On the Geometry of Halley's Method

Abstract

According to Traub [Tra64], Halley's iteration function (I.F.) must share with the secant I.F. the distinction of being the most frequently rediscovered I.F. in the literature. Halley's method is a close relative of Newton's method, an iterative technique depicted as a sequence of tangent lines with zeros converging to a root of a function. The usual derivation of Halley's method, however, lacks any obvious geometric interpretation. We present a derivation of Halley's method having such an interpretation, and give a brief history of Halley's work and the method that bears his name.