Remarks on Heron ’ s cubic root iteration formula

The existence as well as the computation of roots appears in number theory, algebra, numerical analysis and other areas. The present study illustrates the contribution of several authors towards the extraction of different order roots of real numbers. Different methods with number of approaches are studied to extract the roots of real numbers. Some of the methods, described earlier, are equivalent as observed in the present study. Heron developed a general iteration formula to determine the cube root of a real number N i.e. 3 √ N = a+ bd bd+ aD (b − a), where a < N < b, d = N − a and D = b −N . Although the direct proof of the above method is not available in literature, some authors have proved the same with the help of conjectures. In the present investigation, the proof of Heron’s method is explained and is generalized for any odd order roots. Thereafter it is observed that Heron’s method is a particular case of the generalized method.


Introduction
Recently, attempts have been made by many to find the cube and higher order roots of a real number in various methods with different approaches.Heron's iteration formula to determine the cube root of a number N was Deslauriers and Dubuc [13].According to Heath [16], a conjecture on Heron's cubic root iteration formula was made by Wertheim [37] taking b = a + 1.Assuming some elementary considerations, Eneström [14] proved the Wertheim's conjecture.Again Taisbak [34] made a conjecture about Heron's method and provided possible proofs for it with the help of difference operators.Many researchers like Hess [17], Taisbak [35], Crisman and Veatch [11] are also continuing their research on Heron's work.Recently, Gadtia et al. [15] extended the Wertheim [37] conjecture to find the odd order roots of a number and suggested possible proofs for even order roots.Again they showed that Al-Samawal's and Lagrange's method are equivalent for nth root extraction of a real number.A direct proof of Heron's general cubic root iteration formula is provided and it is extended for any odd order roots in the present work.Further, many counterexamples are discussed in support of the work.In addition to above some contributions made by different authors for extraction of roots are described.

Historical Background
In the middle of the tenth century, the book written by al-Udl īdis ī (Book of chapters on Hindu arithmetic, kitāb al-fusūl f ī al-hisāb al-hind ī [2]) described the earliest work in Arabic that treats the computation of cube roots.The oldest description of the extraction of cube roots was also found in China, in the classical work of the Nine chapters on the Mathematical Art which was translated by Chemla and Guo [8] in French and Kangshen et al. [21] in English.Later, Jia Xian used five rows in computation of cube root of a real number in the early eleventh century.Jia's algorithm differs from the Nine chapters in surface structure.Again Jia gave two methods for cube root extraction and one method for fourth root extraction as quoted by Yang Hui in 1261.
The work presented in Principles of Hindu Reckoning (kitāb al-fusūl f ī al-hisāb al-hind ī) written c.1000 by the Persian Kūshyār ibn Labbān, The Sufficient on Hindu Calculation (al-Muqni f ī al-hisāb al-hind ī) by Al ī ibn Ahmad al-Nasaw ī(text written before 1030 [3]) and The Completion of Arithmetic (al-Takmila f ī al-hisāb) by Abd al-Qāhir ibn Tāhir al-Baghdād ī were roughly contemporary with Jia Xian.Kūshyār's work was translated into English by Levey and Petruck [23].Al-Nasaw ī's text on the cube root was translated into German by Paul [28] and the work by Ibn Tāhir was edited by Saidan [3].The generalization of the algorithm for higher order roots was known in the twelfth century by al-Samawal al-Maghribi (Rashed [30]) and also by Nas īr al-D īn al-Tūs ī in the thirteenth century [1].Fifth and fourth order root of a number was formulated by Al-Samawal (1172) and Nasir al-Din al-Tūs ī (1265), respectively using sexagesimal system.The algorithm described by Al-Tūs ī was very close to the earlier proposed algorithm of Kūshyār.Their procedures coincide completely with the methods given by Jia Xian and Al-Samawal.This approach was later followed by N īzām al-D īn al-N īsābūr ī, who described the extraction of cube and higher roots elaborately in "The Epistle on Arithmetic", arround fourteenth century [24].
The earliest works on arithmetic from the Maghreb and Muslim Spain came from the twelfth and thirteenth centuries.During this period Al-Hassār, Ibn al-Yāsam īn and Ibn Mun'im all included cube extraction in some of their treatises.With the help of complete binomial expansions Ibn Mun'im extracted the fifth and seventh roots of a number.The fifth root of a 13 digits number was extracted by a Mathematician in Kairouan, Tunisia, before 1241 as described by Rashed [31] using binomial expansion.According to Lamrabet [22], Ibn al-Yāsam īn proposed an algorithm for the extraction of cube root on the development of (a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3 .In 1427, Jamshid al-Kashi extracted the fifth root of a decimal number and sixth root of a sexagesimal number.The same algorithm was described by al-Qatrawān ī [22] in fifteenth century.
Later, many researchers have studied the cube and other roots of a real number in different ways (Ruffini [32], Horner [18,19]).For the first time Paul [28] studied, the extraction of higher roots in Arabic (Islamic) mathematics, which was focused on the work of Jamshid al-Kashi.Burr [6] developed several iteration methods for computing cube roots, when a fast square root was available.Some methods are superior to the conventional Newton's method in particular situations was observed by him.Padro and Saez [26] generalized the algorithms established by Shanks [33] and Peralta [29] for computing square roots modulo of a prime to algorithms for computing cube roots, which played an important role in cryptosystems.Ahmadi et al. [4] calculated the number of nonzero coefficients (Hamming weight) in the polynomial representation of x 1 3 in F 3 [x]/(f ), where f ∈ F 3 [x] is an irreducible trinomial.Cho et al. [10] found that the shifted polynomial basis and variation of polynomial basis reduces Hamming weight of x

Iteration formulae for odd order roots
In this section, the Heron's cubic root iteration formula is proved and extended this for any odd order roots.

Heron's root iteration formula
Let us state the Heron's cubic root iteration formula as Theorem 3.1.Proof: Let x be the cube root of N .Assume that (x − a) As the value of δ 1 and δ 2 are very small, from eqn. (3.6), we get

Generalization of Heron's root iteration formula
In the Theorem 3.2, the general formula for any odd order root of a number N is discussed. , where d = x 2m+1 − a 2m+1 .

Conclusion
A direct proof of Heron's general cubic root iteration formula is described and extended for any odd order roots.It is observed that the Heron's general cubic root iteration formula is a particular case of the present study.Counterexamples are discussed in support of the present investigation.

3 √
N = a + bd bd + aD (b − a), where a 3 < N < b 3 , d = N − a 3 and D = b 3 − N as stated by

Theorem 3 . 1 3 √
If a 3 < N < b 3 , then cube root of N is defined as N = a + bd bd + aD (b − a), where d = N − a 3 and D = b 3 − N .