Iterative methods for simultaneous inclusion of polynomial zeros
type: Item
instance of: scholarly article
title: Iterative methods for simultaneous inclusion of polynomial zeros
zbMATH Open document ID: 0689.65028
published in: Lecture Notes in Mathematics
publication date: 1993-06-05T00:00:00Z
review text: This book collects, reviews, unifies and augments many interesting developments in the solution of polynomial equations. Galois' famous theorem concerning polynomial zeros states that a general direct method in terms of explicit formulas exist only for polynomials of degree less than five. Because of that, for finding zeros of higher degree polynomials one has to apply numerical methods. These methods, which generally take the form of an iterative process, give rise to the questions: How close is the current iterate to the solution, how fast does the iteration converge, can we trust the computed numbers? Consider a polynomial \(p(z)=a_ 0+a_ 1z...a_ nz^ n\) with \(a_ n\neq 0\) having k zeros \(\xi_ 1,...,\xi_ k\) with multiplicities \(\mu_ 1,...,\mu_ k\). Let us pose the task: For given \(a_ 0,...,a_ n\) and given \(\epsilon >0\) compute the number k, compute complex numbers \(z_ 1,...,z_ k\) such that \(| z_ i-\xi_ i| \leq \epsilon\) for \(i=1,...,k\), and compute \(\mu_ 1,...,\mu_ k\). There seems to exist no software package that does the job, although (1) the mathematics concerning polynomials is well understood, (2) the numerical analysis literature abounds with polynomial zero finders, (3) nowadays PC's and mainframes facilitate tremendous power, and (4) the problem is of far reaching relevance for practical purposes. The book under consideration presents quite a lot of iterative methods aiming at the task stated. The condition \(| z_ i-\xi_ i| \leq \epsilon\) leads to the use of disks. The typical step in the total step methods is: From a collection of k disks \(Z_ 1,...,Z_ k\) with \(\xi_ i\in Z_ i\) a new collection of k disks \(Z^*_ 1,...,Z^*_ k\) is constructed, \[ Z^*_ i:=F_ i(Z_ 1,...,Z_ k) \] for \(i=1,...,k,\) such that \(\xi_ i\in Z^*_ i.\) The corresponding single step methods have the form \[ Z^*_ i:=F_ i(Z^*_ i,...,Z^*_{i-1},Z_ i,...,Z_ k) \] for \(i=1,...,k.\) Convergence conditions and the order of convergence are investigated intensively. Most o…
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 65G30
Mathematics Subject Classification ID: 65-02
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 194795
zbMATH Keywords: interval analysis
zbMATH Keywords: reserach survey
zbMATH Keywords: polynomial equations
zbMATH Keywords: polynomial zeros
zbMATH Keywords: iterative methods
zbMATH Keywords: total step methods
zbMATH Keywords: single step methods
zbMATH Keywords: Convergence conditions
zbMATH Keywords: order of convergence
zbMATH Keywords: rectangular intervals
zbMATH Keywords: clusters of zeros
zbMATH Keywords: point iteration
zbMATH Keywords: disk iteration
zbMATH Keywords: numerical stability
zbMATH Keywords: computational efficiency
zbMATH Keywords: numerical tests
author: M. S. Petković
reviewed by: Herbert Fischer
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0083599
OpenAlex ID: W633035909
DOI: 10.1007/BFB0083599
description: scientific article; zbMATH DE number 194795