On the Chebyshev-Halley family of iteration functions

François Dubeau and Calvin Gnang

Département de mathématiques Faculté des sciences, Université de Sherbrooke 2500, boul. de l’Université, Sherbrooke (Qc), Canada, J1K 2R1 francois.dubeau@usherbrooke.ca

Paper published in the International Journal of Pure and Applied Mathematics, 85.6 (2013), 1051-1059 .

AMS Subject Classification : 65-01, 11B37, 65B99, 65D99, 65H05.

Key Words : Chebyshev, Halley, Super-Halley, iteration function, order 3, n-th root.

Summary

The Chebyshev-Halley family of iteration functions (IFs) to solve f(x) = 0 has been introduced by Werner [12]. It can also also be found in [1] and [5], as reported in [11]. Each member of this family is obtained as an improvement of the Newton’s IF, depends on a real parameter β, and can be written as

(0.1)  G_{β}(x) = x − \frac{f(x)}{f^{(1)}(x)} [\frac{1 − (β − 1/2)L_{f}(x)}{1 − βL_{f}(x)}]

where L_{f}(x) = \frac{f(x)f^{(2)}(x)}{[f^{(1)}(x)]^{2}}. These IFs are of order 3 when we look for an α ∈ ℝ

such that f(α) = 0 and α is a simple root of f(x).

In this paper we obtain this family of IFs from a linear combination of two Newton’s IFs. We will also see how to modify the parameter β, and express it as a function of x, to obtain an IF of order 4.

We also consider the best parameter β for the n-th root computation problem. For this problem the best parameter depends only on n. In particular, when we compare the Halley (β = 1/2) and Super-Halley (β = 1) IFs, we show that Super-Halley IF is the best method to compute the n-th root for n = 2, 3, 4, Halley IF is the best method for n ≥ 6, and they are equivalent for n = 5. Both are better than the Chebyshev (β = 0) IF.

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