Halley’s Method as the First Member of an Infinite Family of Cubic Order Rootfinding Methods

For each natural number m ≥ 3, we give a rootfinding method Hm, with cubic order of convergence for simple roots. However, for quadratic polynomials the order of convergence of Hm is m. Each Hm depends on the input, the corresponding function value, as well as the first two derivatives. We shall refer to this family as Halley Family, since H3 is the well-known method of Halley. For all m ≥ 4, the asymptotic error constant of Hm is the same constant. Each Hm is described in terms of determinants that are computable recursively. The Halley Family and their derivative-free variants offer alternatives to the traditional rootfinding methods, such as secant, Newton, and Muller methods, as well as Halley’s method itself.