On the accuracy of the ellipsoid norm approximation of the joint spectral radius
type: Item
instance of: scholarly article
title: On the accuracy of the ellipsoid norm approximation of the joint spectral radius
zbMATH Open document ID: 1086.15020
author: Yuri Nesterov
author: Jacques Theys
author: Vincent D. Blondel
published in: Linear Algebra and its Applications
publication date: 2005-02-23T00:00:00Z
review text: The authors consider two computable approximations of the joint spectral radius for a finite set \({\mathcal U}\) of matrices. The first approximation satisfies \(\frac{1}{\sqrt{n}}\hat{\rho}\leq \rho\leq \hat{\rho}\) based on ellipsoid norms, where \(\rho\) is the joint spectral radius of \({\mathcal U}\), \(\hat{\rho}=\inf_{P\succ 0}\max_{A_i\in {\mathcal U}}\| A_i\| _p\), \(\| A\| _p\) is induced by the vector norm \(\| x\| _p=\sqrt{x^TPx}\) and \(n\) is the dimension of the matrices. Moreover, for the special case of symmetric matrices, triangular matrices, or for sets of matrices that have a solvable Lie algebra, the equality \(\rho=\hat{\rho}\) is satisfied. The other approximation is for the set of nonnegative matrices. In this case the approximation is proved to be within a factor at most \(m\) of the exact value, where \(m\) is the number of matrices in \({\mathcal U}\).
reviewed by: Shuhuang Xiang
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 2137774
zbMATH Keywords: joint spectral radius
zbMATH Keywords: symmetric matrices
zbMATH Keywords: generalized spectral radius
zbMATH Keywords: ellipsoid norm
zbMATH Keywords: approximation
zbMATH Keywords: switched systems
zbMATH Keywords: stability
zbMATH Keywords: algorithm
zbMATH Keywords: triangular matrices
zbMATH Keywords: solvable Lie algebra
zbMATH Keywords: nonnegative matrices
MaRDI profile type: Publication
full work available at URL: j.laa.2004.06.024
OpenAlex ID: W2055472587
cites work: Simultaneous Contractibility
cites work: Bounded semigroups of matrices
cites work: Computationally Efficient Approximations of the Joint Spectral Radius
cites work: Sets of matrices all infinite products of which converge
cites work: Dynamical systems which undergo switching
cites work: Computing the joint spectral radius
cites work: Q5791470
cites work: Algebraic unsolvability of problem of absolute stability of desynchronized systems
cites work: The finiteness conjecture for the generalized spectral radius of a set of matrices
cites work: Stability of switched systems: a Lie-algebraic condition
cites work: An efficient lower bound for the generalized spectral radius of a set of matrices
cites work: Q4324980
cites work: Q5609479
cites work: Q3272731
cites work: The Lyapunov exponent and joint spectral radius of pairs of matrices are hard - when not impossible - to compute and to approximate
cites work: The generalized spectral radius and extremal norms
DOI: 10.1016/J.LAA.2004.06.024
Recommended article: Computationally Efficient Approximations of the Joint Spectral Radius
Recommended article: Approximation of the joint spectral radius using sum of squares
Recommended article: Optimal norms and the computation of joint spectral radius of matrices
Recommended article: Estimates for the joint spectral radius
Recommended article: On the computational aspects of the theory of joint spectral radius
Recommended article: Q4468953
Recommended article: Approximation of the Joint Spectral Radius of a Set of Matrices Using Sum of Squares
Recommended article: Q3457816
Recommended article: The generalized joint spectral radius. A geometric approach
Recommended article: Q4831827
description: scientific article; zbMATH DE number 2137774