Weierstrass formula and zero-finding methods
type: Item
instance of: scholarly article
title: Weierstrass formula and zero-finding methods
zbMATH Open document ID: 0821.65028
author: M. S. Petković
author: Miroslav Trajković
author: C. Carstensen
published in: Numerische Mathematik
publication date: 1995-09-27T00:00:00Z
review text: For a monic polynomial \(P(z) = \prod_{j=1}^ n (z - \zeta_ j)\) of degree \(n \geq 3\), the identity \[ \zeta_ i = z - {P(z) \over \prod_{j=1,j \neq i}^ n(z - \zeta_ j)}, \quad i = 1, \dots, n, \] gives rise to the iterative process \[ z_ i^{(m+1)} = z_ i^{(m)} - W(z_ i^{(m)}),\quad W(z_ i^{(m)}) = {P(z_ i^{(m)}) \over \prod_{j=1,j \neq i}^ n (z_ i^{(m)} - z_ j^{(m)})},\;i = 1, \dots, n;\;m = 0,1, \dots \] known as Weierstrass' formula. The points \((z_ 1^{(m)}, \dots, z_ n^{(m)})\) simultaneously approximate the zeros of \(P(z)\), and the process presents quadratic convergence. This formula has been subject of several investigations (see the references given in the paper). The authors present some further applications of this method. A new result concerning localization of polynomial zeros, based on Weierstrass' correction \(W(z)\) is proved; this is used for the construction of inclusion disks which are necessary for the application of inclusion methods. Conditions for the convergence of Weierstrass' process, depending only on initial approximations and the degree of \(P(z)\) are given, and two hybrid methods (that combine both ordinary and circular complex arithmetic) for the inclusion of polynomial zeros are constructed. For the interval version of this method, a procedure for finding circular enclosures of the sets containing all zeros of \(P(z)\) is obtained. Finally, an iterative method of Weierstrass' type for the simultaneous finding of the zeros of a class of analytic functions is studied, along with the analysis of its convergence and stability. The algorithms are illustrated with several numerical examples.
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 65E05
Mathematics Subject Classification ID: 65G30
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 12Y05
zbMATH DE Number: 739416
zbMATH Keywords: zero-finding methods
zbMATH Keywords: interval arithmetic
zbMATH Keywords: Weierstrass' formula
zbMATH Keywords: quadratic convergence
zbMATH Keywords: localization of polynomial zeros
zbMATH Keywords: inclusion disks
zbMATH Keywords: inclusion methods
zbMATH Keywords: circular complex arithmetic
zbMATH Keywords: iterative method
zbMATH Keywords: analytic functions
zbMATH Keywords: stability
zbMATH Keywords: numerical examples
reviewed by: Andrei Martínez Finkelshtein
MaRDI profile type: Publication
full work available at URL: s002110050097
OpenAlex ID: W2153196903
Wikidata QID: Q29026800
DOI: 10.1007/S002110050097
description: scientific article; zbMATH DE number 739416