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


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.


Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)


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


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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[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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