ALTERNATING DIRECTION IMPLICIT ITERATION FOR SYSTEMS WITH COMPLEX SPECTRA

The alternating direction implicit (ADI) iteration "model problem" has hitherto been the discretized Dirichlet problem with an SPD matrix splitting for which optimum iteration parameters are obtained as the solution, due to W. B. Jordan, of a rational Chebyshev minimax problem over the reals. Recently disclosed application to a class of Sylvester equations (AX + XB = C, with A and B positive-real matrices) required generalization to complex spectra in the positive-real half plane. Theory is developed here for applying Jordan's parameters to complex domains which remain "close" to the real line, and for predicting associated error reduction. Numerical studies are presented in support of this theory. This analysis provided a basis for analysis of more general complex spectra which is reported in another paper.