Generalizations of Nekrasov matrices and applications

Abstract

In this paper we present a nonsingularity result which is a generalization of Nekrasov property by using two different permutations of the index set. The main motivation comes from the following observation: matrices that are Nekrasov matrices up to the same permutations of rows and columns, are nonsingular. But, testing all the permutations of the index set for the given matrix is too expensive. So, in some cases, our new nonsingularity criterion allows us to use the results already calculated in order to conclude that the given matrix is nonsingular. Also, we present new max-norm bounds for the inverse matrix and illustrate these results by numerical examples, comparing the results to some already known bounds for Nekrasov matrices.

An improvement of the infinity norm bound for the inverse of $\{P_{1},P_{2}\}$-Nekrasov matrices

“…where z is an SDD matrix, where C is given by (7). By the proof of Theorem 3.1 (see [6]), we have that, for a fixed k ∈ {1, 2},…”

“…Motivated by Definition 1, Cvetković et al in [6] presented the following new subclass of H-matrices, called {P 1 , P 2 }-Nekrasov matrices, which contains Nekrasov matrices.…”

“…Next, we recall two upper bounds for the infinity norm of the inverse of {P 1 , P 2 }-Nekrasov matrices which are given by Cvetković et al in [6].…”

“…Moreover, when the class of involved matrices is a wider subclass of H-matrices, such as doubly strictly diagonally dominant (DSDD) matrices, S-SDD matrices, weakly chained diagonally dominant matrices, Nekrasov matrices, S-Nekrasov matrices, and DZ-type matrices, upper bounds for A -1 ∞ are derived, which sometimes are tighter in the SDD case, see [3,5,7,9,10,15,16,21] and the references therein. Recently, Cvetković et al [6] presented two upper bounds for A -1 ∞ involved with {P 1 , P 2 }-Nekrasov matrices, which are only dependent on the entries of the matrix A.…”

Scaling technique for Partition-Nekrasov matrices

“…The original setting was in terms of the convergence of the Gauss-Seidel iteration (see [15]). This class of matrices was further discussed in many papers and it was used to obtain max-norm bounds of the inverse, bounds for determinants, and, also, this class was a starting point for many different generalisations, made in order to expand this nonsingularity result to wider classes of matrices (see [5][6][7]16,19,24]). Because of the way Nekrasov class is defined, involving recursively calculated row sums, finding the whole class of corresponding diagonal scaling matrices (as it is done for Partition-SDD) is not an easy task.…”