The Ostrowski–Reich theorem for SOR iterations: extensions to the rank deficient case Original Research Article

The Ostrowski–Reich theorem gives the necessary and sufficient condition of convergence of the SOR method for hermitian and positive definite matrices. Ortega and Plemmons have generalized the theorem to non-hermitian matrices. For a general nonsingular matrix A, they have given necessary and sufficient conditions of convergence of splitting methods, and have studied the convergence of the SOR method. This note is a generalization of the Ortega and Plemmons theorems to singular matrices. Some necessary and sufficient conditions of semi-convergence for singular matrices are given.