On the analysis of the unsymmetric successive overrelaxation method when applied top-cyclic matrices

Summary A Determinantal Invariance, associated with consistently ordered weakly cyclic matrices, is given. The DI is then used to obtain a new functional equation which relates the eigenvalues of a particular block Jacobi iteration matrix to the eigenvalues of its associated Unsymmetric Successive Overrelaxation (USSOR) iteration matrix. This functional equation as well as the theory of nonnegative matrices and regular splittings are used to obtain convergence and divergence regions of the USSOR method.