New Families of Higher Order Iterative Methods for Solving Equations

Abstract

In this paper, several one-parameter families of root-finding algorithms that have higher order convergence to simple and/or multiple roots have been derived. Specifically, the rth root iterations for simple and multiple zeros are analyzed. The rth root iteration family is an infinite family of rth order methods for every positive integer r, and uses only the first r − 1 derivatives. This family includes Newton's method and the square root iteration as the first and second member, respectively. In addition, this work provides analyses and generalizations of Halley's and Laguerre's iterations, and develops a procedure of deriving higher order methods of any desired order. Many important properties of the rth root iteration family and its variants are established. Some of these variants maintain a high order of convergence for multiple roots whether the multiplicity is known or not. Based on individual methods, disks containing at least one zero are derived.