Volume 60, pp. A1-A14, 2024.

Application of the Schur-Cohn Theorem to the precise convergence domain for a p-cyclic SOR iteration matrix

Apostolos Hadjidimos, Xiezhang Li, and Richard S. Varga


Assume that $A\in{C}^{n\times n}$ is a block $p$-cyclic consistently ordered matrix and that its associated Jacobi iteration matrix $B$, which is weakly cyclic of index $p$, has eigenvalues $\mu$ whose $p$th powers are all real nonpositive (resp. nonnegative). Usually, one is interested only in the relaxation parameter $\omega$ that minimizes the spectral radius of the iteration matrix of the associated SOR iterative method, but here we are interested in all real values for the relaxation parameter $\omega$ for which the SOR iteration matrix is convergent. This will be achieved for the values of $p=2,3,4,\ldots,$ and for $p\rightarrow\infty.$

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Key words

block $p$-cyclic matrix, weakly cyclic of index $p$ matrix, block Jacobi and SOR iteration matrices, Schur–Cohn Algorithm

AMS subject classifications


Links to the cited ETNA articles

[10]Vol. 28 (2007-2008), pp. 78-94 A. Hadjidimos and P. Stratis: Minimization of the spectral norm of the SOR operator in a mixed case

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