Optimal Steffensen type iterative methods obtained by inverse interpolation

Let \(f:I\subset \mathbb{R\rightarrow R}\) be a nonlinear mapping and the equation \(f\left( x\right) =0\) with solution \(x^{\ast}\); consider the equivalent equations \(\varphi_{i}\left( x\right) =x\), \(i=1,…,n+1\). Given \(x_{k}\) an approximation to \(x^{\ast}\), we consider the following nodes for the Hermite interpolation polynomial \[x_{k}^{1}=\varphi_{1}(x_{k}) ,\ \ x_{k}^{2}=\varphi_{2}(x_{k} ^{1}), \ldots, \ \ x_{k}^{n+1}=\varphi_{n+1}(x_{k}^{n}). \] Assume that the nodes \(x_{k}^{i},i=1,…,n+1\) have the multiplicity orders resp. \(\alpha_{i},i=1,…,n+1\). Moreover, the convergence orders of the successive iterations for \(\varphi_{i}\) are resp. \(p_{i},p_{i}\in \mathbb{N},p_{i}\geq1,i=1,…,n+1\). The iterative method obtained from the inverse interpolation Hermite polynomial is a Steffensen type method. If we permute the multiplicity orders of the nodes and the assumed convergence orders, we obtain class of iterative methods. Among this class we determine the methods with the highest convergence orders.

Crăciun Iancu, Ion Păvăloiu, Ioan Şerb

Original title (in French)

Méthodes itératives optimales de type Steffensen obtenues par interpolation invèrse

English translation of the title

Optimal Steffensen type iterative methods obtained by inverse interpolation

Hermite inverse interpolation; Steffensen type methods; iterative methods; nonlinear equations in R; convergence order

[1] C. Iancu, I. Pavaloiu, La resolution des equations par interpolation inverse de type Hemite. Seminar of Functional and Numerical Methods, “Babes-Bolyai” University, Faculty de Matematica, Research Seminaries, Preprint Nr. 4 (1981), 72-84.

[2] I. Pavaloiu, Rezolvarea ecuatiilor prin interpolare. Editura Dacia, Cluj-Napoca, 1981.

[3] S. Popa, Asupra unei probleme a lui E. Erdos si G. Weiss, Studii si cercetari matematice, 33, 5 (1981), 539-542.

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C. Iancu, I. Păvăloiu, I. Şerb, Méthodes itératives optimales de type Steffensen obtenues par interpolation invèrse, Seminar on functional analysis and numerical methods, Preprint no. 1 (1983), pp. 81-88 (in French).

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