Search Advanced

AIMS Mathematics

2023, Volume 8, Issue 5: 10558-10578. doi: 10.3934/math.2023536
Previous Article Next Article
Research article

New matrix splitting iteration method for generalized absolute value equations

  • Department of Mathematics, College of Sciences, Northeastern University, Shenyang 110819, China
  • Received: 06 November 2022 Revised: 09 February 2023 Accepted: 17 February 2022 Published: 02 March 2023

  • MSC : 65F10, 90C05, 90C30

  • In this paper, a relaxed Newton-type matrix splitting (RNMS) iteration method is proposed for solving the generalized absolute value equations, which includes the Picard method, the modified Newton-type (MN) iteration method, the shift splitting modified Newton-type (SSMN) iteration method and the Newton-based matrix splitting (NMS) iteration method. We analyze the sufficient convergence conditions of the RNMS method. Lastly, the efficiency of the RNMS method is analyzed by numerical examples involving symmetric and non-symmetric matrices.

    Citation: Wan-Chen Zhao, Xin-Hui Shao. New matrix splitting iteration method for generalized absolute value equations[J]. AIMS Mathematics, 2023, 8(5): 10558-10578. doi: 10.3934/math.2023536

    Related Papers:

  • In this paper, a relaxed Newton-type matrix splitting (RNMS) iteration method is proposed for solving the generalized absolute value equations, which includes the Picard method, the modified Newton-type (MN) iteration method, the shift splitting modified Newton-type (SSMN) iteration method and the Newton-based matrix splitting (NMS) iteration method. We analyze the sufficient convergence conditions of the RNMS method. Lastly, the efficiency of the RNMS method is analyzed by numerical examples involving symmetric and non-symmetric matrices.


    加载中


    [1] J. Rohn, A theorem of the alternatives for the equation Ax+B|x| = b, Linear Multilinear Algebra, 52 (2004), 421–426. https://doi.org/10.1080/0308108042000220686 doi: 10.1080/0308108042000220686
    [2] Z. Z. Bai, Modulus-based matrix splitting iteration methods for linear complementarity problems, Numer. Linear Algebra Appl., 17 (2010), 917–933. https://doi.org/10.1002/nla.680 doi: 10.1002/nla.680
    [3] O. L. Mangasarian, R. R. Meyer, Absolute value equations, Linear Algebra Appl., 419 (2006), 359–367. https://doi.org/10.1016/j.laa.2006.05.004 doi: 10.1016/j.laa.2006.05.004
    [4] O. L. Mangasarian, Absolute value programming, Comput. Optim. Appl., 36 (2007), 43–53. https://doi.org/10.1007/s10589-006-0395-5 doi: 10.1007/s10589-006-0395-5
    [5] J. Rohn, Systems of linear interval equations, Linear Algebra Appl., 126 (1989), 39–78. https://doi.org/10.1016/0024-3795(89)90004-9 doi: 10.1016/0024-3795(89)90004-9
    [6] L. Abdallah, M. Haddou, T. Migot, Solving absolute value equation using complementarity and smoothing functions, J. Comput. Appl. Math., 327 (2018), 196–207. https://doi.org/10.1016/j.cam.2017.06.019 doi: 10.1016/j.cam.2017.06.019
    [7] L. Caccetta, B. Qu, G. L. Zhou, A globally and quadratically convergent method for absolute value equations, Comput. Optim. Appl., 48 (2011), 45–58. https://doi.org/10.1007/s10589-009-9242-9 doi: 10.1007/s10589-009-9242-9
    [8] A. Wang, Y. Cao, J. X. Chen, Modified Newton-type iteration methods for generalized absolute value equations, J. Optim. Theory Appl., 181 (2019), 216–230. https://doi.org/10.1007/s10957-018-1439-6 doi: 10.1007/s10957-018-1439-6
    [9] O. L. Mangasarian, A generalized Newton method for absolute value equations, Optim. Lett., 3 (2009), 101–108. https://doi.org/10.1007/s11590-008-0094-5 doi: 10.1007/s11590-008-0094-5
    [10] Y. Cao, Q. Shi, S. L. Zhu, A relaxed generalized Newton iteration method for generalized absolute value equations, AIMS Math., 6 (2021), 1258–1275. https://doi.org/10.3934/math.2021078 doi: 10.3934/math.2021078
    [11] H. Y. Zhou, S. L. Wu, C. X. Li, Newton-based matrix splitting method for generalized absolute value equation, J. Comput. Appl. Math., 394 (2021), 113578. https://doi.org/10.1016/j.cam.2021.113578 doi: 10.1016/j.cam.2021.113578
    [12] C. Zhang, Q. J. Wei, Global and finite convergence of a generalized Newton method for absolute value equations, J. Optim. Theory Appl., 143 (2009), 391–403. https://doi.org/10.1007/s10957-009-9557-9 doi: 10.1007/s10957-009-9557-9
    [13] J. Rohn, An algorithm for solving the absolute value equations, Electron. J. Linear Algebra, 18 (2009), 589–599. https://doi.org/10.13001/1081-3810.1332 doi: 10.13001/1081-3810.1332
    [14] P. Guo, S. L. Wu, C. X. Li, On the SOR-like iteration method for solving absolute value equations, Appl. Math. Lett., 97 (2019), 107–113. https://doi.org/10.1016/j.aml.2019.03.033 doi: 10.1016/j.aml.2019.03.033
    [15] Y. F. Ke, C. F. Ma, SOR-like iteration method for solving absolute value equations, Appl. Math. Comput., 311 (2017), 195–202. https://doi.org/10.1016/j.amc.2017.05.035 doi: 10.1016/j.amc.2017.05.035
    [16] X. Dong, X. H. Shao, H. L. Shen, A new SOR-like method for solving absolute value equations, Appl. Numer. Math., 156 (2020), 410–421. https://doi.org/10.1016/j.apnum.2020.05.013 doi: 10.1016/j.apnum.2020.05.013
    [17] A. Mansoori, M. Erfanian, A dynamic model to solve the absolute value equations, J. Comput. Appl. Math., 333 (2018), 28–35. https://doi.org/10.1016/j.cam.2017.09.032 doi: 10.1016/j.cam.2017.09.032
    [18] C. R. Chen, Y. N. Yang, D. M. Yu, D. R. Han, An inverse-free dynamical system for solving the absolute value equations, Appl. Numer. Math., 168 (2021), 170–181. https://doi.org/10.1016/j.apnum.2021.06.002 doi: 10.1016/j.apnum.2021.06.002
    [19] S. L. Hu, Z. H. Huang, Q. Zhang, A generalized Newton method for absolute value equations associated with second order cones, J. Comput. Appl. Math., 235 (2012), 1490–1501. https://doi.org/10.1016/j.cam.2010.08.036 doi: 10.1016/j.cam.2010.08.036
    [20] X. Li, X. X. Yin, A new modified Newton-type iteration methods for solving generalized absolute value equations, arXiv, 2103. https://doi.org/10.48550/arXiv.2103.09452 doi: 10.48550/arXiv.2103.09452
    [21] C. X. Li, S. L. Wu, A shift splitting iteration method for generalized absolute value equations, Comput. Meth. Appl. Math., 21 (2021), 863–872. doi: 10.1515/cmam-2020-0004
    [22] D. F. Han, The majorant method and convergence for solving nondifferentiable equations in Banach space, Appl. Math. Comput., 118 (2001), 73–82. https://doi.org/10.1016/S0096-3003(99)00183-6 doi: 10.1016/S0096-3003(99)00183-6
    [23] J. L. Dong, M. Q. Jiang, A modified modulus method for symmetric positive-definite linear complementarity problems, Numer. Linear Algebra Appl., 16 (2009), 129–143. https://doi.org/10.1002/nla.609 doi: 10.1002/nla.609
    [24] D. K. Salkuyeh, The Picard-HSS iteration method for absolute value equations, Optim. Lett., 8 (2014), 2191–2202. https://doi.org/10.1007/s11590-014-0727-9 doi: 10.1007/s11590-014-0727-9
    [25] G. H. Golub, C. F. Van Loan, Matrix computations, Johns Hopkins University Press, 2013.
    [26] A. Frommer, G. Mayer, Convergence of relaxed parallel multisplitting methods, Linear Algebra Appl., 119 (1989), 141–152. https://doi.org/10.1016/0024-3795(89)90074-8 doi: 10.1016/0024-3795(89)90074-8
    [27] S. L. Wu, C. X. Li, Two-sweep modulus-based matrix splitting iteration methods for linear complementarity problems, J. Comput. Appl. Math., 302 (2016), 327–339. https://doi.org/10.1016/j.cam.2016.02.011 doi: 10.1016/j.cam.2016.02.011
  • Reader Comments
  • © 2023 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
1.8 3.4

Metrics

Article views(1373) PDF downloads(102) Cited by(3)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(8)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog