Regarding the accuracy of optimal eighth-order methods

It is widely known that when the order of root solvers increases, their accuracy comes up as well. In light of this, most of the researchers in this field of study have tried to increase the order of known schemes for obtaining optimal three-step eighth-order methods in which there are four evaluations per iteration. The aim of this article is to challenge this standpoint when the starting points are in the vicinity of the root, but not so close. Toward this end, a novel method of order six with the same number of evaluations per iteration is suggested and demonstrated while its accuracy is better than the accuracy of optimal eighth-order schemes for such initial guesses. The superiority of the developed technique is confirmed by numerical examples.

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