The complex step approximation to the Fréchet derivative of a matrix function (Q849151)

The authors show that the Fréchet derivative of a matrix function \(f\) at \(A\) in the direction \(E\), where \(A\) and \(E\) are real matrices, can be approximatecd by \(\text{Im\,}f(A+ ih\,E)/h\) for some suitably small \(h\). This approach is known in the scalar case but has not been applied previously to matrix functions. Its main advantage is that the stepsize \(h\) is allowed to be chosen as small as necessary to obtain an accurate approximation, without cancellation errors contaminating the result in floating point arithmetic. The implementation of the approximation is ease, assuming the availability of complex arithmetic.