Abstract
We present a semilocal as well as a local convergence analysis of Steffensen’s method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. The novelty of this paper is twofold. On the one hand, we show convergence under general conditions including earlier ones as special cases. In particular, these conditions allow the important study when the operator involved is not differentiable. On the other hand, we use a combination of conditions that allow a more precise computation of the upper bounds on the norms of the inverses involved leading to a tighter convergence analysis. Finally, we provide numerical examples involving nonlinear integral equations on mixed Hammerstein type that appear in chemistry, vehicular traffic theory, biology, and queuing theory.
Similar content being viewed by others
References
Alarcón, V., Amat, S., Busquier, S., López, D. J.: A Steffensen’s type method in Banach spaces with applications on boundary-value problems. J. Comput. Appl. Math. 216(1), 243–250 (2008)
Argyros, I. K.: A new convergence theorem for Steffensen’s method on Banach spaces and applications. Southwest J. Pure Appl. Math. 1, 23–29 (1997)
Argyros, I.: On the Secant method. Publ. Math. Debrecen 43(3-4), 223–238 (1993)
Argyros, I. K., Ren, H: On an improved local convergence analysis for the Secant method. Numer. Algor. 52, 257–271 (2009)
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)
Chen, J., Shen, Z.: Convergence analysis of the secant type methods. Appl. Math. Comput. 188, 514–524 (2007)
Deimling, K.: Nonlinear functional analysis. Springer-Verlag, Berlin (1985)
Ezquerro, J. A., Hernández, M. A., Romero, N., Velasco, A.: On Steffensen’s method on Banach spaces. J. Comput. Appl. Math. 249, 9–23 (2013)
Hernández, M. A., Rubio, M. J., Ezquerro, J. A.: Secant-like methods for solving nonlinear integral equations of the Hammerstein type. J. Comput. Appl. Math. 115(1-2), 245–254 (2000)
Hernández, M. A., Rubio, M. J.: A uniparametric family of iterative processes for solving nondifferentiable equations. J. Math. Anal. Appl. 275, 821–834 (2002)
Hernández-Verón, M. A., Rubio, M. J.: On the ball of convergence of Secant-like methods for non-differentiable operators Applied Mathematics and Computation. doi:10.1016/j.amc.2015.10.007 (2015)
Hongmin, R., Qingiao, W: The convergence ball of the Secant method under Hlder continuous divided differences. J. Comput. Appl. Math. 194, 284–293 (2006)
Kewei, L.: Homocentric convergence ball of the Secant method. Appl. Math. J. Chinese Univ. Ser. B 22(3), 353–365 (2007)
Kung, H. T., Traub, J. F.: Optimal order of one-point and multipoint iteration Computer Science Department. Paper, 1747 (1973)
Potra, F. A., Pták, V.: Nondiscrete induction and iterative methods. Pitman Publishing Limited, London (1984)
Potra, F. A.: A characterisation of the divided differences of an operator which can be represented by Riemann integrals. Anal. Numer. Theory Approx. 9, 251–253 (1980)
Rashidinia, J., Zarebnia, M.: New approach for numerical solution of Hammerstein integral equations. Appl. Math. Comput. 185, 147–154 (2007)
Ren, H., Argyros, I. K.: Local convergence of efficient Secant-type methods for solving nonlinear equations. Appl. Math. Comput. 218, 7655–7664 (2012)
Shakhno, S.: On the Secant method under generalized Lipschitz conditions for the divided difference operator. PAMM-Proc. Appl. Math. Mech. 7, 2060083–2060084 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Argyros, I.K., Hernández-Verón, M.A. & Rubio, M.J. Convergence of Steffensen’s method for non-differentiable operators. Numer Algor 75, 229–244 (2017). https://doi.org/10.1007/s11075-016-0203-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-016-0203-y
Keywords
- Nonlinear equation
- Non-differentiable operator
- Divided difference
- Iterative method
- The Steffensen method
- Local convergence
- Semilocal convergence