Generalizations of Nekrasov matrices and applications

Ljiljana Cvetković; Vladimir Kostić; Maja Nedović

Open Mathematics (2015)

  • Volume: 13, Issue: 1, page 96-105, electronic only
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Ljiljana Cvetković, Vladimir Kostić, and Maja Nedović. "Generalizations of Nekrasov matrices and applications." Open Mathematics 13.1 (2015): 96-105, electronic only. <http://eudml.org/doc/268683>.

@article{LjiljanaCvetković2015,
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.},
author = {Ljiljana Cvetković, Vladimir Kostić, Maja Nedović},
journal = {Open Mathematics},
keywords = {Nekrasov matrices; H-matrices; -matrices; nonsingularity criterion; inverse matrix; numerical example},
language = {eng},
number = {1},
pages = {96-105, electronic only},
title = {Generalizations of Nekrasov matrices and applications},
url = {http://eudml.org/doc/268683},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Ljiljana Cvetković
AU - Vladimir Kostić
AU - Maja Nedović
TI - Generalizations of Nekrasov matrices and applications
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - 96
EP - 105, electronic only
AB - 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.
LA - eng
KW - Nekrasov matrices; H-matrices; -matrices; nonsingularity criterion; inverse matrix; numerical example
UR - http://eudml.org/doc/268683
ER -

References

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  1. [1] Berman, A., Plemmons, R.J.: Nonnegative matrices in the mathematical sciences. Classics in Applied Mathematics, vol. 9, SIAM, Philadelphia, 1994. Zbl0815.15016
  2. [2] Cvetkovi´c, Lj.: H-matrix theory vs. eigenvalue localization. Numer. Algor. 42(2006), 229-245. 
  3. [3] Cvetkovi´c, Lj, Ping-Fan Dai, Doroslovaˇcki, K., Yao-Tang Li: Infinity norm bounds for the inverse of Nekrasov matrices. Appl. Math. Comput. 219, 10 (2013), 5020-5024. Zbl1283.15014
  4. [4] Cvetkovi´c, Lj., Kosti´c, V., Rauški, S., A new subclass of H-matrices. Appl. Math. Comput. 208/1(2009), 206-210.[WoS] 
  5. [5] Gudkov, V.V.: On a certain test for nonsingularity of matrices. Latv. Mat. Ezhegodnik 1965, Zinatne, Riga (1966), 385-390. 
  6. [6] Li, W.: On Nekrasov matrices. Linear Algebra Appl. 281(1998), 87-96. Zbl0937.15019
  7. [7] Robert, F.: Blocs H-matrices et convergence des methodes iteratives classiques par blocs. Linear Algebra Appl. 2(1969), 223-265.[Crossref] Zbl0182.21302
  8. [8] Szulc, T.: Some remarks on a theorem of Gudkov. Linear Algebra Appl. 225(1995), 221-235. Zbl0833.15020
  9. [9] Varah, J. M.: A lower bound for the smallest value of a matrix. Linear Algebra Appl. 11(1975), 3-5. Zbl0312.65028

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