A unified approach to Krylov subspace methods for the Drazin-inverse solution of singular nonsymmetric linear systems

Consider the linear system Ax = b, where A ∈ CN×N is a singular matrix. In the present work we propose a general framework within which Krylov subspace methods for Drazininverse solution of this system can be derived in a convenient way. The Krylov subspace methods known to us to date treat only the cases in which A is hermitian and its index ind(A) is unity necessarily. In the present work A is not required to be hermitian. It can have any type of spectrum and ind(A) is arbitrary. We show that, as is the case with nonsingular systems, the Krylov subspace methods developed here terminate in a finite number of steps that is at most N − ind(A). For one of the methods derived here we also provide an analysis by which we are able to bound the errors, the relevant bounds decreasing with increasing dimension of the Krylov subspaces involved. The results of this paper are applicable to consistent systems as well as to inconsistent ones. An interesting feature of the approach to singular systems presented in this work is that it is formulated as a generalization of the standard Krylov subspace approach to nonsingular systems. Indeed, our approach here reduces to that relevant for nonsingular systems upon setting ind(A) = 0 everywhere. © 1999 Elsevier Science Inc. All rights reserved. AMS classification: 15A06; 15A09; 65F10; 65F50