It is well known that the Steffensen, Aitken, and Aitken-Steffensen methods are obtained from the inverse Lagrange interpolation polynomial of degree one, with controlled nodes [9]–[13], [7]. Consider the equation

where

We also consider the following three equations, each of them equivalent with equation (1.1):

and

The Steffensen method in given by relations

and, analogously, the Aitken method is of the following form:

Finally, the Aitken-Steffensen method is given by the relations:

The order of convergence of all three methods (1.4)–(1.6) is at least two, and this order depends on the functions $g$ and $g_1,$ $g_2$ respectively. Essentially, the methods (1.4)–(1.6) are obtained from the method of chord where the interpolation nodes depend on the functions $g$ and respectively $g_1,g_2.$ In papers [2], [9], [10], [13] some conditions had to be considered in order that all the three methods (1.4)–(1.6) generate two sequences ${(u_n)}_{n\ge 0}$ and ${(v_n)}_{n\ge 0}$ with the properties:

$\alpha$) The sequence ${(u_n)}_{n\ge 0}$ is increasing and the sequence ${(v_n)}_{n\ge 1}$ is decreasing:

$\beta )limu{}_n=limv{}_n=\overline{x},$ where $\overline{x}$ is the root of equation (1.1), $\overline{x}$ $\in [a,b].$

Practically, such sequences are very interesting, because by inequalities

the errors of approximation at every step of iteration may be controlled.

Let $a_1,a_2,$ $a_3\in [a,b]$ be three nodes of interpolation, and let $b_1,$ $b_2,$ $b_3$ the values of the function $f$, i.e.

Suppose that the function $f:[a,b]\to F,$ is bijective where $F=f([a,b]).$

Then there exists $f^{-1}:F\to [a,b]$ and the following equality holds [13], [15]:

for every $y\in F.$

If $\overline{x}\in [a,b]$ is the root of equation (1.1), then $\overline{x}=f^{-1}(0),$ and by (1) one obtains the following representation of $\overline{x}:$

By relations (see [13])

and

using (1), one obtains the following approximation for $\overline{x}:$

and the error:

If $f\in C^3([a,b])$ and $f^{\prime }(x)\ne 0,$ $\forall x\in [a,b],$ then $f^{-1}\in C^3(F)$ and the following equality holds (see [13], [17]):

where $y=f\left(x\right).$

Using the mean value formula for divided differences (see [13]) it follows that there exists $\eta \in F$ such that

Because $f$ is bijective, it follows that there exists $\xi \in [a,b]$ such that $\eta =f(\xi )$, and by (1.11) and (1.12) one obtains:

By (1.9), if one considers particular nodes $a_1,a_2,a_3$ it is possible to obtain different methods of Steffensen type, of Aitken type or of Aitken-Steffensen type.

Let $x_n\in [a,b]$ be an approximation of the root $\overline{x}$ of equation (1.1).

If one considers $a_1=x_n,$ $a_2=g(x_n),$ $a_3=g(g(x_n))$, then it follows the following method of Steffensen type [9], [10], [13]:

$x_0\in [a,b],n=0,1,2,\mathrm{\dots }.$

If $a_1=x_n,$ $a_2=g_1(x_n),$ $a_3=g_2(g_1(x_n))$, then one obtains the following method of Aitken-Steffensen type:

$n=0,1,\mathrm{\dots },$ $x_0\in [a,b],$ and finally, for $a_1=x_n,a_2=g_1(x_n),$ $a_3=g_2(x_n)$ one obtains the following method of Aitken type:

Using the symmetry of Lagrange polynomial with respect to nodes, by permutations of $a_1,$ $a_2,$ $a_3$ in methods (1)–(1), one obtains the same results for $x_{n+1}.$

In [15] conditions are given in order that method (1) generates sequences approximating the root of equation (1.1) bilaterally. In this paper we study methods (1) and (1) and we obtain same conditions in order that the sequences generated by there methods are bilateral approximations of the root $\overline{x}$ of equations (1.1).

In the following we study the method (1) and we search the conditions on the sequences ${(x_n)}_{n\ge 0},$ ${(g_1(x_n))}_{n\ge 0}$ and $(g_2{(g_1(x_n))}_{n\ge 0}$ generated from this method in order that they are monotonic sequences, bilaterally approximating the root $\overline{x}$ of equation (1.1).

We need the following hypothesis:

1. a)

   $g_1$ is increasing on $[a,b]$;

2. b)

   $g_2$ is continuous and decreasing on $[a,b]$;

3. c)

   equation (1.1) has a solution $\overline{x}\in [a,b]$ and $g_1(\overline{x})=g_2(\overline{x})=\overline{x},$

4. d)

   function $f$ is in $C^3([a,b]),$ and for every $x\in [a,b]$ the following relation is fulfilled:

   (2.1)

5. e)

   function $g_1$ satisfies inequality

   $$\left|g_1(x)-g_1(\overline{x})\right|\le L\left|x-\overline{x}\right|,\forall x\in [a,b],$$

   where

Concerning the convergence of sequence ${(x_n)}_{n\ge 0}$ generated by (1), the following Theorem holds:

1. i${}_1{}^{})$

   $f$ is increasing on $[a,b]$;

2. ii${}_1{}^{})$

   $f$ is convex on $[a,b]$;

3. iii${}_1{}^{})$

   functions $g_1,g_2$ and $f$ verify hypotheses $a)-e)$;

4. iv${}_1{}^{})$

   $x_0>\overline{x}$ and $g_2(g_1(x_0))\ge a.$

Then the sequences ${(x_n)}_{n\ge 0},$ ${(g_1(x_n))}_{n\ge 0}$ and $(g_2{(g_1(x_n))}_{n\ge 0}$ generated by (1) verify the properties:

1. j${}_1{}^{})$

   sequences ${(x_n)}_{n\ge 0}$ and ${(g_1(x_n))}_{n\ge 0}$ are decreasing and bounded from below by $\overline{x}$ ;

2. jj${}_1{}^{})$

   sequence $(g_2{(g_1(x_n))}_{n\ge 0}$ is increasing and bounded from above by $\overline{x}$;

3. jjj${}_1{}^{})$

   at every iteration step the following inequalities hold:

   (2.2)

   $$x_n-\overline{x}\le x_n-g_2(g_1(x_n)),n=0,1,\mathrm{\dots };$$

4. jv${}_1{}^{})$

   $lim{x}_n=lim{g}_1(x_n)=lim{g}_2(g_1(x_n)=\overline{x}.$

An analogous proof with that from Theorem 2.1 is valid for the following:

1. i${}_2{}^{})$

   function $f$ is increasing on $[a,b]$;

2. ii${}_2{}^{})$

   function $f$ is concave on $[a,b]$;

3. iii${}_2{}^{})$

   functions $g_1,g_2$ and $f$ verify the hypotheses $a)-e)$;

4. iv${}_2{}^{})$

   and $g_2(g_1(x_0)\le b.$

Then sequences ${(x_n)}_{n\ge 0},$ ${(g_1(x_n))}_{n\ge 0}$ and $(g_2(g_1(x_{n\ge 0})))$ generated by (1) have the following properties:

1. j${}_2{}^{})$

   sequences ${(x_n)}_{n\ge 0}$ and ${(g_1(x_n))}_{n\ge 0}$ are increasing and bounded from above by $\overline{x};$

2. jj${}_2{}^{})$

   sequence $g_2(g_1(x_n))$ is decreasing and bounded from below by $\overline{x};$

3. jjj${}_2{}^{})$

   the following relations hold:

   (2.4)

   $$\overline{x}-x_n\le g_2(g_1(x_n))-x_n,n=0,1,2,\mathrm{\dots },$$

4. jv${}_2{}^{})$

   $lim{x}_n=lim{g}_1(x_n)=lim{g}_2(g_1(x_n))=\overline{x}.$

In order to study the convergence of the sequences generated by (1) we must suppose that functions $f,g_1$ and $g_2$ verify hypotheses a)-e) of section 2.

The following theorem holds:

1. i${}_3{}^{})$

   $f$ is increasing an $[a,b]$;

2. ii${}_3{}^{})$

   $f$ is convex an $[a,b]$;

3. iii${}_3{}^{})$

   $f,$ $g_1$ and $g_2$ verifies the hypotheses $a)-e)$;

4. iv${}_3{}^{})$

   $x_0>\overline{x}$ and $g_2(x_0)>a.$

Then sequences ${(x_n)}_{n\ge 0},$ ${(g_1(x_n))}_{n\ge 0}$ and ${(g_2(x_n))}_{n\ge 0}$ generated by (1) have the properties:

1. j${}_3{}^{})$

   sequences ${(x_n)}_{n\ge 0}$ and ${(g_1(x_n))}_{n\ge 0}$ are decreasing and bounded from below by $\overline{x}$ ;

2. jj${}_3{}^{})$

   sequence ${(g_2(x_n))}_{n\ge 0}$ is increasing and bounded from above by $\overline{x}$;

3. jjj${}_3{}^{})$

   the following relations hold,

   $$x_n-\overline{x}\le x_n-g_2(x_n),n=0,1,\mathrm{\dots };$$

4. jv${}_3{}^{})$

   $lim{x}_n=lim{g}_1(x_n)=lim{g}_2(x_n)=\overline{x}.$

Analogously, the following result can be proved

1. i₄₎

   $f$ is increasing on $[a,b]$;

2. ii${}_4{}^{})$

   $f$ is concave on $[a,b]$;

3. iii${}_4{}^{})$

   $f,g_1,$ and $g_2$ verify hypothesis a)-e);

4. iv${}_4{}^{})$

   and $g_2(x_0)\le b.$

Then sequences ${(x_n)}_{n\ge 0},$ ${(g_1(x_n))}_{n\ge 0}$ and ${(g_2(x_n))}_{n\ge 0}$ generated by (1) verify the properties:

1. j${}_4{}^{})$

   ${(x_n)}_{n\ge 0}$ and ${(g_1(x_n))}_{n\ge 0}$ are increasing and bounded from above by $\overline{x};$

2. jj${}_4{}^{})$

   sequence ${(g_2(x_n))}_{n\ge 0}$ is decreasing and bounded from below by $\overline{x};$

3. jjj${}_4{}^{})$

   the following relations hold:

   $$\overline{x}-x_n\le g_2(x_n)-x_n,n=0,1,2,\mathrm{\dots },$$

4. jv${}_4{}^{})$

   $lim{x}_n=lim{g}_1(x_n)=lim{g}_2(x_n)=\overline{x}.$

In the following, for every situation concerning the monotonicity and convexity of function $f,$ functions $g_1$ and $g_2$ can be determined such that conditions a), b), c) and e) should be verified. In the following we thoroughly present the case in which function $f$ is increasing and convex.

Supposing that $f^{\prime }(x)>0$ for every $x\in [a,b],$ one considers the functions:

and

Then

for every $x\in [a,b]$ and, consequently $g_1$ is increasing. Analogously

for every $x\in [a,b]$ and then $g_2$ is a decreasing function. It follows that $g_1$ and $g_2$ verify hypotheses a) and b). The hypothesis e) is also verified because for every $x\in [a,b].$ In $i{v}_1)$ the inequality $g_2(g_1(x_0)\ge a$ holds. Because and $x_0>\overline{x},$ the above inequality is verified if $g_2(x_0)\ge a.$ This follows, because $g_2(g_1(x_0))>g_2(x_0)$ ($g_2$ is a decreasing functions). This means that the following relation must hold:

i.e.

Because $f(x_0)>0$ and $f^{\prime }(a)>0$, for the veridicity of last inequality it is sufficient that

where $\overline{x}$ is the root of equation (1.1). This last inequality may be realized if $x_0$ is sufficiently close to $\overline{x}.$

Consequently, the hypotheses of theorems 2.1 and 3.1 are realized for $g_1$ and $g_2$ considered above. The other cases may be similarly analyzed.

In the following we prove that every method (1) and (1) have the order of convergence three. The order of convergence for method (1) was treated in [15], and this order is at least three. Assume the following hypotheses:

$\alpha$) function $g_2$ verifies relation

for every $x\in [a,b],$ where $p>0,$ $p\in ℝ;$

$\beta )$ $m\le \left|f^{\prime }(x)\right|\le M,$ for every $x\in [a,b],$ where $m>0$ and $M>0$ are real numbers.

$\gamma )|3[f^{\prime }(x)]{}^2-f{}^{\prime }(x)f{}^{\prime \prime \prime }(x)|\le q,$ for every $x\in [a,b],$ where $q>0,$ $q\in ℝ$ .

In hypotheses $\alpha ),\beta ),\gamma )$ and e), by (1.10) and (1.13), for sequence ${(x_n)}_{n\ge 0}$ generated by (1), one obtains:

and this means that the order of convergence for method (1) is at least three.

With the same hypotheses, for tree sequence ${(x_n)}_{n\ge 0}$ generated by (1) it follows:

i.e. the order of convergence of (1) is also at least three.