Abstract
Consider applying the Conjugate Residual (CR) method to systems of linear equations Ax = b or least squares problems min__<x∈R^2>‖b-Ax‖_2, where A ∈ R^<n×n> is singular and nonsymmetric. First, we prove that the necessary and sufficient condition for the method to converge to a least squares solution without breaking down for arbitrary b and initial approximate solution x_0 is that the symmetric part M(A) of A is semi-definite, rank M(A) = rankA, and R(A)^⊥ = kerA. Next, we derive the necessary and sufficient condition for the CR method to converge to a solution without breaking down for arbitrary b ∈ R(A) and arbitrary x_0.
