From a practical standpoint, in order to approximate the roots of equations it is advantageous to use methods which lead to monotonic sequences. In this paper we shall use a single iterative process to determine an increasing sequence ${\left(x_n\right)}_{n\ge 0}$ and a decreasing one ${\left(y_n\right)}_{n\ge 0}$, both converging to a root $\overline{x}$ of the equation $f\left(x\right)=0.$ Such a procedure has the advantage of allowing, at each iteration step, an approximation error checking, i.e.

As it is well known, such sequences can be generated by applying simultaneously two methods, e.g. Newton’s method and the chord method. In the sequel we shall show that, in conditions similar to those imposed to the above mentioned methods, the Aitken-Steffensen method generates two sequences which fulfil the above inequality.

In Section 2 we give sufficient conditions for the general method of Aitken-Steffensen ((3) below) to generate two monotonic sequences, both converging to the solution $\overline{x}$ of equation (1) below. In this way some similar results obtained in [4] are completed, and the proofs given in [5] are made simpler to a certain extent.

In Section 3 we study some particular cases of the method (3), namely the methods (7) and (9) below. In Section 4 there is indicated a way to construct the auxiliary functions $g_1,g_{2,}$ or $g,$ required by (3) or (7), in relatively simple conditions as the monotonicity and convexity of the function $f.$ In Section 5 the convergence orders and the efficiency indices of the methods (3) and (7) are studied, concluding that the method (7) has a higher efficiency index. This one is the same as that of Newton’s method, but is given conditions, the method (7) as well as (3) and (9), provide in addition bilateral approximations for the root of the equation (1).

So, let be an interval of the real axis, and consider the equation

where $f:I\to ℝ$. Besides (1), consider two more equations

where $g_1,g_2:I\to ℝ$.

The Aitken-Steffensen method consists in constructing the sequences ${\left(g_2\left(g_1(x_n)\right)\right)}_{n\ge 0}$ and ${\left(g_1\left(x_n\right)\right)}_{n\ge 0}$ by means of the following iterative process

where $[u,v;f]$ stands for the first order divided difference of $f$ on the points $u,v\in I.$ We shall denote by $[u,v,w;f]$ the second order divided difference of $f$ on $u,v,w\in I.$

As it is well known [6, pp. 268–269], and will also be shown in Section 5, the convergence order of the sequence ${\left(x_n\right)}_{n\ge 0}$ generated by (3) is at least 2, but it generally depends on the convergence orders of the sequences ${\left(y_n\right)}_{n\ge 0}$ and ${\left(z_n\right)}_{n\ge 0}$ generated by $y_{n+1}=g_1\left(y_n\right),y_0\in I;z_{n+1}=g_2\left(z_n\right),$ $z_0\in I;n=0,1,\mathrm{\dots },;$[2], [5].

Another advantage of the iteration method (3) is the following: if equation (1) is given, then (as we shall see in Section 4) we have at our disposal many possibilities to construct the functions $g_1$ and $g_2$ from equations (2). On the other hand, hypotheses concerning the differentiability of $f$ on the whole interval $I$ are not needed.

If $g:I\to ℝ$ is a function, we shall adopt the following definitions concerning the monotonicity and convexity of $g$:

We shall adopt the following hypotheses concerning the functions $f,g_1$ and $g_2;$

• (a)

  $f,g_1,g_2$ are continuous on $I;$

• (b)

  $g_1$ is increasing on $I;$

• (c)

  $g_2$ is decreasing on $I;$

• (d)

  equations (1) and (2) have only one common root $\overline{x}\in ]a,b[;$

• (e)

  for every $x,y\in I,$ the relation holds.

Concerning the problem stated in Section 1, the following theorem holds:

In can be shown analogously that the following theorems hold, too:

An interesting particular case of the procedure (3) is obtained taking

$g_1\left(x\right)=x$ for every $x\in I.$ In this way one obtains Steffensen’s method, namely

where $g$ stands for $g_2.$

One observes easily that in this case hypotheses (a), (b), (d) and (e) are automatically fulfilled by $g_1.$ For the study of the convergence of the sequence ${\left(x_n\right)}_{n\ge 0}$ generated by (7), we have to adopt the following hypotheses:

• (a₁)

  the functions $f$ and $g$ are continuous on $I;$

• (b₁)

  the function $g$ is decreasing on $I;$

• (c₁)

  equations (1) and $x=g\left(x\right)$ have only one common root $\overline{x}\in ]a,b[.$

With these specifications, the theorems stated in Section 2 yield:

Another interesting particular case is that in which $f$ has the form:

In this case the iterative method (7) will have the form

whose convergence follows easily by Corollaries 3.1–3.4. The following results simplify the conditions imposed to $g$ and $x_0$ in [1]:

The restriction imposed to $f$ in Corollaries 3.3 and 3.4 ($f$ to be decreasing) can no more be reached in this case, because in those corollaries the condition on $g$ to be decreasing is essential, and is easy to see that if $g$ decreases the $f$ is increasing.

In what follows we shall show that, under reasonable hypotheses concerning the monotonicity and convexity of $f,$ we have at our disposal various ways to choose $g_1,g_2$ and $g$ such that the hypotheses of the theorems and corollaries stated above are verified.

1. Let us admit that $f:[a,b]\to ℝ$ is differentiable on $[a,b],$ and denote by $f^{\prime }\left(a\right)$ the right-hand derivative in a and by $f\prime \left(b\right)$ the left-hand derivative in $b;$ we also admit that $f$ is increasing and convex on $[a,b].$ Suppose that the equation $f\left(x\right)=0$ has a root $\overline{x}\in ]a,b[.$ Then we choose $g_1\left(x\right)=x-f\left(x\right)/f^{\prime }\left(b\right)$ and $g_2\left(x\right)=x-f\left(x\right)/f^{\prime }\left(a\right).$

Since $f$ is convex, if follows that $f^{\prime }$ is increasing on $[a,b],$ hence for every $x\in ]a,b[.$ In this case we have $g_1^{\prime }\left(x\right)=1-f^{\prime }\left(x\right)/f^{\prime }\left(b\right)>0$ for every $x\in ]a,b[$ and for every $x\in ]a,b[.$ We choose then a subinterval $[\alpha ,\beta ]\subset ]a,b[$ for which $\overline{x}\in [\alpha ,\beta ].$ If we suppose in addition that there exists $x_0\in [\alpha ,\beta ]$ for which and $g_2\left(g_1\left(x_0\right)\right)\in [\alpha ,\beta ]$ then the above constructed functions $f,g_1$ and $g_2$ verify the hypotheses of Theorem 2.1. It is easy to see that if we choose $g_1\left(x\right)=x-{\lambda }_{1}f\left(x\right)$ and $g_2\left(x\right)=x-{\lambda }_{2}f\left(x\right),$ with ${\lambda }_1,$ ${\lambda }_2\in ℝ$, ${\lambda }_1>f^{\prime }\left(b\right)$ and then $g_1$ and $g_2$ also verify the hypotheses of Theorem 2.1.

2. If $f$ is differentiable, decreasing and convex on $[a,b],$ then $g_1$ and $g_2$ can be chosen as in case $1.$ If there exists $x_0\in [\alpha ,\beta ]$ such that and $g_2\left(g_1\left(x_0\right)\right)\in [\alpha ,\beta ],$ then the hypotheses of Theorem 2.3 are verified.

3. If $f$ is differentiable, decreasing and concave on $[a,b],$ then we may put $g_1\left(x\right)=x-f\left(x\right)/f^{\prime }\left(a\right)$ and $g_2\left(x\right)=x-f\left(x\right)/f^{\prime }\left(b\right).$ If there exists $x_0\in [\alpha ,\beta ]$ for which $f\left(x_0\right)>0$ and $g_2\left(g_1\left(x_0\right)\right)\in [\alpha ,\beta ],$ then $f,g_1$ and $g_2$ fulfil the conditions of Theorem 2.4.

4. If $f$ is differentiable, increasing and concave on $[a,b],$ then $g_1$ and $g_2$ can be chosen as in case 3. If there exists $x_0\in [\alpha ,\beta ]$ such that $f\left(x_0\right)>0$ and $g_2\left(g_1\left(x_0\right)\right)\in I,$ then $f,g_1$ and $g_2$ verify the hypotheses of Theorem 2.2.

5. The function $g$ appearing in Section 3, Procedure (7) can be chosen in the following ways:

a) If $f$ is differentiable, increasing and convex on $[a,b],$ then we can choose $g\left(x\right)=x-f\left(x\right)/f^{\prime }\left(a\right),$ and if $x_0$ verifies and $g\left(x_0\right)\in [\alpha ,\beta ],$ then the hypotheses of Corollary 3.1 are verified; while if $f$ is decreasing and convex and there exists $x_0\in [\alpha ,\beta ]$ for which and $g\left(x_0\right)\in [\alpha ,\beta ],$ the hypotheses of Corollary 3.3 are verified.

b) If $f$ is differentiable, decreasing and concave on $[a,b],$ we choose $g\left(x\right)=x-f\left(x\right)/f^{\prime }\left(b\right).$ If there exists $x_0\in [\alpha ,\beta ]$ such that $f\left(x_0\right)>0$ and $g\left(x_0\right)\in [\alpha ,\beta ],$ then the hypotheses of Corollary 3.4 are verified; while if $f$ is increasing and concave and there exists $x_0\in [a,b]$ such that $f\left(x_0\right)>0$ and $g\left(x_0\right)\in [\alpha ,\beta ],$ the hypotheses of Corollary 3.2 are verified.

To fix the ideas, we shall consider hereafter, besides (1), another equation, equivalent to (1), of the form

where $h:I\to ℝ$. We shall also consider a sequence ${\left(x_n\right)}_{n\ge 0},$ $x_n\in I,$ which, together with $h$ and $f,$ verifies the properties:

• (a)

  $h\left(x_n\right)\in I$ for every $n\in \mathbb{N}$;

• (b)

  ${\left(x_n\right)}_{n\ge 0}$ and ${\left(h\left(x_n\right)\right)}_{n\ge 0}$ are convergent and $lim{x}_n=limh\left(x_n\right)=\overline{x},$ where $\overline{x}$ is the root of equation (1);

• (c)

  $[x,y;f]\ne 0$ for every $x,y\in I;$

• (d)

  $f$ is differentiable at $x=\overline{x}.$

We shall adopt the following definition of the convergence order of the sequence ${\left(x_n\right)}_{n\ge 0}:$

Consider now the functions $g_1,g_2$ appearing in equations (2) and let the function $h$ be given by

Concerning the convergence order of the sequence ${\left(x_n\right)}_{n\ge 0}$ generated by (3), the following theorem holds:

An analogous theorem holds in the case of the procedure (3), too.