Computing the Fréchet Derivative of the Matrix Exponential, with an Application to Condition Number Estimation

The matrix exponential is a much-studied matrix function having many applications. The Fréchet derivative of the matrix exponential describes the first-order sensitivity of $e^A$ to perturbations in A and its norm determines a condition number for $e^A$. Among the numerous methods for computing $e^A$ the scaling and squaring method is the most widely used. We show that the implementation of the method in [N. J. Higham, The scaling and squaring method for the matrix exponential revisited, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 1179–1193] can be extended to compute both $e^A$ and the Fréchet derivative at A in the direction E, denoted by $L(A,E)$, at a cost about three times that for computing $e^A$ alone. The algorithm is derived from the scaling and squaring method by differentiating the Padé approximants and the squaring recurrence, reusing quantities computed during the evaluation of the Padé approximant, and intertwining the recurrences in the squaring phase. To guide the choice of algorithmic parameters, an extension of the existing backward error analysis for the scaling and squaring method is developed which shows that, modulo rounding errors, the approximations obtained are $e^{A+\Delta A}$ and $L(A+\Delta A,E+\Delta E)$, with the same $\Delta A$ in both cases, and with computable bounds on $\|\Delta A\|$ and $\|\Delta E\|$. The algorithm for $L(A,E)$ is used to develop an algorithm that computes $e^A$ together with an estimate of its condition number. In addition to results specific to the exponential, we develop some results and techniques for arbitrary functions. We show how a matrix iteration for $f(A)$ yields an iteration for the Fréchet derivative and show how to efficiently compute the Fréchet derivative of a power series. We also show that a matrix polynomial and its Fréchet derivative can be evaluated at a cost at most three times that of computing the polynomial itself and give a general framework for evaluating a matrix function and its Fréchet derivative via Padé approximation.