On some Aitken-Steffensen-Halley-type methods for approximating the roots of scalar equations

Ion Păvăloiu

doi:10.33993/jnaat302-697

Authors

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

DOI:

https://doi.org/10.33993/jnaat302-697

Abstract views: 205

Abstract

In this note we extend the Aitken-Steffensen method to the Halley transformation. Under some rather simple assumptions we obtain error bounds for each iteration step; moreover, the convergence order of the iterates is 3, i.e. higher than for the Aitken-Steffensen case.

Downloads

Download data is not yet available.

References

A. Ben-Israel, Newton's method with modified functions, Contemp. Math., 204, pp. 39-50, 1997, https://doi.org/10.1090/conm/204/02621 DOI: https://doi.org/10.1090/conm/204/02621

G.H. Brown, Jr., On Halley's variation of Newton's method, Amer. Math. Monthly, 84, pp. 726-728, 1977. DOI: https://doi.org/10.1080/00029890.1977.11994468

V. Candela and A. Marquina, Recurrence relations for rational cubic methods I: The Halley's method, Computing, 44, pp. 169-184, 1990, https://doi.org/10.1007/bf02241866 DOI: https://doi.org/10.1007/BF02241866

G. Deslauries and S. Dubuc, Le calcul de la racine cubique selon Héron, El. Math., 51, pp. 28-34, 1996.

W.F. Ford and J.A. Pennline, Accelerated convergence in Newton method, SIAM Rev., 38, pp. 658-659, 1996, https://doi.org/10.1137/s0036144594292972 DOI: https://doi.org/10.1137/S0036144594292972

J. Gerlach, Accelerated convergence in Newton's method, SIAM Rev., 36, pp. 272-276, 1994, https://doi.org/10.1137/1036057 DOI: https://doi.org/10.1137/1036057

D. Luca and I. Păvăloiu, On the Heron's method for the approximation of the cubic root of a real number, Rev. Anal. Numér. Théor. Approx., 28, pp. 103-108, 1997.

A. Melman, Geometry and convergence of Euler's and Halley's methods, SIAM Rev., 39, pp. 728-735, 1997, https://doi.org/10.1137/s0036144595301140 DOI: https://doi.org/10.1137/S0036144595301140

A.M. Ostrowski, The Solution of Equations and Systems of Equations, Academic Press, New York-London, 1960.

I. Păvăloiu, On the monotonicity of the sequences of approximations obtained by Steffensen method, Mathematica (Cluj), 35 (58), pp. 171-76, 1993.

I. Păvăloiu, Approximation of the roots of equations by Aitken--Steffensen-type monotonic sequences, Calcolo, 32, pp. 69-82, 1995, https://doi.org/10.1007/bf02576543 DOI: https://doi.org/10.1007/BF02576543

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

T. Popoviciu, 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.