Abstract
We study the solving of nonlinear equations by an iterative method of Aitken type, which has the interpolation nodes controlled by the Newton method. We obtain a local convergence result which shows that the q-convergence order of this method is 6 and its efficiency index is \(\sqrt[5]{6},\) which is higher than the efficiency index of the Aitken or Newton methods. Monotone sequences are obtained for initial approximations farther from the solution, if they satisfy the Fourier condition and the nonlinear mapping satisfies monotony and convexity assumptions on the domain.
Similar content being viewed by others
References
Amat, S., Busquier, S.: A two step Steffenssen’s method under modified convergence conditions. J. Math. Anal. Appl. 324, 1084–1092 (2006)
Beltyukov, B.A.: An analogue of the Aitken–Steffensen method with controlled step. URSS Comput. Math. Math. Phys. 27(3), 103–112 (1987)
Chun, C.: A geometric construction of iterative formulas of order three. Appl. Math. Lett. 23, 512–516 (2010)
Cordero, A., Torregrosa, J.R.: Variants of Newton’s method for functions of several variables. Appl. Math. Comput. 183, 199–2008 (2006)
Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)
Cordero A., Torregrosa R.J.: A class of Steffenssen type method with optimal order of convergence. Appl. Math. Comput. 217, 7653–7659 (2011)
Hongmin, R., Qingbio, W., Welhong, B.: A class of two-step Steffensen type methods with fourth order convergence. Appl. Math. Comput. 209(2), 206–210 (2009)
Jain, P.: Steffensen type method for solving non-linear equations. Appl. Math. Comput. 194, 527–533 (2007)
Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)
Ostrowski, M.A.: Solution of Equations in Euclidian and Banach Spaces. Academic Press, New York and London (1973)
Păvăloiu, I.: Approximation of the root of equations by Aitken–Steffensen-type monotonic sequences. Calcolo 32, 69–82 (1995)
Păvăloiu, I., Cătinaş, E.: On a Steffensen–Hermite method of order three. Appl. Math. Comput. 215(7), 2663–2672 (2009)
Păvăloiu, I., Cătinaş, E.: On a Steffensen type method. In: Ninth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, Proceedings, pp. 369–375 (2007)
Păvăloiu, I.: Optimal algorithms, concerning the solving of equations by interpolation. In: Popoviciu, E. (ed.) Research on Theory of Allure, Approximation, Convexity and Optimization, Editura Srima, Cluj-Napoca, pp. 222–248 (1999)
Păvăloiu, I.: Aitken–Steffensen-type method for nondifferentiable functions (I). Rev. Anal. Numér. Theor. Approx. 31(1), 109–114 (2002)
Păvăloiu, I.: Aitken–Steffensen-type method for nonsmooth functions (II). Rev. Anal. Numér. Théor. Approx. 31(2), 195–198 (2002)
Păvăloiu, I.: Aitken–Steffensen-type method for nonsmooth functions (III). Rev. Anal. Numér. Théor. Approx. 32(1), 73–78 (2003)
Quan, Z., Peng, Z., Li, Z., Wenchao, M.: Variants of Steffensen secant method and applications. Appl. Math. Comput. 216(12), 3486–3496 (2010)
Sharma, R.J.: A composite thid order Newton–Steffensen method for solving nonlinear equations. Appl. Math. Comput. 169, 242–246 (2005)
Traub, Y.F.: Iterative Method for Solution of Equations. Prentice-Hall, Englewood Cliffs, NJ (1964)
Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13(8), 87–93 (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Păvăloiu, I., Cătinaş, E. On an Aitken–Newton type method. Numer Algor 62, 253–260 (2013). https://doi.org/10.1007/s11075-012-9577-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-012-9577-7