Let $X$ be a Banach space, $D\subseteq X$ a subset, and $f:D\to X$ a nonlinear mapping. Consider the equation

where $\theta \in X$ is the zero vector of $X.$

Regarding $f$ we make the following assumptions:

• a)

  $f:D\to f(D)$ is a one to one mapping;

• b)

  equation (1) has a solution $x^{\ast }\in D;$

• c)

  the operator $f$ is Fréchet differentiable on $D$ and $f^{\prime }(x)\ne {\theta }_1,$ where ${\theta }_1$ is the null linear operator.

In order to accelerate the convergence of the iterative methods of interpolatory type, we also consider an equation, equivalent to (1), of the form

where $\phi :D\to D.$

We make the following assumptions regarding $\phi$:

• a^′)

  $\phi$ is $p$ times differentiable on the whole set $D,$ for some $p\in \mathbb{N};$

• b^′)

  we have ${\phi }^{(i)}(\theta )={\theta }_i,$ $i=1,p-1,$ but ${\phi }^{(p)}(\theta )\ne {\theta }_p,$ where ${\theta }_i$ denotes the $i$-linear operator. Also, $\Vert {\phi }^{(p)}(x)\Vert \le L,$ $\forall x\in D,$ for some $L\ge 0.$

Let $x_0,x_1,\mathrm{\dots },x_n\in D$ and $y_0=f(x_0),y_1=f(x_1),\mathrm{\dots },y_n=f(x_n).$ For $f^{-1}:f(D)\to D,$ the Newton identity holds (see, e.g., [4], [5]):

By hypotheses a) and b) we get the relation:

The relation above and (3) for $y=\theta$ attract that

whence we deduce an approximation $u$ for $x^{\ast },$ of the form:

The error for this approximation is bounded by

where the norm of $[\theta ,y_0,\mathrm{\dots }{y}_n;f^{-1}]$ is considered in the space of $n+1$-linear operators.

In [3] it is shown that the convergence order of the iterative methods given by (6) cannot be greater than $2$, even if the number of the interpolation nodes is arbitrarily increased. However, the convergence order can be increased if we use the auxilliary function $\phi$ considered above.

Let $x_0\in D$ be an initial approximation to $x^{\ast }.$ Denote $u_0^0=x_0,u_1^0=\phi (u_0^0),\mathrm{\dots },u_n^0=\phi (u_{n-1}^0)$ and $y_0^0=f(u_0^0),y_1^0=f(u_1^0),\mathrm{\dots },y_n^0=f(u_n^0).$ Replacing in (6) the interpolation nodes by $y_i^0,$ $i=0,n,$ we obtain for $x^{\ast }$ a first approximation, denoted by $x_1$:

If $x_i\in D$ is an approximation for $x^{\ast },$ $i\in \mathbb{N},$ $n\ge 1,$ then we obtain the next approximation in the following way. Denote $u_0^i=x_i,u_1^i=\phi (u_0^i),\mathrm{\dots },u_n^i=\phi (u_{n-1}^i)$ and $y_0^i=f(u_0^i),\mathrm{\dots },y_n^i=f(u_n^i).$ Analogously to (8) we get

with the error

In the following we shall analyze some particular instance of (9).

If we take $n=1$ in (9), we get

Taking into account the identities $[y_0^i,y_1^i;f^{-1}]={[x_i,\phi (x_i);f]}^{-1}$ and $y_0^i=f(x_i),$ $y_1^i=f(\phi (x_i)),$ we notice that we are lead to the Steffensen method:

If we take $n=2$ and recall that

$i=0,1,\mathrm{\dots },$ and we get

a corrected Steffensen method.

In [6] we have studied the local convergence of the Steffensen method (12). In the following we shall study the local convergence of the general method (9). We shall show that the convergence order considerably grows even for $p=1,$ under condition a^′).

From (10), using the finite growth formula (see, e.g., [5]), we get

where we have assumed that there exist $M,K>0$ such that

From the hypotheses a^′) and b^′), using the Taylor formula we get

where $l=\frac{L}{p!}$ and ${\delta }_s={\sum }_{j=0}^{s-1}{p}^j.$

We make the following notations:

Using (18), relations (15) become

Multiplying these inequalities by $q^{\frac{1}{\beta -1}}$ and denoting ${\eta }_i=q^{\frac{1}{\beta -1}}\Vert x^{\ast }-x_i\Vert ,$ $i=0,1,\mathrm{\dots },$ we get:

Taking into account the above considerations, we obtain the following result.

In case of the particular methods (12) and (14) we get the following results.

We conclude that the iterative methods of interpolatory type may attain a substantially higher convergence order if the nodes are properly controlled.