Evaluating the Fréchet derivative of the matrix exponential
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 \(\| L(X,\cdot)\|\).
scientific article; zbMATH DE number 86130