  Matrix functions play an increasingly important role in many areas of
scientific computing and engineering disciplines. In such real-world
applications, algorithms working in floating-point arithmetic are used for
computing matrix functions and additionally, input data might be unreliable,
e.g., due to measurement errors. Therefore, it is crucial to understand the
sensitivity of matrix functions to perturbations, which is measured by
condition numbers. However, the condition number itself might not be computed
exactly as well due to round-off and errors in the input. The sensitivity of
the condition number is measured by the so-called level-2 condition number. For
the usual (level-1) condition number, it is well-known that structured
condition numbers (i.e., where only perturbations are taken into account that
preserve the structure of the input matrix) might be much smaller than
unstructured ones, which, e.g., suggests that structure-preserving algorithms
for matrix functions might yield much more accurate results than
general-purpose algorithms. In this work, we examine structured level-2
condition numbers in the particular case of restricting the perturbation matrix
to an automorphism group, a Lie or Jordan algebra or the space of
quasi-triangular matrices. In numerical experiments, we then compare the
unstructured level-2 condition number with the structured one for some specific
matrix functions such as the matrix logarithm, matrix square root, and matrix
exponential.
