ON THE CONVERGENCE ORDER OF SOME AITKEN–STEFFENSEN TYPE METHODS

. In this note we make a comparative study of the convergence orders for the Steﬀensen, Aitken and Aitken–Steﬀensen methods. We provide some conditions ensuring their local convergence. We study the case when the auxiliary operators used have convergence orders r 1 , r 2 ∈ N respectively. We show that the Steﬀensen, Aitken and Aitken–Steﬀensen methods have the convergence orders r 1 + 1, r 1 + r 2 and r 1 r 2 + r 1 respectively.

Let X be a Banach space and F : D ⊆ X → X a nonlinear mapping.Consider the equation (1.1) where θ is the null element of X.
Additionally, consider the equations x = ϕ 2 (x) , (1.3) which are assumed to be equivalent to (1.1), i.e., they have the same solutions.
As usually, L (X) stands for the set of linear operators from X into itself.For x, y, z ∈ X denote by [x, y; F ] ∈ L (X) the first order divided difference of F at the nodes x and y and by [x, y, z; F ] the second order divided difference of F at x, y, z ([7]- [9]).
In this note we show that the Steffensen method has the convergence order r 1 + 1, the Aitken method r 1 + r 2 , while the Aitken-Steffensen method r 1 (r 2 + 1).We also provide conditions ensuring the local convergence of these sequences.

LOCAL CONVERGENCE
Let S = {x ∈ X : x − x * ≤ r} be the ball with center at x * and with radius r, and suppose S ⊆ D.
The mappings ϕ 1 , ϕ 2 are assumed to obey the following hypotheses: i) the mapping ϕ 1 admits Fréchet derivatives up to the order r 1 ≥ 1 on S, and ii) the mapping ϕ 2 admits Fréchet derivatives up to the order r 2 ≥ 1 on S, and The mapping F is assumed to obey i 1 ) the linear mapping [x, y; F ] is invertible for all x, y ∈ S, and sup ii 1 ) the bilinear mapping [x, y, z; F ] is bounded for all x, y, z ∈ S: Regarding the convergence of the Steffensen method, we obtain the following result.
We obtain the following result regarding the Aitken method.
The Taylor formula leads to the relations By (2.23), we get x * − x k+1 ≤ x * − x 0 and so, from a ) and b ), and, analogously According to the induction principle, relations (2.19) are true for all i ∈ N, so Finally, the following result holds for the Aitken-Steffensen method.
The numbers r 1 , resp.r 2 represent as we have already specified, the convergence orders of the iteration methods given by (1.9), resp.(1.10).
We also notice the following facts.