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Optimal eighth-order iterative methods for approximating multiple zeros of nonlinear functions

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Abstract

It is well known that the optimal iterative methods are of more significance than the non-optimal ones in view of their efficiency and convergence speed. There are only a few number of optimal iterative methods for finding multiple zeros with eighth order of convergence. In this paper, we propose a new family of optimal eighth order convergent iterative methods for multiple roots with known multiplicity. We present an extensive convergence analysis which confirms theoretically eighth-order convergence of the presented scheme. Moreover, we consider several real life problems that contain simple as well as multiple zeros in order to compare our proposed methods with the existing eighth-order iterative schemes. Some dynamical aspects of the presented methods are also discussed. Finally, we conclude on the basis of obtained numerical results that the proposed family of iterative methods perform better than the existing methods in terms of residual error, computational order of convergence and difference between the two consecutive iterations.

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Acknowledgements

The authors would like to thank the anonymous reviewers for their comments and suggestions that have improved the final version of this manuscript.

Author information

Correspondence to Juan R. Torregrosa.

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This research was partially supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER, UE), Generalitat Valenciana PROMETEO/2016/089 and Schlumberger Foundation-Faculty for Future Program.

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Zafar, F., Cordero, A., Junjua, M. et al. Optimal eighth-order iterative methods for approximating multiple zeros of nonlinear functions. RACSAM 114, 64 (2020) doi:10.1007/s13398-020-00794-7

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Keywords

  • Nonlinear equations
  • Order of convergence
  • Multiple roots
  • Optimality

Mathematics Subject Classification

  • 65H05