Local convergence for a Chebyshev-type method in Banach space free of derivatives

Year 2018, , 62 - 69, 25.03.2018

Santhosh George , Ioannis K Argyros

https://doi.org/10.31197/atnaa.400459

Abstract

This  paper is devoted to the study of a Chebyshev-type method  free of derivatives  for solving nonlinear equations in Banach spaces. Using the idea of restricted convergence domain, we extended the applicability of the Chebyshev-type methods.   Our convergence conditions are weaker than the  conditions used in earlier studies. Therefore the applicability of the method is extended.   Numerical examples  where earlier results cannot apply to solve equations but our results can apply are also given in this study.

Keywords

Chebyshev-type method, restricted convergence domain, radius of convergence

References

• S. Amat, M.A. Hern\'{a}ndez, N. Romero, Semilocal convergence of a sixth order iterative method for quadratic equations, Applied Numerical Mathematics, 62 (2012), 833-841.
• 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.
• I. KArgyros, A semilocal convergence analysis for directional Newton methods. Math. Comput. 80 (2011), 327--343. I.K. Argyros, S. Hilout, Weaker conditions for the convergence of Newton's method. J. Complexity 28 (2012) 364--387.
• I. K. Argyros and Said Hilout, Computational methods in nonlinear analysis. Efficient algorithms, fixed point theory and applications, World Scientific, 2013.
• I.K. Argyros and H. Ren, Improved local analysis for certain class of iterative methods with cubic convergence, Numerical Algorithms, 59(2012), 505-521.
• J. M. Guti\'{e}rrez, A.A. Magre\={n}\'{a}n and N. Romero, On the semi-local convergence of Newton-Kantorovich method under center-Lipschitz conditions, Applied Mathematics and Computation, 221 (2013), 79-88.
• L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982. A. A. Magrenan, Different anomalies in a Jarratt family of iterative root finding methods, Appl. Math. Comput. 233, (2014), 29-38. A. A. Magrenan, A new tool to study real dynamics: The convergence plane, Appl. Math. Comput. 248, (2014), 29-38. A.N .Romero, J.A. Ezquerro, M .A. Hernandez, Approximacion de soluciones de algunas equacuaciones integrals de Hammerstein mediante metodos iterativos tipo. Newton, XXI Congresode ecuaciones diferenciales y aplicaciones Universidad de Castilla-La Mancha (2009) J.R. Sharma, P.K. Guha and R. Sharma, An efficient fourth order weighted-Newton method for systems of nonlinear equations, Numerical Algorithms, 62, 2, (2013), 307-323.
• J.R. Sharma, H. Arora, A novel derivative free algorithm with seventh order convergence for solving systems of nonlinear equations, Numer. Algor., 67, (2014), 917--933.
• S. Weerakoon, T.G.I. Fernando, A variant of Newton's method with accelerated third-order convergence, Appl. Math. Lett. 13, (2000), 87-93.