Optimized mean based second derivative-free families of Chebyshev–Halley type methods

In this paper, we present new interesting fourth-order optimal families of Chebyshev–Halley type methods free from second-order derivatives. In terms of computational cost, each member of the families requires two functions and one first-order derivative evaluation per iteration, so that their efficiency indices are 1.587. It is found by way of illustration that the proposed methods are useful in high precision computing environment. Moreover, it is also observed that larger basins of attraction belong to our methods, whereas the other methods are slow and have darker basins, while some of the methods are too sensitive to the choice of the initial guess.