Practical Use of Polynomial Preconditionings for the Conjugate Gradient Method

Abstract

This paper presents some practical ways of using polynomial preconditions for solving large sparse linear systems of equations issued from discretizations of partial differential equations. For a symmetric positive definite matrix A these techniques are based on least squares polynomials on the interval $[0,b]$ where b is the Gershgorin estimate of the largest eigenvalue. Therefore, as opposed to previous work in the field, there is no need for computing eigenvalues of A. We formulate a version of the conjugate gradient algorithm that is more suitable for parallel architectures and discuss the advantages of polynomial preconditioning in the context of these architectures.