Skip to main content

Advertisement

Log in

Error Bounds for Linear Complementarity Problems of Nekrasov and Generalized Nekrasov Matrices

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

We first propose a new error bound for the linear complementarity problems when the involved matrices are generalized Nekrasov matrices, which generalizes the recent result obtained by Li et al. (Numer. Algorithms 74:997–1009, 2017). Then we present two new error bounds for the linear complementarity problems when the involved matrices are Nekrasov matrices. Numerical examples are given to illustrate the effectiveness of the proposed results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. Classics in Applied Mathematics, vol. 9. SIAM, Philadelphia (1994)

    Book  Google Scholar 

  2. Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, San Diego (1992)

    Google Scholar 

  3. Murty, K.G.: Linear Complementarity, Linear and Nonlinear Programming. Heldermann, Berlin (1988)

    Google Scholar 

  4. Chen, X.J., Xiang, S.H.: Perturbation bounds of P-matrix linear complementarity problems. SIAM J. Optim. 18, 1250–1265 (2008)

    Article  MathSciNet  Google Scholar 

  5. Chen, X.J., Xiang, S.H.: Computation of error bounds for P-matrix linear complementarity problems. Math. Program., Ser. A 106, 513–525 (2006)

    Article  MathSciNet  Google Scholar 

  6. Hou, Z.W., Jing, X., Gao, L.: New error bounds for linear complementarity problems of \({\Sigma}\)-SDD matrices and SB-matrices. Open Math. 17, 1599–1614 (2019). https://doi.org/10.1515/math-2019-0127

    Article  MathSciNet  Google Scholar 

  7. García-Esnaola, M., Peña, J.M.: Error bounds for linear complementarity problems of Nekrasov matrices. Numer. Algorithms 67, 655–667 (2014)

    Article  MathSciNet  Google Scholar 

  8. Gao, L., Li, C.Q., Li, Y.T.: An improvement of the error bounds for linear complementarity problems of Nekrasov matrices. Linear Multilinear Algebra 66(8), 1505–1519 (2018)

    Article  MathSciNet  Google Scholar 

  9. Li, C.Q., Dai, P.F., Li, Y.T.: New error bounds for linear complementarity problems of Nekrasov matrices and B-Nekrasov matrices. Numer. Algorithms 74, 997–1009 (2017)

    Article  MathSciNet  Google Scholar 

  10. Dai, P.F., Li, J., Bai, J., Dong, L.Q.: Notes on new error bounds for linear complementarity problems of Nekrasov matrices, B-Nekrasov matrices and QN-matrices. Numer. Math., Theory Methods Appl. 12, 1191–1212 (2019)

    MathSciNet  Google Scholar 

  11. Li, C.Q., Yang, S., Huang, H., et al.: Note on error bounds for linear complementarity problems of Nekrasov matrices. Numer. Algorithms 83, 355–372 (2020)

    Article  MathSciNet  Google Scholar 

  12. Gao, L., Wang, Y.Q., Li, C.Q., Li, Y.T.: Error bounds for the linear complementarity problem of S-Nekrasov matrices and B-S-Nekrasov matrices. J. Comput. Appl. Math. 336, 147–159 (2018)

    Article  MathSciNet  Google Scholar 

  13. Dai, P.F., Li, J.C., Li, Y.T., Zhang, C.Y.: Error bounds for the linear complementarity problem of QN-matrices. Calcolo 53, 647–657 (2016)

    Article  MathSciNet  Google Scholar 

  14. Gao, L., Wang, Y.Q., Li, C.Q.: New error bounds for the linear complementarity problem of QN-matrices. Numer. Algorithms 77, 229–242 (2018)

    Article  MathSciNet  Google Scholar 

  15. Nedović, M., Cvetković, L.: Norm bounds for the inverse and error bounds for linear complementarity problems for \(\{P_{1}, P_{2}\}\)-Nekrasov matrices. Filomat 35(1), 239–250 (2021)

    Article  MathSciNet  Google Scholar 

  16. 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)

    Article  MathSciNet  Google Scholar 

  17. Gao, L., Gu, X.M., Jia, X.D., Li, C.Q.: Upper triangulation-based infinity norm bounds for the inverse of Nekrasov matrices with applications. Numer. Algorithms (2024). https://doi.org/10.1007/s11075-024-01758-3

    Article  Google Scholar 

  18. Orera, H., Peña, J.M.: Infinity norm bounds for the inverse of Nekrasov matrices using scaling matrices. Appl. Math. Comput. 358, 119–127 (2019)

    Article  MathSciNet  Google Scholar 

  19. Wang, S., Liang, N., Zhou, Y., Lyu, Z.: Two infinity norm bounds for the inverse of Nekrasov matrices. Linear Multilinear Algebra (2023). https://doi.org/10.1080/03081087.2023.2195150

    Article  Google Scholar 

  20. Liu, J., Zhang, J., Zhou, L., Tu, G.: The Nekrasov diagonally dominant degree on the Schur complement of Nekrasov matrices and its applications. Appl. Math. Comput. 320, 251–263 (2018)

    Article  MathSciNet  Google Scholar 

  21. Wang, S.Y., Li, Q., Sun, X., Lyu, Z.H.: Diagonal-Schur complements of Nekrasov matrices. Electron. J. Linear Algebra 39, 539–555 (2023)

    Article  MathSciNet  Google Scholar 

  22. Xue, J., Li, C., Li, Y.: On subdirect sums of Nekrasov matrices. Linear Multilinear Algebra (2023). https://doi.org/10.1080/03081087.2023.2172378

    Article  Google Scholar 

  23. Lyu, Z., Wang, X., Wen, L.: K-subdirect sums of Nekrasov matrices. Electron. J. Linear Algebra 38, 339–346 (2022)

    Article  MathSciNet  Google Scholar 

  24. Varah, J.M.: A lower bound for the smallest singular value of a matrix. Linear Algebra Appl. 11(1), 3–5 (1975)

    Article  MathSciNet  Google Scholar 

  25. Kolotilina, L.Y.: On bounding inverse to Nekrasov matrices in the infinity norm. Zap. Nauč. Semin. POMI 419, 111–120 (2013)

    Google Scholar 

  26. 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 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Funding

The work was supported in part by National Science Fund Project (12171323, 11701390), Liaoning Provincial Department of Education Program (JYTMS20230281), and LiaoNing Revitalization Talents Program (XLYC2002017).

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to Shiyun Wang or Zhen-Hua Lyu.

Ethics declarations

Competing Interests

The authors declare that there is no conflict of interest in the manuscript.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, S., Liu, D., Tian, W. et al. Error Bounds for Linear Complementarity Problems of Nekrasov and Generalized Nekrasov Matrices. Acta Appl Math 191, 9 (2024). https://doi.org/10.1007/s10440-024-00659-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10440-024-00659-w

Keywords

Mathematics Subject Classification

Navigation