Extension and further development of the differential calculus for matrix norms with applications.
type: Item
instance of: scholarly article
title: Extension and further development of the differential calculus for matrix norms with applications.
zbMATH Open document ID: 1042.65051
DOI: 10.1016/S0377-0427(03)00385-6
published in: Journal of Computational and Applied Mathematics
publication date: 2003-08-07T00:00:00Z
review text: In an earlier work [J. Comput. Appl. Math. 135, 1--21 (2001; Zbl 1025.34004)], the author has discussed differential calculus for the operator norms \(\| \cdot \|_p\), \(p\in \{ 1, 2, \infty\}\), of the fundamental matrix \(\Phi(t)=e^{At}\), \(t\geq0\), where \(A\) is a complex \(n\times n\) matrix. In this paper, he extends the earlier study to \(m\) times continuously differentiable matrix functions \(\chi(t)\), \(t>0\), and other \(p\)-norms \(| \cdot |_p\), \(1<p<\infty\). In particular, formulae for the first two logarithmic derivatives \(D^1_+|\Phi(0)|_p\) and \(D^2_+|\Phi(0)|_p\), \(1<p<\infty\), of the function \(\Phi(t)\) are derived. Further, upper bounds on the discrete evolution \(\Psi(t)\), \(t\geq0\) and on the difference \(R(t)=\Phi(t)-\Psi(t)\), \(t\geq0\), are derived. The results obtained are applied to the computation of the optimal upper bounds on \(\| R(t)\|_{\infty}\), \(\| R(t)\|_2\) and \(| R(t)|_2\).
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L70
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 1961630
zbMATH Keywords: differential calculus for matrix norms
zbMATH Keywords: best upper bound
zbMATH Keywords: logarithmic derivative
zbMATH Keywords: fundamental matrix
zbMATH Keywords: matrix functions
author: Ludwig Kohaupt
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1961630