A note on the evaluation of matrix polynomials

The problem of evaluating a polynomial
p(x)
in a matrix
A
arises in many applications, e.g., the Taylor approximation of e
^A
. The
0(\sqrt{q} n^{3})
algorithm of Paterson and Stockmeyer has the drawback that it requires
0(\sqrt{q}q n^{2})
storage, where
q
is the degree of
p
and
n
is the dimension of
A
. An algorithm which greatly reduces this storage requirement without undue loss of speed is presented.