Bilateral approximations of the roots of scalar equations by Lagrange-Aitken-Steffensen method of order three

Ion Păvăloiu

doi:10.33993/jnaat352-843

Authors

Ion Păvăloiu Tiberiu Popoviciu, Institute of Numerical Analysis, Romanian Academy, Romania

DOI:

https://doi.org/10.33993/jnaat352-843

Keywords:

Aitken-Steffenssen methods, Lagrange inverse interpolation

Abstract views: 234

Abstract

We study the monotone convergence of two general methods of Aitken-Steffenssen type. These methods are obtained from the Lagrange inverse interpolation polynomial of degree two, having controlled nodes. The obtained results provide information on controlling the errors at each iteration step.

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References

Balázs, M., A bilateral approximating method for finding the real roots of real equations, Rev. Anal. Numér. Théor. Approx., 21 (2), pp. 111-117, 1992.

Casulli, V., Trigiante, D., The convergence order for iterative multipoint procedures, Calcolo, 13 (1), pp. 25-44, 1997, https://doi.org/10.1007/bf02576646 DOI: https://doi.org/10.1007/BF02576646

Costabile, F., Gualtieri, I. M., 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 Euclidian 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 sequences, 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 Mathématique, 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.

Păvăloiu, I. and Pop, N., Interpolation and Applications, Risoprint, Cluj-Napoca, 2005 (in Romanian).

Păvăloiu, I., On a Steffensen-Hermite-type Method for approximating the solution of nonlinear equations, Rev. Anal. Numér. Théor. Approx., 25 1, pp. 87-94, 2006.

Păvăloiu, I., Bilateral approximation of solutions of equations by order-three Steffensen type methods, Studia Univ. "Babeş-Bolyai", Mathematica, Vol. LI, no. 3, pp. 105-114, 2006.

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 supérieur 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