Alternating direction implicit iteration for iteration matrices with some complex spectra

Alternating Direction Implicit iteration was intro duced by Peaceman and Rachford to solve parabolic and elliptic linear systems. The theory of optimum iteration parameters for iteration matrices with real spectra was developed using Chebyshev minimax theory. Using this method to solve the Lyapunov matrix equation arising from the computations by Hurwitz of impedence boundary conditions for finite element problems leads to consideration of the case of certain spectra with small complex components. Rouche's theorem provides the basis for this generalization. A theory to find the optimum set of parameters for a complex spectrum among all sets optimum for real spectra is developed. Theoretical error bounds are derived much as for the real spectral case. Numerical tests indicate that this theory is appropriate only for very small complex spectral components. Then actual convergence rates are slightly improved but theoretical error bounds are much more valid. These optimum iteration parameters converge towards one with increasing complex spectral component. Since Wachspress has shown the optimum iteration parameters for circular spectra to be one, repeated, it is expected that further investigation will show optimum iteration parameters converging as spectra vary from strictly real to complex and circular.