A family of Steffensen type methods with seventh-order convergence
type: Item
instance of: scholarly article
title: A family of Steffensen type methods with seventh-order convergence
zbMATH Open document ID: 1276.65028
author: Xiaofeng Wang
author: Tie Zhang
published in: Numerical Algorithms
publication date: 2013-03-27T00:00:00Z
review text: The authors develop a new family of seventh-order derivative-free methods for finding the root of nonlinear equations \(f(x)=0\). They consider a new iteration scheme \(y_n=x_n-f(x_n)/f[x_n,z_n]\), \(u_n=g_4(x_n)\), \(x_{n+1}=u_n-f(u_n)/(f[u_n,x_n]+f[u_n,y_n]-f[x_n,y_n])\), where \(g_4(x_n)\) is an optimal fourth-order Steffensen type method, and the third step is constructed by using suitable Taylor expansions and divided difference approximations. The new method is proven to be seventh-order. Finally, the authors give some numerical examples to illustrate the convergence of the method.
Mathematics Subject Classification ID: 65H05
zbMATH DE Number: 6149002
zbMATH Keywords: Steffensen's method
zbMATH Keywords: Newton's method
zbMATH Keywords: derivative free
zbMATH Keywords: seventh-order convergence
zbMATH Keywords: root-finding
zbMATH Keywords: nonlinear equation
zbMATH Keywords: Taylor expansion
zbMATH Keywords: divided difference
zbMATH Keywords: numerical example
reviewed by: Przemysław Stpiczyński
MaRDI profile type: Publication
full work available at URL: s11075-012-9597-3
OpenAlex ID: W1978037138
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DOI: 10.1007/S11075-012-9597-3
description: scientific article; zbMATH DE number 6149002