Determination of the $D^ {1/2}$-norm of the SOR iterative matrix for the unsymmetric case

Abstract:

This paper is concerned with the determination of the Jordan canonical form and ${D^{1/2}}$-norm of the SOR iterative matrix derived from the coefficient matrix A having the form \[ A = \left ( {\begin {array}{*{20}{c}} {D1} \hfill & { - H} \hfill \\ {{H^T}} \hfill & {{D_2}} \hfill \\ \end {array} } \right )\] with ${D_1}$ and ${D_2}$ symmetric and positive definite. The theoretical results show that the Jordan form is not diagonal, but has only q principal vectors of grade 2 and that the ${D^{1/2}}$-norm of ${\mathcal {L}_{{\omega _b}}}$ (${\omega _b}$, the optimum parameter) is less than unity if and only if $\bar \mu = \rho (B)$, the spectral radius of the associated Jacobi iterative matrix, is less than unity. Here q is the multiplicity of the eigenvalue $i\bar \mu$ of B.