• Received : 2014.01.30
  • Published : 2015.01.01


We present a semilocal convergence analysis of a third order method for approximating a locally unique solution of an equation in a Banach space setting. Recently, this method was studied by Chun, Stanica and Neta. These authors extended earlier results by Kou, Li and others. Our convergence analysis extends the applicability of these methods under less computational cost and weaker convergence criteria. Numerical examples are also presented to show that the earlier results cannot apply to solve these equations.


  1. S. Amat, A. A. Magrenan, and N. Romero, On a two-step relaxed Newton-type method, App. Math. Comput. 219 (2013), no. 24, 11341-11347.
  2. I. K. Argyros, Computational theory of iterative methods, Series: Studies in Computational Mathematics, 15, Editors: C. K. Chui and L. Wuytack, Elsevier Publ. Co. New York, U.S.A, 2007.
  3. I. K. Argyros and S. Hilout, Weaker conditions for the convergence of Newton's method, J. Complexity 28 (2012), no. 3, 364-387.
  4. I. K. Argyros and S. Hilout, Computational Methods in Nonlinear Analysis, World Scientific Publ. Comp., New Jersey, 2013.
  5. V. Candela and A. Marquina, Recurrence relations for rational cubic methods I: The Halley method, Computing 44 (1990), no. 2, 169-184.
  6. V. Candela and A. Marquina, Recurrence relations for rational cubic methods II: The Chebyshev method, Computing 45 (1990), no. 4, 355-367.
  7. C. Chun, P. Stanica, and B. Neta, Third order family of methods in Banach spaces, Comput. Math. Appl. 61 (2011), no. 6, 1665-1675.
  8. J. A. Ezquerro and M. A. Hernandez, Recurrence relations for Chebyshev-type methods, Appl. Math. Optim. 41 (2000), no. 2, 227-236.
  9. J. M. Gutierrez and M. A. Hernandez, Recurrence relations for the super-Halley method, Computers Math. Appl. 36 (1998), no. 7, 1-8.
  10. J. M. Gutierrez and M. A. Hernandez, Third-order iterative methods for operators with bounded second derivative, J. Comput. Appl. Math. 82 (1997), no. 1-2, 171-183.
  11. J. M. Gutierrez, A. A. Magrenan, and N. Romero, On the semilocal convergence of Newton-Kantorovich method under center-Lipschitz conditions, Appl. Math. Comput. 221 (2013), 79-88.
  12. L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982.
  13. J. S. Kou and T. Li, Modified Chebyshev's method free from second derivative for non-linear equations, Appl. Math. Comput. 187 (2007), no. 2, 1027-1032.
  14. J. S. Kou, T. Li, and X. H. Wang, A modification of Newton method with third-order convergence, Appl. Math. comput. 181 (2006), no. 2, 1106-1111.
  15. A. A. Magrenan, Estudio de la dinamica del metodo de Newton amortiguado, PhD Thesis, Servicio de Publicaciones, Universidad de La Rioja, 2013.
  16. A. A. Magrenan and I. K. Argyros, Two-step Newton methods, J. Complexity 30 (2014), no. 4, 533-553.
  17. J. M. Ortega andW. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic press, New York, 1970.
  18. P. K. Parida, Study of some third order methods for nonlinear equations in Banach spaces, Ph.D. Dessertation, Indian Institute of Technology, Department of Mathematics, Kharagpur, India, 2007.
  19. P. K. Parida and D. K. Gupta, Recurrence relations for semilocal convergence of a Newton-like method in Banach spaces, J. Math. Anal. Appl. 345 (2008), 350-361.
  20. F. A. Potra and V. Ptra, Nondiscrete induction and iterative processes, in Research Notes in Mathematics, Vol. 103, Pitman, Boston, 1984.
  21. L. B. Rall, Computational solution of nonlinear operator equations, Robert E. Krieger, New York, 1979.
  22. Q. Wu and Y. Zhao, Third order convergence theorem by using majorizing function for a modified Newton method in Banach space, Appl. Math. Comput. 175 (2006), no. 2, 1515-1524.