On the eigenvalue distribution of a class of preconditioning methods (Q1058260)

A class of preconditioning methods depending on a relaxation parameter is presented for the solution of large linear systems of equations \(Ax=b\), where A is a symmetric positive definite matrix. The methods are based on an incomplete factorization of the matrix A and include both pointwise and blockwise factorizations. We study the dependence of the rate of convergence of the preconditioned conjugate gradient method on the distribution of eigenvalues of \(C^{-1}A\), where C is the preconditioning matrix. We also show graphic representations of the eigenvalues and present numerical tests of the methods.