New Subclasses of the Class of $\mathrm{\mathscr{H}}$-Matrices and Related Bounds for the Inverses

The paper introduces new subclasses, called P$\mathrm{\mathscr{H}}$N(π) and P$\mathrm{\mathscr{H}}$QN(π), of (nonsingular) $\mathrm{\mathscr{H}}$-matrices of order n dependent on a partition π of the index set {1, . . ., n}, which generalize the classes P$\mathrm{\mathscr{H}}$(π), introduced previously, and contain, in particular, such subclasses as those of strictly diagonally dominant (SDD), Nekrasov, S-SDD, S-Nekrasov, QN, and P$\mathrm{\mathscr{H}}$(π) matrices. Properties of the matrices introduced are studied, and upper bounds on their inverses in l _∞ norm are obtained. Block generalizations of the classes P$\mathrm{\mathscr{H}}$N(π) and P$\mathrm{\mathscr{H}}$QN(π) in the sense of Robert are considered.

Also a general approach to defining subclasses ${\mathcal{K}}^{\pi }$ of the class $\mathrm{\mathscr{H}}$ containing a given subclass $\mathcal{K}$ ⊂ $\mathrm{\mathscr{H}}$ and dependent on a partition π is presented.