In this paper we shall specify and go deeply into some problems presented in [2], concerning the Heron’s method for approximating the cubic root of a positive number.

The authors of [2] construct a method based on the Heron’s algorithm for computing the cubic root of 100.

The method works as follows: Given two real numbers $a$ and $b$ satisfying the Heron’s method for approximating $\sqrt[3]{N}$ is

where $d_1=N-a^3$ and $d_2=b^3-N.$ We shall show that the approximation (1) of $\sqrt[3]{N}$ follows from the regula falsi applied to the equation $x^2-\frac{N}{x}=0$ [3]. This will given a rigorous interpretation of (1), and the results from [2] will be reached again.

Using results from [5], we shall give other error bounds than those in [2]. On the other hand, the method is generalized to the case $\sqrt[p]{N},p\in \mathbb{N}$, $p\ge 2,$ the method also offering bilateral approximations. Some remarks on applying the results in [5] for the error bounds will lead us to the generalization of the Heron’s method.

A. In order to approximate the cubic root of $N>0$ by (1), consider the function $f:[a,b]\to ℝ,$ and the function $g:[a,b]\to ℝ$, $g\left(x\right)=\frac{f\left(x\right)}{x}.$

It is well known that regula falsi applied to the equation $g\left(x\right)=0$ leads to the following approximation of its root

$[u,v;g]$denoting the first-order divided difference of $g$ on the nodes $u$ and $v.$ It can be easily verified that $c=\varphi (N,a,b).$

Taking into account that $c\in (a,b)$ and denoting by $[u,v,w;g]$ the second-order divided difference of $g$ on the points $u,v,w,$ we get

for all $x\in (a,b).$

For $x=\sqrt[3]{N}$ in (3) we obtain

from which, by dividing $[a,b;g]$ it follows

An elementary calculation on $\frac{[a,b,\sqrt[3]{N};g]}{[a,b;g]}$ shows that

which gives Theorem 3, [2].

Taking into account the above remarks and using the evaluations obtained by T. Popoviciu in [5], (4) gives the following error bounds

where

Note that (6) leads to a very good error evalutation; since $a$ and $b$ are close to $N,$ then $m_2$ and $M_2$ are close to zero. This is implied by the fact that the function $g$ and its second derivative vanish at the same point $x=\sqrt[3]{N}.$

B. It can be easily seen that the method presented at A can be generalized. For the approximation of the root of order $p$ of the real number $N,p\in \mathbb{N}$, $p\ge 2,$ consider the function $f_1:[a_1,b_1]\to ℝ$,

and the function $g_1:[a_1,b_1]\to ℝ$,

The function $g_1$satisfies $g_1(\sqrt[p]{N})=g_1^{\prime \prime }(\sqrt[p]{N}).$

Applying regula falsi to the equation $g_1\left(x\right)=0,$ we obtain

Similarly to A, we obtain

which gives

where

Let be an interval of the real axis.

Consider the equation

where $F:I\to ℝ$. Suppose that equation (10)has a root $\overline{x}\in (\alpha ,\beta ).$ Consider also a function $h:I\to ℝ$ such that equation

is equivalent to (10).

The Steffensen’s method consists in the generation of two sequences $\left(x_n\right)$ and $\left(h\left(x_n\right)\right)$ by the relations

As we shall see, this method offers the possibility to obtain better both upper and lower approximations, by starting with a lower approximation of $\sqrt[p]{N}.$ Then, by applying only once the regula falsi (7), the precision can be increased.

As concerns the convergence of $\left(x_n\right)$ and $\left(h\left(x_n\right)\right)$ in (12), in [4] the following theorem is proved.

Applying this Theorem for $F:[\alpha ,\beta ]\to ℝ$, $F\left(x\right)=x^p-N,h:[\alpha ,\beta ]\to ℝ$,

where and $p\in \mathbb{N}$, $p\ge 2,$ we obtain

Since $F$ is increasing and convex $[\alpha ,\beta ],$ it follows that $h$ is decreasing on $[\alpha ,\beta ],$ and the equations $F\left(x\right)=0$ and $h\left(x\right)-x=0$ are equivalent. So the conclusion of Theorem 3.1 follows.

The sequences $\left(x_n\right)$ and $\left(h\left(x_n\right)\right)$ being convergent, it follows that for all $\epsilon >0$ there exists $n_0\in \mathbb{N}$ such that for $n\ge n_0$ we have

which implies and

If we use (7) for $a_1=x_n$ and $b_1=h\left(x_n\right)$ and $g_1\left(x\right)=\frac{F\left(x\right)}{x^{\frac{p-1}{2}}},$ and we denote the approximation obtained by $c_n,$ then by (9) we have

where

We intend to apply the method described in Section 3 for the approximation of the number $\sqrt[5]{100}$, i.e., for solving the equation $x^3-100=0.$ In this case we have

and, taking $\alpha =2,$ for the function $h$ we have

Considering $x_0=\alpha =2$ and using (12), with $F$ and $h$ given above, we obtain for the sequences ${\left(x_n\right)}_{n\ge 0}$ and ${\left(h\left(x_n\right)\right)}_{n\ge 0}$ the following values: