On a global descent method for polynomials
type: Item
instance of: scholarly article
title: On a global descent method for polynomials
zbMATH Open document ID: 0553.65028
DOI: 10.1007/BF01389468
published in: Numerische Mathematik
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/132965
review text: For a given complex polynomial p of degree n the author derives a function \(f=f_ p\) with the property \(| p(f(z))| \leq \kappa (z)| p(z)|,\) \(\kappa (z)<1\) in \({\mathbb{C}}\), and then uses the descent algorithm \(z_{k+1}=f(z_ k)\) with an arbitrary starting point \(z_ 0\) for the construction of a zero of p. A key role in the underlying theory is played by the ''Cauchy radius of p at z'' which is defined as the non-negative solution of the equation \[ \sum^{n}_{k=1}| p^{(k)}(z)| x^ k/k!=| p(z)|. \] There is a detailed discussion of numerical aspects of the associated root-finding algorithm (which is based on a combination of the global descent method with the local Newton method) as well as a report on results of some numerical experiments.
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 3881777
zbMATH Keywords: zeros of polynomials
zbMATH Keywords: complex polynomial
zbMATH Keywords: descent algorithm
zbMATH Keywords: Cauchy radius
zbMATH Keywords: root-finding algorithm
zbMATH Keywords: Newton method
zbMATH Keywords: numerical experiments
author: Stephan Ruscheweyh
reviewed by: Gerhard Merz
MaRDI profile type: Publication
cites work: Q5577953
cites work: Q4072022
cites work: Q3040409
cites work: Ein konvergentes Iterationsverfahren zur Bestimmung der Nullstellen eines Polynoms
cites work: Die numerische Berechnung der Wurzeln eines Polynoms
cites work: Q5599275
cites work: On the Number of Multiplications for the Evaluation of a Polynomial and Some of Its Derivatives
description: scientific article; zbMATH DE number 3881777