On a Newton-Type method under weak conditions with dynamics

Abstract

In this paper, we present new cubically convergent Newton-Type iterative methods with dynamics for solving nonlinear algebraic equations under weak conditions. The proposed methods are free from second-order derivative and work well when f′(x) = 0. Numerical results show that the proposed method performs better when Newton's method fails or diverges and competes well with same order existing method. Fractal patterns of different methods also support the numerical results and explain the compactness regarding the convergence, divergence, and stability of the methods to different roots. © 2021 World Scientific Publishing Company.