the convergence order is the price to pay for the replacement of the Fréchet differential with the generalized abstract divided difference. In this paper we consider the issue of the improvement of the convergence order with respect to the method of Steffensen and Aitken-Steffensen or their generalizations through the method of the auxiliary sequences. This method will be presented in the paper together with the specification of the convergence order of the main sequence and the auxiliary sequences.