As it is well known, the Steffensen, Aitken and Aitken-Steffensen methods are interpolatory methods, with controlled nodes. More precisely, they can be obtained from the first degree Lagrange inverse interpolatory polynomial, for which the two interpolation nodes are controlled by one or two auxiliary functions suitably chosen [1]–[6], [9]–[12], [17].

Some generalizations of these methods have been obtained in [12], [15], [16], by controlling the interpolation nodes in the Lagrange, respectively Hermite inverse interpolatory polynomials of degree 2; the resulted methods have convergence order 3.

Moreover, in [13], [15], [16], were obtained conditions under which the Steffensen, Aitken or Aitken-Steffensen type methods lead to sequences which approximate bilaterally the solution.

In this paper we propose two iterative methods of Aitken-Steffensen type, which are obtained from the inverse interpolatory Hermite polynomial of degree 2, and which have the q-convergence order 3. The interpolation nodes are obtained using two auxiliary functions.

In the first section we present the two methods and we show that they have the convergence order 3. In Section 2 we obtain convergence results depending on the monotony of the function (increasing/decreasing) and on its convexity (convex/concave). In Section 3 we show how the auxiliary functions may be constructed such that they verify the convergence results obtained in Section 2, while the last section contains some numerical example which illustrate the results.

Consider the equation

where $f:[a,b]\to ℝ,$ $a,b\in ℝ,$ and the additional two equations, equivalent to the above one:

where $p,q:[a,b]\to [a,b].$ Denote $h:[a,b]\to [a,b]$ given by

Here we shall study the method obtained when the nodes are controlled by $p$ and $h.$

Denote $F=f([a,b])$ and let $s,k\in \mathbb{N}$ and $n=s+k.$ We shall make the following hypotheses on $f:$

$\alpha )$ $f\in C^n([a,b])$

$\beta )$ $f^{\prime }(x)\ne 0,\forall x\in [a,b].$

It can be easily seen that $\alpha )$ and $\beta )$ imply that $f$ is continuous on $[a,b]$ and is invertible, therefore there exists the inverse $f^{-1}:F\to [a,b]$ and the following result holds.

We recall below some particular cases of (4), which will be subsequently used, i.e., $j=1,2,3$:

where $y=f(x).$

Let $x_1,x_2\in [a,b]$ be two interpolation nodes and $y_1=f(x_1),$ $y_2=f(x_2).$ Obviously $f^{-1}(y_1)=x_1$ and $f^{-1}(y_2)=x_2.$ Moreover under the hypotheses of Theorem 1.1 we can compute the successive derivatives of $f^{-1}$ at $y_1,$ $y_2$: ${\left(f^{-1}(y_1)\right)}^{\prime }$, ${\left(f^{-1}(y_1)\right)}^{\prime \prime },\mathrm{\dots },{(f^{-1}(y_1))}^{(s-1)}$ resp. ${(f^{-1}(y_2))}^{\prime }$, ${(f^{-1}(y_2))}^{\prime \prime },\mathrm{\dots },{(f^{-1}(y_2))}^{(k-1)}.$ These values determine the unique Hermite polynomial of degree $n-1,$ denoted by $H(y),$ satisfying

We denote this polynomial by $H(y_1,s;y_2,k;f^{-1}|y).$

It can be easily seen that under the hypotheses of Theorem 1.1 we have that

where $\eta \in \mathrm{int}(F).$

If $x^{\ast }\in [a,b]$ is a solution of (1), then $x^{\ast }=f^{-1}(0)$ and by (6) we get

where ${\eta }_0$is an interior point of the smallest interval containing $0,y_1,y_2.$

If in (7) we neglect the remainder, we obtain for $x^{\ast }$ an approximation $x_3$ given by

If $x_3\in [a,b],$ then we can take $y_2=f(x_2)$ and $y_3=f(x_3)$ as the new interpolation nodes, and continue the process.

In general, if $x_{m-1},x_m\in [a,b]\text{and }{y}_{m-1}=f(x_{m-1}),y_m=f(x_m)$ then we take

By (7) we have that

The above relation shows that if all the elements of the sequence ${(x_m)}_{m\ge 1}$ generated by (8) remain in $[a,b]$ and converge to $x^{\ast },$ then the $r$-convergence order $\omega$ is given by the positive root of the following equation [7], [8], [12], [14]:

i.e.

The convergence order can be higher than $\omega$ if the interpolation nodes in (8) are controlled with the aid of $p$ and $h.$

If, given $x_m\in [a,b],$ we take as interpolation nodes the values $y_m=f(p(x_m)),$ ${\overline{y}}_m=f(h(x_m)),$ $m\ge 1,$ then we obtain the iterations

with error

where ${\overline{\eta }}_m\in \mathrm{int}(F).$ We call (10) as an Aitken-Steffensen-Hermite type method with two steps.

In the following result we show that under some reasonable assumptions on functions $f,$ $p$ and $q,$ the sequence (10) has the $q$-convergence order at least $n.$

The proof is immediately obtained from (11).

Next we shall study the convergence of the method (10) in the particular cases $s=1,$ $k=2$ resp. $s=2,$ $k=1,$ which both lead by (13) to $q$-convergence orders at least 3.

Under reasonable assumptions regarding mainly the monotony and convexity of $f$ on $[a,b],$ we shall show that the functions $p$ and $q$ may be determined such that the convergence order of (10) is 3 and, moreover, we obtain sequences which approximate bilaterally the solution.

If $x,y,z\in [a,b]$ and $u=f(x),$ $v=f(y),$ $w=f(z)$ then it is known that the following relations hold for the divided differences of $f$ and $f^{-1}$

where

and $[x,x;f]:=f^{\prime }(x).$

Using the divided differences for the Hermite polynomial in the case $s=1$ and $k=2,$ from (10) we get for $x_{m+1}$ the following expression

$x_1\in [a,b],m=1,2,\mathrm{\dots }.$

In this case, the error verifies

The mean value formula for the divided differences attracts the existence of a point ${\theta }_m\in \mathrm{int}(F)$ such that:

Since $f$ is bijective there exists ${\xi }_m\in [a,b]$ such that ${\theta }_m=f({\xi }_m).$ Taking into account (5), by (16) and (17) it follows

Analogously, for $s=2$ and $k=1$ we obtain the iterations

$m=1,2,\mathrm{\dots },x_1\in [a,b].$

In this case, for the error one holds an equality analogous to formula (18):

$m=1,2,\mathrm{\dots }$

In this section we shall provide some conditions under which methods (15) and (19) generate sequences which approximate bilaterally the solution.

We consider the following assumptions on $f,$ $p$ and $q:$

(a) equation (1) has at least one solution $x^{\ast }\in ]a,b[;$

(b) $f\in C^3([a,b]);$

(c) equations (1) and (2) are equivalent;

(d) function $p$ is derivable on $]a,b[$ and $\forall x\in ]a,b[;$

(e) function $q$ is decreasing and continuous on $[a,b];$

Denote $E_f:[a,b]\to ℝ,$

It can be easily seen that

We obtain the following result when $f$ is increasing and convex.

The fact that the elements of ${(x_m)}_{m\ge 1},$ ${(p(x_m))}_{m\ge 1}$ and ${(h(x_m))}_{m\ge 1}$ remain in $[a,b]$ follows from j${}_1.$ Moreover, these sequences are monotone and bounded, and therefore they converge. Letting $\mathrm{\ell }=lim{x}_m,$ then by (15) we have $\mathrm{\ell }=x^{\ast }.$ Since $p$ and $q$ are continuous functions, we obtain jjj${}_1.$

The theorem is proved.  

Next we consider the case when $f$ is decreasing and concave. If instead of equation (1) we consider equation

and we take into account (22) and the fact that $x_{m+1}$ given by (15) is the same if we replace $f$ by $-f,$ then from Theorem 2.1 we deduce

The following result refers to the case when $f$ is increasing and concave.

We obtain the following consequence of the above theorem, which is similar to the one obtained for Theorem 2.1. Now $f$ is decreasing and convex.

Now we study the convergence of the sequences ${(x_m)}_{m\ge 1}$ given by (19). We notice that in (19) $x_{m+1}$ may also be written as:

We have seen in the proof of the previous results that the hypothesis $E_f(x)\ge 0,$ $\forall x\in [a,b]$ was essential. The iterates (15) and the results concerning their convergence cannot be applied if $E_f(x)\le 0,$ $\forall x\in [a,b].$ We shall see in the following that in the study of the convergence of iterations (19) or (23), the hypothesis $E_f(x)\le 0,$ $\forall x\in [a,b]$ turns out to be essential. Therefore the previous and the subsequent results allow us to choose in practice either method (15) or method (19), (23) depending on the sign of $E_f(x).$

The following result is similar to Corollary 2.2, we state it without proof.

The next result is analogous to Theorem 2.3.