Two accelerated gradient-based iteration methods for solving the Sylvester matrix equation AX + XB = C

Huiling Wang; Nian-Ci Wu; Yufeng Nie; Huiling Wang; Nian-Ci Wu; Yufeng Nie

doi:10.3934/math.20241654

AIMS Mathematics

2024, Volume 9, Issue 12: 34734-34752. doi: 10.3934/math.20241654

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Research article

Two accelerated gradient-based iteration methods for solving the Sylvester matrix equation AX + XB = C

1.
College of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, China
2.
School of Mathematics and Statistics, South-Central Minzu University, Wuhan 430074, China
3.
School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710072, China

Received: 25 September 2024 Revised: 17 November 2024 Accepted: 29 November 2024 Published: 12 December 2024
MSC : 15A24, 65F30

In this paper, combining the precondition technique and momentum item with the gradient-based iteration algorithm, two accelerated iteration algorithms are presented for solving the Sylvester matrix equation $ AX+XB = C $. Sufficient conditions to guarantee the convergence properties of the proposed algorithms are analyzed in detail. Varying the parameters of these algorithms in each iteration, the corresponding adaptive iteration algorithms are also provided, and the adaptive parameters can be explicitly obtained by the minimum residual technique. Several numerical examples are implemented to illustrate the effectiveness of the proposed algorithms.
- Sylvester matrix equation,
- gradient-based iteration,
- momentum term,
- precondition technique,
- minimum residual technique
Citation: Huiling Wang, Nian-Ci Wu, Yufeng Nie. Two accelerated gradient-based iteration methods for solving the Sylvester matrix equation AX + XB = C[J]. AIMS Mathematics, 2024, 9(12): 34734-34752. doi: 10.3934/math.20241654

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Abstract

In this paper, combining the precondition technique and momentum item with the gradient-based iteration algorithm, two accelerated iteration algorithms are presented for solving the Sylvester matrix equation $ AX+XB = C $. Sufficient conditions to guarantee the convergence properties of the proposed algorithms are analyzed in detail. Varying the parameters of these algorithms in each iteration, the corresponding adaptive iteration algorithms are also provided, and the adaptive parameters can be explicitly obtained by the minimum residual technique. Several numerical examples are implemented to illustrate the effectiveness of the proposed algorithms.

References

[1]	A. L. Andrew, Eigenvectors of certain matrices, Linear Algebra Appl., 7 (1973), 151–162. http://dx.doi.org/10.1016/0024-3795(73)90049-9 doi: 10.1016/0024-3795(73)90049-9
[2]	Z. Z. Bai, G. H. Golub, J. Y. Pan, Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math., 98 (2004), 1–32. http://dx.doi.org/10.1007/s00211-004-0521-1 doi: 10.1007/s00211-004-0521-1
[3]	Z. Z. Bai, On hermitian and skew-hermitian splitting iteration methods for continuous sylvester equations, J. Comput. Math., 29 (2011), 185–198. http://dx.doi.org/10.4208/jcm.1009-m3152 doi: 10.4208/jcm.1009-m3152
[4]	Z. Z. Bai, M. Benzi, F. Chen, Z. Q. Wang, Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems, IMA J. Numer. Anal., 33 (2013), 343–369. http://dx.doi.org/10.1093/imanum/drs001 doi: 10.1093/imanum/drs001
[5]	A. Bhaya, E. Kaszkurewicz, Steepest descent with momentum for quadratic functions is a version of the conjugate gradient method, Neural Networks, 17 (2004), 65–71. http://dx.doi.org/10.1016/S0893-6080(03)00170-9 doi: 10.1016/S0893-6080(03)00170-9
[6]	Z. B. Chen, X. S. Chen, Conjugate gradient-based iterative algorithm for solving generalized periodic coupled Sylvester matrix equation, J. Franklin I., 359 (2022), 9925–9951. http://dx.doi.org/10.1016/j.jfranklin.2022.09.049 doi: 10.1016/j.jfranklin.2022.09.049
[7]	M. Dehghan, A. Shirilord, The double-step scale splitting method for solving complex Sylvester matrix equation, Comp. Appl. Math., 38 (2019), 146. http://dx.doi.org/10.1007/s40314-019-0921-6 doi: 10.1007/s40314-019-0921-6
[8]	F. Ding, T. W. Chen, Gradient based iterative algorithms for solving a class of matrix equations, IEEE T. Automat. Contr., 50 (2005), 1216–1221. http://dx.doi.org/10.1109/TAC.2005.852558 doi: 10.1109/TAC.2005.852558
[9]	F. Ding, P. X. Liu, J. Ding, Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle, Appl. Math. Comput., 197 (2008), 41–50. http://dx.doi.org/10.1016/j.amc.2007.07.040 doi: 10.1016/j.amc.2007.07.040
[10]	F. Ding, X. H. Wang, Q. J. Chen, Y. S. Xiao, Recursive least squares parameter estimation for a class of output nonlinear systems based on the model decompositions, Circuits Syst. Signal Process., 35 (2016), 3323–3338. http://dx.doi.org/10.1007/s00034-015-0190-6 doi: 10.1007/s00034-015-0190-6
[11]	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. http://dx.doi.org/10.1016/j.apnum.2020.05.013 doi: 10.1016/j.apnum.2020.05.013
[12]	K. Du, C. C. Ruan, X. H. Sun, On the convergence of a randomized block coordinate descent algorithm for a matrix least squares problem, Appl. Math. Lett., 124 (2022), 107689. http://dx.doi.org/10.1016/j.aml.2021.107689 doi: 10.1016/j.aml.2021.107689
[13]	W. Fan, C. Gu, Z. Tian, Jacobi-gradient iterative algorithms for Sylvester matrix equations, 14th Conference of the International Linear Algebra Society, Shanghai, China, 2007, 16–20.
[14]	C. Q. Gu, H. Y. Xue, A shift-splitting hierarchical identification method for solving Lyapunov matrix equations, Linear Algebra Appl., 430 (2009), 1517–1530. http://dx.doi.org/10.1016/j.laa.2008.01.010 doi: 10.1016/j.laa.2008.01.010
[15]	B. H. Huang, W. Li, A modified SOR-like method for absolute value equations associated with second order cones, J. Comput. Appl. Math., 400 (2022), 113745. http://dx.doi.org/10.1016/j.cam.2021.113745 doi: 10.1016/j.cam.2021.113745
[16]	X. Li, H. F. Huo, A. L. Yang, Preconditioned HSS iteration method and its non-alternating variant for continuous Sylvester equations, Comput. Math. Appl., 75 (2018), 1095–1106. http://dx.doi.org/10.1016/j.camwa.2017.10.028 doi: 10.1016/j.camwa.2017.10.028
[17]	M. S. Mehany, Q. W. Wang, Three symmetrical systems of coupled Sylvester-like quaternion matrix equations, Symmetry, 14 (2022), 550. http://dx.doi.org/10.3390/sym14030550 doi: 10.3390/sym14030550
[18]	Q. Niu, X. Wang, L. Z. Lu, A relaxed gradient based algorithm for solving Sylvester equations, Asian J. Control, 13 (2011), 461–464. http://dx.doi.org/10.1002/asjc.328 doi: 10.1002/asjc.328
[19]	B. T. Polyak, Some methods of speeding up the convergence of iteration methods, Comp. Math. Math. Phys., 4 (1964), 1–17. http://dx.doi.org/10.1016/0041-5553(64)90137-5 doi: 10.1016/0041-5553(64)90137-5
[20]	S. G. Shafiei, M. Hajarian, An iterative method based on ADMM for solving generalized Sylvester matrix equations, J. Franklin I., 359 (2022), 8155–8170. http://dx.doi.org/10.1016/j.jfranklin.2022.07.049 doi: 10.1016/j.jfranklin.2022.07.049
[21]	Z. L. Tian, M. Y. Tian, C. Q. Gu, X. N. Hao, An accelerated Jacobi-gradient based iterative algorithm for solving Sylvester matrix equations, Filomat, 31 (2017), 2381–2390. http://dx.doi.org/10.2298/FIL1708381T doi: 10.2298/FIL1708381T
[22]	Z. L. Tian, Y. D. Wang, Y. H. Dong, S. Y. Wang, New results of the IO iteration algorithm for solving Sylvester matrix equation, J. Franklin I., 359 (2022), 8201–8217. http://dx.doi.org/10.1016/j.jfranklin.2022.08.018 doi: 10.1016/j.jfranklin.2022.08.018
[23]	Q. W. Wang, R. Y. Lv, Y. Zhang, The least-squares solution with the least norm to a system of tensor equations over the quaternion algebra, Linear Multilinear A., 70 (2022), 1942–1962. http://dx.doi.org/10.1080/03081087.2020.1779172 doi: 10.1080/03081087.2020.1779172
[24]	Q. W. Wang, X. Wang, A system of coupled two-sided Sylvester-type tensor equations over the quaternion algebra, Taiwanese J. Math., 24 (2020), 1399–1416. http://dx.doi.org/10.11650/tjm/200504 doi: 10.11650/tjm/200504
[25]	Y. J. Xie, C. F. Ma, The accelerated gradient based iterative algorithm for solving a class of generalized Sylvester-transpose matrix equation, Appl. Math. Comput., 273 (2016), 1257–1269. http://dx.doi.org/10.1016/j.amc.2015.07.022 doi: 10.1016/j.amc.2015.07.022
[26]	A. L. Yang, Y. Cao, Y. J. Wu, Minimum residual Hermitian and skew-Hermitian splitting iteration method for non Hermitian positive definite linear systems, BIT Numer. Math., 59 (2019), 299–319. http://dx.doi.org/10.1007/s10543-018-0729-6 doi: 10.1007/s10543-018-0729-6
[27]	A. L. Yang, On the convergence of the minimum residual HSS iteration method, Appl. Math. Lett., 94 (2019), 210–216. http://dx.doi.org/10.1016/j.aml.2019.02.031 doi: 10.1016/j.aml.2019.02.031
[28]	J. F. Yin, Q. Y. Dou, Generalized preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive-definite linear systems, J. Comput. Math., 30 (2012), 404–417. http://dx.doi.org/10.4208/jcm.1201-m3209 doi: 10.4208/jcm.1201-m3209
[29]	X. F. Zhang, Q. W. Wang, Developing iterative algorithms to solve Sylvester tensor equations, Appl. Math. Comput., 409 (2021), 126403. http://dx.doi.org/10.1016/j.amc.2021.126403 doi: 10.1016/j.amc.2021.126403

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