On generalized multipoint root-solvers with memory

The improved versions of the Kung–Traub family and the Zheng–Li–Huang family of n$n$-point derivative free methods for solving nonlinear equations are proposed. The convergence speed of the modified families is considerably accelerated by employing a self-correcting parameter. This parameter is calculated in each iteration using information from the current and previous iteration so that the proposed families can be regarded as the families with memory. The increase of convergence order is attained without any additional function evaluations meaning that these families with memory possess high computational efficiency. Numerical examples are included to confirm theoretical results and demonstrate convergence behaviour of the proposed methods.