Newton’s Method and an Exact Opposite That Average into Halley’s Method

Abstract

This note is mainly concerned with the creation of oppositely converging and alternatingly converging iterative methods that have the added advantage of providing ever tighter bounds on the targeted root. By a slight parametric perturbation of Newton’s method we create an oscillating super-linear method approaching the targeted root alternatingly from above and from below. Further extension of Newton’s method creates an oppositely converging quadratic counterpart to it. This new method requires a second derivative, but for it, the average of the two opposite methods rises to become a cubic method. This note examines also the creation of high order iterative methods by a repeated specification of undetermined coefficients.