On some simultaneous methods based on Weierstrass' correction
type: Item
instance of: scholarly article
title: On some simultaneous methods based on Weierstrass' correction
zbMATH Open document ID: 0859.65048
DOI: 10.1016/0377-0427(95)00278-2
author: Tetsuya Sakurai
author: M. S. Petković
published in: Journal of Computational and Applied Mathematics
publication date: 1997-04-07T00:00:00Z
review text: The authors consider iteration methods for the simultaneous approximation of all zeros of a polynomial \(f(z)\) with complex coefficients. All methods are based on the Weierstrass correction \(W_i(z)=f(z)/\prod^n_{\substack{ k=1\\ k\neq i}} (z-z_k)\), which appears in the iteration formula \(\widehat z_i=z_i- W_i(z_i)\), \(i=1,2,\dots,n\). The \(z_i\) are approximations to the zeros of \(f(z)\). Simultaneous methods of order \(q+1\) are derived from a one point method of order \(q\) by replacing \(f(z)\) by \(W_i(z)\), \(f'(z)\) by \(W_i'(z)\), etc. For example, for Newton's method \((q=2)\), the corresponding simultaneous method is the third-order Maehly-Ehrlich-Aberth method. Simultaneous multistep methods, based on rational interpolation, are then discussed, and the paper concludes by presenting a new combined interval method based on the secant method and the \(W\)-correction. Convergence analysis is included and some numerical examples presented.
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 12Y05
zbMATH DE Number: 932485
zbMATH Keywords: zeros of polynomials
zbMATH Keywords: convergence
zbMATH Keywords: simultaneous methods
zbMATH Keywords: iteration methods
zbMATH Keywords: Weierstrass correction
zbMATH Keywords: Newton's method
zbMATH Keywords: Maehly-Ehrlich-Aberth method
zbMATH Keywords: multistep methods
zbMATH Keywords: rational interpolation
zbMATH Keywords: interval method
zbMATH Keywords: secant method
zbMATH Keywords: numerical examples
reviewed by: Adrian Swift
describes a project that uses: PASCAL-XSC
MaRDI profile type: Publication
full work available at URL: 0377-0427(95)00278-2
OpenAlex ID: W2024328969
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description: scientific article; zbMATH DE number 932485