On the Behaviour of the Conjugate Residual Method for Singular Systems

Consider applying the Conjugate Residual (CR) method, which is a Krylov subspace type iterative solver, to systems of linear equations Ax = b or least squares problems min x∈Rn ‖b −Ax‖2, where A is singular and nonsymmetric. We will show that when R(A)⊥ = kerA, the CR method can be decomposed into the R(A) and kerA components, and the necessary and sufficient condition for the CR method to converge to the least squares solution without breaking down for arbitrary b and initial approximate solution x0, is that the symmetric partM(A) ofA is semi-definite and rankM(A) = rankA. Furthermore, when x0 ∈ R(A), the approximate solution converges to the pseudo inverse solution. Next, we will also derive the necessary and sufficient condition for the CR method to converge to the least squares solution without breaking down for arbitrary initial approximate solutions, for the case when R(A)⊕kerA = Rn and b ∈ R(A).