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

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.

Author

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Keywords

Aitken-Steffenssen methods; Lagrange inverse interpolation

References

[1] Balazs, M., A bilateral approximating method for finding the real roots of real equations, Rev. Anal. Numer. Theor. Approx., 21 (2), pp. 111-117, 1992.

[2] Casulli, V., Trigiante, D.,  The convergence order for iterative multipoint procedures, Calcolo, 13 (1), pp. 25-44, 1997.

[3] 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.

[4] 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.

[5] Grau, M., An improvement to the computing of nonlinear equation solutions, Numer. Algorithms., 34, pp. 1-12, 2003.

[6] Ostrowski, A., Solution of Equations in Euclidian and Banach Spaces, Academic Press, New York and London, 1973.

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

[8] Păvăloiu I.,  Approximation of the roots of equations by Aitken-Steffensen-type monotonic sequences, Calcolo, 32 (1-2), pp. 69-82, 1995.

[9] Păvăloiu I., Optimal problems concerning interpolation methods of solution of equations, Publications de L’Institut Mathematique, 52 (66), pp. 113-126, 1992.

[10] Păvăloiu I., Optimal effiency index of a class of Hermite iterative methods, with two steps, Rev. Anal. Numer. Theor. Approx., 29 (1), pp. 83-89, 2000.

[11] Păvăloiu I., Local convergence of general Steffensen type methods, Rev. Anal. Numer. Theor. Approx., 33 (1), pp. 79-86, 2004.

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

[13] Păvăloiu I., On a Steffensen-Hermite-type Method for approximating the solution of nonlinear equations, Rev. Anal. Numer. Theor. Approx., 25 1, pp. 87-94, 2006.

[14] Păvăloiu I.,  Bilateral approximation of solutions of equations by order-three Steffensen type methods, Studia Univ. “Babeș-Bolyai”, Mathematica, Vol. LI, no. 3, pp. 105-114, 2006.

[15] Traub, J.F., Iterative Methods for Solutions of Equations, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1964.

[16] Turowicz, B. A., Sur les derivees d’ordre superieur d’une function inverse, Ann. Polon. Math., 8, pp. 265-269, 1960.

PDF

Scanned paper.

PDF-LaTeX version of the paper (soon).

About this paper

Cite this paper as:

I. Păvăloiu, Bilateral approximations of the roots of scalar equations by Lagrange-Aitken-Steffensen method of order three, Rev. Anal. Numér. Théor. Approx., 35 (2006) no. 2, pp. 173-182.

Print ISSN

1222-9024

Online ISSN

2457-8126

Google Scholar Profile

Related Posts

Menu