Optimal Semi-Iterative Methods for Complex SOR with Results from Potential Theory

We consider the application of semi-iterative methods (SIM) to the standard (SOR) method with complex relaxation parameter ω, under the following two assumptions: (1) the associated Jacobi matrix J is consistently ordered and weakly cyclic of index 2, and (2) the spectrum σ(J ) of J belongs to a compact subset of the complex plane C, which is symmetric with respect to the origin. By using results from potential theory, we determine the region of optimal choice of ω ∈ C for the combination SIM–SOR and settle, for a large class of compact sets , the classical problem of characterising completely all the cases for which the use of the SIM-SOR is advantageous over the sole use of SOR, under the hypothesis that σ(J ) ⊂ . In particular, our results show that, unless the outer boundary of is an ellipse, SIM–SOR is always better and, furthermore, one of the best possible choices is an asymptotically optimal SIM applied to the Gauss–Seidel method. In addition, we derive the optimal complex SOR parameters for all ellipses which are symmetric with respect to the origin. Our work was motivated by recent results of M. Eiermann and R.S. Varga. Dedicated to Professor Richard S. Varga in recognition of his substantial contributions to the subject of the paper. A. Hadjidimos Department of Mathematics, University of Crete, P.O. Box 2208, 714 09 Heraklion, Greece E-mail: [email protected] N.S. Stylianopoulos (B) Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus E-mail: [email protected] 592 A. Hadjidimos, N.S. Stylianopoulos