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 $O(\sqrt{q}n^{3})$ algorithm of Paterson and Stockmeyer has the drawback that it requires $O(\sqrt{q}n^{2})$ 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.