Infinity norm bounds for the inverse of $ SDD_1^{+} $ matrices with applications

Lanlan Liu; Yuxue Zhu; Feng Wang; Yuanjie Geng; Lanlan Liu; Yuxue Zhu; Feng Wang; Yuanjie Geng

doi:10.3934/math.20241034

AIMS Mathematics

2024, Volume 9, Issue 8: 21294-21320. doi: 10.3934/math.20241034

Previous Article Next Article

Research article Special Issues

Infinity norm bounds for the inverse of $ SDD_1^{+} $ matrices with applications

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

Received: 08 May 2024 Revised: 21 June 2024 Accepted: 24 June 2024 Published: 02 July 2024
MSC : 15A48, 65G50, 90C33, 90C31

A new subclass of nonsingular $ H $-matrix named $ SDD_1^{+} $ matrices is studied in this paper. The relationships between $ SDD_1^{+} $ matrices and other subclasses of nonsingular $ H $-matrices are analyzed by numerical examples. Moreover, the infinity norm bounds of the inverse for $ SDD_1^{+} $ matrices are derived in two different methods. As applications, two error bounds of the linear complementarity problems ($ LCP $) for $ SDD_1^{+} $ matrices are given. Finally, the effectiveness of corresponding results is illustrated by numerical examples.
- $ H $-matrices,
- $ SDD_1^{+} $ matrices,
- infinity norm bounds,
- linear complementarity problems,
- error bounds
Citation: Lanlan Liu, Yuxue Zhu, Feng Wang, Yuanjie Geng. Infinity norm bounds for the inverse of $ SDD_1^{+} $ matrices with applications[J]. AIMS Mathematics, 2024, 9(8): 21294-21320. doi: 10.3934/math.20241034

Related Papers:

Abstract

A new subclass of nonsingular $ H $-matrix named $ SDD_1^{+} $ matrices is studied in this paper. The relationships between $ SDD_1^{+} $ matrices and other subclasses of nonsingular $ H $-matrices are analyzed by numerical examples. Moreover, the infinity norm bounds of the inverse for $ SDD_1^{+} $ matrices are derived in two different methods. As applications, two error bounds of the linear complementarity problems ($ LCP $) for $ SDD_1^{+} $ matrices are given. Finally, the effectiveness of corresponding results is illustrated by numerical examples.

References

[1]	A. Berman, R. Plemmons, Nonnegative Matrices in the Mathematical Sciences, New York: SIAM, 1994.
[2]	C. Zhang, New Advances in Research on H-matrices, Beijing: Science Press, 2017.
[3]	L. Cvetković, $H$-matrix theory vs. eigenvalue localization, Numer. Algor., 42 (2006), 229–245. http://doi.org/10.1007/s11075-006-9029-3 doi: 10.1007/s11075-006-9029-3
[4]	X. Gu, S. Wu, A parallel-in-time iterative algorithm for Volterra partial integro-differential problems with weakly singular kernel, J. Comput. Phys., 417 (2020), 109576. http://doi.org/10.1016/j.jcp.2020.109576 doi: 10.1016/j.jcp.2020.109576
[5]	A. Berman, R. Plemmons, Nonnegative Matrix in the Mathematical Sciences, New York: Academic Press, 1979.
[6]	R. Cottle, J. Pang, R. Stone, The Linear Complementarity Problem, New York: SIAM, 2009.
[7]	X. Chen, S. Xiang, Computation of error bounds for $P$-matrix linear complementarity problems, Math. Program., 106 (2006), 513–525. http://doi.org/10.1007/s10107-005-0645-9 doi: 10.1007/s10107-005-0645-9
[8]	P. Dai, J. Li, S. Zhao, Infinity norm bounds for the inverse for $GSDD_1$ matrices using scaling matrices, Comput. Appl. Math., 42 (2023), 121–141. http://doi.org/10.1007/s40314-022-02165-x doi: 10.1007/s40314-022-02165-x
[9]	D. Cvetković, L. Cvetković, C. Li, $CKV$-type matrices with applications, Linear Algebra Appl., 608 (2020), 158–184. http://doi.org/10.1016/j.laa.2020.08.028 doi: 10.1016/j.laa.2020.08.028
[10]	L. Kolotilina, Some bounds for inverses involving matrix sparsity pattern, J. Math. Sci., 249 (2020), 242–255. http://doi.org/10.1007/s10958-020-04938-3 doi: 10.1007/s10958-020-04938-3
[11]	L. Cvetković, Kostić, S. Rauški, A new subclass of $H$-matrices, Appl. Math. Comput., 208 (2009), 206–210. http://dx.doi.org/10.1016/j.amc.2008.11.037 doi: 10.1016/j.amc.2008.11.037
[12]	L. Cvetković, V. Kostic, R. Varga, A new Geršgorin-type eigenvalue inclusion set, Electron. Trans. Numer. Anal., 18 (2004), 73–80.
[13]	H. Orera, J. Peña, Infinity norm bounds for the inverse of Nekrasov matrices using scaling matrices, Appl. Math. Comput., 358 (2019), 119–127. http://doi.org/10.1016/j.amc.2019.04.027 doi: 10.1016/j.amc.2019.04.027
[14]	P. Dai, C. Lu, Y. Li, New error bounds for the linear complementarity problem with an $SB$-matrix, Numer. Algor., 64 (2013), 741–757. http://doi.org/10.1007/s11075-012-9691-6 doi: 10.1007/s11075-012-9691-6
[15]	P. Dai, Y. Li, C. Lu, Error bounds for linear complementarity problems for $SB$-matrices, Numer. Algor., 61 (2012), 121–139. http://doi.org/10.1007/s11075-012-9580-z doi: 10.1007/s11075-012-9580-z
[16]	C. Li, Y. Li, Note on error bounds for linear complementarity problems for $B$-matrices, Appl. Math. Lett., 57 (2016), 108–113. http://doi.org/10.1016/j.aml.2016.01.013 doi: 10.1016/j.aml.2016.01.013
[17]	R. Cottle, J. Pang, R. E. Stone, The Linear Complementarity Problem, San Diego: Academic Press, 1992.
[18]	X. Song, L. Gao, $CKV$-type $B$-matrices and error bounds for linear complementarity problems, AIMS Math., 6 (2021), 10846–10860. http://doi.org/10.3934/math.2021630 doi: 10.3934/math.2021630
[19]	J. Peña, Diagonal dominance, Schur complements and some classes of $H$-matrices and $P$-matrices, Adv. Comput. Math., 35 (2011), 357–373. http://doi.org/10.1007/s10444-010-9160-5 doi: 10.1007/s10444-010-9160-5
[20]	J. Varah, A lower bound for the smallest singular value of a matrix, Linear Algebra Appl., 11 (1975), 3–5. http://doi.org/10.1016/0024-3795(75)90112-3 doi: 10.1016/0024-3795(75)90112-3
[21]	R. S. Varga, Geršgorin and His Circles, Berlin: Springer-Verlag, 2004.
[22]	V. Kostić, L. Cvetković, D. Cvetković, Pseudospectra localizations and their applications, Numer. Linear Algebra Appl., 23 (2016), 356–372. http://doi.org/10.1002/nla.2028 doi: 10.1002/nla.2028
[23]	J. Peña, A class of $P$-matrices with applications to the localization of the eigenvalues of a real matrix, SIAM J. Matrix Anal. Appl., 22 (2001), 1027–1037. http://doi.org/10.1137/s0895479800370342 doi: 10.1137/s0895479800370342
[24]	M. Neumann, J. Peña, O. Pryporova, Some classes of nonsingular matrices and applications, Linear Algebra Appl., 438 (2013), 1936–1945. http://doi.org/10.1016/j.laa.2011.10.041 doi: 10.1016/j.laa.2011.10.041
[25]	J. Peña, On an alternative to Gerschgorin circles and ovals of Cassini, Numer. Math., 95 (2003), 337–345. http://doi.org/10.1007/s00211-002-0427-8 doi: 10.1007/s00211-002-0427-8
[26]	P. Dai, J. Li, Y. Li, C. Zhang, Error bounds for linear complementarity problems of $QN$-matrices, Calcolo, 53 (2016), 647–657. http://doi.org/10.1007/s10092-015-0167-7 doi: 10.1007/s10092-015-0167-7
[27]	J. Li, G. Li, Error bounds for linear complementarity problems of $S$-$QN$ matrices, Numer. Algor., 83 (2020), 935–955. http://doi.org/10.1007/s11075-019-00710-0 doi: 10.1007/s11075-019-00710-0
[28]	C. Li, P. Dai, Y. Li, New error bounds for linear complementarity problems of Nekrasov matrices and $B$-Nekrasov matrices, Numer. Algor., 74 (2016), 997–1009. http://doi.org/10.1007/s11075-016-0181-0 doi: 10.1007/s11075-016-0181-0
[29]	M. García-Esnaola, J. Peña, A comparison of error bounds for linear complementarity problems of $H$-matrices, Linear Algebra Appl., 433 (2010), 956–964. http://doi.org/10.1016/j.laa.2010.04.024 doi: 10.1016/j.laa.2010.04.024

Reader Comments

Your name:*

Email:*
© 2024 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)