Next generation computational methods

Abstract

Ram U. Verma (2013) invented several classes of the next generation computational methods, which are also referred to as "next genberation Newton-type method" in the literature. In this pape, we consider the following method. For n = 0, 1, 2, ·, a class of next generation computational Methods is described as follows: xn+1 = xn-6f(xn)f?(xn)/[6f?(xn)f?(xn)-3f?(xn)f(xn) + (f(xn))2], where x0 is an initial point and and f is a real-valued continuously differentiable function. The convergence rate for this method has been established, which indicates that the convergence rate is higher than super quadratic or 3, that is based on the analytical methods as well as the numerical calculations. We shall use more frequently an abbreviation of "Next generation Computational method" as "NGC method" throughout this paper and beyond.