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Numerical Algorithms

, Volume 72, Issue 4, pp 937–958 | Cite as

A stable class of improved second-derivative free Chebyshev-Halley type methods with optimal eighth order convergence

  • Alicia CorderoEmail author
  • Munish Kansal
  • Vinay Kanwar
  • Juan R. Torregrosa
Original Paper

Abstract

In this paper, we present a uniparametric family of modified Chebyshev-Halley type methods with optimal eighth-order of convergence. In terms of computational cost, each member of the family requires only four functional evaluations per step, and hence is optimal in the sense of Kung-Traub conjecture. Moreover, in order to have additional information to choose some elements of the class, in particular some stable enough, we use complex dynamics tools to analyze their stability. Then, some ranges of values of the parameter are found to be avoided but we show that the region of stable members of this family is vast. It is found by way of illustration that these proposed methods are very useful in high precision computations.

Keywords

Nonlinear equations Optimal iterative schemes Newton method Chebyshev-Halley scheme Efficiency index Complex dynamics Stability functions Dynamical planes 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Alicia Cordero
    • 1
    Email author
  • Munish Kansal
    • 2
  • Vinay Kanwar
    • 2
  • Juan R. Torregrosa
    • 1
  1. 1.Multidisciplinary Institute of MathematicsUniversitat Politècnica de ValènciaValenciaSpain
  2. 2.University Institute of Engineering and TechnologyPanjab UniversityChandigarhIndia

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