Advertisement

Numerical Algorithms

, Volume 59, Issue 4, pp 623–638 | Cite as

Recurrence relations for semilocal convergence of a fifth-order method in Banach spaces

  • Lin Zheng
  • Chuanqing Gu
Original Paper

Abstract

In this paper, we study the semilocal convergence for a fifth-order method for solving nonlinear equations in Banach spaces. The semilocal convergence of this method is established by using recurrence relations. We prove an existence-uniqueness theorem and give a priori error bounds which demonstrates the R-order of the method. As compared with the Jarratt method in Hernández and Salanova (Southwest J Pure Appl Math 1:29–40, 1999) and the Multi-super-Halley method in Wang et al. (Numer Algorithms 56:497–516, 2011), the differentiability conditions of the convergence of the method in this paper are mild and the R-order is improved. Finally, we give some numerical applications to demonstrate our approach.

Keywords

Nonlinear equations in Banach spaces Recurrence relations Semilocal convergence Newton-super-Halley method 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Candela, V., Marquina, A.: Recurrence relations for rational cubic methods I: the Halley method. Computing. 44, 169–184 (1990)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Candela, V., Marquina, A.: Recurrence relations for rational cubic methods II: the Chebyshev method. Computing. 45, 355–367 (1990)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Hernández, M.A., Salanova, M.A.: Sufficient conditions for semilocal convergence of a fourth order multipoint iterative method for solving equations in Banach spaces. Southwest J. Pure Appl. Math. 1, 29–40 (1999)MATHGoogle Scholar
  4. 4.
    Parida, P.K., Gupta, D.K.: Recurrence relations for a Newton-like method in Banach spaces. J. Comput. Appl. Math. 206, 873–887 (2007)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Wang, X.H., Gu, C.Q., Kou, J.S.: Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces. Numer. Algorithms 56, 497–516 (2011)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Parhi, S.K., Gupta, D.K.: A third order method for fixed points in Banach spaces. J. Math. Anal. Appl. 359, 642–652 (2009)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Argyros, I.K.: Convergence and Applications of Newton-Type Iterations. Springer, New York (2008)MATHGoogle Scholar
  8. 8.
    Argyros, I.K., Cho, Y.J., Hilout, S.: On the semilocal convergence of the Halley method using recurrent functions. J. Appl. Math. Comput. doi: 10.1007/s12190-010-0431-6 (2010)MathSciNetGoogle Scholar
  9. 9.
    Kou, J.S., Li, Y.T., Wang, X.H.: A family of fifth-order iterations composed of Newton and third-order methods. Appl. Math. Comput. 186, 50–55 (2007)Google Scholar
  10. 10.
    Wu, Q.B., Zhao, Y.Q.: Third-order convergence theorem by using majorizing function for a modified Newton method in Banach space. Appl. Math. Comput. 175, 1515–1524 (2006)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Parida, P.K., Gupta, D.K.: Recurrence relations for semilocal convergence of a Newton-like method in Banach spaces. J. Math. Anal. Appl. 345, 350–361 (2008)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Gutiérrez, J.M., Hernández, M.A.: Recurrence relations for the super-Halley method. Comput. Math. Appl. 36, 1–8 (1998)MATHCrossRefGoogle Scholar
  13. 13.
    Ezquerro, J.A., Hernández, M.A.: Avoiding the computation of the second Fréchet-derivative in the convex accelerations of Newton’s method. J. Comput. Appl. Math. 96, 1–12 (1998)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Amat, S., Bermúdez, C., Busquier, S., Plaza, S.: On a third-order Newton-type method free of bilinear operators. Numer. Linear Algebr. Appl. 17, 639–653 (2010)MATHGoogle Scholar
  15. 15.
    Amat, S., Busquier, S., Gutiérrez, J.M.: Third-order iterative methods with applications to Hammerstein equations: a unified approach. J. Comput. Appl. Math. 235, 2936–2943 (2011)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Department of MathematicsShanghai UniversityShanghaiChina

Personalised recommendations