Abstract:
The paper introduces and considers two new classes of the so-called $S$-SDD$_k(\sigma)$ and $S$-GSDD$_k(\sigma)$ matrices, where $S$ is a nonempty subset of the subset $R_A$ of strictly diagonally dominant rows of $A$, $k\ge 1$, and $\sigma\in (0, 1]$ is a real parameter. Properties of these matrices and their interrelations with some other matrix classes are considered. In particular, it is shown that $S$-SDD$_k(\sigma)$ and $S$-GSDD$_k(\sigma)$ matrices are nonsingular $\mathcal{H}$-matrices and, moreover, $SD$-SDD and $S$-SSDD (Schur SDD) matrices. Based on the latter result, a general parameter-free upper bound for the $l_\infty$-norms of the inverses to $S$-SDD$_k(\sigma)$ and $S$-GSDD$_k(\sigma)$ matrices is obtained. Also upper bounds for $\|A^{-1}\|_\infty$ based on a specific diagonal scaling of $S$-SDD$_k(\sigma)$ and $S$-SDD$_k(\sigma)$ matrices $A$, directly related to their definitions, are provided.
Key words and phrases:$S$-SDD$_k(\sigma)$ matrices, $S$-GSDD$_k(\sigma)$ matrices, SDD matrices, $S$-SDD matrices, $SD$-SDD matrices, $\mathcal{H}$-matrices, $l_\infty$-norm of the inverse, upper bounds.
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