Nonlinear Dynamics

, Volume 91, Issue 1, pp 81–112 | Cite as

Multiplicity anomalies of an optimal fourth-order class of iterative methods for solving nonlinear equations

  • Ramandeep Behl
  • Alicia Cordero
  • Sandile S. Motsa
  • Juan R. TorregrosaEmail author
Original Paper


There is a few number of optimal fourth-order iterative methods for obtaining the multiple roots of nonlinear equations. But, in most of the earlier studies, scholars gave the flexibility in their proposed schemes only at the second step (not at the first step) in order to explore new schemes. Unlike what happens in existing methods, the main aim of this manuscript is to construct a new fourth-order optimal scheme which will give the flexibility to the researchers at both steps as well as faster convergence, smaller residual errors and asymptotic error constants. The construction of the proposed scheme is based on the mid-point formula and weight function approach. From the computational point of view, the stability of the resulting class of iterative methods is studied by means of the conjugacy maps and the analysis of strange fixed points. Their basins of attractions and parameter planes are also given to show their dynamical behavior around the multiple roots. Finally, we consider a real-life problem and a concrete variety of standard test functions for numerical experiments and relevant results are extensively treated to confirm the theoretical development.


Nonlinear equations Multiple roots Complex dynamics Dynamical and parameter plane Stability 



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


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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of MathematicsKing Abdulaziz UniversityJeddahSaudi Arabia
  2. 2.School of Mathematical SciencesUniversity of KwaZulu-NatalPietermaritzburgSouth Africa
  3. 3.Instituto Universitario de Matemática MultidisciplinarUniversitat Politècnica de ValènciaValenciaSpain

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