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

Abstract

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.