On a Steffensen-Hermite type method for approximating the solutions of nonlinear equations
DOI:
https://doi.org/10.33993/jnaat351-1015Keywords:
nonlinear equations in \({\mathbb R}\), Steffensen and Aitken-Steffensen methods, inverse interpolatory polynomial of Hermite typeAbstract
It is well known that the Steffensen and Aitken-Steffensen type methods are obtained from the chord method, using controlled nodes. The chord method is an interpolatory method, with two distinct nodes. Using this remark, the Steffensen and Aitken-Steffensen methods have been generalized using interpolatory methods obtained from the inverse interpolation polynomial of Lagrange or Hermite type. In this paper we study the convergence and efficiency of some Steffensen type methods which are obtained from the inverse interpolatory polynomial of Hermite type with two controlled nodes.Downloads
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Costabile, F., Gualtieri, I.M. and Luceri, R., A new iterative method for the computation of the solution of nonlinear equations, Numer. Algorithms, 28, pp. 87-100, 2001, https://doi.org/10.1023/a:1014078328575 DOI: https://doi.org/10.1023/A:1014078328575
Frontini, M., Hermite interpolation and a new iterative method for the computation of the roots of non-linear equations, Calcolo, 40, pp. 109-119, 2003. DOI: https://doi.org/10.1007/s100920300006
Grau, M. An improvement to the computing of nonlinear equation solutions, Numer. Algorithms, 34, pp. 1-12, 2003, https://doi.org/10.1023/a:1026100500306 DOI: https://doi.org/10.1023/A:1026100500306
Ostrowski, A., Solution of Equations in Euclidean and Banach Spaces, Academic Press, New York and London, 1973.
Păvăloiu, I., Optimal efficiency index for iterative methods of interpolatory type, Computer Science Journal of Moldova, 1, 5, pp. 20-43, 1997.
Păvăloiu, I., Approximation of the roots of equations by Aitken-Steffensen-type monotonic wequences, Calcolo, 32, 1-2, pp. 69-82, 1995,https://doi.org/10.1007/bf02576543 . DOI: https://doi.org/10.1007/BF02576543
Păvăloiu, I., Optimal problems concerning interpolation methods of solution of equations, Publications de L'Institut Mathematique, 52 (66), pp. 113-126, 1992.
Păvăloiu, I., Optimal effiency index of a class of Hermite iterative methods, with two steps, Rev. Anal. Numér. Théor. Approx., 29, 1, pp. 83-89, 2000.
Păvăloiu, I., Local convergence of general Steffensen type methods, Rev. Anal. Numér. Théor. Approx., 33, 1, pp. 79-86, 2004, http://ictp.acad.ro/jnaat/journal/article/view/2004-vol33-no1-art10
Traub, J.F., Iterative Methods for Solutions of Equations, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1964.
Turowicz, B.A., Sur les derivées d'ordre superieur d'une function inverse, Ann. Polon. Math., 8, pp. 265-269, 1960, https://doi.org/10.4064/ap-8-3-265-269 DOI: https://doi.org/10.4064/ap-8-3-265-269
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