Efficient families of Newton’s method and its variants suitable for non-convergent cases

Abstract

In this paper, we derive new one-parameter family of Newton's method, Schröder's method, super-Halley method and Halley's method respectively for finding simple zeros of nonlinear functions, permitting f (x n ) = 0 at some points in the vicinity of required root. Using the newly derived family of super-Halley method, we further obtain new interesting families of famous quartically convergent Traub-Ostrowski's and Jarratt's methods respectively. Further, the approach has been extended to solve a system of nonlinear equations. It is found by way of illustration that the proposed methods are very useful in high-precision computing environment and non-convergent cases.