Extended convergence of a derivative free fourth order method for equations or systems

Abstract. In many areas of science and engineering, solving equations or systems of equations is really important. Instead of finding exact solutions, which can be very hard or impossible in some cases, people often use iterative methods to attain the desired solutions. This article is dedicated to introducing a highly efficient derivative-free fourth order iterative technique renowned for its exceptional convergence properties. The analysis within this article deals with scrutinizing both local and semi-local convergence characteristics, taking into account the
\(\varphi\), \(\psi\)-continuity constraints imposed on the operators that are present in these methods. It's worth noting that the innovative methodology proposed herein isn't confined to specific techniques but has broader applicability, encompassing a wide spectrum of approaches involving the utilization of inverses of linear operators or matrices.