Numerical Algorithms

, Volume 77, Issue 4, pp 1249–1272 | Cite as

An eighth-order family of optimal multiple root finders and its dynamics

  • Ramandeep Behl
  • Alicia CorderoEmail author
  • Sandile S. Motsa
  • Juan R. Torregrosa
Original Paper


There is a very small number of higher-order iteration functions for multiple zeros whose order of convergence is greater than four. Some scholars have tried to propose optimal eighth-order methods for multiple zeros. But, unfortunately, they did not get success in this direction and attained only sixth-order convergence. So, as far as we know, there is not a single optimal eighth-order iteration function in the available literature that will work for multiple zeros. Motivated and inspired by this fact, we present an optimal eighth-order iteration function for multiple zeros. An extensive convergence study is discussed in order to demonstrate the optimal eighth-order convergence of the proposed scheme. In addition, we also demonstrate the applicability of our proposed scheme on real-life problems and illustrate that the proposed methods are more efficient among the available multiple root finding techniques. Finally, dynamical study of the proposed schemes also confirms the theoretical results.


Nonlinear equations Optimal iterative methods Multiple roots Efficiency index Kung-Traub conjecture 


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The authors thank to the anonymous referees for their useful comments and suggestions to improve the final version of the manuscript.


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.School of Mathematics, Statistics and Computer SciencesUniversity of KwaZulu-NatalScottsvilleSouth Africa
  2. 2.Instituto Universitario de Matemática MultidisciplinarUniversitat Politècnica de ValènciaValènciaSpain
  3. 3.Mathematics DepartmentUniversity of SwazilandKwaluseniSwaziland

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