The Complex Step Approximation to the Fréchet Derivative of a Matrix Function

We show that the Fr\'echet derivative of a matrix function $f$ at $A$ in the direction $E$, where $A$ and $E$ are real matrices, can be approximated by $\Im f(A+ihE)/h$ for some suitably small $h$. This approximation, requiring a single function evaluation at a complex argument, generalizes the complex step approximation known in the scalar case. The approximation is proved to be of second order in $h$ for analytic functions $f$ and also for the matrix sign function. It is shown that it does not suffer the inherent cancellation that limits the accuracy of finite difference approximations in floating point arithmetic. However, cancellation does nevertheless vitiate the approximation when the underlying method for evaluating $f$ employs complex arithmetic. The ease of implementation of the approximation, and its superiority over finite differences, make it attractive when specialized methods for evaluating the Fr\'echet derivative are not available, and in particular for condition number estimation when used in conjunction with a block 1-norm estimation algorithm