Abstract
The present investigation deals with the critical study of the works of Lancaster and Traub, who have developed $n$th root extraction methods of a real number. It is found that their developed methods are equivalent and the particular cases of Halley's and Householder's methods. Again the methods presented by them are easily obtained from simple modifications of Newton's method, which is the extension of Heron's square root iteration formula. Further, the rate of convergency of their reported methods are studied.
Highlights
The aim of this paper is to focus on the works of Lancaster [7] and Traub [12]
Taking x3 = N, this formula can be extended for the cube root extraction of the real number N as xi+1
The recurrence relation for square root extraction is expressed as which has cubic convergence
Summary
A connection between Newton’s formula, Halley’s method, Householder’s method with the methods of Lancaster [7] and Traub [12] is described. Which is the machine method for nth root extraction as proposed by Lancaster [7] and Traub [12]. Comparison with Halley’s Method Let us consider the Halley’s method for cube root extraction. In this case f (x) = x3 − N, f ′(x) = 3x2, f ′′(x) = 6x x. Consider the Halley’s method for nth root extraction, in this case f (x) = xn − N, f ′(x) = nxn−1, f ′′(x) = n(n − 1)xn−2 x. The nth root extraction methods in Lancaster [7] and Traub [12] are the particular cases of Halley’s formula
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