Numerical Algorithms

, Volume 70, Issue 2, pp 377–392 | Cite as

On the semilocal convergence of a three steps Newton-type iterative process under mild convergence conditions

Original Paper

Abstract

In this paper the semilocal convergence for an alternative to the three steps Newton’s method with frozen derivative is presented. We analyze the generalization of convergence conditions given by w-conditioned non-decreasing functions instead of the first derivative Lipschitz or Holder continuous given by other authors. A nonlinear integral equation of mixed Hammerstein type is considered for illustrating the new theoretical results obtained in this paper, where previous results can not be satisfied.

Keywords

Nonlinear equations Order of convergence Iterative methods Semilocal convergence Hammerstein equation 

Mathematics Subject Classification (2010)

47H99 65H10 

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References

  1. 1.
    Amat, S., Busquier, S., Bermúdez, C., Plaza, S.: On two families of high order Newton type methods. Appl. Math. Comput. 25, 2209–2217 (2012)MathSciNetMATHGoogle Scholar
  2. 2.
    Argyros, I.K.: The Newton-Kantorovich method under mild differentiability conditions and the Pták error estimates. Monatsh. Math. 101, 175–193 (1990)CrossRefMATHGoogle Scholar
  3. 3.
    Argyros, I.K.: Remarks on the convergence of Newton’s method under Hölder continuity conditions. Tamkang J. Math. 23(4), 269–277 (1992)MathSciNetMATHGoogle Scholar
  4. 4.
    Bruns, D.D., Bailey, J.E.: Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chem. Eng. Sci. 32, 257–264 (1977)CrossRefGoogle Scholar
  5. 5.
    Ezquerro, J.A., Hernández, M.A.: Generalized differentiability conditions for Newton’s method. IMA J. Numer. Anal. 22(2), 187–205 (2002)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ganesh, M., Joshi, M.C.: Numerical solvability of Hammerstein integral equations of mixed type. IMA J. Numer. Anal. 11, 21–31 (1991)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Hueso, J.L., Martínez, E., Torregrosa, J.R.: Third and fourth order iterative methods free from second derivative for nonlinear systems. Appl. Math. Comput. 211, 190–197 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hueso, J.L., Martínez, E.: Semilocal convergence of a family of iterative methods in Banach spaces. Numer. Algor. In Press. doi:10.1007/s11075-013-9795-7
  9. 9.
    Kantorovich, L.V., Akilov, G.P.: Functional analysis. Pergamon Press, Oxford (1982)MATHGoogle Scholar
  10. 10.
    Ostrowski, A.M.: Solutions of equations and system of equations. Academic Press, New York (1960)MATHGoogle Scholar
  11. 11.
    Traub, J.F.: Iterative methods for the solution of equations. Prentice-Hall, Englewood Cliffs (1964)MATHGoogle Scholar
  12. 12.
    Yamamoto, T.: A method for finding sharp error bounds for Newton’s method under the Kantorovich assumptions. Numer. Math. 49, 203–220 (1986)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematics and ComputationUniversity of La RiojaLogroñoSpain
  2. 2.Instituto de Matemática Pura y AplicadaUniversidad Politécnica de ValenciaValenciaSpain

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