On the centroidal mean Newton's method for simple and multiple roots of nonlinear equations

Abstract

In this paper, the convergence behaviour of a variant of Newton's method based on the centroidal mean is considered. The convergence properties of this method for solving equations which have simple or multiple roots have been discussed. It is shown that it converges cubically to simple roots with efficiency index is 1.442 and linearly to multiple roots. Moreover, the values of the corresponding asymptotic error constants of convergence are determined. Theoretical results have been verified on the relevant numerical problems. The proposed new method has the advantage of evaluating only the first derivative and less number of iterations to achieve third order accuracy. A comparison of the efficiency of this method with other mean-based Newton's methods, based on the arithmetic, geometric and harmonic means, is also included. Convergences to the root and error propagation with iteration are exhibited graphically with iterations.