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Optimized Steffensen-type methods with eighth-order convergence and high efficiency index. (English) Zbl 1253.65073
Summary: Steffensen-type methods are practical in solving nonlinear equations, since such schemes do not need derivative evaluation per iteration. Hence, this work contributes two new multistep classes of Steffensen-type methods for finding the solution of the nonlinear equation f(x)=0. New techniques can be taken into account as the generalizations of the one-step method of Steffensen. Theoretical proofs of the main theorems are furnished to reveal the eighth-order convergence. Per computing step, the derived methods require only four function evaluations. Experimental results are also given to add more supports on the underlying theory of this paper as well as they lead us to draw a conclusion on the efficiency of the developed classes.
65H05Single nonlinear equations (numerical methods)