On the convergence order of accelerated root iterations
type: Item
instance of: scholarly article
title: On the convergence order of accelerated root iterations
zbMATH Open document ID: 0537.65041
DOI: 10.1007/BF01405575
published in: Numerische Mathematik
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/132949
review text: A Gauß-Seidel procedure for accelerating the convergence of a generalized method of the root iterations type of \((k+2)\)-th order (\(k\in N)\) for finding polynomial complex zeros, introduced by the first author [Computing 27, 37-55 (1981; Zbl 0442.65030)], is considered. It is shown that the R-order of convergence of the accelerated method is at least \(k+1+\sigma_ n(k)\), where \(\sigma_ n(k)>1\) is the unique positive zero of the equation \(\sigma^ n-\sigma -k-1=0\) and n is the polynomial degree. The convergence analysis is performed using circular arithmetic and it is applicable to the interval simultaneous methods in terms of circular regions as well as to the ordinary (noninterval) simultaneous methods. In the second case, for \(k=1\), the results due to \textit{G. Alefeld} and \textit{J. Herzberger} [SIAM J. Numer. Anal. 11, 237-243 (1974; Zbl 0282.65038)] are obtained. The cases of multiple zeros and clusters of zeros are also discussed. Examples of algebraic equations in ordinary and circular arithmetic are given.
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 65G30
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 3854315
zbMATH Keywords: convergence acceleration
zbMATH Keywords: Gauß-Seidel procedure
zbMATH Keywords: root iterations
zbMATH Keywords: polynomial complex zeros
zbMATH Keywords: order of convergence
zbMATH Keywords: circular arithmetic
zbMATH Keywords: multiple zeros
zbMATH Keywords: clusters of zeros
author: M. S. Petković
author: Lidija V. Stefanović
MaRDI profile type: Publication
cites work: Q4099123
cites work: On the Convergence Speed of Some Algorithms for the Simultaneous Approximation of Polynomial Roots
cites work: Circular arithmetic and the determination of polynomial zeros
cites work: Parallel Laguerre iterations: The complex case
cites work: Q4060267
cites work: Q5652137
cites work: On a generalisation of the root iterations for polynomial complex zeros in circular interval arithmetic
cites work: Generalised Root Iterations for the Simultaneous Determination of Multiple Complex Zeros
OpenAlex ID: W1986516087
description: scientific article; zbMATH DE number 3854315