  In this note we exploit polynomial preconditioners for the Conjugate Gradient
method to solve large symmetric positive definite linear systems in a parallel
environment. We put in connection a specialized Newton method to solve the
matrix equation X^{-1} = A and the Chebyshev polynomials for preconditioning.
We propose a simple modification of one parameter which avoids clustering of
extremal eigenvalues in order to speed-up convergence. We provide results on
very large matrices (up to 8 billion unknowns) in a parallel environment
showing the efficiency of the proposed class of preconditioners.
