A note on the "Evaluation of an analytical function of a companion matrix with distinct eigenvalues"

It was recently shown that companion matrices possess a recursive property which allows their function to be computed efficiently and accurately. In this letter, an additional recursive property of companion matrices is derived. These two properties are then combined to suggest a substantially improved algorithm for the evaluation of functions of companion matrices. It is also indicated that by using this algorithm, the complex arithmetic operations stemming from the presence of complex eigenvalues in real matrices are reduced to a bare minimum.