Efficient and stable derivative-free Steffensen algorithm for root finding

We explore a family of numerical methods, based on the Steffensen divided difference iterative algorithm, that do not evaluate the derivative of the objective functions. The family of methods achieves second-order convergence with two function evaluations per iteration with marginal additional computational cost. An important side benefit of the method is the improvement in stability for different initial conditions compared to the vanilla Steffensen method. We present numerical results for scalar functions, fields, and scalar fields. This family of methods outperforms the Steffensen method with respect to standard quantitative metrics in most cases.