Entrywise relative perturbation bounds for exponentials of essentially non-negative matrices
type: Item
instance of: scholarly article
title: Entrywise relative perturbation bounds for exponentials of essentially non-negative matrices
zbMATH Open document ID: 1163.65021
author: Jungong Xue
author: Qiang Ye
published in: Numerische Mathematik
publication date: 2008-12-02T00:00:00Z
review text: The authors consider the perturbation behavior of essentially non-negative matrix exponentials. A matrix is said to be essentially non-negative if \(a_{ij} \geq 0\) for all \(i \neq j\). The generator matrix \(Q\) of a continuous-time Markov chain is used as an example with the mentioned property. The \((i,j)\) entry of the matrix exponential exp\((Qt)\) is the probability that the chain has at state \(j\) at time \(t\) if it is at state \(i\) initially. Especially, taking a sensitivity analysis one is interested in entries of the transition matrix exp\((Qt)\) which represent small probability events. Because normwise perturbation bounds do not reflect these small entries sharp entrywise relative perturbation bounds for the exponential matrices are established that are determined entrywise by the absolute accuracy of the diagonal entries and by the relative accuracy of the off-diagonal elements.
reviewed by: Georg Hebermehl
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15A54
Mathematics Subject Classification ID: 15A12
Mathematics Subject Classification ID: 65C40
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60J22
zbMATH DE Number: 5377099
zbMATH Keywords: essentially non-negative matrices
zbMATH Keywords: matrix exponentials
zbMATH Keywords: perturbation bounds
zbMATH Keywords: spectral radius
zbMATH Keywords: condition number
zbMATH Keywords: continuous-time Markov chain
describes a project that uses: Expokit
describes a project that uses: MATLAB expm
MaRDI profile type: Publication
full work available at URL: s00211-008-0167-5
OpenAlex ID: W2020051491
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DOI: 10.1007/S00211-008-0167-5
description: scientific article; zbMATH DE number 5377099