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International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 932420, 18 pages
Research Article

Optimized Steffensen-Type Methods with Eighth-Order Convergence and High Efficiency Index

Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

Received 21 March 2012; Revised 23 May 2012; Accepted 6 June 2012

Academic Editor: V. R. Khalilov

Copyright © 2012 F. Soleymani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 lead us to draw a conclusion on the efficiency of the developed classes.