SSDD matrices and relations with other subclasses of nonsingular $\mathcal H$-matrices

L. Yu. Kolotilina

Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXVII, Tome 534 (2024), pp. 57-88

Résumé

The paper introduces into consideration a new matrix class of the so-called SSDD (Schur SDD) matrices, which contains the class of SDD (strictly diagonally dominant) matrices and is itself contained in the class of nonsingular $\mathcal{H}$-matrices. The definition of an SSDD matrix $A$ is based on distinguishing a subset $S$ of its strictly diagonally dominant rows and requiring that the Schur complement $\mathcal{M}(A)/S$ of its comparison matrix be an SDD matrix. Properties of SSDD matrices and their relations with other subclasses of the class of $\mathcal{H}$-matrices are considered. In particular, it is shown that such known matrix classes as those of ОВ, SOB, DZ, DZT (DZ-type), CKV-type, $S$-SDD, SDD$_1$, SDD$_k$, GSDD$_1$, and also GSDD$_1^*$ matrices all are contained in the class of SSDD matrices. On the other hand, the SSDD matrices themselves are simultaneously РН- and $SD$-SDD matrices and, up to symmetric row and column permutations, they coincide with the block $2\times 2$ generalized Nekrasov matrices, the so-called GN matrices. Also some upper bounds for the $l_\infty$-norm of the inverse to an SSDD matrix are established.