## Abstract

We show that the Steffensen method for solving the scalar equation \(f(x)=0\), applied to equation \(h(x)=\frac{f(x)}{\sqrt{f'(x)}}=0\), leads to bilateral approximations for the solution. Moreover, the convergence order is at least 3, i.e. as in the case of the Halley method.

## Author

## Keywords

nonlinear equations in R; Steffensen method; Halley method; monotone iterations.

## References

[1] Adi Ben-Israel,

*Newton’s method with modi ed functions*, Contemp. Math., 204 (1997),pp. 39-50.[2] G. H. Brown, Jr.,

*On Halley’s variation of Newton’s method*, Amer. Math. Monthly, 84 (1977), 726{728.[3] V. Candela and A. Marquina,

*Recurrence relations for rational cubic methods I: the Halley’s method*, Computing 44 (1990), 169{184.[4] G. Deslauries and S. Dubuc,

*Le calcul de la racine cubique selon Heron*, El. Math., 51 (1996), 28{34.[5] D. Luca and I. Pavaloiu,

*On the Heron’s method for the approximation of the cubic root of a real number*, Rev. Anal. Numer. Theor. Approx., 28 (1997), 103{108.[6] A. M. Ostrowski,

*The Solution of Equations and Systems of Equations*, Academic Press, New York-London, 1960.[7] I. Pavaloiu,

*On the monotonicity of the sequences of approximations obtained by Steffensen’s method*, Mathematica (Cluj) 35 (58) (1993), 71-76.[8] I. Pavaloiu,

*Approximation of the roots of equations by Aitken-Ste ensen-type monotonic sequences*, Calcolo, 32 (1995), 69{82.[9] T. Popoviciu,

*Sur la delimitation de l’erreur dans l’approximation des racines d’une equation par interpolation lineaire ou quadratique*, Rev. Roumaine Math. Pures Appl., 13 (1968), 75-78.Scanned paper.

PDF-LaTeX version of the paper (soon).

## About this paper

##### Cite this paper as:

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

##### Publisher Name

##### Article on the journal website

##### Print ISSN

1222-9024

##### Online ISSN

2457-8126