Search Advanced

AIMS Mathematics

2023, Volume 8, Issue 9: 19876-19891. doi: 10.3934/math.20231013
Previous Article Next Article
Research article

SOR-based alternately linearized implicit iteration method for nonsymmetric algebraic Riccati equations

  • School of Computer Science and Mathematics, Fujian University of Technology, Fuzhou 350118, China
  • Received: 18 April 2023 Revised: 29 May 2023 Accepted: 06 June 2023 Published: 14 June 2023

  • MSC : 15A24, 15A57

  • In this paper, we propose a class of successive over relaxation-based alternately linearized implicit iteration method for computing the minimal nonnegative solution of nonsymmetric algebraic Riccati equations. Under certain conditions, we prove the convergence of the iterative method. Finally, numerical examples are given to show the iterative method is efficient.

    Citation: Chunjuan Du, Tongxin Yan. SOR-based alternately linearized implicit iteration method for nonsymmetric algebraic Riccati equations[J]. AIMS Mathematics, 2023, 8(9): 19876-19891. doi: 10.3934/math.20231013

    Related Papers:

  • In this paper, we propose a class of successive over relaxation-based alternately linearized implicit iteration method for computing the minimal nonnegative solution of nonsymmetric algebraic Riccati equations. Under certain conditions, we prove the convergence of the iterative method. Finally, numerical examples are given to show the iterative method is efficient.


    加载中


    [1] A. Berman, R. J. Plemmons, Nonnegative matrices in the mathematical sciences, Academic Press, 1994. https://doi.org/10.1137/1.9781611971262
    [2] N. G. Bean, M. M. O'Reilly, P. G. Taylor, Algorithms for return probabilities for stochastic fluid flows, Stoch. Models, 21 (2005), 149–184. https://doi.org/10.1081/STM-200046511 doi: 10.1081/STM-200046511
    [3] R. Bellman, G. M. Wing, An introduction to invariant iembedding, Academic Press, 1975. http://doi.org/10.1137/1.9781611971279
    [4] Z. Z. Bai, X. X. Guo, S. F. Xu, Alternately linearized implicit iteration methods for the minimal nonnegative solutions of the nonsymmetric algebraic Riccati equations, Numer. Linear Algebra Appl., 13 (2006), 655–674. https://doi.org/10.1002/nla.500 doi: 10.1002/nla.500
    [5] F. Coron, Computation of the asymptotic states for linear half space kinetic problem, Transp. Theory Stat. Phys., 19 (1990), 89–114. https://doi.org/10.1080/00411459008214506 doi: 10.1080/00411459008214506
    [6] B. D. Ganapol, An investigation of a simple transport model, Transp. Theory Stat. Phys., 21 (1991), 1–37. https://doi.org/10.1080/00411459208203520 doi: 10.1080/00411459208203520
    [7] C. H. Guo, Nonsymmetric algebraic Riccati equations and Wiener-Hopf factorization for $M$-matrices, SIAM J. Matrix Anal. Appl., 23 (2001), 225–242. https://doi.org/10.1137/S0895479800375680 doi: 10.1137/S0895479800375680
    [8] C. H. Guo, A new class of nonsymmetric algebraic Riccati equations, Linear Algebra Appl., 426 (2007), 636–649. https://doi.org/10.1016/j.laa.2007.05.044 doi: 10.1016/j.laa.2007.05.044
    [9] C. H. Guo, On algebraic Riccati equations associated with $M$-matrices, Linear Algebra Appl., 439 (2013), 2800–2814. https://doi.org/10.1016/j.laa.2013.08.018 doi: 10.1016/j.laa.2013.08.018
    [10] C. H. Guo, D. Lu, On algebraic Riccati equations associated with regular singular $M$-matrices, Linear Algebra Appl., 493 (2016), 108–119. https://doi.org/10.1016/j.laa.2015.11.024 doi: 10.1016/j.laa.2015.11.024
    [11] J. R. Guan, Modified alternately linearized implicit iteration method for $M$-matrix algebraic Riccati equations, Appl. Math. Comput., 347 (2019), 442–448. https://doi.org/10.1016/j.amc.2018.11.025 doi: 10.1016/j.amc.2018.11.025
    [12] P. C. Guo, X. X. Guo, A modified structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equations from transport theory, J. Comput. Appl. Math., 261 (2014), 213–220. https://doi.org/10.1016/j.cam.2013.09.058 doi: 10.1016/j.cam.2013.09.058
    [13] X. X. Guo, W. W. Lin, S. F. Xu, A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation, Numer. Math., 103 (2006), 393–412. https://doi.org/10.1007/s00211-005-0673-7 doi: 10.1007/s00211-005-0673-7
    [14] J. Juang, W. W. Lin, Nonsymmetric algebraic Riccati equations and Hamiltonian-like matrices, SIAM J. Matrix Anal. Appl., 20 (1998), 228–243. https://doi.org/10.1137/S0895479897318253 doi: 10.1137/S0895479897318253
    [15] H. Z. Lu, C. F. Ma, A new linearized implicit iteration method for nonsymmetric algebraic Riccati equations, J. Appl. Math. Comput., 50 (2016), 227–241. https://doi.org/10.1007/s12190-015-0867-9 doi: 10.1007/s12190-015-0867-9
    [16] L. Z. Lu, Solution form and simple iteration of a nonsymmetric algebraic Riccati equation arising in transport theory, SIAM J. Matrix Anal. Appl., 26 (2005), 679–685. https://doi.org/10.1137/S0895479801397275 doi: 10.1137/S0895479801397275
    [17] Y. Q. Lin, L. Bao, Convergence analysis of the Newton-Shamanskii method for a nonsymmetric algebraic Riccati equation, Numer Linear Algebra Appl., 15 (2008), 535–546. https://doi.org/10.1002/nla.582 doi: 10.1002/nla.582
    [18] J. A. Meijerink, H. A. van der Vorst, An iterative solution method for linear systems of which the coefficient matrix is a symmetric $M$-matrix, Math. Comput., 31 (1977), 148–162. https://doi.org/10.1090/S0025-5718-1977-0438681-4 doi: 10.1090/S0025-5718-1977-0438681-4
    [19] S. Miyajima, Fast verified computation for solutions of algebraic Riccati equations arising in transport theory, Numer. Linear Algebra Appl., 24 (2017), e2098. https://doi.org/10.1002/nla.2098 doi: 10.1002/nla.2098
    [20] L. C. G. Rogers, Fluid models in queueing theory and Wiener-Hopf factorization of Markov chains, Ann. Appl. Probab., 4 (1994), 390–413.
    [21] Y. Peretz, On applications of Schauder's fixed-point theorem for the solution of the non-symmetric algebraic Riccati equation, Linear Algebra Appl., 445 (2014), 29–55. https://doi.org/10.1016/j.laa.2013.12.012 doi: 10.1016/j.laa.2013.12.012
  • 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(1044) PDF downloads(95) Cited by(0)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(18)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog