On a conjecture by LeVeque and Trefethen related to the Kreiss matrix theorem
type: Item
instance of: scholarly article
title: On a conjecture by LeVeque and Trefethen related to the Kreiss matrix theorem
zbMATH Open document ID: 0736.15015
DOI: 10.1007/BF01933268
published in: BIT
publication date: 1992-06-25T00:00:00Z
review text: Let \(A\) be an \(N\times N\) complex matrix. The condition (1) \(\| A^n\|\leq C_0\) for \(n=1,2,3,\ldots\), for a positive constant \(C_0\) implies the condition (2) \(zI-A\) is regular with \(\|(zI-A)^{- 1}\|\leq C_1(| z|-1)^{-1}\) for all complex \(z\) with \(| z|>1\). \textit{H.-O. Kreiss} BIT, Nord. Tidskr. Inf.-behandl. 2, 153--181 (1962; Zbl 0109.34702)]\ proved that (2) implies (1) with \(C_0\) depending on \(C_1\) and \(N\) only. \textit{R. J. LeVeque} and \textit{L. N. Trefethen} proved [BIT 24, 584--591 (1984; Zbl 0559.15018)]\ that (2) implies (1) if \(C_0=2eNC_1\) and they conjectured that the implication holds for the optimal value \(C_0=eNC_1\). The author proves this conjecture, his proof relying on a lemma which gives an upper bound for the arc-length of the image of the unit circle in the complex plane under a rational function.
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 65F35
zbMATH DE Number: 14225
zbMATH Keywords: Kreiss matrix theorem
zbMATH Keywords: power boundedness of matrices
zbMATH Keywords: resolvent condition
zbMATH Keywords: norm inequality
zbMATH Keywords: stability analysis
Wikidata QID: Q123121000
author: Marc N. Spijker
reviewed by: Fergus J. Gaines
MaRDI profile type: Publication
cites work: Über Die Stabilitätsdefinition Für Differenzengleichungen Die Partielle Differentialgleichungen Approximieren
cites work: On the resolvent condition in the Kreiss matrix theorem
cites work: Q5536978
cites work: Numerical Methods in Fluid Dynamics
cites work: The equivalence of \(L^ q-\)stability, the resolvent condition, and strict H-stability
full work available at URL: https://doi.org/10.1007/bf01933268
OpenAlex ID: W1980886588
description: scientific article; zbMATH DE number 14225