A variant of Steffensen–King’s type family with accelerated sixth-order convergence and high efficiency index: Dynamic study and approach

First, it is attempted to derive an optimal derivative-free Steffensen–King’s type family without memory for computing a simple zero of a nonlinear function with efficiency index 41/3≈1.58741/3≈1.587. Next, since our without memory family includes a parameter in which it is still possible to increase the convergence order without any new function evaluations. Therefore, we extract a new method with memory so that the convergence order rises to six without any new function evaluation and therefore reaches efficiency index 61/3≈1.81761/3≈1.817. Consequently, derivative-free and high efficiency index would be the substantial contributions of this work as opposed to the classical Steffensen’s and King’s methods. Finally, we compare some of the convergence planes with different weight functions in order to show which are the best ones.