NII Technical Report (NII-2004-009E)：On the Convergence of the GCR(k) Method for Singular Systems

Consider applying the restarted Generalized Conjugate Residual (GCR(k)) method to systems of linear equations A x = b or least squares problems min_x || b - A x ||_2, where A is a n x n real matrix which may be singular and/or nonsymmetric and x, b are real vectors of size n. Let R(A) and N(A) be the range and null space of A, respectively. First, we prove that the necessary and sufficient condition for the method to converge to a least squares solution without breakdown for arbitrary b and initial approximate solution x_0, is that A is definite in R(A), and that R(A) and N(A) are orthogonal to each other. Next, we show that the necessary and sufficient condition for the method to converge to a solution without breakdown for arbitrary b in R(A) and arbitrary x_0, is that A is definite in R(A). The main idea of the proofs is to decompose the algorithm into the R(A) and its orthogonal complement components. Finally, we will give examples arising in the finite difference discretization of two-point boundary value problems of an ordinary differential equation, corresponding to the above two cases.