Some modifications of King’s family with optimal eighth order of convergence

Kung and Traub (1974) [4] conjectured that an iteration method without memory based on n+1n+1 evaluations could achieve optimal convergence order 2n2n. Hence, based on this conjecture, we derive two optimal three-step eighth-order classes of methods in which there are only four evaluations per full cycle. Analytical proofs of the presented derivative-involved classes are provided. Finally, a number of numerical examples are also proposed to illustrate the accuracy of the contributed methods by comparing with the new existing optimal eighth-order methods without memory in the literature.