Skip to content
Licensed Unlicensed Requires Authentication Published online by De Gruyter January 21, 2022

An improvement of derivative-free point-to-point iterative processes with central divided differences

Miguel Ángel Hernández-Verón, Ángel Alberto Magreñán, Eulalia Martínez and Sukhjit Singh


In this article, we introduce a new Steffensen-type method with the advantage that its behavior is very similar to Newton’s method; therefore, it is a very remarkable way of avoiding the drawback that Newton’s method presents for nondifferentiable operators. In our study, we perform an exhaustive comparative study between the semilocal convergence of Newton’s method and the derivative-free point-to-point iterative process considered; in the case of differentiable operators, we use the majoring sequences and the majorant principle. In the nondifferentiable case, we impose conditions on the starting point and on the nonlinear operator to obtain a semilocal convergence result for the iterative process considered. In both cases, we complete the theoretical convergence proofs with a dynamical study and a numerical test. In the case of differentiable operators, this study confirms that the accessibility and numerical behavior of both iterative processes, Newton’s method and the derivative-free point-to-point iterative process considered, are very similar.

MSC 2010: 45G10; 47H17; 65J15

Corresponding author: Ángel Alberto Magreñán, Dept. of Mathematics and Computation, University of La Rioja, Logroño, Spain, E-mail:

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: This research was partially supported by Ministerio de Economía y Competitividad under grant PGC2018-095896-B-C21-C22 and by the project EEQ/2018/000720 under Science and Engineering Research Board.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.


[1] S. Amat, J. A. Ezquerro, and M. A. Hernández-Verón, “On a Steffensen-like method for solving nonlinear equations,” Calcolo, vol. 53, pp. 171–188, 2016. in Google Scholar

[2] I. K. Argyros, “On the Secant method,” Publ. Math. Debrecen, vol. 43, pp. 223–238, 1993.Search in Google Scholar

[3] I. K. Argyros, Y. J. Cho, and S. Hilout, Numerical Methods for Equations and its Applications, Boca Raton, Florida, USA, CRC Press, Taylor and Francis, 2012.10.1201/b12297Search in Google Scholar

[4] I. K. Argyros and S. Hilout, Numerical Methods in Nonlinear Analysis, New Jersey, World Scientific Publ. Comp., 2013.10.1142/8475Search in Google Scholar

[5] I. K. Argyros and S. Regmi, UndergraduateResearch at Cameron University on Iterative Procedures in Banach and Other Spaces, New York, Nova Science, 2019.Search in Google Scholar

[6] F. Awawdeh, A. Adawi, and S. Al-Shara, “A numerical method for solving nonlinear integral equations,” Int. Math. Forum, vol. 4, pp. 805–817, 2009.Search in Google Scholar

[7] M. Balazs and G. Goldner, “On existence of divided differences in linear spaces,” Rev. Anal. Numer. Theor. Approximation, vol. 2, pp. 3–6, 1973.Search in Google Scholar

[8] B. M. Barbashov, V. V. Nesterenko, and A. M. Chervyakov, “General solutions of nonlinear equations in the geometric theory of the relativistic string,” Commun. Math. Phys., vol. 84, pp. 471–481, 1982. in Google Scholar

[9] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge, Cambridge University Press, 2004.10.1017/CBO9780511543234Search in Google Scholar

[10] J. A. Ezquerro and M. A. Hernández-Verón, “Newton’s method: an updated approach of Kantorovich’s theory,” in Frontiers in Mathematics, Cham, Birkhäuser, Springer, 2017.Search in Google Scholar

[11] J. A. Ezquerro and M. A. Hernández-Verón, “Mild differentiability conditions for Newton’s method in Banach spaces,” in Frontiers in Mathematics, Cham, Birkhäuser, Springer, 2020.10.1007/978-3-030-48702-7Search in Google Scholar

[12] J. A. Ezquerro, D. González, and M. A. Hernández, “Majorizing sequences for Newton’s method from initial value problems,” J. Comput. Appl. Math., vol. 236, pp. 2246–2258, 2012. in Google Scholar

[13] M. Grau-Sánchez, M. Noguera, and S. Amat, “On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods,” J. Comput. Appl. Math., vol. 237, pp. 363–372, 2013. in Google Scholar

[14] J. M. Gutiérrez, A. A. Magreñán, and J. L. Varona, “The ‘Gauss–Seidelization’ of iterative methods for solving nonlinear equations in the complex plane,” Appl. Math. Comput., vol. 218, no. 6, pp. 2467–2479, 2011. in Google Scholar

[15] M. A. Hernández and M. J. Rubio, “A uniparametric family of iterative processes for solving nondifferentiable equations,” J. Math. Anal. Appl., vol. 275, pp. 821–834, 2002. in Google Scholar

[16] M. A. Hernández-Verón, Á. A. Magreñán, and M. J. Rubio, “Dynamics and local convergence of a family of derivative-free iterative processes,” J. Comput. Appl. Math., vol. 354, pp. 414–430, 2019. in Google Scholar

[17] L. V. Kantorovich and G. P. Akilov, Functional Analysis, Oxford, Pergamon Press, 1982.Search in Google Scholar

[18] T. Lotfi, Á. A. Magreñán, K. Mahdiani, and J. J. Rainer, “A variant of Steffensen–King’s type family with accelerated sixth-order convergence and high efficiency index: dynamic study and approach,” Appl. Math. Comput., vol. 252, pp. 347–353, 2015. in Google Scholar

[19] Á. A. Magreñán and I. K. Argyros, “Ball convergence theorems and the convergence planes of an iterative method for nonlinear equations,” SeMA J., vol. 71, no. 1, pp. 39–55, 2015. in Google Scholar

[20] J. M. Ortega, “The Newton–Kantorovich theorem,” Am. Math. Mon., vol. 75, pp. 658–660, 1968. in Google Scholar

[21] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, New York, Academic Press, 1970.Search in Google Scholar

[22] M. A. Ostrowski, Solution of Equations and Systems of Equations, New York, Academic Press, 1966.Search in Google Scholar

[23] S. Regmi, Optimized iterative Methods with Applications in Diverse Disciplines, New York, Nova Science, 2021.Search in Google Scholar

[24] S. M. Shakhno, “On a Kurchatov’s method of linear interpolation for solving nonlinear equations,” Proc. Appl. Math. Mech., vol. 4, pp. 650–651, 2004. in Google Scholar

[25] J. F. Traub, Iterative Methods for the Solution of Equations, New Jersey, Prentice-Hall, 1964.Search in Google Scholar

Received: 2021-06-16
Revised: 2021-11-03
Accepted: 2021-12-24
Published Online: 2022-01-21

© 2022 Walter de Gruyter GmbH, Berlin/Boston