Evaluating the Fréchet derivative of the matrix exponential
type: Item
instance of: scholarly article
title: Evaluating the Fréchet derivative of the matrix exponential
zbMATH Open document ID: 0756.65068
DOI: 10.1007/BF01385857
author: Roy Mathias
published in: Numerische Mathematik
publication date: 1993-02-02T00:00:00Z
full work available at URL: https://eudml.org/doc/133678
review text: Let \(M_ n\) denote the space of \(n\times n\) matrices. Given \(X,Z\in M_ n\) define \[ L(X,Z)=\int^ 1_ 0 e^{sX}Ze^{(1-s)X}ds, \] and given a norm \(\| \cdot\|\) on \(M_ n\) define \[ \| L(X,\cdot)\|=\max\{\| L(X,Y)\| : \| Y\| \leq 1,\;Y\in M_ n\}. \] The quantity \(\| L(X,\cdot)\|\) is important in the study of the sensitivity of the matrix exponential. Given an integration rule \(R\) and a positive integer \(m\) let \(L_{R,m}(X,Z)\) denote the approximation to \(L(X,Z)\) given by the composite rule \(R\) applied with \(m\) subintervals. We give bounds on \(\| L(X,\cdot)- L_{R,m}(X,\cdot)\|\) that are valid for any unitarily invariant \(\| \cdot\|\) and any integration rule \(R\). We show that in a particular situation, that arises in practice, the composite Simpson's rule has a better error bound than the composite trapezoidal rule, and that it can be evaluated with the same number of matrix multiplications. We also note a symmetry property of the function \(L(X,\cdot)\) and compare the performance of the power method and two variants of the Lanczos algorithm for estimating \(\| L(X,\cdot)\|\).
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 65F35
Mathematics Subject Classification ID: 15A12
zbMATH DE Number: 86130
zbMATH Keywords: condition estimation
zbMATH Keywords: Fréchet derivative
zbMATH Keywords: matrix exponential
zbMATH Keywords: Simpson's rule
zbMATH Keywords: error bound
zbMATH Keywords: trapezoidal rule
zbMATH Keywords: Lanczos algorithm
MaRDI profile type: Publication
cites work: Q5737099
cites work: Singular value decompositions of complex symmetric matrices
cites work: Q3960030
cites work: Q3359644
cites work: Condition Estimates for Matrix Functions
cites work: Q3868672
cites work: On Estimating the Largest Eigenvalue with the Lanczos Algorithm
description: scientific article; zbMATH DE number 86130