Consider the equation

$f:[a,b]\to ℝ,$ and another equivalent equation:

$g:[a,b]\to [a,b].$

The Steffensen type methods are interpolatory methods (see, e.g., [8], [5]), with the nodes controlled at each step by the auxiliary function $g$ [1]–[4], [6]–[7], [9]–[11], [14]. The method generates a sequence ${\{x_n\}}_{n\ge 0}$ which approximates a solution $x^{\ast }$ of (1):

In the papers [9], [10] is studied the convergence of the Steffensen method when $g$ is given by

$\lambda \in ℝ.$ The parameter $\lambda$ is determined such that the sequences ${\{x_n\}}_{n\ge 0}$ and ${\{g(x_n)\}}_{n\ge 0}$ approximate bilaterally the solution, and the convergence order of these methods is 2.

In [11] Sharma introduces a Steffensen type method of order three, with $g$ given by

which we shall call as the Newton-Steffensen (N-S) method. Since it requires 3 function evaluations at each iteration step, its efficiency index is $I_3=\sqrt[3]{3},$ and is larger than $I_2=\sqrt{2},$ which corresponds to the Newton method or the Steffensen method (see, e.g., [9], [8], [12]), and therefore the method is worth to be considered.

In this paper we shall show in Section 2 that the convergence order 3 can be obtained under some weaker smoothness assumptions on $f$ than in [11] and [13]. Moreover, in Section 3 we shall study the convergence of the (N-S) method in hypotheses regarding the monotony and convexity of $f$ on $[a,b]$. Under some simple assumptions on $f$, including the Fourier conditions, we shall show that the (N-S) method leads to monotonic sequences which approximate the solution. In Section 4 we illustrate the theory by some numerical examples, which confirm the theoretical results.

Consider the following hypotheses:

$\alpha )$ equation (1) has a solution $x^{\ast }\in ]a,b[.$

$\beta )$ $f\in C^2[a,b],$

$\gamma )$ $f^{\prime }\left(x\right)\ne 0$ on $[a,b].$

Sharma proved the following result.

Of course that the fact that the sequence is well defined must be implicitly assumed.

We show that the requirement that $f$ is three times differentiable can be dropped.

In [13], the authors studied the semilocal and local convergence of the (N-S) iterates in the setting of Banach spaces. Local convergence with order 3 was established under the assumption of Lipschitz continuity of $f^{\prime \prime }$. We show here that even the Lipschitz continuity may be relaxed, and we consider the following assumption:

${\beta }^{\prime })$ $f\in C^1[a,b],$ and $\exists f^{\prime \prime }(x),\forall x\in (a,b)$, which is continuous at $x^{\ast }$.

By $\alpha ),\beta )$ and the Taylor formula we obtain

where ${\xi }_n$ belongs to the open interval determined by $x_n$ and $x^{\ast }.$

Taking into account that $g(x_n)=x_n-\frac{f(x_n)}{f^{\prime }(x_n)},$ by (7) we infer

which together with (6) attracts

which shows that the (N-S) method has q-order 3. The Mean Value Theorem for divided differences ensures that $[x^{\ast },x_n,g(x_n);f]\to \frac{1}{2}{f}^{\prime \prime }(x^{\ast })$ and $[x_n,g(x_n);f]\to f^{\prime }(x^{\ast })$, as $n\to \mathrm{\infty }$, so we are led to the same asymptotic constant as in (5).  

As it is well known, when $f\in C^2[a,b]$ preserves its monotonicity and convexity on $[a,b]$, if the initial approximation $x_0\in [a,b]$ verifies the Fourier condition, then the Newton method yields a monotone sequence which converges to the solution $x^{\ast }\in [a,b].$ We shall show that under same such hypotheses, we obtain the same conclusions, but for a method with superior efficiency index.

We assume that the initial approximation $x_0$ verifies the Fourier condition (see [8]):

$\delta )$

$f^{\prime \prime }(x_0)f(x_0)>0,x_0\in [a,b].$

We obtain the following results.

Using the mean value formulas for divided differences and taking into account $i_1,$ $i{i}_1$ and (6), it follows that

We notice that relation (3) can be written as

whence, since $f(g(x_n))>0$ by $i_1$ it follows

It is obvious that the elements of ${\{x_n\}}_{n\ge 0}$ and ${\{g(x_n)\}}_{n\ge 0}$ belong to $]x^{\ast },x_0]\subset [a,b].$ It remains to show the convergence to the solution. Indeed, if $\mathrm{\ell }=lim{x}_n$ then by (10) $limg(x_n)=g(\mathrm{\ell })=\mathrm{\ell },$ whence $f(\mathrm{\ell })=0$ i.e., $\mathrm{\ell }=x^{\ast }.$  

We consider four numerical examples, and we compute the iterations using double precision in Matlab.

It is interesting to note that we get $f(x_3)=0$ in Example 3, which suggests that $x_3=x^{\ast }$. However, if instead of the standard double precision we use higher precision (e.g., variable precision arithmetic) for representing $x_3$ from Table 3, the computation of $f(x_3)$ no longer yields $0$ (of course, we obtain values with the magnitude comparable to the machine epsilon from double precision, $ϵ\approx 2.22$e$-16$). This means that $x_3$ from Table 3 is just a very good approximation (in double precision arithmetic) to the solution.

Acknowledgement. We are grateful to an anonymous referee for his constructive remarks in improving the presentation of our paper.