Abstract
We present a local convergence analysis of an eighth order- iterative method in order to approximate a locally unique solution of an equation in Banach space setting. Earlier studies have used hypotheses up to the fourth derivative although only the first derivative appears in the definition of these methods. In this study we only use the hypothesis on the first derivative. This way we expand the applicability of these methods. Moreover, we provide a radius of convergence, a uniqueness ball and computable error bounds based on Lipschitz constants. Numerical examples computing the radii of the convergence balls as well as examples where earlier results cannot apply to solve equations but our results can apply are also given in this study.
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Abad, M.F., Cordero, A., Torregrosa, J.R.: Fourth and Fifth-order methods for solving nonlinear systems of equations: an application to the global positioning system abstract and applied analysis (2013)
Amat, S., Hernández, M.A., Romero, N.: Semilocal convergence of a sixth order iterative method for quadratic equations. Appl. Numer. Math. 62, 833–841 (2012)
Argyros, I.K.: Computational theory of iterative methods. In: Chui, C.K., Wuytack, L. (eds.) Series: studies in computational mathematics, 15. Elsevier, New York (2007)
Argyros, I.K.: A semilocal convergence analysis for directional Newton methods. Math. Comput. 80, 327–343 (2011)
Argyros, I.K., Hilout, S.: Weaker conditions for the convergence of Newton’s method. J. Complex. 28, 364–387 (2012)
Argyros, I.K., Hilout, S.: Computational Methods in Nonlinear Analysis. Efficient Algorithms, Fixed Point Theory and Applications. World Scientific (2013)
Aryros, I.K., Ren, H.: Improved local analysis for certain class of iterative methods with cubic convergence. Numer Algorithm 59, 505–521 (2012)
Candela, V., Marquina, A.: Recurrence relations for rational cubic methods I: The Halley method. Computing 44, 169–184 (1990)
Candela, V., Marquina, A.: Recurrence relations for rational cubic methods II: The Chebyshev method. Computing 45(4), 355–367 (1990)
Chun, C., Stănică, P., Neta, B.: Third-order family of methods in Banach spaces. Comput. Math. Appl. 61, 1665–1675 (2011)
Cordero, A., Martinez, F., Torregrosa, J.R.: Iterative methods of order four and five for systems of nonlinear equations. J. Comput. Appl. Math. 231, 541–551 (2009)
Cordero, A., Hueso, J., Martinez, E., Torregrosa, J.R.: A modified Newton–Tarratt’s composition. Numer. Algor. 55, 87–99 (2010)
Cordero, A., Torregrosa, J.R., Vasileva, M.P.: Increasing the order of convergence of iterative schemes for solving nonlinear systems. J. Comput. Appl. Math. 252, 86–94 (2013)
Ezzati, R., Azandegan, E.: A simple iterative method with fifth order convergence by using Potra and Ptak’s method. Math. Sci. 3(2), 191–200 (2009)
Gutiérrez, J.M., Magren̄án, A.A., Romero, N.: On the semi-local convergence of Newton–Kantorovich method under center-Lipschitz conditions. Appl. Math. Comput. 221, 79–88 (2013)
Hasanov, V.I., Ivanov, I.G., Nebzhibov, F.: A new modification of Newton’s method. Appl. Math. Eng. 27, 278–286 (2002)
Hernández, M.A., Salanova, M.A.: Modification of the Kantorovich assumptions for semi-local convergence of the Chebyshev method. J. Comput. Appl. Math. 126, 131–143 (2000)
Jaiswal, J.P.: Semilocal convergence of an eighth-order method in Banach spaces and its computational efficiency. Numer. Algor. 71, 933–951 (2016)
Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon Press, Oxford (1982)
Kou, J.S., Li, Y.T., Wang, X.H.: A modification of Newton method with third-order convergence. Appl. Math. Comput. 181, 1106–1111 (2006)
Magrenan, A.A.: Different anomalies in a Jarratt family of iterative root finding methods. Appl. Math. Comput. 233, 29–38 (2014)
Magrenan, A.A.: A new tool to study real dynamics: the convergence plane. Appl. Math. Comput. 248, 29–38 (2014)
Petkovic, M.S., Neta, B., Petkovic, L., Džunič, J.: Multipoint methods for solving nonlinear equations. Elsevier, New York (2013)
Potra, F.A., Pták, V.: Nondiscrete induction and iterative processes. In: Research Notes in Mathematics, vol. 103. Pitman, Boston (1984)
Ren, H., Wu, Q.: Convergence ball and error analysis of a family of iterative methods with cubic convergence. Appl. Math. Comput. 209, 369–378 (2009)
Rheinboldt, W.C.: An adaptive continuation process for solving systems of nonlinear equations. In: Tikhonov, A.N. et al. (eds.) Mathematical Models and Numerical Methods, pub.3, pp. 129–142. Banach Center, Warsaw (1977)
Sharma, J.R., Guha, P.K., Sharma, R.: An efficient fourth order weighted-Newton method for systems of nonlinear equations. Numer. Algorithms 62(2), 307–323 (2013)
Traub, J.F.: Iterative methods for the solution of equations. AMS Chelsea Publishing (1982)
Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13, 87–93 (2000)
Zhanlar, T., Chulunbaatar, O., Ankhbayar, G.: On Newton-type methods with fourth and fifth-order convergence. Bull. PFUR Ser. Math. Inf. Sci. Phys. 2(2), 29–34 (2010)
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Argyros, I.K., George, S. & Erappa, S.M. Ball convergence for an eighth order efficient method under weak conditions in Banach spaces. SeMA 74, 513–521 (2017). https://doi.org/10.1007/s40324-016-0098-5
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DOI: https://doi.org/10.1007/s40324-016-0098-5