Abstract

We study a general Steffensen type method based on the inverse interpolation Lagrange polynomial of second degree. We show how the auxiliary functions may be constructed and we analyze some conditions on them which lead to monotone approximations. We obtain some local convergence results, which are illustrated by some numerical examples.

Author

Keywords

nonlinear equations in R; Steffensen type method; inverse interpolation Lagrange polynomial of second degree; monotone iterations; local convergence; numerical examples.

Cite this paper as:

I. Păvăloiu, E. Cătinaş, On a Steffensen type method, IEEE Proceedings, 2007, pp. 369-375.

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About this paper

Journal

Ninth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC 2007)

Publisher Name

IEEE

ISBN

978-0-7695-3078-8

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References

[1] I. K. Argyros. An error analysis for Steffensen’s method. Panamer. Math. J., 10(4):27–33, 2000.

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

[3] B. A. Bel’tyukov. An analogue of the Aitken-Steffensen method with a controllable step (in russian). Zh. Vychisl. Mat. i Mat. Fiz., 27(6):803–817, 1987.

[4] C. Iancu, I. Pavaloiu, and I. Serb. Methodes it erative optimales de type Steffensen obtenues par interpolation inverse. Research Seminar on Functional Analysis and Numerical Methods, Preprint, 1:81–88, 1983.

[5] W. L. Johnson and R. D. Scholz. On Steffensen’s method. SIAM J. Numer. Anal., 5(2):296–302, 1968.

[6] M. A. Ostrowski. Solution of Equations and Systems of Equations. Academic Press, New York, 1980.

[7] I. Pavaloiu. Solutions of Equations by Interpolation (in Romanian). Dacia, Cluj-Napoca, Romania, 1981.

[8] I. Pavaloiu. Optimal problems concerning interpolation methods of solution of equations. Publ. l’Inst. Math. Beograd, 52 (66):113–126, 1992.

[9] I. Pavaloiu. On the monotonicity of the sequences of approximations obtained by Steffensen’s method. Mathematica (Cluj), 35 (58)(1):71–76, 1993.

[10] I. Pavaloiu. Bilateral approximations for the solutions of scalar equations. Rev. Anal. Numer. Theor. Approx. , 23(1):95–100, 1994.

[11] I. Pavaloiu. Approximation of the roots of equations by Aitken-Steffensen-type monotonic sequences. Calcolo, 32(1-2):69–82, 1995.

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

[13] J. R. Sharma. A composite third order Newton-Steffensen method for solving nonlinear equations. Appl. Math. Comput., 169(1):242–246, 2005.

[14] J. F. Traub. Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs, N.J., 1964.

[15] B. A. Turowicz. Sur les derivees d’ordre superieur d’une fonction inverse. Ann. Polon. Math., 8:265–269, 1960.

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