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 is an element of C-NxN is a sing ular matrix. In the present work we propose a general framework within whic h Krylov subspace methods for Drazin-inverse solution of this system can be derived in a convenient way. The Krylov subspace methods known to us to da te treat only the cases in which A is hermitian and its index ind(A) is uni ty necessarily. In the present work A is not required to be hermitian. It c an have any type of spectrum and ind(A) is arbitrary. We show that, as is t he case with nonsingular systems, the Krylov subspace methods developed her e 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 ab le to bound the errors, the relevant bounds decreasing with increasing dime nsion of the Krylov subspaces involved. The results of this paper are appli cable 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 appro ach to nonsingular systems. Indeed, our approach here reduces to that relev ant for nonsingular systems upon setting ind(A) = 0 everywhere. (C) 1999 El sevier Science Inc. All rights reserved.