  In this paper, we present an iterative three-point method with memory based
on the family of King's methods to solve nonlinear equations. This proposed
method has eighth order convergence and costs only four function evaluations
per iteration which supports the Kung-Traub conjecture on the optimal order of
convergence. An acceleration of the convergence speed is achieved by an
appropriate variation of a free parameter in each step. This self accelerator
parameter is estimated using Newton's interpolation polynomial of fourth
degree. The order of convergence is increased from 8 to 12 without any extra
function evaluation. Consequently, this method, possesses a high computational
efficiency. Finally, a numerical comparison of the proposed method with related
methods shows its effectiveness and performance in high precision computations.
