Error bounds for linear complementarity problems of $ SD{{D}_{1}} $ matrices and $ SD{{D}_{1}} $-$ B $ matrices

Yingxia Zhao; Lanlan Liu; Feng Wang; Yingxia Zhao; Lanlan Liu; Feng Wang

doi:10.3934/math.2022662

AIMS Mathematics

2022, Volume 7, Issue 7: 11862-11878. doi: 10.3934/math.2022662

Previous Article Next Article

Research article

Error bounds for linear complementarity problems of $ SD{{D}_{1}} $ matrices and $ SD{{D}_{1}} $-$ B $ matrices

College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, Guizhou 550025, China

Received: 10 December 2021 Revised: 25 March 2022 Accepted: 06 April 2022 Published: 20 April 2022
MSC : 15A48, 65G50, 90C31, 90C33

An upper bound of the infinity norm for the inverse of $ SD{D_1} $ matrix is presented. We apply the new bound to linear complementarity problems (LCPs) and obtain an alternative error bound for LCPs of $ SD{D_1} $ matrices and $ SD{{D}_{1}} $-$ B $ matrices. In addition, a new lower bound for the smallest singular value is also given. Numerical examples show the validity of the results.
- error bounds,
- linear complementarity problems,
- infinity norm,
- $ SD{D_1} $ matrices,
- $ SD{{D}_{1}} $-$ B $ matrices,
- singular values
Citation: Yingxia Zhao, Lanlan Liu, Feng Wang. Error bounds for linear complementarity problems of $ SD{{D}_{1}} $ matrices and $ SD{{D}_{1}} $-$ B $ matrices[J]. AIMS Mathematics, 2022, 7(7): 11862-11878. doi: 10.3934/math.2022662

Related Papers:

Abstract

An upper bound of the infinity norm for the inverse of $ SD{D_1} $ matrix is presented. We apply the new bound to linear complementarity problems (LCPs) and obtain an alternative error bound for LCPs of $ SD{D_1} $ matrices and $ SD{{D}_{1}} $-$ B $ matrices. In addition, a new lower bound for the smallest singular value is also given. Numerical examples show the validity of the results.

References

[1]	A. Berman, R. J. Plemmons, Nonnegative matrix in the mathematical sciences, Society for Industrial and Applied Mathematics, 1994.
[2]	R. W. Cottle, J. S. Pang, R. E. Stone, The linear complementarity problem, SIAM, 1992.
[3]	K. G. Murty, F. T. Yu, Linear Complementarity, Linear and nonlinear Programming, Berlin: Heldermann Verlag, 1998.
[4]	X. J. Chen, S. H. Xiang, Computation of error bounds for $P$-matrix linear complementarity problem, Math. Program., 106 (2006), 513–525. https://doi.org/10.1007/s10107-005-0645-9 doi: 10.1007/s10107-005-0645-9
[5]	J. C. Li, G. Li, Error bounds for linear complementarity problems of $S$-$QN$ matrices, Numer. Algor., 83 (2020), 935–955. https://doi.org/10.1007/s11075-019-00710-0 doi: 10.1007/s11075-019-00710-0
[6]	L. Cvetkovi$\acute{c}$, V. Kosti$\acute{c}$, S. Rau$\check{s}$ki, A new subclass of $H$-matrices, Appl. Math. Comput., 208 (2009), 20–210. https://doi.org/10.1016/j.amc.2008.11.037 doi: 10.1016/j.amc.2008.11.037
[7]	L. Y. Kolotilina, Some bounds for inverses involving matrix sparsity pattern, J. Math. Sci., 249 (2020), 242–255. https://doi.org/10.1007/s10958-020-04938-3 doi: 10.1007/s10958-020-04938-3
[8]	J. X. Zhao, Q. L. Liu, C. Q. Li, Y. T. Li, Dashnic-Zusmanovich type matrices: A new subclass of nonsingular $H$-matrices, Linear Algebra Appl., 552 (2018), 277–287. https://doi.org/10.1016/j.laa.2018.04.028 doi: 10.1016/j.laa.2018.04.028
[9]	L. Gao, Y. Q. Wang, C. Q. Li, Y. T. Li, Error bounds for linear complementarity problems of $S$-Nekrasov matrices and $B$-$S$-Nekrasov matrices, J. Comput. Appl. Math., 336 (2018), 147–159. https://doi.org/10.1016/j.cam.2017.12.032 doi: 10.1016/j.cam.2017.12.032
[10]	P. F. Dai, J. C. Li, J. C. Bai, L. Q. Dong, New error bounds for linear complementarity problems of $S$-Nekrasovmatrices and $B$-$S$-Nekrasovmatrices, Comput. Appl. Math., 38 (2019), 61. https://doi.org/10.1007/s40314-019-0818-4 doi: 10.1007/s40314-019-0818-4
[11]	M. García-Esnaola, J. M. Pe$\tilde{n}$a, $B^{R}_{\pi}$-matrices and error bounds for linear complementarity problems, Calcolo, 54 (2017), 813–822. https://doi.org/10.1007/s10092-016-0209-9 doi: 10.1007/s10092-016-0209-9
[12]	C. Q. Li, P. F. Dai, Y. T. Li, New error bounds for linear complementarity problems of Nekrasov matrices and $B$-Nekrasov matrices, Numer. Algor., 74 (2017), 997–1009. https://doi.org/10.1007/s11075-016-0181-0 doi: 10.1007/s11075-016-0181-0
[13]	M. García-Esnaola, J. M. Pe$\tilde{n}$a, $B$-Nekrasov matrices and error bounds for linear complementarity problems, Numer Algor., 72 (2016), 435–445. https://doi.org/10.1007/s11075-015-0054-y doi: 10.1007/s11075-015-0054-y
[14]	R. J. Zhao, B. Zheng, M. L. Liang, A new error bound for linear complementarity problems with weakly chained diagonally dominant $B$-matrices, Appl. Math. Comput., 367 (2020), 124788. https://doi.org/10.1016/j.amc.2019.124788 doi: 10.1016/j.amc.2019.124788
[15]	X. Song, L. Gao, $CKV$-type $B$-matrices and error bounds for linear complementarity problems. AIMS Math., 6 (2020), 10846–10860. https://doi.org/10.3934/math.2021630 doi: 10.3934/math.2021630
[16]	P. F. Dai, Error bounds for linear complementarity problems of $DB$-matrices, Linear Algebra Appl., 434 (2011), 830–840. https://doi.org/10.1016/j.laa.2010.09.049 doi: 10.1016/j.laa.2010.09.049
[17]	T. T. Chen, W. Li, X. P. Wu, S. Vong, Error bounds for linear complementarity problems of $MB$-matrices, Numer. Algor., 70 (2015), 341–356. https://doi.org/10.1007/s11075-014-9950-9 doi: 10.1007/s11075-014-9950-9
[18]	C. Q. Li, Y. T. Li, Note on error bounds for linear complementarity problems for $B$-matrices, Appl. Math. Lett., 57 (2016), 108–113. https://doi.org/10.1016/j.aml.2016.01.013 doi: 10.1016/j.aml.2016.01.013
[19]	D. S. Sun, F. Wang, New error bounds for linear complementarity problem of weakly chained diagonally dominant $B$-matrices, Open Math., 15 (2017), 978–986. https://doi.org/10.1515/math-2017-0080 doi: 10.1515/math-2017-0080
[20]	R. Bru, F. Pedroche, D. B. Szyld, Subdirect sums of $S$-strictly diagonally dominant matrices, Electron. J. Linear Al., 15 (2006), 201–209. https://doi.org/10.13001/1081-3810.1230 doi: 10.13001/1081-3810.1230
[21]	A. Berman, R. J. Plemmons, Nonnegative matrices in the mathematical sciences, Academic Press, 1979.
[22]	J. M. Pe$\tilde{n}$a, Diagonal dominance, Schur complements and some classes of $H$-matrices and $P$-matrices, Adv. Comput. Math., 35 (2011), 357–373. https://doi.org/10.1007/s10444-010-9160-5 doi: 10.1007/s10444-010-9160-5
[23]	L. Gao, An alternative error bound for linear complementarily problems involving ${B^S}$-matrices, J. Inequal. Appl., 2018 (2018), 28. https://doi.org/10.1186/s13660-018-1618-x doi: 10.1186/s13660-018-1618-x
[24]	P. Wang, An upper bound for ${\left\\| {{A^{ - 1}}} \right\\|_\infty }$ of strictly diagonally dominant $M$-matrices, Linear Algebra Appl., 431 (2009), 511–517. https://doi.org/10.1016/j.laa.2009.02.037 doi: 10.1016/j.laa.2009.02.037
[25]	L. Zou, A lower bound for the smallest singular value, J. Math. Inequal., 6 (2012), 625–629. https://doi.org/10.7153/jmi-06-60 doi: 10.7153/jmi-06-60
[26]	R. A. Horn, C. R. Johnson, Matrix analysis, Cambridge: Cambridge University Press, 1985.
[27]	R. A. Horn, C. R. Johnson, Topics in matrix analysis, Cambridge: Cambridge University Press, 1991.
[28]	J. M. Varah, A lower bound for the smallest singular value of a matrix, Linear Algebra Appl., 11 (1975), 3–5. https://doi.org/10.1016/0024-3795(75)90112-3 doi: 10.1016/0024-3795(75)90112-3

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)