A sixth-order bi-parametric iterative method for nonlinear systems: Theory, stability and computational complexity

In this paper, we propose a new bi-parametric three-step iterative method with sixth-order convergence for solving systems of nonlinear equations. The method is formulated using the composition technique, which we uniquely apply twice within a single method - an approach that, to our knowledge, is the first of its kind in the literature. This allows us to incorporate two free disposable parameters, offering flexibility with enhanced performance, stability and adaptability. The local convergence behaviour is rigorously analysed within the framework of Banach spaces, where we establish theoretical bounds for the convergence radius and demonstrate uniqueness conditions under Lipschitz-continuous Fréchet derivative assumptions. The theoretical outcomes are supported by numerical experiments. Lastly, we evaluate the method's efficiency by exploring its basins of attraction and applying it to systems of nonlinear equations and boundary value problems.