Evaluating the Frechet derivative of the matrix exponential

Summary

LetM _n denote the space ofn×n matrices. GivenX, Z∈M _n define

$$L\left( {X,Z} \right) = \int\limits_0^1 {e^{sX} } Ze^{\left( {1 - s} \right)X} ds,$$

and given a norm ‖·‖ onM _n define

$$\left\| {L\left( {X, \cdot } \right)} \right\| = \max \left\{ {\left\| {L\left( {X,Y} \right)} \right\|:\left\| Y \right\| \leqq 1,Y \in M_n } \right\}.$$

The quantity ‖L(X,·)‖ is important in the study of the sensitivity of the matrix exponential. Given an integration ruleR and a positive integerm letL _R.m (X, Z) denote the approximation toL(X, Z) given by the composite ruleR applied withm subintervals. We give bounds on ‖L(X,·)−L _R.m (X,·)‖ that are valid for any unitarily invariant norm ‖·‖ and any integration ruleR.

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 functionL(X,·) and compare the performance of the power method and two variants of the Lanczos algorithm for estimating ‖L(X,·)‖.