On the stability of a two-step method for a fourth-degree family by computer designs along with applications

In this paper, some important features of Traub’s method are studied: Analysis of the stability behavior, obtaining the 4th root of a matrix, semi-local convergence, and local convergence. The stability of Traub’s method is studied by using the dynamic behavior of a family of 4th-degree polynomials. The obtained equations are very complex and do not solve with the software. Therefore, we find the results by plotting diagrams and pictures, and then we show the very stable behavior of Traub’s method. Then Traub’s method is extended to a matrix iterative method for calculating the 4th root of a square matrix. We also present the local and semi-local convergence of the method based on the divided differences, and therefore, the benefits of our approach are more precise error estimation in semi-local convergence and a large ball of convergence in local convergence. We confirm our theoretical results by some numerical examples such as the nonlinear integral equation of mixed Hammerstein type.