Logarithmic Mean Newton's Method for Simple and Multiple Roots of Nonlinear Equations

In this paper the convergence behavior of a variant of Newton's method based on the Logarithmic mean for solving nonlinear equations is proposed. The convergence properties of this method for solving equations which have simple or multiple ischolar_mains have been discussed and it is shown that it converges cubically to simple ischolar_mains and linearly to multiple ischolar_mains. 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 others means, is also included.