Generalized Augmented Matrix Preconditioning Approach and its Application to Iterative Solution of Ill-Conditioned Algebraic Systems

Abstract

The present work is devoted to a class of preconditioners based on the augmented matrix approach considered earlier by two of the present authors. It presents some generalizations of the subspace-correction schemes studied earlier and gives a brief comparison of the developed technique with a somewhat similar "deflation" algorithm. The developed preconditioners are able to improve significantly an eigenvalue distribution of certain severely ill-conditioned algebraic systems by using properly chosen projection matrices, which correct the low-frequency components in the spectrum. One of the main advantages of the proposed approach is the possibility of using inexact solvers within the projectors. Another attractive feature of the developed method is that it can be easily combined with other preconditioners, for instance, those which correct the high-frequency eigenmodes.