The semi-tensor product method for special least squares solutions of the complex generalized Sylvester matrix equation

Fengxia Zhang; Ying Li; Jianli Zhao; Fengxia Zhang; Ying Li; Jianli Zhao

doi:10.3934/math.2023261

AIMS Mathematics

2023, Volume 8, Issue 3: 5200-5215. doi: 10.3934/math.2023261

Previous Article Next Article

Research article

The semi-tensor product method for special least squares solutions of the complex generalized Sylvester matrix equation

College of Mathematical Science, Liaocheng University, Liaocheng 252000, China

Received: 30 July 2022 Revised: 10 November 2022 Accepted: 28 November 2022 Published: 13 December 2022
MSC : 15A06, 15A24

In this paper, we are interested in the minimal norm of least squares Hermitian solution and the minimal norm of least squares anti-Hermitian solution for the complex generalized Sylvester matrix equation $ CXD+EXF = G $. By utilizing of the real vector representations of complex matrices and the semi-tensor product of matrices, we first transform solving special least squares solutions of the above matrix equation into solving the general least squares solutions of the corresponding real matrix equations, and then obtain the expressions of the minimal norm of least squares Hermitian solution and the minimal norm of least squares anti-Hermitian solution. Further, we give two numerical algorithms and two numerical examples, and numerical examples illustrate that our proposed algorithms are more efficient and accurate.
Citation: Fengxia Zhang, Ying Li, Jianli Zhao. The semi-tensor product method for special least squares solutions of the complex generalized Sylvester matrix equation[J]. AIMS Mathematics, 2023, 8(3): 5200-5215. doi: 10.3934/math.2023261

Related Papers:

Abstract

In this paper, we are interested in the minimal norm of least squares Hermitian solution and the minimal norm of least squares anti-Hermitian solution for the complex generalized Sylvester matrix equation $ CXD+EXF = G $. By utilizing of the real vector representations of complex matrices and the semi-tensor product of matrices, we first transform solving special least squares solutions of the above matrix equation into solving the general least squares solutions of the corresponding real matrix equations, and then obtain the expressions of the minimal norm of least squares Hermitian solution and the minimal norm of least squares anti-Hermitian solution. Further, we give two numerical algorithms and two numerical examples, and numerical examples illustrate that our proposed algorithms are more efficient and accurate.

References

[1]	X. P. Sheng, A relaxed gradient based algorithm for solving generalized coupled Sylvester matrix equations, J. Franklin Inst., 355 (2018), 4282–4297. https://doi.org/10.1016/j.jfranklin.2018.04.008 doi: 10.1016/j.jfranklin.2018.04.008
[2]	M. Dehghan, M. Hajarian, On the generalized bisymmetric and skew-symmetric solutions of the system of generalized Sylvester matrix equations, Linear Multilinear Algebra, 59 (2011), 1281–1309. https://doi.org/10.1080/03081087.2010.524363 doi: 10.1080/03081087.2010.524363
[3]	M. Dehghan, M. Dehghani-Madiseh, M. Hajarian, A generalized preconditioned MHSS method for a class of complex symmetric linear systems, Math. Model. Anal., 18 (2013), 561–576. https://doi.org/10.3846/13926292.2013.839964 doi: 10.3846/13926292.2013.839964
[4]	M. Dehghani-Madiseh, M. Dehghan, Parametric AE-solution sets to the parametric linear systems with multiple right-hand sides and parametric matrix equation $A(p)X = B(p)$, Numer. Algor., 73 (2016), 245–279. https://doi.org/10.1007/s11075-015-0094-3 doi: 10.1007/s11075-015-0094-3
[5]	Y. B. Deng, Z. Z. Bai, Y. H. Gao, Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations, Numer. Linear Algebra Appl., 13 (2006), 801–823. https://doi.org/10.1002/nla.496 doi: 10.1002/nla.496
[6]	B. H. Huang, C. F. Ma, On the least squares generalized Hamiltonian solution of generalized coupled Sylvester-conjugate matrix equations, Comput. Math. Appl., 74 (2017), 532–555. https://doi.org/10.1016/j.camwa.2017.04.035 doi: 10.1016/j.camwa.2017.04.035
[7]	F. X. Zhang, M. S. Wei, Y. Li, J. L. Zhao, The minimal norm least squares Hermitian solution of the complex matrix equation $AXB+CXD = E$, J. Franklin Inst., 355 (2018), 1296–1310. https://doi.org/10.1016/j.jfranklin.2017.12.023 doi: 10.1016/j.jfranklin.2017.12.023
[8]	Y. Gu, Y. Z. Song, Global Hessenberg and CMRH methods for a class of complex matrix equations, J. Comput. Appl. Math., 404 (2022), 113868. https://doi.org/10.1016/j.cam.2021.113868 doi: 10.1016/j.cam.2021.113868
[9]	F. X. Zhang, Y. Li, J. L. Zhao, A real representation method for special least squares solutions of the quaternion matrix equation $(AXB, DXE) = (C, F)$, AIMS Mathematics, 7 (2022), 14595–14613. https://doi.org/10.3934/math.2022803 doi: 10.3934/math.2022803
[10]	Q. W. Wang, Z. H. He, Solvability conditions and general solution for mixed Sylvester equations, Automatica, 49 (2013), 2713–2719. https://doi.org/10.1016/j.automatica.2013.06.009 doi: 10.1016/j.automatica.2013.06.009
[11]	F. X. Zhang, W. S. Mu, Y. Li, J. L. Zhao, Special least squares solutions of the quaternion matrix equation $AXB+CXD = E$, Comput. Math. Appl., 72 (2016), 1426–1435. https://doi.org/10.1016/j.camwa.2016.07.019 doi: 10.1016/j.camwa.2016.07.019
[12]	H. T. Zhang, L. N. Liu, H. Liu, Y. X. Yuan, The solution of the matrix equation $AXB = D$ and the system of matrix equations $AX = C, XB = D$ with $X^X = I_p$, Appl. Math. Comput., 418* (2022), 126789. https://doi.org/10.1016/j.amc.2021.126789 doi: 10.1016/j.amc.2021.126789
[13]	G. J. Song, Q. W. Wang, S. W. Yu, Cramer's rule for a system of quaternion matrix equations with applications, Appl. Math. Comput., 336 (2018), 490–499. https://doi.org/10.1016/j.amc.2018.04.056 doi: 10.1016/j.amc.2018.04.056
[14]	F. X. Zhang, M. S. Wei, Y. Li, J. L. Zhao, An efficient real representation method for least squares problem of the quaternion constrained matrix equation $AXB+CYD = E$, Int. J. Comput. Math., 98 (2021), 1408–1419. https://doi.org/10.1080/00207160.2020.1821001 doi: 10.1080/00207160.2020.1821001
[15]	X. Peng, X. X. Guo, Real iterative algorithms for a common solution to the complex conjugate matrix equation system, Appl. Math. Comput., 270 (2015), 472–482. https://doi.org/10.1016/j.amc.2015.07.105 doi: 10.1016/j.amc.2015.07.105
[16]	X. P. Sheng, W. W. Sun, The relaxed gradient based iterative algorithm for solving matrix equations $A_iXB_i = F_i$, Comput. Math. Appl., 74 (2017), 597–604. https://doi.org/10.1016/j.camwa.2017.05.008 doi: 10.1016/j.camwa.2017.05.008
[17]	M. Dehghan, A. Shirilord, A new approximation algorithm for solving generalized Lyapunov matrix equations, J. Comput. Appl. Math., 404 (2022), 113898. https://doi.org/10.1016/j.cam.2021.113898 doi: 10.1016/j.cam.2021.113898
[18]	M. Dehghan, R. Mohammadi-Arani, Generalized product-type methods based on bi-conjugate gradient (GPBiCG) for solving shifted linear systems, Comput. Appl. Math., 36 (2017), 1591–1606. https://doi.org/10.1007/s40314-016-0315-y doi: 10.1007/s40314-016-0315-y
[19]	B. H. Huang, C. F. Ma, On the least squares generalized Hamiltonian solution of generalized coupled Sylvester-conjugate matrix equations, Comput. Math. Appl., 74 (2017), 532–555. https://doi.org/10.1016/j.camwa.2017.04.035 doi: 10.1016/j.camwa.2017.04.035
[20]	T. X. Yan, C. F. Ma, An iterative algorithm for generalized Hamiltonian solution of a class of generalized coupled Sylvester-conjugate matrix equations, Appl. Math. Comput., 411 (2021), 126491. https://doi.org/10.1016/j.amc.2021.126491 doi: 10.1016/j.amc.2021.126491
[21]	M. Hajarian, Developing BiCOR and CORS methods for coupled Sylvester-transpose and periodic Sylvester matrix equations, Appl. Math. Model., 39 (2015), 6073–6084. https://doi.org/10.1016/j.apm.2015.01.026 doi: 10.1016/j.apm.2015.01.026
[22]	H. M. Zhang, A finite iterative algorithm for solving the complex generalized coupled Sylvester matrix equations by using the linear operators, J. Frankl. Inst., 354 (2017), 1856–1874. https://doi.org/10.1016/j.jfranklin.2016.12.011 doi: 10.1016/j.jfranklin.2016.12.011
[23]	L. L. Lv, Z. Zhang, Finite iterative solutions to periodic Sylvester matrix equations, J. Frankl. Inst., 354 (2017), 2358–2370. https://doi.org/10.1016/j.jfranklin.2017.01.004 doi: 10.1016/j.jfranklin.2017.01.004
[24]	N. Huang, C. F. Ma, Modified conjugate gradient method for obtaining the minimum-norm solution of the generalized coupled Sylvester-conjugate matrix equations, Appl. Math. Model., 40 (2016), 1260–1275. https://doi.org/10.1016/j.apm.2015.07.017 doi: 10.1016/j.apm.2015.07.017
[25]	F. P. A. Beik, D. K. Salkuyeh, On the global Krylov subspace methods for solving general coupled matrix equations, Comput. Math. Appl., 62 (2011), 4605–4613. https://doi.org/10.1016/j.camwa.2011.10.043 doi: 10.1016/j.camwa.2011.10.043
[26]	M. Dehghan, A. Shirilord, Solving complex Sylvester matrix equation by accelerated double-step scale splitting (ADSS) method, Eng. Comput., 37 (2021), 489–508. https://doi.org/10.1007/s00366-019-00838-6 doi: 10.1007/s00366-019-00838-6
[27]	S. F. Yuan, A. P. Liao, Least squares Hermitian solution of the complex matrix equation $AXB+CXD = E$ with least norm, J. Frankl. Inst., 351 (2014), 4978–4997. https://doi.org/10.1016/j.jfranklin.2014.08.003 doi: 10.1016/j.jfranklin.2014.08.003
[28]	F. X. Zhang, M. S. Wei, Y. Li, J. L. Zhao, The minimal norm least squares Hermitian solution of the complex matrix equation $AXB+CXD = E$, J. Frankl. Inst., 355 (2018), 1296–1310. https://doi.org/10.1016/j.jfranklin.2017.12.023 doi: 10.1016/j.jfranklin.2017.12.023
[29]	S. F. Yuan, Least squares pure imaginary solution and real solution of the quaternion matrix equation $AXB+CXD = E$ with the least norm, J. Appl. Math., 2014 (2014), 857081. https://doi.org/10.1155/2014/857081 doi: 10.1155/2014/857081
[30]	Q. W. Wang, A. Rehman, Z. H. He, Y. Zhang, Constraint generalized sylvester matrix equations, Automatica, 69 (2016), 60–64. https://doi.org/10.1016/j.automatica.2016.02.024 doi: 10.1016/j.automatica.2016.02.024
[31]	Y. X. Yuan, Solving the mixed Sylvester matrix equations by matrix decompositions, C. R. Math., 353 (2015), 1053–1059. https://doi.org/10.1016/j.crma.2015.08.010 doi: 10.1016/j.crma.2015.08.010
[32]	S. F. Yuan, Q. W. Wang, Y. B. Yu, Y. Tian, On Hermitian solutions of the split quaternion matrix equation $AXB+CXD = E$, Adv. Appl. Clifford Algebras, 27 (2017), 3235–3252. https://doi.org/10.1007/s00006-017-0806-y doi: 10.1007/s00006-017-0806-y
[33]	I. Kyrchei, Determinantal representations of solutions to systems of two-sided quaternion matrix equations, Linear Multilinear Algebra, 69 (2021), 648–672. https://doi.org/10.1080/03081087.2019.1614517 doi: 10.1080/03081087.2019.1614517
[34]	D. Z. Cheng, H. S. Qi, Y. Zhao, An introduction to semi-tensor product of matrices and its application, Singapore: World Scientific Publishing Company, 2012.
[35]	J. Feng, J. Yao, P. Cui, Singular boolean networks: Semi-tensor product approach, Sci. China Inf. Sci., 56 (2013), 1–14. https://doi.org/10.1007/s11432-012-4666-8 doi: 10.1007/s11432-012-4666-8
[36]	M. R. Xu, Y. Z. Wang, Conflict-free coloring problem with appliction to frequency assignment, J. Shandong Univ., 45 (2015), 64–69.
[37]	D. Z. Cheng, Q. S. Qi, Z. Q. Liu, From STP to game-based control, Sci. China Inf. Sci., 61 (2018), 010201. https://doi.org/10.1007/s11432-017-9265-2 doi: 10.1007/s11432-017-9265-2
[38]	J. Q. Lu, H. T. Li, Y. Liu, F. F. Li, Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems, IET Control Theory Appl., 11 (2017), 2040–2047. https://doi.org/10.1049/iet-cta.2016.1659 doi: 10.1049/iet-cta.2016.1659
[39]	D. Z. Cheng, H. S. Qi, Z. Q. Li, Analysis and control of Boolean networks: A semi-tensor product approach, London: Springer, 2011.
[40]	W. X. Ding, Y. Li, D. Wang, T. Wang, The application of the semi-tensor product in solving special Toeplitz solution of complex linear system, J. Liaocheng Univ. (Nat. Sci.), 34 (2021), 1–6.

Reader Comments

Your name:*

Email:*
© 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)