It is well known that the Lagrange-Hermite type inverse interpolation offers a lot of iterative methods for solving equations. The preference for one method or for another in obtaining a suitable approximation of a solution must take into account, among other facts, not only the convergence order of the chosen method, but also the amount of calculations needed at each iteration step. While the convergence order may be determined exactly, the amount of calculations from each iteration may be difficult to determine exactly.

The problem we are interested in this note consists in considering a certain class of iterative methods and in determining the methods of that class which have an optimal efficiency index.

Concerning the efficiency index, we shall adopt the definition given by A.M. Ostrowski in [6]. As we shall see in the following, the essential elements in defining the efficiency index of an iterative method are the convergence order and the number of function evaluations needed at each iteration step. As it is well known, the Lagrange-Hermite type interpolatory methods use the values of the considered function, as well as the values of its derivatives up to a certain order. The efficiency index introduced by Ostrowski, as we shall see, seems to be well suited to our purposes.

We shall restrict here to considering the class of methods given by the Chebyshev-type methods, which was also studied in [8], [9].

We shall show that the results obtained in [8] and [9] remain valid under more general hypotheses than those considered in those papers. More exactly, we shall

The factor proportionality $\delta >0$ is considered constant for a given function.

Let be an interval of the real axis and consider the equation

where $f:I\to ℝ$. We assume that equation (1) has a unique solution $\overline{x}\in ]a,b[.$ For solving this equation we consider the function $g:I\to I$ and we suppose that $\overline{x}$ is a fixed point for $g.$ We also assume that the sequence ${\left(x_p\right)}_{p\ge 0}$ given by the iterations

converges. Obviously, if $g$ is continuous at $\overline{x}$, then $\underset{p\to \mathrm{\infty }}{lim}{x}_p=\overline{x}.$

Concerning the function $f,$ we shall also assume the following.

• a)

  the function $f$ is differentiable at $\overline{x};$

• b)

  for any $x,y\in I,$ we have that for some $m>0,$ where $[x,y;f]$ is the divided difference of $f$ on the points $x$ and $y.$

The following lemma can be easily proved.

We shall denote in the following by $m_p$ the number of functions whose values must be computed at each step $p$ in the iterations (2).

Taking into account lemma 2 and according to the efficiency index proposed by Ostrowski, we shall admit the following definition.

Let $q\in \mathbb{N}$ be a natural number, and assume that the function $f$ obeys

• $\alpha )$

  the function $f$ is $q$ times differentiable on $]a,b[.$

• $\beta )$

  the derivative satisfies $f^{\prime }\left(x\right)\ne 0,$ for any $x\in ]a,b[.$

In the above hypotheses it follows that the function $f$ admits an inverse $f^{-1}:D\to I,$ where $D=f\left(I\right),$ and the solution $\overline{x}$ of $\left(l\right)$ verifies

Moreover, the function $f^{-1}$ is $q$ times differentiable on each point from the interior of $D,$ and the $k$-th derivative, where $1\le k\le q,$ is given by

where $y=f\left(x\right),$ and the above sum extends on all nonnegative integer solutions of the system

Let $x_0\in I$ be given. The class of Chebyshev-type iterative methods is given by

where $y_0=f\left(x_p\right).$

For different values of $q,$ relation (6) generates a class of iterative methods of Chebyshev type. We shall denote by (CM) this class.

Taking into account remarks 3 and 4, by (3) we get the following expression for the efficiency index of the class (CM):

Considering the function $\phi \left(t\right)=t^{\frac{1}{bt}},\phi :(0,\mathrm{\infty })\to ℝ$, it can be easily verified that it attains its maximum value at the unique point $l=e.$ It is clear then that $E=\phi \left(q\right),q\in \mathbb{N}$ has the maximum value at $q=3.$ The following result holds.