A modified Newton method in parallel circular iteration of single-step and double-step
type: Item
instance of: scholarly article
title: A modified Newton method in parallel circular iteration of single-step and double-step
zbMATH Open document ID: 1088.65046
author: Ling Zhu
publication date: 2006-02-02T00:00:00Z
review text: The Newton method is a classical method for finding zeros of a monic polynomial, \(f(z)\), of degree \(n>1\) with complex coefficients. The well-known iteration \(z^{k+1}=z^k-f(z^k)/f'(z^k)\) is of order \(2\) if the zero is simple. Another higher-order iteration which can be applied to find simultaneously all zeros of \(f(z)\) is constructed as \(z_i^{k+1}=z_i^k-1/[f'(z_i^k)/f(z_i^k)-\sum_{j\not=i}1/(z_i^k-z_j^k)].\) Furthermore, a circular iteration corresponding to this construction can be formed as \(Z_i^{k+1}=z_i^k-1/[f'(z_i^k)/f(z_i^k)-\sum_{j\not=i}1/(z_i^k-Z_j^k)].\) This construction is called modified Newton method in parallel circular iteration. In this paper, two theorems for the convergence of this method are given. The convergence condition of single-step method's circular iteration is relaxed compared with the classical theorem in the same field, while the one of the first proposed double-step method of the fifth order is obtained accurately as well.
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 12Y05
Mathematics Subject Classification ID: 65Y05
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 5003140
zbMATH Keywords: Zeros of a polynomial
zbMATH Keywords: Convergence condition
zbMATH Keywords: Single-step method
zbMATH Keywords: Double-step method
zbMATH Keywords: Complex coefficients
reviewed by: Sonia Pérez Díaz
MaRDI profile type: Publication
full work available at URL: j.camwa.2005.08.023
OpenAlex ID: W2032559632
cites work: A modified Newton method for polynomials
cites work: Further Applications of Circular Arithmetic: Schroeder-Like Algorithms with Error Bounds for Finding Zeros of Polynomials
DOI: 10.1016/J.CAMWA.2005.08.023
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 5003140