TITLE:
Newton’s Method and an Exact Opposite That Average into Halley’s Method
AUTHORS:
Isaac Fried
KEYWORDS:
Iterative Methods, Alternating Methods, Opposite Methods, Root Bounds, Undetermined Coefficients
JOURNAL NAME:
Applied Mathematics,
Vol.8 No.10,
October
23,
2017
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.