Power means based modification of Newton’s method for solving nonlinear equations with cubic convergence

Abstract

In this paper, a class of Newton-type methods known as trapezoidal power means Newton method for solving nonlinear equation is proposed.  The new methods incorporate power means in the trapezoidal integration rule along with midpoint, thus replacing  in the classical Newton method.  Some known variants can be regarded as particular cases of this method.  The order of convergence of these methods is shown to be three.  Numerical examples and their results are provided to compare the efficiency of the new methods with few other similar methods.