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Numerical Algorithms

, Volume 68, Issue 2, pp 261–288 | Cite as

A new class of three-point methods with optimal convergence order eight and its dynamics

  • Taher Lotfi
  • Somayeh Sharifi
  • Mehdi Salimi
  • Stefan SiegmundEmail author
Original Paper

Abstract

We establish a new class of three-point methods for the computation of simple zeros of a scalar function. Based on the two-point optimal method by Ostrowski (1966), we construct a family of order eight methods which use three evaluations of f and one of f′ and therefore have an efficiency index equal to \(\sqrt [4]{8}\approx 1.682\) and are optimal in the sense of the Kung and Traub conjecture (Kung and Traub J. Assoc. Comput. Math. 21, 634–651, 1974). Moreover, the dynamics of the proposed methods are shown with some comparisons to other existing methods. Numerical comparison with existing optimal schemes suggests that the new class provides a valuable alternative for solving nonlinear equations.

Keywords

Simple root Three-step iterative method Kung and Traub conjecture Optimal order of convergence Computational efficiency 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Taher Lotfi
    • 1
  • Somayeh Sharifi
    • 1
  • Mehdi Salimi
    • 2
  • Stefan Siegmund
    • 2
    Email author
  1. 1.Department of Mathematics, Hamedan BranchIslamic Azad UniversityHamedanIran
  2. 2.Center for Dynamics, Department of MathematicsTechnische Universität DresdenDresdenGermany

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