On using the complex step method for the approximation of Fréchet derivatives of matrix functions in automorphism groups

We show, that the complex step approximation $\mathrm{Im}(f(A+ihE))/h$ to the Fr\'echet derivative of matrix functions $f:\mathbb{R}^{m,n}\rightarrow\mathbb{R}^{m,n}$ is applicable to the matrix sign, square root and polar mapping using iterative schemes. While this property was already discovered for the matrix sign using Newtons method, we extend the research to the family of Pad\'e iterations, that allows us to introduce iterative schemes for finding function and derivative values while approximately preserving automorphism group structure.