A NOVEL ITERATIVE METHOD FOR SOLVING THE COUPLED SYLVESTER-CONJUGATE MATRIX EQUATIONS AND ITS APPLICATION IN ANTILINEAR SYSTEM

Wenli Wang; Caiqin Song

doi:10.11948/20220032

2023 Volume 13 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2023 > 13(1): 249-274

Next Article Previous Article

Wenli Wang, Caiqin Song. A NOVEL ITERATIVE METHOD FOR SOLVING THE COUPLED SYLVESTER-CONJUGATE MATRIX EQUATIONS AND ITS APPLICATION IN ANTILINEAR SYSTEM[J]. Journal of Applied Analysis & Computation, 2023, 13(1): 249-274. doi: 10.11948/20220032

Citation:

Wenli Wang, Caiqin Song. A NOVEL ITERATIVE METHOD FOR SOLVING THE COUPLED SYLVESTER-CONJUGATE MATRIX EQUATIONS AND ITS APPLICATION IN ANTILINEAR SYSTEM[J]. Journal of Applied Analysis & Computation, 2023, 13(1): 249-274. doi: 10.11948/20220032

A NOVEL ITERATIVE METHOD FOR SOLVING THE COUPLED SYLVESTER-CONJUGATE MATRIX EQUATIONS AND ITS APPLICATION IN ANTILINEAR SYSTEM

Wenli Wang¹,
Caiqin Song^1,2, ,

1.
School of Mathematical Science, University of Jinan, Jinan 250022, China
2.
Department of Mathematics and Statistics, University of Nevada, Reno 89503, USA

Corresponding author: Email: songcaiqin1983@163.com (C. Song)

Abstract

Abstract

This paper is devoted to constructing a modified relaxed gradient based iterative (MRGI) algorithm to solve the coupled Sylvester-conjugate matrix equations (CSCMEs) based on the hierarchical identification principle. Convergence analysis shows that the proposed algorithm is effective for arbitrary initial matrices. Further, we apply the MRGI algorithm to study a more general coupled Sylvester conjugate matrix equations and give a sufficient condition to guarantee that the iterative solution converges to the exact solution. Two numerical experiments are provided to demonstrate that the MRGI algorithm has better efficiency and accuracy than the three existing algorithms, which are presented by Wu et al. (2010) and Huang and Ma (2018). Finally, we derive an application of MRGI algorithm in discrete-time antilinear system.
MSC: 15A24, 15A21, 65F10

FullText(HTML)

References

[1]	K. Adisorn and C. Pattrawut, Approximated least-squares solutions of a generalized Sylvester-transpose matrix equation via gradient-descent iterative algorithm, Advances in Difference Equations, 2021, 2021, 266. doi: 10.1186/s13662-021-03427-4 CrossRef Google Scholar
[2]	K. Adison, C. Pattrawut and L. Wicharn, Convergence analysis of gradient-based iterative algorithms for a class of rectangular Sylvester matrix equations based on Banach contraction principle, Advances in Difference Equations, 2021, 2021, 17. doi: 10.1186/s13662-020-03185-9 CrossRef Google Scholar
[3]	R. Agarwal, S. Hristova, D. O'Regan and K. Stefanova, Iterative Algorithm for Solving Scalar Fractional Differential Equations with Riemann-Liouville Derivative and Supremum, Algorithms, 2020, 13(8), 184. doi: 10.3390/a13080184 CrossRef Google Scholar
[4]	M. Dehghan and M. Hajarian, The generalised Sylvester matrix equations over the generalised bisymmetric and skew-symmetric matrices, International Journal of Systems Science, 2012, 43(8), 1580–1590. doi: 10.1080/00207721.2010.549584 CrossRef Google Scholar
[5]	M. Dehghan and R. Mohammadi-Arani, Generalized product-type methods based on bi-conjugate gradient (GPBiCG) for solving shifted linear systems, Comput. Appl. Math., 2016, 36(4), 1–16. Google Scholar
[6]	F. Ding, X. Liu and J. Ding, Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle, Appl. Math. Comput., 2008, 197(1), 41–50. Google Scholar
[7]	F. Ding, F. Wang, L. Xu and M. Wu, Decomposition based least squares iterative identification algorithm for multivariate pseudo-linear ARMA systems using the data filtering, J. Frankl. Inst., 2017, 354(3), 1321–1339. doi: 10.1016/j.jfranklin.2016.11.030 CrossRef Google Scholar
[8]	M. Hajarian, Solving the general Sylvester discrete-time periodic matrix equations via the gradient based iterative method, Appl. Math. Lett., 2016, 52, 87–95. doi: 10.1016/j.aml.2015.08.017 CrossRef Google Scholar
[9]	M. Hajarian, Gradient based iterative algorithm to solve general coupled discretetime periodic matrix equations over generalized reflexive matrices, Math. Model. Anal., 2016, 21, 533–549. doi: 10.3846/13926292.2016.1186119 CrossRef Google Scholar
[10]	M. Hajarian, New finite algorithm for solving the generalized nonhomogeneous Yakubovich-transpose matrix equation, Asian J. Control, 2017, 19, 164–172. doi: 10.1002/asjc.1343 CrossRef Google Scholar
[11]	M. Hajarian, Convergence of HS version of BCR algorithm to solve the generalized Sylvester matrix equation over generalized reflexive matrices, J. Frankl. Inst., 2017, 354, 2340–2357. doi: 10.1016/j.jfranklin.2017.01.008 CrossRef Google Scholar
[12]	M. Hajarian, Reflexive periodic solutions of general periodic matrix equations, Mathematical Methods in the Applied Sciences, 2019, 42(10), 3527–3548. doi: 10.1002/mma.5596 CrossRef Google Scholar
[13]	M. Hajarian, Three types of biconjugate residual method for general periodic matrix equations over generalized bisymmetric periodic matrices, Transactions of the Institute of Measurement and Control, 2019, 41(10), 2708–2725. doi: 10.1177/0142331218808859 CrossRef Google Scholar
[14]	B. Huang and C. Ma, Gradient-based iterative algorithms for generalized coupled Sylvester-conjugate matrix equations, Comput. Math. Appl., 2018, 75, 2295–2310. doi: 10.1016/j.camwa.2017.12.011 CrossRef Google Scholar
[15]	B. Huang and C. Ma, The relaxed gradient-based iterative algorithms for a class of generalized coupled Sylvester-conjugate matrix equations, J. Frankl. Inst., 2018, 355, 3168–3195. doi: 10.1016/j.jfranklin.2018.02.014 CrossRef Google Scholar
[16]	Y. Ji, H. J. Chizeck, X. Feng and K. A. Loparo, Stability and control of discrete-time jump linear systems, Control Theory Adv. Technol., 1991, 7(2), 247–270. Google Scholar
[17]	T. Jiang and M. Wei, On solutions of the matrix equations $X-AXB=C$ and $X-A\overline{X}B=C$, Linear Algebra and its Applications, 2003, 367, 225–233. doi: 10.1016/S0024-3795(02)00633-X CrossRef Google Scholar
[18]	Q. Niu, X. Wang and L. Lu, A relaxed gradient based algorithm for solving Sylvester equations, Asian J. Control, 2011, 13, 461–464. doi: 10.1002/asjc.328 CrossRef Google Scholar
[19]	X. Sheng, A relaxed gradient based algorithm for solving generalized coupled Sylvester matrix equations, J. Frankl. Inst., 2018, 355, 4282–4297. doi: 10.1016/j.jfranklin.2018.04.008 CrossRef Google Scholar
[20]	C. Song, G. Chen and L. Zhao, Iterative solutions to coupled Sylvester-transpose matrix equations, Appl. Math. Model., 2011, 35(10), 4675–4683. doi: 10.1016/j.apm.2011.03.038 CrossRef Google Scholar
[21]	C. Vanhille, A note on the convergence of the irrational Halley¡¯s iterative algorithm for solving nonlinear equations, International Journal of Computer Mathematics, 2020, 97(9), 1840–1848. doi: 10.1080/00207160.2019.1664736 CrossRef Google Scholar
[22]	X. Wang, L. Dai and D. Liao, A modified gradient based algorithm for solving Sylvester equations, Appl. Math. Comput., 2012, 218, 5620–5628. Google Scholar
[23]	X. Wang and D. Liao, The optimal convergence factor of the gradient based iterative algorithm for linear matrix equations, Filomat, 2012, 26, 607–613. doi: 10.2298/FIL1203607W CrossRef Google Scholar
[24]	A. Wu, G. Feng, G. Duan and W. Wu, Iterative solutions to coupled Sylvester-conjugate matrix equations, Comput. Math. Appl., 2010, 60(1), 54–66. doi: 10.1016/j.camwa.2010.04.029 CrossRef Google Scholar
[25]	A. Wu, L. Lv and G. Duan, Iterative algorithms for solving a class of complex conjugate and transpose matrix equations, Appl. Math. Comput., 2011, 217(21), 8343–8353. Google Scholar
[26]	A. Wu, X. Zeng, G. Duan and W. Wu, Iterative solutions to the extended Sylvester-conjugate matrix equatinos, Appl. Math. Comput., 2010, 217(1), 130–142. Google Scholar
[27]	Y. Xie and C. Ma, The accelerated gradient based iterative algorithm for solving a class of generalized Sylvester-transpose matrix equation, Appl. Math. Comput., 2016, 273, 1257–1269. Google Scholar
[28]	T. Yan and C. Ma, The BCR algorithms for solving the reflexive or anti-reflexive solutions of generalized coupled Sylvester matrix equations, Journal of the Franklin Institute, 2020, 357, 12787–12807. doi: 10.1016/j.jfranklin.2020.09.030 CrossRef Google Scholar
[29]	L. Zhang, B. Huang and J. Lam, H_∞ model reduction of Markovian jump linear systems, Syst. Control Lett., 2003, 50, 103–118. doi: 10.1016/S0167-6911(03)00133-6 CrossRef Google Scholar
[30]	H. Zhang, Reduced-rank gradient-based algorithms for generalized coupled Sylvester matrix equations and its applications, Comput. Math. Appl., 2015, 70, 2049–2062. doi: 10.1016/j.camwa.2015.08.013 CrossRef Google Scholar
[31]	H. Zhang and F. Ding, A property of the eigenvalues of the symmetric positive definite matrix and the iterative algorithm for coupled Sylvester matrix equations, J. Frankl. Inst., 2014, 351, 340–357. doi: 10.1016/j.jfranklin.2013.08.023 CrossRef Google Scholar
[32]	H. Zhang and H. Yin, New proof of the gradient-based iterative algorithm for a complex conjugate and transpose matrix equation, J. Frankl. Inst., 2017, 354, 7585–7603. doi: 10.1016/j.jfranklin.2017.09.005 CrossRef Google Scholar
[33]	B. Zhou and G. Duan, Periodic Lyapunov equation based approaches to the stabilization of continuous-time periodic linear systems, IEEE Trans. Autom. Control, 2012, 57, 2139–2146. doi: 10.1109/TAC.2011.2181796 CrossRef Google Scholar
[34]	B. Zhou, J. Lam and G. Duan, Gradient-based maximal convergence rate iterative method for solving linear matrix equations, International Journal of Computer Mathematics, 2010, 87(3), 515–527. doi: 10.1080/00207160802123458 CrossRef Google Scholar