As it is well known, the Steffensen method for approximating the solutions of equations is an interpolatory type method with controlled nodes [3], [4], [6], [8], [9].

More precisely, if we generate in the Lagrange polynomial of inverse interpolation of degree 1, the nodes of interpolation, in a particular way, we obtain one of the known variants of the Steffensen’s method [1], [5], [12], [9], [13], [14].

Consider the equation

where $f:[a,b]\to ℝ,$ $a,b\in ℝ,$ In a sufficiently general case, in order to obtain approximations of a root $\overline{x}\in [a,b]$ of equation (1), we shall consider another equation, equivalent to (1), of the form

Let $F=f([a,b])$ be the set of values of the function $f$ for $x\in [a,b].$ We suppose that $f$ is one to one, i.e. there exists $f^{-1}:F\to [a,b].$ We consider the interpolation nodes $a_i\in [a,b],$ $i=1,2,$ and $b_i=f(a_i),$ $i=1,2$ the values of function $f$ at these nodes. The Lagrange interpolation polynomial of first degree for the function $f^{-1},$ on the nodes $b_i,$ $i=1,2,$ has the form:

Taking into account relation $\overline{x}=f^{-1}(0),$ for $y=0$ from (3), we obtain an approximation of root $\overline{x}$, given by relation

from which, if we take into account equality

we obtain:

i.e. the regula falsi, which leads us to the chord method.

Let $x_n\in [a,b]$ $n\in {\mathbb{N}}^{\ast }$ be an approximation of root $\overline{x}$ of equation (1). If in (5) we take $a_1=x_n$ and $a_2=g(x_n)$, we obtain the Steffensen’s method, which is written as:

In the present work we shall study a Steffensen type method, more general than (6), which will rely on the Lagrange polynomial of inverse interpolation of second degree. Accordingly, let $a_i\in [a,b],i=1,2,3$ be three nodes of interpolation, and $b_i=f(a_i),i=1,2,3.$ The Lagrange interpolation polynomial of second degree, for the inverse function $f^{-1},$ has the form:

with equality

for every $y\in F.$

For $y=0$ from (7) and (8) we obtain:

It is easy to see that the following equality takes place:

From (4), (9) and (10) we obtain for $\overline{x}$ the approximation

with the error given by relation

Further on, we shall suppose that $f\in C^3[a,b]$ and $f^{\prime }(x)\ne 0$ for every $x\in [a,b].$ Hence $f^{-1}\in C^3(F)$, and the following relation takes place

where $y=f(x)$, [6], [8], [7], [15]. From the mean value theorem for divided differences, it results that there exists $\eta \in \mathrm{int}(F)$ so that

If we take into account that $f$ is one to one and onto, it results that for $\eta \in \mathrm{int}(F),$ there exists $\xi \in ]a,b[,$ so that $\eta =f(\xi )$ and from (14) and (13) we have:

Using as in (6) function $g$ for the control of interpolation nodes, we shall obtain, from (11), a generalized method of Steffensen type.

Thus, let $x_n\in [a,b],$ $n\in {\mathbb{N}}^{\ast },$ an approximation of solution $\overline{x}$ of equation (1); then, we shall obtain approximation $x_{n+1}$ from (11) considering $a_1=x_n,$ $a_2=g(x_n),$ $a_3=g(g(x_n)),$ i.e.

We shall name the method (16), the Steffensen’s method of degree three.

For the study of convergence of sequences ${(x_n)}_{n\ge 0}$ and ${(g(x_n))}_{n\ge 0}$ generated by (16), we shall analyze first the conditions on functions $f$ and $g$, as well as on initial value $x_0\in [a,b],$ so that the two considered sequences to be monotonically. This fact will give us the possibility to control the error at each iteration step. We also study the local convergence, and we shall show that, under certain assumptions, function $g$ can be chosen to assure assumptions regarding monotonous convergence [2], [10], [11], [12].

In the following we shall study the convergence of sequences ${(x_n)}_{n\ge 0}$ and ${(g(x_n))}_{n\ge 0}$ given by (16).

We shall consider the following assumptions on the functions $f$ and $g$:

• $({\alpha }_1$)

  equation (1) has a unique root $\overline{x}\in ]a,b[;$

• $({\alpha }_2)$

  equation (2) is equivalent to (1);

• $({\alpha }_3)$

  function $g$ is decreasing on $[a,b];$

• $({\alpha }_4)$

  there exists $\mathrm{\ell }\in ℝ,$ so that, for every $x\in [a,b]$ the following relation takes place:

  $$\left|g(x)-g(\overline{x})\right|\le \mathrm{\ell }\left|x-\overline{x}\right|;$$

  (17)

• $({\alpha }_5)$

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

• $({\alpha }_6)$

  function $E_f:[a,b]\to ℝ,$ given by relation:

  $$E_f(x)=3{(f^{\prime \prime }(x))}^2-f^{\prime }(x)f^{\prime \prime \prime }(x)$$

  (18)

  verifies condition $E_f(x)\le 0$ for every $x\in [a,b].$

The following theorems take place, depending on the properties of monotonicity and convexity of function $f$:

From (17) it is easy to see that relation $g(g(x_n))\ge x_n$ takes place, i.e. we have:

From the above relation it results that $f(x_n)>0$ and $f(g(g(x_n)))>0$ and from (25) and (16) it results From (24) for ${\xi }_n\in ]g(x_n),x{}_n[$ it results $x_{n+1}\ge \overline{x}$ which, together with $g(x_{n+1})>g(x_n)$ leads us to relations (20).

The consequence ${\mathrm{𝐣𝐣𝐣}}_1$ is a result of relation (19) or (20). For ${\mathrm{𝐣𝐯}}_1,$ from (19), respectively (20), it results that it exists $u=lim{x}_n.$ Getting to the limit for $n\to \mathrm{\infty }$ in (16) we obtain $f(u)=0$, and from assumption ${\alpha }_2)$ it results $u=g(u),$ i.e. $u=\overline{x},$ and $limg(x_n)=g(\overline{x})=\overline{x}.$ Thus, Theorem 1 is proven.  

Obviously, assumptions ${\mathrm{𝐢𝐢}}_1$ and ${\mathrm{𝐢𝐢𝐢}}_1$ are verified by $h$ $=-$ $f.$ Also, relations (16) do not change if we replace $f$ by $-f.$ Taking into account Theorem 1, it is obvious that the consequences from Theorem 2 take place.  

In the following, in order to point out the convergence order of method (16), we shall provide a result concerning the local convergence of this method. Accordingly, we shall admit that functions $f$ and $g$ verify the following assumptions:

${\beta }_1)$

there exists $m,$ $M\in ℝ$, $m>0,$ $M>0,$ so that $m\le \left|f^{\prime }(x)\right|\le M,$ for every $x\in [a,b];$

${\beta }_2)$

there exists $E>0,$ $E\in ℝ$ so that $\left|E_f(x)\right|\le E$ for every $x\in [a,b].$

The following theorem holds:

Further on, we shall show that, within supplementary assumptions upon $f,$ depending on its monotony and convexity, we can construct the functions $g,$ which fulfill, respectively, the assumptions of Theorems 1–4. More precisely, the essential assumptions upon function $g,$ are given by ${\alpha }_2),$ ${\alpha }_3),$ ${\alpha }_4)$ and ${𝐢}_1$ or, analogously, ${𝐢}_2,$ ${𝐢}_3,$ ${𝐢}_4.$

The following theorems take place:

It is obvious that function $g$ given by (32) verifies ${\alpha }_2).$

For ${\alpha }_3)$ and ${\alpha }_4)$ it is sufficient that function $g^{\prime }(x)$ should verify relations

where

From (33) ${\mathrm{𝐢𝐢}}_1$ and ${\mathrm{𝐢𝐢𝐢}}_1$ it results:

It is not difficult to show that if $\lambda$ verifies (34), then $g^{\prime }(x)$ verifies

i.e. assumptions ${\alpha }_3)$ and ${\alpha }_4)$ are verified.