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An improvement of Ostrowski’s and King’s techniques with optimal convergence order eight. (English) Zbl 1237.90229
Summary: In this paper, we first establish a new class of three-point methods based on the two-point optimal method of Ostrowski. Analysis of convergence shows that any method of our class arrives at eighth order of convergence by using three evaluations of the function and one evaluation of the first derivative per iteration. Thus, this order agrees with the conjecture of H. T. Kung and J. F. Traub [J. Assoc. Comput. Mach. 21, 643–651 (1974; Zbl 0289.65023)] for constructing multipoint optimal iterations without memory. We second present another optimal eighth-order class based on the King’s fourth-order family and the first attained class. To support the underlying theory developed in this work, we examine some methods of the proposed classes by comparison with some of the existing optimal eighth-order methods in literature. Numerical experience suggests that the new classes would be valuable alternatives for solving nonlinear equations.
90C30Nonlinear programming
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