Infinity norm bounds for the inverse of Nekrasov matrices

Abstract

From the application point of view, it is important to have a good upper bound for the maximum norm of the inverse of a given matrix A. In this paper we will give two simple and practical upper bounds for the maximum norm of the inverse of a Nekrasov matrix.

Bounds for the Inverses of Generalized Nekrasov Matrices

“…As indicated in [2], in the particular case of Nekrasov matrices, the bounds (2.5) and (2.6) improve the corresponding bounds presented in [1].…”

“…As was shown in [9], the bound (1.5) generally improves the earlier bounds proposed by Cvetković et al in [1] and, for an SDD matrix A = (a ij ), the bound (1.5) is at least as good as the classical Varah bound [14]…”

Generalizations of Nekrasov matrices and applications

“…Matrix A 3 is neither SDD nor Nekrasov, but it does satisfy our new fP 1 ; P 2 g-Nekrasov condition, where P 1 is the identical permutation of order 6 and P 2 is counteridentical permutation of order 6. In the following table, we compare the results for max-norm bounds of the inverse matrix obtained using Theorem 3.1 and Theorem 3.2 of this paper to those of Varah, for SDD matrices, and to the bounds for Nekrasov matrices presented in [3] (in Table 1 we call them Nekrasov I and Nekrasov II). Exact values for the max-norm of the inverse matrix are as follows: As one can see from Table 1, our bounds are better than Varah for some SDD matrices, and, in some cases, they are better than bounds for Nekrasov matrices presented in [3].…”

“…In the following table, we compare the results for max-norm bounds of the inverse matrix obtained using Theorem 3.1 and Theorem 3.2 of this paper to those of Varah, for SDD matrices, and to the bounds for Nekrasov matrices presented in [3] (in Table 1 we call them Nekrasov I and Nekrasov II). Exact values for the max-norm of the inverse matrix are as follows: As one can see from Table 1, our bounds are better than Varah for some SDD matrices, and, in some cases, they are better than bounds for Nekrasov matrices presented in [3]. If the matrix is neither SDD nor Nekrasov, like, for example, A 3 , the only bounds that can be applied are bounds (8) and (10).…”

“…In [3], new max-norm bounds are given for the inverse of Nekrasov matrices using the well-known bound for the inverse of an SDD matrix by Varah (see [9]). -Varah bound for SDD matrices:…”