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

Solving nonlinear equations by a derivative-free form of the King's family with memory