Convergence Analysis of Optimal SOR for a Class of Consistently Ordered 2-Cyclic Matrices with Complex Spectra

TL;DR

This paper extends the convergence analysis of optimal SOR methods from real spectra to complex spectra for a specific class of matrices, revealing how phase impacts convergence rates.

Contribution

It generalizes the convergence analysis of optimal SOR to matrices with complex spectra, providing explicit rates considering phase effects.

Findings

Convergence rates depend on the distance to 1 and the phase of the spectral line segment.

Optimal relaxation parameters are identified for this class of matrices.

Complex analysis techniques are used to derive the new convergence bounds.

Abstract

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 eigenvalues are are 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.