Derivatives of the matrix exponential and their computation
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.
scientific article; zbMATH DE number 820051