Convergence properties of Krylov subspace methods for singular linear systems with arbitrary index

Krylov subspace methods have been recently considered to solve singular linear systems Ax = b. In this paper, we derive the necessary and sufficient conditions guaranteeing that a Krylov subspace method converges to a vector ADb+Px0, where AD is the Drazin inverse of A and P is the projection P = I−ADA. Let k be the index of A. We further show that ADb+Px0, x0 ∈ R (Ak−1)+N (A), is a generalized least-squares solution of Ax = b in R (Ak)+N (A). Finally, we present the convergence bounds for the quasi-minimal residual algorithm (QMR) and transpose-free quasi-minimal residual algorithm (TFQMR). The index k of A in this paper can be arbitrary, which extends to the main results of Freund and Hochbruck (Numer. Linear Algebra Appl. 1 (1994) 403–420) that only considers the case k = 1.