Derivatives of the matrix exponential and their computation
type: Item
instance of: scholarly article
title: Derivatives of the matrix exponential and their computation
zbMATH Open document ID: 0839.15004
author: Igor Najfeld
author: T. F. Havel
published in: Advances in Applied Mathematics
publication date: 1996-06-19T00:00:00Z
review text: Properties, representations and computation of various derivatives of the matrix exponential are studied. For the first directional derivative, evaluated at \(A\) in the direction \(V\), defined by \(D_V (t,A) = \lim_{h \to 0} {1 \over h} (e^{t(A + hV)} - e^{tA})\), the integral representation \(D_V (t,A) = \int^t_0 e^{(t - \sigma) A} Ve^{\sigma A} d \sigma\) is used to derive some properties, along with spectral, series and exact representations. The close relationship between the Kronecker sum and commutator series is emphasized and a new convergence criterion for commutator series is derived. Several compact spectral representations of \(D_V (t,A)\) in terms of the eigensystem of \(A\) and the Hadamard product are presented. The Hausdorff commutator series is summed to obtain a new closed form representation; another representation is obtained in terms of a faster converging ``sinch'' commutator series. This is used to derive two new ``scaling and squaring'' algorithms for \(D_V (t,A)\). Many of these results are extended to arbitrary analytic functions of a matrix argument. A relation between the gradients of the entries of an analytic matrix function \(F(A)\) and its directional derivatives in the elementary directions is emphasized. A new algorithm for computing the matrix exponential is developed, which is based on a rational representation of the exponential in terms of the hyperbolic function \(A\text{coth} (A)\). These results are illustrated by an application to a parametric estimation problem which arises in nuclear magnetic spectroscopy.
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 15A90
zbMATH DE Number: 820051
zbMATH Keywords: scaling and squaring algorithm
zbMATH Keywords: matrix exponential
zbMATH Keywords: directional derivative
zbMATH Keywords: Kronecker sum
zbMATH Keywords: convergence
zbMATH Keywords: spectral representations
zbMATH Keywords: Hadamard product
zbMATH Keywords: Hausdorff commutator series
zbMATH Keywords: gradients
zbMATH Keywords: analytic matrix function
zbMATH Keywords: parametric estimation
zbMATH Keywords: nuclear magnetic spectroscopy
reviewed by: Valeriu Prepeliţă
MaRDI profile type: Publication
full work available at URL: aama.1995.1017
OpenAlex ID: W2079780979
DOI: 10.1006/AAMA.1995.1017
description: scientific article; zbMATH DE number 820051