In this note we investigate the Chebyshev iteration and the conjugate cradient method applied to the system of linear equations Ax = f where 0 ,4 is a symmetric, positive definite matrix. For both methods we present algorithms which approximate during the iteration process the kth error 

Ek =II x - zk l[A . The algorithms are based on the theory of modified mo- ments and Gaussian quadrature. The proposed schemes are also applicable for other polynomial iteration schemes. Several examples, illustrating the performance of the described methods, axe presented.