Two general higher-order derivative free iterative techniques having optimal convergence order

  • Ramandeep Behl
  • Ali Saleh AlshomraniEmail author
  • Á. A. Magreñán
Original Paper


The principal objective of this study is to propose two derivative free iteration functions. Both are applicable to each earlier optimal multi-point derivative free scheme of order four and eight whose first sub step should be Steffensen’s type method to develop more advanced optimal iteration techniques of order eight and sixteen, respectively. Both schemes satisfy the Kung–Traub optimality conjecture. In addition, the theoretical convergence properties of our schemes are fully explore with the help of main theorem that demonstrate the convergence order. The performance and effectiveness of our optimal iteration functions have compared with the existing competitors on some standard academic problems. Finally, on the account of results obtained, our methods are find to be more efficient as compared to some standard and robust iteration functions of same order.


Computational order of convergence Simple zeros Kung–Traub conjecture Scalar equations Steffensen’s type method 



This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. D-228-130-1439. The authors, therefore, gratefully acknowledge with thanks DSR for technical and financial support.


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ramandeep Behl
    • 1
  • Ali Saleh Alshomrani
    • 1
    Email author
  • Á. A. Magreñán
    • 2
  1. 1.Department of MathematicsKing Abdulaziz UniversityJeddahSaudi Arabia
  2. 2.Departamento de Matemáticas y ComputaciónUniversidad de La RiojaLogroñoSpain

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