Abstract
We present a local convergence analysis of a sixth order iterative method for approximate a locally unique solution of an equation defined on the real line. Earlier studies such as Sharma et al. (Appl Math Comput 190:111–115, 2007) have shown convergence of these methods under hypotheses up to the fifth derivative of the function although only the first derivative appears in the method. In this study we expand the applicability of these methods using only hypotheses up to the first derivative of the function. Numerical examples are also presented in this study.
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Amat, S., Busquier, S., Plaza, S.: Dynamics of the King’s and Jarratt iterations. Aequ. Math. 69, 212–213 (2005)
Argyros, I.K.: Convergence and Application of Newton-type Iterations. Springer, New York (2008)
Argyros, I.K., Hilout, Said: Computational Methods in Nonlinear Analysis. World Scientific Publ. Co., New Jersey (2013)
Argyros, I.K., Chen, D., Quian, Q.: The Jarratt method in Banach space setting. J. Comput. Appl. Math. 51, 103–106 (1994)
Argyros, I.K., George, S., Alberto, A.: Magrenan, high convergence order. J. Comput. Appl. Math. 282, 215–224 (2015)
Argyros, I.K., George, S.: Ball comparison between two optimal eight-order methods under weak conditions. SeMA Journal Boletin de la Sociedad Espaola de Matemtica Aplicada (2015). doi:10.1007/s40324-015-0035-z
Argyros, I.K., George, S.: Local convergence for an efficient eighth order iterative method with a parameter for solving equations under weak conditions. Int. J. Appl. Comput. Math. doi:10.1007/s40819-015-0078-y
Candela, V., Marquina, A.: Recurrence relations for rational cubic methods I: the Halley method. Computing 44, 169–184 (1990)
Chun, C., Neta, B., Scott, M.: Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equations. Appl. Math. Comput. 227, 567–592 (2014)
Cordero, A., Maimo, J., Torregrosa, J., Vassileva, M.P., Vindel, P.: Chaos in King’s iterative family. Appl. Math. Lett. 26, 842–848 (2013)
Frontini, M., Sormani, E.: Some variants of Newton’s method with third order convergence. Appl. Math. Comput. 140, 419–426 (2003)
King, R.F.: A family of fourth-order methods for nonlinear equations. SIAM. J. Numer. Anal. 10, 876–879 (1973)
Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iterations. J. Appl. Comput. Math. 21, 643–651 (1974)
Noor, M.A.: Some applications of quadrature formulas for solving nonlinear equations. Nonlinear Anal. Forum 12(1), 91–96 (2007)
Ortega, J.M., Rheinbolt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)
Petkovic, M.S., Neta, B., Petkovic, L., Džunič, J.: Multipoint Methods for Solving Nonlinear Equations. Elsevier, Amsterdam (2013)
Potra, F.A., Ptak, V.: Nondiscrete Induction and Iterative Processes, Research Notes in Mathematics, vol. 103. Pitman Publ., Boston (1984)
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 Publications, vol. 3(19), pp. 129–142. Banach Center, Warsaw
Sharma, J.R., Guha, R.K.: A family of modified Ostrowski methods with accelerated sixth order convergence. Appl. Math. Comput. 190, 111–115 (2007)
Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, Englewood Cliffs (1964)
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Argyros, I.K., George, S. Ball convergence of a sixth order iterative method with one parameter for solving equations under weak conditions. Calcolo 53, 585–595 (2016). https://doi.org/10.1007/s10092-015-0163-y
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DOI: https://doi.org/10.1007/s10092-015-0163-y