Parallel Newton–Chebyshev polynomial preconditioners for the conjugate gradient method

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 8.6 billion unknowns) in a parallel environment showing the efficiency of the proposed class of preconditioners.