homotopy continuation method

This study explores the efficacy of the homotopy function, which combines the classical iterative technique with the homotopy continuation method for solving zeros of nonlinear equations and approximating their roots.  While numerous homotopy functions have been examined in previous research, this paper focuses on employing a cubic parameter as the homotopy function within the Newton-HCM framework.  Through a series of numerical examples, we demonstrate that our proposed homotopy function, named cubic parameter homotopy function, consistently outperforms existing altern