Geometric mean Newton’s method for simple and multiple roots

In this work we consider the convergence behavior of a variant of Newton’s method based on the geometric mean. The convergence properties of this method for solving equations which have simple or multiple roots have been discussed and it has been shown that it converges cubically to simple roots 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. A comparison of the efficiency of this method with other mean-based Newton’s methods, based on the arithmetic and harmonic means, is also included.