ON A HALLEY-STEFFENSEN METHOD FOR APPROXIMATING THE SOLUTIONS OF SCALAR EQUATIONS

. In the present paper we show that the Steﬀensen method for solving the scalar equation f ( x ) = 0, applied to equation h ( x ) = f ( x ) √ f (cid:48) ( x ) = 0 , leads to bilateral approximations for the solution. Moreover, the convergence order is at least 3 , i

→ R given by h (x) = f (x) / f (x).As it is well known (see e.g.[2]), the Halley method for solving (1.1) f (x) = 0 consists in constructing the sequence (x n ) n≥0 by (1.2) As it can be easily noticed, (1.2) is the Newton method applied to equation h (x) = 0.The advantage of using the function h instead of f in (1.2) consists in the fact that h obeys h (x) = 0, where x is a solution of (1.1).It is well known that h (x) = 0 ensures for (1.2) the convergence order 3 (see [2]).
Starting from an algorithm proposed by Heron for approximating 3 √ 100, the authors of [4] establish the following relation: where The Heron algorithm is obtained from (1.3) for α = 4 and β = 5.In the paper [5], the authors noticed that the approximation given by (1.3) for 3  √ N is obtained by applying one step of the chord method to equation ϕ (x) = 0, where ϕ (x) = x 2 − N/x, x > 0.
In [5] it is noticed that if one considers equation f (x) = x 3 − N , then, apart of a constant factor, equation ϕ (x) = 0 is equivalent to i.e., 1/ √ 3 x 2 − N/x = 0. Therefore there exists a connection between the Halley method (1.2) and the Heron algorithm concerning the function h.In the case of the Halley method there is applied the Newton method to h (x) = 0 whereas in the Heron method there is applied the chord method to h (x) = 0.The both methods benefit from the advantages implied by h (x) = 0.These remarks have led in [5] to generalizations of the results from [4].A brief analysis of the convergence order of the chord method applied to h (x) = 0 in the general case (quasi-Halley method) is given in [1].
In this note we shall study the convergence of an iterative method obtained by applying the Steffensen algorithm to equation h (x) = 0, where h (x) = f (x) / f (x).As we shall see, this method has some advantages over the Halley and chord methods applied to h.The most important one is the fact that the method we propose allows the control of the absolute error at each iteration step.Its convergence order is the same as for the Halley method, being higher than the order of the chord method.
For solving (1.1) we shall consider the sequence where ϕ will be suitably chosen, and [x, the first order divided difference of f on x and y.We shall call this method the Halley-Steffensen method.

LOCAL CONVERGENCE AND ERROR BOUNDS
Concerning the function f we shall assume the following conditions: We notice in the beginning that which shows that the Halley-Steffensen sequence obeys (2.2) while for the first and second order derivatives we obtain We obtain the following result: Theorem 1. Assume that the function f and the initial approximation x 0 satisfy: i 1 .the number x 0 is sufficiently close to x and ϕ (x 0 ) ∈ [α, β], with α and β determined above; ii 2 .the function f obeys (i)-(vi); Then the Halley-Steffensen sequence (1.4) converges to the solution x and, moreover, where K is a constant which does not depend on n.
Let I s denote the closed intervals determined by the points x s and ϕ(x s ), s = 0, k.Suppose that (2.8) and x ∈ I k .As we have shown for x 0 , we can prove that the interval I k+1 , determined by x k+1 and ϕ(x k+1 ), obeys and x ∈ I k+1 .From the above reasons, it follows that relations (2.5) are true.
It remains to show that (2.6) holds, which implies the convergence of (x n ) n≥0 generated by (1.4).For this purpose we shall use the identity whence, taking into account (1.4) and h (x) = 0, it follows (see [9]) By (ii) and (v) we get |ϕ (x)| < 1, ∀x ∈ I 0 , and so On the other hand, from the mean value theorems for divided differences one obtains (2.12) From (2.11) and from h (x) = 0 we get where δ n ∈ I n .Further, by (2.10) it follows i.e., (2.6) with K = m 3 /(2m 1 ).The proof is complete.