Efficient derivative-free with memory variants of King's family for solving nonlinear equations

In this paper, we present several new two-step derivative-free iterative methods with and without memory for solving nonlinear equations. The convergence order of the proposed class without memory is four requiring only three functional evaluations per step. We further increase the convergence order from four to six by suitable variation of a free parameter in each iterative step without any additional functional evaluation. This self-accelerating parameter is calculated using Newton's interpolation polynomial of third degree. Numerical experiments and the comparison with the existing robust methods are included to confirm the theoretical results and high computational efficiency.