Preconditioned Krylov subspace methods for the solution of least-squares problems

and Kk(BA,Br) = span{Br, (BA)Br, . . . , (BA)k−1Br}, (3) where B ∈ Rn×m is the mapping and preconditioning matrix, and apply Krylov subspace iteration methods on these subspaces. For overdetermined problems, applying the standard CG method to Kk(BA,Br) leads to the preconditioned CGLS [3] or CGNR [9] method while for underdetermined problems it leads to preconditioned CGNE [9] method. The GMRES method applied to Kk(AB, r) and Kk(BA,Br) is respectively called AB-GMRES and BAGMRES method [4, 5, 7]. Theoretical analysis [4, 5, 7] shows that a sufficient and necessary condition for the above GMRES methods to give a least squares solution without breakdown for arbitrary b, for over-determined, under-determined and possibly rank-deficient problems is