The iterative methods play a crucial role in approximating the solutions of nonlinear equations. The methods with superlinear convergence offer good approximations with a reduced number of steps. In a series of papers ([1]–[11]) the authors obtain different methods or modifications of some known methods, in order to achieve iterative methods with higher convergence orders.

The Steffensen, Aitken or Aitken-Steffensen methods lead to sequences having at least order 2 of convergence. A natural approach to generalize such methods can be obtained with the aid of inverse polynomial interpolation (Lagrange, Hermite, Taylor, etc), with controlled interpolation nodes [12]–[17]. One of the advantages of such methods is the fact that the interpolation nodes may be controlled such that the methods offer sequences with bilateral approximations (both from above and from below) of the solutions. This aspect offers the control of the error at each step [14], [16].

In this paper we shall extend a Steffensen type method using the Hermite inverse interpolatory polynomial of degree 2 with 2 nodes. In [13] we have shown that among all the Steffensen-Hermite methods with two nodes of arbitrary orders, the optimal efficiency index is attained in the case when one node is simple and the other one is double (see [18] for definitions of efficiency index); we have also shown there that the convergence order of this method is 3. Here we provide new convergence conditions, which offer bilateral approximations of the solution; these are very useful for controlling the error at each iteration step. In Section 2, we shall study the convergence of this method, and in Section 3 we shall indicate a method of constructing the auxiliary functions used for controlling the interpolations nodes. Some numerical examples will be shown in Section 4.

Let $c,d\in ℝ$, $f:[c,d]\to ℝ$, $g:[c,d]\to [c,d]$ and consider the following equivalent equations

As usually, the first order divided difference of $f$ at $a,b\in [c,d]$ will be denoted by $[a,b;f]$; if $a$ is double, then $[a,a;f]=f^{\prime }(a)$. For the second order divided differences we have

and if $a$ is double, then

Let $F=f([c,d])$ and assume the following conditions hold:

A.

$f\in C^3([c,d])$ and $f^{\prime }(x)\ne 0$ $\forall x\in [c,d].$

By A it follows that $f:[c,d]\to F$ is invertible so there exists $f^{-1}:F\to [c,d].$

Let $a_i\in [c,d],$ $i=1,2$ and $b_i=f(a_i),$ $i=1,2,$ i.e. $a_i=f^{-1}(b_i),$ and denote $a_1^{\prime }={(f^{-1}(b_1))}^{{}^{\prime }}=\frac{1}{f^{\prime }(a_1)}.$ Consider now the inverse interpolatory Hermite polynomial having $b_1$ as double node and $b_2$ as simple node, i.e. the second degree polynomial $H$ determined such that

Using the divided differences on multiple nodes, the resulted Hermite polynomial may be expressed in one of the following equivalent ways [17]:

or

The remainder is given by

It can be easily seen that the representations given by (4), (5) and (6) verify condition (3).

B.

Assume that equation (1) has a solution $\overline{x}\in [c,d].$

By A it follows that the solution $\overline{x}$ is unique in $[c,d].$

One has $\overline{x}=f^{-1}(0),$ whence, by (4)–(7), one obtains the following representations for $\overline{x}$:

or

where

If in (8), (9) or (10) we neglect the remainder $r$, one may obtain an approximation for $\overline{x},$ denoted by $a_3$:

or

or

It can be easily seen that

For the divided difference $[0,b_1,b_1,b_2;f^{-1}]$ we take into account hypothesis A, i.e. $f^{-1}\in C^3(F)$ and apply the mean value formula. There exists $\eta \in \mathrm{int}{E}_0$ such that

where $E_0$ is the smallest interval containing $b_1,b_2$ and $0.$

Since$f$ is invertible it follows that there exists $\xi \in \mathrm{int}{e}_0$ such that $\eta =f(\xi ),$ where $e_0$ is the smallest interval containing $a_1,a_2$ and $\overline{x}.$

It can be easily seen that [15], [17], [19]

Relations (15) and (16) lead to the following representation for the third order divided difference

Let $x_n\in [c,d]$ be an approximation of the solution $\overline{x}.$ If we set in (12)–(14) $a_1=x_n,$ $a_2=g(x_n)$ and we take into account relations (15)–(17) then we obtain a new approximation $x_{n+1}$ for $\overline{x}$ [16], [13]:

It can be easily seen that (21)–(23) yield in fact a same approximation $x_{n+1}.$

Analogously, if in (12)–(14) we set $a_1=g(x_n),$ $a_2=x_n$ we obtain the approximation $x_{n+1}$ in one of the following (equivalent) forms

Obviously, (24)–(26) represent the same approximation of $\overline{x}.$

The approximation $x_{n+1}$ given by (21)–(23) obeys

Analogously, for $x_{n+1}$ given by (24)–(26) one obtains:

In this paper we shall study both the convergence of the sequence ${(x_n)}_{n\ge 0}$ given by any of relations (21)–(23) and of the sequence given by any of relations (24)–(26).

Under some reasonable conditions on $f$ regarding monotonicity and convexity on $[c,d],$ we shall show that de above methods yield monotone sequences. Moreover, these iterations provide bilateral sequences, approximating the solution both from above and from below, which lead so a more precise control of the error [13]–[15].

For the study of method given by (21)–(23), besides A and B we consider the following hypotheses:

1)

equations (1) and (2) are equivalent;

2)

function $g$ is continuous and decreasing on $[c,d];$

3)

function $E_f(x)=3{(f^{\prime \prime }(x))}^2-f^{\prime }(x)f^{\prime \prime \prime }(x)\le 0,$ $\forall x\in [c,d].$

In the following we shall study the 4 situations when $f$ does not change the monotony and convexity on $[c,d].$

Let $x_n\in [c,d]$ such that $f(x_n)>0$ and $g(x_n)\in [c,d].$ Obviously, $x_n>\overline{x}$ and i.e. By ${𝐢}_1,$ ${\mathrm{𝐢𝐢}}_1$ and (22) it follows From (29) we get i.e. These prove ${\mathrm{𝐣𝐣}}_1.$

For proving ${\mathrm{𝐣𝐣𝐣}}_1,$ let ${lim}_{n\to \mathrm{\infty }}{x}_n=\mathrm{\ell }.$ Passing to limit $n\to \mathrm{\infty }$ in (21) and taking into account the continuity of $f$ and $g$ leads to $f(\mathrm{\ell })=0.$ Since the solution $\overline{x}$ is unique on $[c,d],$ it follows $\mathrm{\ell }=\overline{x}.$ Relation ${lim}_{n\to \mathrm{\infty }}g(x_n)=\overline{x}$ is obvious, as well as property ${\mathrm{𝐢𝐯}}_1.$  

Let $x_n\in [c,d]$ such that and $g(x_n)\in [c,d].$ Obviously, from ${𝐢}_2$ it follows $x_n>\overline{x}$ and from 2) it follows i.e. $f(g(x_n))>0.$ By ${𝐢}_2,$ ${\mathrm{𝐢𝐢}}_2$ and (21) we get By (29), taking into account 3) and ${𝐢}_2$ it follows $x_{n+1}>\overline{x},$ i.e. Property ${\mathrm{𝐣𝐣}}_2$ is proved.

Property ${\mathrm{𝐣𝐣𝐣}}_2$ is obvious.  

In the following, instead of equation (1) we shall consider equation

where $h:[c,d]\to ℝ$, $h(x)=-f(x).$

We notice that if function $E_f(x)$ verifies 3), then function $E_h(x)=3{(h^{\prime \prime }(x))}^2-h^{\prime }(x)h^{\prime \prime \prime }(x)=E_f(x)$ verifies hypothesis 3).

Taking into account the above relations, the following assertions are consequences of Theorems 1 respectively 2:

In the following we shall study the convergence of the sequence ${(x_n)}_{n\ge 0}$ generated by (24)–(26). Instead of hypothesis 3) we shall assume further hypothesis

3^′)

$E_f(x)\ge 0,$ $\forall x\in [c,d].$

This equality, together with 3^′), ${𝐢}_5$ and inequality lead to $\overline{x}>x_{n+1}$ and $g(x_{n+1})>\overline{x}.$ From $x_{n+1}>x_n$ and 2) we get Property ${𝐣}_5$ is proved. Property ${\mathrm{𝐣𝐣}}_5$ is obvious if we take into account the proof of Theorem 1.  

If instead of equation (1) we consider equation (30) then the following consequences of Theorem 5 and respectively 6 are true:

The convergence order of the sequences studied above is given in the following results.

Now, by hypotheses of Theorem 1 there exist ${\xi }_n,{\nu }_n,{\mu }_n\in ]x{}_n,g(x_n)[$ such that