Numerical Algorithms

, Volume 43, Issue 3, pp 273–278 | Cite as

A variant of Chebyshev’s method with sixth-order convergence

Original Paper


In this paper, we present a new variant of Chebyshev’s method for solving non-linear equations. Analysis of convergence shows that the new method has sixth-order convergence. Per iteration the new method requires two evaluations of the function, one of its first derivative and one of its second derivative. Thus the efficiency, in term of function evaluations, of the new method is better than that of Chebyshev’s method. Numerical examples verifying the theory are given.


Chebyshev’s method non-linear equations root-finding iterative method 

Mathematics Subject Classification (2000)



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  1. 1.
    Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs, NJ (1964)MATHGoogle Scholar
  2. 2.
    Amat, S., Busquier, S., Gutiérrez, J.M.: Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157, 197–205 (2003)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Grau, M., Díaz-Barrero, J.L.: An improvement of the Euler-Chebyshev iterative method. J. Math. Anal. Appl. 315, 1–7 (2006)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Gautschi, W.: Numerical Analysis: An Introduction. Birkhäuser, Cambridge, MA (1997)MATHGoogle Scholar
  5. 5.
    Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13, 87–93 (2000)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Grau, M.: An improvement to the computing of nonlinear equation solutions. Numer. Algor. 34, 1–12 (2003)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.State Key Laboratory of Water Resources and Hydropower Engineering SciencesWuhan UniversityWuhanChina
  2. 2.Department of MathematicsShanghai UniversityShanghaiChina

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