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The Scientific World Journal
Volume 2015, Article ID 614612, 11 pages
Research Article

Geometric Construction of Eighth-Order Optimal Families of Ostrowski’s Method

School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg 3209, South Africa

Received 15 July 2014; Accepted 28 August 2014

Academic Editor: Juan R. Torregrosa

Copyright © 2015 Ramandeep Behl and S. S. Motsa. 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.


Based on well-known fourth-order Ostrowski’s method, we proposed many new interesting optimal families of eighth-order multipoint methods without memory for obtaining simple roots. Its geometric construction consists in approximating at zn in such a way that its average with the known tangent slopes at xn and yn is the same as the known weighted average of secant slopes and then we apply weight function approach. The adaptation of this strategy increases the convergence order of Ostrowski's method from four to eight and its efficiency index from 1.587 to 1.682. Finally, a number of numerical examples are also proposed to illustrate their accuracy by comparing them with the new existing optimal eighth-order methods available in the literature. It is found that they are very useful in high precision computations. Further, it is also noted that larger basins of attraction belong to our methods although the other methods are slow and have darker basins while some of the methods are too sensitive upon the choice of the initial value.