On the convergence domains of the \(p\)-cyclic SOR
The authors consider the block successive overrelaxation (SOR) method for complex matrices \(A\) partitioned into \(p\times p\) blocks. Furthermore, the \(p\)-cyclic matrices \(A\) have the properties that \(A^T\) are consistently ordered and the associated block Jacobi matrices \(B\) have eigenvalues \(\mu\) with \(\mu^p\) real. Throughout the paper, the authors concentrate their attention on values of \(p\leq 5\), although many proofs are given for arbitrary \(p\). Denoting the spectral radii of the Jacobi matrices \(\rho\) and the under- or overrelaxation factor \(\omega\), the authors construct the regions in \((\rho,\omega)\) space for which SOR converges. The method of proof is an expansion of earlier, unpublished work of Hadjidimos, Li and Varga and is based on application of the Schur-Cohn algorithm. This algorithm relates the location of roots of a polynomial within the unit disk to positivity of some coefficients of the iterated Schur transform of the polynomial. One surprising conclusion reached is that there are \((\rho,\omega)\) pairs for which SOR applied to the matrix \(A\) converges, but for which SOR applied to the consistently-ordered matrix \(A^T\) does not converge. The authors also apply the Schur-Cohn algorithm to establish optimal values of the overrelaxation parameter \(\omega\).
scientific article; zbMATH DE number 933154
scientific article