On an approximation formula

Not available

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 $r_{1},r_{2}\in \mathbb{N}$ 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

On a Steffensen-Hermite type method for approximating the solutions of nonlinear equations

It is well known that the Steffensen and Aitken-Steffensen type methods are obtained from the chord method, using controlled nodes. The chord method is an interpolatory method, with two distinct nodes. Using this remark, the Steffensen and Aitken-Steffensen methods have been generalized using interpolatory methods obtained from the inverse interpolation polynomial of Lagrange or Hermite type. In this paper we study the convergence and efficiency of some Steffensen type methods which are obtained from the inverse interpolatory polynomial of Hermite type with two controlled nodes

On some Aitken-Steffensen-Halley-type methods for approximating the roots of scalar equations

In this note we extend the Aitken-Steffensen method to the Halley transformation. Under some rather simple assumptions we obtain error bounds for each iteration step; moreover, the convergence order of the iterates is 3, i.e. higher than for the Aitken-Steffensen case

Aitken-Steffensen type methods for nonsmooth functions (III)

We provide sufficient conditions for the convergence of the Steffensen method for solving the scalar equation $f(x)=0$, without assuming differentiability of $f$ at other points than the solution $x^\ast$. We analyze the cases when the Steffensen method generates two sequences which approximate bilaterally the solution

Aitken-Steffensen type methods for nondifferentiable functions (I)

We study the convergence of the Aitken-Steffensen method for solving a scalar equation $f(x)=0$. Under reasonable conditions (without assuming the differentiability of $f$) we construct some auxilliary functions used in these iterations, which generate bilateral sequences approximating the solution of the considered equation

Aitken-Steffensen-type methods for nonsmooth functions (II)

We present some new conditions which assure that the Aitken-Steffensen method yields bilateral approximation for the solution of a nonlinear scalar equation. The auxiliary functions appearing in the method are constructed under the hypothesis that the nonlinear application is not differentiable on an interval containing the solution

Bilateral approximations of the roots of scalar equations by Lagrange-Aitken-Steffensen method of order three

We study the monotone convergence of two general methods of Aitken-Steffenssen type. These methods are obtained from the Lagrange inverse interpolation polynomial of degree two, having controlled nodes. The obtained results provide information on controlling the errors at each iteration step

The optimal efficiency index of a class of Hermite iterative methods with two steps

Not available

Local convergence of general Steffensen type methods

We study the local convergence of a generalized Steffensen method. We show that this method substantially improves the convergence order of the classical Steffensen method. The convergence order of our method is greater or equal to the number of the controlled nodes used in the Lagrange-type inverse interpolation, which, in its turn, is substantially higher than the convergence orders of the Lagrange type inverse interpolation with uncontrolled nodes (since their convergence order is at most $2$)