Abstract
The semilocal convergence of a family of Chebyshev-Halley like iterations for nonlinear operator equations is studied under the hypothesis that the first derivative satisfies a mild differentiability condition. This condition includes the usual Lipschitz condition and the Hölder condition as special cases. The method employed in the present paper is based on a family of recurrence relations. The R-order of convergence of the methods is also analyzed. As well, an application to a nonlinear Hammerstein integral equation of the second kind is provided. Furthermore, two numerical examples are presented to demonstrate the applicability and efficiency of the convergence results.
Similar content being viewed by others
References
Candela, V., Marquina, A.: Recurrence relations for rational cubic methods I: the Halley method. Computing 44, 169–184 (1990)
Candela, V., Marquina, A.: Recurrence relations for rational cubic methods II: the Chebyshev method. Computing 45, 355–367 (1990)
Ezquerro, J.A.: A modification of the Chebyshev method. IMA J. Numer. Anal. 17, 511–525 (1997)
Ezquerro, J.A., Hernández, M.A.: Generalized differentiability conditions for Newton’s method. IMA J. Numer. Anal. 22, 519–530 (2002)
Ezquerro, J.A., Hernández, M.A.: On an application of Newton’s method to nonlinear operators with ω-conditioned second derivative. BIT Numer. Math. 42, 519–530 (2002)
Ezquerro, J.A., Hernández, M.A.: On the R-order of the Halley method. J. Math. Anal. Appl. 303, 591–601 (2005)
Ezquerro, J.A., Hernández, M.A.: On the R-order of convergence of Newton’s method under mild differentiability conditions. J. Comput. Appl. Math. 197, 53–61 (2006)
Ganesh, M., Joshi, M.C.: Numerical solvability of Hammerstein integral equation of mixed type. IMA J. Numer. Anal. 11, 21–31 (1991)
Gutiérrez, J.M., Hernández, M.A.: A family of Chebyshev-Halley type methods in Banach spaces. Bull. Aust. Math. Soc. 55, 113–130 (1997)
Gutiérrez, J.M., Hernández, M.A.: Recurrence relations for the super-Halley method. Comput. Math. Appl. 36, 1–8 (1998)
Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon, Oxford (1982)
Melman, A.: Geometry and convergence of Euler’s and Halley’s methods. SIAM Rev. 39, 728–735 (1997)
Potra, F.A.: On Q-order and R-order of convergence. J. Optim. Theory Appl. 63, 415–431 (1989)
Wang, X.: Convergence of Newton’s method and uniqueness of the solution of equations in Banach space. IMA J. Numer. Anal. 20, 123–134 (2000)
Wang, X., Li, C., Lai, M.: A unified convergence theory for Newton-type methods for zeros of nonlinear operators in Banach spaces. BIT Numer. Math. 42, 206–213 (2002)
Xu, X., Li, C.: Convergence of Newton’s method for systems of equations with constant rank derivatives. J. Comput. Math. 25, 705–718 (2007)
Xu, X., Li, C.: Convergence criterion of Newton’s method for singular systems with constant rank derivatives. J. Math. Anal. Appl. 345, 689–701 (2008)
Yao, Q.: On Halley iteration. Numer. Math. 81, 647–677 (1999)
Ye, X., Li, C.: Convergence of the family of the deformed Euler-Hally iterations under the Hölder condition of the second derivative. J. Comput. Appl. Math. 194, 294–308 (2006)
Ye, X., Li, C., Shen, W.: Convergence of the variants of the Chebyshev-Halley iteration family under the Hölder condition of the first derivative. J. Comput. Appl. Math. 203, 279–288 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported in part by the National Natural Science Foundation of China (Grants No. 61170109 and No. 10971194) and Zhejiang Innovation Project (Grant No. T200905).
Rights and permissions
About this article
Cite this article
Xu, X., Ling, Y. Semilocal convergence for a family of Chebyshev-Halley like iterations under a mild differentiability condition. J. Appl. Math. Comput. 40, 627–647 (2012). https://doi.org/10.1007/s12190-012-0557-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-012-0557-9
Keywords
- Nonlinear operator equation
- Variant Chebyshev-Halley iteration family
- Semilocal convergence
- R-order of convergence
- Hammerstein integral equation