Parallel Newton–Chebyshev polynomial preconditioners for the conjugate gradient method

Abstract

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.