04078628

j

04078628 Galanis, S. Hadjidimos, A. Noustos, D. On the equivalence of the k-step iterative Euler methods and successive overrelaxation (SOR) methods for k-cyclic matrices. Math. Comput. Simul. 30, No.3, 213-230 (1988). 1988 Elsevier Science B.V. (North-Holland), Amsterdam EN consistently ordered k-cyclic matrices successive overrelaxation Jacobi matrix k-cyclic SOR method stationary k-step iterative method optimal relaxation factor region of convergence Zbl 0487.65018

doi:10.1016/0378-4754(88)90001-8

The authors establish the relationship $\omega\sp kT\sp k{\cal L}\sb{\omega}\sp{k-1}=[{\cal L}\sb{\omega}+(\omega +1)I]\sp k$ connecting the successive overrelaxation (SOR) matrix ${\cal L}\sb{\omega}$ and the Jacobi matrix T associated with a linear system $x=Tx+c$, where T is weakly cyclic of index k. Based on this result, they derive an equivalence between the k-cyclic SOR method and a certain stationary k- step iterative method [cf. {\it W. Niethammer} and {\it R. S. Varga}, Numer. Math. 41, 177-206 (1983; Zbl 0487.65018)]. By applying the theory of these stationary k-step methods, old and new results on the k-cyclic SOR method are shown, e.g., the optimal relaxation factor and the region of convergence are determined. M.Eiermann