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Classes of general \(H\)-matrices. (English) Zbl 1171.15026

Three classes of \(H\)-matrices \(A \in \mathbb C ^{ n \times n}\), one of them with non-singular and the other two with singular comparison \(M\)-matrices \({\mathcal M}(A)\) are examined. In the third mixed class there are both singular and non-singular matrices in the set of equimodular matrices of \(A\). Necessary or sufficient conditions are obtained to conclude as to whether a given \(A\) is an \(H\)-matrix, and if so, to which class in question it belongs. A classification of the \(H\)-matrices considered then follows.

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices
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[1] Alanelli, M.; Hadjidimos, A., A new iterative criterion for H-matrices, SIAM J. matrix anal. appl., 29, 1, 160-176, (2006) · Zbl 1140.65031
[2] Berman, A.; Plemmons, R.J., Nonnegative matrices in the mathematical sciences, (1979), Academic Press London, (reprinted and updated, SIAM, Philadelphia, 1994) · Zbl 0484.15016
[3] Boman, E.G.; Chen, D.; Parekh, O.; Toledo, S., On factor width and symmetric H-matrices, Linear algebra appl., 405, 239-248, (2005) · Zbl 1098.15014
[4] C. Corral, I. Giménez, J. Mas, Algorithms based on diagonal dominance: H-matrix, Perron vector and reducibility, in: Proceedings of the Fifth Int. Conf. on Eng. Comp. Tech., Las Palmas de Gran Canaria, 2006.
[5] Cvetkovic, L.; Kostic, V., New criteria for identifying H-matrices, J. comput. appl. math., 180, 265-278, (2005) · Zbl 1073.65038
[6] Cvetkovic, L.; Kostic, V.; Varga, R.S., A new geršgoring-type eigenvalue inclusion set, Electron. trans. numer. anal., 18, 73-80, (2004) · Zbl 1069.15016
[7] Elsner, L.; Mehrmann, V., Convergence of block iterative methods for linear systems arising in the numerical solution of Euler equations, Numer. math., 59, 541-559, (1991) · Zbl 0744.65026
[8] Fiedler, M.; Markham, T.L., A classification of matrices of class Z, Linear algebra appl., 173, 115-124, (1992) · Zbl 0754.15016
[9] Gan, T.-B.; Huang, T.-Z., Simple criteria for nonsingular H-matrices, Linear algebra appl., 374, 317-326, (2003) · Zbl 1033.15019
[10] Gao, Y.-M.; Wang, X.-H., Criteria for generalized diagonally dominant matrices and M-matrices, Linear algebra appl., 169, 257-268, (1992) · Zbl 0757.15010
[11] Huang, T.; Shen, S.; Li, H., On generalized H-matrices, Linear algebra appl., 396, 81-90, (2005) · Zbl 1065.65050
[12] Kohno, T.; Niki, H.; Sawami, H.; Gao, Y.-M., An iterative test for H-matrix, J. comput. appl. math., 115, 349-355, (2000) · Zbl 0946.65021
[13] Li, B.; Li, L.; Harada, M.; Niki, H.; Tsatsomeros, M.J., An iterative criterion for H-matrices, Linear algebra appl., 271, 179-190, (1998) · Zbl 0891.15021
[14] Nabben, R., On a class of matrices which arise in the numerical solution of Euler equations, Numer. math., 63, 411-431, (1992) · Zbl 0764.65019
[15] Nabben, R., Z-matrices and inverse Z-matrices, Linear algebra appl., 256, 31-48, (1997) · Zbl 0874.15003
[16] Ostrowski, A.M., Über die determinanten mit überwiegender hauptdiagonale, Comment. math. helv., 10, 69-96, (1937) · JFM 63.0035.01
[17] Robert, F., Blocs H-matrices et convergence des methodes iteratives classiques par blocs, Linear algebra appl., 2, 223-265, (1969) · Zbl 0182.21302
[18] Varga, R.S., On recurring theorems on diagonal dominance, Linear algebra appl., 13, 1-9, (1976) · Zbl 0336.15007
[19] Varga, R.S., Matrix iterative analysis, (1962), Prentice Hall Englewoods Cliffs, New Jersey, (reprinted and updated, Springer, Berlin, 2000) · Zbl 0133.08602
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