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Zbl 1227.65044
Zheng, Quan; Li, Jingya; Huang, Fengxi
An optimal Steffensen-type family for solving nonlinear equations. (English)
[J] Appl. Math. Comput. 217, No. 23, 9592-9597 (2011). ISSN 0096-3003

The paper is devoted to the numerical solution of nonlinear equations $f(x) = 0$. The authors use Newton's iteration for the direct Newtonian interpolation of the function to construct optimal Steffensen-type methods of second-, fourth- and eighth-order, which use only two, three and four evaluations of the function, respectively. Moreover, they deduce the corresponding error equations and asymptotic convergence constants. The proposed general optimal Steffensen-type family only uses $n$ evaluations of $f$ to achieve the optimal $2^{n-1}$th order of convergence for solving the simple root of nonlinear functions, and the authors compare this family with Newton's method, Steffensen's method, Ren-Wu-Bi's method, Kung-Traub's method and Neta-Petković's method for solving nonlinear equations in numerical examples.
[Luis Filipe Pinheiro de Castro (Aveiro)]

MSC 2000:: *65H05 Single nonlinear equations (numerical methods)

Keywords: nonlinear equation; iterative method; Newton's method; Steffensen's method; derivative free method; optimal convergence; comparison of methods; asymptotic convergence constants; Ren-Wu-Bi's method; Kung-Traub's method; Neta-Petković's method; numerical examples

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Zentralblatt MATH Berlin [Germany]