  The computation of matrix functions is a well-studied problem. Of special
importance are the exponential and the logarithm of a matrix, where the latter
also raises existence and uniqueness questions. This is particularly relevant
in the context of matrix semigroups and their generators. Here, we look at
matrix functions of triangular matrices, where a recursive approach is possible
when the matrix has simple spectrum. The special feature is that no knowledge
of eigenvectors is required, and that the same recursion applies to the
computation of multiple functions or semigroups simultaneously.
