A new class of derivative-free root solvers with increasing optimal convergence order and their complex dynamics

We present a general class of derivative free iterative methods with optimal order of convergence for solving nonlinear equations. The methodology is based on quadratically convergent Traub–Steffensen scheme and inverse Padé approximation. Unlike that of existing higher order techniques the proposed technique is attractive since it leads to a simple implementation. Numerical examples are provided to confirm the theoretical results and to show the feasibility and efficacy of the new methods. The performance is compared with well established methods in literature. Computational results, including the elapsed CPU-time, confirm the accurate and efficient character of proposed techniques. Moreover, the stability of the methods is checked through complex geometry shown by drawing basins of attraction. © 2022, The Author(s), under exclusive licence to Sociedad Española de Matemática Aplicada.