Abstract
We introduce strong B-matrices and strong B-Nekrasov matrices, for which some error bounds for linear complementarity problems are analyzed. In particular, it is proved that the bounds of García-Esnaola and Peña (Appl. Math. Lett. 22, 1071–1075, 2009) and of (Numer. Algor. 72, 435–445, 2016) are asymptotically optimal for strong B-matrices and strong B-Nekrasov matrices, respectively. Other comparisons with a bound of Li and Li (Appl. Math. Lett. 57, 108–113, 2016) are performed.
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Chen, X., Xiang, S.: Computation of error bounds for P-matrix linear complementarity problems. Math. Program., Ser. A 106, 513–525 (2006)
Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problems. Academic Press, Boston MA (1992)
Cvetković, L., Dai, P.-F., Doroslovaški, K., Li, Y.-T.: Infinity norm bounds for the inverse of Nekrasov matrices. Appl. Math. Comput. 219, 5020–5024 (2013)
Dai, P.-F.: Error bounds for linear complementarity problems of DB-matrices. Linear Algebra Appl. 434, 830–840 (2011)
García-Esnaola, M., Peña, J.M.: Error bounds for linear complementarity problems for B-matrices. Appl. Math. Lett. 22, 1071–1075 (2009)
García-Esnaola, M., Peña, J.M.: A comparison of error bounds for linear complementarity problems of H-matrices. Linear Algebra Appl. 433, 956–964 (2010)
García-Esnaola, M., Peña, J.M.: Error bounds for linear complementarity problems of B S-matrices. Appl. Math. Lett. 25, 1379–1383 (2012)
García-Esnaola, M., Peña, J.M.: B-Nekrasov matrices and error bounds for linear complementarity problems. Numer. Algor. 72, 435–445 (2016)
García-Esnaola, M., Peña, J.M.: \(B_{\pi }^{R}\)-matrices and error bounds for linear complementarity problems. Calcolo 54, 813–822 (2017)
Golub, G.H., Van Loan, C.F.: Matrix Computations, 4Th Edn. The Johns Hopkins University Press, Baltimore (2013)
Li, C., Li, Y.: Note on error bounds for linear complementarity problems for B-matrices. Appl. Math. Lett. 57, 108–113 (2016)
Mathias, R., Pang, J.-S.: Error bounds for the linear complementarity problem with a P-matrix. Linear Algebra Appl. 132, 123–136 (1990)
Peña, J.M.: A class of P-matrices with applications to the localization of the eigenvalues of a real matrix. SIAM J. Matrix Anal. Appl. 22, 1027–1037 (2001)
Peña, J.M.: On an alternative to Gerschgorin circles and ovals of Cassini. Numer. Math. 95, 337–345 (2003)
Schäfer, U.: An enclosure method for free boundary problems based on a linear complementarity problem with interval data. Numer. Funct. Anal. Optim. 22, 991–1011 (2001)
Szulc, T.: Some remarks on a theorem of Gudkov. Linear Algebra Appl. 225, 221–235 (1995)
Acknowledgements
The authors wish to thank the anonymous referees for their valuable suggestions to improve the paper. This work was partially supported by the Spanish Research Grant MTM2015-65433-P (MINECO/FEDER), Gobierno de Aragón and Fondo Social Europeo.
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This work was partially supported by the Spanish Research Grant MTM2015-65433-P (MINECO/FEDER), Gobierno de Aragón and Fondo Social Europeo.
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Research partially supported by the Spanish Research Grant MTM2015-65433-P (MINECO/FEDER) and by Gobierno de Aragón and Fondo Social Europeo.
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García-Esnaola, M., Peña, J.M. On the asymptotic optimality of error bounds for some linear complementarity problems. Numer Algor 80, 521–532 (2019). https://doi.org/10.1007/s11075-018-0495-1
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DOI: https://doi.org/10.1007/s11075-018-0495-1