Relationships between order and efficiency of a class of methods for multiple zeros of polynomials
type: Item
instance of: scholarly article
title: Relationships between order and efficiency of a class of methods for multiple zeros of polynomials
zbMATH Open document ID: 0837.65048
DOI: 10.1016/0377-0427(94)00086-G
author: Masao Igarashi
author: Tjalling J. Ypma
published in: Journal of Computational and Applied Mathematics
publication date: 1995-01-01T00:00:00Z
review text: Newton's method is a well-known iterative method for solving polynomial equations. Let \(p(z)\) be a complex polynomial of degree \(n= \deg(p)\). Newton's method is given by the function \(N_1(z)= z- p(z)/p'(z)\). For an initial approximation \(z_0\in \mathbb{C}\) of some root \(w\) of \(p\) the iterative algorithm \(z_{\nu+ 1}= N_1(z_\nu)\) generates a sequence of successive approximations \(z_\nu\), \(\nu= 0, 1,\dots\), converging to \(w\). The authors study the efficiency of Newton's method and its higher order analogs. Let \(m\) denote the multiplicity of \(w\) and \(k\) the order of the iterative method. If \(k\leq m\) then convergence analysis gives \(|z_{\nu+ 1}- w|\approx (1- {k- 1\over m+ k- 2})|z_\nu- w|\). From this the authors derive that for \(k\geq 5\) the number of required iterations decreases as the multiplicity \(m\) increases. Furthermore, the analysis as well as practical experience indicate the following optimal choices for \(k: k\approx 1.5\sqrt m\) when \(z_0\) is close to a root of multiplicity \(m> 1\) and \(k\approx 1.5\sqrt n\) when far from any root of \(p\).
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 12Y05
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 808365
zbMATH Keywords: complex polynomials
zbMATH Keywords: efficiency
zbMATH Keywords: multiple roots
zbMATH Keywords: Newton's method
zbMATH Keywords: order
zbMATH Keywords: convergence
reviewed by: Hartje Kriete
MaRDI profile type: Publication
cites work: Iteration Methods for Finding all Zeros of a Polynomial Simultaneously
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description: scientific article; zbMATH DE number 808365