The sharpness of stability estimates corresponding to a general resolvent condition
type: Item
instance of: scholarly article
title: The sharpness of stability estimates corresponding to a general resolvent condition
zbMATH Open document ID: 0961.15016
DOI: 10.1016/S0024-3795(00)00085-9
published in: Linear Algebra and its Applications
publication date: 2001-05-28T00:00:00Z
review text: In a previous article published by the authors, upper bounds were given for the norm of the \(n\)-th power of matrices \(B\in\mathbb{C}^{s\times s}\) under the following generalized version of the Kreiss resolvent condition: \[ r (B) \leq 1,\;\bigl\|(\zeta I-B)^{-1}\bigr\|\leq{L\over \bigl(|\zeta |-1 \bigr)^k|\zeta-1|^l },\;1<|\zeta |<\rho, \] where \(L\) is a positive constant, \(\rho\) is a constant with \(1<\rho \leq\infty\) and \(k,l\) are fixed integers satisfying \(k\geq 0\), \(l\geq 0\), \(k+l>1\) (here \(r(B)\) and \(\|B \|\) denote respectively the spectral radius and the operator norm of the matrix \(B)\). In the present paper the sharpness of these upper bounds is examined. The main result (Theorem 2.1) shows that these upper bounds are the best possible for \((k,l)\neq(0,1)\) in the sense that no general bounds exist which grow more slowly, as \(n\to\infty\) and \(s\to\infty\), than the given bounds.
reviewed by: Valeriu Prepeliţă
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 65F35
zbMATH DE Number: 1462182
zbMATH Keywords: stability estimates
zbMATH Keywords: powers of matrices
zbMATH Keywords: resolvent condition
zbMATH Keywords: Kreiss matrix theorem
zbMATH Keywords: spectral radius
zbMATH Keywords: operator norm
zbMATH Keywords: upper bounds
author: Natalia Borovykh
author: Marc N. Spijker
MaRDI profile type: Publication
cites work: A note about ritt's condition, related resolvent conditions and power bounded operators
cites work: Resolvent conditions and bounds on the powers of matrices, with relevance to numerical stability of initial value problems
cites work: Q3139557
cites work: On the stability and convergence of discretizations of initial value p.d.e.s
cites work: Two counterexamples related to the Kreiss matrix theorem
cites work: Über Die Stabilitätsdefinition Für Differenzengleichungen Die Partielle Differentialgleichungen Approximieren
cites work: On the resolvent condition in the Kreiss matrix theorem
cites work: On the norm of a finite boolean algebra of projections, and applications to theorems of Kreiss and Morton
cites work: Q4352376
cites work: Stability of the method of lines
cites work: On a conjecture by LeVeque and Trefethen related to the Kreiss matrix theorem
cites work: Numerical stability, resolvent conditions and delay differential equations
cites work: Stability estimates for families of matrices of nonuniformly bounded order
cites work: Q3993078
cites work: Q4352383
cites work: The Kreiss Matrix Theorem on a General Complex Domain
cites work: From the Buffon Needle Problem to the Kreiss Matrix Theorem
description: scientific article; zbMATH DE number 1462182