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ETNA - Electronic Transactions on Numerical Analysis
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |
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DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)
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ETNA - Electronic Transactions on Numerical Analysis, pp. A1-A14, 2024/01/15
Assume that $A\in{C}^{n\times n}$ is a block $p$-cyclic consistently ordered matrixand that its associated Jacobiiteration matrix $B$, which is weakly cyclic of index $p$, haseigenvalues $\mu$ whose $p$th powers are all realnonpositive (resp. nonnegative). Usually, one is interestedonly in the relaxation parameter $\omega$ that minimizes the spectral radiusof the iteration matrix of the associated SOR iterative method, but here weare interested in all real values for the relaxation parameter$\omega$ for which the SOR iteration matrix is convergent. This will beachieved for the values of $p=2,3,4,\ldots,$ and for $p\rightarrow\infty.$
Keywords: block p-cyclic matrix, weakly cyclic of index p matrix, block Jacobi and SOR iteration matrices, Schur–Cohn Algorithm