Can we approximate zeros of functions with nonzero topological degree?
type: Item
instance of: scholarly article
title: Can we approximate zeros of functions with nonzero topological degree?
zbMATH Open document ID: 0671.65037
DOI: 10.1016/0885-064X(88)90013-1
author: S. H. Smith
published in: Journal of Complexity
publication date: 1988-01-01T00:00:00Z
review text: The topological degree of a scalar continuous function f, defined on an interval [a,b] of the real-axis, decides whether we can or cannot solve the problem of finding an \(\epsilon\)-approximation \(x^*\), \(| x^*- \alpha (f)| \leq \epsilon\), to a zero \(\alpha\) (f) of f, since if we assume that \(f(a)f(a)<0\) then \(\deg (f,[a,b],0)=(sgn(f(b))- sgn(f(a)))/2\neq 0\) and, as the second author has proved [Numer. Math. 40, 111-117 (1982; Zbl 0492.65027)], the bisection method for finding such an approximation is optimal. If the degree is zero then, in general, there exists no algorithm using linear information of f to find \(x^*.\) The authors are able to show that the situation drastically changes when we add just one more dimension. They prove that, in general, it is impossible to find an \(\epsilon\)-approximation to a zero of a smooth bivariate function with nonzero topological degree.
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 55M25
zbMATH DE Number: 4098625
zbMATH Keywords: topological degree
zbMATH Keywords: continuous function
zbMATH Keywords: bisection method
zbMATH Keywords: smooth bivariate function
reviewed by: Dimitrie D. Stancu
MaRDI profile type: Publication
cites work: Simplicial and Continuation Methods for Approximating Fixed Points and Solutions to Systems of Equations
cites work: Q4112523
cites work: Q4744214
cites work: Q5652137
cites work: Bisection is optimal
cites work: For which error criteria can we solve nonlinear equations?
cites work: Extensions of Lemke's algorithm for the linear complementarity problem
cites work: Q3883494
cites work: Q3874718
description: scientific article; zbMATH DE number 4098625