Constructing two-step iterative methods with and without memory

Some new iterative methods without memory are constructed to reach the optimal convergence order four using only three function evaluations to solve nonlinear equations. Per full cycle, each method from the derived classes is derivative-free. We further extend the derived classes of methods to contribute some schemes with memory, without any additional evaluation of the function. Hence, with a same cost, the contributed methods with memory possess higher $R$-orders and better efficiency indices. In order to attest the efficiency of the obtained methods, we employ numerical comparisons.