Numerical Algorithms

, Volume 58, Issue 4, pp 513–527

Accurate fourteenth-order methods for solving nonlinear equations

Authors

  • Parviz Sargolzaei
    • Department of MathematicsUniversity of Sistan and Baluchestan
    • Young Researchers Club, Zahedan BranchIslamic Azad University
Original Paper

DOI: 10.1007/s11075-011-9467-4

Cite this article as:
Sargolzaei, P. & Soleymani, F. Numer Algor (2011) 58: 513. doi:10.1007/s11075-011-9467-4
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Abstract

We establish new iterative methods of local order fourteen to approximate the simple roots of nonlinear equations. The considered three-step eighth-order construction can be viewed as a variant of Newton’s method in which the concept of Hermite interpolation is used at the third step to reduce the number of evaluations. This scheme includes three evaluations of the function and one evaluation of the first derivative per iteration, hence its efficiency index is 1.6817. Next, the obtained approximation for the derivative of the Newton’s iteration quotient is again taken into consideration to furnish novel fourteenth-order techniques consuming four function and one first derivative evaluations per iteration. In providing such new fourteenth-order methods, we also take a special heed to the computational burden. The contributed four-step methods have 1.6952 as their efficiency index. Finally, various numerical examples are given to illustrate the accuracy of the developed techniques.

Keywords

Nonlinear equationsThree-step methodsFour-step methodsEfficiency indexOrder of convergenceSimple root

Mathematics Subject Classifications (2010)

65H0565B99
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Copyright information

© Springer Science+Business Media, LLC. 2011