Minimum asymptotic error of algorithms for solving ODE
type: Item
instance of: scholarly article
title: Minimum asymptotic error of algorithms for solving ODE
zbMATH Open document ID: 0662.65066
DOI: 10.1016/0885-064X(88)90016-7
published in: Journal of Complexity
publication date: 1988-01-01T00:00:00Z
review text: This paper extends previous results of the author [Numer. Math. 45, 93- 104 (1984; Zbl 0527.65055) and J. Complexity 3, 451-465 (1987; Zbl 0643.65033)] concerned with the error of some algorithms for the numerical solution ordinary differential operations. Here the error is to be understood as the largest value of the difference between the real and the computed solution of the initial value problem \(y'=f(y,t)\), \(y(t_ 0)=y_ 0\), \(t\in [0,c]\) when f belongs to a space of functions with r continuous bounded derivatives. First, assuming that the algorithms are dependent on f through the values of n linear continuous functionals at f, it is proved that the error cannot converge to zero faster than \(n^{-r}\) as \(n\to +\infty\) and this minimal error is achieved by the Taylor algorithm. Finally, for arbitrary linear functionals, the optimal error bounds are of order \(n^{-(r+1)}\) when \(n\to +\infty\).
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65G50
zbMATH DE Number: 4082871
zbMATH Keywords: minimum asymptotic error
zbMATH Keywords: algorithmic complexity
zbMATH Keywords: optimal error bounds
author: Bolesław Kacewicz
MaRDI profile type: Publication
cites work: Q5719651
cites work: How to increase the order to get minimal-error algorithms for systems of ODE
cites work: Asymptotic error of algorithms for solving nonlinear problems
cites work: Q4002473
cites work: Q5680530
description: scientific article; zbMATH DE number 4082871