On the convergence order of some Aitken-Steffensen type methods

Ion Păvăloiu

doi:10.33993/jnaat322-748

Authors

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

DOI:

https://doi.org/10.33993/jnaat322-748

Keywords:

Steffensen, Aitken and Aitken-Steffensen iterations

Abstract views: 259

Abstract

In this note we make a comparative study of the convergence orders for the Steffensen, Aitken and Aitken-Steffensen methods. We provide some conditions ensuring their local convergence. We study the case when the auxiliary operators used have convergence orders \(r_{1},r_{2}\in \mathbb{N}\) respectively. We show that the Steffensen, Aitken and Aitken-Steffensen methods have the convergence orders \(r_{1}+1\), \(r_{1}+r_{2}\) and \(r_{1}r_{2}+r_{1}\) respectively.

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References

Argyros, I. K., A new convergence theorem for the Steffensen method in Banach space and applications, Rev. Anal. Numér. Théor. Approx., 29, no. 2, pp. 119-127, 2000, http://ictp.acad.ro/jnaat/journal/article/view/2000-vol29-no2-art1

Argyros, I. K., An error analysis for the Steffensen method under generalized Zabrejko-Nguen-type assumptions, Rev. Anal. Numér. Théor. Approx., 25, nos. 1-2, pp. 11-22, 1996, http://ictp.acad.ro/jnaat/journal/article/view/1996-vol25-nos1-2-art2

Argyros, I. K., Polynomial Operator Equations in Abstract Spaces and Applications, CRC Press LLC, Boca Raton, Florida, 1998.

Argyros, I. K. and Szidarovszky, F., The theory and Applications of Iteration Methods, C.R.C. Press, Boca Raton, Florida, 1993.

Balazs, M. and Goldner, G., On the approximate solutions of equations in Hilbert space by a Steffensen-type method, Rev. Anal. Numér. Théor. Approx., 17, no. 1, pp. 19-23, 1998, http://ictp.acad.ro/jnaat/journal/article/view/1988-vol17-no1-art3

Cătinaş, E., On some Steffensen-type iterative methods for a class of nonlinear equations, Rev. Anal. Numér. Théor. Approx., 24, nos. 1-2, pp. 37-43, 1995, http://ictp.acad.ro/jnaat/journal/article/view/1995-vol24-nos1-2-art4

Cătinaş, E. and Păvăloiu, I., On some interpolatory iterative methods for the second degree polynomial operators (I), Rev. Anal. Numér. Théor. Approx., 27, no. 1, pp. 33-45, 1998, http://ictp.acad.ro/jnaat/journal/article/view/1998-vol27-no1-art5

Cătinaş, E. and Păvăloiu, I., On some interpolatory iterative methods for the second degree polynomial operators (II), Rev. Anal. Numér. Théor. Approx., 28, no. 2, pp. 133-144, 1999, http://ictp.acad.ro/jnaat/journal/article/view/1999-vol28-no2-art4

Păvăloiu, I., Introduction to the Theory of Approximating the Solutions of Equations, Ed. Dacia, Cluj, 1986 (in Romanian).

Păvăloiu, I., Optimal efficiency index for iterative methods of interpolatory type, Computer Science Journal of Moldova, 5, 1 (13), pp. 20-43, 1997.

Păvăloiu, I., Approximation of the roots of equations by Aitken-Steffensen-type monotonic sequence, Calcolo, 32, nos. 1-2, pp. 69-82, 1995, https://doi.org/10.1007/BF02576543. DOI: https://doi.org/10.1007/BF02576543

Păvăloiu, I., Sur la méthode de Steffensen pour la résolution des équations opèrationnelles non linéaires, Rev. Roum. Math. Pure et Appl., XIII, no. 6, pp. 857-861, 1968.

Păvăloiu, I., Sur une généralisation de la methode de Steffensen, Rev. Anal. Numér. Théor. Approx., 21, no. 1, pp. 56-65, 1992, http://ictp.acad.ro/jnaat/journal/article/view/1992-vol21-no1-art8

Păvăloiu, I., Bilateral approximations for the solutions of scalar equations, Rev. Anal. Numér. Théor. Approx., 23, no. 1, pp. 95-100, 1994, http://ictp.acad.ro/jnaat/journal/article/view/1994-vol23-no1-art10

Păvăloiu, I., On the monotonicity of the sequence of approximations obtained by Steffensen method, Mathematica (Cluj), 35 (58), pp. 171-176, 1993.

Păvăloiu, I., On a Halley--Steffensen method for approximating the solutions of scalar equations, Rev. Anal. Numér. Théor. Approx., 30, no. 1, pp. 69-74, 2001.

Popoviciu, T., Sur la délimitation de l'erreur dans l'approximation des racines d'une équation par interpolation linéaire ou quadratique, Rev. Roumaine Math. Pures Appl., 13, pp. 75-78, 1968.