Abstract
(soon)
Authors
Dan Luca
Tiberiu Popoviciu Institute of Numerical Analysis
Ion Păvăloiu
Tiberiu Popoviciu Institute of Numerical Analysis
Keywords
Scanned paper: on the journal website.
PDF-LaTeX version of the paper.
Cite this paper as:
D. Luca, I. Păvăloiu, On the Heron’s method for approximating the cubic root of a real number, Rev. Anal. Numér. Théor. Approx., 26 (1997) nos. 1-2, pp. 103-108.
About this paper
Publisher Name
Article on the journal website
Print ISSN
1222-9024
Online ISSN
2457-8126
References
[1] G. Deslauries and S. ubuc, Le calcul de la racine cubique selon Heron, Elemente der 51, I (1996), pp. 28-34.
[2] M. Ostrowski, A Solution of Equations and Systems of Equation, Academic Press, New York-London, 1960.
[3] I. Păvăloiu I., On the monotonicity of the sequences of approximations obtained by Steffensen,s method, Mathematica (Cluj) 35 (58),1 (1993), pp. 71-76.
[4] T. Popoviciu, Sur la delimiation de l’erreur dans l’approximation des racines d’une equation par interpolation lineaire ou quadratique, Rev. Roumaine Math. Pures Appl. XIII, 1 (1968), pp. 75-78.
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On the Heron’s method for approximating the cubic root of a real number
On the Heron’s method for approximating
Table 1.
On the Heron’s method for approximating
the cubic root of a real number
1991 Mathematics Subject Classification:
65G05, 65B051. Introduction
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 and satisfying the Heron’s method for approximating is
| (1) |
where and We shall show that the approximation (1) of follows from the regula falsi applied to the equation [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 , 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.
2. Herons’s method and regula falsi
A. In order to approximate the cubic root of by (1), consider the function and the function ,
It is well known that regula falsi applied to the equation leads to the following approximation of its root
| (2) |
denoting the first-order divided difference of on the nodes and It can be easily verified that
Taking into account that and denoting by the second-order divided difference of on the points we get
| (3) |
for all
For in (3) we obtain
from which, by dividing it follows
| (4) |
An elementary calculation on shows that
| (5) |
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
| (6) |
where
Note that (6) leads to a very good error evalutation; since and are close to then and are close to zero. This is implied by the fact that the function and its second derivative vanish at the same point
B. It can be easily seen that the method presented at A can be generalized. For the approximation of the root of order of the real number , consider the function ,
and the function ,
The function satisfies
Applying regula falsi to the equation we obtain
| (7) |
Similarly to A, we obtain
| (8) |
which gives
| (9) |
where
3. Steffensen’s method for approximating the th-order root
Let be an interval of the real axis.
Consider the equation
| (10) |
where . Suppose that equation (10)has a root Consider also a function such that equation
| (11) |
is equivalent to (10).
The Steffensen’s method consists in the generation of two sequences and by the relations
| (12) |
As we shall see, this method offers the possibility to obtain better both upper and lower approximations, by starting with a lower approximation of Then, by applying only once the regula falsi (7), the precision can be increased.
Theorem 3.1.
[4]. If the functions and are continuous and satisfy the following conditions:
-
i)
the function is decreasing on
-
ii)
the function is increasing and convex on
-
iii)
there exists such that and
- iv)
then the following properties hold:
-
j)
the sequence is increasing,
-
jj)
the sequence is decreasing,
-
jjj)
-
jv)
the relations hold for all
-
v)
Applying this Theorem for , ,
where and , we obtain
Since is increasing and convex it follows that is decreasing on and the equations and are equivalent. So the conclusion of Theorem 3.1 follows.
The sequences and being convergent, it follows that for all there exists such that for we have
which implies and
4. A Numerical example
We intend to apply the method described in Section 3 for the approximation of the number , i.e., for solving the equation In this case we have
and, taking for the function we have
Considering and using (12), with and given above, we obtain for the sequences and the following values:
References
- [1]
- [2] G. Deslauries and S. Dubuc, Le calcul de la racine cubique selon Héron, Elemente der Mathematik 51, (1996) 1, 28–34.
- [3] M. Ostrowski, The solution of Equations and Systems of Equations, Academic Press, New York-London, 1960.
- [4] ††margin: clickable I. Păvăloiu, On the monotonicity of the sequences of approximations obtained by Steffensen’s method, Mathematica (Cluj) 35 (58), (1993) 1, 71–76.
-
[5]
T. Popoviciu, Sur la dèlimitation de
††margin:
available soon,
refresh and click here l’erreur dans l’approximation des racines d’une équation par interpolation linéaire ou quadratique, Rev. Roumaine Math. Pures Appl. XIII, (1968) 1, 75–78.