New Subclasses of the Class of \( \mathrm{\mathscr{H}} \)-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.