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

Abstract 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−A D A . Let k be the index of A. We further show that A D b+Px 0 , x 0 ∈ R (A k−1 )+ N (A) , is a generalized least-squares solution of Ax = b in R (A k )+ 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 .