The relationship between the Jacobi and the successive overrelaxation (SOR) matrices of a k-cyclic matrix
In the SOR theory [cf. e.g. \textit{D. Young}, Iterative solution of large linear systems (1971; Zbl 0231.65034)] the relation \([\lambda +\omega - 1]^ k=\omega^ k\mu^ k\lambda^{k-1}\) between the eigenvalues of the Jacobi and the SOR matrix \(\mu\), \(\lambda\) and the relaxation parameter \(\omega\) play an important role. The authors show that for a certain subclass of consistently ordered matrices the same relation is true for the iteration matrices. This implies that the SOR method in this case is equivalent to a monoparametric k-step iterative method in the sense of \textit{W. Niethammer} and \textit{D. S. Varga} [Numer. Math. 41, 177- 206 (1983; Zbl 0487.65018)].
scientific article; zbMATH DE number 4108010
scientific article