Hybrid type new generation Newton methods with cubic convergence rates

Abstract

Recently, Verma [3] introduced several classes of hybrid-type Newton’s methods, which outperform most of the traditional Newton’s methods with convergence rate higher than at least quadratic. In this communication, we proved successfully the cubic convergence rate by an analytical method, which also is quite consistent with numerical calculations. We consider a class of hybrid-type Newton’s methods, for n = 0, 1, 2, · · ·, 6f(xn)f(xn) xn+1 = xn - [6f(xn)f(xn) - 3f(xn)f(xn) + (f(xn))2 ] , where x0 is an initial point with f(xn), f(xn) and f(xn) non-zero. An alternate model is as follows: xn+1 = xn - f(xn)-1 f(xn) + f(xn)-1 f(xn)1 + An)-1 An, where x0 is an initial point with 1 f(xn)f(xn) An = - 2 f(xn)f(xn) + 1 f(xn)2 6 f(xn)f(xn) , and f a real-valued continuously differentiable function. To the best of our knowledge, these findings are new in the available literature.