Local convergence of a multi-step high order method with divided differences under hypotheses on the first derivative

Ioannis K. Argyros, Santhosh George


This paper is devoted to the study of a multi-step method with divided differences for solving nonlinear equations in Banach spaces. In earlier studies, hypotheses on the Fréchet derivative up to the sixth order of the operator under consideration is used to prove the convergence of the method. That restricts the applicability of the method. In this paper we extended the applicability of the sixth-order multi-step method by using only hypotheses on the first derivative of the operator involved. Our convergence conditions are weaker than the conditions used in earlier studies. Numerical examples where earlier results cannot be applied to solve equations but our results can be applied are also given in this study.


Multi-step method; restricted convergence domain; radius of convergence; local convergence

Mathematics Subject Classification (2010)

65D10; 49M15; 74G20; 41A25.


Amat, Sergio, and Miguel A. Hernández, and Natalia Romero. "Semilocal convergence of a sixth order iterative method for quadratic equations." Appl. Numer. Math. 62, no. 7 (2012): 833–841.

Argyros, Ioannis K. Computational theory of iterative methods. Vol. 15 of Studies in Computational Mathematics. Amsterdam: Elsevier B. V., 2007.

Argyros, Ioannis K. "A semilocal convergence analysis for directional Newton methods." Math. Comp. 80, no. 273 (2011): 327–343.

Argyros, Ioannis K., and Saïd Hilout. "Weaker conditions for the convergence of Newton’s method." J. Complexity 28, no. 3 (2012): 364–387.

Argyros, Ioannis K., and Saïd Hilout. Computational methods in nonlinear analysis. Efficient algorithms, fixed point theory and applications. Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., 2013.

Cordero, Alicia, and Eulalia Martínez, and Juan R. Torregrosa. "Iterative methods of order four and five for systems of nonlinear equations." J. Comput. Appl. Math. 231, no. 2 (2009): 541–551.

Cordero, Alicia et al. "A modified Newton-Jarratt’s composition." Numer. Algorithms 55, no. 1 (2010): 87–99.

Cordero, Alicia, and Juan R. Torregrosa, and María P. Vassileva. "Increasing the order of convergence of iterative schemes for solving nonlinear systems." J. Comput. Appl. Math. 252 (2013): 86–94.

Ezquerro, J.A., and M.A. Hernández, and A.N. Romero. "Aproximación de soluciones de algunas ecuaciones integrales de Hammerstein mediante métodos iterativos tipo Newton." In XXI Congreso de Ecuaciones Diferenciales y Aplicaciones, XI Congreso de Matemática Aplicada, Ciudad Real, 21-25 septiembre 2009, 1-8. Universidad de Castilla-La Mancha, 2009.

Grau, Miquel, and José Luis Díaz-Barrero. "An improvement of the Euler-Chebyshev iterative method." J. Math. Anal. Appl. 315, no. 1 (2006): 1–7.

Grau-Sánchez, Miquel, and Ángela Grau, and Miquel Noguera. "Ostrowski type methods for solving systems of nonlinear equations." Appl. Math. Comput. 218,

no. 6 (2011): 2377–2385.

Grau-Sánchez, Miquel, and Miquel Noguera, and Sergio Amat. "On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods." J. Comput. Appl. Math. 237, no. 1 (2013): 363–372.

Gutiérrez, José Manuel, and Ángel A. Magreñán, and Natalia Romero. "On the semilocal convergence of Newton-Kantorovich method under center-Lipschitz conditions." Appl. Math. Comput. 221 (2013): 79–88.

Kantorovich, Leonid V., and Gleb P. Akilov. Functional analysis. Second edition. Oxford-Elmsford, N.Y.: Pergamon Press, 1982.

Magreñán, Ángel A. "Different anomalies in a Jarratt family of iterative rootfindingmethods." Appl. Math. Comput. 233 (2014): 29–38.

Magreñán, Ángel A. "A new tool to study real dynamics: the convergence plane." Appl. Math. Comput. 248 (2014): 215–224.

Petkovic, Miodrag S.et al. Multipoint methods for solving nonlinear equations. Amsterdam: Elsevier/Academic Press, 2013.

Ren, Hongmin, and Ioannis K. Argyros. "Improved local analysis for a certain class of iterative methods with cubic convergence." Numer. Algorithms 59, no. 4 (2012): 505–521.

Rheinboldt, Werner C. "An adaptive continuation process for solving systems of nonlinear equations." In Mathematical models and numerical methods, 129–142. Warsaw: Banach Center Publ. 3, 1978.

Sharma, Janak Raj, and Puneet Gupta. "An efficient fifth order method for solving systems of nonlinear equations." Comput. Math. Appl. 67, no. 3 (2014): 591–601.

Traub, Joseph F. Iterative methods for the solution of equations. New-York: AMS Chelsea Publishing, 1982.

Weerakoon, S., and T.G.I. Fernando. "A variant of Newton’s method with accelerated third-order convergence." Appl. Math. Lett. 13, no. 8 (2000): 87–93.

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