Evaluation of an analytical function of a companion matrix with distinct eigenvalues

This letter derives an explicit formula for the evaluation of an arbitrary analytical function of a companion matrix with distinct eigenvalues. The derivation presented here is simple and meaningful. It is also shown herein that this formula has a remarkable recursive property, which makes it extremely attractive for use in digital computations. In fact, the commonly used diagonalization procedure involves, in general, far more computational effort than the use of the procedure derived in this letter.