For which error criteria can we solve nonlinear equations?
type: Item
instance of: scholarly article
title: For which error criteria can we solve nonlinear equations?
zbMATH Open document ID: 0609.65035
DOI: 10.1016/0885-064X(86)90017-8
author: B. George
published in: Journal of Complexity
publication date: 1986-01-01T00:00:00Z
review text: For the solution of an equation \(f(x)=0\), involving a real function f on an interval [a,b], the information on f is assumed to consist of sequential (adaptive) evaluations of arbitrary functionals. The error of an algorithm based on these evaluations is defined by its worst performance. For the root criterion as error measure it is shown that there does not exist an algorithm for finding a point x such that \(| x-\alpha | \leq \epsilon\) where \(\alpha\) is a zero of f and \(\epsilon <(b-a)/2\). This holds for arbitrary information and infinitely differentiable f with simple zeros, but without assuming f(a)f(b)\(\leq 0\). For the residual criterion some almost optimal information and algorithm is given and it is shown that for n functional evaluations the residual at the computed point x satisfies \(f(x)=O(n^{-r})\) as \(n\to \infty\), where r defines the smoothness class of f. These results are then generalized to a more general error criterion.
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 68Q25
zbMATH DE Number: 3986541
zbMATH Keywords: adaptive information
zbMATH Keywords: worst performance
zbMATH Keywords: root criterion
zbMATH Keywords: almost optimal information
zbMATH Keywords: error criterion
reviewed by: Werner C. Rheinboldt
MaRDI profile type: Publication
cites work: Q4078976
cites work: The optimal recovery of smooth functions
cites work: Bisection is optimal
cites work: Optimal solution of nonlinear equations satisfying a Lipschitz condition
cites work: Optimal solution of nonlinear equations
cites work: Q5535819
cites work: Q3883494
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cites work: Average case optimality
full work available at URL: 0885-064x(86)90017-8
OpenAlex ID: W2135596970
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description: scientific article; zbMATH DE number 3986541