An optimal three-point eighth-order iterative method without memory for solving nonlinear equations with its dynamics

  • Gunar Matthies
  • Mehdi SalimiEmail author
  • Somayeh Sharifi
  • Juan Luis Varona
Original Paper Area 2


We present a three-point iterative method without memory for solving nonlinear equations in one variable. The proposed method provides convergence order eight with four function evaluations per iteration. Hence, it possesses a very high computational efficiency and supports Kung–Traub’s conjecture. The construction, the convergence analysis, and the numerical implementation of the method will be presented. Using several test problems, the proposed method will be compared with existing methods of convergence order eight concerning accuracy and basins of attraction. Furthermore, some measures are used to judge methods with respect to their performance in finding the basins of attraction.


Optimal multi-point iterative methods Simple root Order of convergence Kung–Traub’s conjecture Basins of attraction 

Mathematics Subject Classification

65H05 37F10 



The research of the fourth author is supported by Grant MTM2015-65888-C4-4-P (MINECO/FEDER).


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Copyright information

© The JJIAM Publishing Committee and Springer Japan 2016

Authors and Affiliations

  1. 1.Institut für Numerische MathematikTechnische Universität DresdenDresdenGermany
  2. 2.Center for Dynamics, Department of MathematicsTechnische Universität DresdenDresdenGermany
  3. 3.Department of MathematicsUniversiti Putra MalaysiaSerdangMalaysia
  4. 4.MEDAlics, Research Center at Università per Stranieri Dante AlighieriReggio CalabriaItaly
  5. 5.Departamento de Matemáticas y ComputaciónUniversidad de La RiojaLogroñoSpain

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