Convergence analysis of optimal SOR for a class of consistently ordered 2-cyclic matrices with complex spectra

Asymptotic rates of convergence of optimal SOR applied to linear systems with consistently ordered 2-cyclic matrices have been extensively studied in the case where the Jacobi spectrum is real and contained in an interval centered at the origin. It is well known that as the rightmost endpoint of the interval approaches 1 from below, optimal SOR converges an order of magnitude faster than Jacobi. We generalize this to the situation where the Jacobi spectrum is contained in a line segment in the complex plane that is symmetric about the origin. This is an important class of linear systems, which arise often in various physical applications; complex-shifted linear systems are included in this family. Optimal relaxation parameters are known in this case, but a detailed convergence analysis does not seem to exist in the literature. Using techniques of complex analysis, we derive convergence rates, finding that in the complex case they are affected not only by the distance to 1 of the right-hand endpoint of the line segment as in the real case, but also by its phase. We then exploit a useful convergence property that our analysis reveals to generate a hybrid SOR–multigrid scheme and demonstrate its merits as a shifted Laplacian preconditioner for GMRES applied to the Helmholtz equation.