# On a Halley-Steffensen method for approximating the solutions of scalar equations

## 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.

## 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.

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##### 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.

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