A geometric view of Krylov subspace methods on singular systems

Abstract

We give a geometric framework for analysing iterative methods on singular linear systems Ax=b and apply them to Krylov subspace methods. The idea is to decompose the method into the ℛ(A) component and its orthogonal complement ℛ(A)⟂, where ℛ(A) is the range of A. We apply the framework to GMRES, GMRES(k) and GCR(k), and derive conditions for convergence without breakdown for inconsistent and consistent singular systems. The approach also gives a geometric interpretation and different proofs of the conditions obtained by Brown and Walker for GMRES. We also give examples arising in the finite difference discretization of two-point boundary value problems of an ordinary differential equation. Copyright © 2010 John Wiley & Sons, Ltd.