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A sixth order method for nonlinear equations. (English) Zbl 1154.65327

Summary: A sixth order method is developed by extending a third order method of S. Weerakoon and T. G. I. Fernando [Appl. Math. Lett. 13, No. 8, 87–93 (2000; Zbl 0973.65037)] for finding the real roots of nonlinear equations in \(R\). Starting with a suitably chosen \(x_{0}\), the method generates a sequence of iterates converging to the root. The convergence analysis is provided to establish its sixth order of convergence. In terms of computational cost, it requires evaluations of only two functions and two first derivatives per iteration. This implies that efficiency index of our method is 1.565. Our method is comparable with the methods of B. Neta [Int. J. Comput. Math. 7, 158–161 (1979; Zbl 0397.65032)] and that of J. Kou and Y. Li [Appl. Math. Comput. 189, No. 2, 1816–1821 (2007; Zbl 1122.65338)]. It does not require the evaluation of the second order derivative of the given function as required in the family of Chebyshev-Halley type methods [J. Kou and X. Wang, Appl. Math. Comput. 190, No. 2, 1839–1843 (2007; Zbl 1122.65339); J. Kou, Appl. Math. Comput. 190, No. 1, 126–131 (2007; Zbl 1122.65334)]. The efficacy of the method is tested on a number of numerical examples. It is observed that our method takes less number of iterations than Newton’s method and the method of Weerakoon and Fernando. On comparison with the other sixth order methods, it behaves either similarly or better for the examples considered.

MSC:

65H05 Numerical computation of solutions to single equations
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