Convergence of block iterative methods applied to sparse least-squares problems

Recently, special attention has been given, in the mathematical literature, to the problems of accurately computing the least-squares solutions of very large-scale overdetermined systems of linear equations, such as those arising in geodetical network problems. In particular, it has been suggested that one solve such problems, iteratively by applying the block-SOR (successive overrelaxation) iterative method to a consistently ordered block-Jacobi matrix that is weakly cyclic of index 3. Here, we obtain new results (Theorem 1), giving the exact convergence and divergence domains for such iterative applications. It is then shown how these results extend, and correct, the literature on such applications. In addition, analogous results (Theorem 2) are given for the case when the eigenvalues of the associated block-Jacobi matrix are nonnegative.