Numerical Algorithms

, Volume 81, Issue 3, pp 947–981 | Cite as

Stability analysis of a family of optimal fourth-order methods for multiple roots

  • Fiza Zafar
  • Alicia Cordero
  • Juan R. TorregrosaEmail author
Original Paper


Complex dynamics tools applied on the rational functions resulting from a parametric family of roots solvers for nonlinear equations provide very useful results that have been stated in the last years. These qualitative properties allow the user to select the most efficient members from the family of iterative schemes, in terms of stability and wideness of the sets of convergent initial guesses. These tools have been widely used in the case of iterative procedures for finding simple roots and only recently are being applied on the case of multiplicity m > 1. In this paper, by using weight function procedure, we design a general class of iterative methods for calculating multiple roots that includes some known methods. In this class, conditions on the weight function are not very restrictive, so a large number of different subfamilies can be generated, all of them are optimal with fourth-order of convergence. Their dynamical analysis gives us enough information to select those with better properties and test them on different numerical experiments, showing their numerical properties.


Nonlinear equations Multiple zeros Optimal methods Weight functions Complex dynamics Parameter and dynamical planes 


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The authors would like to thank the anonymous reviewers for their help to improve the final version of this manuscript.

Compliance with Ethical Standards

Conflict of interests

The authors declare that there are no conflicts of interest.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Centre for Advanced Studies in Pure and Applied MathematicsBahauddin Zakariya UniversityMultanPakistan
  2. 2.Instituto Universitario de Matemática MultidisciplinarUniversitat Politècnica de ValènciaValènciaSpain

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