An iterative method, with convergence of any desired finite order, for computing the generalized inverseA+and the associated projectionsAA+andA+Afor any matrixAis developed. The method uses an upper bound on the maximum eigen-value ofAAast; (orA*A). It is also shown that the trace of a sequence of approximations toAA+(orA+A) converges to the rank ofA. Finally, examples are given illustrating the computation of the generalized inverse and rank.