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Numerical Algorithms

, Volume 59, Issue 4, pp 505–521 | Cite as

Improved local analysis for a certain class of iterative methods with cubic convergence

  • Hongmin Ren
  • Ioannis K. Argyros
Original Paper

Abstract

We use Lipschitz and center-Lipschitz conditions to provide an improved local convergence analysis for a certain class of iterative methods with cubic order of convergence. It turns out that under the same computational cost as before, we obtain a larger radius of convergence and tighter error bounds. Numerical examples are also provided in this study.

Keywords

Banach space Iterative methods Local convergence Radius of convergence Convergence ball 

Mathematics Subject Classifications (2010)

65G99 65K10 65B05 47H17 49M15 

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References

  1. 1.
    Argyros, I.K.: A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space. J. Math. Anal. Appl. 298, 374–397 (2004)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Argyros, I.K.: Convergence and Applications of Newton-type Iterations. Springer-Verlag, New York (2009)Google Scholar
  3. 3.
    Babajee, D.K.R., Dauhoo, M.Z.: An analysis of the properties of the variants of Newton’s method with third order convergence. Appl. Math. Comput. 183, 659–684 (2006)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bi, W.H., Wu, Q.B., Ren, H.M.: Convergence ball and error analysis of Ostrowski–Traubs method. Appl. Math. J. Chinese Univ. 25, 374–378 (2010)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Hasanov, V.I., Ivanov, I.G., Nedzhibov, G.: A new modification of Newton method. Appl. Math. Eng. 27, 278–286 (2002)MathSciNetGoogle Scholar
  6. 6.
    Huang, Z.D.: The convergence ball of Newton’s method and the uniqueness ball of equations under Hölder-type continuous derivatives. Comput. Math. Appl. 25, 247–251 (2004)CrossRefGoogle Scholar
  7. 7.
    Guo, X.P.: Convergence ball of iterations with one parameter. Appl. Math. J. Chinese Univ. Ser. B 20, 462–468 (2005)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Ortega, J.M., Rheinbolt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)MATHGoogle Scholar
  9. 9.
    Ren, H.M., Wu, Q.B.: The convergence ball of the Secant method under Hölder continuous divided differences. J. Comput. Appl. Math. 194, 284–293 (2006)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Ren, H.M., Wu, Q.B.: Convergence ball and error anlysis of a family of iterative methods with cubic convergence. Appl. Math. Comput. 209, 369–378 (2009)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Ren, H.M., Argyros, I.K.: On convergence of the modified Newton’s method under Hölder continuous Fréchet derivative. Appl. Math. Comput. 213, 440–448 (2009)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Ren, H.M., Argyros, I.K.: Convergence radius of the modified Newton method for multiple zeros under Hölder continuous derivative. Appl. Math. Comput. 217, 612–621 (2010)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, Englewood Cliffs, NJ (1964)MATHGoogle Scholar
  14. 14.
    Traub, J.F., Wozniakowski, H.: Convergence and complexity of Newton iteration for operator equation. J. Assoc. Comput. Mech. 26, 250–258 (1979)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Wang, X.H.: The convergence ball on Newton’s method. Chin. Sci. Bull., A Special Issue of Mathematics. Phys. Chem. 25, 36–37 (1980) (in Chinese)Google Scholar
  16. 16.
    Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third order convergence. Appl. Math. Lett. 13, 87–93 (2000)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Wu, Q.B., Ren, H.M.: Convergence ball of a modified secant method for finding zero of derivatives. Appl. Math. Comput. 174, 24–33 (2006)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Wu, Q.B., Ren, H.M., Bi, W.H.: Convergence ball and error analysis of Müller’s method. Appl. Math. Comput. 184, 464–470 (2007)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Wu, Q.B., Ren, H.M., Bi, W.H.: The convergence ball of Wang’s method for finding a zero of a derivative. Math. Comput. Model. 49, 740–744 (2009)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.College of Information and ElectronicsHangzhou PolytechnicHangzhouPeoples Republic of China
  2. 2.Department of Mathematical SciencesCameron UniversityLawtonUSA

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