Generalized Power means Modification of Newton’s Method for Simple Roots of Nonlinear Equation

Abstract: In this paper, a class of Newton-type methods known as generalized power means Newton method for solving nonlinear equations is proposed. The new method includes power means instead of arithmetic mean when trapezoidal integration rule is used, thus replacing f′(x) in the classical Newton method. Also, three arbitrary parameters are introduced to generalize the method and to obtain higher order convergence. Many known variants can be regarded as particular cases of this method. The order of convergence of these methods is shown to be three and five for an additional condition. Numerical examples and their results are provided to show the efficiency of the new class of methods compared to Newton’s method